transport processes emission – [mass/time] pollutants released into the environment imission –...
TRANSCRIPT
Transport processesTransport processes
bull Emission ndash [masstime]pollutants released into the environment
bull Imission ndash [massvolume]amount of pollutants received by a living
being at a given location Also the state of the environment with respect to the pollutant
bull TransmissionEverything between emission and
immission
Transport processesTransport processes
What is a transport processbull Describes the fate of any material in the
environmentbull The continuous steps of transportation
reactions and interactions which happen to the pollutants
bull It determines the concentration-distribution of the observed material in time and in space
bull Transmission
Transport processesTransport processes
Transport processes includebull Physical transportation due to the
movement of the mediumbull Advectionbull Diffusion and dispersion
bull Physical chemical biochemical conversion processes
bull Settlingbull Ad- and desorptionbull Reactionsbull Degradation decomposition
Transport processesTransport processes
What is our aimbull Defend our values
bull Decrease the immission by controlling the emission ndash EFFORT MONEY
bull Describe transport processes
Transport processesTransport processesImportant conceptsbull State variable concentration density temperaturehellip
bull Conservative material no reactions no settling
bull Non-conservative material opposite realistic
bull Flow processes 3D u(xyzt) h(xyzt) V(xyzt)hellip
bull Steady state dudt=0 dCdt=0hellip
bull Homogeneous distribution ndash totally mixed reactor
Transport processesTransport processesThis presentation deals only with transport
processes in surface waters So the transporting medium is water
Classical water quality state variables arebull Dissolved oxygen (DO)bull Organic material - biochemical oxygen
demand (BOD)bull Nutrients ndash N- P-forms (NO3-N PO4-Phellip)
bull Suspended solids (SS)bull Algae
Mass balanceMass balance
General mass balance equationbull Expresses the conversation of massbull Differential equations
IN ndash OUT + SOURCES ndash SINKS = CHANGE
VIN
(1)
OUT
(2)
Controlling surface
We work with mass fluxes [gs]
Mass balanceMass balanceBy solving the general mass balance equation
we can describe the transport processes for a given substance
Solutionbull Analyticalbull Numerical
Discretizationbull Temporalbull Spatial
Simplificationsbull Temporal ndash steady-statebull Spatial ndash 2D 1Dbull Other
Boundary initial conditionsbull Geometrical DATAbull Hydrologic DATAbull Water quality DATA
Simplified case ISimplified case IIN ndash OUT + SOURCES ndash SINKS = CHANGE
Assumptionsbull Steady statebull Conservative material
Solutionbull IN=OUTbull SOURCES=0 SINKS=0bull CHANGE=0
Simplified case IISimplified case IIAssumptionsbull Steady state ndash Q(t) E(t) are constants dC(t)dt=0bull Non-conservative settling material (vs)bull Prismatic river bed ndash A B H are constantsbull 1D
vs
A ~ B middot H [m2]
H
Simplified case IISimplified case II
Q
(1) (2)x
Q
IN
OUT
LOSS settled matter
C(x) is linear (assumption)
Av
v
vs
If x = O C = Co
Exponential decrease
Simplified case IISimplified case IIThe calculation
C0 concentration under the inlet
Determination of C0 value
Q
E = q middot c emission
Cbg background concentration
1D ndash Complete mixing (two water mixing with each other)
Increment
Dilution ratio
E
Simplified case IISimplified case II
The solution
Transmission coefficient
Dilution Sedimentation Conservative matter
Simplified case IISimplified case II
Clim = 15 gm3
concentration limit
Q = 15 m3s
q = 1 m3sc = 300 gm3
Cbg = 10 gm3
L = 5 km
B = 75 m
vS = 03 mh
H = 2 m
EXAMPLE
Complete mixing
3bg0 gm28125
11530011015
cqCQC
3lim
33
s0
gm15Cgm22836)10513600
0321
( exp28125
L)vv
H1
( expC L)(xC
Concentration at distance L from the emission
Flow velocity
ms10275
15HB
Qv
Maximum possible C0 value
Maximum allowed waste water concentration
Allowed maximum concentration
C (x = L)max = Clim
3
3s
limmax0
gm18474
)10513600
0321
( exp
15
L)vv
H1
( exp
C C
3
bgmax0max
gm145584
11015-11518474
q
CQ-qQCc
Transmission coefficient
Necessary reduction of emission
gs154416145584)(3001)c(cqΔE max
33
s
sm0051)10513600
0321
( exp115
1
L)vv
H1
( expqQ
1 L)(xa
The general situationThe general situation
bull Unsteadybull Inhomogeneousbull Non-conservativebull 3D
General transport equation
Advection-Dispersion Equation (ADE)
Advection-dispersion EquationAdvection-dispersion Equationbull Can be derived from the mass balancebull Expresses the temporal and spatial changes of
the observed state variablebull Itrsquos solution is the concentration-distribution over
timebull Differential equation
Change in concentration with time
Change due to advection
=Change due to diffusion or dispersion
Change due to conversion processes
+ +
Molecular diffusionMolecular diffusion
DIFFUSION
CONVECTION
v
Molecular Molecular diffusiondiffusion
FickrsquosFickrsquos LawLaw
c1 c2x
bull c1 c2 separated tanks
bull open valve
Mass flux (kgs)
D - molecular diffusioncoefficient [m2s]
bull equalization and mixing (Brown-movement)
through area unit
bull depends on temperature
IN conv + diff
dxdy
dzOUT conv + diff
x direction
IN OUT
bull convection
CHANGE
bull diffusion
Mass balance Mass balance equationequation
x direction
IN conv + diff
dxdy
dzOUT conv + diff
Mass balance Mass balance equationequation
Convection Diffusion
If D(x) = const in x direction
Convection - Diffusion 1D Equation
The other directions are similar
Convection transfering Diffusion spreading
Mass balance Mass balance equationequation
x y z directions (3D)
D ndash molecular diffusion coefficient
material specific water 10-4 cm2s
space independent
slow mixing and equalization
Convection The water particles with differentconcentrations of pollutant move according to their differing flow velocity
Diffusion The adjacent water particles mix with each otherwhich causes the equalization of concentration
Mass balance equationMass balance equation
bull Laminar flow ordered with parallel flow linesbull Turbulent flow disordered random (flow and direction)
because of surface roughness (friction) rarr causes intensive mixing
v(c)
t
v deviation pulsation
T Time step of the turbulence
averagev~
0
bull Natural streams always turbulent
We usually can use the average only
TurbulenceTurbulence
Convection v middot c [ kgm2 middot s ]
0
0
Turbulent transport Turbulent transport descriptiondescription
v
turbulent diffusion
molecular diffusion
Dtx Dty Dtz gtgt D
Depends on the direction
Function of the velocity space
X direction (the other are similar)
Analogy with Fickrsquos Law rarr the original transport equation does not change only the D parameter
Turbulent diffusionTurbulent diffusion
Dx = D + Dtx Dy = D + Dty Dz = D + Dtz
Convection
Diffusion
Turbulent diffusion
Stochastic fluctuations of the flow velocity (pulsations)
Diffusive process in mathematical sense (~ Fickrsquos Law)
Convective transport (temporal differences)
Temporal averaging (T)
3D transport equation in turbulent flow3D transport equation in turbulent flow
Spatial simplifications (2D)Spatial simplifications (2D) - Dispersion - Dispersion
Depth averaged flow velocity v
0v
H
After the expansion of the convectiv part (v middot c) in x direction
Depth Integration (3D2D)
Turbulent dispersion (~ diffusion)
z
x
Velocity depth profile
Dx Dy
turbulent dispersion coefficients in 2D equation
Depth averaging (H)
11D tD transport equation in turbulent flow (cross-section avransport equation in turbulent flow (cross-section av))
Dx turbulent dispersion coefficient in 1D equation
Cross-section averaging (A)
2D transport equation in turbulent flow (depth averaged)2D transport equation in turbulent flow (depth averaged)
Turbulent dispersionTurbulent dispersion
Convectiv transport arising from the spatial differences of the velocity profile (faster and slower particles compared to the average)
v
In the 2D and 1D equations only
Dispersion coefficient a function of the velocity space (hydraulic parameters channel geometry)
Dx = Ddx + Dtx + D Dy
= Ddy + Dty + D Dx = Ddxx + Ddx + Dtx + D
Dx Dy
gtgt Dx Dx gtgt Dx
In case of more irregular channel and higher depth or larger cross-section area its value is bigger
Also present in laminar flow
Order of magnitude of Order of magnitude of diffusion dispersion diffusion dispersion coefficientscoefficients
10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s
Pore water
Mol diff
Vertical turbulent diff
High depth Shallow depth
Cross disp (2D)
Longitudinal disp (2D)
Horizontal turbulent diff
Longitudinal disp (1D)
Evaluation of dispersion coefficients measuring
Measuring with tracing matter (eg paint)
Inverse calculation from the measured concentration
2D Cross dispersion coefficient (Fischer)
Dy = dy u R (m2s)
dy - dimensionless cross dispersion coefficient
Straight regular channel dy 015
Moderately winding channel dy 02 ndash 06
Hard winding irregular channel dy gt 06 (1-2)
R ndash hydraulic radius (m)
u - shearing velocity at the channel bottom (ms)
u = (g R I)05Longitudinal dispersion coefficient dx 6 (2D) dxx = 100 ndash 1000 (1D)
Velocity space turbulent pulsation unequal distribution
Estimation of dispersion coefficients empirical forms
TRANSPORT EQUATION FOR NON-CONSERVATIVE TRANSPORT EQUATION FOR NON-CONSERVATIVE MATTERSMATTERS
bull There are sources and losses in the flow space
bull Physical chemical biochemical transformations take place
bull Non-conservative pollutants reaction kinetics ( R(C) )
bull They are taken into account in a linear way
dCdt = -C where is the coefficient of reaction kinetics (usually first-order kinetics)
cαxc
DAxx
c)v(Atc)(A
xx
1D equation in this case
ANALYTICAL SOLUTIONS OF THE TRANSPORT ANALYTICAL SOLUTIONS OF THE TRANSPORT EQUATIONEQUATION
Permanent mixing of the pollutants
Polluntant-wave transmission
Estimation of the geometric and hydraulic parameters
Exacter calculations based on measurments using numerical methods
The analitical solutions can be obtained in simpler cases only (approximate calculations)
Main steps
2
2
y
cD
x
cv yx
PERMANENT MIXING
The pollutant emission is steady in time
Permanent river discharge (low flow condition)
Constant flow velocity flow depth and dispersion coef
2D equation negligible vertical changes (shallow river)
Convection transferring
Dispersion spreading
Initial condition M0 (x0 y0) - emission
Boundary cond cy = 0 at the bank
)()()(
cvhy
cvhxt
chyx )()(
y
cDh
yx
cDh
xyx
x
yy
v
xD2
INLET AT THE MAIN FLOW PATH
Longitudinal according to x-frac12 function
Cross direction Gauss-distribution
cmax
M [kgs]cmax
)4
exp(2
2
xD
yv
xvDh
Mc (x t)y
x
xy
x
yb
v
xDB
234
Bb at value 01 cmax 1522bBBrand width
B ~ Bb2
1 0270 BD
vL
y
x
First distance of the mixing
M
1L
Bb
C (x1 y)
x1
M
x
yb
v
xDB
2152
21 110 B
D
vL
y
x
INLET AT THE RIVER BANK
)4
exp(2
xD
yv
xvDh
Mc
y
x
xy
cmax
C (x1 y)
x1
M
INLET NEAR THE RIVER BANK
)4
(exp ( xDy
( y-y0 )2-v
xvD2h
Mc x
xy
cmax
))4
+exp ( xDy
( y+y0 )2-vx
y0 C (x1 y)
x1
IMPACTS OF THE RIVER BANKS (total river section)
Boundary condition reflection theory (infinite series)
Complete mixing the change along the cross-section is less than 10
L2 ~ 3L1 second distance of the mixing
Inlet at a point in y0 distance from the bank
xvD2h
Mc
xy
)4
exp ( xDy
( y-y0 +2nB)2-vx
)4
+ exp ( xDy
( y+y0 -2nB)2-vx
sumn=infin
n=minusinfin(
)
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
Transport processesTransport processes
What is a transport processbull Describes the fate of any material in the
environmentbull The continuous steps of transportation
reactions and interactions which happen to the pollutants
bull It determines the concentration-distribution of the observed material in time and in space
bull Transmission
Transport processesTransport processes
Transport processes includebull Physical transportation due to the
movement of the mediumbull Advectionbull Diffusion and dispersion
bull Physical chemical biochemical conversion processes
bull Settlingbull Ad- and desorptionbull Reactionsbull Degradation decomposition
Transport processesTransport processes
What is our aimbull Defend our values
bull Decrease the immission by controlling the emission ndash EFFORT MONEY
bull Describe transport processes
Transport processesTransport processesImportant conceptsbull State variable concentration density temperaturehellip
bull Conservative material no reactions no settling
bull Non-conservative material opposite realistic
bull Flow processes 3D u(xyzt) h(xyzt) V(xyzt)hellip
bull Steady state dudt=0 dCdt=0hellip
bull Homogeneous distribution ndash totally mixed reactor
Transport processesTransport processesThis presentation deals only with transport
processes in surface waters So the transporting medium is water
Classical water quality state variables arebull Dissolved oxygen (DO)bull Organic material - biochemical oxygen
demand (BOD)bull Nutrients ndash N- P-forms (NO3-N PO4-Phellip)
bull Suspended solids (SS)bull Algae
Mass balanceMass balance
General mass balance equationbull Expresses the conversation of massbull Differential equations
IN ndash OUT + SOURCES ndash SINKS = CHANGE
VIN
(1)
OUT
(2)
Controlling surface
We work with mass fluxes [gs]
Mass balanceMass balanceBy solving the general mass balance equation
we can describe the transport processes for a given substance
Solutionbull Analyticalbull Numerical
Discretizationbull Temporalbull Spatial
Simplificationsbull Temporal ndash steady-statebull Spatial ndash 2D 1Dbull Other
Boundary initial conditionsbull Geometrical DATAbull Hydrologic DATAbull Water quality DATA
Simplified case ISimplified case IIN ndash OUT + SOURCES ndash SINKS = CHANGE
Assumptionsbull Steady statebull Conservative material
Solutionbull IN=OUTbull SOURCES=0 SINKS=0bull CHANGE=0
Simplified case IISimplified case IIAssumptionsbull Steady state ndash Q(t) E(t) are constants dC(t)dt=0bull Non-conservative settling material (vs)bull Prismatic river bed ndash A B H are constantsbull 1D
vs
A ~ B middot H [m2]
H
Simplified case IISimplified case II
Q
(1) (2)x
Q
IN
OUT
LOSS settled matter
C(x) is linear (assumption)
Av
v
vs
If x = O C = Co
Exponential decrease
Simplified case IISimplified case IIThe calculation
C0 concentration under the inlet
Determination of C0 value
Q
E = q middot c emission
Cbg background concentration
1D ndash Complete mixing (two water mixing with each other)
Increment
Dilution ratio
E
Simplified case IISimplified case II
The solution
Transmission coefficient
Dilution Sedimentation Conservative matter
Simplified case IISimplified case II
Clim = 15 gm3
concentration limit
Q = 15 m3s
q = 1 m3sc = 300 gm3
Cbg = 10 gm3
L = 5 km
B = 75 m
vS = 03 mh
H = 2 m
EXAMPLE
Complete mixing
3bg0 gm28125
11530011015
cqCQC
3lim
33
s0
gm15Cgm22836)10513600
0321
( exp28125
L)vv
H1
( expC L)(xC
Concentration at distance L from the emission
Flow velocity
ms10275
15HB
Qv
Maximum possible C0 value
Maximum allowed waste water concentration
Allowed maximum concentration
C (x = L)max = Clim
3
3s
limmax0
gm18474
)10513600
0321
( exp
15
L)vv
H1
( exp
C C
3
bgmax0max
gm145584
11015-11518474
q
CQ-qQCc
Transmission coefficient
Necessary reduction of emission
gs154416145584)(3001)c(cqΔE max
33
s
sm0051)10513600
0321
( exp115
1
L)vv
H1
( expqQ
1 L)(xa
The general situationThe general situation
bull Unsteadybull Inhomogeneousbull Non-conservativebull 3D
General transport equation
Advection-Dispersion Equation (ADE)
Advection-dispersion EquationAdvection-dispersion Equationbull Can be derived from the mass balancebull Expresses the temporal and spatial changes of
the observed state variablebull Itrsquos solution is the concentration-distribution over
timebull Differential equation
Change in concentration with time
Change due to advection
=Change due to diffusion or dispersion
Change due to conversion processes
+ +
Molecular diffusionMolecular diffusion
DIFFUSION
CONVECTION
v
Molecular Molecular diffusiondiffusion
FickrsquosFickrsquos LawLaw
c1 c2x
bull c1 c2 separated tanks
bull open valve
Mass flux (kgs)
D - molecular diffusioncoefficient [m2s]
bull equalization and mixing (Brown-movement)
through area unit
bull depends on temperature
IN conv + diff
dxdy
dzOUT conv + diff
x direction
IN OUT
bull convection
CHANGE
bull diffusion
Mass balance Mass balance equationequation
x direction
IN conv + diff
dxdy
dzOUT conv + diff
Mass balance Mass balance equationequation
Convection Diffusion
If D(x) = const in x direction
Convection - Diffusion 1D Equation
The other directions are similar
Convection transfering Diffusion spreading
Mass balance Mass balance equationequation
x y z directions (3D)
D ndash molecular diffusion coefficient
material specific water 10-4 cm2s
space independent
slow mixing and equalization
Convection The water particles with differentconcentrations of pollutant move according to their differing flow velocity
Diffusion The adjacent water particles mix with each otherwhich causes the equalization of concentration
Mass balance equationMass balance equation
bull Laminar flow ordered with parallel flow linesbull Turbulent flow disordered random (flow and direction)
because of surface roughness (friction) rarr causes intensive mixing
v(c)
t
v deviation pulsation
T Time step of the turbulence
averagev~
0
bull Natural streams always turbulent
We usually can use the average only
TurbulenceTurbulence
Convection v middot c [ kgm2 middot s ]
0
0
Turbulent transport Turbulent transport descriptiondescription
v
turbulent diffusion
molecular diffusion
Dtx Dty Dtz gtgt D
Depends on the direction
Function of the velocity space
X direction (the other are similar)
Analogy with Fickrsquos Law rarr the original transport equation does not change only the D parameter
Turbulent diffusionTurbulent diffusion
Dx = D + Dtx Dy = D + Dty Dz = D + Dtz
Convection
Diffusion
Turbulent diffusion
Stochastic fluctuations of the flow velocity (pulsations)
Diffusive process in mathematical sense (~ Fickrsquos Law)
Convective transport (temporal differences)
Temporal averaging (T)
3D transport equation in turbulent flow3D transport equation in turbulent flow
Spatial simplifications (2D)Spatial simplifications (2D) - Dispersion - Dispersion
Depth averaged flow velocity v
0v
H
After the expansion of the convectiv part (v middot c) in x direction
Depth Integration (3D2D)
Turbulent dispersion (~ diffusion)
z
x
Velocity depth profile
Dx Dy
turbulent dispersion coefficients in 2D equation
Depth averaging (H)
11D tD transport equation in turbulent flow (cross-section avransport equation in turbulent flow (cross-section av))
Dx turbulent dispersion coefficient in 1D equation
Cross-section averaging (A)
2D transport equation in turbulent flow (depth averaged)2D transport equation in turbulent flow (depth averaged)
Turbulent dispersionTurbulent dispersion
Convectiv transport arising from the spatial differences of the velocity profile (faster and slower particles compared to the average)
v
In the 2D and 1D equations only
Dispersion coefficient a function of the velocity space (hydraulic parameters channel geometry)
Dx = Ddx + Dtx + D Dy
= Ddy + Dty + D Dx = Ddxx + Ddx + Dtx + D
Dx Dy
gtgt Dx Dx gtgt Dx
In case of more irregular channel and higher depth or larger cross-section area its value is bigger
Also present in laminar flow
Order of magnitude of Order of magnitude of diffusion dispersion diffusion dispersion coefficientscoefficients
10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s
Pore water
Mol diff
Vertical turbulent diff
High depth Shallow depth
Cross disp (2D)
Longitudinal disp (2D)
Horizontal turbulent diff
Longitudinal disp (1D)
Evaluation of dispersion coefficients measuring
Measuring with tracing matter (eg paint)
Inverse calculation from the measured concentration
2D Cross dispersion coefficient (Fischer)
Dy = dy u R (m2s)
dy - dimensionless cross dispersion coefficient
Straight regular channel dy 015
Moderately winding channel dy 02 ndash 06
Hard winding irregular channel dy gt 06 (1-2)
R ndash hydraulic radius (m)
u - shearing velocity at the channel bottom (ms)
u = (g R I)05Longitudinal dispersion coefficient dx 6 (2D) dxx = 100 ndash 1000 (1D)
Velocity space turbulent pulsation unequal distribution
Estimation of dispersion coefficients empirical forms
TRANSPORT EQUATION FOR NON-CONSERVATIVE TRANSPORT EQUATION FOR NON-CONSERVATIVE MATTERSMATTERS
bull There are sources and losses in the flow space
bull Physical chemical biochemical transformations take place
bull Non-conservative pollutants reaction kinetics ( R(C) )
bull They are taken into account in a linear way
dCdt = -C where is the coefficient of reaction kinetics (usually first-order kinetics)
cαxc
DAxx
c)v(Atc)(A
xx
1D equation in this case
ANALYTICAL SOLUTIONS OF THE TRANSPORT ANALYTICAL SOLUTIONS OF THE TRANSPORT EQUATIONEQUATION
Permanent mixing of the pollutants
Polluntant-wave transmission
Estimation of the geometric and hydraulic parameters
Exacter calculations based on measurments using numerical methods
The analitical solutions can be obtained in simpler cases only (approximate calculations)
Main steps
2
2
y
cD
x
cv yx
PERMANENT MIXING
The pollutant emission is steady in time
Permanent river discharge (low flow condition)
Constant flow velocity flow depth and dispersion coef
2D equation negligible vertical changes (shallow river)
Convection transferring
Dispersion spreading
Initial condition M0 (x0 y0) - emission
Boundary cond cy = 0 at the bank
)()()(
cvhy
cvhxt
chyx )()(
y
cDh
yx
cDh
xyx
x
yy
v
xD2
INLET AT THE MAIN FLOW PATH
Longitudinal according to x-frac12 function
Cross direction Gauss-distribution
cmax
M [kgs]cmax
)4
exp(2
2
xD
yv
xvDh
Mc (x t)y
x
xy
x
yb
v
xDB
234
Bb at value 01 cmax 1522bBBrand width
B ~ Bb2
1 0270 BD
vL
y
x
First distance of the mixing
M
1L
Bb
C (x1 y)
x1
M
x
yb
v
xDB
2152
21 110 B
D
vL
y
x
INLET AT THE RIVER BANK
)4
exp(2
xD
yv
xvDh
Mc
y
x
xy
cmax
C (x1 y)
x1
M
INLET NEAR THE RIVER BANK
)4
(exp ( xDy
( y-y0 )2-v
xvD2h
Mc x
xy
cmax
))4
+exp ( xDy
( y+y0 )2-vx
y0 C (x1 y)
x1
IMPACTS OF THE RIVER BANKS (total river section)
Boundary condition reflection theory (infinite series)
Complete mixing the change along the cross-section is less than 10
L2 ~ 3L1 second distance of the mixing
Inlet at a point in y0 distance from the bank
xvD2h
Mc
xy
)4
exp ( xDy
( y-y0 +2nB)2-vx
)4
+ exp ( xDy
( y+y0 -2nB)2-vx
sumn=infin
n=minusinfin(
)
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
Transport processesTransport processes
Transport processes includebull Physical transportation due to the
movement of the mediumbull Advectionbull Diffusion and dispersion
bull Physical chemical biochemical conversion processes
bull Settlingbull Ad- and desorptionbull Reactionsbull Degradation decomposition
Transport processesTransport processes
What is our aimbull Defend our values
bull Decrease the immission by controlling the emission ndash EFFORT MONEY
bull Describe transport processes
Transport processesTransport processesImportant conceptsbull State variable concentration density temperaturehellip
bull Conservative material no reactions no settling
bull Non-conservative material opposite realistic
bull Flow processes 3D u(xyzt) h(xyzt) V(xyzt)hellip
bull Steady state dudt=0 dCdt=0hellip
bull Homogeneous distribution ndash totally mixed reactor
Transport processesTransport processesThis presentation deals only with transport
processes in surface waters So the transporting medium is water
Classical water quality state variables arebull Dissolved oxygen (DO)bull Organic material - biochemical oxygen
demand (BOD)bull Nutrients ndash N- P-forms (NO3-N PO4-Phellip)
bull Suspended solids (SS)bull Algae
Mass balanceMass balance
General mass balance equationbull Expresses the conversation of massbull Differential equations
IN ndash OUT + SOURCES ndash SINKS = CHANGE
VIN
(1)
OUT
(2)
Controlling surface
We work with mass fluxes [gs]
Mass balanceMass balanceBy solving the general mass balance equation
we can describe the transport processes for a given substance
Solutionbull Analyticalbull Numerical
Discretizationbull Temporalbull Spatial
Simplificationsbull Temporal ndash steady-statebull Spatial ndash 2D 1Dbull Other
Boundary initial conditionsbull Geometrical DATAbull Hydrologic DATAbull Water quality DATA
Simplified case ISimplified case IIN ndash OUT + SOURCES ndash SINKS = CHANGE
Assumptionsbull Steady statebull Conservative material
Solutionbull IN=OUTbull SOURCES=0 SINKS=0bull CHANGE=0
Simplified case IISimplified case IIAssumptionsbull Steady state ndash Q(t) E(t) are constants dC(t)dt=0bull Non-conservative settling material (vs)bull Prismatic river bed ndash A B H are constantsbull 1D
vs
A ~ B middot H [m2]
H
Simplified case IISimplified case II
Q
(1) (2)x
Q
IN
OUT
LOSS settled matter
C(x) is linear (assumption)
Av
v
vs
If x = O C = Co
Exponential decrease
Simplified case IISimplified case IIThe calculation
C0 concentration under the inlet
Determination of C0 value
Q
E = q middot c emission
Cbg background concentration
1D ndash Complete mixing (two water mixing with each other)
Increment
Dilution ratio
E
Simplified case IISimplified case II
The solution
Transmission coefficient
Dilution Sedimentation Conservative matter
Simplified case IISimplified case II
Clim = 15 gm3
concentration limit
Q = 15 m3s
q = 1 m3sc = 300 gm3
Cbg = 10 gm3
L = 5 km
B = 75 m
vS = 03 mh
H = 2 m
EXAMPLE
Complete mixing
3bg0 gm28125
11530011015
cqCQC
3lim
33
s0
gm15Cgm22836)10513600
0321
( exp28125
L)vv
H1
( expC L)(xC
Concentration at distance L from the emission
Flow velocity
ms10275
15HB
Qv
Maximum possible C0 value
Maximum allowed waste water concentration
Allowed maximum concentration
C (x = L)max = Clim
3
3s
limmax0
gm18474
)10513600
0321
( exp
15
L)vv
H1
( exp
C C
3
bgmax0max
gm145584
11015-11518474
q
CQ-qQCc
Transmission coefficient
Necessary reduction of emission
gs154416145584)(3001)c(cqΔE max
33
s
sm0051)10513600
0321
( exp115
1
L)vv
H1
( expqQ
1 L)(xa
The general situationThe general situation
bull Unsteadybull Inhomogeneousbull Non-conservativebull 3D
General transport equation
Advection-Dispersion Equation (ADE)
Advection-dispersion EquationAdvection-dispersion Equationbull Can be derived from the mass balancebull Expresses the temporal and spatial changes of
the observed state variablebull Itrsquos solution is the concentration-distribution over
timebull Differential equation
Change in concentration with time
Change due to advection
=Change due to diffusion or dispersion
Change due to conversion processes
+ +
Molecular diffusionMolecular diffusion
DIFFUSION
CONVECTION
v
Molecular Molecular diffusiondiffusion
FickrsquosFickrsquos LawLaw
c1 c2x
bull c1 c2 separated tanks
bull open valve
Mass flux (kgs)
D - molecular diffusioncoefficient [m2s]
bull equalization and mixing (Brown-movement)
through area unit
bull depends on temperature
IN conv + diff
dxdy
dzOUT conv + diff
x direction
IN OUT
bull convection
CHANGE
bull diffusion
Mass balance Mass balance equationequation
x direction
IN conv + diff
dxdy
dzOUT conv + diff
Mass balance Mass balance equationequation
Convection Diffusion
If D(x) = const in x direction
Convection - Diffusion 1D Equation
The other directions are similar
Convection transfering Diffusion spreading
Mass balance Mass balance equationequation
x y z directions (3D)
D ndash molecular diffusion coefficient
material specific water 10-4 cm2s
space independent
slow mixing and equalization
Convection The water particles with differentconcentrations of pollutant move according to their differing flow velocity
Diffusion The adjacent water particles mix with each otherwhich causes the equalization of concentration
Mass balance equationMass balance equation
bull Laminar flow ordered with parallel flow linesbull Turbulent flow disordered random (flow and direction)
because of surface roughness (friction) rarr causes intensive mixing
v(c)
t
v deviation pulsation
T Time step of the turbulence
averagev~
0
bull Natural streams always turbulent
We usually can use the average only
TurbulenceTurbulence
Convection v middot c [ kgm2 middot s ]
0
0
Turbulent transport Turbulent transport descriptiondescription
v
turbulent diffusion
molecular diffusion
Dtx Dty Dtz gtgt D
Depends on the direction
Function of the velocity space
X direction (the other are similar)
Analogy with Fickrsquos Law rarr the original transport equation does not change only the D parameter
Turbulent diffusionTurbulent diffusion
Dx = D + Dtx Dy = D + Dty Dz = D + Dtz
Convection
Diffusion
Turbulent diffusion
Stochastic fluctuations of the flow velocity (pulsations)
Diffusive process in mathematical sense (~ Fickrsquos Law)
Convective transport (temporal differences)
Temporal averaging (T)
3D transport equation in turbulent flow3D transport equation in turbulent flow
Spatial simplifications (2D)Spatial simplifications (2D) - Dispersion - Dispersion
Depth averaged flow velocity v
0v
H
After the expansion of the convectiv part (v middot c) in x direction
Depth Integration (3D2D)
Turbulent dispersion (~ diffusion)
z
x
Velocity depth profile
Dx Dy
turbulent dispersion coefficients in 2D equation
Depth averaging (H)
11D tD transport equation in turbulent flow (cross-section avransport equation in turbulent flow (cross-section av))
Dx turbulent dispersion coefficient in 1D equation
Cross-section averaging (A)
2D transport equation in turbulent flow (depth averaged)2D transport equation in turbulent flow (depth averaged)
Turbulent dispersionTurbulent dispersion
Convectiv transport arising from the spatial differences of the velocity profile (faster and slower particles compared to the average)
v
In the 2D and 1D equations only
Dispersion coefficient a function of the velocity space (hydraulic parameters channel geometry)
Dx = Ddx + Dtx + D Dy
= Ddy + Dty + D Dx = Ddxx + Ddx + Dtx + D
Dx Dy
gtgt Dx Dx gtgt Dx
In case of more irregular channel and higher depth or larger cross-section area its value is bigger
Also present in laminar flow
Order of magnitude of Order of magnitude of diffusion dispersion diffusion dispersion coefficientscoefficients
10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s
Pore water
Mol diff
Vertical turbulent diff
High depth Shallow depth
Cross disp (2D)
Longitudinal disp (2D)
Horizontal turbulent diff
Longitudinal disp (1D)
Evaluation of dispersion coefficients measuring
Measuring with tracing matter (eg paint)
Inverse calculation from the measured concentration
2D Cross dispersion coefficient (Fischer)
Dy = dy u R (m2s)
dy - dimensionless cross dispersion coefficient
Straight regular channel dy 015
Moderately winding channel dy 02 ndash 06
Hard winding irregular channel dy gt 06 (1-2)
R ndash hydraulic radius (m)
u - shearing velocity at the channel bottom (ms)
u = (g R I)05Longitudinal dispersion coefficient dx 6 (2D) dxx = 100 ndash 1000 (1D)
Velocity space turbulent pulsation unequal distribution
Estimation of dispersion coefficients empirical forms
TRANSPORT EQUATION FOR NON-CONSERVATIVE TRANSPORT EQUATION FOR NON-CONSERVATIVE MATTERSMATTERS
bull There are sources and losses in the flow space
bull Physical chemical biochemical transformations take place
bull Non-conservative pollutants reaction kinetics ( R(C) )
bull They are taken into account in a linear way
dCdt = -C where is the coefficient of reaction kinetics (usually first-order kinetics)
cαxc
DAxx
c)v(Atc)(A
xx
1D equation in this case
ANALYTICAL SOLUTIONS OF THE TRANSPORT ANALYTICAL SOLUTIONS OF THE TRANSPORT EQUATIONEQUATION
Permanent mixing of the pollutants
Polluntant-wave transmission
Estimation of the geometric and hydraulic parameters
Exacter calculations based on measurments using numerical methods
The analitical solutions can be obtained in simpler cases only (approximate calculations)
Main steps
2
2
y
cD
x
cv yx
PERMANENT MIXING
The pollutant emission is steady in time
Permanent river discharge (low flow condition)
Constant flow velocity flow depth and dispersion coef
2D equation negligible vertical changes (shallow river)
Convection transferring
Dispersion spreading
Initial condition M0 (x0 y0) - emission
Boundary cond cy = 0 at the bank
)()()(
cvhy
cvhxt
chyx )()(
y
cDh
yx
cDh
xyx
x
yy
v
xD2
INLET AT THE MAIN FLOW PATH
Longitudinal according to x-frac12 function
Cross direction Gauss-distribution
cmax
M [kgs]cmax
)4
exp(2
2
xD
yv
xvDh
Mc (x t)y
x
xy
x
yb
v
xDB
234
Bb at value 01 cmax 1522bBBrand width
B ~ Bb2
1 0270 BD
vL
y
x
First distance of the mixing
M
1L
Bb
C (x1 y)
x1
M
x
yb
v
xDB
2152
21 110 B
D
vL
y
x
INLET AT THE RIVER BANK
)4
exp(2
xD
yv
xvDh
Mc
y
x
xy
cmax
C (x1 y)
x1
M
INLET NEAR THE RIVER BANK
)4
(exp ( xDy
( y-y0 )2-v
xvD2h
Mc x
xy
cmax
))4
+exp ( xDy
( y+y0 )2-vx
y0 C (x1 y)
x1
IMPACTS OF THE RIVER BANKS (total river section)
Boundary condition reflection theory (infinite series)
Complete mixing the change along the cross-section is less than 10
L2 ~ 3L1 second distance of the mixing
Inlet at a point in y0 distance from the bank
xvD2h
Mc
xy
)4
exp ( xDy
( y-y0 +2nB)2-vx
)4
+ exp ( xDy
( y+y0 -2nB)2-vx
sumn=infin
n=minusinfin(
)
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
