convective mass flows iii
DESCRIPTION
In this lecture, we shall concern ourselves once more with convective mass and heat flows, as we still have not gained a comprehensive understanding of the physics behind such phenomena. We shall start by looking once more at the capacitive field . - PowerPoint PPT PresentationTRANSCRIPT
Start Presentation
Mathematical Modeling of Physical Systems
© Prof. Dr. François E. CellierNovember 22, 2012
Convective Mass Flows III
• In this lecture, we shall concern ourselves once more with convective mass and heat flows, as we still have not gained a comprehensive understanding of the physics behind such phenomena.
• We shall start by looking once more at the capacitive field.
• We shall then study the internal energy of matter.
• Finally, we shall look at general energy transport phenomena, which by now include mass flows as an integral aspect of general energy flows.
Start Presentation
Mathematical Modeling of Physical Systems
© Prof. Dr. François E. CellierNovember 22, 2012
Table of Contents
• Capacitive Fields• Internal energy of matter• Bus-bonds and bus-junctions• Heat conduction• Volume work• General mass transport• Multi-phase systems• Evaporation and condensation• Thermodynamics of mixtures• Multi-element systems
Start Presentation
Mathematical Modeling of Physical Systems
© Prof. Dr. François E. CellierNovember 22, 2012
Capacitive Fields III
• Let us briefly consider the following electrical circuit:
C1
C2
C3
i1 i2i3
i1-i3 i2+i3u1 u2
i1 – i3 = C1 · du1 /dt
i2 + i3 = C3 · du2 /dt
i3 = C2 · (du1 /dt – du2 /dt )
i1 = ( C1 + C2 ) · du1 /dt – C2 · du2 /dt
i2 = – C2 · du1 /dt + ( C2 + C3 ) · du2 /dt
0 0 1 0 0
C1 C2 C3
i1
i2
i3
i3 i3
i1-i3i2+ i3
u1 u1
u1
u2
u2
u2u1-u2
Start Presentation
Mathematical Modeling of Physical Systems
© Prof. Dr. François E. CellierNovember 22, 2012
Capacitive Fields IVi1 = ( C1 + C2 ) · du1 /dt – C2 · du2 /dt
i2 = – C2 · du1 /dt + ( C2 + C3 ) · du2 /dt
i1
i2
=( C1 + C2 ) – C2
– C2 ( C2 + C3 )
·du1 /dt
du2 /dt
i1
i2
=
( C2 + C3 ) C2
C2 ( C1 + C2 ) ·du1 /dt
du2 /dt C1 C2 + C1 C3 + C2 C3
Symmetric capacity matrix
0 0 1 0 0
C1 C2 C3
i1
i2
i3
i3 i3
i1-i3i2+ i3
u1 u1
u1
u2
u2
u2u1-u2 0 CF
i1
u1 0i2
u2
Start Presentation
Mathematical Modeling of Physical Systems
© Prof. Dr. François E. CellierNovember 22, 2012
Volume and Entropy Storage• Let us consider once more the situation discussed in the
previous lecture.
0 1 0
C
I
CCth
0 Sf 0
CthS/V
It was no accident that I drew the two capacitors so close to each other. In reality, the two capacitors together form a two-port capacitive field. After all, heat and volume are only two different properties of one and the same material.
Start Presentation
Mathematical Modeling of Physical Systems
© Prof. Dr. François E. CellierNovember 22, 2012
The Internal Energy of Matter I
• As we have already seen, there are three different (though inseparable) storages of matter:
• These three storage elements represent different storage properties of one and the same material.
• Consequently, we are dealing with a storage field.
• This storage field is of a capacitive nature.
• The capacitive field stores the internal energy of matter.
Mass Volume Heat
Start Presentation
Mathematical Modeling of Physical Systems
© Prof. Dr. François E. CellierNovember 22, 2012
The Internal Energy of Matter II
• Change of the internal energy in a system, i.e. the total power flow into or out of the capacitive field, can be described as follows :
• This is the Gibbs equation.
