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
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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.
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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
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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
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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
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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.
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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
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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
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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
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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.
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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.
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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
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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
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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
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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
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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.
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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.
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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
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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
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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).
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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
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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
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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.
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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.
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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)
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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}
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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... ...
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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.
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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.