Transport processesTransport processes
What is our aimbull Defend our values
bull Decrease the immission by controlling the emission ndash EFFORT MONEY
bull Describe transport processes
Transport processesTransport processesImportant conceptsbull State variable concentration density temperaturehellip
bull Conservative material no reactions no settling
bull Non-conservative material opposite realistic
bull Flow processes 3D u(xyzt) h(xyzt) V(xyzt)hellip
bull Steady state dudt=0 dCdt=0hellip
bull Homogeneous distribution ndash totally mixed reactor
Transport processesTransport processesThis presentation deals only with transport
processes in surface waters So the transporting medium is water
Classical water quality state variables arebull Dissolved oxygen (DO)bull Organic material - biochemical oxygen
demand (BOD)bull Nutrients ndash N- P-forms (NO3-N PO4-Phellip)
bull Suspended solids (SS)bull Algae
Mass balanceMass balance
General mass balance equationbull Expresses the conversation of massbull Differential equations
IN ndash OUT + SOURCES ndash SINKS = CHANGE
VIN
(1)
OUT
(2)
Controlling surface
We work with mass fluxes [gs]
Mass balanceMass balanceBy solving the general mass balance equation
we can describe the transport processes for a given substance
Solutionbull Analyticalbull Numerical
Discretizationbull Temporalbull Spatial
Simplificationsbull Temporal ndash steady-statebull Spatial ndash 2D 1Dbull Other
Boundary initial conditionsbull Geometrical DATAbull Hydrologic DATAbull Water quality DATA
Simplified case ISimplified case IIN ndash OUT + SOURCES ndash SINKS = CHANGE
Assumptionsbull Steady statebull Conservative material
Solutionbull IN=OUTbull SOURCES=0 SINKS=0bull CHANGE=0
Simplified case IISimplified case IIAssumptionsbull Steady state ndash Q(t) E(t) are constants dC(t)dt=0bull Non-conservative settling material (vs)bull Prismatic river bed ndash A B H are constantsbull 1D
vs
A ~ B middot H [m2]
H
Simplified case IISimplified case II
Q
(1) (2)x
Q
IN
OUT
LOSS settled matter
C(x) is linear (assumption)
Av
v
vs
If x = O C = Co
Exponential decrease
Simplified case IISimplified case IIThe calculation
C0 concentration under the inlet
Determination of C0 value
Q
E = q middot c emission
Cbg background concentration
1D ndash Complete mixing (two water mixing with each other)
Increment
Dilution ratio
E
Simplified case IISimplified case II
The solution
Transmission coefficient
Dilution Sedimentation Conservative matter
Simplified case IISimplified case II
Clim = 15 gm3
concentration limit
Q = 15 m3s
q = 1 m3sc = 300 gm3
Cbg = 10 gm3
L = 5 km
B = 75 m
vS = 03 mh
H = 2 m
EXAMPLE
Complete mixing
3bg0 gm28125
11530011015
cqCQC
3lim
33
s0
gm15Cgm22836)10513600
0321
( exp28125
L)vv
H1
( expC L)(xC
Concentration at distance L from the emission
Flow velocity
ms10275
15HB
Qv
Maximum possible C0 value
Maximum allowed waste water concentration
Allowed maximum concentration
C (x = L)max = Clim
3
3s
limmax0
gm18474
)10513600
0321
( exp
15
L)vv
H1
( exp
C C
3
bgmax0max
gm145584
11015-11518474
q
CQ-qQCc
Transmission coefficient
Necessary reduction of emission
gs154416145584)(3001)c(cqΔE max
33
s
sm0051)10513600
0321
( exp115
1
L)vv
H1
( expqQ
1 L)(xa
The general situationThe general situation
bull Unsteadybull Inhomogeneousbull Non-conservativebull 3D
General transport equation
Advection-Dispersion Equation (ADE)
Advection-dispersion EquationAdvection-dispersion Equationbull Can be derived from the mass balancebull Expresses the temporal and spatial changes of
the observed state variablebull Itrsquos solution is the concentration-distribution over
timebull Differential equation
Change in concentration with time
Change due to advection
=Change due to diffusion or dispersion
Change due to conversion processes
+ +
Molecular diffusionMolecular diffusion
DIFFUSION
CONVECTION
v
Molecular Molecular diffusiondiffusion
FickrsquosFickrsquos LawLaw
c1 c2x
bull c1 c2 separated tanks
bull open valve
Mass flux (kgs)
D - molecular diffusioncoefficient [m2s]
bull equalization and mixing (Brown-movement)
through area unit
bull depends on temperature
IN conv + diff
dxdy
dzOUT conv + diff
x direction
IN OUT
bull convection
CHANGE
bull diffusion
Mass balance Mass balance equationequation
x direction
IN conv + diff
dxdy
dzOUT conv + diff
Mass balance Mass balance equationequation
Convection Diffusion
If D(x) = const in x direction
Convection - Diffusion 1D Equation
The other directions are similar
Convection transfering Diffusion spreading
Mass balance Mass balance equationequation
x y z directions (3D)
D ndash molecular diffusion coefficient
material specific water 10-4 cm2s
space independent
slow mixing and equalization
Convection The water particles with differentconcentrations of pollutant move according to their differing flow velocity
Diffusion The adjacent water particles mix with each otherwhich causes the equalization of concentration
Mass balance equationMass balance equation
bull Laminar flow ordered with parallel flow linesbull Turbulent flow disordered random (flow and direction)
because of surface roughness (friction) rarr causes intensive mixing
v(c)
t
v deviation pulsation
T Time step of the turbulence
averagev~
0
bull Natural streams always turbulent
We usually can use the average only
TurbulenceTurbulence
Convection v middot c [ kgm2 middot s ]
0
0
Turbulent transport Turbulent transport descriptiondescription
v
turbulent diffusion
molecular diffusion
Dtx Dty Dtz gtgt D
Depends on the direction
Function of the velocity space
X direction (the other are similar)
Analogy with Fickrsquos Law rarr the original transport equation does not change only the D parameter
Turbulent diffusionTurbulent diffusion
Dx = D + Dtx Dy = D + Dty Dz = D + Dtz
Convection
Diffusion
Turbulent diffusion
Stochastic fluctuations of the flow velocity (pulsations)
Diffusive process in mathematical sense (~ Fickrsquos Law)
Convective transport (temporal differences)
Temporal averaging (T)
3D transport equation in turbulent flow3D transport equation in turbulent flow
Spatial simplifications (2D)Spatial simplifications (2D) - Dispersion - Dispersion
Depth averaged flow velocity v
0v
H
After the expansion of the convectiv part (v middot c) in x direction
Depth Integration (3D2D)
Turbulent dispersion (~ diffusion)
z
x
Velocity depth profile
Dx Dy
turbulent dispersion coefficients in 2D equation
Depth averaging (H)
11D tD transport equation in turbulent flow (cross-section avransport equation in turbulent flow (cross-section av))
Dx turbulent dispersion coefficient in 1D equation
Cross-section averaging (A)
2D transport equation in turbulent flow (depth averaged)2D transport equation in turbulent flow (depth averaged)
Turbulent dispersionTurbulent dispersion
Convectiv transport arising from the spatial differences of the velocity profile (faster and slower particles compared to the average)
v
In the 2D and 1D equations only
Dispersion coefficient a function of the velocity space (hydraulic parameters channel geometry)
Dx = Ddx + Dtx + D Dy
= Ddy + Dty + D Dx = Ddxx + Ddx + Dtx + D
Dx Dy
gtgt Dx Dx gtgt Dx
In case of more irregular channel and higher depth or larger cross-section area its value is bigger
Also present in laminar flow
Order of magnitude of Order of magnitude of diffusion dispersion diffusion dispersion coefficientscoefficients
10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s
Pore water
Mol diff
Vertical turbulent diff
High depth Shallow depth
Cross disp (2D)
Longitudinal disp (2D)
Horizontal turbulent diff
Longitudinal disp (1D)
Evaluation of dispersion coefficients measuring
Measuring with tracing matter (eg paint)
Inverse calculation from the measured concentration
2D Cross dispersion coefficient (Fischer)
Dy = dy u R (m2s)
dy - dimensionless cross dispersion coefficient
Straight regular channel dy 015
Moderately winding channel dy 02 ndash 06
Hard winding irregular channel dy gt 06 (1-2)
R ndash hydraulic radius (m)
u - shearing velocity at the channel bottom (ms)
u = (g R I)05Longitudinal dispersion coefficient dx 6 (2D) dxx = 100 ndash 1000 (1D)
Velocity space turbulent pulsation unequal distribution
Estimation of dispersion coefficients empirical forms
TRANSPORT EQUATION FOR NON-CONSERVATIVE TRANSPORT EQUATION FOR NON-CONSERVATIVE MATTERSMATTERS
bull There are sources and losses in the flow space
bull Physical chemical biochemical transformations take place
bull Non-conservative pollutants reaction kinetics ( R(C) )
bull They are taken into account in a linear way
dCdt = -C where is the coefficient of reaction kinetics (usually first-order kinetics)
cαxc
DAxx
c)v(Atc)(A
xx
1D equation in this case
ANALYTICAL SOLUTIONS OF THE TRANSPORT ANALYTICAL SOLUTIONS OF THE TRANSPORT EQUATIONEQUATION
Permanent mixing of the pollutants
Polluntant-wave transmission
Estimation of the geometric and hydraulic parameters
Exacter calculations based on measurments using numerical methods
The analitical solutions can be obtained in simpler cases only (approximate calculations)
Main steps
2
2
y
cD
x
cv yx
PERMANENT MIXING
The pollutant emission is steady in time
Permanent river discharge (low flow condition)
Constant flow velocity flow depth and dispersion coef
2D equation negligible vertical changes (shallow river)
Convection transferring
Dispersion spreading
Initial condition M0 (x0 y0) - emission
Boundary cond cy = 0 at the bank
)()()(
cvhy
cvhxt
chyx )()(
y
cDh
yx
cDh
xyx
x
yy
v
xD2
INLET AT THE MAIN FLOW PATH
Longitudinal according to x-frac12 function
Cross direction Gauss-distribution
cmax
M [kgs]cmax
)4
exp(2
2
xD
yv
xvDh
Mc (x t)y
x
xy
x
yb
v
xDB
234
Bb at value 01 cmax 1522bBBrand width
B ~ Bb2
1 0270 BD
vL
y
x
First distance of the mixing
M
1L
Bb
C (x1 y)
x1
M
x
yb
v
xDB
2152
21 110 B
D
vL
y
x
INLET AT THE RIVER BANK
)4
exp(2
xD
yv
xvDh
Mc
y
x
xy
cmax
C (x1 y)
x1
M
INLET NEAR THE RIVER BANK
)4
(exp ( xDy
( y-y0 )2-v
xvD2h
Mc x
xy
cmax
))4
+exp ( xDy
( y+y0 )2-vx
y0 C (x1 y)
x1
IMPACTS OF THE RIVER BANKS (total river section)
Boundary condition reflection theory (infinite series)
Complete mixing the change along the cross-section is less than 10
L2 ~ 3L1 second distance of the mixing
Inlet at a point in y0 distance from the bank
xvD2h
Mc
xy
)4
exp ( xDy
( y-y0 +2nB)2-vx
)4
+ exp ( xDy
( y+y0 -2nB)2-vx
sumn=infin
n=minusinfin(
)
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
Transport processesTransport processesImportant conceptsbull State variable concentration density temperaturehellip
bull Conservative material no reactions no settling
bull Non-conservative material opposite realistic
bull Flow processes 3D u(xyzt) h(xyzt) V(xyzt)hellip
bull Steady state dudt=0 dCdt=0hellip
bull Homogeneous distribution ndash totally mixed reactor
Transport processesTransport processesThis presentation deals only with transport
processes in surface waters So the transporting medium is water
Classical water quality state variables arebull Dissolved oxygen (DO)bull Organic material - biochemical oxygen
demand (BOD)bull Nutrients ndash N- P-forms (NO3-N PO4-Phellip)
bull Suspended solids (SS)bull Algae
Mass balanceMass balance
General mass balance equationbull Expresses the conversation of massbull Differential equations
IN ndash OUT + SOURCES ndash SINKS = CHANGE
VIN
(1)
OUT
(2)
Controlling surface
We work with mass fluxes [gs]
Mass balanceMass balanceBy solving the general mass balance equation
we can describe the transport processes for a given substance
Solutionbull Analyticalbull Numerical
Discretizationbull Temporalbull Spatial
Simplificationsbull Temporal ndash steady-statebull Spatial ndash 2D 1Dbull Other
Boundary initial conditionsbull Geometrical DATAbull Hydrologic DATAbull Water quality DATA
Simplified case ISimplified case IIN ndash OUT + SOURCES ndash SINKS = CHANGE
Assumptionsbull Steady statebull Conservative material
Solutionbull IN=OUTbull SOURCES=0 SINKS=0bull CHANGE=0
Simplified case IISimplified case IIAssumptionsbull Steady state ndash Q(t) E(t) are constants dC(t)dt=0bull Non-conservative settling material (vs)bull Prismatic river bed ndash A B H are constantsbull 1D
vs
A ~ B middot H [m2]
H
Simplified case IISimplified case II
Q
(1) (2)x
Q
IN
OUT
LOSS settled matter
C(x) is linear (assumption)
Av
v
vs
If x = O C = Co
Exponential decrease
Simplified case IISimplified case IIThe calculation
C0 concentration under the inlet
Determination of C0 value
Q
E = q middot c emission
Cbg background concentration
1D ndash Complete mixing (two water mixing with each other)
Increment
Dilution ratio
E
Simplified case IISimplified case II
The solution
Transmission coefficient
Dilution Sedimentation Conservative matter
Simplified case IISimplified case II
Clim = 15 gm3
concentration limit
Q = 15 m3s
q = 1 m3sc = 300 gm3
Cbg = 10 gm3
L = 5 km
B = 75 m
vS = 03 mh
H = 2 m
EXAMPLE
Complete mixing
3bg0 gm28125
11530011015
cqCQC
3lim
33
s0
gm15Cgm22836)10513600
0321
( exp28125
L)vv
H1
( expC L)(xC
Concentration at distance L from the emission
Flow velocity
ms10275
15HB
Qv
Maximum possible C0 value
Maximum allowed waste water concentration
Allowed maximum concentration
C (x = L)max = Clim
3
3s
limmax0
gm18474
)10513600
0321
( exp
15
L)vv
H1
( exp
C C
3
bgmax0max
gm145584
11015-11518474
q
CQ-qQCc
Transmission coefficient
Necessary reduction of emission
gs154416145584)(3001)c(cqΔE max
33
s
sm0051)10513600
0321
( exp115
1
L)vv
H1
( expqQ
1 L)(xa
The general situationThe general situation
bull Unsteadybull Inhomogeneousbull Non-conservativebull 3D
General transport equation
Advection-Dispersion Equation (ADE)
Advection-dispersion EquationAdvection-dispersion Equationbull Can be derived from the mass balancebull Expresses the temporal and spatial changes of
the observed state variablebull Itrsquos solution is the concentration-distribution over
timebull Differential equation
Change in concentration with time
Change due to advection
=Change due to diffusion or dispersion
Change due to conversion processes
+ +
Molecular diffusionMolecular diffusion
DIFFUSION
CONVECTION
v
Molecular Molecular diffusiondiffusion
FickrsquosFickrsquos LawLaw
c1 c2x
bull c1 c2 separated tanks
bull open valve
Mass flux (kgs)
D - molecular diffusioncoefficient [m2s]
bull equalization and mixing (Brown-movement)
through area unit
bull depends on temperature
IN conv + diff
dxdy
dzOUT conv + diff
x direction
IN OUT
bull convection
CHANGE
bull diffusion
Mass balance Mass balance equationequation
x direction
IN conv + diff
dxdy
dzOUT conv + diff
Mass balance Mass balance equationequation
Convection Diffusion
If D(x) = const in x direction
Convection - Diffusion 1D Equation
The other directions are similar
Convection transfering Diffusion spreading
Mass balance Mass balance equationequation
x y z directions (3D)
D ndash molecular diffusion coefficient
material specific water 10-4 cm2s
space independent
slow mixing and equalization
Convection The water particles with differentconcentrations of pollutant move according to their differing flow velocity
Diffusion The adjacent water particles mix with each otherwhich causes the equalization of concentration
Mass balance equationMass balance equation
bull Laminar flow ordered with parallel flow linesbull Turbulent flow disordered random (flow and direction)
because of surface roughness (friction) rarr causes intensive mixing
v(c)
t
v deviation pulsation
T Time step of the turbulence
averagev~
0
bull Natural streams always turbulent
We usually can use the average only
TurbulenceTurbulence
Convection v middot c [ kgm2 middot s ]
0
0
Turbulent transport Turbulent transport descriptiondescription
v
turbulent diffusion
molecular diffusion
Dtx Dty Dtz gtgt D
Depends on the direction
Function of the velocity space
X direction (the other are similar)
Analogy with Fickrsquos Law rarr the original transport equation does not change only the D parameter
Turbulent diffusionTurbulent diffusion
Dx = D + Dtx Dy = D + Dty Dz = D + Dtz
Convection
Diffusion
Turbulent diffusion
Stochastic fluctuations of the flow velocity (pulsations)
Diffusive process in mathematical sense (~ Fickrsquos Law)
Convective transport (temporal differences)
Temporal averaging (T)
3D transport equation in turbulent flow3D transport equation in turbulent flow
Spatial simplifications (2D)Spatial simplifications (2D) - Dispersion - Dispersion
Depth averaged flow velocity v
0v
H
After the expansion of the convectiv part (v middot c) in x direction
Depth Integration (3D2D)
Turbulent dispersion (~ diffusion)
z
x
Velocity depth profile
Dx Dy
turbulent dispersion coefficients in 2D equation
Depth averaging (H)
11D tD transport equation in turbulent flow (cross-section avransport equation in turbulent flow (cross-section av))
Dx turbulent dispersion coefficient in 1D equation
Cross-section averaging (A)
2D transport equation in turbulent flow (depth averaged)2D transport equation in turbulent flow (depth averaged)
Turbulent dispersionTurbulent dispersion
Convectiv transport arising from the spatial differences of the velocity profile (faster and slower particles compared to the average)
v
In the 2D and 1D equations only
Dispersion coefficient a function of the velocity space (hydraulic parameters channel geometry)
Dx = Ddx + Dtx + D Dy
= Ddy + Dty + D Dx = Ddxx + Ddx + Dtx + D
Dx Dy
gtgt Dx Dx gtgt Dx
In case of more irregular channel and higher depth or larger cross-section area its value is bigger
Also present in laminar flow
Order of magnitude of Order of magnitude of diffusion dispersion diffusion dispersion coefficientscoefficients
10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s
Pore water
Mol diff
Vertical turbulent diff
High depth Shallow depth
Cross disp (2D)
Longitudinal disp (2D)
Horizontal turbulent diff
Longitudinal disp (1D)
Evaluation of dispersion coefficients measuring
Measuring with tracing matter (eg paint)
Inverse calculation from the measured concentration
2D Cross dispersion coefficient (Fischer)
Dy = dy u R (m2s)
dy - dimensionless cross dispersion coefficient
Straight regular channel dy 015
Moderately winding channel dy 02 ndash 06
Hard winding irregular channel dy gt 06 (1-2)
R ndash hydraulic radius (m)
u - shearing velocity at the channel bottom (ms)
u = (g R I)05Longitudinal dispersion coefficient dx 6 (2D) dxx = 100 ndash 1000 (1D)
Velocity space turbulent pulsation unequal distribution
Estimation of dispersion coefficients empirical forms
TRANSPORT EQUATION FOR NON-CONSERVATIVE TRANSPORT EQUATION FOR NON-CONSERVATIVE MATTERSMATTERS
bull There are sources and losses in the flow space
bull Physical chemical biochemical transformations take place
bull Non-conservative pollutants reaction kinetics ( R(C) )
bull They are taken into account in a linear way
dCdt = -C where is the coefficient of reaction kinetics (usually first-order kinetics)
cαxc
DAxx
c)v(Atc)(A
xx
1D equation in this case
ANALYTICAL SOLUTIONS OF THE TRANSPORT ANALYTICAL SOLUTIONS OF THE TRANSPORT EQUATIONEQUATION
Permanent mixing of the pollutants
Polluntant-wave transmission
Estimation of the geometric and hydraulic parameters
Exacter calculations based on measurments using numerical methods
The analitical solutions can be obtained in simpler cases only (approximate calculations)
Main steps
2
2
y
cD
x
cv yx
PERMANENT MIXING
The pollutant emission is steady in time
Permanent river discharge (low flow condition)
Constant flow velocity flow depth and dispersion coef
2D equation negligible vertical changes (shallow river)
Convection transferring
Dispersion spreading
Initial condition M0 (x0 y0) - emission
Boundary cond cy = 0 at the bank
)()()(
cvhy
cvhxt
chyx )()(
y
cDh
yx
cDh
xyx
x
yy
v
xD2
INLET AT THE MAIN FLOW PATH
Longitudinal according to x-frac12 function
Cross direction Gauss-distribution
cmax
M [kgs]cmax
)4
exp(2
2
xD
yv
xvDh
Mc (x t)y
x
xy
x
yb
v
xDB
234
Bb at value 01 cmax 1522bBBrand width
B ~ Bb2
1 0270 BD
vL
y
x
First distance of the mixing
M
1L
Bb
C (x1 y)
x1
M
x
yb
v
xDB
2152
21 110 B
D
vL
y
x
INLET AT THE RIVER BANK
)4
exp(2
xD
yv
xvDh
Mc
y
x
xy
cmax
C (x1 y)
x1
M
INLET NEAR THE RIVER BANK
)4
(exp ( xDy
( y-y0 )2-v
xvD2h
Mc x
xy
cmax
))4
+exp ( xDy
( y+y0 )2-vx
y0 C (x1 y)
x1
IMPACTS OF THE RIVER BANKS (total river section)
Boundary condition reflection theory (infinite series)
Complete mixing the change along the cross-section is less than 10
L2 ~ 3L1 second distance of the mixing
Inlet at a point in y0 distance from the bank
xvD2h
Mc
xy
)4
exp ( xDy
( y-y0 +2nB)2-vx
)4
+ exp ( xDy
( y+y0 -2nB)2-vx
sumn=infin
n=minusinfin(
)
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
Transport processesTransport processesThis presentation deals only with transport
processes in surface waters So the transporting medium is water
Classical water quality state variables arebull Dissolved oxygen (DO)bull Organic material - biochemical oxygen
demand (BOD)bull Nutrients ndash N- P-forms (NO3-N PO4-Phellip)
bull Suspended solids (SS)bull Algae
Mass balanceMass balance
General mass balance equationbull Expresses the conversation of massbull Differential equations
IN ndash OUT + SOURCES ndash SINKS = CHANGE
VIN
(1)
OUT
(2)
Controlling surface
We work with mass fluxes [gs]
Mass balanceMass balanceBy solving the general mass balance equation
we can describe the transport processes for a given substance
Solutionbull Analyticalbull Numerical
Discretizationbull Temporalbull Spatial
Simplificationsbull Temporal ndash steady-statebull Spatial ndash 2D 1Dbull Other
Boundary initial conditionsbull Geometrical DATAbull Hydrologic DATAbull Water quality DATA
Simplified case ISimplified case IIN ndash OUT + SOURCES ndash SINKS = CHANGE
Assumptionsbull Steady statebull Conservative material
Solutionbull IN=OUTbull SOURCES=0 SINKS=0bull CHANGE=0
Simplified case IISimplified case IIAssumptionsbull Steady state ndash Q(t) E(t) are constants dC(t)dt=0bull Non-conservative settling material (vs)bull Prismatic river bed ndash A B H are constantsbull 1D
vs
A ~ B middot H [m2]
H
Simplified case IISimplified case II
Q
(1) (2)x
Q
IN
OUT
LOSS settled matter
C(x) is linear (assumption)
Av
v
vs
If x = O C = Co
Exponential decrease
Simplified case IISimplified case IIThe calculation
C0 concentration under the inlet
Determination of C0 value
Q
E = q middot c emission
Cbg background concentration
1D ndash Complete mixing (two water mixing with each other)
Increment
Dilution ratio
E
Simplified case IISimplified case II
The solution
Transmission coefficient
Dilution Sedimentation Conservative matter
Simplified case IISimplified case II
Clim = 15 gm3
concentration limit
Q = 15 m3s
q = 1 m3sc = 300 gm3
Cbg = 10 gm3
L = 5 km
B = 75 m
vS = 03 mh
H = 2 m
EXAMPLE
Complete mixing
3bg0 gm28125
11530011015
cqCQC
3lim
33
s0
gm15Cgm22836)10513600
0321
( exp28125
L)vv
H1
( expC L)(xC
Concentration at distance L from the emission
Flow velocity
ms10275
15HB
Qv
Maximum possible C0 value
Maximum allowed waste water concentration
Allowed maximum concentration
C (x = L)max = Clim
3
3s
limmax0
gm18474
)10513600
0321
( exp
15
L)vv
H1
( exp
C C
3
bgmax0max
gm145584
11015-11518474
q
CQ-qQCc
Transmission coefficient
Necessary reduction of emission
gs154416145584)(3001)c(cqΔE max
33
s
sm0051)10513600
0321
( exp115
1
L)vv
H1
( expqQ
1 L)(xa
The general situationThe general situation
bull Unsteadybull Inhomogeneousbull Non-conservativebull 3D
General transport equation
Advection-Dispersion Equation (ADE)
Advection-dispersion EquationAdvection-dispersion Equationbull Can be derived from the mass balancebull Expresses the temporal and spatial changes of
the observed state variablebull Itrsquos solution is the concentration-distribution over
timebull Differential equation
Change in concentration with time
Change due to advection
=Change due to diffusion or dispersion
Change due to conversion processes
+ +
Molecular diffusionMolecular diffusion
DIFFUSION
CONVECTION
v
Molecular Molecular diffusiondiffusion
FickrsquosFickrsquos LawLaw
c1 c2x
bull c1 c2 separated tanks
bull open valve
Mass flux (kgs)
D - molecular diffusioncoefficient [m2s]
bull equalization and mixing (Brown-movement)
through area unit
bull depends on temperature
IN conv + diff
dxdy
dzOUT conv + diff
x direction
IN OUT
bull convection
CHANGE
bull diffusion
Mass balance Mass balance equationequation
x direction
IN conv + diff
dxdy
dzOUT conv + diff
Mass balance Mass balance equationequation
Convection Diffusion
If D(x) = const in x direction
Convection - Diffusion 1D Equation
The other directions are similar
Convection transfering Diffusion spreading
Mass balance Mass balance equationequation
x y z directions (3D)
D ndash molecular diffusion coefficient
material specific water 10-4 cm2s
space independent
slow mixing and equalization
Convection The water particles with differentconcentrations of pollutant move according to their differing flow velocity
Diffusion The adjacent water particles mix with each otherwhich causes the equalization of concentration
Mass balance equationMass balance equation
bull Laminar flow ordered with parallel flow linesbull Turbulent flow disordered random (flow and direction)
because of surface roughness (friction) rarr causes intensive mixing
v(c)
t
v deviation pulsation
T Time step of the turbulence
averagev~
0
bull Natural streams always turbulent
We usually can use the average only
TurbulenceTurbulence
Convection v middot c [ kgm2 middot s ]
0
0
Turbulent transport Turbulent transport descriptiondescription
v
turbulent diffusion
molecular diffusion
Dtx Dty Dtz gtgt D
Depends on the direction
Function of the velocity space
X direction (the other are similar)
Analogy with Fickrsquos Law rarr the original transport equation does not change only the D parameter
Turbulent diffusionTurbulent diffusion
Dx = D + Dtx Dy = D + Dty Dz = D + Dtz
Convection
Diffusion
Turbulent diffusion
Stochastic fluctuations of the flow velocity (pulsations)
Diffusive process in mathematical sense (~ Fickrsquos Law)
Convective transport (temporal differences)
Temporal averaging (T)
3D transport equation in turbulent flow3D transport equation in turbulent flow
Spatial simplifications (2D)Spatial simplifications (2D) - Dispersion - Dispersion
Depth averaged flow velocity v
0v
H
After the expansion of the convectiv part (v middot c) in x direction
Depth Integration (3D2D)
Turbulent dispersion (~ diffusion)
z
x
Velocity depth profile
Dx Dy
turbulent dispersion coefficients in 2D equation
Depth averaging (H)
11D tD transport equation in turbulent flow (cross-section avransport equation in turbulent flow (cross-section av))
Dx turbulent dispersion coefficient in 1D equation
Cross-section averaging (A)
2D transport equation in turbulent flow (depth averaged)2D transport equation in turbulent flow (depth averaged)
Turbulent dispersionTurbulent dispersion
Convectiv transport arising from the spatial differences of the velocity profile (faster and slower particles compared to the average)
v
In the 2D and 1D equations only
Dispersion coefficient a function of the velocity space (hydraulic parameters channel geometry)
Dx = Ddx + Dtx + D Dy
= Ddy + Dty + D Dx = Ddxx + Ddx + Dtx + D
Dx Dy
gtgt Dx Dx gtgt Dx
In case of more irregular channel and higher depth or larger cross-section area its value is bigger
Also present in laminar flow
Order of magnitude of Order of magnitude of diffusion dispersion diffusion dispersion coefficientscoefficients
10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s
Pore water
Mol diff
Vertical turbulent diff
High depth Shallow depth
Cross disp (2D)
Longitudinal disp (2D)
Horizontal turbulent diff
Longitudinal disp (1D)
Evaluation of dispersion coefficients measuring
Measuring with tracing matter (eg paint)
Inverse calculation from the measured concentration
2D Cross dispersion coefficient (Fischer)
Dy = dy u R (m2s)
dy - dimensionless cross dispersion coefficient
Straight regular channel dy 015
Moderately winding channel dy 02 ndash 06
Hard winding irregular channel dy gt 06 (1-2)
R ndash hydraulic radius (m)
u - shearing velocity at the channel bottom (ms)
u = (g R I)05Longitudinal dispersion coefficient dx 6 (2D) dxx = 100 ndash 1000 (1D)
Velocity space turbulent pulsation unequal distribution
Estimation of dispersion coefficients empirical forms
TRANSPORT EQUATION FOR NON-CONSERVATIVE TRANSPORT EQUATION FOR NON-CONSERVATIVE MATTERSMATTERS
bull There are sources and losses in the flow space
bull Physical chemical biochemical transformations take place
bull Non-conservative pollutants reaction kinetics ( R(C) )
bull They are taken into account in a linear way
dCdt = -C where is the coefficient of reaction kinetics (usually first-order kinetics)
cαxc
DAxx
c)v(Atc)(A
xx
1D equation in this case
ANALYTICAL SOLUTIONS OF THE TRANSPORT ANALYTICAL SOLUTIONS OF THE TRANSPORT EQUATIONEQUATION
Permanent mixing of the pollutants
Polluntant-wave transmission
Estimation of the geometric and hydraulic parameters
Exacter calculations based on measurments using numerical methods
The analitical solutions can be obtained in simpler cases only (approximate calculations)
Main steps
2
2
y
cD
x
cv yx
PERMANENT MIXING
The pollutant emission is steady in time
Permanent river discharge (low flow condition)
Constant flow velocity flow depth and dispersion coef
2D equation negligible vertical changes (shallow river)
Convection transferring
Dispersion spreading
Initial condition M0 (x0 y0) - emission
Boundary cond cy = 0 at the bank
)()()(
cvhy
cvhxt
chyx )()(
y
cDh
yx
cDh
xyx
x
yy
v
xD2
INLET AT THE MAIN FLOW PATH
Longitudinal according to x-frac12 function
Cross direction Gauss-distribution
cmax
M [kgs]cmax
)4
exp(2
2
xD
yv
xvDh
Mc (x t)y
x
xy
x
yb
v
xDB
234
Bb at value 01 cmax 1522bBBrand width
B ~ Bb2
1 0270 BD
vL
y
x
First distance of the mixing
M
1L
Bb
C (x1 y)
x1
M
x
yb
v
xDB
2152
21 110 B
D
vL
y
x
INLET AT THE RIVER BANK
)4
exp(2
xD
yv
xvDh
Mc
y
x
xy
cmax
C (x1 y)
x1
M
INLET NEAR THE RIVER BANK
)4
(exp ( xDy
( y-y0 )2-v
xvD2h
Mc x
xy
cmax
))4
+exp ( xDy
( y+y0 )2-vx
y0 C (x1 y)
x1
IMPACTS OF THE RIVER BANKS (total river section)
Boundary condition reflection theory (infinite series)
Complete mixing the change along the cross-section is less than 10
L2 ~ 3L1 second distance of the mixing
Inlet at a point in y0 distance from the bank
xvD2h
Mc
xy
)4
exp ( xDy
( y-y0 +2nB)2-vx
)4
+ exp ( xDy
( y+y0 -2nB)2-vx
sumn=infin
n=minusinfin(
)
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
Mass balanceMass balance
General mass balance equationbull Expresses the conversation of massbull Differential equations
IN ndash OUT + SOURCES ndash SINKS = CHANGE
VIN
(1)
OUT
(2)
Controlling surface
We work with mass fluxes [gs]
Mass balanceMass balanceBy solving the general mass balance equation
we can describe the transport processes for a given substance
Solutionbull Analyticalbull Numerical
Discretizationbull Temporalbull Spatial
Simplificationsbull Temporal ndash steady-statebull Spatial ndash 2D 1Dbull Other
Boundary initial conditionsbull Geometrical DATAbull Hydrologic DATAbull Water quality DATA
Simplified case ISimplified case IIN ndash OUT + SOURCES ndash SINKS = CHANGE
Assumptionsbull Steady statebull Conservative material
Solutionbull IN=OUTbull SOURCES=0 SINKS=0bull CHANGE=0
Simplified case IISimplified case IIAssumptionsbull Steady state ndash Q(t) E(t) are constants dC(t)dt=0bull Non-conservative settling material (vs)bull Prismatic river bed ndash A B H are constantsbull 1D
vs
A ~ B middot H [m2]
H
Simplified case IISimplified case II
Q
(1) (2)x
Q
IN
OUT
LOSS settled matter
C(x) is linear (assumption)
Av
v
vs
If x = O C = Co
Exponential decrease
Simplified case IISimplified case IIThe calculation
C0 concentration under the inlet
Determination of C0 value
Q
E = q middot c emission
Cbg background concentration
1D ndash Complete mixing (two water mixing with each other)
Increment
Dilution ratio
E
Simplified case IISimplified case II
The solution
Transmission coefficient
Dilution Sedimentation Conservative matter
Simplified case IISimplified case II
Clim = 15 gm3
concentration limit
Q = 15 m3s
q = 1 m3sc = 300 gm3
Cbg = 10 gm3
L = 5 km
B = 75 m
vS = 03 mh
H = 2 m
EXAMPLE
Complete mixing
3bg0 gm28125
11530011015
cqCQC
3lim
33
s0
gm15Cgm22836)10513600
0321
( exp28125
L)vv
H1
( expC L)(xC
Concentration at distance L from the emission
Flow velocity
ms10275
15HB
Qv
Maximum possible C0 value
Maximum allowed waste water concentration
Allowed maximum concentration
C (x = L)max = Clim
3
3s
limmax0
gm18474
)10513600
0321
( exp
15
L)vv
H1
( exp
C C
3
bgmax0max
gm145584
11015-11518474
q
CQ-qQCc
Transmission coefficient
Necessary reduction of emission
gs154416145584)(3001)c(cqΔE max
33
s
sm0051)10513600
0321
( exp115
1
L)vv
H1
( expqQ
1 L)(xa
The general situationThe general situation
bull Unsteadybull Inhomogeneousbull Non-conservativebull 3D
General transport equation
Advection-Dispersion Equation (ADE)
Advection-dispersion EquationAdvection-dispersion Equationbull Can be derived from the mass balancebull Expresses the temporal and spatial changes of
the observed state variablebull Itrsquos solution is the concentration-distribution over
timebull Differential equation
Change in concentration with time
Change due to advection
=Change due to diffusion or dispersion