U = T · S - p · V + i · Nii
· · · ·
Heat flow Mass flow
Volume flow
Flow of internal energy
Chemical potential
Molar mass flow
Start Presentation
Mathematical Modeling of Physical Systems
© Prof. Dr. François E. CellierNovember 22, 2012
The Internal Energy of Matter III
• The internal energy is proportional to the the total mass n.
• By normalizing with n, all extensive variables can be made intensive.
• Therefore:
u = Un s = S
n v = Vn ni =
Nin
id
dt(n·u) = T · d
dt (n·s) - p · ddt
(n·v) + i · (n· ni )d
dt
id
dt(n·u) - T · d
dt (n·s) + p · ddt
(n·v) - i · (n· ni ) = 0ddt
Start Presentation
Mathematical Modeling of Physical Systems
© Prof. Dr. François E. CellierNovember 22, 2012
The Internal Energy of Matter IV
id
dt(n·u) - T · d
dt (n·s) + p · ddt
(n·v) - i · (n· ni ) = 0ddt
i
dudt - T · + p · - i · n ·[ ds
dtdvdt
dni
dt ]
= 0+dndt ·[u - T · s + p · v - i ·
ni
i ]
This equation must be valid independently of the amount n, therefore:
= 0u - T · s + p · v - i · ni
i
i
dudt - T · + p · - i ·
dsdt
dvdt
dni
dt = 0Flow of internal energy
Internal energy
Finally, here is an explanation, why it was okay to compute with funny derivatives.
Start Presentation
Mathematical Modeling of Physical Systems
© Prof. Dr. François E. CellierNovember 22, 2012
The Internal Energy of Matter V
U = T · S - p · V + i ·Nii
U = T · S - p · V + i · Ni + T · S - p · V + i · Ni i
· · · ·i
· · ·
= T · S - p · V + i · Nii· · ·
T · S - p · V + i · Ni = 0· · ·
This is the Gibbs-Duhem equation.
Start Presentation
Mathematical Modeling of Physical Systems
© Prof. Dr. François E. CellierNovember 22, 2012
The Capacitive Field of Matter
C C
C
GY
GY GY
TS·
pq
i i
Vp·
p·VT·
T·
S
S
i·
i· nini
CF
Start Presentation
Mathematical Modeling of Physical Systems
© Prof. Dr. François E. CellierNovember 22, 2012
Simplifications• In the case that no chemical reactions take place, it is
possible to replace the molar mass flows by conventional mass flows.
• In this case, the chemical potential is replaced by the Gibbs potential.
MgVpSTdt
dU
Start Presentation
Mathematical Modeling of Physical Systems
© Prof. Dr. François E. CellierNovember 22, 2012
Bus-Bond and Bus-0-Junction• The three outer legs of the CF-element can be grouped
together.
pq
TS
g
M
.
.
0
0 0
CFCF
C C
C
3Ø CF
Start Presentation
Mathematical Modeling of Physical Systems
© Prof. Dr. François E. CellierNovember 22, 2012
Once Again Heat Conduction
CFCF CFCF
11
T
2T
S.
T
1
1
1 S.
1
S.
1
2
0 mGSmGS
2T
Ø Ø
TT S
.1
2
S.
12
S.
1x S.
2xT1
3 3
CFCF1CFCF2
3 3HE
Start Presentation
Mathematical Modeling of Physical Systems
© Prof. Dr. François E. CellierNovember 22, 2012
Volume Pressure Exchange
pq q11
1 p2
0 GSGS
2T1T
Ø Ø33
CFCF1 CFCF2
pq
p p
2q
2q
S1x
. S2x
.
CFCF1CFCF2
3 3PVE
Pressure is being equilibrated just like temperature. It is assumed that the inertia of the mass may be neglected (relatively small masses and/or velocities), and that the equilibration occurs without friction.
The model makes sense if the exchange occurs locally, and if not too large masses get moved in the process.
