transport processes – part 1 - startsidausers.abo.fi/rzevenho/trp-slides-1-2018.pdf · transport...
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Transport processes – Part 1
Ron ZevenhovenÅbo Akademi University
Thermal and Flow Engineering / Värme- och strömningstekniktel. 3223 ; [email protected]
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Introduction / re-wrap of concepts
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Physical transport phenomena /1
• Transfer of mass and/or energy in a system that is not in thermodynamic equilibrium, towards such equilibrium.
• Systems are usually not very far away from equilbrium, which results in (practically) lineardriving forces:transport = coefficient × driving force– heat flux (W/m2)=
conductivity (W/m2.K)×temperature gradient (K/m)
et cetera.
Theat "
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Physical transport phenomena /2
• Continuum approach:
a small volume dV where system propertiesare constant– For example dx = 0.1 µm dV = 10-21 m3 still
contains in liquid water ~106 molecules
• Not considered here: cross-correlations suchas – Mass transfer = coefficient × temperature gradient
(“thermal diffusion”)
– For example Seebeck effect, Peltier effect
– See also so-called ”irreversible thermodynamics”
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Fourier’s Law /1• In a non-moving medium (i.e. a solid, or
stagnant fluid) in the presence of a
temperature gradient, heat is transferred from
high to low temperature as a result of
molecular movement: heat conduction(sv: värmeledning)
• For a one-dimensional temperature gradient ΔT/Δx or dT/dx, Fourier’s Law gives the conductive heat transfer rate Q through a cross-sectional area A (m2). If λ is a constant:
with thermal conductivity λ, unit: W/(mK)
(sv: termisk konduktivitet eller värmeledningsförmåga)
)(W/m (W) 2
dx
dT
A
Q"Q
dx
dTAQ
Pictures: T06
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Fourier’s Law /2• For a general case with a 3-dimensional temperature
gradient T = (∂T/∂x,∂T/∂y,∂T/∂z), Fourier’s Law gives (for constant λ) for the heat flux Q” = - λ T
• The temperature field inside the conducting medium can be written as T = T(t, x) with time t and 3-dimensional location vector x
• For stationary (sv: stationärt, tidsinvariant)
heat transfer ∂T/∂t = 0 at each position x
• The heat transfer vector is perpendicular (sv: vinkelrätt) to the isothermal surfaces
• Note that material property λ is, in fact, a function of temperature:
more accurately Q” = - λ(T)T
∆∆.
Figure: KJ05
∆ Q is a vectorwith direction - T
∆..
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Non-steady heat conduction
where in principle heat Q is a 3-dimensional vector Q that creates (or is the result of !) a vector temperature gradient:
(in Cartesian coordinates)
• Non-steady or transient (sv: övergående) heat conduction through a stagnant medium depends not only on heat conductivity λ but also on heat capacity c (or cp, cv). A general energy balance for mass m gives
t
TcmQQ outin
z
T,
y
T,
x
TT Picture: ÖS96
.
.
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Transient heat conduction 1-D /1
• For 1-dimensional transient heat conduction in a balance volume dV with mass dm = ρ·dV = ρ·A·dx :
2
2
2
2
x
Ta
t
T
x
TA
x
xT
A
t
TAc
x
TA
x
Q
t
TcA
dxx
Q
t
TcdmQQ outin
-
-Q Laws Fourier' with
Aρdm/dx with
x
w
L
dx
Q.
A = L·w
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Transient heat conduction 1-D /2
• The initial and boundary conditions(sv: start- och randvillkor) determine a heat transfer process
• The three most important cases are:
0 t for t))T(0,- (Th x
t)T(0,- and0 t for T T(x,0)
:0t at h0 convection surface of change Sudden 3.
0t for Q x
t)T(0,- and0 t for T T(x,0)
:0t at Q0 flux heat surface of change Sudden 2.
0t for Tt)T(0, and0 t for T T(x,0)
:0t at T T etemperatur surface of change Sudden 1.
surr0
"0x0
"0x
10
10
x
w
L
dx
Q.
A = L·w
x
w
L
dx
Q.
Q.