Change due to conversion processes
+ +
Molecular diffusionMolecular diffusion
DIFFUSION
CONVECTION
v
Molecular Molecular diffusiondiffusion
FickrsquosFickrsquos LawLaw
c1 c2x
bull c1 c2 separated tanks
bull open valve
Mass flux (kgs)
D - molecular diffusioncoefficient [m2s]
bull equalization and mixing (Brown-movement)
through area unit
bull depends on temperature
IN conv + diff
dxdy
dzOUT conv + diff
x direction
IN OUT
bull convection
CHANGE
bull diffusion
Mass balance Mass balance equationequation
x direction
IN conv + diff
dxdy
dzOUT conv + diff
Mass balance Mass balance equationequation
Convection Diffusion
If D(x) = const in x direction
Convection - Diffusion 1D Equation
The other directions are similar
Convection transfering Diffusion spreading
Mass balance Mass balance equationequation
x y z directions (3D)
D ndash molecular diffusion coefficient
material specific water 10-4 cm2s
space independent
slow mixing and equalization
Convection The water particles with differentconcentrations of pollutant move according to their differing flow velocity
Diffusion The adjacent water particles mix with each otherwhich causes the equalization of concentration
Mass balance equationMass balance equation
bull Laminar flow ordered with parallel flow linesbull Turbulent flow disordered random (flow and direction)
because of surface roughness (friction) rarr causes intensive mixing
v(c)
t
v deviation pulsation
T Time step of the turbulence
averagev~
0
bull Natural streams always turbulent
We usually can use the average only
TurbulenceTurbulence
Convection v middot c [ kgm2 middot s ]
0
0
Turbulent transport Turbulent transport descriptiondescription
v
turbulent diffusion
molecular diffusion
Dtx Dty Dtz gtgt D
Depends on the direction
Function of the velocity space
X direction (the other are similar)
Analogy with Fickrsquos Law rarr the original transport equation does not change only the D parameter
Turbulent diffusionTurbulent diffusion
Dx = D + Dtx Dy = D + Dty Dz = D + Dtz
Convection
Diffusion
Turbulent diffusion
Stochastic fluctuations of the flow velocity (pulsations)
Diffusive process in mathematical sense (~ Fickrsquos Law)
Convective transport (temporal differences)
Temporal averaging (T)
3D transport equation in turbulent flow3D transport equation in turbulent flow
Spatial simplifications (2D)Spatial simplifications (2D) - Dispersion - Dispersion
Depth averaged flow velocity v
0v
H
After the expansion of the convectiv part (v middot c) in x direction
Depth Integration (3D2D)
Turbulent dispersion (~ diffusion)
z
x
Velocity depth profile
Dx Dy
turbulent dispersion coefficients in 2D equation
Depth averaging (H)
11D tD transport equation in turbulent flow (cross-section avransport equation in turbulent flow (cross-section av))
Dx turbulent dispersion coefficient in 1D equation
Cross-section averaging (A)
2D transport equation in turbulent flow (depth averaged)2D transport equation in turbulent flow (depth averaged)
Turbulent dispersionTurbulent dispersion
Convectiv transport arising from the spatial differences of the velocity profile (faster and slower particles compared to the average)
v
In the 2D and 1D equations only
Dispersion coefficient a function of the velocity space (hydraulic parameters channel geometry)
Dx = Ddx + Dtx + D Dy
= Ddy + Dty + D Dx = Ddxx + Ddx + Dtx + D
Dx Dy
gtgt Dx Dx gtgt Dx
In case of more irregular channel and higher depth or larger cross-section area its value is bigger
Also present in laminar flow
Order of magnitude of Order of magnitude of diffusion dispersion diffusion dispersion coefficientscoefficients
10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s
Pore water
Mol diff
Vertical turbulent diff
High depth Shallow depth
Cross disp (2D)
Longitudinal disp (2D)
Horizontal turbulent diff
Longitudinal disp (1D)
Evaluation of dispersion coefficients measuring
Measuring with tracing matter (eg paint)
Inverse calculation from the measured concentration
2D Cross dispersion coefficient (Fischer)
Dy = dy u R (m2s)
dy - dimensionless cross dispersion coefficient
Straight regular channel dy 015
Moderately winding channel dy 02 ndash 06
Hard winding irregular channel dy gt 06 (1-2)
R ndash hydraulic radius (m)
u - shearing velocity at the channel bottom (ms)
u = (g R I)05Longitudinal dispersion coefficient dx 6 (2D) dxx = 100 ndash 1000 (1D)
Velocity space turbulent pulsation unequal distribution
Estimation of dispersion coefficients empirical forms
TRANSPORT EQUATION FOR NON-CONSERVATIVE TRANSPORT EQUATION FOR NON-CONSERVATIVE MATTERSMATTERS
bull There are sources and losses in the flow space
bull Physical chemical biochemical transformations take place
bull Non-conservative pollutants reaction kinetics ( R(C) )
bull They are taken into account in a linear way
dCdt = -C where is the coefficient of reaction kinetics (usually first-order kinetics)
cαxc
DAxx
c)v(Atc)(A
xx
1D equation in this case
ANALYTICAL SOLUTIONS OF THE TRANSPORT ANALYTICAL SOLUTIONS OF THE TRANSPORT EQUATIONEQUATION
Permanent mixing of the pollutants
Polluntant-wave transmission
Estimation of the geometric and hydraulic parameters
Exacter calculations based on measurments using numerical methods
The analitical solutions can be obtained in simpler cases only (approximate calculations)
Main steps
2
2
y
cD
x
cv yx
PERMANENT MIXING
The pollutant emission is steady in time
Permanent river discharge (low flow condition)
Constant flow velocity flow depth and dispersion coef
2D equation negligible vertical changes (shallow river)
Convection transferring
Dispersion spreading
Initial condition M0 (x0 y0) - emission
Boundary cond cy = 0 at the bank
)()()(
cvhy
cvhxt
chyx )()(
y
cDh
yx
cDh
xyx
x
yy
v
xD2
INLET AT THE MAIN FLOW PATH
Longitudinal according to x-frac12 function
Cross direction Gauss-distribution
cmax
M [kgs]cmax
)4
exp(2
2
xD
yv
xvDh
Mc (x t)y
x
xy
x
yb
v
xDB
234
Bb at value 01 cmax 1522bBBrand width
B ~ Bb2
1 0270 BD
vL
y
x
First distance of the mixing
M
1L
Bb
C (x1 y)
x1
M
x
yb
v
xDB
2152
21 110 B
D
vL
y
x
INLET AT THE RIVER BANK
)4
exp(2
xD
yv
xvDh
Mc
y
x
xy
cmax
C (x1 y)
x1
M
INLET NEAR THE RIVER BANK
)4
(exp ( xDy
( y-y0 )2-v
xvD2h
Mc x
xy
cmax
))4
+exp ( xDy
( y+y0 )2-vx
y0 C (x1 y)
x1
IMPACTS OF THE RIVER BANKS (total river section)
Boundary condition reflection theory (infinite series)
Complete mixing the change along the cross-section is less than 10
L2 ~ 3L1 second distance of the mixing
Inlet at a point in y0 distance from the bank
xvD2h
Mc
xy
)4
exp ( xDy
( y-y0 +2nB)2-vx
)4
+ exp ( xDy
( y+y0 -2nB)2-vx
sumn=infin
n=minusinfin(
)
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
Mass balanceMass balanceBy solving the general mass balance equation
we can describe the transport processes for a given substance
Solutionbull Analyticalbull Numerical
Discretizationbull Temporalbull Spatial
Simplificationsbull Temporal ndash steady-statebull Spatial ndash 2D 1Dbull Other
Boundary initial conditionsbull Geometrical DATAbull Hydrologic DATAbull Water quality DATA
Simplified case ISimplified case IIN ndash OUT + SOURCES ndash SINKS = CHANGE
Assumptionsbull Steady statebull Conservative material
Solutionbull IN=OUTbull SOURCES=0 SINKS=0bull CHANGE=0
Simplified case IISimplified case IIAssumptionsbull Steady state ndash Q(t) E(t) are constants dC(t)dt=0bull Non-conservative settling material (vs)bull Prismatic river bed ndash A B H are constantsbull 1D
vs
A ~ B middot H [m2]
H
Simplified case IISimplified case II
Q
(1) (2)x
Q
IN
OUT
LOSS settled matter
C(x) is linear (assumption)
Av
v
vs
If x = O C = Co
Exponential decrease
Simplified case IISimplified case IIThe calculation
C0 concentration under the inlet
Determination of C0 value
Q
E = q middot c emission
Cbg background concentration
1D ndash Complete mixing (two water mixing with each other)
Increment
Dilution ratio
E
Simplified case IISimplified case II
The solution
Transmission coefficient
Dilution Sedimentation Conservative matter
Simplified case IISimplified case II
Clim = 15 gm3
concentration limit
Q = 15 m3s
q = 1 m3sc = 300 gm3
Cbg = 10 gm3
L = 5 km
B = 75 m
vS = 03 mh
H = 2 m
EXAMPLE
Complete mixing
3bg0 gm28125
11530011015
cqCQC
3lim
33
s0
gm15Cgm22836)10513600
0321
( exp28125
L)vv
H1
( expC L)(xC
Concentration at distance L from the emission
Flow velocity
ms10275
15HB
Qv
Maximum possible C0 value
Maximum allowed waste water concentration
Allowed maximum concentration
C (x = L)max = Clim
3
3s
limmax0
gm18474
)10513600
0321
( exp
15
L)vv
H1
( exp
C C
3
bgmax0max
gm145584
11015-11518474
q
CQ-qQCc
Transmission coefficient
Necessary reduction of emission
gs154416145584)(3001)c(cqΔE max
33
s
sm0051)10513600
0321
( exp115
1
L)vv
H1
( expqQ
1 L)(xa
The general situationThe general situation
bull Unsteadybull Inhomogeneousbull Non-conservativebull 3D
General transport equation
Advection-Dispersion Equation (ADE)
Advection-dispersion EquationAdvection-dispersion Equationbull Can be derived from the mass balancebull Expresses the temporal and spatial changes of
the observed state variablebull Itrsquos solution is the concentration-distribution over
timebull Differential equation
Change in concentration with time
Change due to advection
=Change due to diffusion or dispersion
Change due to conversion processes
+ +
Molecular diffusionMolecular diffusion
DIFFUSION
CONVECTION
v
Molecular Molecular diffusiondiffusion
FickrsquosFickrsquos LawLaw
c1 c2x
bull c1 c2 separated tanks
bull open valve
Mass flux (kgs)
D - molecular diffusioncoefficient [m2s]
bull equalization and mixing (Brown-movement)
through area unit
bull depends on temperature
IN conv + diff
dxdy
dzOUT conv + diff
x direction
IN OUT
bull convection
CHANGE
bull diffusion
Mass balance Mass balance equationequation
x direction
IN conv + diff
dxdy
dzOUT conv + diff
Mass balance Mass balance equationequation
Convection Diffusion
If D(x) = const in x direction
Convection - Diffusion 1D Equation
The other directions are similar
Convection transfering Diffusion spreading
Mass balance Mass balance equationequation
x y z directions (3D)
D ndash molecular diffusion coefficient
material specific water 10-4 cm2s
space independent
slow mixing and equalization
Convection The water particles with differentconcentrations of pollutant move according to their differing flow velocity
Diffusion The adjacent water particles mix with each otherwhich causes the equalization of concentration
Mass balance equationMass balance equation
bull Laminar flow ordered with parallel flow linesbull Turbulent flow disordered random (flow and direction)
because of surface roughness (friction) rarr causes intensive mixing
v(c)
t
v deviation pulsation
T Time step of the turbulence
averagev~
0
bull Natural streams always turbulent
We usually can use the average only
TurbulenceTurbulence
Convection v middot c [ kgm2 middot s ]
0
0
Turbulent transport Turbulent transport descriptiondescription
v
turbulent diffusion
molecular diffusion
Dtx Dty Dtz gtgt D
Depends on the direction
Function of the velocity space
X direction (the other are similar)
Analogy with Fickrsquos Law rarr the original transport equation does not change only the D parameter
Turbulent diffusionTurbulent diffusion
Dx = D + Dtx Dy = D + Dty Dz = D + Dtz
Convection
Diffusion
Turbulent diffusion
Stochastic fluctuations of the flow velocity (pulsations)
Diffusive process in mathematical sense (~ Fickrsquos Law)
Convective transport (temporal differences)
Temporal averaging (T)
3D transport equation in turbulent flow3D transport equation in turbulent flow
Spatial simplifications (2D)Spatial simplifications (2D) - Dispersion - Dispersion
Depth averaged flow velocity v
0v
H
After the expansion of the convectiv part (v middot c) in x direction
Depth Integration (3D2D)
Turbulent dispersion (~ diffusion)
z
x
Velocity depth profile
Dx Dy
turbulent dispersion coefficients in 2D equation
Depth averaging (H)
11D tD transport equation in turbulent flow (cross-section avransport equation in turbulent flow (cross-section av))
Dx turbulent dispersion coefficient in 1D equation
Cross-section averaging (A)
2D transport equation in turbulent flow (depth averaged)2D transport equation in turbulent flow (depth averaged)
Turbulent dispersionTurbulent dispersion
Convectiv transport arising from the spatial differences of the velocity profile (faster and slower particles compared to the average)
v
In the 2D and 1D equations only
Dispersion coefficient a function of the velocity space (hydraulic parameters channel geometry)
Dx = Ddx + Dtx + D Dy
= Ddy + Dty + D Dx = Ddxx + Ddx + Dtx + D
Dx Dy
gtgt Dx Dx gtgt Dx
In case of more irregular channel and higher depth or larger cross-section area its value is bigger
Also present in laminar flow
Order of magnitude of Order of magnitude of diffusion dispersion diffusion dispersion coefficientscoefficients
10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s
Pore water
Mol diff
Vertical turbulent diff
High depth Shallow depth
Cross disp (2D)
Longitudinal disp (2D)
Horizontal turbulent diff
Longitudinal disp (1D)
Evaluation of dispersion coefficients measuring
Measuring with tracing matter (eg paint)
Inverse calculation from the measured concentration
2D Cross dispersion coefficient (Fischer)
Dy = dy u R (m2s)
dy - dimensionless cross dispersion coefficient
Straight regular channel dy 015
Moderately winding channel dy 02 ndash 06
Hard winding irregular channel dy gt 06 (1-2)
R ndash hydraulic radius (m)
u - shearing velocity at the channel bottom (ms)
u = (g R I)05Longitudinal dispersion coefficient dx 6 (2D) dxx = 100 ndash 1000 (1D)
Velocity space turbulent pulsation unequal distribution
Estimation of dispersion coefficients empirical forms
TRANSPORT EQUATION FOR NON-CONSERVATIVE TRANSPORT EQUATION FOR NON-CONSERVATIVE MATTERSMATTERS
bull There are sources and losses in the flow space
bull Physical chemical biochemical transformations take place
bull Non-conservative pollutants reaction kinetics ( R(C) )
bull They are taken into account in a linear way
dCdt = -C where is the coefficient of reaction kinetics (usually first-order kinetics)
cαxc
DAxx
c)v(Atc)(A
xx
1D equation in this case
ANALYTICAL SOLUTIONS OF THE TRANSPORT ANALYTICAL SOLUTIONS OF THE TRANSPORT EQUATIONEQUATION
Permanent mixing of the pollutants
Polluntant-wave transmission
Estimation of the geometric and hydraulic parameters
Exacter calculations based on measurments using numerical methods
The analitical solutions can be obtained in simpler cases only (approximate calculations)
Main steps
2
2
y
cD
x
cv yx
PERMANENT MIXING
The pollutant emission is steady in time
Permanent river discharge (low flow condition)
Constant flow velocity flow depth and dispersion coef
2D equation negligible vertical changes (shallow river)
Convection transferring
Dispersion spreading
Initial condition M0 (x0 y0) - emission
Boundary cond cy = 0 at the bank
)()()(
cvhy
cvhxt
chyx )()(
y
cDh
yx
cDh
xyx
x
yy
v
xD2
INLET AT THE MAIN FLOW PATH
Longitudinal according to x-frac12 function
Cross direction Gauss-distribution
cmax
M [kgs]cmax
)4
exp(2
2
xD
yv
xvDh
Mc (x t)y
x
xy
x
yb
v
xDB
234
Bb at value 01 cmax 1522bBBrand width
B ~ Bb2
1 0270 BD
vL
y
x
First distance of the mixing
M
1L
Bb
C (x1 y)
x1
M
x
yb
v
xDB
2152
21 110 B
D
vL
y
x
INLET AT THE RIVER BANK
)4
exp(2
xD
yv
xvDh
Mc
y
x
xy
cmax
C (x1 y)
x1
M
INLET NEAR THE RIVER BANK
)4
(exp ( xDy
( y-y0 )2-v
xvD2h
Mc x
xy
cmax
))4
+exp ( xDy
( y+y0 )2-vx
y0 C (x1 y)
x1
IMPACTS OF THE RIVER BANKS (total river section)
Boundary condition reflection theory (infinite series)
Complete mixing the change along the cross-section is less than 10
L2 ~ 3L1 second distance of the mixing
Inlet at a point in y0 distance from the bank
xvD2h
Mc
xy
)4
exp ( xDy
( y-y0 +2nB)2-vx
)4
+ exp ( xDy
( y+y0 -2nB)2-vx
sumn=infin
n=minusinfin(
)
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
Simplified case ISimplified case IIN ndash OUT + SOURCES ndash SINKS = CHANGE
Assumptionsbull Steady statebull Conservative material
Solutionbull IN=OUTbull SOURCES=0 SINKS=0bull CHANGE=0
Simplified case IISimplified case IIAssumptionsbull Steady state ndash Q(t) E(t) are constants dC(t)dt=0bull Non-conservative settling material (vs)bull Prismatic river bed ndash A B H are constantsbull 1D
vs
A ~ B middot H [m2]
H
Simplified case IISimplified case II
Q
(1) (2)x
Q
IN
OUT
LOSS settled matter
C(x) is linear (assumption)
Av
v
vs
If x = O C = Co
Exponential decrease
Simplified case IISimplified case IIThe calculation
C0 concentration under the inlet
Determination of C0 value
Q
E = q middot c emission
Cbg background concentration
1D ndash Complete mixing (two water mixing with each other)
Increment
Dilution ratio
E
Simplified case IISimplified case II
The solution
Transmission coefficient
Dilution Sedimentation Conservative matter
Simplified case IISimplified case II
Clim = 15 gm3
concentration limit
Q = 15 m3s
q = 1 m3sc = 300 gm3
Cbg = 10 gm3
L = 5 km
B = 75 m
vS = 03 mh
H = 2 m
EXAMPLE
Complete mixing
3bg0 gm28125
11530011015
cqCQC
3lim
33
s0
gm15Cgm22836)10513600
0321
( exp28125
L)vv
H1
( expC L)(xC
Concentration at distance L from the emission
Flow velocity
ms10275
15HB
Qv
Maximum possible C0 value
Maximum allowed waste water concentration
Allowed maximum concentration
C (x = L)max = Clim
3
3s
limmax0
gm18474
)10513600
0321
( exp
15
L)vv
H1
( exp
C C
3
bgmax0max
gm145584
11015-11518474
q
CQ-qQCc
Transmission coefficient
Necessary reduction of emission
gs154416145584)(3001)c(cqΔE max
33
s
sm0051)10513600
0321
( exp115
1
L)vv
H1
( expqQ
1 L)(xa
The general situationThe general situation
bull Unsteadybull Inhomogeneousbull Non-conservativebull 3D
General transport equation
Advection-Dispersion Equation (ADE)
Advection-dispersion EquationAdvection-dispersion Equationbull Can be derived from the mass balancebull Expresses the temporal and spatial changes of
the observed state variablebull Itrsquos solution is the concentration-distribution over
timebull Differential equation
Change in concentration with time
Change due to advection
=Change due to diffusion or dispersion
Change due to conversion processes
+ +
Molecular diffusionMolecular diffusion
DIFFUSION
CONVECTION
v
Molecular Molecular diffusiondiffusion
FickrsquosFickrsquos LawLaw
c1 c2x
bull c1 c2 separated tanks
bull open valve
Mass flux (kgs)
D - molecular diffusioncoefficient [m2s]
bull equalization and mixing (Brown-movement)
through area unit
bull depends on temperature
IN conv + diff
dxdy
dzOUT conv + diff
x direction
IN OUT
bull convection
CHANGE
bull diffusion
Mass balance Mass balance equationequation
x direction
IN conv + diff
dxdy
dzOUT conv + diff
Mass balance Mass balance equationequation
Convection Diffusion
If D(x) = const in x direction
Convection - Diffusion 1D Equation
The other directions are similar
Convection transfering Diffusion spreading
Mass balance Mass balance equationequation
x y z directions (3D)
D ndash molecular diffusion coefficient
material specific water 10-4 cm2s
space independent
slow mixing and equalization
Convection The water particles with differentconcentrations of pollutant move according to their differing flow velocity
Diffusion The adjacent water particles mix with each otherwhich causes the equalization of concentration
Mass balance equationMass balance equation
bull Laminar flow ordered with parallel flow linesbull Turbulent flow disordered random (flow and direction)
because of surface roughness (friction) rarr causes intensive mixing
v(c)
t
v deviation pulsation
T Time step of the turbulence
averagev~
0
bull Natural streams always turbulent
We usually can use the average only
TurbulenceTurbulence
Convection v middot c [ kgm2 middot s ]
0
0
Turbulent transport Turbulent transport descriptiondescription
v
turbulent diffusion
molecular diffusion
Dtx Dty Dtz gtgt D
Depends on the direction
Function of the velocity space
X direction (the other are similar)
Analogy with Fickrsquos Law rarr the original transport equation does not change only the D parameter
Turbulent diffusionTurbulent diffusion
Dx = D + Dtx Dy = D + Dty Dz = D + Dtz
Convection
Diffusion
Turbulent diffusion
Stochastic fluctuations of the flow velocity (pulsations)
Diffusive process in mathematical sense (~ Fickrsquos Law)
Convective transport (temporal differences)
Temporal averaging (T)
3D transport equation in turbulent flow3D transport equation in turbulent flow
Spatial simplifications (2D)Spatial simplifications (2D) - Dispersion - Dispersion
Depth averaged flow velocity v
0v
H
After the expansion of the convectiv part (v middot c) in x direction
Depth Integration (3D2D)
Turbulent dispersion (~ diffusion)
z
x
Velocity depth profile
Dx Dy
turbulent dispersion coefficients in 2D equation
Depth averaging (H)
11D tD transport equation in turbulent flow (cross-section avransport equation in turbulent flow (cross-section av))
Dx turbulent dispersion coefficient in 1D equation
Cross-section averaging (A)
2D transport equation in turbulent flow (depth averaged)2D transport equation in turbulent flow (depth averaged)
Turbulent dispersionTurbulent dispersion
Convectiv transport arising from the spatial differences of the velocity profile (faster and slower particles compared to the average)
v
In the 2D and 1D equations only
Dispersion coefficient a function of the velocity space (hydraulic parameters channel geometry)
Dx = Ddx + Dtx + D Dy
= Ddy + Dty + D Dx = Ddxx + Ddx + Dtx + D
Dx Dy
gtgt Dx Dx gtgt Dx
In case of more irregular channel and higher depth or larger cross-section area its value is bigger
Also present in laminar flow
Order of magnitude of Order of magnitude of diffusion dispersion diffusion dispersion coefficientscoefficients
10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s
Pore water
Mol diff
Vertical turbulent diff
High depth Shallow depth
Cross disp (2D)
Longitudinal disp (2D)
Horizontal turbulent diff
Longitudinal disp (1D)
Evaluation of dispersion coefficients measuring
Measuring with tracing matter (eg paint)
Inverse calculation from the measured concentration
2D Cross dispersion coefficient (Fischer)
Dy = dy u R (m2s)
dy - dimensionless cross dispersion coefficient
Straight regular channel dy 015
Moderately winding channel dy 02 ndash 06
Hard winding irregular channel dy gt 06 (1-2)
R ndash hydraulic radius (m)
u - shearing velocity at the channel bottom (ms)
u = (g R I)05Longitudinal dispersion coefficient dx 6 (2D) dxx = 100 ndash 1000 (1D)
Velocity space turbulent pulsation unequal distribution
Estimation of dispersion coefficients empirical forms
TRANSPORT EQUATION FOR NON-CONSERVATIVE TRANSPORT EQUATION FOR NON-CONSERVATIVE MATTERSMATTERS
bull There are sources and losses in the flow space
bull Physical chemical biochemical transformations take place
bull Non-conservative pollutants reaction kinetics ( R(C) )
bull They are taken into account in a linear way
dCdt = -C where is the coefficient of reaction kinetics (usually first-order kinetics)
cαxc
DAxx
c)v(Atc)(A
xx
1D equation in this case
ANALYTICAL SOLUTIONS OF THE TRANSPORT ANALYTICAL SOLUTIONS OF THE TRANSPORT EQUATIONEQUATION
Permanent mixing of the pollutants
Polluntant-wave transmission
Estimation of the geometric and hydraulic parameters
Exacter calculations based on measurments using numerical methods
The analitical solutions can be obtained in simpler cases only (approximate calculations)
Main steps
2
2
y
cD
x
cv yx
PERMANENT MIXING
The pollutant emission is steady in time
Permanent river discharge (low flow condition)
Constant flow velocity flow depth and dispersion coef
2D equation negligible vertical changes (shallow river)
Convection transferring
Dispersion spreading
Initial condition M0 (x0 y0) - emission
Boundary cond cy = 0 at the bank
)()()(
cvhy
cvhxt
chyx )()(
y
cDh
yx
cDh
xyx
x
yy
v
xD2
INLET AT THE MAIN FLOW PATH
Longitudinal according to x-frac12 function
Cross direction Gauss-distribution
cmax
M [kgs]cmax
)4
exp(2
2
xD
yv
xvDh
Mc (x t)y
x
xy
x
yb
v
xDB
234
Bb at value 01 cmax 1522bBBrand width
B ~ Bb2
1 0270 BD
vL
y
x
First distance of the mixing
M
1L
Bb
C (x1 y)
x1
M
x
yb
v
xDB
2152
21 110 B
D
vL
y
x
INLET AT THE RIVER BANK
)4
exp(2
xD
yv
xvDh
Mc
y
x
xy
cmax
C (x1 y)
x1
M
INLET NEAR THE RIVER BANK
)4
(exp ( xDy
( y-y0 )2-v
xvD2h
Mc x
xy
cmax
))4
+exp ( xDy
( y+y0 )2-vx
y0 C (x1 y)
x1
IMPACTS OF THE RIVER BANKS (total river section)
Boundary condition reflection theory (infinite series)
Complete mixing the change along the cross-section is less than 10
L2 ~ 3L1 second distance of the mixing
Inlet at a point in y0 distance from the bank
xvD2h
Mc
xy
)4
exp ( xDy
( y-y0 +2nB)2-vx
)4
+ exp ( xDy
( y+y0 -2nB)2-vx
sumn=infin
n=minusinfin(
)
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
Simplified case IISimplified case IIAssumptionsbull Steady state ndash Q(t) E(t) are constants dC(t)dt=0bull Non-conservative settling material (vs)bull Prismatic river bed ndash A B H are constantsbull 1D
vs
A ~ B middot H [m2]
H
Simplified case IISimplified case II
Q
(1) (2)x
Q
IN
OUT
LOSS settled matter
C(x) is linear (assumption)
Av
v
vs
If x = O C = Co
Exponential decrease
Simplified case IISimplified case IIThe calculation
C0 concentration under the inlet
Determination of C0 value
Q
E = q middot c emission
Cbg background concentration
1D ndash Complete mixing (two water mixing with each other)
Increment
Dilution ratio
E
Simplified case IISimplified case II
The solution
Transmission coefficient
Dilution Sedimentation Conservative matter
Simplified case IISimplified case II
Clim = 15 gm3
concentration limit
Q = 15 m3s
q = 1 m3sc = 300 gm3
Cbg = 10 gm3
L = 5 km
B = 75 m
vS = 03 mh
H = 2 m
EXAMPLE
Complete mixing
3bg0 gm28125
11530011015
cqCQC
3lim
33
s0
gm15Cgm22836)10513600
0321
( exp28125
L)vv
H1
( expC L)(xC
Concentration at distance L from the emission
Flow velocity
ms10275
15HB
Qv
Maximum possible C0 value
Maximum allowed waste water concentration
Allowed maximum concentration
C (x = L)max = Clim
3
3s
limmax0
gm18474
)10513600
0321
( exp
15
L)vv
H1
( exp
C C
3
bgmax0max
gm145584
11015-11518474
q
CQ-qQCc
Transmission coefficient
Necessary reduction of emission
gs154416145584)(3001)c(cqΔE max
33
s
sm0051)10513600
0321
( exp115
1
L)vv
H1
( expqQ
1 L)(xa
The general situationThe general situation
bull Unsteadybull Inhomogeneousbull Non-conservativebull 3D
General transport equation
Advection-Dispersion Equation (ADE)
Advection-dispersion EquationAdvection-dispersion Equationbull Can be derived from the mass balancebull Expresses the temporal and spatial changes of
the observed state variablebull Itrsquos solution is the concentration-distribution over
timebull Differential equation
Change in concentration with time
Change due to advection
=Change due to diffusion or dispersion
Change due to conversion processes
+ +
Molecular diffusionMolecular diffusion
DIFFUSION
CONVECTION
v
Molecular Molecular diffusiondiffusion
FickrsquosFickrsquos LawLaw
c1 c2x
bull c1 c2 separated tanks
bull open valve
Mass flux (kgs)
D - molecular diffusioncoefficient [m2s]
bull equalization and mixing (Brown-movement)
through area unit
bull depends on temperature
IN conv + diff
dxdy
dzOUT conv + diff
x direction
IN OUT
bull convection
CHANGE
bull diffusion
Mass balance Mass balance equationequation
x direction
IN conv + diff
dxdy
dzOUT conv + diff
Mass balance Mass balance equationequation
Convection Diffusion
If D(x) = const in x direction
Convection - Diffusion 1D Equation
The other directions are similar
Convection transfering Diffusion spreading
Mass balance Mass balance equationequation
x y z directions (3D)
D ndash molecular diffusion coefficient
material specific water 10-4 cm2s
space independent
slow mixing and equalization
Convection The water particles with differentconcentrations of pollutant move according to their differing flow velocity
Diffusion The adjacent water particles mix with each otherwhich causes the equalization of concentration
Mass balance equationMass balance equation
bull Laminar flow ordered with parallel flow linesbull Turbulent flow disordered random (flow and direction)
because of surface roughness (friction) rarr causes intensive mixing
v(c)
t
v deviation pulsation
T Time step of the turbulence
averagev~
0
bull Natural streams always turbulent
We usually can use the average only
TurbulenceTurbulence
Convection v middot c [ kgm2 middot s ]
0
0
Turbulent transport Turbulent transport descriptiondescription
v
turbulent diffusion
molecular diffusion
Dtx Dty Dtz gtgt D
Depends on the direction
Function of the velocity space
X direction (the other are similar)
Analogy with Fickrsquos Law rarr the original transport equation does not change only the D parameter
Turbulent diffusionTurbulent diffusion
Dx = D + Dtx Dy = D + Dty Dz = D + Dtz
Convection
Diffusion
Turbulent diffusion
Stochastic fluctuations of the flow velocity (pulsations)
Diffusive process in mathematical sense (~ Fickrsquos Law)
Convective transport (temporal differences)
Temporal averaging (T)
3D transport equation in turbulent flow3D transport equation in turbulent flow
Spatial simplifications (2D)Spatial simplifications (2D) - Dispersion - Dispersion
Depth averaged flow velocity v
0v
H
After the expansion of the convectiv part (v middot c) in x direction
Depth Integration (3D2D)
Turbulent dispersion (~ diffusion)
z
x
Velocity depth profile
Dx Dy
turbulent dispersion coefficients in 2D equation
Depth averaging (H)
11D tD transport equation in turbulent flow (cross-section avransport equation in turbulent flow (cross-section av))
Dx turbulent dispersion coefficient in 1D equation
Cross-section averaging (A)
2D transport equation in turbulent flow (depth averaged)2D transport equation in turbulent flow (depth averaged)
Turbulent dispersionTurbulent dispersion
Convectiv transport arising from the spatial differences of the velocity profile (faster and slower particles compared to the average)
v
In the 2D and 1D equations only
Dispersion coefficient a function of the velocity space (hydraulic parameters channel geometry)
Dx = Ddx + Dtx + D Dy
= Ddy + Dty + D Dx = Ddxx + Ddx + Dtx + D
Dx Dy
gtgt Dx Dx gtgt Dx
In case of more irregular channel and higher depth or larger cross-section area its value is bigger
Also present in laminar flow
Order of magnitude of Order of magnitude of diffusion dispersion diffusion dispersion coefficientscoefficients
10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s
Pore water
Mol diff
Vertical turbulent diff
High depth Shallow depth
Cross disp (2D)
Longitudinal disp (2D)
Horizontal turbulent diff
Longitudinal disp (1D)
Evaluation of dispersion coefficients measuring
Measuring with tracing matter (eg paint)
Inverse calculation from the measured concentration
2D Cross dispersion coefficient (Fischer)
Dy = dy u R (m2s)
dy - dimensionless cross dispersion coefficient
Straight regular channel dy 015
Moderately winding channel dy 02 ndash 06
Hard winding irregular channel dy gt 06 (1-2)
R ndash hydraulic radius (m)
u - shearing velocity at the channel bottom (ms)
u = (g R I)05Longitudinal dispersion coefficient dx 6 (2D) dxx = 100 ndash 1000 (1D)
Velocity space turbulent pulsation unequal distribution
Estimation of dispersion coefficients empirical forms
TRANSPORT EQUATION FOR NON-CONSERVATIVE TRANSPORT EQUATION FOR NON-CONSERVATIVE MATTERSMATTERS
bull There are sources and losses in the flow space
bull Physical chemical biochemical transformations take place
bull Non-conservative pollutants reaction kinetics ( R(C) )
bull They are taken into account in a linear way
dCdt = -C where is the coefficient of reaction kinetics (usually first-order kinetics)
cαxc
DAxx
c)v(Atc)(A
xx
1D equation in this case
ANALYTICAL SOLUTIONS OF THE TRANSPORT ANALYTICAL SOLUTIONS OF THE TRANSPORT EQUATIONEQUATION
Permanent mixing of the pollutants
Polluntant-wave transmission
Estimation of the geometric and hydraulic parameters
Exacter calculations based on measurments using numerical methods
The analitical solutions can be obtained in simpler cases only (approximate calculations)
Main steps
2
2
y
cD
x
cv yx
PERMANENT MIXING
The pollutant emission is steady in time
Permanent river discharge (low flow condition)
Constant flow velocity flow depth and dispersion coef
2D equation negligible vertical changes (shallow river)
Convection transferring
Dispersion spreading
Initial condition M0 (x0 y0) - emission
Boundary cond cy = 0 at the bank
)()()(
cvhy
cvhxt
chyx )()(
y
cDh
yx
cDh
xyx
x
yy
v
xD2
INLET AT THE MAIN FLOW PATH
Longitudinal according to x-frac12 function
Cross direction Gauss-distribution
cmax
M [kgs]cmax
)4
exp(2
2
xD
yv
xvDh
Mc (x t)y
x
xy
x
yb
v
xDB
234
Bb at value 01 cmax 1522bBBrand width
B ~ Bb2
1 0270 BD
vL
y
x
First distance of the mixing
M
1L
Bb
C (x1 y)
x1
M
x
yb
v
xDB
2152
21 110 B
D
vL
y
x
INLET AT THE RIVER BANK
)4
exp(2
xD
yv
xvDh
Mc
y
x
xy
cmax
C (x1 y)
x1
M
INLET NEAR THE RIVER BANK
)4
(exp ( xDy
( y-y0 )2-v
xvD2h
Mc x
xy
cmax
))4
+exp ( xDy
( y+y0 )2-vx
y0 C (x1 y)
x1
IMPACTS OF THE RIVER BANKS (total river section)
Boundary condition reflection theory (infinite series)
Complete mixing the change along the cross-section is less than 10
L2 ~ 3L1 second distance of the mixing