Start Presentation
Mathematical Modeling of Physical Systems
© Prof. Dr. François E. CellierNovember 22, 2012
General Exchange Element I
pq
11
GS
0
1 pq
211
GS
11 11
0
mGS
0
mGS
TS. 1 T
S. 2
1 2
g
M. 1 g
M. 2
SwSw
The three flows are coupled through RS-elements.
This is a switching element used to encode the direction of positive flow.
Start Presentation
Mathematical Modeling of Physical Systems
© Prof. Dr. François E. CellierNovember 22, 2012
General Exchange Element II
• In the general exchange element, the temperatures, the pressures, and the Gibbs potentials of neighboring media are being equilibrated.
• This process can be interpreted as a resistive field.
Ø Ø3RF
3
33
CFCF1 CFCF2
, S1 1 , S2 2
Start Presentation
Mathematical Modeling of Physical Systems
© Prof. Dr. François E. CellierNovember 22, 2012
Multi-phase Systems
• We may also wish to study phenomena such as evaporation and condensation.
CFCFgas
3
ØHE, PVE,
Evaporation,Condensation
3
CFCFliq
3
Ø3
Start Presentation
Mathematical Modeling of Physical Systems
© Prof. Dr. François E. CellierNovember 22, 2012
Evaporation (Boiling)• Mass and energy exchange between capacitive storages of
matter (CF-elements) representing different phases is accomplished by means of special resistive fields (RF-elements).
• The mass flows are calculated as functions of the pressure and the corresponding saturation pressure.
• The volume flows are computed as the product of the mass flows with the saturation volume at the given temperature.
• The entropy flows are superposed with the enthalpy of evaporation (in the process of evaporation, the thermal domain loses heat latent heat).
Start Presentation
Mathematical Modeling of Physical Systems
© Prof. Dr. François E. CellierNovember 22, 2012
Condensation On Cold Surfaces• Here, a boundary layer must be introduced.
CFCFliq
CFCFgas
CFCF surface
3
3
Ø
Ø
Rand-schicht
3
3
Heat conduction (HE)Volume work (PVE)
Condensation and Evaporation
3 3HEPVERF
3 3HEPVERF
CFCFliq
CFCF gas
3
3
Ø
Ø
3
3TS
.
Heat conduction (HE)Volume work (PVE)
Condensation and Evaporation
HE
gas
s
T S
HE
liq
s
.
Boundary layer
Ø 3
Start Presentation
Mathematical Modeling of Physical Systems
© Prof. Dr. François E. CellierNovember 22, 2012
Thermodynamics of Mixtures• When fluids (gases or liquids) are being mixed, additional
entropy is generated.
• This mixing entropy must be distributed among the participating component fluids.
• The distribution is a function of the partial masses.
• Usually, neighboring CF-elements are not supposed to know anything about each other. In the process of mixing, this rule cannot be maintained. The necessary information is being exchanged.
CF1 CF2MIMI{M1}
{x1
{M2}
{x2
Start Presentation
Mathematical Modeling of Physical Systems
© Prof. Dr. François E. CellierNovember 22, 2012
Entropy of Mixing• The mixing entropy is taken out of the Gibbs potential.
TS.
pq 11
1
1
11
g1(T,p)11M
.1
TS.
pq
1
1
g1 (T,p)
M.
1
mix
TRSM.
1
g1
Sid
mix
1
CFCF11 CFCF12
TS.
pq 11
2
2
11
g2(T,p)11M
.2
TS.
pq
2
2
g2 (T,p)
M.
2
mix
TRSM.
2
g2
Sid
mix
2
CFCF21 CFCF22
MIMI
x21
x11
M21
M11
HEPVE.
.
It was assumed here that the fluids to be mixed are at the same temperature and pressure.
Start Presentation
Mathematical Modeling of Physical Systems
© Prof. Dr. François E. CellierNovember 22, 2012
T2
S.
p2
q 112
2
11
g2(T2,p2) 11M.
2
T2mix
S.
p2mix
q
2
2
g2 (T2,p2)
M2
.mix
CFCF 21 CFCF 22
MIMI HEPVE
RS
M.
2g2
S2
.RS mRS
0
p2
T2
q2
S2
.