A = L·w
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Transient heat conduction 1-D /3
• Case 1: Assume a material with flat boundary at x=0, infinite length in x-direction, with T=T0 at all x
• At time t≥0 the temperature at x=0 is increased to T=T1 and heat starts to enter (diffuse into) the material. At x→∞, T stays at T0.
conditions initial
and boundary
2
2
x
Ta
t
T
Picture: BMH99
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Transient heat conduction 1-D /4
• With dimensionless variables
θ = (T-T0)/(T1-T0)
and
ξ = x / (4at)½
this gives the following solution:
)(yerfde
deTT
TT
y
at
x
0
4
001
1
2
2
2
21
with
ÖS96: erf(x) ≈ 1 - exp(- 1.128x - 0.655x2 - 0.063x3)
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Transient heat conduction 1-D /5
• At x = 0 the slope of the penetration profile lines equals
∂T/∂x = -(T1-T0)/(πat)½
where x = (πat)½
is referred to as penetration depth.
• Fourier number Fo is (for heat transfer) defined as
Fo = at/d2 = t /(d2/a)) for a medium with thickness d • Fo gives the ratio between time t and the penetration time d2/a
• The penetration depth concept is valid for Fo < 0.1 Picture: BMH99
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Diffusion and heat conduction
• Heat conduction is in principle diffusion of heat
• Since a ”temperature balance” does not exist, an energy balance must be used: T → ρcpT (unit: J/m3)
Fick’s Law Fourier’s Law
p
pp
px,heat cρ
λa
dx
Tcρda
dx
Tcρd
cρ
λ"Φ ydiffusivit thermal with --
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Internal friction in fluid flow /1
• Diffusion of momentum subscript ”xy” means in y-direction in
plane of fixed x
• Kinematic viscosity = dynamic viscosity/density, ν = η/ρ
xyyyy
xymomentum dx
vd-
dx
vd-
dx
dv-
,"
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Internal friction in fluid flow /2
• Concentration, c, temperature, T, and energy, E, are scalars, and their gradient is a vector dc/dx or c = (∂c/ ∂x, ∂c/ ∂y, ∂c/ ∂z), etc.
• Velocity is a vector v, for example v = (vx, vy, vz) and it’s gradient is a (second order) tensor: dvx/dy (gradient of vx in y-direction)
z
v
z
v
z
vy
v
y
v
y
vx
v
x
v
x
v
v
zyx
zyx
zyx
)(.z
v
y
v
x
vv
:note
zyx
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Internal friction in fluid flow /3
• v results in compressive stresses xx, yy and zz and shear stresses xy, xz, yz, zx, yx and zy:
etc. dy
vd
dy
dv
dy
vd
dy
dv zzyz
xxyx ;;
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The course book;Chapters 1 – 6
are used for thiscourse
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Note here: W > 0 if work is doneBY the system.
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Note: mass = density · volumem = ρ·V dm = ρ·dV + dρ·V
thus: dm = 0 ≠ dV =0
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Left: Cartesian (x,y,z)
Centre: Cylindrical (r,θ,z)
with r2 = x2+y2
Right: Spherical (r,φ,θ)
with r2 = x2+y2+z2
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Sources used(besides course book Hanjalić et al.)
• Beek, W.J., Muttzall, K.M.K., van Heuven, J.W. ”Transport phenomena” Wiley, 2nd edition (1999)
• R.B. Bird, W.E. Stewart, E.N. Lightfoot ”Transport phenomena” Wiley, New York (1960)
• * C.J. Hoogendoorn ”Fysische Transportverschijnselen II”, TU Delft / D.U.M., the Netherlands 2nd. ed. (1985)
• * C.J. Hoogendoorn, T.H. van der Meer ”Fysische Transport-verschijnselen II”, TU Delft /VSSD, the Netherlands 3nd. ed. (1991)
• D. Kaminski, M. Jensen ”Introduction to Thermal and Fluids Engineering”, Wiley (2005)
• S.R. Turns ”Thermal – Fluid Sciences”, Cambridge Univ. Press (2006)• R. Zevenhoven ”Principles of process engineering” (Processteknikens
grunder), course material ÅA (compendium Aug. 2013, 214 pp.):http://users.abo.fi/rzevenho/PTG%20Aug2013.pdf
• R. Zevenhoven ”Massöverföring & separationsteknik” (2016), ”Processteknik” (2017) course material ÅA
* Earlier versions of Hanjalić et al. book but in Dutch
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