Inlet at a point in y0 distance from the bank
xvD2h
Mc
xy
)4
exp ( xDy
( y-y0 +2nB)2-vx
)4
+ exp ( xDy
( y+y0 -2nB)2-vx
sumn=infin
n=minusinfin(
)
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
Simplified case IISimplified case II
Q
(1) (2)x
Q
IN
OUT
LOSS settled matter
C(x) is linear (assumption)
Av
v
vs
If x = O C = Co
Exponential decrease
Simplified case IISimplified case IIThe calculation
C0 concentration under the inlet
Determination of C0 value
Q
E = q middot c emission
Cbg background concentration
1D ndash Complete mixing (two water mixing with each other)
Increment
Dilution ratio
E
Simplified case IISimplified case II
The solution
Transmission coefficient
Dilution Sedimentation Conservative matter
Simplified case IISimplified case II
Clim = 15 gm3
concentration limit
Q = 15 m3s
q = 1 m3sc = 300 gm3
Cbg = 10 gm3
L = 5 km
B = 75 m
vS = 03 mh
H = 2 m
EXAMPLE
Complete mixing
3bg0 gm28125
11530011015
cqCQC
3lim
33
s0
gm15Cgm22836)10513600
0321
( exp28125
L)vv
H1
( expC L)(xC
Concentration at distance L from the emission
Flow velocity
ms10275
15HB
Qv
Maximum possible C0 value
Maximum allowed waste water concentration
Allowed maximum concentration
C (x = L)max = Clim
3
3s
limmax0
gm18474
)10513600
0321
( exp
15
L)vv
H1
( exp
C C
3
bgmax0max
gm145584
11015-11518474
q
CQ-qQCc
Transmission coefficient
Necessary reduction of emission
gs154416145584)(3001)c(cqΔE max
33
s
sm0051)10513600
0321
( exp115
1
L)vv
H1
( expqQ
1 L)(xa
The general situationThe general situation
bull Unsteadybull Inhomogeneousbull Non-conservativebull 3D
General transport equation
Advection-Dispersion Equation (ADE)
Advection-dispersion EquationAdvection-dispersion Equationbull Can be derived from the mass balancebull Expresses the temporal and spatial changes of
the observed state variablebull Itrsquos solution is the concentration-distribution over
timebull Differential equation
Change in concentration with time
Change due to advection
=Change due to diffusion or dispersion
Change due to conversion processes
+ +
Molecular diffusionMolecular diffusion
DIFFUSION
CONVECTION
v
Molecular Molecular diffusiondiffusion
FickrsquosFickrsquos LawLaw
c1 c2x
bull c1 c2 separated tanks
bull open valve
Mass flux (kgs)
D - molecular diffusioncoefficient [m2s]
bull equalization and mixing (Brown-movement)
through area unit
bull depends on temperature
IN conv + diff
dxdy
dzOUT conv + diff
x direction
IN OUT
bull convection
CHANGE
bull diffusion
Mass balance Mass balance equationequation
x direction
IN conv + diff
dxdy
dzOUT conv + diff
Mass balance Mass balance equationequation
Convection Diffusion
If D(x) = const in x direction
Convection - Diffusion 1D Equation
The other directions are similar
Convection transfering Diffusion spreading
Mass balance Mass balance equationequation
x y z directions (3D)
D ndash molecular diffusion coefficient
material specific water 10-4 cm2s
space independent
slow mixing and equalization
Convection The water particles with differentconcentrations of pollutant move according to their differing flow velocity
Diffusion The adjacent water particles mix with each otherwhich causes the equalization of concentration
Mass balance equationMass balance equation
bull Laminar flow ordered with parallel flow linesbull Turbulent flow disordered random (flow and direction)
because of surface roughness (friction) rarr causes intensive mixing
v(c)
t
v deviation pulsation
T Time step of the turbulence
averagev~
0
bull Natural streams always turbulent
We usually can use the average only
TurbulenceTurbulence
Convection v middot c [ kgm2 middot s ]
0
0
Turbulent transport Turbulent transport descriptiondescription
v
turbulent diffusion
molecular diffusion
Dtx Dty Dtz gtgt D
Depends on the direction
Function of the velocity space
X direction (the other are similar)
Analogy with Fickrsquos Law rarr the original transport equation does not change only the D parameter
Turbulent diffusionTurbulent diffusion
Dx = D + Dtx Dy = D + Dty Dz = D + Dtz
Convection
Diffusion
Turbulent diffusion
Stochastic fluctuations of the flow velocity (pulsations)
Diffusive process in mathematical sense (~ Fickrsquos Law)
Convective transport (temporal differences)
Temporal averaging (T)
3D transport equation in turbulent flow3D transport equation in turbulent flow
Spatial simplifications (2D)Spatial simplifications (2D) - Dispersion - Dispersion
Depth averaged flow velocity v
0v
H
After the expansion of the convectiv part (v middot c) in x direction
Depth Integration (3D2D)
Turbulent dispersion (~ diffusion)
z
x
Velocity depth profile
Dx Dy
turbulent dispersion coefficients in 2D equation
Depth averaging (H)
11D tD transport equation in turbulent flow (cross-section avransport equation in turbulent flow (cross-section av))
Dx turbulent dispersion coefficient in 1D equation
Cross-section averaging (A)
2D transport equation in turbulent flow (depth averaged)2D transport equation in turbulent flow (depth averaged)
Turbulent dispersionTurbulent dispersion
Convectiv transport arising from the spatial differences of the velocity profile (faster and slower particles compared to the average)
v
In the 2D and 1D equations only
Dispersion coefficient a function of the velocity space (hydraulic parameters channel geometry)
Dx = Ddx + Dtx + D Dy
= Ddy + Dty + D Dx = Ddxx + Ddx + Dtx + D
Dx Dy
gtgt Dx Dx gtgt Dx
In case of more irregular channel and higher depth or larger cross-section area its value is bigger
Also present in laminar flow
Order of magnitude of Order of magnitude of diffusion dispersion diffusion dispersion coefficientscoefficients
10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s
Pore water
Mol diff
Vertical turbulent diff
High depth Shallow depth
Cross disp (2D)
Longitudinal disp (2D)
Horizontal turbulent diff
Longitudinal disp (1D)
Evaluation of dispersion coefficients measuring
Measuring with tracing matter (eg paint)
Inverse calculation from the measured concentration
2D Cross dispersion coefficient (Fischer)
Dy = dy u R (m2s)
dy - dimensionless cross dispersion coefficient
Straight regular channel dy 015
Moderately winding channel dy 02 ndash 06
Hard winding irregular channel dy gt 06 (1-2)
R ndash hydraulic radius (m)
u - shearing velocity at the channel bottom (ms)
u = (g R I)05Longitudinal dispersion coefficient dx 6 (2D) dxx = 100 ndash 1000 (1D)
Velocity space turbulent pulsation unequal distribution
Estimation of dispersion coefficients empirical forms
TRANSPORT EQUATION FOR NON-CONSERVATIVE TRANSPORT EQUATION FOR NON-CONSERVATIVE MATTERSMATTERS
bull There are sources and losses in the flow space
bull Physical chemical biochemical transformations take place
bull Non-conservative pollutants reaction kinetics ( R(C) )
bull They are taken into account in a linear way
dCdt = -C where is the coefficient of reaction kinetics (usually first-order kinetics)
cαxc
DAxx
c)v(Atc)(A
xx
1D equation in this case
ANALYTICAL SOLUTIONS OF THE TRANSPORT ANALYTICAL SOLUTIONS OF THE TRANSPORT EQUATIONEQUATION
Permanent mixing of the pollutants
Polluntant-wave transmission
Estimation of the geometric and hydraulic parameters
Exacter calculations based on measurments using numerical methods
The analitical solutions can be obtained in simpler cases only (approximate calculations)
Main steps
2
2
y
cD
x
cv yx
PERMANENT MIXING
The pollutant emission is steady in time
Permanent river discharge (low flow condition)
Constant flow velocity flow depth and dispersion coef
2D equation negligible vertical changes (shallow river)
Convection transferring
Dispersion spreading
Initial condition M0 (x0 y0) - emission
Boundary cond cy = 0 at the bank
)()()(
cvhy
cvhxt
chyx )()(
y
cDh
yx
cDh
xyx
x
yy
v
xD2
INLET AT THE MAIN FLOW PATH
Longitudinal according to x-frac12 function
Cross direction Gauss-distribution
cmax
M [kgs]cmax
)4
exp(2
2
xD
yv
xvDh
Mc (x t)y
x
xy
x
yb
v
xDB
234
Bb at value 01 cmax 1522bBBrand width
B ~ Bb2
1 0270 BD
vL
y
x
First distance of the mixing
M
1L
Bb
C (x1 y)
x1
M
x
yb
v
xDB
2152
21 110 B
D
vL
y
x
INLET AT THE RIVER BANK
)4
exp(2
xD
yv
xvDh
Mc
y
x
xy
cmax
C (x1 y)
x1
M
INLET NEAR THE RIVER BANK
)4
(exp ( xDy
( y-y0 )2-v
xvD2h
Mc x
xy
cmax
))4
+exp ( xDy
( y+y0 )2-vx
y0 C (x1 y)
x1
IMPACTS OF THE RIVER BANKS (total river section)
Boundary condition reflection theory (infinite series)
Complete mixing the change along the cross-section is less than 10
L2 ~ 3L1 second distance of the mixing
Inlet at a point in y0 distance from the bank
xvD2h
Mc
xy
)4
exp ( xDy
( y-y0 +2nB)2-vx
)4
+ exp ( xDy
( y+y0 -2nB)2-vx
sumn=infin
n=minusinfin(
)
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
v
vs
If x = O C = Co
Exponential decrease
Simplified case IISimplified case IIThe calculation
C0 concentration under the inlet
Determination of C0 value
Q
E = q middot c emission
Cbg background concentration
1D ndash Complete mixing (two water mixing with each other)
Increment
Dilution ratio
E
Simplified case IISimplified case II
The solution
Transmission coefficient
Dilution Sedimentation Conservative matter
Simplified case IISimplified case II
Clim = 15 gm3
concentration limit
Q = 15 m3s
q = 1 m3sc = 300 gm3
Cbg = 10 gm3
L = 5 km
B = 75 m
vS = 03 mh
H = 2 m
EXAMPLE
Complete mixing
3bg0 gm28125
11530011015
cqCQC
3lim
33
s0
gm15Cgm22836)10513600
0321
( exp28125
L)vv
H1
( expC L)(xC
Concentration at distance L from the emission
Flow velocity
ms10275
15HB
Qv
Maximum possible C0 value
Maximum allowed waste water concentration
Allowed maximum concentration
C (x = L)max = Clim
3
3s
limmax0
gm18474
)10513600
0321
( exp
15
L)vv
H1
( exp
C C
3
bgmax0max
gm145584
11015-11518474
q
CQ-qQCc
Transmission coefficient
Necessary reduction of emission
gs154416145584)(3001)c(cqΔE max
33
s
sm0051)10513600
0321
( exp115
1
L)vv
H1
( expqQ
1 L)(xa
The general situationThe general situation
bull Unsteadybull Inhomogeneousbull Non-conservativebull 3D
General transport equation
Advection-Dispersion Equation (ADE)
Advection-dispersion EquationAdvection-dispersion Equationbull Can be derived from the mass balancebull Expresses the temporal and spatial changes of
the observed state variablebull Itrsquos solution is the concentration-distribution over
timebull Differential equation
Change in concentration with time
Change due to advection
=Change due to diffusion or dispersion
Change due to conversion processes
+ +
Molecular diffusionMolecular diffusion
DIFFUSION
CONVECTION
v
Molecular Molecular diffusiondiffusion
FickrsquosFickrsquos LawLaw
c1 c2x
bull c1 c2 separated tanks
bull open valve
Mass flux (kgs)
D - molecular diffusioncoefficient [m2s]
bull equalization and mixing (Brown-movement)
through area unit
bull depends on temperature
IN conv + diff
dxdy
dzOUT conv + diff
x direction
IN OUT
bull convection
CHANGE
bull diffusion
Mass balance Mass balance equationequation
x direction
IN conv + diff
dxdy
dzOUT conv + diff
Mass balance Mass balance equationequation
Convection Diffusion
If D(x) = const in x direction
Convection - Diffusion 1D Equation
The other directions are similar
Convection transfering Diffusion spreading
Mass balance Mass balance equationequation
x y z directions (3D)
D ndash molecular diffusion coefficient
material specific water 10-4 cm2s
space independent
slow mixing and equalization
Convection The water particles with differentconcentrations of pollutant move according to their differing flow velocity
Diffusion The adjacent water particles mix with each otherwhich causes the equalization of concentration
Mass balance equationMass balance equation
bull Laminar flow ordered with parallel flow linesbull Turbulent flow disordered random (flow and direction)
because of surface roughness (friction) rarr causes intensive mixing
v(c)
t
v deviation pulsation
T Time step of the turbulence
averagev~
0
bull Natural streams always turbulent
We usually can use the average only
TurbulenceTurbulence
Convection v middot c [ kgm2 middot s ]
0
0
Turbulent transport Turbulent transport descriptiondescription
v
turbulent diffusion
molecular diffusion
Dtx Dty Dtz gtgt D
Depends on the direction
Function of the velocity space
X direction (the other are similar)
Analogy with Fickrsquos Law rarr the original transport equation does not change only the D parameter
Turbulent diffusionTurbulent diffusion
Dx = D + Dtx Dy = D + Dty Dz = D + Dtz
Convection
Diffusion
Turbulent diffusion
Stochastic fluctuations of the flow velocity (pulsations)
Diffusive process in mathematical sense (~ Fickrsquos Law)
Convective transport (temporal differences)
Temporal averaging (T)
3D transport equation in turbulent flow3D transport equation in turbulent flow
Spatial simplifications (2D)Spatial simplifications (2D) - Dispersion - Dispersion
Depth averaged flow velocity v
0v
H
After the expansion of the convectiv part (v middot c) in x direction
Depth Integration (3D2D)
Turbulent dispersion (~ diffusion)
z
x
Velocity depth profile
Dx Dy
turbulent dispersion coefficients in 2D equation
Depth averaging (H)
11D tD transport equation in turbulent flow (cross-section avransport equation in turbulent flow (cross-section av))
Dx turbulent dispersion coefficient in 1D equation
Cross-section averaging (A)
2D transport equation in turbulent flow (depth averaged)2D transport equation in turbulent flow (depth averaged)
Turbulent dispersionTurbulent dispersion
Convectiv transport arising from the spatial differences of the velocity profile (faster and slower particles compared to the average)
v
In the 2D and 1D equations only
Dispersion coefficient a function of the velocity space (hydraulic parameters channel geometry)
Dx = Ddx + Dtx + D Dy
= Ddy + Dty + D Dx = Ddxx + Ddx + Dtx + D
Dx Dy
gtgt Dx Dx gtgt Dx
In case of more irregular channel and higher depth or larger cross-section area its value is bigger
Also present in laminar flow
Order of magnitude of Order of magnitude of diffusion dispersion diffusion dispersion coefficientscoefficients
10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s
Pore water
Mol diff
Vertical turbulent diff
High depth Shallow depth
Cross disp (2D)
Longitudinal disp (2D)
Horizontal turbulent diff
Longitudinal disp (1D)
Evaluation of dispersion coefficients measuring
Measuring with tracing matter (eg paint)
Inverse calculation from the measured concentration
2D Cross dispersion coefficient (Fischer)
Dy = dy u R (m2s)
dy - dimensionless cross dispersion coefficient
Straight regular channel dy 015
Moderately winding channel dy 02 ndash 06
Hard winding irregular channel dy gt 06 (1-2)
R ndash hydraulic radius (m)
u - shearing velocity at the channel bottom (ms)
u = (g R I)05Longitudinal dispersion coefficient dx 6 (2D) dxx = 100 ndash 1000 (1D)
Velocity space turbulent pulsation unequal distribution
Estimation of dispersion coefficients empirical forms
TRANSPORT EQUATION FOR NON-CONSERVATIVE TRANSPORT EQUATION FOR NON-CONSERVATIVE MATTERSMATTERS
bull There are sources and losses in the flow space
bull Physical chemical biochemical transformations take place
bull Non-conservative pollutants reaction kinetics ( R(C) )
bull They are taken into account in a linear way
dCdt = -C where is the coefficient of reaction kinetics (usually first-order kinetics)
cαxc
DAxx
c)v(Atc)(A
xx
1D equation in this case
ANALYTICAL SOLUTIONS OF THE TRANSPORT ANALYTICAL SOLUTIONS OF THE TRANSPORT EQUATIONEQUATION
Permanent mixing of the pollutants
Polluntant-wave transmission
Estimation of the geometric and hydraulic parameters
Exacter calculations based on measurments using numerical methods
The analitical solutions can be obtained in simpler cases only (approximate calculations)
Main steps
2
2
y
cD
x
cv yx
PERMANENT MIXING
The pollutant emission is steady in time
Permanent river discharge (low flow condition)
Constant flow velocity flow depth and dispersion coef
2D equation negligible vertical changes (shallow river)
Convection transferring
Dispersion spreading
Initial condition M0 (x0 y0) - emission
Boundary cond cy = 0 at the bank
)()()(
cvhy
cvhxt
chyx )()(
y
cDh
yx
cDh
xyx
x
yy
v
xD2
INLET AT THE MAIN FLOW PATH
Longitudinal according to x-frac12 function
Cross direction Gauss-distribution
cmax
M [kgs]cmax
)4
exp(2
2
xD
yv
xvDh
Mc (x t)y
x
xy
x
yb
v
xDB
234
Bb at value 01 cmax 1522bBBrand width
B ~ Bb2
1 0270 BD
vL
y
x
First distance of the mixing
M
1L
Bb
C (x1 y)
x1
M
x
yb
v
xDB
2152
21 110 B
D
vL
y
x
INLET AT THE RIVER BANK
)4
exp(2
xD
yv
xvDh
Mc
y
x
xy
cmax
C (x1 y)
x1
M
INLET NEAR THE RIVER BANK
)4
(exp ( xDy
( y-y0 )2-v
xvD2h
Mc x
xy
cmax
))4
+exp ( xDy
( y+y0 )2-vx
y0 C (x1 y)
x1
IMPACTS OF THE RIVER BANKS (total river section)
Boundary condition reflection theory (infinite series)
Complete mixing the change along the cross-section is less than 10
L2 ~ 3L1 second distance of the mixing
Inlet at a point in y0 distance from the bank
xvD2h
Mc
xy
)4
exp ( xDy
( y-y0 +2nB)2-vx
)4
+ exp ( xDy
( y+y0 -2nB)2-vx
sumn=infin
n=minusinfin(
)
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
C0 concentration under the inlet
Determination of C0 value
Q
E = q middot c emission
Cbg background concentration
1D ndash Complete mixing (two water mixing with each other)
Increment
Dilution ratio
E
Simplified case IISimplified case II
The solution
Transmission coefficient
Dilution Sedimentation Conservative matter
Simplified case IISimplified case II
Clim = 15 gm3
concentration limit
Q = 15 m3s
q = 1 m3sc = 300 gm3
Cbg = 10 gm3
L = 5 km
B = 75 m
vS = 03 mh
H = 2 m
EXAMPLE
Complete mixing
3bg0 gm28125
11530011015
cqCQC
3lim
33
s0
gm15Cgm22836)10513600
0321
( exp28125
L)vv
H1
( expC L)(xC
Concentration at distance L from the emission
Flow velocity
ms10275
15HB
Qv
Maximum possible C0 value
Maximum allowed waste water concentration
Allowed maximum concentration
C (x = L)max = Clim
3
3s
limmax0
gm18474
)10513600
0321
( exp
15
L)vv
H1
( exp
C C
3
bgmax0max
gm145584
11015-11518474
q
CQ-qQCc
Transmission coefficient
Necessary reduction of emission
gs154416145584)(3001)c(cqΔE max
33
s
sm0051)10513600
0321
( exp115
1
L)vv
H1
( expqQ
1 L)(xa
The general situationThe general situation
bull Unsteadybull Inhomogeneousbull Non-conservativebull 3D
General transport equation
Advection-Dispersion Equation (ADE)
Advection-dispersion EquationAdvection-dispersion Equationbull Can be derived from the mass balancebull Expresses the temporal and spatial changes of
the observed state variablebull Itrsquos solution is the concentration-distribution over
timebull Differential equation
Change in concentration with time
Change due to advection
=Change due to diffusion or dispersion
Change due to conversion processes
+ +
Molecular diffusionMolecular diffusion
DIFFUSION
CONVECTION
v
Molecular Molecular diffusiondiffusion
FickrsquosFickrsquos LawLaw
c1 c2x
bull c1 c2 separated tanks
bull open valve
Mass flux (kgs)
D - molecular diffusioncoefficient [m2s]
bull equalization and mixing (Brown-movement)
through area unit
bull depends on temperature
IN conv + diff
dxdy
dzOUT conv + diff
x direction
IN OUT
bull convection
CHANGE
bull diffusion
Mass balance Mass balance equationequation
x direction
IN conv + diff
dxdy
dzOUT conv + diff
Mass balance Mass balance equationequation
Convection Diffusion
If D(x) = const in x direction
Convection - Diffusion 1D Equation
The other directions are similar
Convection transfering Diffusion spreading
Mass balance Mass balance equationequation
x y z directions (3D)
D ndash molecular diffusion coefficient
material specific water 10-4 cm2s
space independent
slow mixing and equalization
Convection The water particles with differentconcentrations of pollutant move according to their differing flow velocity
Diffusion The adjacent water particles mix with each otherwhich causes the equalization of concentration
Mass balance equationMass balance equation
bull Laminar flow ordered with parallel flow linesbull Turbulent flow disordered random (flow and direction)
because of surface roughness (friction) rarr causes intensive mixing
v(c)
t
v deviation pulsation
T Time step of the turbulence
averagev~
0
bull Natural streams always turbulent
We usually can use the average only
TurbulenceTurbulence
Convection v middot c [ kgm2 middot s ]
0
0
Turbulent transport Turbulent transport descriptiondescription
v
turbulent diffusion
molecular diffusion
Dtx Dty Dtz gtgt D
Depends on the direction
Function of the velocity space
X direction (the other are similar)
Analogy with Fickrsquos Law rarr the original transport equation does not change only the D parameter
Turbulent diffusionTurbulent diffusion
Dx = D + Dtx Dy = D + Dty Dz = D + Dtz
Convection
Diffusion
Turbulent diffusion
Stochastic fluctuations of the flow velocity (pulsations)
Diffusive process in mathematical sense (~ Fickrsquos Law)
Convective transport (temporal differences)
Temporal averaging (T)
3D transport equation in turbulent flow3D transport equation in turbulent flow
Spatial simplifications (2D)Spatial simplifications (2D) - Dispersion - Dispersion
Depth averaged flow velocity v
0v
H
After the expansion of the convectiv part (v middot c) in x direction
Depth Integration (3D2D)
Turbulent dispersion (~ diffusion)
z
x
Velocity depth profile
Dx Dy
turbulent dispersion coefficients in 2D equation
Depth averaging (H)
11D tD transport equation in turbulent flow (cross-section avransport equation in turbulent flow (cross-section av))
Dx turbulent dispersion coefficient in 1D equation
Cross-section averaging (A)
2D transport equation in turbulent flow (depth averaged)2D transport equation in turbulent flow (depth averaged)
Turbulent dispersionTurbulent dispersion
Convectiv transport arising from the spatial differences of the velocity profile (faster and slower particles compared to the average)
v
In the 2D and 1D equations only
Dispersion coefficient a function of the velocity space (hydraulic parameters channel geometry)
Dx = Ddx + Dtx + D Dy
= Ddy + Dty + D Dx = Ddxx + Ddx + Dtx + D
Dx Dy
gtgt Dx Dx gtgt Dx
In case of more irregular channel and higher depth or larger cross-section area its value is bigger
Also present in laminar flow
Order of magnitude of Order of magnitude of diffusion dispersion diffusion dispersion coefficientscoefficients
10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s
Pore water
Mol diff
Vertical turbulent diff
High depth Shallow depth
Cross disp (2D)
Longitudinal disp (2D)
Horizontal turbulent diff
Longitudinal disp (1D)
Evaluation of dispersion coefficients measuring
Measuring with tracing matter (eg paint)
Inverse calculation from the measured concentration
2D Cross dispersion coefficient (Fischer)
Dy = dy u R (m2s)
dy - dimensionless cross dispersion coefficient
Straight regular channel dy 015
Moderately winding channel dy 02 ndash 06
Hard winding irregular channel dy gt 06 (1-2)
R ndash hydraulic radius (m)
u - shearing velocity at the channel bottom (ms)
u = (g R I)05Longitudinal dispersion coefficient dx 6 (2D) dxx = 100 ndash 1000 (1D)
Velocity space turbulent pulsation unequal distribution
Estimation of dispersion coefficients empirical forms
TRANSPORT EQUATION FOR NON-CONSERVATIVE TRANSPORT EQUATION FOR NON-CONSERVATIVE MATTERSMATTERS
bull There are sources and losses in the flow space
bull Physical chemical biochemical transformations take place
bull Non-conservative pollutants reaction kinetics ( R(C) )
bull They are taken into account in a linear way
dCdt = -C where is the coefficient of reaction kinetics (usually first-order kinetics)
cαxc
DAxx
c)v(Atc)(A
xx
1D equation in this case
ANALYTICAL SOLUTIONS OF THE TRANSPORT ANALYTICAL SOLUTIONS OF THE TRANSPORT EQUATIONEQUATION
Permanent mixing of the pollutants
Polluntant-wave transmission
Estimation of the geometric and hydraulic parameters
Exacter calculations based on measurments using numerical methods
The analitical solutions can be obtained in simpler cases only (approximate calculations)
Main steps
2
2
y
cD
x
cv yx
PERMANENT MIXING
The pollutant emission is steady in time
Permanent river discharge (low flow condition)
Constant flow velocity flow depth and dispersion coef
2D equation negligible vertical changes (shallow river)
Convection transferring
Dispersion spreading
Initial condition M0 (x0 y0) - emission
Boundary cond cy = 0 at the bank
)()()(
cvhy
cvhxt
chyx )()(
y
cDh
yx
cDh
xyx
x
yy
v
xD2
INLET AT THE MAIN FLOW PATH
Longitudinal according to x-frac12 function
Cross direction Gauss-distribution
cmax
M [kgs]cmax
)4
exp(2
2
xD
yv
xvDh
Mc (x t)y
x
xy
x
yb
v
xDB
234
Bb at value 01 cmax 1522bBBrand width
B ~ Bb2
1 0270 BD
vL
y
x
First distance of the mixing
M
1L
Bb
C (x1 y)
x1
M
x
yb
v
xDB
2152
21 110 B
D
vL
y
x
INLET AT THE RIVER BANK
)4
exp(2
xD
yv
xvDh
Mc
y
x
xy
cmax
C (x1 y)
x1
M
INLET NEAR THE RIVER BANK
)4
(exp ( xDy
( y-y0 )2-v
xvD2h
Mc x
xy
cmax
))4
+exp ( xDy
( y+y0 )2-vx
y0 C (x1 y)
x1
IMPACTS OF THE RIVER BANKS (total river section)
Boundary condition reflection theory (infinite series)
Complete mixing the change along the cross-section is less than 10
L2 ~ 3L1 second distance of the mixing
Inlet at a point in y0 distance from the bank
xvD2h
Mc
xy
)4
exp ( xDy
( y-y0 +2nB)2-vx
)4
+ exp ( xDy
( y+y0 -2nB)2-vx
sumn=infin
n=minusinfin(
)
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
The solution
Transmission coefficient
Dilution Sedimentation Conservative matter
Simplified case IISimplified case II
Clim = 15 gm3
concentration limit
Q = 15 m3s
q = 1 m3sc = 300 gm3
Cbg = 10 gm3
L = 5 km
B = 75 m
vS = 03 mh
H = 2 m
EXAMPLE
Complete mixing
3bg0 gm28125
11530011015
cqCQC
3lim
33
s0
gm15Cgm22836)10513600
0321
( exp28125
L)vv
H1
( expC L)(xC
Concentration at distance L from the emission
Flow velocity
ms10275
15HB
Qv
Maximum possible C0 value
Maximum allowed waste water concentration
Allowed maximum concentration
C (x = L)max = Clim
3
3s
limmax0
gm18474
)10513600
0321
( exp
15
L)vv
H1
( exp
C C
3
bgmax0max
gm145584
11015-11518474
q
CQ-qQCc
Transmission coefficient
Necessary reduction of emission
gs154416145584)(3001)c(cqΔE max
33
s
sm0051)10513600
0321
( exp115
1
L)vv
H1
( expqQ
1 L)(xa
The general situationThe general situation
bull Unsteadybull Inhomogeneousbull Non-conservativebull 3D
General transport equation
Advection-Dispersion Equation (ADE)
Advection-dispersion EquationAdvection-dispersion Equationbull Can be derived from the mass balancebull Expresses the temporal and spatial changes of
the observed state variablebull Itrsquos solution is the concentration-distribution over
timebull Differential equation
Change in concentration with time
Change due to advection
=Change due to diffusion or dispersion
Change due to conversion processes
+ +
Molecular diffusionMolecular diffusion
DIFFUSION
CONVECTION
v
Molecular Molecular diffusiondiffusion
FickrsquosFickrsquos LawLaw
c1 c2x
bull c1 c2 separated tanks
bull open valve
Mass flux (kgs)
D - molecular diffusioncoefficient [m2s]
bull equalization and mixing (Brown-movement)
through area unit
bull depends on temperature
IN conv + diff
dxdy
dzOUT conv + diff
x direction
IN OUT
bull convection
CHANGE
bull diffusion
Mass balance Mass balance equationequation
x direction
IN conv + diff
dxdy
dzOUT conv + diff
Mass balance Mass balance equationequation
Convection Diffusion
If D(x) = const in x direction
Convection - Diffusion 1D Equation
The other directions are similar
Convection transfering Diffusion spreading
Mass balance Mass balance equationequation
x y z directions (3D)
D ndash molecular diffusion coefficient
material specific water 10-4 cm2s
space independent
slow mixing and equalization
Convection The water particles with differentconcentrations of pollutant move according to their differing flow velocity
Diffusion The adjacent water particles mix with each otherwhich causes the equalization of concentration
Mass balance equationMass balance equation
bull Laminar flow ordered with parallel flow linesbull Turbulent flow disordered random (flow and direction)
because of surface roughness (friction) rarr causes intensive mixing
v(c)
t
v deviation pulsation
T Time step of the turbulence
averagev~
0
bull Natural streams always turbulent
We usually can use the average only
TurbulenceTurbulence
Convection v middot c [ kgm2 middot s ]
0
0
Turbulent transport Turbulent transport descriptiondescription
v
turbulent diffusion
molecular diffusion
Dtx Dty Dtz gtgt D
Depends on the direction
Function of the velocity space
X direction (the other are similar)
Analogy with Fickrsquos Law rarr the original transport equation does not change only the D parameter
Turbulent diffusionTurbulent diffusion
Dx = D + Dtx Dy = D + Dty Dz = D + Dtz
Convection
Diffusion
Turbulent diffusion
Stochastic fluctuations of the flow velocity (pulsations)
Diffusive process in mathematical sense (~ Fickrsquos Law)
Convective transport (temporal differences)
Temporal averaging (T)
3D transport equation in turbulent flow3D transport equation in turbulent flow
Spatial simplifications (2D)Spatial simplifications (2D) - Dispersion - Dispersion
Depth averaged flow velocity v
0v
H
After the expansion of the convectiv part (v middot c) in x direction
Depth Integration (3D2D)
Turbulent dispersion (~ diffusion)
z
x
Velocity depth profile
Dx Dy
turbulent dispersion coefficients in 2D equation
Depth averaging (H)
11D tD transport equation in turbulent flow (cross-section avransport equation in turbulent flow (cross-section av))
Dx turbulent dispersion coefficient in 1D equation
Cross-section averaging (A)
2D transport equation in turbulent flow (depth averaged)2D transport equation in turbulent flow (depth averaged)
Turbulent dispersionTurbulent dispersion
Convectiv transport arising from the spatial differences of the velocity profile (faster and slower particles compared to the average)
v
In the 2D and 1D equations only
Dispersion coefficient a function of the velocity space (hydraulic parameters channel geometry)
Dx = Ddx + Dtx + D Dy
= Ddy + Dty + D Dx = Ddxx + Ddx + Dtx + D
Dx Dy
gtgt Dx Dx gtgt Dx
In case of more irregular channel and higher depth or larger cross-section area its value is bigger
Also present in laminar flow
Order of magnitude of Order of magnitude of diffusion dispersion diffusion dispersion coefficientscoefficients
10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s
Pore water
Mol diff
Vertical turbulent diff
High depth Shallow depth
Cross disp (2D)
Longitudinal disp (2D)
Horizontal turbulent diff
Longitudinal disp (1D)
Evaluation of dispersion coefficients measuring
Measuring with tracing matter (eg paint)
Inverse calculation from the measured concentration
2D Cross dispersion coefficient (Fischer)
Dy = dy u R (m2s)
dy - dimensionless cross dispersion coefficient
Straight regular channel dy 015
Moderately winding channel dy 02 ndash 06
Hard winding irregular channel dy gt 06 (1-2)
R ndash hydraulic radius (m)
u - shearing velocity at the channel bottom (ms)
u = (g R I)05Longitudinal dispersion coefficient dx 6 (2D) dxx = 100 ndash 1000 (1D)
Velocity space turbulent pulsation unequal distribution
Estimation of dispersion coefficients empirical forms
TRANSPORT EQUATION FOR NON-CONSERVATIVE TRANSPORT EQUATION FOR NON-CONSERVATIVE MATTERSMATTERS
bull There are sources and losses in the flow space
bull Physical chemical biochemical transformations take place
bull Non-conservative pollutants reaction kinetics ( R(C) )
bull They are taken into account in a linear way
dCdt = -C where is the coefficient of reaction kinetics (usually first-order kinetics)
cαxc
DAxx
c)v(Atc)(A
xx
1D equation in this case
ANALYTICAL SOLUTIONS OF THE TRANSPORT ANALYTICAL SOLUTIONS OF THE TRANSPORT EQUATIONEQUATION
Permanent mixing of the pollutants
Polluntant-wave transmission
Estimation of the geometric and hydraulic parameters
Exacter calculations based on measurments using numerical methods
The analitical solutions can be obtained in simpler cases only (approximate calculations)
Main steps
2
2
y
cD
x
cv yx
PERMANENT MIXING
The pollutant emission is steady in time
Permanent river discharge (low flow condition)
Constant flow velocity flow depth and dispersion coef
2D equation negligible vertical changes (shallow river)
Convection transferring
Dispersion spreading
Initial condition M0 (x0 y0) - emission
Boundary cond cy = 0 at the bank
)()()(
cvhy
cvhxt
chyx )()(
y
cDh
yx
cDh
xyx
x
yy
v
xD2
INLET AT THE MAIN FLOW PATH
Longitudinal according to x-frac12 function
Cross direction Gauss-distribution
cmax
M [kgs]cmax
)4
exp(2
2
xD
yv
xvDh
Mc (x t)y
x
xy
x
yb
v
xDB
234
Bb at value 01 cmax 1522bBBrand width
B ~ Bb2
1 0270 BD
vL
y
x
First distance of the mixing
M
1L
Bb
C (x1 y)
x1
M
x
yb
v
xDB
2152
21 110 B
D
vL
y
x
INLET AT THE RIVER BANK
)4
exp(2
xD
yv
xvDh
Mc
y
x
xy
cmax
C (x1 y)
x1
M
INLET NEAR THE RIVER BANK
)4
(exp ( xDy
( y-y0 )2-v
xvD2h
Mc x
xy
cmax
))4
+exp ( xDy
( y+y0 )2-vx
y0 C (x1 y)
x1
IMPACTS OF THE RIVER BANKS (total river section)
Boundary condition reflection theory (infinite series)
Complete mixing the change along the cross-section is less than 10
L2 ~ 3L1 second distance of the mixing
Inlet at a point in y0 distance from the bank
xvD2h
Mc
xy
)4
exp ( xDy
( y-y0 +2nB)2-vx
)4
+ exp ( xDy
( y+y0 -2nB)2-vx
sumn=infin
n=minusinfin(
)
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
Clim = 15 gm3
concentration limit
Q = 15 m3s
q = 1 m3sc = 300 gm3