T2mix
T1
S.
p1
q 111
1
11
g1 (T1,p1) 11M.
1
T1mix
S.
p1mix
q
1
1g1 (T1,p1)
M1
.mix
CFCF 11
RS
M.
1
g1
S1
.RS mRS
0
p1
T1
q1
S1
.
T1mix
CFCF 12
It is also possible that the fluids to be mixed are initially at different temperature or pressure values.
Start Presentation
Mathematical Modeling of Physical Systems
© Prof. Dr. François E. CellierNovember 22, 2012
Convection in Multi-element Systems
CF12
CF13
CF11
Ø
3
3
3
3
3
Ø
Ø
3
PVEHE
3
3
HEPVE CF22
CF23
CF21
3
3
3
3
3
Ø
Ø
Ø
3
PVEHE
3
3PVEHE
3
HEPVE
3
PVEHE
3RF
PVEHE
3
3 3RFPVEHE
3 3RFPVEHE
horizontalexchange(transport)
verticalexchange(mixture)
Start Presentation
Mathematical Modeling of Physical Systems
© Prof. Dr. François E. CellierNovember 22, 2012
Two-element, Two-phase, Two-compartmentConvective System
Gas
CF11
Fl.
CF11
Fl.
CF21
Gas
CF21
Ø
3
3
3
3
3
Ø
Ø
Ø
3
PVEHE
3
HECondensation/Evaporation
PVE
3
3
3
3
HECondensation/Evaporation
PVE
Gas
CF12
Fl.
CF12
Fl.
CF22
Gas
CF22
Ø3
3
3
3
3
3
Ø
Ø
Ø
3
PVEHE
3
HECondensation/Evaporation
PVE
3
3
3PVEHE
3
HECondensation/Evaporation
PVE
3
PVEHE
3RF
PVEHE
3
3 3
3 3
HEPVERF
HEPVERF
3 3RF
PVEHE
phase-boundary
3
PVEHE
3
3
PVEHE
3
3
PVEHE
3
3
PVEHE
3
MIMI{x21, S
E
21, VE
21}
{M21, T21, p 21} MIMI1 2
+
Vge
s
+
Vge
s
{M11, T
11, p
11}
{x21,
SE
21,
VE
21}
{M12, T
12, p
12}
{x12,
SE
12,
VE
12}
{M22, T22, p 22}
{x22, SE
22, VE
22}
Start Presentation
Mathematical Modeling of Physical Systems
© Prof. Dr. François E. CellierNovember 22, 2012
Concentration Exchange
• It may happen that neighboring compartments are not completely homogeneous. In that case, also the concentrations must be exchanged.
CFCFii
3
Ø 3 3HEPVECE
3
Ø
CFCFi+1i+1
33... ...
Start Presentation
Mathematical Modeling of Physical Systems
© Prof. Dr. François E. CellierNovember 22, 2012
References I
• Cellier, F.E. (1991), Continuous System Modeling, Springer-Verlag, New York, Chapter 9.
• Greifeneder, J. and F.E. Cellier (2001), “Modeling convective flows using bond graphs,” Proc. ICBGM’01, Intl. Conference on Bond Graph Modeling and Simulation, Phoenix, Arizona, pp. 276 – 284.
• Greifeneder, J. and F.E. Cellier (2001), “Modeling multi-phase systems using bond graphs,” Proc. ICBGM’01, Intl. Conference on Bond Graph Modeling and Simulation, Phoenix, Arizona, pp. 285 – 291.
Start Presentation
Mathematical Modeling of Physical Systems
© Prof. Dr. François E. CellierNovember 22, 2012
References II
• Greifeneder, J. and F.E. Cellier (2001), “Modeling multi-element systems using bond graphs,” Proc. ESS’01, European Simulation Symposium, Marseille, France, pp. 758 – 766.
• Greifeneder, J. (2001), Modellierung thermodynamischer Phänomene mittels Bondgraphen, Diploma Project, Institut für Systemdynamik und Regelungstechnik, University of Stuttgart, Germany.