Cbg = 10 gm3
L = 5 km
B = 75 m
vS = 03 mh
H = 2 m
EXAMPLE
Complete mixing
3bg0 gm28125
11530011015
cqCQC
3lim
33
s0
gm15Cgm22836)10513600
0321
( exp28125
L)vv
H1
( expC L)(xC
Concentration at distance L from the emission
Flow velocity
ms10275
15HB
Qv
Maximum possible C0 value
Maximum allowed waste water concentration
Allowed maximum concentration
C (x = L)max = Clim
3
3s
limmax0
gm18474
)10513600
0321
( exp
15
L)vv
H1
( exp
C C
3
bgmax0max
gm145584
11015-11518474
q
CQ-qQCc
Transmission coefficient
Necessary reduction of emission
gs154416145584)(3001)c(cqΔE max
33
s
sm0051)10513600
0321
( exp115
1
L)vv
H1
( expqQ
1 L)(xa
The general situationThe general situation
bull Unsteadybull Inhomogeneousbull Non-conservativebull 3D
General transport equation
Advection-Dispersion Equation (ADE)
Advection-dispersion EquationAdvection-dispersion Equationbull Can be derived from the mass balancebull Expresses the temporal and spatial changes of
the observed state variablebull Itrsquos solution is the concentration-distribution over
timebull Differential equation
Change in concentration with time
Change due to advection
=Change due to diffusion or dispersion
Change due to conversion processes
+ +
Molecular diffusionMolecular diffusion
DIFFUSION
CONVECTION
v
Molecular Molecular diffusiondiffusion
FickrsquosFickrsquos LawLaw
c1 c2x
bull c1 c2 separated tanks
bull open valve
Mass flux (kgs)
D - molecular diffusioncoefficient [m2s]
bull equalization and mixing (Brown-movement)
through area unit
bull depends on temperature
IN conv + diff
dxdy
dzOUT conv + diff
x direction
IN OUT
bull convection
CHANGE
bull diffusion
Mass balance Mass balance equationequation
x direction
IN conv + diff
dxdy
dzOUT conv + diff
Mass balance Mass balance equationequation
Convection Diffusion
If D(x) = const in x direction
Convection - Diffusion 1D Equation
The other directions are similar
Convection transfering Diffusion spreading
Mass balance Mass balance equationequation
x y z directions (3D)
D ndash molecular diffusion coefficient
material specific water 10-4 cm2s
space independent
slow mixing and equalization
Convection The water particles with differentconcentrations of pollutant move according to their differing flow velocity
Diffusion The adjacent water particles mix with each otherwhich causes the equalization of concentration
Mass balance equationMass balance equation
bull Laminar flow ordered with parallel flow linesbull Turbulent flow disordered random (flow and direction)
because of surface roughness (friction) rarr causes intensive mixing
v(c)
t
v deviation pulsation
T Time step of the turbulence
averagev~
0
bull Natural streams always turbulent
We usually can use the average only
TurbulenceTurbulence
Convection v middot c [ kgm2 middot s ]
0
0
Turbulent transport Turbulent transport descriptiondescription
v
turbulent diffusion
molecular diffusion
Dtx Dty Dtz gtgt D
Depends on the direction
Function of the velocity space
X direction (the other are similar)
Analogy with Fickrsquos Law rarr the original transport equation does not change only the D parameter
Turbulent diffusionTurbulent diffusion
Dx = D + Dtx Dy = D + Dty Dz = D + Dtz
Convection
Diffusion
Turbulent diffusion
Stochastic fluctuations of the flow velocity (pulsations)
Diffusive process in mathematical sense (~ Fickrsquos Law)
Convective transport (temporal differences)
Temporal averaging (T)
3D transport equation in turbulent flow3D transport equation in turbulent flow
Spatial simplifications (2D)Spatial simplifications (2D) - Dispersion - Dispersion
Depth averaged flow velocity v
0v
H
After the expansion of the convectiv part (v middot c) in x direction
Depth Integration (3D2D)
Turbulent dispersion (~ diffusion)
z
x
Velocity depth profile
Dx Dy
turbulent dispersion coefficients in 2D equation
Depth averaging (H)
11D tD transport equation in turbulent flow (cross-section avransport equation in turbulent flow (cross-section av))
Dx turbulent dispersion coefficient in 1D equation
Cross-section averaging (A)
2D transport equation in turbulent flow (depth averaged)2D transport equation in turbulent flow (depth averaged)
Turbulent dispersionTurbulent dispersion
Convectiv transport arising from the spatial differences of the velocity profile (faster and slower particles compared to the average)
v
In the 2D and 1D equations only
Dispersion coefficient a function of the velocity space (hydraulic parameters channel geometry)
Dx = Ddx + Dtx + D Dy
= Ddy + Dty + D Dx = Ddxx + Ddx + Dtx + D
Dx Dy
gtgt Dx Dx gtgt Dx
In case of more irregular channel and higher depth or larger cross-section area its value is bigger
Also present in laminar flow
Order of magnitude of Order of magnitude of diffusion dispersion diffusion dispersion coefficientscoefficients
10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s
Pore water
Mol diff
Vertical turbulent diff
High depth Shallow depth
Cross disp (2D)
Longitudinal disp (2D)
Horizontal turbulent diff
Longitudinal disp (1D)
Evaluation of dispersion coefficients measuring
Measuring with tracing matter (eg paint)
Inverse calculation from the measured concentration
2D Cross dispersion coefficient (Fischer)
Dy = dy u R (m2s)
dy - dimensionless cross dispersion coefficient
Straight regular channel dy 015
Moderately winding channel dy 02 ndash 06
Hard winding irregular channel dy gt 06 (1-2)
R ndash hydraulic radius (m)
u - shearing velocity at the channel bottom (ms)
u = (g R I)05Longitudinal dispersion coefficient dx 6 (2D) dxx = 100 ndash 1000 (1D)
Velocity space turbulent pulsation unequal distribution
Estimation of dispersion coefficients empirical forms
TRANSPORT EQUATION FOR NON-CONSERVATIVE TRANSPORT EQUATION FOR NON-CONSERVATIVE MATTERSMATTERS
bull There are sources and losses in the flow space
bull Physical chemical biochemical transformations take place
bull Non-conservative pollutants reaction kinetics ( R(C) )
bull They are taken into account in a linear way
dCdt = -C where is the coefficient of reaction kinetics (usually first-order kinetics)
cαxc
DAxx
c)v(Atc)(A
xx
1D equation in this case
ANALYTICAL SOLUTIONS OF THE TRANSPORT ANALYTICAL SOLUTIONS OF THE TRANSPORT EQUATIONEQUATION
Permanent mixing of the pollutants
Polluntant-wave transmission
Estimation of the geometric and hydraulic parameters
Exacter calculations based on measurments using numerical methods
The analitical solutions can be obtained in simpler cases only (approximate calculations)
Main steps
2
2
y
cD
x
cv yx
PERMANENT MIXING
The pollutant emission is steady in time
Permanent river discharge (low flow condition)
Constant flow velocity flow depth and dispersion coef
2D equation negligible vertical changes (shallow river)
Convection transferring
Dispersion spreading
Initial condition M0 (x0 y0) - emission
Boundary cond cy = 0 at the bank
)()()(
cvhy
cvhxt
chyx )()(
y
cDh
yx
cDh
xyx
x
yy
v
xD2
INLET AT THE MAIN FLOW PATH
Longitudinal according to x-frac12 function
Cross direction Gauss-distribution
cmax
M [kgs]cmax
)4
exp(2
2
xD
yv
xvDh
Mc (x t)y
x
xy
x
yb
v
xDB
234
Bb at value 01 cmax 1522bBBrand width
B ~ Bb2
1 0270 BD
vL
y
x
First distance of the mixing
M
1L
Bb
C (x1 y)
x1
M
x
yb
v
xDB
2152
21 110 B
D
vL
y
x
INLET AT THE RIVER BANK
)4
exp(2
xD
yv
xvDh
Mc
y
x
xy
cmax
C (x1 y)
x1
M
INLET NEAR THE RIVER BANK
)4
(exp ( xDy
( y-y0 )2-v
xvD2h
Mc x
xy
cmax
))4
+exp ( xDy
( y+y0 )2-vx
y0 C (x1 y)
x1
IMPACTS OF THE RIVER BANKS (total river section)
Boundary condition reflection theory (infinite series)
Complete mixing the change along the cross-section is less than 10
L2 ~ 3L1 second distance of the mixing
Inlet at a point in y0 distance from the bank
xvD2h
Mc
xy
)4
exp ( xDy
( y-y0 +2nB)2-vx
)4
+ exp ( xDy
( y+y0 -2nB)2-vx
sumn=infin
n=minusinfin(
)
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
Complete mixing
3bg0 gm28125
11530011015
cqCQC
3lim
33
s0
gm15Cgm22836)10513600
0321
( exp28125
L)vv
H1
( expC L)(xC
Concentration at distance L from the emission
Flow velocity
ms10275
15HB
Qv
Maximum possible C0 value
Maximum allowed waste water concentration
Allowed maximum concentration
C (x = L)max = Clim
3
3s
limmax0
gm18474
)10513600
0321
( exp
15
L)vv
H1
( exp
C C
3
bgmax0max
gm145584
11015-11518474
q
CQ-qQCc
Transmission coefficient
Necessary reduction of emission
gs154416145584)(3001)c(cqΔE max
33
s
sm0051)10513600
0321
( exp115
1
L)vv
H1
( expqQ
1 L)(xa
The general situationThe general situation
bull Unsteadybull Inhomogeneousbull Non-conservativebull 3D
General transport equation
Advection-Dispersion Equation (ADE)
Advection-dispersion EquationAdvection-dispersion Equationbull Can be derived from the mass balancebull Expresses the temporal and spatial changes of
the observed state variablebull Itrsquos solution is the concentration-distribution over
timebull Differential equation
Change in concentration with time
Change due to advection
=Change due to diffusion or dispersion
Change due to conversion processes
+ +
Molecular diffusionMolecular diffusion
DIFFUSION
CONVECTION
v
Molecular Molecular diffusiondiffusion
FickrsquosFickrsquos LawLaw
c1 c2x
bull c1 c2 separated tanks
bull open valve
Mass flux (kgs)
D - molecular diffusioncoefficient [m2s]
bull equalization and mixing (Brown-movement)
through area unit
bull depends on temperature
IN conv + diff
dxdy
dzOUT conv + diff
x direction
IN OUT
bull convection
CHANGE
bull diffusion
Mass balance Mass balance equationequation
x direction
IN conv + diff
dxdy
dzOUT conv + diff
Mass balance Mass balance equationequation
Convection Diffusion
If D(x) = const in x direction
Convection - Diffusion 1D Equation
The other directions are similar
Convection transfering Diffusion spreading
Mass balance Mass balance equationequation
x y z directions (3D)
D ndash molecular diffusion coefficient
material specific water 10-4 cm2s
space independent
slow mixing and equalization
Convection The water particles with differentconcentrations of pollutant move according to their differing flow velocity
Diffusion The adjacent water particles mix with each otherwhich causes the equalization of concentration
Mass balance equationMass balance equation
bull Laminar flow ordered with parallel flow linesbull Turbulent flow disordered random (flow and direction)
because of surface roughness (friction) rarr causes intensive mixing
v(c)
t
v deviation pulsation
T Time step of the turbulence
averagev~
0
bull Natural streams always turbulent
We usually can use the average only
TurbulenceTurbulence
Convection v middot c [ kgm2 middot s ]
0
0
Turbulent transport Turbulent transport descriptiondescription
v
turbulent diffusion
molecular diffusion
Dtx Dty Dtz gtgt D
Depends on the direction
Function of the velocity space
X direction (the other are similar)
Analogy with Fickrsquos Law rarr the original transport equation does not change only the D parameter
Turbulent diffusionTurbulent diffusion
Dx = D + Dtx Dy = D + Dty Dz = D + Dtz
Convection
Diffusion
Turbulent diffusion
Stochastic fluctuations of the flow velocity (pulsations)
Diffusive process in mathematical sense (~ Fickrsquos Law)
Convective transport (temporal differences)
Temporal averaging (T)
3D transport equation in turbulent flow3D transport equation in turbulent flow
Spatial simplifications (2D)Spatial simplifications (2D) - Dispersion - Dispersion
Depth averaged flow velocity v
0v
H
After the expansion of the convectiv part (v middot c) in x direction
Depth Integration (3D2D)
Turbulent dispersion (~ diffusion)
z
x
Velocity depth profile
Dx Dy
turbulent dispersion coefficients in 2D equation
Depth averaging (H)
11D tD transport equation in turbulent flow (cross-section avransport equation in turbulent flow (cross-section av))
Dx turbulent dispersion coefficient in 1D equation
Cross-section averaging (A)
2D transport equation in turbulent flow (depth averaged)2D transport equation in turbulent flow (depth averaged)
Turbulent dispersionTurbulent dispersion
Convectiv transport arising from the spatial differences of the velocity profile (faster and slower particles compared to the average)
v
In the 2D and 1D equations only
Dispersion coefficient a function of the velocity space (hydraulic parameters channel geometry)
Dx = Ddx + Dtx + D Dy
= Ddy + Dty + D Dx = Ddxx + Ddx + Dtx + D
Dx Dy
gtgt Dx Dx gtgt Dx
In case of more irregular channel and higher depth or larger cross-section area its value is bigger
Also present in laminar flow
Order of magnitude of Order of magnitude of diffusion dispersion diffusion dispersion coefficientscoefficients
10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s
Pore water
Mol diff
Vertical turbulent diff
High depth Shallow depth
Cross disp (2D)
Longitudinal disp (2D)
Horizontal turbulent diff
Longitudinal disp (1D)
Evaluation of dispersion coefficients measuring
Measuring with tracing matter (eg paint)
Inverse calculation from the measured concentration
2D Cross dispersion coefficient (Fischer)
Dy = dy u R (m2s)
dy - dimensionless cross dispersion coefficient
Straight regular channel dy 015
Moderately winding channel dy 02 ndash 06
Hard winding irregular channel dy gt 06 (1-2)
R ndash hydraulic radius (m)
u - shearing velocity at the channel bottom (ms)
u = (g R I)05Longitudinal dispersion coefficient dx 6 (2D) dxx = 100 ndash 1000 (1D)
Velocity space turbulent pulsation unequal distribution
Estimation of dispersion coefficients empirical forms
TRANSPORT EQUATION FOR NON-CONSERVATIVE TRANSPORT EQUATION FOR NON-CONSERVATIVE MATTERSMATTERS
bull There are sources and losses in the flow space
bull Physical chemical biochemical transformations take place
bull Non-conservative pollutants reaction kinetics ( R(C) )
bull They are taken into account in a linear way
dCdt = -C where is the coefficient of reaction kinetics (usually first-order kinetics)
cαxc
DAxx
c)v(Atc)(A
xx
1D equation in this case
ANALYTICAL SOLUTIONS OF THE TRANSPORT ANALYTICAL SOLUTIONS OF THE TRANSPORT EQUATIONEQUATION
Permanent mixing of the pollutants
Polluntant-wave transmission
Estimation of the geometric and hydraulic parameters
Exacter calculations based on measurments using numerical methods
The analitical solutions can be obtained in simpler cases only (approximate calculations)
Main steps
2
2
y
cD
x
cv yx
PERMANENT MIXING
The pollutant emission is steady in time
Permanent river discharge (low flow condition)
Constant flow velocity flow depth and dispersion coef
2D equation negligible vertical changes (shallow river)
Convection transferring
Dispersion spreading
Initial condition M0 (x0 y0) - emission
Boundary cond cy = 0 at the bank
)()()(
cvhy
cvhxt
chyx )()(
y
cDh
yx
cDh
xyx
x
yy
v
xD2
INLET AT THE MAIN FLOW PATH
Longitudinal according to x-frac12 function
Cross direction Gauss-distribution
cmax
M [kgs]cmax
)4
exp(2
2
xD
yv
xvDh
Mc (x t)y
x
xy
x
yb
v
xDB
234
Bb at value 01 cmax 1522bBBrand width
B ~ Bb2
1 0270 BD
vL
y
x
First distance of the mixing
M
1L
Bb
C (x1 y)
x1
M
x
yb
v
xDB
2152
21 110 B
D
vL
y
x
INLET AT THE RIVER BANK
)4
exp(2
xD
yv
xvDh
Mc
y
x
xy
cmax
C (x1 y)
x1
M
INLET NEAR THE RIVER BANK
)4
(exp ( xDy
( y-y0 )2-v
xvD2h
Mc x
xy
cmax
))4
+exp ( xDy
( y+y0 )2-vx
y0 C (x1 y)
x1
IMPACTS OF THE RIVER BANKS (total river section)
Boundary condition reflection theory (infinite series)
Complete mixing the change along the cross-section is less than 10
L2 ~ 3L1 second distance of the mixing
Inlet at a point in y0 distance from the bank
xvD2h
Mc
xy
)4
exp ( xDy
( y-y0 +2nB)2-vx
)4
+ exp ( xDy
( y+y0 -2nB)2-vx
sumn=infin
n=minusinfin(
)
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
Maximum possible C0 value
Maximum allowed waste water concentration
Allowed maximum concentration
C (x = L)max = Clim
3
3s
limmax0
gm18474
)10513600
0321
( exp
15
L)vv
H1
( exp
C C
3
bgmax0max
gm145584
11015-11518474
q
CQ-qQCc
Transmission coefficient
Necessary reduction of emission
gs154416145584)(3001)c(cqΔE max
33
s
sm0051)10513600
0321
( exp115
1
L)vv
H1
( expqQ
1 L)(xa
The general situationThe general situation
bull Unsteadybull Inhomogeneousbull Non-conservativebull 3D
General transport equation
Advection-Dispersion Equation (ADE)
Advection-dispersion EquationAdvection-dispersion Equationbull Can be derived from the mass balancebull Expresses the temporal and spatial changes of
the observed state variablebull Itrsquos solution is the concentration-distribution over
timebull Differential equation
Change in concentration with time
Change due to advection
=Change due to diffusion or dispersion
Change due to conversion processes
+ +
Molecular diffusionMolecular diffusion
DIFFUSION
CONVECTION
v
Molecular Molecular diffusiondiffusion
FickrsquosFickrsquos LawLaw
c1 c2x
bull c1 c2 separated tanks
bull open valve
Mass flux (kgs)
D - molecular diffusioncoefficient [m2s]
bull equalization and mixing (Brown-movement)
through area unit
bull depends on temperature
IN conv + diff
dxdy
dzOUT conv + diff
x direction
IN OUT
bull convection
CHANGE
bull diffusion
Mass balance Mass balance equationequation
x direction
IN conv + diff
dxdy
dzOUT conv + diff
Mass balance Mass balance equationequation
Convection Diffusion
If D(x) = const in x direction
Convection - Diffusion 1D Equation
The other directions are similar
Convection transfering Diffusion spreading
Mass balance Mass balance equationequation
x y z directions (3D)
D ndash molecular diffusion coefficient
material specific water 10-4 cm2s
space independent
slow mixing and equalization
Convection The water particles with differentconcentrations of pollutant move according to their differing flow velocity
Diffusion The adjacent water particles mix with each otherwhich causes the equalization of concentration
Mass balance equationMass balance equation
bull Laminar flow ordered with parallel flow linesbull Turbulent flow disordered random (flow and direction)
because of surface roughness (friction) rarr causes intensive mixing
v(c)
t
v deviation pulsation
T Time step of the turbulence
averagev~
0
bull Natural streams always turbulent
We usually can use the average only
TurbulenceTurbulence
Convection v middot c [ kgm2 middot s ]
0
0
Turbulent transport Turbulent transport descriptiondescription
v
turbulent diffusion
molecular diffusion
Dtx Dty Dtz gtgt D
Depends on the direction
Function of the velocity space
X direction (the other are similar)
Analogy with Fickrsquos Law rarr the original transport equation does not change only the D parameter
Turbulent diffusionTurbulent diffusion
Dx = D + Dtx Dy = D + Dty Dz = D + Dtz
Convection
Diffusion
Turbulent diffusion
Stochastic fluctuations of the flow velocity (pulsations)
Diffusive process in mathematical sense (~ Fickrsquos Law)
Convective transport (temporal differences)
Temporal averaging (T)
3D transport equation in turbulent flow3D transport equation in turbulent flow
Spatial simplifications (2D)Spatial simplifications (2D) - Dispersion - Dispersion
Depth averaged flow velocity v
0v
H
After the expansion of the convectiv part (v middot c) in x direction
Depth Integration (3D2D)
Turbulent dispersion (~ diffusion)
z
x
Velocity depth profile
Dx Dy
turbulent dispersion coefficients in 2D equation
Depth averaging (H)
11D tD transport equation in turbulent flow (cross-section avransport equation in turbulent flow (cross-section av))
Dx turbulent dispersion coefficient in 1D equation
Cross-section averaging (A)
2D transport equation in turbulent flow (depth averaged)2D transport equation in turbulent flow (depth averaged)
Turbulent dispersionTurbulent dispersion
Convectiv transport arising from the spatial differences of the velocity profile (faster and slower particles compared to the average)
v
In the 2D and 1D equations only
Dispersion coefficient a function of the velocity space (hydraulic parameters channel geometry)
Dx = Ddx + Dtx + D Dy
= Ddy + Dty + D Dx = Ddxx + Ddx + Dtx + D
Dx Dy
gtgt Dx Dx gtgt Dx
In case of more irregular channel and higher depth or larger cross-section area its value is bigger
Also present in laminar flow
Order of magnitude of Order of magnitude of diffusion dispersion diffusion dispersion coefficientscoefficients
10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s
Pore water
Mol diff
Vertical turbulent diff
High depth Shallow depth
Cross disp (2D)
Longitudinal disp (2D)
Horizontal turbulent diff
Longitudinal disp (1D)
Evaluation of dispersion coefficients measuring
Measuring with tracing matter (eg paint)
Inverse calculation from the measured concentration
2D Cross dispersion coefficient (Fischer)
Dy = dy u R (m2s)
dy - dimensionless cross dispersion coefficient
Straight regular channel dy 015
Moderately winding channel dy 02 ndash 06
Hard winding irregular channel dy gt 06 (1-2)
R ndash hydraulic radius (m)
u - shearing velocity at the channel bottom (ms)
u = (g R I)05Longitudinal dispersion coefficient dx 6 (2D) dxx = 100 ndash 1000 (1D)
Velocity space turbulent pulsation unequal distribution
Estimation of dispersion coefficients empirical forms
TRANSPORT EQUATION FOR NON-CONSERVATIVE TRANSPORT EQUATION FOR NON-CONSERVATIVE MATTERSMATTERS
bull There are sources and losses in the flow space
bull Physical chemical biochemical transformations take place
bull Non-conservative pollutants reaction kinetics ( R(C) )
bull They are taken into account in a linear way
dCdt = -C where is the coefficient of reaction kinetics (usually first-order kinetics)
cαxc
DAxx
c)v(Atc)(A
xx
1D equation in this case
ANALYTICAL SOLUTIONS OF THE TRANSPORT ANALYTICAL SOLUTIONS OF THE TRANSPORT EQUATIONEQUATION
Permanent mixing of the pollutants
Polluntant-wave transmission
Estimation of the geometric and hydraulic parameters
Exacter calculations based on measurments using numerical methods
The analitical solutions can be obtained in simpler cases only (approximate calculations)
Main steps
2
2
y
cD
x
cv yx
PERMANENT MIXING
The pollutant emission is steady in time
Permanent river discharge (low flow condition)
Constant flow velocity flow depth and dispersion coef
2D equation negligible vertical changes (shallow river)
Convection transferring
Dispersion spreading
Initial condition M0 (x0 y0) - emission
Boundary cond cy = 0 at the bank
)()()(
cvhy
cvhxt
chyx )()(
y
cDh
yx
cDh
xyx
x
yy
v
xD2
INLET AT THE MAIN FLOW PATH
Longitudinal according to x-frac12 function
Cross direction Gauss-distribution
cmax
M [kgs]cmax
)4
exp(2
2
xD
yv
xvDh
Mc (x t)y
x
xy
x
yb
v
xDB
234
Bb at value 01 cmax 1522bBBrand width
B ~ Bb2
1 0270 BD
vL
y
x
First distance of the mixing
M
1L
Bb
C (x1 y)
x1
M
x
yb
v
xDB
2152
21 110 B
D
vL
y
x
INLET AT THE RIVER BANK
)4
exp(2
xD
yv
xvDh
Mc
y
x
xy
cmax
C (x1 y)
x1
M
INLET NEAR THE RIVER BANK
)4
(exp ( xDy
( y-y0 )2-v
xvD2h
Mc x
xy
cmax
))4
+exp ( xDy
( y+y0 )2-vx
y0 C (x1 y)
x1
IMPACTS OF THE RIVER BANKS (total river section)
Boundary condition reflection theory (infinite series)
Complete mixing the change along the cross-section is less than 10
L2 ~ 3L1 second distance of the mixing
Inlet at a point in y0 distance from the bank
xvD2h
Mc
xy
)4
exp ( xDy
( y-y0 +2nB)2-vx
)4
+ exp ( xDy
( y+y0 -2nB)2-vx
sumn=infin
n=minusinfin(
)
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
Transmission coefficient
Necessary reduction of emission
gs154416145584)(3001)c(cqΔE max
33
s
sm0051)10513600
0321
( exp115
1
L)vv
H1
( expqQ
1 L)(xa
The general situationThe general situation
bull Unsteadybull Inhomogeneousbull Non-conservativebull 3D
General transport equation
Advection-Dispersion Equation (ADE)
Advection-dispersion EquationAdvection-dispersion Equationbull Can be derived from the mass balancebull Expresses the temporal and spatial changes of
the observed state variablebull Itrsquos solution is the concentration-distribution over
timebull Differential equation
Change in concentration with time
Change due to advection
=Change due to diffusion or dispersion
Change due to conversion processes
+ +
Molecular diffusionMolecular diffusion
DIFFUSION
CONVECTION
v
Molecular Molecular diffusiondiffusion
FickrsquosFickrsquos LawLaw
c1 c2x
bull c1 c2 separated tanks
bull open valve
Mass flux (kgs)
D - molecular diffusioncoefficient [m2s]
bull equalization and mixing (Brown-movement)
through area unit
bull depends on temperature
IN conv + diff
dxdy
dzOUT conv + diff
x direction
IN OUT
bull convection
CHANGE
bull diffusion
Mass balance Mass balance equationequation
x direction
IN conv + diff
dxdy
dzOUT conv + diff
Mass balance Mass balance equationequation
Convection Diffusion
If D(x) = const in x direction
Convection - Diffusion 1D Equation
The other directions are similar
Convection transfering Diffusion spreading
Mass balance Mass balance equationequation
x y z directions (3D)
D ndash molecular diffusion coefficient
material specific water 10-4 cm2s
space independent
slow mixing and equalization
Convection The water particles with differentconcentrations of pollutant move according to their differing flow velocity
Diffusion The adjacent water particles mix with each otherwhich causes the equalization of concentration
Mass balance equationMass balance equation
bull Laminar flow ordered with parallel flow linesbull Turbulent flow disordered random (flow and direction)
because of surface roughness (friction) rarr causes intensive mixing
v(c)
t
v deviation pulsation
T Time step of the turbulence
averagev~
0
bull Natural streams always turbulent
We usually can use the average only
TurbulenceTurbulence
Convection v middot c [ kgm2 middot s ]
0
0
Turbulent transport Turbulent transport descriptiondescription
v
turbulent diffusion
molecular diffusion
Dtx Dty Dtz gtgt D
Depends on the direction
Function of the velocity space
X direction (the other are similar)
Analogy with Fickrsquos Law rarr the original transport equation does not change only the D parameter
Turbulent diffusionTurbulent diffusion
Dx = D + Dtx Dy = D + Dty Dz = D + Dtz
Convection
Diffusion
Turbulent diffusion
Stochastic fluctuations of the flow velocity (pulsations)
Diffusive process in mathematical sense (~ Fickrsquos Law)
Convective transport (temporal differences)
Temporal averaging (T)
3D transport equation in turbulent flow3D transport equation in turbulent flow
Spatial simplifications (2D)Spatial simplifications (2D) - Dispersion - Dispersion
Depth averaged flow velocity v
0v
H
After the expansion of the convectiv part (v middot c) in x direction
Depth Integration (3D2D)
Turbulent dispersion (~ diffusion)
z
x
Velocity depth profile
Dx Dy
turbulent dispersion coefficients in 2D equation
Depth averaging (H)
11D tD transport equation in turbulent flow (cross-section avransport equation in turbulent flow (cross-section av))
Dx turbulent dispersion coefficient in 1D equation
Cross-section averaging (A)
2D transport equation in turbulent flow (depth averaged)2D transport equation in turbulent flow (depth averaged)
Turbulent dispersionTurbulent dispersion
Convectiv transport arising from the spatial differences of the velocity profile (faster and slower particles compared to the average)
v
In the 2D and 1D equations only
Dispersion coefficient a function of the velocity space (hydraulic parameters channel geometry)
Dx = Ddx + Dtx + D Dy
= Ddy + Dty + D Dx = Ddxx + Ddx + Dtx + D
Dx Dy
gtgt Dx Dx gtgt Dx
In case of more irregular channel and higher depth or larger cross-section area its value is bigger
Also present in laminar flow
Order of magnitude of Order of magnitude of diffusion dispersion diffusion dispersion coefficientscoefficients
10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s
Pore water
Mol diff
Vertical turbulent diff
High depth Shallow depth
Cross disp (2D)
Longitudinal disp (2D)
Horizontal turbulent diff
Longitudinal disp (1D)
Evaluation of dispersion coefficients measuring
Measuring with tracing matter (eg paint)
Inverse calculation from the measured concentration
2D Cross dispersion coefficient (Fischer)
Dy = dy u R (m2s)
dy - dimensionless cross dispersion coefficient
Straight regular channel dy 015
Moderately winding channel dy 02 ndash 06
Hard winding irregular channel dy gt 06 (1-2)
R ndash hydraulic radius (m)
u - shearing velocity at the channel bottom (ms)
u = (g R I)05Longitudinal dispersion coefficient dx 6 (2D) dxx = 100 ndash 1000 (1D)
Velocity space turbulent pulsation unequal distribution
Estimation of dispersion coefficients empirical forms
TRANSPORT EQUATION FOR NON-CONSERVATIVE TRANSPORT EQUATION FOR NON-CONSERVATIVE MATTERSMATTERS
bull There are sources and losses in the flow space
bull Physical chemical biochemical transformations take place
bull Non-conservative pollutants reaction kinetics ( R(C) )
bull They are taken into account in a linear way
dCdt = -C where is the coefficient of reaction kinetics (usually first-order kinetics)
cαxc
DAxx
c)v(Atc)(A
xx
1D equation in this case
ANALYTICAL SOLUTIONS OF THE TRANSPORT ANALYTICAL SOLUTIONS OF THE TRANSPORT EQUATIONEQUATION
Permanent mixing of the pollutants
Polluntant-wave transmission
Estimation of the geometric and hydraulic parameters
Exacter calculations based on measurments using numerical methods
The analitical solutions can be obtained in simpler cases only (approximate calculations)
Main steps
2
2
y
cD
x
cv yx
PERMANENT MIXING
The pollutant emission is steady in time
Permanent river discharge (low flow condition)
Constant flow velocity flow depth and dispersion coef
2D equation negligible vertical changes (shallow river)
Convection transferring
Dispersion spreading
Initial condition M0 (x0 y0) - emission
Boundary cond cy = 0 at the bank
)()()(
cvhy
cvhxt
chyx )()(
y
cDh
yx
cDh
xyx
x
yy
v
xD2
INLET AT THE MAIN FLOW PATH
Longitudinal according to x-frac12 function
Cross direction Gauss-distribution
cmax
M [kgs]cmax
)4
exp(2
2
xD
yv
xvDh
Mc (x t)y
x
xy
x
yb
v
xDB
234
Bb at value 01 cmax 1522bBBrand width
B ~ Bb2
1 0270 BD
vL
y
x
First distance of the mixing
M
1L
Bb
C (x1 y)
x1
M
x
yb
v
xDB
2152
21 110 B
D
vL
y
x
INLET AT THE RIVER BANK
)4
exp(2
xD
yv
xvDh
Mc
y
x
xy
cmax
C (x1 y)
x1
M
INLET NEAR THE RIVER BANK
)4
(exp ( xDy
( y-y0 )2-v
xvD2h
Mc x
xy
cmax
))4
+exp ( xDy
( y+y0 )2-vx
y0 C (x1 y)
x1
IMPACTS OF THE RIVER BANKS (total river section)
Boundary condition reflection theory (infinite series)
Complete mixing the change along the cross-section is less than 10
L2 ~ 3L1 second distance of the mixing
Inlet at a point in y0 distance from the bank
xvD2h
Mc
xy
)4
exp ( xDy
( y-y0 +2nB)2-vx
)4
+ exp ( xDy
( y+y0 -2nB)2-vx
sumn=infin
n=minusinfin(
)
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
The general situationThe general situation
bull Unsteadybull Inhomogeneousbull Non-conservativebull 3D
General transport equation
Advection-Dispersion Equation (ADE)
Advection-dispersion EquationAdvection-dispersion Equationbull Can be derived from the mass balancebull Expresses the temporal and spatial changes of
the observed state variablebull Itrsquos solution is the concentration-distribution over
timebull Differential equation
Change in concentration with time
Change due to advection
=Change due to diffusion or dispersion
Change due to conversion processes
+ +
Molecular diffusionMolecular diffusion
DIFFUSION
CONVECTION
v
Molecular Molecular diffusiondiffusion
FickrsquosFickrsquos LawLaw
c1 c2x
bull c1 c2 separated tanks
bull open valve
Mass flux (kgs)
D - molecular diffusioncoefficient [m2s]
bull equalization and mixing (Brown-movement)
through area unit
bull depends on temperature
IN conv + diff
dxdy
dzOUT conv + diff
x direction
IN OUT
bull convection
CHANGE
bull diffusion
Mass balance Mass balance equationequation
x direction
IN conv + diff
dxdy
dzOUT conv + diff
Mass balance Mass balance equationequation
Convection Diffusion
If D(x) = const in x direction
Convection - Diffusion 1D Equation
The other directions are similar
Convection transfering Diffusion spreading
Mass balance Mass balance equationequation
x y z directions (3D)
D ndash molecular diffusion coefficient
material specific water 10-4 cm2s
space independent
slow mixing and equalization
Convection The water particles with differentconcentrations of pollutant move according to their differing flow velocity
Diffusion The adjacent water particles mix with each otherwhich causes the equalization of concentration
Mass balance equationMass balance equation
bull Laminar flow ordered with parallel flow linesbull Turbulent flow disordered random (flow and direction)
because of surface roughness (friction) rarr causes intensive mixing
v(c)
t
v deviation pulsation
T Time step of the turbulence
averagev~
0
bull Natural streams always turbulent
We usually can use the average only
TurbulenceTurbulence
Convection v middot c [ kgm2 middot s ]
0
0
Turbulent transport Turbulent transport descriptiondescription
v
turbulent diffusion
molecular diffusion
Dtx Dty Dtz gtgt D
Depends on the direction
Function of the velocity space
X direction (the other are similar)
Analogy with Fickrsquos Law rarr the original transport equation does not change only the D parameter
Turbulent diffusionTurbulent diffusion
Dx = D + Dtx Dy = D + Dty Dz = D + Dtz
Convection
Diffusion
Turbulent diffusion
Stochastic fluctuations of the flow velocity (pulsations)
Diffusive process in mathematical sense (~ Fickrsquos Law)
Convective transport (temporal differences)
Temporal averaging (T)
3D transport equation in turbulent flow3D transport equation in turbulent flow
Spatial simplifications (2D)Spatial simplifications (2D) - Dispersion - Dispersion
Depth averaged flow velocity v
0v
H
After the expansion of the convectiv part (v middot c) in x direction
Depth Integration (3D2D)
Turbulent dispersion (~ diffusion)
z
x
Velocity depth profile
Dx Dy
turbulent dispersion coefficients in 2D equation
Depth averaging (H)
11D tD transport equation in turbulent flow (cross-section avransport equation in turbulent flow (cross-section av))
Dx turbulent dispersion coefficient in 1D equation
Cross-section averaging (A)
2D transport equation in turbulent flow (depth averaged)2D transport equation in turbulent flow (depth averaged)
Turbulent dispersionTurbulent dispersion
Convectiv transport arising from the spatial differences of the velocity profile (faster and slower particles compared to the average)
v
In the 2D and 1D equations only
Dispersion coefficient a function of the velocity space (hydraulic parameters channel geometry)
Dx = Ddx + Dtx + D Dy
= Ddy + Dty + D Dx = Ddxx + Ddx + Dtx + D
Dx Dy
gtgt Dx Dx gtgt Dx
In case of more irregular channel and higher depth or larger cross-section area its value is bigger
Also present in laminar flow
Order of magnitude of Order of magnitude of diffusion dispersion diffusion dispersion coefficientscoefficients
10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s
Pore water
Mol diff
Vertical turbulent diff
High depth Shallow depth
Cross disp (2D)
Longitudinal disp (2D)
Horizontal turbulent diff
Longitudinal disp (1D)
Evaluation of dispersion coefficients measuring
Measuring with tracing matter (eg paint)
Inverse calculation from the measured concentration
2D Cross dispersion coefficient (Fischer)
Dy = dy u R (m2s)
dy - dimensionless cross dispersion coefficient
Straight regular channel dy 015
Moderately winding channel dy 02 ndash 06
Hard winding irregular channel dy gt 06 (1-2)
R ndash hydraulic radius (m)
u - shearing velocity at the channel bottom (ms)
u = (g R I)05Longitudinal dispersion coefficient dx 6 (2D) dxx = 100 ndash 1000 (1D)
Velocity space turbulent pulsation unequal distribution
Estimation of dispersion coefficients empirical forms
TRANSPORT EQUATION FOR NON-CONSERVATIVE TRANSPORT EQUATION FOR NON-CONSERVATIVE MATTERSMATTERS
bull There are sources and losses in the flow space
bull Physical chemical biochemical transformations take place
bull Non-conservative pollutants reaction kinetics ( R(C) )
bull They are taken into account in a linear way
dCdt = -C where is the coefficient of reaction kinetics (usually first-order kinetics)
cαxc
DAxx
c)v(Atc)(A
xx
1D equation in this case
ANALYTICAL SOLUTIONS OF THE TRANSPORT ANALYTICAL SOLUTIONS OF THE TRANSPORT EQUATIONEQUATION
Permanent mixing of the pollutants
Polluntant-wave transmission
Estimation of the geometric and hydraulic parameters
Exacter calculations based on measurments using numerical methods
The analitical solutions can be obtained in simpler cases only (approximate calculations)
Main steps
2
2
y
cD
x
cv yx
PERMANENT MIXING
The pollutant emission is steady in time
Permanent river discharge (low flow condition)
Constant flow velocity flow depth and dispersion coef
2D equation negligible vertical changes (shallow river)
Convection transferring
Dispersion spreading
Initial condition M0 (x0 y0) - emission
Boundary cond cy = 0 at the bank
)()()(
cvhy
cvhxt
chyx )()(
y
cDh
yx
cDh
xyx
x
yy
v
xD2
INLET AT THE MAIN FLOW PATH
Longitudinal according to x-frac12 function
Cross direction Gauss-distribution
cmax
M [kgs]cmax
)4
exp(2
2
xD
yv
xvDh
Mc (x t)y
x
xy
x
yb
v
xDB
234
Bb at value 01 cmax 1522bBBrand width
B ~ Bb2
1 0270 BD
vL
y
x
First distance of the mixing
M
1L
Bb
C (x1 y)
x1
M
x
yb
v
xDB
2152
21 110 B
D
vL
y
x
INLET AT THE RIVER BANK
)4
exp(2
xD
yv
xvDh
Mc
y
x
xy
cmax
C (x1 y)
x1
M
INLET NEAR THE RIVER BANK
)4
(exp ( xDy
( y-y0 )2-v
xvD2h
Mc x
xy
cmax
))4
+exp ( xDy
( y+y0 )2-vx
y0 C (x1 y)
x1
IMPACTS OF THE RIVER BANKS (total river section)
Boundary condition reflection theory (infinite series)
Complete mixing the change along the cross-section is less than 10
L2 ~ 3L1 second distance of the mixing
Inlet at a point in y0 distance from the bank
xvD2h
Mc
xy
)4
exp ( xDy
( y-y0 +2nB)2-vx
)4
+ exp ( xDy
( y+y0 -2nB)2-vx
sumn=infin
n=minusinfin(
)
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
Advection-dispersion EquationAdvection-dispersion Equationbull Can be derived from the mass balancebull Expresses the temporal and spatial changes of
the observed state variablebull Itrsquos solution is the concentration-distribution over
timebull Differential equation
Change in concentration with time
Change due to advection
=Change due to diffusion or dispersion
Change due to conversion processes
+ +
Molecular diffusionMolecular diffusion
DIFFUSION
CONVECTION
v
Molecular Molecular diffusiondiffusion
FickrsquosFickrsquos LawLaw
c1 c2x
bull c1 c2 separated tanks
bull open valve
Mass flux (kgs)
D - molecular diffusioncoefficient [m2s]
bull equalization and mixing (Brown-movement)
through area unit
bull depends on temperature
IN conv + diff
dxdy
dzOUT conv + diff
x direction
IN OUT
bull convection
CHANGE
bull diffusion
Mass balance Mass balance equationequation
x direction
IN conv + diff
dxdy
dzOUT conv + diff
Mass balance Mass balance equationequation
Convection Diffusion
If D(x) = const in x direction
Convection - Diffusion 1D Equation
The other directions are similar
Convection transfering Diffusion spreading
Mass balance Mass balance equationequation
x y z directions (3D)
D ndash molecular diffusion coefficient
material specific water 10-4 cm2s
space independent
slow mixing and equalization
Convection The water particles with differentconcentrations of pollutant move according to their differing flow velocity
Diffusion The adjacent water particles mix with each otherwhich causes the equalization of concentration
Mass balance equationMass balance equation
bull Laminar flow ordered with parallel flow linesbull Turbulent flow disordered random (flow and direction)
because of surface roughness (friction) rarr causes intensive mixing
v(c)
t
v deviation pulsation
T Time step of the turbulence
averagev~
0
bull Natural streams always turbulent
We usually can use the average only
TurbulenceTurbulence
Convection v middot c [ kgm2 middot s ]
0
0
Turbulent transport Turbulent transport descriptiondescription
v
turbulent diffusion
molecular diffusion
Dtx Dty Dtz gtgt D
Depends on the direction
Function of the velocity space
X direction (the other are similar)
Analogy with Fickrsquos Law rarr the original transport equation does not change only the D parameter
Turbulent diffusionTurbulent diffusion
Dx = D + Dtx Dy = D + Dty Dz = D + Dtz
Convection
Diffusion
Turbulent diffusion
Stochastic fluctuations of the flow velocity (pulsations)
Diffusive process in mathematical sense (~ Fickrsquos Law)
Convective transport (temporal differences)
Temporal averaging (T)
3D transport equation in turbulent flow3D transport equation in turbulent flow
Spatial simplifications (2D)Spatial simplifications (2D) - Dispersion - Dispersion
Depth averaged flow velocity v
0v
H
After the expansion of the convectiv part (v middot c) in x direction
Depth Integration (3D2D)
Turbulent dispersion (~ diffusion)
z
x
Velocity depth profile
Dx Dy
turbulent dispersion coefficients in 2D equation
Depth averaging (H)
11D tD transport equation in turbulent flow (cross-section avransport equation in turbulent flow (cross-section av))
Dx turbulent dispersion coefficient in 1D equation
Cross-section averaging (A)
2D transport equation in turbulent flow (depth averaged)2D transport equation in turbulent flow (depth averaged)
Turbulent dispersionTurbulent dispersion
Convectiv transport arising from the spatial differences of the velocity profile (faster and slower particles compared to the average)
v
In the 2D and 1D equations only
Dispersion coefficient a function of the velocity space (hydraulic parameters channel geometry)
Dx = Ddx + Dtx + D Dy
= Ddy + Dty + D Dx = Ddxx + Ddx + Dtx + D
Dx Dy
gtgt Dx Dx gtgt Dx
In case of more irregular channel and higher depth or larger cross-section area its value is bigger
Also present in laminar flow
Order of magnitude of Order of magnitude of diffusion dispersion diffusion dispersion coefficientscoefficients
10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s
Pore water
Mol diff
Vertical turbulent diff
High depth Shallow depth
Cross disp (2D)
Longitudinal disp (2D)
Horizontal turbulent diff
Longitudinal disp (1D)
Evaluation of dispersion coefficients measuring
Measuring with tracing matter (eg paint)
Inverse calculation from the measured concentration
2D Cross dispersion coefficient (Fischer)
Dy = dy u R (m2s)
dy - dimensionless cross dispersion coefficient
Straight regular channel dy 015
Moderately winding channel dy 02 ndash 06
Hard winding irregular channel dy gt 06 (1-2)
R ndash hydraulic radius (m)
u - shearing velocity at the channel bottom (ms)
u = (g R I)05Longitudinal dispersion coefficient dx 6 (2D) dxx = 100 ndash 1000 (1D)
Velocity space turbulent pulsation unequal distribution
Estimation of dispersion coefficients empirical forms
TRANSPORT EQUATION FOR NON-CONSERVATIVE TRANSPORT EQUATION FOR NON-CONSERVATIVE MATTERSMATTERS
bull There are sources and losses in the flow space
bull Physical chemical biochemical transformations take place
bull Non-conservative pollutants reaction kinetics ( R(C) )
bull They are taken into account in a linear way
dCdt = -C where is the coefficient of reaction kinetics (usually first-order kinetics)
cαxc
DAxx
c)v(Atc)(A
xx
1D equation in this case
ANALYTICAL SOLUTIONS OF THE TRANSPORT ANALYTICAL SOLUTIONS OF THE TRANSPORT EQUATIONEQUATION
Permanent mixing of the pollutants
Polluntant-wave transmission
Estimation of the geometric and hydraulic parameters
Exacter calculations based on measurments using numerical methods
The analitical solutions can be obtained in simpler cases only (approximate calculations)
Main steps
2
2
y
cD
x
cv yx
PERMANENT MIXING
The pollutant emission is steady in time
Permanent river discharge (low flow condition)
Constant flow velocity flow depth and dispersion coef
2D equation negligible vertical changes (shallow river)
Convection transferring
Dispersion spreading
Initial condition M0 (x0 y0) - emission
Boundary cond cy = 0 at the bank
)()()(
cvhy
cvhxt
chyx )()(
y
cDh
yx
cDh
xyx
x
yy
v
xD2
INLET AT THE MAIN FLOW PATH
Longitudinal according to x-frac12 function
Cross direction Gauss-distribution
cmax
M [kgs]cmax
)4
exp(2
2
xD
yv
xvDh
Mc (x t)y
x
xy
x
yb
v
xDB
234
Bb at value 01 cmax 1522bBBrand width
B ~ Bb2
1 0270 BD
vL
y
x
First distance of the mixing
M
1L
Bb
C (x1 y)
x1
M
x
yb
v
xDB
2152
21 110 B
D
vL
y
x
INLET AT THE RIVER BANK
)4
exp(2
xD
yv
xvDh
Mc
y
x
xy
cmax
C (x1 y)
x1
M
INLET NEAR THE RIVER BANK
)4
(exp ( xDy
( y-y0 )2-v
xvD2h
Mc x
xy
cmax
))4
+exp ( xDy
( y+y0 )2-vx
y0 C (x1 y)
x1
IMPACTS OF THE RIVER BANKS (total river section)
Boundary condition reflection theory (infinite series)
Complete mixing the change along the cross-section is less than 10
L2 ~ 3L1 second distance of the mixing
Inlet at a point in y0 distance from the bank
xvD2h
Mc
xy
)4
exp ( xDy
( y-y0 +2nB)2-vx
)4
+ exp ( xDy
( y+y0 -2nB)2-vx
sumn=infin
n=minusinfin(
)
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
Molecular diffusionMolecular diffusion
DIFFUSION
CONVECTION
v
Molecular Molecular diffusiondiffusion
FickrsquosFickrsquos LawLaw
c1 c2x
bull c1 c2 separated tanks
bull open valve
Mass flux (kgs)
D - molecular diffusioncoefficient [m2s]
bull equalization and mixing (Brown-movement)
through area unit
bull depends on temperature
IN conv + diff
dxdy
dzOUT conv + diff
x direction
IN OUT
bull convection
CHANGE
bull diffusion
Mass balance Mass balance equationequation
x direction
IN conv + diff
dxdy
dzOUT conv + diff
Mass balance Mass balance equationequation
Convection Diffusion
If D(x) = const in x direction
Convection - Diffusion 1D Equation
The other directions are similar
Convection transfering Diffusion spreading
Mass balance Mass balance equationequation
x y z directions (3D)
D ndash molecular diffusion coefficient
material specific water 10-4 cm2s
space independent
slow mixing and equalization
Convection The water particles with differentconcentrations of pollutant move according to their differing flow velocity
Diffusion The adjacent water particles mix with each otherwhich causes the equalization of concentration
Mass balance equationMass balance equation
bull Laminar flow ordered with parallel flow linesbull Turbulent flow disordered random (flow and direction)
because of surface roughness (friction) rarr causes intensive mixing
v(c)
t
v deviation pulsation
T Time step of the turbulence
averagev~
0
bull Natural streams always turbulent
We usually can use the average only
TurbulenceTurbulence
Convection v middot c [ kgm2 middot s ]
0
0
Turbulent transport Turbulent transport descriptiondescription
v
turbulent diffusion
molecular diffusion
Dtx Dty Dtz gtgt D
Depends on the direction
Function of the velocity space
X direction (the other are similar)
Analogy with Fickrsquos Law rarr the original transport equation does not change only the D parameter
Turbulent diffusionTurbulent diffusion
Dx = D + Dtx Dy = D + Dty Dz = D + Dtz
Convection
Diffusion
Turbulent diffusion
Stochastic fluctuations of the flow velocity (pulsations)
Diffusive process in mathematical sense (~ Fickrsquos Law)
Convective transport (temporal differences)
Temporal averaging (T)
3D transport equation in turbulent flow3D transport equation in turbulent flow
Spatial simplifications (2D)Spatial simplifications (2D) - Dispersion - Dispersion
Depth averaged flow velocity v
0v
H
After the expansion of the convectiv part (v middot c) in x direction
Depth Integration (3D2D)
Turbulent dispersion (~ diffusion)
z
x
Velocity depth profile
Dx Dy
turbulent dispersion coefficients in 2D equation
Depth averaging (H)
11D tD transport equation in turbulent flow (cross-section avransport equation in turbulent flow (cross-section av))
Dx turbulent dispersion coefficient in 1D equation
Cross-section averaging (A)
2D transport equation in turbulent flow (depth averaged)2D transport equation in turbulent flow (depth averaged)
Turbulent dispersionTurbulent dispersion
Convectiv transport arising from the spatial differences of the velocity profile (faster and slower particles compared to the average)
v
In the 2D and 1D equations only
Dispersion coefficient a function of the velocity space (hydraulic parameters channel geometry)
Dx = Ddx + Dtx + D Dy
= Ddy + Dty + D Dx = Ddxx + Ddx + Dtx + D
Dx Dy
gtgt Dx Dx gtgt Dx
In case of more irregular channel and higher depth or larger cross-section area its value is bigger
Also present in laminar flow
Order of magnitude of Order of magnitude of diffusion dispersion diffusion dispersion coefficientscoefficients
10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s
Pore water
Mol diff
Vertical turbulent diff
High depth Shallow depth
Cross disp (2D)
Longitudinal disp (2D)
Horizontal turbulent diff
Longitudinal disp (1D)
Evaluation of dispersion coefficients measuring
Measuring with tracing matter (eg paint)
Inverse calculation from the measured concentration
2D Cross dispersion coefficient (Fischer)
Dy = dy u R (m2s)
dy - dimensionless cross dispersion coefficient
Straight regular channel dy 015
Moderately winding channel dy 02 ndash 06
Hard winding irregular channel dy gt 06 (1-2)
R ndash hydraulic radius (m)
u - shearing velocity at the channel bottom (ms)
u = (g R I)05Longitudinal dispersion coefficient dx 6 (2D) dxx = 100 ndash 1000 (1D)
Velocity space turbulent pulsation unequal distribution
Estimation of dispersion coefficients empirical forms
TRANSPORT EQUATION FOR NON-CONSERVATIVE TRANSPORT EQUATION FOR NON-CONSERVATIVE MATTERSMATTERS
bull There are sources and losses in the flow space
bull Physical chemical biochemical transformations take place
bull Non-conservative pollutants reaction kinetics ( R(C) )
bull They are taken into account in a linear way
dCdt = -C where is the coefficient of reaction kinetics (usually first-order kinetics)
cαxc
DAxx
c)v(Atc)(A
xx
1D equation in this case
ANALYTICAL SOLUTIONS OF THE TRANSPORT ANALYTICAL SOLUTIONS OF THE TRANSPORT EQUATIONEQUATION
Permanent mixing of the pollutants
Polluntant-wave transmission
Estimation of the geometric and hydraulic parameters
Exacter calculations based on measurments using numerical methods
The analitical solutions can be obtained in simpler cases only (approximate calculations)
Main steps
2
2
y
cD
x
cv yx
PERMANENT MIXING
The pollutant emission is steady in time
Permanent river discharge (low flow condition)
Constant flow velocity flow depth and dispersion coef
2D equation negligible vertical changes (shallow river)
Convection transferring
Dispersion spreading
Initial condition M0 (x0 y0) - emission
Boundary cond cy = 0 at the bank
)()()(
cvhy
cvhxt
chyx )()(
y
cDh
yx
cDh
xyx
x
yy
v
xD2
INLET AT THE MAIN FLOW PATH
Longitudinal according to x-frac12 function
Cross direction Gauss-distribution
cmax
M [kgs]cmax
)4
exp(2
2
xD
yv
xvDh
Mc (x t)y
x
xy
x
yb
v
xDB
234
Bb at value 01 cmax 1522bBBrand width
B ~ Bb2
1 0270 BD
vL
y
x
First distance of the mixing
M
1L
Bb
C (x1 y)
x1
M
x
yb
v
xDB
2152
21 110 B
D
vL
y
x
INLET AT THE RIVER BANK
)4
exp(2
xD
yv
xvDh
Mc
y
x
xy
cmax
C (x1 y)
x1
M
INLET NEAR THE RIVER BANK
)4
(exp ( xDy
( y-y0 )2-v
xvD2h
Mc x
xy
cmax
))4
+exp ( xDy
( y+y0 )2-vx
y0 C (x1 y)
x1
IMPACTS OF THE RIVER BANKS (total river section)
Boundary condition reflection theory (infinite series)
Complete mixing the change along the cross-section is less than 10
L2 ~ 3L1 second distance of the mixing
Inlet at a point in y0 distance from the bank
xvD2h
Mc
xy
)4
exp ( xDy
( y-y0 +2nB)2-vx
)4
+ exp ( xDy
( y+y0 -2nB)2-vx
sumn=infin
n=minusinfin(
)
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
Molecular Molecular diffusiondiffusion
FickrsquosFickrsquos LawLaw
c1 c2x
bull c1 c2 separated tanks
bull open valve
Mass flux (kgs)
D - molecular diffusioncoefficient [m2s]
bull equalization and mixing (Brown-movement)
through area unit
bull depends on temperature
IN conv + diff
dxdy
dzOUT conv + diff
x direction
IN OUT
bull convection
CHANGE
bull diffusion
Mass balance Mass balance equationequation
x direction
IN conv + diff
dxdy
dzOUT conv + diff
Mass balance Mass balance equationequation
Convection Diffusion
If D(x) = const in x direction
Convection - Diffusion 1D Equation
The other directions are similar
Convection transfering Diffusion spreading
Mass balance Mass balance equationequation
x y z directions (3D)
D ndash molecular diffusion coefficient
material specific water 10-4 cm2s
space independent
slow mixing and equalization
Convection The water particles with differentconcentrations of pollutant move according to their differing flow velocity
Diffusion The adjacent water particles mix with each otherwhich causes the equalization of concentration
Mass balance equationMass balance equation
bull Laminar flow ordered with parallel flow linesbull Turbulent flow disordered random (flow and direction)
because of surface roughness (friction) rarr causes intensive mixing
v(c)
t
v deviation pulsation
T Time step of the turbulence
averagev~
0
bull Natural streams always turbulent
We usually can use the average only
TurbulenceTurbulence
Convection v middot c [ kgm2 middot s ]
0
0
Turbulent transport Turbulent transport descriptiondescription
v
turbulent diffusion
molecular diffusion
Dtx Dty Dtz gtgt D
Depends on the direction
Function of the velocity space
X direction (the other are similar)
Analogy with Fickrsquos Law rarr the original transport equation does not change only the D parameter
Turbulent diffusionTurbulent diffusion
Dx = D + Dtx Dy = D + Dty Dz = D + Dtz
Convection
Diffusion
Turbulent diffusion
Stochastic fluctuations of the flow velocity (pulsations)
Diffusive process in mathematical sense (~ Fickrsquos Law)
Convective transport (temporal differences)
Temporal averaging (T)
3D transport equation in turbulent flow3D transport equation in turbulent flow
Spatial simplifications (2D)Spatial simplifications (2D) - Dispersion - Dispersion
Depth averaged flow velocity v
0v
H
After the expansion of the convectiv part (v middot c) in x direction
Depth Integration (3D2D)
Turbulent dispersion (~ diffusion)
z
x
Velocity depth profile
Dx Dy
turbulent dispersion coefficients in 2D equation
Depth averaging (H)
11D tD transport equation in turbulent flow (cross-section avransport equation in turbulent flow (cross-section av))
Dx turbulent dispersion coefficient in 1D equation
Cross-section averaging (A)
2D transport equation in turbulent flow (depth averaged)2D transport equation in turbulent flow (depth averaged)
Turbulent dispersionTurbulent dispersion
Convectiv transport arising from the spatial differences of the velocity profile (faster and slower particles compared to the average)
v
In the 2D and 1D equations only
Dispersion coefficient a function of the velocity space (hydraulic parameters channel geometry)
Dx = Ddx + Dtx + D Dy
= Ddy + Dty + D Dx = Ddxx + Ddx + Dtx + D
Dx Dy
gtgt Dx Dx gtgt Dx
In case of more irregular channel and higher depth or larger cross-section area its value is bigger
Also present in laminar flow
Order of magnitude of Order of magnitude of diffusion dispersion diffusion dispersion coefficientscoefficients
10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s
Pore water
Mol diff
Vertical turbulent diff
High depth Shallow depth
Cross disp (2D)
Longitudinal disp (2D)
Horizontal turbulent diff
Longitudinal disp (1D)
Evaluation of dispersion coefficients measuring
Measuring with tracing matter (eg paint)
Inverse calculation from the measured concentration
2D Cross dispersion coefficient (Fischer)
Dy = dy u R (m2s)
dy - dimensionless cross dispersion coefficient
Straight regular channel dy 015
Moderately winding channel dy 02 ndash 06
Hard winding irregular channel dy gt 06 (1-2)
R ndash hydraulic radius (m)
u - shearing velocity at the channel bottom (ms)
u = (g R I)05Longitudinal dispersion coefficient dx 6 (2D) dxx = 100 ndash 1000 (1D)
Velocity space turbulent pulsation unequal distribution
Estimation of dispersion coefficients empirical forms
TRANSPORT EQUATION FOR NON-CONSERVATIVE TRANSPORT EQUATION FOR NON-CONSERVATIVE MATTERSMATTERS
bull There are sources and losses in the flow space
bull Physical chemical biochemical transformations take place
bull Non-conservative pollutants reaction kinetics ( R(C) )
bull They are taken into account in a linear way
dCdt = -C where is the coefficient of reaction kinetics (usually first-order kinetics)
cαxc
DAxx
c)v(Atc)(A
xx
1D equation in this case
ANALYTICAL SOLUTIONS OF THE TRANSPORT ANALYTICAL SOLUTIONS OF THE TRANSPORT EQUATIONEQUATION
Permanent mixing of the pollutants
Polluntant-wave transmission
Estimation of the geometric and hydraulic parameters
Exacter calculations based on measurments using numerical methods
The analitical solutions can be obtained in simpler cases only (approximate calculations)
Main steps
2
2
y
cD
x
cv yx
PERMANENT MIXING
The pollutant emission is steady in time
Permanent river discharge (low flow condition)
Constant flow velocity flow depth and dispersion coef
2D equation negligible vertical changes (shallow river)
Convection transferring
Dispersion spreading
Initial condition M0 (x0 y0) - emission
Boundary cond cy = 0 at the bank
)()()(
cvhy
cvhxt
chyx )()(
y
cDh
yx
cDh
xyx
x
yy
v
xD2
INLET AT THE MAIN FLOW PATH
Longitudinal according to x-frac12 function
Cross direction Gauss-distribution
cmax
M [kgs]cmax
)4
exp(2
2
xD
yv
xvDh
Mc (x t)y
x
xy
x
yb
v
xDB
234
Bb at value 01 cmax 1522bBBrand width
B ~ Bb2
1 0270 BD
vL
y
x
First distance of the mixing
M
1L
Bb
C (x1 y)
x1
M
x
yb
v
xDB
2152
21 110 B
D
vL
y
x
INLET AT THE RIVER BANK
)4
exp(2
xD
yv
xvDh
Mc
y
x
xy
cmax
C (x1 y)
x1
M
INLET NEAR THE RIVER BANK
)4
(exp ( xDy
( y-y0 )2-v
xvD2h
Mc x
xy
cmax
))4
+exp ( xDy
( y+y0 )2-vx
y0 C (x1 y)
x1
IMPACTS OF THE RIVER BANKS (total river section)
Boundary condition reflection theory (infinite series)
Complete mixing the change along the cross-section is less than 10
L2 ~ 3L1 second distance of the mixing
Inlet at a point in y0 distance from the bank
xvD2h
Mc
xy
)4
exp ( xDy
( y-y0 +2nB)2-vx
)4
+ exp ( xDy
( y+y0 -2nB)2-vx
sumn=infin
n=minusinfin(
)
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
IN conv + diff
dxdy
dzOUT conv + diff
x direction
IN OUT
bull convection
CHANGE
bull diffusion
Mass balance Mass balance equationequation
x direction
IN conv + diff
dxdy
dzOUT conv + diff
Mass balance Mass balance equationequation
Convection Diffusion
If D(x) = const in x direction
Convection - Diffusion 1D Equation
The other directions are similar
Convection transfering Diffusion spreading
Mass balance Mass balance equationequation
x y z directions (3D)
D ndash molecular diffusion coefficient
material specific water 10-4 cm2s
space independent
slow mixing and equalization
Convection The water particles with differentconcentrations of pollutant move according to their differing flow velocity
Diffusion The adjacent water particles mix with each otherwhich causes the equalization of concentration
Mass balance equationMass balance equation
bull Laminar flow ordered with parallel flow linesbull Turbulent flow disordered random (flow and direction)
because of surface roughness (friction) rarr causes intensive mixing
v(c)
t
v deviation pulsation
T Time step of the turbulence
averagev~
0
bull Natural streams always turbulent
We usually can use the average only
TurbulenceTurbulence
Convection v middot c [ kgm2 middot s ]
0
0
Turbulent transport Turbulent transport descriptiondescription
v
turbulent diffusion
molecular diffusion
Dtx Dty Dtz gtgt D
Depends on the direction
Function of the velocity space
X direction (the other are similar)
Analogy with Fickrsquos Law rarr the original transport equation does not change only the D parameter
Turbulent diffusionTurbulent diffusion
Dx = D + Dtx Dy = D + Dty Dz = D + Dtz
Convection
Diffusion
Turbulent diffusion
Stochastic fluctuations of the flow velocity (pulsations)
Diffusive process in mathematical sense (~ Fickrsquos Law)
Convective transport (temporal differences)
Temporal averaging (T)
3D transport equation in turbulent flow3D transport equation in turbulent flow
Spatial simplifications (2D)Spatial simplifications (2D) - Dispersion - Dispersion
Depth averaged flow velocity v
0v
H
After the expansion of the convectiv part (v middot c) in x direction
Depth Integration (3D2D)
Turbulent dispersion (~ diffusion)
z
x
Velocity depth profile
Dx Dy
turbulent dispersion coefficients in 2D equation
Depth averaging (H)
11D tD transport equation in turbulent flow (cross-section avransport equation in turbulent flow (cross-section av))
Dx turbulent dispersion coefficient in 1D equation
Cross-section averaging (A)
2D transport equation in turbulent flow (depth averaged)2D transport equation in turbulent flow (depth averaged)
Turbulent dispersionTurbulent dispersion
Convectiv transport arising from the spatial differences of the velocity profile (faster and slower particles compared to the average)
v
In the 2D and 1D equations only
Dispersion coefficient a function of the velocity space (hydraulic parameters channel geometry)
Dx = Ddx + Dtx + D Dy
= Ddy + Dty + D Dx = Ddxx + Ddx + Dtx + D
Dx Dy
gtgt Dx Dx gtgt Dx
In case of more irregular channel and higher depth or larger cross-section area its value is bigger
Also present in laminar flow
Order of magnitude of Order of magnitude of diffusion dispersion diffusion dispersion coefficientscoefficients
10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s
Pore water
Mol diff
Vertical turbulent diff
High depth Shallow depth
Cross disp (2D)
Longitudinal disp (2D)
Horizontal turbulent diff
Longitudinal disp (1D)
Evaluation of dispersion coefficients measuring
Measuring with tracing matter (eg paint)
Inverse calculation from the measured concentration
2D Cross dispersion coefficient (Fischer)
Dy = dy u R (m2s)
dy - dimensionless cross dispersion coefficient
Straight regular channel dy 015
Moderately winding channel dy 02 ndash 06
Hard winding irregular channel dy gt 06 (1-2)
R ndash hydraulic radius (m)
u - shearing velocity at the channel bottom (ms)
u = (g R I)05Longitudinal dispersion coefficient dx 6 (2D) dxx = 100 ndash 1000 (1D)
Velocity space turbulent pulsation unequal distribution
Estimation of dispersion coefficients empirical forms
TRANSPORT EQUATION FOR NON-CONSERVATIVE TRANSPORT EQUATION FOR NON-CONSERVATIVE MATTERSMATTERS
bull There are sources and losses in the flow space
bull Physical chemical biochemical transformations take place
bull Non-conservative pollutants reaction kinetics ( R(C) )
bull They are taken into account in a linear way
dCdt = -C where is the coefficient of reaction kinetics (usually first-order kinetics)
cαxc
DAxx
c)v(Atc)(A
xx
1D equation in this case
ANALYTICAL SOLUTIONS OF THE TRANSPORT ANALYTICAL SOLUTIONS OF THE TRANSPORT EQUATIONEQUATION
Permanent mixing of the pollutants
Polluntant-wave transmission
Estimation of the geometric and hydraulic parameters
Exacter calculations based on measurments using numerical methods
The analitical solutions can be obtained in simpler cases only (approximate calculations)
Main steps
2
2
y
cD
x
cv yx
PERMANENT MIXING
The pollutant emission is steady in time
Permanent river discharge (low flow condition)
Constant flow velocity flow depth and dispersion coef
2D equation negligible vertical changes (shallow river)
Convection transferring
Dispersion spreading
Initial condition M0 (x0 y0) - emission
Boundary cond cy = 0 at the bank
)()()(
cvhy
cvhxt
chyx )()(
y
cDh
yx
cDh
xyx
x
yy
v
xD2
INLET AT THE MAIN FLOW PATH
Longitudinal according to x-frac12 function
Cross direction Gauss-distribution
cmax
M [kgs]cmax
)4
exp(2
2
xD
yv
xvDh
Mc (x t)y
x
xy
x
yb
v
xDB
234
Bb at value 01 cmax 1522bBBrand width
B ~ Bb2
1 0270 BD
vL
y
x
First distance of the mixing
M
1L
Bb
C (x1 y)
x1
M
x
yb
v
xDB
2152
21 110 B
D
vL
y
x
INLET AT THE RIVER BANK
)4
exp(2
xD
yv
xvDh
Mc
y
x
xy
cmax
C (x1 y)
x1
M
INLET NEAR THE RIVER BANK
)4
(exp ( xDy
( y-y0 )2-v
xvD2h
Mc x
xy
cmax
))4
+exp ( xDy
( y+y0 )2-vx
y0 C (x1 y)
x1
IMPACTS OF THE RIVER BANKS (total river section)
Boundary condition reflection theory (infinite series)
Complete mixing the change along the cross-section is less than 10
L2 ~ 3L1 second distance of the mixing
Inlet at a point in y0 distance from the bank
xvD2h
Mc
xy
)4
exp ( xDy
( y-y0 +2nB)2-vx
)4
+ exp ( xDy
( y+y0 -2nB)2-vx
sumn=infin
n=minusinfin(
)
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
x direction
IN conv + diff
dxdy
dzOUT conv + diff
Mass balance Mass balance equationequation
Convection Diffusion
If D(x) = const in x direction
Convection - Diffusion 1D Equation
The other directions are similar
Convection transfering Diffusion spreading
Mass balance Mass balance equationequation
x y z directions (3D)
D ndash molecular diffusion coefficient
material specific water 10-4 cm2s
space independent
slow mixing and equalization
Convection The water particles with differentconcentrations of pollutant move according to their differing flow velocity
Diffusion The adjacent water particles mix with each otherwhich causes the equalization of concentration
Mass balance equationMass balance equation
bull Laminar flow ordered with parallel flow linesbull Turbulent flow disordered random (flow and direction)
because of surface roughness (friction) rarr causes intensive mixing
v(c)
t
v deviation pulsation
T Time step of the turbulence
averagev~
0
bull Natural streams always turbulent
We usually can use the average only
TurbulenceTurbulence
Convection v middot c [ kgm2 middot s ]
0
0
Turbulent transport Turbulent transport descriptiondescription
v
turbulent diffusion
molecular diffusion
Dtx Dty Dtz gtgt D
Depends on the direction
Function of the velocity space
X direction (the other are similar)
Analogy with Fickrsquos Law rarr the original transport equation does not change only the D parameter
Turbulent diffusionTurbulent diffusion
Dx = D + Dtx Dy = D + Dty Dz = D + Dtz
Convection
Diffusion
Turbulent diffusion
Stochastic fluctuations of the flow velocity (pulsations)
Diffusive process in mathematical sense (~ Fickrsquos Law)
Convective transport (temporal differences)
Temporal averaging (T)
3D transport equation in turbulent flow3D transport equation in turbulent flow
Spatial simplifications (2D)Spatial simplifications (2D) - Dispersion - Dispersion
Depth averaged flow velocity v
0v
H
After the expansion of the convectiv part (v middot c) in x direction
Depth Integration (3D2D)
Turbulent dispersion (~ diffusion)
z
x
Velocity depth profile
Dx Dy
turbulent dispersion coefficients in 2D equation
Depth averaging (H)
11D tD transport equation in turbulent flow (cross-section avransport equation in turbulent flow (cross-section av))
Dx turbulent dispersion coefficient in 1D equation
Cross-section averaging (A)
2D transport equation in turbulent flow (depth averaged)2D transport equation in turbulent flow (depth averaged)
Turbulent dispersionTurbulent dispersion
Convectiv transport arising from the spatial differences of the velocity profile (faster and slower particles compared to the average)
v
In the 2D and 1D equations only
Dispersion coefficient a function of the velocity space (hydraulic parameters channel geometry)
Dx = Ddx + Dtx + D Dy
= Ddy + Dty + D Dx = Ddxx + Ddx + Dtx + D
Dx Dy
gtgt Dx Dx gtgt Dx
In case of more irregular channel and higher depth or larger cross-section area its value is bigger
Also present in laminar flow
Order of magnitude of Order of magnitude of diffusion dispersion diffusion dispersion coefficientscoefficients
10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s
Pore water
Mol diff
Vertical turbulent diff
High depth Shallow depth
Cross disp (2D)
Longitudinal disp (2D)
Horizontal turbulent diff
Longitudinal disp (1D)
Evaluation of dispersion coefficients measuring
Measuring with tracing matter (eg paint)
Inverse calculation from the measured concentration
2D Cross dispersion coefficient (Fischer)
Dy = dy u R (m2s)
dy - dimensionless cross dispersion coefficient
Straight regular channel dy 015
Moderately winding channel dy 02 ndash 06
Hard winding irregular channel dy gt 06 (1-2)
R ndash hydraulic radius (m)
u - shearing velocity at the channel bottom (ms)
u = (g R I)05Longitudinal dispersion coefficient dx 6 (2D) dxx = 100 ndash 1000 (1D)
Velocity space turbulent pulsation unequal distribution
Estimation of dispersion coefficients empirical forms
TRANSPORT EQUATION FOR NON-CONSERVATIVE TRANSPORT EQUATION FOR NON-CONSERVATIVE MATTERSMATTERS
bull There are sources and losses in the flow space
bull Physical chemical biochemical transformations take place
bull Non-conservative pollutants reaction kinetics ( R(C) )
bull They are taken into account in a linear way
dCdt = -C where is the coefficient of reaction kinetics (usually first-order kinetics)
cαxc
DAxx
c)v(Atc)(A
xx
1D equation in this case
ANALYTICAL SOLUTIONS OF THE TRANSPORT ANALYTICAL SOLUTIONS OF THE TRANSPORT EQUATIONEQUATION
Permanent mixing of the pollutants
Polluntant-wave transmission
Estimation of the geometric and hydraulic parameters
Exacter calculations based on measurments using numerical methods
The analitical solutions can be obtained in simpler cases only (approximate calculations)
Main steps
2
2
y
cD
x
cv yx
PERMANENT MIXING
The pollutant emission is steady in time
Permanent river discharge (low flow condition)
Constant flow velocity flow depth and dispersion coef
2D equation negligible vertical changes (shallow river)
Convection transferring
Dispersion spreading
Initial condition M0 (x0 y0) - emission
Boundary cond cy = 0 at the bank
)()()(
cvhy
cvhxt
chyx )()(
y
cDh
yx
cDh
xyx
x
yy
v
xD2
INLET AT THE MAIN FLOW PATH
Longitudinal according to x-frac12 function
Cross direction Gauss-distribution
cmax
M [kgs]cmax
)4
exp(2
2
xD
yv
xvDh
Mc (x t)y
x
xy
x
yb
v
xDB
234
Bb at value 01 cmax 1522bBBrand width
B ~ Bb2
1 0270 BD
vL
y
x
First distance of the mixing
M
1L
Bb
C (x1 y)
x1
M
x
yb
v
xDB
2152
21 110 B
D
vL
y
x
INLET AT THE RIVER BANK
)4
exp(2
xD
yv
xvDh
Mc
y
x
xy
cmax
C (x1 y)
x1
M
INLET NEAR THE RIVER BANK
)4
(exp ( xDy
( y-y0 )2-v
xvD2h
Mc x
xy
cmax
))4
+exp ( xDy
( y+y0 )2-vx
y0 C (x1 y)
x1
IMPACTS OF THE RIVER BANKS (total river section)
Boundary condition reflection theory (infinite series)
Complete mixing the change along the cross-section is less than 10
L2 ~ 3L1 second distance of the mixing
Inlet at a point in y0 distance from the bank
xvD2h
Mc
xy
)4
exp ( xDy
( y-y0 +2nB)2-vx
)4
+ exp ( xDy
( y+y0 -2nB)2-vx
sumn=infin
n=minusinfin(
)
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
Convection Diffusion
If D(x) = const in x direction
Convection - Diffusion 1D Equation
The other directions are similar
Convection transfering Diffusion spreading
Mass balance Mass balance equationequation
x y z directions (3D)
D ndash molecular diffusion coefficient
material specific water 10-4 cm2s
space independent
slow mixing and equalization
Convection The water particles with differentconcentrations of pollutant move according to their differing flow velocity
Diffusion The adjacent water particles mix with each otherwhich causes the equalization of concentration
Mass balance equationMass balance equation
bull Laminar flow ordered with parallel flow linesbull Turbulent flow disordered random (flow and direction)
because of surface roughness (friction) rarr causes intensive mixing
v(c)
t
v deviation pulsation
T Time step of the turbulence
averagev~
0
bull Natural streams always turbulent
We usually can use the average only
TurbulenceTurbulence
Convection v middot c [ kgm2 middot s ]
0
0
Turbulent transport Turbulent transport descriptiondescription
v
turbulent diffusion
molecular diffusion
Dtx Dty Dtz gtgt D
Depends on the direction
Function of the velocity space
X direction (the other are similar)
Analogy with Fickrsquos Law rarr the original transport equation does not change only the D parameter
Turbulent diffusionTurbulent diffusion
Dx = D + Dtx Dy = D + Dty Dz = D + Dtz
Convection
Diffusion
Turbulent diffusion
Stochastic fluctuations of the flow velocity (pulsations)
Diffusive process in mathematical sense (~ Fickrsquos Law)
Convective transport (temporal differences)
Temporal averaging (T)
3D transport equation in turbulent flow3D transport equation in turbulent flow
Spatial simplifications (2D)Spatial simplifications (2D) - Dispersion - Dispersion
Depth averaged flow velocity v
0v
H
After the expansion of the convectiv part (v middot c) in x direction
Depth Integration (3D2D)
Turbulent dispersion (~ diffusion)
z
x
Velocity depth profile
Dx Dy
turbulent dispersion coefficients in 2D equation
Depth averaging (H)
11D tD transport equation in turbulent flow (cross-section avransport equation in turbulent flow (cross-section av))
Dx turbulent dispersion coefficient in 1D equation
Cross-section averaging (A)
2D transport equation in turbulent flow (depth averaged)2D transport equation in turbulent flow (depth averaged)
Turbulent dispersionTurbulent dispersion
Convectiv transport arising from the spatial differences of the velocity profile (faster and slower particles compared to the average)
v
In the 2D and 1D equations only
Dispersion coefficient a function of the velocity space (hydraulic parameters channel geometry)
Dx = Ddx + Dtx + D Dy
= Ddy + Dty + D Dx = Ddxx + Ddx + Dtx + D
Dx Dy
gtgt Dx Dx gtgt Dx
In case of more irregular channel and higher depth or larger cross-section area its value is bigger
Also present in laminar flow
Order of magnitude of Order of magnitude of diffusion dispersion diffusion dispersion coefficientscoefficients
10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s
Pore water
Mol diff
Vertical turbulent diff
High depth Shallow depth
Cross disp (2D)
Longitudinal disp (2D)
Horizontal turbulent diff
Longitudinal disp (1D)
Evaluation of dispersion coefficients measuring
Measuring with tracing matter (eg paint)
Inverse calculation from the measured concentration
2D Cross dispersion coefficient (Fischer)
Dy = dy u R (m2s)
dy - dimensionless cross dispersion coefficient
Straight regular channel dy 015
Moderately winding channel dy 02 ndash 06
Hard winding irregular channel dy gt 06 (1-2)
R ndash hydraulic radius (m)
u - shearing velocity at the channel bottom (ms)
u = (g R I)05Longitudinal dispersion coefficient dx 6 (2D) dxx = 100 ndash 1000 (1D)
Velocity space turbulent pulsation unequal distribution
Estimation of dispersion coefficients empirical forms
TRANSPORT EQUATION FOR NON-CONSERVATIVE TRANSPORT EQUATION FOR NON-CONSERVATIVE MATTERSMATTERS
bull There are sources and losses in the flow space
bull Physical chemical biochemical transformations take place
bull Non-conservative pollutants reaction kinetics ( R(C) )
bull They are taken into account in a linear way
dCdt = -C where is the coefficient of reaction kinetics (usually first-order kinetics)
cαxc
DAxx
c)v(Atc)(A
xx
1D equation in this case
ANALYTICAL SOLUTIONS OF THE TRANSPORT ANALYTICAL SOLUTIONS OF THE TRANSPORT EQUATIONEQUATION
Permanent mixing of the pollutants
Polluntant-wave transmission
Estimation of the geometric and hydraulic parameters
Exacter calculations based on measurments using numerical methods
The analitical solutions can be obtained in simpler cases only (approximate calculations)
Main steps
2
2
y
cD
x
cv yx
PERMANENT MIXING
The pollutant emission is steady in time
Permanent river discharge (low flow condition)
Constant flow velocity flow depth and dispersion coef
2D equation negligible vertical changes (shallow river)
Convection transferring
Dispersion spreading
Initial condition M0 (x0 y0) - emission
Boundary cond cy = 0 at the bank
)()()(
cvhy
cvhxt
chyx )()(
y
cDh
yx
cDh
xyx
x
yy
v
xD2
INLET AT THE MAIN FLOW PATH
Longitudinal according to x-frac12 function
Cross direction Gauss-distribution
cmax
M [kgs]cmax
)4
exp(2
2
xD
yv
xvDh
Mc (x t)y
x
xy
x
yb
v
xDB
234
Bb at value 01 cmax 1522bBBrand width
B ~ Bb2
1 0270 BD
vL
y
x
First distance of the mixing
M
1L
Bb
C (x1 y)
x1
M
x
yb
v
xDB
2152
21 110 B
D
vL
y
x
INLET AT THE RIVER BANK
)4
exp(2
xD
yv
xvDh
Mc
y
x
xy
cmax
C (x1 y)
x1
M
INLET NEAR THE RIVER BANK
)4
(exp ( xDy
( y-y0 )2-v
xvD2h
Mc x
xy
cmax
))4
+exp ( xDy
( y+y0 )2-vx
y0 C (x1 y)
x1
IMPACTS OF THE RIVER BANKS (total river section)
Boundary condition reflection theory (infinite series)
Complete mixing the change along the cross-section is less than 10
L2 ~ 3L1 second distance of the mixing
Inlet at a point in y0 distance from the bank
xvD2h
Mc
xy
)4
exp ( xDy
( y-y0 +2nB)2-vx
)4
+ exp ( xDy
( y+y0 -2nB)2-vx
sumn=infin
n=minusinfin(
)
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
x y z directions (3D)
D ndash molecular diffusion coefficient
material specific water 10-4 cm2s
space independent
slow mixing and equalization
Convection The water particles with differentconcentrations of pollutant move according to their differing flow velocity
Diffusion The adjacent water particles mix with each otherwhich causes the equalization of concentration
Mass balance equationMass balance equation
bull Laminar flow ordered with parallel flow linesbull Turbulent flow disordered random (flow and direction)
because of surface roughness (friction) rarr causes intensive mixing
v(c)
t
v deviation pulsation
T Time step of the turbulence
averagev~
0
bull Natural streams always turbulent
We usually can use the average only
TurbulenceTurbulence
Convection v middot c [ kgm2 middot s ]
0
0
Turbulent transport Turbulent transport descriptiondescription
v
turbulent diffusion
molecular diffusion
Dtx Dty Dtz gtgt D
Depends on the direction
Function of the velocity space
X direction (the other are similar)
Analogy with Fickrsquos Law rarr the original transport equation does not change only the D parameter
Turbulent diffusionTurbulent diffusion
Dx = D + Dtx Dy = D + Dty Dz = D + Dtz
Convection
Diffusion
Turbulent diffusion
Stochastic fluctuations of the flow velocity (pulsations)
Diffusive process in mathematical sense (~ Fickrsquos Law)
Convective transport (temporal differences)
Temporal averaging (T)
3D transport equation in turbulent flow3D transport equation in turbulent flow
Spatial simplifications (2D)Spatial simplifications (2D) - Dispersion - Dispersion
Depth averaged flow velocity v
0v
H
After the expansion of the convectiv part (v middot c) in x direction
Depth Integration (3D2D)
Turbulent dispersion (~ diffusion)
z
x
Velocity depth profile
Dx Dy
turbulent dispersion coefficients in 2D equation
Depth averaging (H)
11D tD transport equation in turbulent flow (cross-section avransport equation in turbulent flow (cross-section av))
Dx turbulent dispersion coefficient in 1D equation
Cross-section averaging (A)
2D transport equation in turbulent flow (depth averaged)2D transport equation in turbulent flow (depth averaged)
Turbulent dispersionTurbulent dispersion
Convectiv transport arising from the spatial differences of the velocity profile (faster and slower particles compared to the average)
v
In the 2D and 1D equations only
Dispersion coefficient a function of the velocity space (hydraulic parameters channel geometry)
Dx = Ddx + Dtx + D Dy
= Ddy + Dty + D Dx = Ddxx + Ddx + Dtx + D
Dx Dy
gtgt Dx Dx gtgt Dx
In case of more irregular channel and higher depth or larger cross-section area its value is bigger
Also present in laminar flow
Order of magnitude of Order of magnitude of diffusion dispersion diffusion dispersion coefficientscoefficients
10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s
Pore water
Mol diff
Vertical turbulent diff
High depth Shallow depth
Cross disp (2D)
Longitudinal disp (2D)
Horizontal turbulent diff
Longitudinal disp (1D)
Evaluation of dispersion coefficients measuring
Measuring with tracing matter (eg paint)
Inverse calculation from the measured concentration
2D Cross dispersion coefficient (Fischer)
Dy = dy u R (m2s)
dy - dimensionless cross dispersion coefficient
Straight regular channel dy 015
Moderately winding channel dy 02 ndash 06
Hard winding irregular channel dy gt 06 (1-2)
R ndash hydraulic radius (m)
u - shearing velocity at the channel bottom (ms)
u = (g R I)05Longitudinal dispersion coefficient dx 6 (2D) dxx = 100 ndash 1000 (1D)
Velocity space turbulent pulsation unequal distribution
Estimation of dispersion coefficients empirical forms
TRANSPORT EQUATION FOR NON-CONSERVATIVE TRANSPORT EQUATION FOR NON-CONSERVATIVE MATTERSMATTERS
bull There are sources and losses in the flow space
bull Physical chemical biochemical transformations take place
bull Non-conservative pollutants reaction kinetics ( R(C) )
bull They are taken into account in a linear way
dCdt = -C where is the coefficient of reaction kinetics (usually first-order kinetics)
cαxc
DAxx
c)v(Atc)(A
xx
1D equation in this case
ANALYTICAL SOLUTIONS OF THE TRANSPORT ANALYTICAL SOLUTIONS OF THE TRANSPORT EQUATIONEQUATION
Permanent mixing of the pollutants
Polluntant-wave transmission
Estimation of the geometric and hydraulic parameters
Exacter calculations based on measurments using numerical methods
The analitical solutions can be obtained in simpler cases only (approximate calculations)
Main steps
2
2
y
cD
x
cv yx
PERMANENT MIXING
The pollutant emission is steady in time
Permanent river discharge (low flow condition)
Constant flow velocity flow depth and dispersion coef
2D equation negligible vertical changes (shallow river)
Convection transferring
Dispersion spreading
Initial condition M0 (x0 y0) - emission
Boundary cond cy = 0 at the bank
)()()(
cvhy
cvhxt
chyx )()(
y
cDh
yx
cDh
xyx
x
yy
v
xD2
INLET AT THE MAIN FLOW PATH
Longitudinal according to x-frac12 function
Cross direction Gauss-distribution
cmax
M [kgs]cmax
)4
exp(2
2
xD
yv
xvDh
Mc (x t)y
x
xy
x
yb
v
xDB
234
Bb at value 01 cmax 1522bBBrand width
B ~ Bb2
1 0270 BD
vL
y
x
First distance of the mixing
M
1L
Bb
C (x1 y)
x1
M
x
yb
v
xDB
2152
21 110 B
D
vL
y
x
INLET AT THE RIVER BANK
)4
exp(2
xD
yv
xvDh
Mc
y
x
xy
cmax
C (x1 y)
x1
M
INLET NEAR THE RIVER BANK
)4
(exp ( xDy
( y-y0 )2-v
xvD2h
Mc x
xy
cmax
))4
+exp ( xDy
( y+y0 )2-vx
y0 C (x1 y)
x1
IMPACTS OF THE RIVER BANKS (total river section)
Boundary condition reflection theory (infinite series)
Complete mixing the change along the cross-section is less than 10
L2 ~ 3L1 second distance of the mixing
Inlet at a point in y0 distance from the bank
xvD2h
Mc
xy
)4
exp ( xDy
( y-y0 +2nB)2-vx
)4
+ exp ( xDy
( y+y0 -2nB)2-vx
sumn=infin
n=minusinfin(
)
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
bull Laminar flow ordered with parallel flow linesbull Turbulent flow disordered random (flow and direction)
because of surface roughness (friction) rarr causes intensive mixing
v(c)
t
v deviation pulsation
T Time step of the turbulence
averagev~
0
bull Natural streams always turbulent
We usually can use the average only
TurbulenceTurbulence
Convection v middot c [ kgm2 middot s ]
0
0
Turbulent transport Turbulent transport descriptiondescription
v
turbulent diffusion
molecular diffusion
Dtx Dty Dtz gtgt D
Depends on the direction
Function of the velocity space
X direction (the other are similar)
Analogy with Fickrsquos Law rarr the original transport equation does not change only the D parameter
Turbulent diffusionTurbulent diffusion
Dx = D + Dtx Dy = D + Dty Dz = D + Dtz
Convection
Diffusion
Turbulent diffusion
Stochastic fluctuations of the flow velocity (pulsations)
Diffusive process in mathematical sense (~ Fickrsquos Law)
Convective transport (temporal differences)
Temporal averaging (T)
3D transport equation in turbulent flow3D transport equation in turbulent flow
Spatial simplifications (2D)Spatial simplifications (2D) - Dispersion - Dispersion
Depth averaged flow velocity v
0v
H
After the expansion of the convectiv part (v middot c) in x direction
Depth Integration (3D2D)
Turbulent dispersion (~ diffusion)
z
x
Velocity depth profile
Dx Dy
turbulent dispersion coefficients in 2D equation
Depth averaging (H)
11D tD transport equation in turbulent flow (cross-section avransport equation in turbulent flow (cross-section av))
Dx turbulent dispersion coefficient in 1D equation
Cross-section averaging (A)
2D transport equation in turbulent flow (depth averaged)2D transport equation in turbulent flow (depth averaged)
Turbulent dispersionTurbulent dispersion
Convectiv transport arising from the spatial differences of the velocity profile (faster and slower particles compared to the average)
v
In the 2D and 1D equations only
Dispersion coefficient a function of the velocity space (hydraulic parameters channel geometry)
Dx = Ddx + Dtx + D Dy
= Ddy + Dty + D Dx = Ddxx + Ddx + Dtx + D
Dx Dy
gtgt Dx Dx gtgt Dx
In case of more irregular channel and higher depth or larger cross-section area its value is bigger
Also present in laminar flow
Order of magnitude of Order of magnitude of diffusion dispersion diffusion dispersion coefficientscoefficients
10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s
Pore water
Mol diff
Vertical turbulent diff
High depth Shallow depth
Cross disp (2D)
Longitudinal disp (2D)
Horizontal turbulent diff
Longitudinal disp (1D)
Evaluation of dispersion coefficients measuring
Measuring with tracing matter (eg paint)
Inverse calculation from the measured concentration
2D Cross dispersion coefficient (Fischer)
Dy = dy u R (m2s)
dy - dimensionless cross dispersion coefficient
Straight regular channel dy 015
Moderately winding channel dy 02 ndash 06
Hard winding irregular channel dy gt 06 (1-2)
R ndash hydraulic radius (m)
u - shearing velocity at the channel bottom (ms)
u = (g R I)05Longitudinal dispersion coefficient dx 6 (2D) dxx = 100 ndash 1000 (1D)
Velocity space turbulent pulsation unequal distribution
Estimation of dispersion coefficients empirical forms
TRANSPORT EQUATION FOR NON-CONSERVATIVE TRANSPORT EQUATION FOR NON-CONSERVATIVE MATTERSMATTERS
bull There are sources and losses in the flow space
bull Physical chemical biochemical transformations take place
bull Non-conservative pollutants reaction kinetics ( R(C) )
bull They are taken into account in a linear way
dCdt = -C where is the coefficient of reaction kinetics (usually first-order kinetics)
cαxc
DAxx
c)v(Atc)(A
xx
1D equation in this case
ANALYTICAL SOLUTIONS OF THE TRANSPORT ANALYTICAL SOLUTIONS OF THE TRANSPORT EQUATIONEQUATION
Permanent mixing of the pollutants
Polluntant-wave transmission
Estimation of the geometric and hydraulic parameters
Exacter calculations based on measurments using numerical methods
The analitical solutions can be obtained in simpler cases only (approximate calculations)
Main steps
2
2
y
cD
x
cv yx
PERMANENT MIXING
The pollutant emission is steady in time
Permanent river discharge (low flow condition)
Constant flow velocity flow depth and dispersion coef
2D equation negligible vertical changes (shallow river)
Convection transferring
Dispersion spreading
Initial condition M0 (x0 y0) - emission
Boundary cond cy = 0 at the bank
)()()(
cvhy
cvhxt
chyx )()(
y
cDh
yx
cDh
xyx
x
yy
v
xD2
INLET AT THE MAIN FLOW PATH
Longitudinal according to x-frac12 function
Cross direction Gauss-distribution
cmax
M [kgs]cmax
)4
exp(2
2
xD
yv
xvDh
Mc (x t)y
x
xy
x
yb
v
xDB
234
Bb at value 01 cmax 1522bBBrand width
B ~ Bb2
1 0270 BD
vL
y
x
First distance of the mixing
M
1L
Bb
C (x1 y)
x1
M
x
yb
v
xDB
2152
21 110 B
D
vL
y
x
INLET AT THE RIVER BANK
)4
exp(2
xD
yv
xvDh
Mc
y
x
xy
cmax
C (x1 y)
x1
M
INLET NEAR THE RIVER BANK
)4
(exp ( xDy
( y-y0 )2-v
xvD2h
Mc x
xy
cmax
))4
+exp ( xDy
( y+y0 )2-vx
y0 C (x1 y)
x1
IMPACTS OF THE RIVER BANKS (total river section)
Boundary condition reflection theory (infinite series)
Complete mixing the change along the cross-section is less than 10
L2 ~ 3L1 second distance of the mixing
Inlet at a point in y0 distance from the bank
xvD2h
Mc
xy
)4
exp ( xDy
( y-y0 +2nB)2-vx
)4
+ exp ( xDy
( y+y0 -2nB)2-vx
sumn=infin
n=minusinfin(
)
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
Convection v middot c [ kgm2 middot s ]
0
0
Turbulent transport Turbulent transport descriptiondescription
v
turbulent diffusion
molecular diffusion
Dtx Dty Dtz gtgt D
Depends on the direction
Function of the velocity space
X direction (the other are similar)
Analogy with Fickrsquos Law rarr the original transport equation does not change only the D parameter
Turbulent diffusionTurbulent diffusion
Dx = D + Dtx Dy = D + Dty Dz = D + Dtz
Convection
Diffusion
Turbulent diffusion
Stochastic fluctuations of the flow velocity (pulsations)
Diffusive process in mathematical sense (~ Fickrsquos Law)
Convective transport (temporal differences)
Temporal averaging (T)
3D transport equation in turbulent flow3D transport equation in turbulent flow
Spatial simplifications (2D)Spatial simplifications (2D) - Dispersion - Dispersion
Depth averaged flow velocity v
0v
H
After the expansion of the convectiv part (v middot c) in x direction
Depth Integration (3D2D)
Turbulent dispersion (~ diffusion)
z
x
Velocity depth profile
Dx Dy
turbulent dispersion coefficients in 2D equation
Depth averaging (H)
11D tD transport equation in turbulent flow (cross-section avransport equation in turbulent flow (cross-section av))
Dx turbulent dispersion coefficient in 1D equation
Cross-section averaging (A)
2D transport equation in turbulent flow (depth averaged)2D transport equation in turbulent flow (depth averaged)
Turbulent dispersionTurbulent dispersion
Convectiv transport arising from the spatial differences of the velocity profile (faster and slower particles compared to the average)
v
In the 2D and 1D equations only
Dispersion coefficient a function of the velocity space (hydraulic parameters channel geometry)
Dx = Ddx + Dtx + D Dy
= Ddy + Dty + D Dx = Ddxx + Ddx + Dtx + D
Dx Dy
gtgt Dx Dx gtgt Dx
In case of more irregular channel and higher depth or larger cross-section area its value is bigger
Also present in laminar flow
Order of magnitude of Order of magnitude of diffusion dispersion diffusion dispersion coefficientscoefficients
10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s
Pore water
Mol diff
Vertical turbulent diff
High depth Shallow depth
Cross disp (2D)
Longitudinal disp (2D)
Horizontal turbulent diff
Longitudinal disp (1D)
Evaluation of dispersion coefficients measuring
Measuring with tracing matter (eg paint)
Inverse calculation from the measured concentration
2D Cross dispersion coefficient (Fischer)
Dy = dy u R (m2s)
dy - dimensionless cross dispersion coefficient
Straight regular channel dy 015
Moderately winding channel dy 02 ndash 06
Hard winding irregular channel dy gt 06 (1-2)
R ndash hydraulic radius (m)
u - shearing velocity at the channel bottom (ms)
u = (g R I)05Longitudinal dispersion coefficient dx 6 (2D) dxx = 100 ndash 1000 (1D)
Velocity space turbulent pulsation unequal distribution
Estimation of dispersion coefficients empirical forms
TRANSPORT EQUATION FOR NON-CONSERVATIVE TRANSPORT EQUATION FOR NON-CONSERVATIVE MATTERSMATTERS
bull There are sources and losses in the flow space
bull Physical chemical biochemical transformations take place
bull Non-conservative pollutants reaction kinetics ( R(C) )
bull They are taken into account in a linear way
dCdt = -C where is the coefficient of reaction kinetics (usually first-order kinetics)
cαxc
DAxx
c)v(Atc)(A
xx
1D equation in this case
ANALYTICAL SOLUTIONS OF THE TRANSPORT ANALYTICAL SOLUTIONS OF THE TRANSPORT EQUATIONEQUATION
Permanent mixing of the pollutants
Polluntant-wave transmission
Estimation of the geometric and hydraulic parameters
Exacter calculations based on measurments using numerical methods
The analitical solutions can be obtained in simpler cases only (approximate calculations)
Main steps
2
2
y
cD
x
cv yx
PERMANENT MIXING
The pollutant emission is steady in time
Permanent river discharge (low flow condition)
Constant flow velocity flow depth and dispersion coef
2D equation negligible vertical changes (shallow river)
Convection transferring
Dispersion spreading
Initial condition M0 (x0 y0) - emission
Boundary cond cy = 0 at the bank
)()()(
cvhy
cvhxt
chyx )()(
y
cDh
yx
cDh
xyx
x
yy
v
xD2
INLET AT THE MAIN FLOW PATH
Longitudinal according to x-frac12 function
Cross direction Gauss-distribution
cmax
M [kgs]cmax
)4
exp(2
2
xD
yv
xvDh
Mc (x t)y
x
xy
x
yb
v
xDB
234
Bb at value 01 cmax 1522bBBrand width
B ~ Bb2
1 0270 BD
vL
y
x
First distance of the mixing
M
1L
Bb
C (x1 y)
x1
M
x
yb
v
xDB
2152
21 110 B
D
vL
y
x
INLET AT THE RIVER BANK
)4
exp(2
xD
yv
xvDh
Mc
y
x
xy
cmax
C (x1 y)
x1
M
INLET NEAR THE RIVER BANK
)4
(exp ( xDy
( y-y0 )2-v
xvD2h
Mc x
xy
cmax
))4
+exp ( xDy
( y+y0 )2-vx
y0 C (x1 y)
x1
IMPACTS OF THE RIVER BANKS (total river section)
Boundary condition reflection theory (infinite series)
Complete mixing the change along the cross-section is less than 10
L2 ~ 3L1 second distance of the mixing
Inlet at a point in y0 distance from the bank
xvD2h
Mc
xy
)4
exp ( xDy
( y-y0 +2nB)2-vx
)4
+ exp ( xDy
( y+y0 -2nB)2-vx
sumn=infin
n=minusinfin(
)
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
v
turbulent diffusion
molecular diffusion
Dtx Dty Dtz gtgt D
Depends on the direction
Function of the velocity space
X direction (the other are similar)
Analogy with Fickrsquos Law rarr the original transport equation does not change only the D parameter
Turbulent diffusionTurbulent diffusion
Dx = D + Dtx Dy = D + Dty Dz = D + Dtz
Convection
Diffusion
Turbulent diffusion
Stochastic fluctuations of the flow velocity (pulsations)
Diffusive process in mathematical sense (~ Fickrsquos Law)
Convective transport (temporal differences)
Temporal averaging (T)
3D transport equation in turbulent flow3D transport equation in turbulent flow
Spatial simplifications (2D)Spatial simplifications (2D) - Dispersion - Dispersion
Depth averaged flow velocity v
0v
H
After the expansion of the convectiv part (v middot c) in x direction
Depth Integration (3D2D)
Turbulent dispersion (~ diffusion)
z
x
Velocity depth profile
Dx Dy
turbulent dispersion coefficients in 2D equation
Depth averaging (H)
11D tD transport equation in turbulent flow (cross-section avransport equation in turbulent flow (cross-section av))
Dx turbulent dispersion coefficient in 1D equation
Cross-section averaging (A)
2D transport equation in turbulent flow (depth averaged)2D transport equation in turbulent flow (depth averaged)
Turbulent dispersionTurbulent dispersion
Convectiv transport arising from the spatial differences of the velocity profile (faster and slower particles compared to the average)
v
In the 2D and 1D equations only
Dispersion coefficient a function of the velocity space (hydraulic parameters channel geometry)
Dx = Ddx + Dtx + D Dy
= Ddy + Dty + D Dx = Ddxx + Ddx + Dtx + D
Dx Dy
gtgt Dx Dx gtgt Dx
In case of more irregular channel and higher depth or larger cross-section area its value is bigger
Also present in laminar flow
Order of magnitude of Order of magnitude of diffusion dispersion diffusion dispersion coefficientscoefficients
10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s
Pore water
Mol diff
Vertical turbulent diff
High depth Shallow depth
Cross disp (2D)
Longitudinal disp (2D)
Horizontal turbulent diff
Longitudinal disp (1D)
Evaluation of dispersion coefficients measuring
Measuring with tracing matter (eg paint)
Inverse calculation from the measured concentration
2D Cross dispersion coefficient (Fischer)
Dy = dy u R (m2s)
dy - dimensionless cross dispersion coefficient
Straight regular channel dy 015
Moderately winding channel dy 02 ndash 06
Hard winding irregular channel dy gt 06 (1-2)
R ndash hydraulic radius (m)
u - shearing velocity at the channel bottom (ms)
u = (g R I)05Longitudinal dispersion coefficient dx 6 (2D) dxx = 100 ndash 1000 (1D)
Velocity space turbulent pulsation unequal distribution
Estimation of dispersion coefficients empirical forms
TRANSPORT EQUATION FOR NON-CONSERVATIVE TRANSPORT EQUATION FOR NON-CONSERVATIVE MATTERSMATTERS
bull There are sources and losses in the flow space
bull Physical chemical biochemical transformations take place
bull Non-conservative pollutants reaction kinetics ( R(C) )
bull They are taken into account in a linear way
dCdt = -C where is the coefficient of reaction kinetics (usually first-order kinetics)
cαxc
DAxx
c)v(Atc)(A
xx
1D equation in this case
ANALYTICAL SOLUTIONS OF THE TRANSPORT ANALYTICAL SOLUTIONS OF THE TRANSPORT EQUATIONEQUATION
Permanent mixing of the pollutants
Polluntant-wave transmission
Estimation of the geometric and hydraulic parameters
Exacter calculations based on measurments using numerical methods
The analitical solutions can be obtained in simpler cases only (approximate calculations)
Main steps
2
2
y
cD
x
cv yx
PERMANENT MIXING
The pollutant emission is steady in time
Permanent river discharge (low flow condition)
Constant flow velocity flow depth and dispersion coef
2D equation negligible vertical changes (shallow river)
Convection transferring
Dispersion spreading
Initial condition M0 (x0 y0) - emission
Boundary cond cy = 0 at the bank
)()()(
cvhy
cvhxt
chyx )()(
y
cDh
yx
cDh
xyx
x
yy
v
xD2
INLET AT THE MAIN FLOW PATH
Longitudinal according to x-frac12 function
Cross direction Gauss-distribution
cmax
M [kgs]cmax
)4
exp(2
2
xD
yv
xvDh
Mc (x t)y
x
xy
x
yb
v
xDB
234
Bb at value 01 cmax 1522bBBrand width
B ~ Bb2
1 0270 BD
vL
y
x
First distance of the mixing
M
1L
Bb
C (x1 y)
x1
M
x
yb
v
xDB
2152
21 110 B
D
vL
y
x
INLET AT THE RIVER BANK
)4
exp(2
xD
yv
xvDh
Mc
y
x
xy
cmax
C (x1 y)
x1
M
INLET NEAR THE RIVER BANK
)4
(exp ( xDy
( y-y0 )2-v
xvD2h
Mc x
xy
cmax
))4
+exp ( xDy
( y+y0 )2-vx
y0 C (x1 y)
x1
IMPACTS OF THE RIVER BANKS (total river section)
Boundary condition reflection theory (infinite series)
Complete mixing the change along the cross-section is less than 10
L2 ~ 3L1 second distance of the mixing
Inlet at a point in y0 distance from the bank
xvD2h
Mc
xy
)4
exp ( xDy
( y-y0 +2nB)2-vx
)4
+ exp ( xDy
( y+y0 -2nB)2-vx
sumn=infin
n=minusinfin(
)
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
Dx = D + Dtx Dy = D + Dty Dz = D + Dtz
Convection
Diffusion
Turbulent diffusion
Stochastic fluctuations of the flow velocity (pulsations)
Diffusive process in mathematical sense (~ Fickrsquos Law)
Convective transport (temporal differences)
Temporal averaging (T)
3D transport equation in turbulent flow3D transport equation in turbulent flow
Spatial simplifications (2D)Spatial simplifications (2D) - Dispersion - Dispersion
Depth averaged flow velocity v
0v
H
After the expansion of the convectiv part (v middot c) in x direction
Depth Integration (3D2D)
Turbulent dispersion (~ diffusion)
z
x
Velocity depth profile
Dx Dy
turbulent dispersion coefficients in 2D equation
Depth averaging (H)
11D tD transport equation in turbulent flow (cross-section avransport equation in turbulent flow (cross-section av))
Dx turbulent dispersion coefficient in 1D equation
Cross-section averaging (A)
2D transport equation in turbulent flow (depth averaged)2D transport equation in turbulent flow (depth averaged)
Turbulent dispersionTurbulent dispersion
Convectiv transport arising from the spatial differences of the velocity profile (faster and slower particles compared to the average)
v
In the 2D and 1D equations only
Dispersion coefficient a function of the velocity space (hydraulic parameters channel geometry)
Dx = Ddx + Dtx + D Dy
= Ddy + Dty + D Dx = Ddxx + Ddx + Dtx + D
Dx Dy
gtgt Dx Dx gtgt Dx
In case of more irregular channel and higher depth or larger cross-section area its value is bigger
Also present in laminar flow
Order of magnitude of Order of magnitude of diffusion dispersion diffusion dispersion coefficientscoefficients
10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s
Pore water
Mol diff
Vertical turbulent diff
High depth Shallow depth
Cross disp (2D)
Longitudinal disp (2D)
Horizontal turbulent diff
Longitudinal disp (1D)
Evaluation of dispersion coefficients measuring
Measuring with tracing matter (eg paint)
Inverse calculation from the measured concentration
2D Cross dispersion coefficient (Fischer)
Dy = dy u R (m2s)
dy - dimensionless cross dispersion coefficient
Straight regular channel dy 015
Moderately winding channel dy 02 ndash 06
Hard winding irregular channel dy gt 06 (1-2)
R ndash hydraulic radius (m)
u - shearing velocity at the channel bottom (ms)
u = (g R I)05Longitudinal dispersion coefficient dx 6 (2D) dxx = 100 ndash 1000 (1D)
Velocity space turbulent pulsation unequal distribution
Estimation of dispersion coefficients empirical forms
TRANSPORT EQUATION FOR NON-CONSERVATIVE TRANSPORT EQUATION FOR NON-CONSERVATIVE MATTERSMATTERS
bull There are sources and losses in the flow space
bull Physical chemical biochemical transformations take place
bull Non-conservative pollutants reaction kinetics ( R(C) )
bull They are taken into account in a linear way
dCdt = -C where is the coefficient of reaction kinetics (usually first-order kinetics)
cαxc
DAxx
c)v(Atc)(A
xx
1D equation in this case
ANALYTICAL SOLUTIONS OF THE TRANSPORT ANALYTICAL SOLUTIONS OF THE TRANSPORT EQUATIONEQUATION
Permanent mixing of the pollutants
Polluntant-wave transmission
Estimation of the geometric and hydraulic parameters
Exacter calculations based on measurments using numerical methods
The analitical solutions can be obtained in simpler cases only (approximate calculations)
Main steps
2
2
y
cD
x
cv yx
PERMANENT MIXING
The pollutant emission is steady in time
Permanent river discharge (low flow condition)
Constant flow velocity flow depth and dispersion coef
2D equation negligible vertical changes (shallow river)
Convection transferring
Dispersion spreading
Initial condition M0 (x0 y0) - emission
Boundary cond cy = 0 at the bank
)()()(
cvhy
cvhxt
chyx )()(
y
cDh
yx
cDh
xyx
x
yy
v
xD2
INLET AT THE MAIN FLOW PATH
Longitudinal according to x-frac12 function
Cross direction Gauss-distribution
cmax
M [kgs]cmax
)4
exp(2
2
xD
yv
xvDh
Mc (x t)y
x
xy
x
yb
v
xDB
234
Bb at value 01 cmax 1522bBBrand width
B ~ Bb2
1 0270 BD
vL
y
x
First distance of the mixing
M
1L
Bb
C (x1 y)
x1
M
x
yb
v
xDB
2152
21 110 B
D
vL
y
x
INLET AT THE RIVER BANK
)4
exp(2
xD
yv
xvDh
Mc
y
x
xy
cmax
C (x1 y)
x1
M
INLET NEAR THE RIVER BANK
)4
(exp ( xDy
( y-y0 )2-v
xvD2h
Mc x
xy
cmax
))4
+exp ( xDy
( y+y0 )2-vx
y0 C (x1 y)
x1
IMPACTS OF THE RIVER BANKS (total river section)
Boundary condition reflection theory (infinite series)
Complete mixing the change along the cross-section is less than 10
L2 ~ 3L1 second distance of the mixing
Inlet at a point in y0 distance from the bank
xvD2h
Mc
xy
)4
exp ( xDy
( y-y0 +2nB)2-vx
)4
+ exp ( xDy
( y+y0 -2nB)2-vx
sumn=infin
n=minusinfin(
)
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
Spatial simplifications (2D)Spatial simplifications (2D) - Dispersion - Dispersion
Depth averaged flow velocity v
0v
H
After the expansion of the convectiv part (v middot c) in x direction
Depth Integration (3D2D)
Turbulent dispersion (~ diffusion)
z
x
Velocity depth profile
Dx Dy
turbulent dispersion coefficients in 2D equation
Depth averaging (H)
11D tD transport equation in turbulent flow (cross-section avransport equation in turbulent flow (cross-section av))
Dx turbulent dispersion coefficient in 1D equation
Cross-section averaging (A)
2D transport equation in turbulent flow (depth averaged)2D transport equation in turbulent flow (depth averaged)
Turbulent dispersionTurbulent dispersion
Convectiv transport arising from the spatial differences of the velocity profile (faster and slower particles compared to the average)
v
In the 2D and 1D equations only
Dispersion coefficient a function of the velocity space (hydraulic parameters channel geometry)
Dx = Ddx + Dtx + D Dy
= Ddy + Dty + D Dx = Ddxx + Ddx + Dtx + D
Dx Dy
gtgt Dx Dx gtgt Dx
In case of more irregular channel and higher depth or larger cross-section area its value is bigger
Also present in laminar flow
Order of magnitude of Order of magnitude of diffusion dispersion diffusion dispersion coefficientscoefficients
10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s
Pore water
Mol diff
Vertical turbulent diff
High depth Shallow depth
Cross disp (2D)
Longitudinal disp (2D)
Horizontal turbulent diff
Longitudinal disp (1D)
Evaluation of dispersion coefficients measuring
Measuring with tracing matter (eg paint)
Inverse calculation from the measured concentration
2D Cross dispersion coefficient (Fischer)
Dy = dy u R (m2s)
dy - dimensionless cross dispersion coefficient
Straight regular channel dy 015
Moderately winding channel dy 02 ndash 06
Hard winding irregular channel dy gt 06 (1-2)
R ndash hydraulic radius (m)
u - shearing velocity at the channel bottom (ms)
u = (g R I)05Longitudinal dispersion coefficient dx 6 (2D) dxx = 100 ndash 1000 (1D)
Velocity space turbulent pulsation unequal distribution
Estimation of dispersion coefficients empirical forms
TRANSPORT EQUATION FOR NON-CONSERVATIVE TRANSPORT EQUATION FOR NON-CONSERVATIVE MATTERSMATTERS
bull There are sources and losses in the flow space
bull Physical chemical biochemical transformations take place
bull Non-conservative pollutants reaction kinetics ( R(C) )
bull They are taken into account in a linear way
dCdt = -C where is the coefficient of reaction kinetics (usually first-order kinetics)
cαxc
DAxx
c)v(Atc)(A
xx
1D equation in this case
ANALYTICAL SOLUTIONS OF THE TRANSPORT ANALYTICAL SOLUTIONS OF THE TRANSPORT EQUATIONEQUATION
Permanent mixing of the pollutants
Polluntant-wave transmission
Estimation of the geometric and hydraulic parameters
Exacter calculations based on measurments using numerical methods
The analitical solutions can be obtained in simpler cases only (approximate calculations)
Main steps
2
2
y
cD
x
cv yx
PERMANENT MIXING
The pollutant emission is steady in time
Permanent river discharge (low flow condition)
Constant flow velocity flow depth and dispersion coef
2D equation negligible vertical changes (shallow river)
Convection transferring
Dispersion spreading
Initial condition M0 (x0 y0) - emission
Boundary cond cy = 0 at the bank
)()()(
cvhy
cvhxt
chyx )()(
y
cDh
yx
cDh
xyx
x
yy
v
xD2
INLET AT THE MAIN FLOW PATH
Longitudinal according to x-frac12 function
Cross direction Gauss-distribution
cmax
M [kgs]cmax
)4
exp(2
2
xD
yv
xvDh
Mc (x t)y
x
xy
x
yb
v
xDB
234
Bb at value 01 cmax 1522bBBrand width
B ~ Bb2
1 0270 BD
vL
y
x
First distance of the mixing
M
1L
Bb
C (x1 y)
x1
M
x
yb
v
xDB
2152
21 110 B
D
vL
y
x
INLET AT THE RIVER BANK
)4
exp(2
xD
yv
xvDh
Mc
y
x
xy
cmax
C (x1 y)
x1
M
INLET NEAR THE RIVER BANK
)4
(exp ( xDy
( y-y0 )2-v
xvD2h
Mc x
xy
cmax
))4
+exp ( xDy
( y+y0 )2-vx
y0 C (x1 y)
x1
IMPACTS OF THE RIVER BANKS (total river section)
Boundary condition reflection theory (infinite series)
Complete mixing the change along the cross-section is less than 10
L2 ~ 3L1 second distance of the mixing
Inlet at a point in y0 distance from the bank
xvD2h
Mc
xy
)4
exp ( xDy
( y-y0 +2nB)2-vx
)4
+ exp ( xDy
( y+y0 -2nB)2-vx
sumn=infin
n=minusinfin(
)
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
Dx Dy
turbulent dispersion coefficients in 2D equation
Depth averaging (H)
11D tD transport equation in turbulent flow (cross-section avransport equation in turbulent flow (cross-section av))
Dx turbulent dispersion coefficient in 1D equation
Cross-section averaging (A)
2D transport equation in turbulent flow (depth averaged)2D transport equation in turbulent flow (depth averaged)
Turbulent dispersionTurbulent dispersion
Convectiv transport arising from the spatial differences of the velocity profile (faster and slower particles compared to the average)
v
In the 2D and 1D equations only
Dispersion coefficient a function of the velocity space (hydraulic parameters channel geometry)
Dx = Ddx + Dtx + D Dy
= Ddy + Dty + D Dx = Ddxx + Ddx + Dtx + D
Dx Dy
gtgt Dx Dx gtgt Dx
In case of more irregular channel and higher depth or larger cross-section area its value is bigger
Also present in laminar flow
Order of magnitude of Order of magnitude of diffusion dispersion diffusion dispersion coefficientscoefficients
10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s
Pore water
Mol diff
Vertical turbulent diff
High depth Shallow depth
Cross disp (2D)
Longitudinal disp (2D)
Horizontal turbulent diff
Longitudinal disp (1D)
Evaluation of dispersion coefficients measuring
Measuring with tracing matter (eg paint)
Inverse calculation from the measured concentration
2D Cross dispersion coefficient (Fischer)
Dy = dy u R (m2s)
dy - dimensionless cross dispersion coefficient
Straight regular channel dy 015
Moderately winding channel dy 02 ndash 06
Hard winding irregular channel dy gt 06 (1-2)
R ndash hydraulic radius (m)
u - shearing velocity at the channel bottom (ms)
u = (g R I)05Longitudinal dispersion coefficient dx 6 (2D) dxx = 100 ndash 1000 (1D)
Velocity space turbulent pulsation unequal distribution
Estimation of dispersion coefficients empirical forms
TRANSPORT EQUATION FOR NON-CONSERVATIVE TRANSPORT EQUATION FOR NON-CONSERVATIVE MATTERSMATTERS
bull There are sources and losses in the flow space
bull Physical chemical biochemical transformations take place
bull Non-conservative pollutants reaction kinetics ( R(C) )
bull They are taken into account in a linear way
dCdt = -C where is the coefficient of reaction kinetics (usually first-order kinetics)
cαxc
DAxx
c)v(Atc)(A
xx
1D equation in this case
ANALYTICAL SOLUTIONS OF THE TRANSPORT ANALYTICAL SOLUTIONS OF THE TRANSPORT EQUATIONEQUATION
Permanent mixing of the pollutants
Polluntant-wave transmission
Estimation of the geometric and hydraulic parameters
Exacter calculations based on measurments using numerical methods
The analitical solutions can be obtained in simpler cases only (approximate calculations)
Main steps
2
2
y
cD
x
cv yx
PERMANENT MIXING
The pollutant emission is steady in time
Permanent river discharge (low flow condition)
Constant flow velocity flow depth and dispersion coef
2D equation negligible vertical changes (shallow river)
Convection transferring
Dispersion spreading
Initial condition M0 (x0 y0) - emission
Boundary cond cy = 0 at the bank
)()()(
cvhy
cvhxt
chyx )()(
y
cDh
yx
cDh
xyx
x
yy
v
xD2
INLET AT THE MAIN FLOW PATH
Longitudinal according to x-frac12 function
Cross direction Gauss-distribution
cmax
M [kgs]cmax
)4
exp(2
2
xD
yv
xvDh
Mc (x t)y
x
xy
x
yb
v
xDB
234
Bb at value 01 cmax 1522bBBrand width
B ~ Bb2
1 0270 BD
vL
y
x
First distance of the mixing
M
1L
Bb
C (x1 y)
x1
M
x
yb
v
xDB
2152
21 110 B
D
vL
y
x
INLET AT THE RIVER BANK
)4
exp(2
xD
yv
xvDh
Mc
y
x
xy
cmax
C (x1 y)
x1
M
INLET NEAR THE RIVER BANK
)4
(exp ( xDy
( y-y0 )2-v
xvD2h
Mc x
xy
cmax
))4
+exp ( xDy
( y+y0 )2-vx
y0 C (x1 y)
x1
IMPACTS OF THE RIVER BANKS (total river section)
Boundary condition reflection theory (infinite series)
Complete mixing the change along the cross-section is less than 10
L2 ~ 3L1 second distance of the mixing
Inlet at a point in y0 distance from the bank
xvD2h
Mc
xy
)4
exp ( xDy
( y-y0 +2nB)2-vx
)4
+ exp ( xDy
( y+y0 -2nB)2-vx
sumn=infin
n=minusinfin(
)
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
Turbulent dispersionTurbulent dispersion
Convectiv transport arising from the spatial differences of the velocity profile (faster and slower particles compared to the average)
v
In the 2D and 1D equations only
Dispersion coefficient a function of the velocity space (hydraulic parameters channel geometry)
Dx = Ddx + Dtx + D Dy
= Ddy + Dty + D Dx = Ddxx + Ddx + Dtx + D
Dx Dy
gtgt Dx Dx gtgt Dx
In case of more irregular channel and higher depth or larger cross-section area its value is bigger
Also present in laminar flow
Order of magnitude of Order of magnitude of diffusion dispersion diffusion dispersion coefficientscoefficients
10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s
Pore water
Mol diff
Vertical turbulent diff
High depth Shallow depth
Cross disp (2D)
Longitudinal disp (2D)
Horizontal turbulent diff
Longitudinal disp (1D)
Evaluation of dispersion coefficients measuring
Measuring with tracing matter (eg paint)
Inverse calculation from the measured concentration
2D Cross dispersion coefficient (Fischer)
Dy = dy u R (m2s)
dy - dimensionless cross dispersion coefficient
Straight regular channel dy 015
Moderately winding channel dy 02 ndash 06
Hard winding irregular channel dy gt 06 (1-2)
R ndash hydraulic radius (m)
u - shearing velocity at the channel bottom (ms)
u = (g R I)05Longitudinal dispersion coefficient dx 6 (2D) dxx = 100 ndash 1000 (1D)
Velocity space turbulent pulsation unequal distribution
Estimation of dispersion coefficients empirical forms
TRANSPORT EQUATION FOR NON-CONSERVATIVE TRANSPORT EQUATION FOR NON-CONSERVATIVE MATTERSMATTERS
bull There are sources and losses in the flow space
bull Physical chemical biochemical transformations take place
bull Non-conservative pollutants reaction kinetics ( R(C) )
bull They are taken into account in a linear way
dCdt = -C where is the coefficient of reaction kinetics (usually first-order kinetics)
cαxc
DAxx
c)v(Atc)(A
xx
1D equation in this case
ANALYTICAL SOLUTIONS OF THE TRANSPORT ANALYTICAL SOLUTIONS OF THE TRANSPORT EQUATIONEQUATION
Permanent mixing of the pollutants
Polluntant-wave transmission
Estimation of the geometric and hydraulic parameters
Exacter calculations based on measurments using numerical methods
The analitical solutions can be obtained in simpler cases only (approximate calculations)
Main steps
2
2
y
cD
x
cv yx
PERMANENT MIXING
The pollutant emission is steady in time
Permanent river discharge (low flow condition)
Constant flow velocity flow depth and dispersion coef
2D equation negligible vertical changes (shallow river)
Convection transferring
Dispersion spreading
Initial condition M0 (x0 y0) - emission
Boundary cond cy = 0 at the bank
)()()(
cvhy
cvhxt
chyx )()(
y
cDh
yx
cDh
xyx
x
yy
v
xD2
INLET AT THE MAIN FLOW PATH
Longitudinal according to x-frac12 function
Cross direction Gauss-distribution
cmax
M [kgs]cmax
)4
exp(2
2
xD
yv
xvDh
Mc (x t)y
x
xy
x
yb
v
xDB
234
Bb at value 01 cmax 1522bBBrand width
B ~ Bb2
1 0270 BD
vL
y
x
First distance of the mixing
M
1L
Bb
C (x1 y)
x1
M
x
yb
v
xDB
2152
21 110 B
D
vL
y
x
INLET AT THE RIVER BANK
)4
exp(2
xD
yv
xvDh
Mc
y
x
xy
cmax
C (x1 y)
x1
M
INLET NEAR THE RIVER BANK
)4
(exp ( xDy
( y-y0 )2-v
xvD2h
Mc x
xy
cmax
))4
+exp ( xDy
( y+y0 )2-vx
y0 C (x1 y)
x1
IMPACTS OF THE RIVER BANKS (total river section)
Boundary condition reflection theory (infinite series)
Complete mixing the change along the cross-section is less than 10
L2 ~ 3L1 second distance of the mixing
Inlet at a point in y0 distance from the bank
xvD2h
Mc
xy
)4
exp ( xDy
( y-y0 +2nB)2-vx
)4
+ exp ( xDy
( y+y0 -2nB)2-vx
sumn=infin
n=minusinfin(
)
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
Order of magnitude of Order of magnitude of diffusion dispersion diffusion dispersion coefficientscoefficients
10-8 10-6 10-4 10-2 1 102 104 106 108 cm2s
Pore water
Mol diff
Vertical turbulent diff
High depth Shallow depth
Cross disp (2D)
Longitudinal disp (2D)
Horizontal turbulent diff
Longitudinal disp (1D)
Evaluation of dispersion coefficients measuring
Measuring with tracing matter (eg paint)
Inverse calculation from the measured concentration
2D Cross dispersion coefficient (Fischer)
Dy = dy u R (m2s)
dy - dimensionless cross dispersion coefficient
Straight regular channel dy 015
Moderately winding channel dy 02 ndash 06
Hard winding irregular channel dy gt 06 (1-2)
R ndash hydraulic radius (m)
u - shearing velocity at the channel bottom (ms)
u = (g R I)05Longitudinal dispersion coefficient dx 6 (2D) dxx = 100 ndash 1000 (1D)
Velocity space turbulent pulsation unequal distribution
Estimation of dispersion coefficients empirical forms
TRANSPORT EQUATION FOR NON-CONSERVATIVE TRANSPORT EQUATION FOR NON-CONSERVATIVE MATTERSMATTERS
bull There are sources and losses in the flow space
bull Physical chemical biochemical transformations take place
bull Non-conservative pollutants reaction kinetics ( R(C) )
bull They are taken into account in a linear way
dCdt = -C where is the coefficient of reaction kinetics (usually first-order kinetics)
cαxc
DAxx
c)v(Atc)(A
xx
1D equation in this case
ANALYTICAL SOLUTIONS OF THE TRANSPORT ANALYTICAL SOLUTIONS OF THE TRANSPORT EQUATIONEQUATION
Permanent mixing of the pollutants
Polluntant-wave transmission
Estimation of the geometric and hydraulic parameters
Exacter calculations based on measurments using numerical methods
The analitical solutions can be obtained in simpler cases only (approximate calculations)
Main steps
2
2
y
cD
x
cv yx
PERMANENT MIXING
The pollutant emission is steady in time
Permanent river discharge (low flow condition)
Constant flow velocity flow depth and dispersion coef
2D equation negligible vertical changes (shallow river)
Convection transferring
Dispersion spreading
Initial condition M0 (x0 y0) - emission
Boundary cond cy = 0 at the bank
)()()(
cvhy
cvhxt
chyx )()(
y
cDh
yx
cDh
xyx
x
yy
v
xD2
INLET AT THE MAIN FLOW PATH
Longitudinal according to x-frac12 function
Cross direction Gauss-distribution
cmax
M [kgs]cmax
)4
exp(2
2
xD
yv
xvDh
Mc (x t)y
x
xy
x
yb
v
xDB
234
Bb at value 01 cmax 1522bBBrand width
B ~ Bb2
1 0270 BD
vL
y
x
First distance of the mixing
M
1L
Bb
C (x1 y)
x1
M
x
yb
v
xDB
2152
21 110 B
D
vL
y
x
INLET AT THE RIVER BANK
)4
exp(2
xD
yv
xvDh
Mc
y
x
xy
cmax
C (x1 y)
x1
M
INLET NEAR THE RIVER BANK
)4
(exp ( xDy
( y-y0 )2-v
xvD2h
Mc x
xy
cmax
))4
+exp ( xDy
( y+y0 )2-vx
y0 C (x1 y)
x1
IMPACTS OF THE RIVER BANKS (total river section)
Boundary condition reflection theory (infinite series)
Complete mixing the change along the cross-section is less than 10
L2 ~ 3L1 second distance of the mixing
Inlet at a point in y0 distance from the bank
xvD2h
Mc
xy
)4
exp ( xDy
( y-y0 +2nB)2-vx
)4
+ exp ( xDy
( y+y0 -2nB)2-vx
sumn=infin
n=minusinfin(
)
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
Evaluation of dispersion coefficients measuring
Measuring with tracing matter (eg paint)
Inverse calculation from the measured concentration
2D Cross dispersion coefficient (Fischer)
Dy = dy u R (m2s)
dy - dimensionless cross dispersion coefficient
Straight regular channel dy 015
Moderately winding channel dy 02 ndash 06
Hard winding irregular channel dy gt 06 (1-2)
R ndash hydraulic radius (m)
u - shearing velocity at the channel bottom (ms)
u = (g R I)05Longitudinal dispersion coefficient dx 6 (2D) dxx = 100 ndash 1000 (1D)
Velocity space turbulent pulsation unequal distribution
Estimation of dispersion coefficients empirical forms
TRANSPORT EQUATION FOR NON-CONSERVATIVE TRANSPORT EQUATION FOR NON-CONSERVATIVE MATTERSMATTERS
bull There are sources and losses in the flow space
bull Physical chemical biochemical transformations take place
bull Non-conservative pollutants reaction kinetics ( R(C) )
bull They are taken into account in a linear way
dCdt = -C where is the coefficient of reaction kinetics (usually first-order kinetics)
cαxc
DAxx
c)v(Atc)(A
xx
1D equation in this case
ANALYTICAL SOLUTIONS OF THE TRANSPORT ANALYTICAL SOLUTIONS OF THE TRANSPORT EQUATIONEQUATION
Permanent mixing of the pollutants
Polluntant-wave transmission
Estimation of the geometric and hydraulic parameters
Exacter calculations based on measurments using numerical methods
The analitical solutions can be obtained in simpler cases only (approximate calculations)
Main steps
2
2
y
cD
x
cv yx
PERMANENT MIXING
The pollutant emission is steady in time
Permanent river discharge (low flow condition)
Constant flow velocity flow depth and dispersion coef
2D equation negligible vertical changes (shallow river)
Convection transferring
Dispersion spreading
Initial condition M0 (x0 y0) - emission
Boundary cond cy = 0 at the bank
)()()(
cvhy
cvhxt
chyx )()(
y
cDh
yx
cDh
xyx
x
yy
v
xD2
INLET AT THE MAIN FLOW PATH
Longitudinal according to x-frac12 function
Cross direction Gauss-distribution
cmax
M [kgs]cmax
)4
exp(2
2
xD
yv
xvDh
Mc (x t)y
x
xy
x
yb
v
xDB
234
Bb at value 01 cmax 1522bBBrand width
B ~ Bb2
1 0270 BD
vL
y
x
First distance of the mixing
M
1L
Bb
C (x1 y)
x1
M
x
yb
v
xDB
2152
21 110 B
D
vL
y
x
INLET AT THE RIVER BANK
)4
exp(2
xD
yv
xvDh
Mc
y
x
xy
cmax
C (x1 y)
x1
M
INLET NEAR THE RIVER BANK
)4
(exp ( xDy
( y-y0 )2-v
xvD2h
Mc x
xy
cmax
))4
+exp ( xDy
( y+y0 )2-vx
y0 C (x1 y)
x1
IMPACTS OF THE RIVER BANKS (total river section)
Boundary condition reflection theory (infinite series)
Complete mixing the change along the cross-section is less than 10
L2 ~ 3L1 second distance of the mixing
Inlet at a point in y0 distance from the bank
xvD2h
Mc
xy
)4
exp ( xDy
( y-y0 +2nB)2-vx
)4
+ exp ( xDy
( y+y0 -2nB)2-vx
sumn=infin
n=minusinfin(
)
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
2D Cross dispersion coefficient (Fischer)
Dy = dy u R (m2s)
dy - dimensionless cross dispersion coefficient
Straight regular channel dy 015
Moderately winding channel dy 02 ndash 06
Hard winding irregular channel dy gt 06 (1-2)
R ndash hydraulic radius (m)
u - shearing velocity at the channel bottom (ms)
u = (g R I)05Longitudinal dispersion coefficient dx 6 (2D) dxx = 100 ndash 1000 (1D)
Velocity space turbulent pulsation unequal distribution
Estimation of dispersion coefficients empirical forms
TRANSPORT EQUATION FOR NON-CONSERVATIVE TRANSPORT EQUATION FOR NON-CONSERVATIVE MATTERSMATTERS
bull There are sources and losses in the flow space
bull Physical chemical biochemical transformations take place
bull Non-conservative pollutants reaction kinetics ( R(C) )
bull They are taken into account in a linear way
dCdt = -C where is the coefficient of reaction kinetics (usually first-order kinetics)
cαxc
DAxx
c)v(Atc)(A
xx
1D equation in this case
ANALYTICAL SOLUTIONS OF THE TRANSPORT ANALYTICAL SOLUTIONS OF THE TRANSPORT EQUATIONEQUATION
Permanent mixing of the pollutants
Polluntant-wave transmission
Estimation of the geometric and hydraulic parameters
Exacter calculations based on measurments using numerical methods
The analitical solutions can be obtained in simpler cases only (approximate calculations)
Main steps
2
2
y
cD
x
cv yx
PERMANENT MIXING
The pollutant emission is steady in time
Permanent river discharge (low flow condition)
Constant flow velocity flow depth and dispersion coef
2D equation negligible vertical changes (shallow river)
Convection transferring
Dispersion spreading
Initial condition M0 (x0 y0) - emission
Boundary cond cy = 0 at the bank
)()()(
cvhy
cvhxt
chyx )()(
y
cDh
yx
cDh
xyx
x
yy
v
xD2
INLET AT THE MAIN FLOW PATH
Longitudinal according to x-frac12 function
Cross direction Gauss-distribution
cmax
M [kgs]cmax
)4
exp(2
2
xD
yv
xvDh
Mc (x t)y
x
xy
x
yb
v
xDB
234
Bb at value 01 cmax 1522bBBrand width
B ~ Bb2
1 0270 BD
vL
y
x
First distance of the mixing
M
1L
Bb
C (x1 y)
x1
M
x
yb
v
xDB
2152
21 110 B
D
vL
y
x
INLET AT THE RIVER BANK
)4
exp(2
xD
yv
xvDh
Mc
y
x
xy
cmax
C (x1 y)
x1
M
INLET NEAR THE RIVER BANK
)4
(exp ( xDy
( y-y0 )2-v
xvD2h
Mc x
xy
cmax
))4
+exp ( xDy
( y+y0 )2-vx
y0 C (x1 y)
x1
IMPACTS OF THE RIVER BANKS (total river section)
Boundary condition reflection theory (infinite series)
Complete mixing the change along the cross-section is less than 10
L2 ~ 3L1 second distance of the mixing
Inlet at a point in y0 distance from the bank
xvD2h
Mc
xy
)4
exp ( xDy
( y-y0 +2nB)2-vx
)4
+ exp ( xDy
( y+y0 -2nB)2-vx
sumn=infin
n=minusinfin(
)
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
TRANSPORT EQUATION FOR NON-CONSERVATIVE TRANSPORT EQUATION FOR NON-CONSERVATIVE MATTERSMATTERS
bull There are sources and losses in the flow space
bull Physical chemical biochemical transformations take place
bull Non-conservative pollutants reaction kinetics ( R(C) )
bull They are taken into account in a linear way
dCdt = -C where is the coefficient of reaction kinetics (usually first-order kinetics)
cαxc
DAxx
c)v(Atc)(A
xx
1D equation in this case
ANALYTICAL SOLUTIONS OF THE TRANSPORT ANALYTICAL SOLUTIONS OF THE TRANSPORT EQUATIONEQUATION
Permanent mixing of the pollutants
Polluntant-wave transmission
Estimation of the geometric and hydraulic parameters
Exacter calculations based on measurments using numerical methods
The analitical solutions can be obtained in simpler cases only (approximate calculations)
Main steps
2
2
y
cD
x
cv yx
PERMANENT MIXING
The pollutant emission is steady in time
Permanent river discharge (low flow condition)
Constant flow velocity flow depth and dispersion coef
2D equation negligible vertical changes (shallow river)
Convection transferring
Dispersion spreading
Initial condition M0 (x0 y0) - emission
Boundary cond cy = 0 at the bank
)()()(
cvhy
cvhxt
chyx )()(
y
cDh
yx
cDh
xyx
x
yy
v
xD2
INLET AT THE MAIN FLOW PATH
Longitudinal according to x-frac12 function
Cross direction Gauss-distribution
cmax
M [kgs]cmax
)4
exp(2
2
xD
yv
xvDh
Mc (x t)y
x
xy
x
yb
v
xDB
234
Bb at value 01 cmax 1522bBBrand width
B ~ Bb2
1 0270 BD
vL
y
x
First distance of the mixing
M
1L
Bb
C (x1 y)
x1
M
x
yb
v
xDB
2152
21 110 B
D
vL
y
x
INLET AT THE RIVER BANK
)4
exp(2
xD
yv
xvDh
Mc
y
x
xy
cmax
C (x1 y)
x1
M
INLET NEAR THE RIVER BANK
)4
(exp ( xDy
( y-y0 )2-v
xvD2h
Mc x
xy
cmax
))4
+exp ( xDy
( y+y0 )2-vx
y0 C (x1 y)
x1
IMPACTS OF THE RIVER BANKS (total river section)
Boundary condition reflection theory (infinite series)
Complete mixing the change along the cross-section is less than 10
L2 ~ 3L1 second distance of the mixing
Inlet at a point in y0 distance from the bank
xvD2h
Mc
xy
)4
exp ( xDy
( y-y0 +2nB)2-vx
)4
+ exp ( xDy
( y+y0 -2nB)2-vx
sumn=infin
n=minusinfin(
)
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
ANALYTICAL SOLUTIONS OF THE TRANSPORT ANALYTICAL SOLUTIONS OF THE TRANSPORT EQUATIONEQUATION
Permanent mixing of the pollutants
Polluntant-wave transmission
Estimation of the geometric and hydraulic parameters
Exacter calculations based on measurments using numerical methods
The analitical solutions can be obtained in simpler cases only (approximate calculations)
Main steps
2
2
y
cD
x
cv yx
PERMANENT MIXING
The pollutant emission is steady in time
Permanent river discharge (low flow condition)
Constant flow velocity flow depth and dispersion coef
2D equation negligible vertical changes (shallow river)
Convection transferring
Dispersion spreading
Initial condition M0 (x0 y0) - emission
Boundary cond cy = 0 at the bank
)()()(
cvhy
cvhxt
chyx )()(
y
cDh
yx
cDh
xyx
x
yy
v
xD2
INLET AT THE MAIN FLOW PATH
Longitudinal according to x-frac12 function
Cross direction Gauss-distribution
cmax
M [kgs]cmax
)4
exp(2
2
xD
yv
xvDh
Mc (x t)y
x
xy
x
yb
v
xDB
234
Bb at value 01 cmax 1522bBBrand width
B ~ Bb2
1 0270 BD
vL
y
x
First distance of the mixing
M
1L
Bb
C (x1 y)
x1
M
x
yb
v
xDB
2152
21 110 B
D
vL
y
x
INLET AT THE RIVER BANK
)4
exp(2
xD
yv
xvDh
Mc
y
x
xy
cmax
C (x1 y)
x1
M
INLET NEAR THE RIVER BANK
)4
(exp ( xDy
( y-y0 )2-v
xvD2h
Mc x
xy
cmax
))4
+exp ( xDy
( y+y0 )2-vx
y0 C (x1 y)
x1
IMPACTS OF THE RIVER BANKS (total river section)
Boundary condition reflection theory (infinite series)
Complete mixing the change along the cross-section is less than 10
L2 ~ 3L1 second distance of the mixing
Inlet at a point in y0 distance from the bank
xvD2h
Mc
xy
)4
exp ( xDy
( y-y0 +2nB)2-vx
)4
+ exp ( xDy
( y+y0 -2nB)2-vx
sumn=infin
n=minusinfin(
)
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
2
2
y
cD
x
cv yx
PERMANENT MIXING
The pollutant emission is steady in time
Permanent river discharge (low flow condition)
Constant flow velocity flow depth and dispersion coef
2D equation negligible vertical changes (shallow river)
Convection transferring
Dispersion spreading
Initial condition M0 (x0 y0) - emission
Boundary cond cy = 0 at the bank
)()()(
cvhy
cvhxt
chyx )()(
y
cDh
yx
cDh
xyx
x
yy
v
xD2
INLET AT THE MAIN FLOW PATH
Longitudinal according to x-frac12 function
Cross direction Gauss-distribution
cmax
M [kgs]cmax
)4
exp(2
2
xD
yv
xvDh
Mc (x t)y
x
xy
x
yb
v
xDB
234
Bb at value 01 cmax 1522bBBrand width
B ~ Bb2
1 0270 BD
vL
y
x
First distance of the mixing
M
1L
Bb
C (x1 y)
x1
M
x
yb
v
xDB
2152
21 110 B
D
vL
y
x
INLET AT THE RIVER BANK
)4
exp(2
xD
yv
xvDh
Mc
y
x
xy
cmax
C (x1 y)
x1
M
INLET NEAR THE RIVER BANK
)4
(exp ( xDy
( y-y0 )2-v
xvD2h
Mc x
xy
cmax
))4
+exp ( xDy
( y+y0 )2-vx
y0 C (x1 y)
x1
IMPACTS OF THE RIVER BANKS (total river section)
Boundary condition reflection theory (infinite series)
Complete mixing the change along the cross-section is less than 10
L2 ~ 3L1 second distance of the mixing
Inlet at a point in y0 distance from the bank
xvD2h
Mc
xy
)4
exp ( xDy
( y-y0 +2nB)2-vx
)4
+ exp ( xDy
( y+y0 -2nB)2-vx
sumn=infin
n=minusinfin(
)
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
x
yy
v
xD2
INLET AT THE MAIN FLOW PATH
Longitudinal according to x-frac12 function
Cross direction Gauss-distribution
cmax
M [kgs]cmax
)4
exp(2
2
xD
yv
xvDh
Mc (x t)y
x
xy
x
yb
v
xDB
234
Bb at value 01 cmax 1522bBBrand width
B ~ Bb2
1 0270 BD
vL
y
x
First distance of the mixing
M
1L
Bb
C (x1 y)
x1
M
x
yb
v
xDB
2152
21 110 B
D
vL
y
x
INLET AT THE RIVER BANK
)4
exp(2
xD
yv
xvDh
Mc
y
x
xy
cmax
C (x1 y)
x1
M
INLET NEAR THE RIVER BANK
)4
(exp ( xDy
( y-y0 )2-v
xvD2h
Mc x
xy
cmax
))4
+exp ( xDy
( y+y0 )2-vx
y0 C (x1 y)
x1
IMPACTS OF THE RIVER BANKS (total river section)
Boundary condition reflection theory (infinite series)
Complete mixing the change along the cross-section is less than 10
L2 ~ 3L1 second distance of the mixing
Inlet at a point in y0 distance from the bank
xvD2h
Mc
xy
)4
exp ( xDy
( y-y0 +2nB)2-vx
)4
+ exp ( xDy
( y+y0 -2nB)2-vx
sumn=infin
n=minusinfin(
)
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
x
yb
v
xDB
234
Bb at value 01 cmax 1522bBBrand width
B ~ Bb2
1 0270 BD
vL
y
x
First distance of the mixing
M
1L
Bb
C (x1 y)
x1
M
x
yb
v
xDB
2152
21 110 B
D
vL
y
x
INLET AT THE RIVER BANK
)4
exp(2
xD
yv
xvDh
Mc
y
x
xy
cmax
C (x1 y)
x1
M
INLET NEAR THE RIVER BANK
)4
(exp ( xDy
( y-y0 )2-v
xvD2h
Mc x
xy
cmax
))4
+exp ( xDy
( y+y0 )2-vx
y0 C (x1 y)
x1
IMPACTS OF THE RIVER BANKS (total river section)
Boundary condition reflection theory (infinite series)
Complete mixing the change along the cross-section is less than 10
L2 ~ 3L1 second distance of the mixing
Inlet at a point in y0 distance from the bank
xvD2h
Mc
xy
)4
exp ( xDy
( y-y0 +2nB)2-vx
)4
+ exp ( xDy
( y+y0 -2nB)2-vx
sumn=infin
n=minusinfin(
)
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
M
x
yb
v
xDB
2152
21 110 B
D
vL
y
x
INLET AT THE RIVER BANK
)4
exp(2
xD
yv
xvDh
Mc
y
x
xy
cmax
C (x1 y)
x1
M
INLET NEAR THE RIVER BANK
)4
(exp ( xDy
( y-y0 )2-v
xvD2h
Mc x
xy
cmax
))4
+exp ( xDy
( y+y0 )2-vx
y0 C (x1 y)
x1
IMPACTS OF THE RIVER BANKS (total river section)
Boundary condition reflection theory (infinite series)
Complete mixing the change along the cross-section is less than 10
L2 ~ 3L1 second distance of the mixing
Inlet at a point in y0 distance from the bank
xvD2h
Mc
xy
)4
exp ( xDy
( y-y0 +2nB)2-vx
)4
+ exp ( xDy
( y+y0 -2nB)2-vx
sumn=infin
n=minusinfin(
)
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
M
INLET NEAR THE RIVER BANK
)4
(exp ( xDy
( y-y0 )2-v
xvD2h
Mc x
xy
cmax
))4
+exp ( xDy
( y+y0 )2-vx
y0 C (x1 y)
x1
IMPACTS OF THE RIVER BANKS (total river section)
Boundary condition reflection theory (infinite series)
Complete mixing the change along the cross-section is less than 10
L2 ~ 3L1 second distance of the mixing
Inlet at a point in y0 distance from the bank
xvD2h
Mc
xy
)4
exp ( xDy
( y-y0 +2nB)2-vx
)4
+ exp ( xDy
( y+y0 -2nB)2-vx
sumn=infin
n=minusinfin(
)
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
IMPACTS OF THE RIVER BANKS (total river section)
Boundary condition reflection theory (infinite series)
Complete mixing the change along the cross-section is less than 10
L2 ~ 3L1 second distance of the mixing
Inlet at a point in y0 distance from the bank
xvD2h
Mc
xy
)4
exp ( xDy
( y-y0 +2nB)2-vx
)4
+ exp ( xDy
( y+y0 -2nB)2-vx
sumn=infin
n=minusinfin(
)
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
M2
More inlets or diffuser-row theory of the superposition
Separated calculations for each inlet point and addition
M1 C = C1 + C2
C2
C1
SUPERPOSITION
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
)( cvxt
cx )()(
y
cD
yx
cD
xyx
TRANSMISSION OF NON-PERMANENT EMISSIONS
Suddenly jerky pollutant-waves
In time intensively changing emissions
2D Equation
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
febr03
febr04
febr05
febr06
febr07
febr08
febr09
febr10
febr11
febr12
febr13
0
1
2
3
4
5
6
Balsa (szaacutemiacutetott) Balsa (meacutert) Kiskoumlre (szaacutemiacutetott) Kiskoumlre (meacutert)
Csongraacuted (szaacutemiacutetott) Csongraacuted (meacutert) Taacutepeacute (szaacutemiacutetott) Taacutepeacute (meacutert)
CIANID(mgl)
DISPERSION-WAVE
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
)44
)(exp(
4
22
tD
y
tD
tvx
DDht
Gc
yx
x
xy
tDxx 2 tDyy 2
xcL 34 ycB 34
G [kg]
c2B
c2L
x2=vt2
cmax
x1=vt1
C (t2 x 0)
C (t2 x2 y)
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
C
xv
t
Cx 2
2
x
CDx
2
)4
)(exp(
2 tD
tvx
tDA
GC
x
x
x
1D equation ndash narrow and shallow rivers
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
2 tDA
GCmax
x At the fixed time moment (xvx)
x
C C (t1x) C (t2x)
xcL 34 tDxx 2
x1 = vx t1 x2 = vx t2
Lc1 Lc2
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-
)))1((4
)))1(((exp(
))1(((2
2
121 titD
titvx
titDA
tMC
x
xn
i x
i
TIME-CHANGING EMISSIONS
][ skgM i
tti=1 i=n
Dividing into discrete units with constant values
Superposition
Gi ~ Mi Δt t - (i-1) Δt ge 0
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Slide 52
-