Transcript Convective Mass Flows III
M athematical M odeling of Physical S ystems
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.
November 22, 2012
© Prof. Dr. François E. Cellier
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Table of Contents
November 22, 2012
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Capacitive Fields III
• Let us briefly consider the following electrical circuit:
i 1 i 3
C 2
i 2 u 1
C 1
i 1 -i 3 i 2 +i 3
C 3
u 2
0
u 1 i 1
C 1 C 2
u 1
0
u 1 -u 2 i 1 -i 3 1
1
i 3 i 3
C 3
u 2 i 3 u 2
0
i 2 + i 3 i 2 u 2
0
i 1 i 2 i 3 – i 3 + i 3 = C 1 · du 1 /dt = C 3 · du 2 /dt = C 2 · (du 1 /dt – du 2 /dt )
November 22, 2012
i 1 i 2 = ( C 1 + C 2 ) · du 1 /dt – C 2 · du 2 /dt = – C 2 · du 1 /dt + ( C 2 + C 3 ) · du 2 /dt
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Capacitive Fields IV
i 1 i 2 = ( C 1 + C 2 ) · du 1 /dt – C 2 · du 2 /dt = – C 2 · du 1 /dt + ( C 2 + C 3 ) · du 2 /dt Symmetric capacity matrix
i 1 i 2 = ( C 1 + C 2 ) – C 2 – C 2 ( C 2 + C 3 ) · du 1 /dt du 2 /dt
du 1 /dt du 2 /dt = ( C 2 + C 3 ) C 2 C 2 ( C 1 + C 2 ) · C 1 C 2 + C 1 C 3 + C 2 C 3 i 1 i 2
0
u 1 i 1
C 1 C 2
u 1
0
u i 1 -i 3 1 1 -u 2
1
i 3 i 3 u 2 i 3
C 3
u 2
0
i 2 + i 3 i 2 u 2
0
0
u 1 i 1
CF
i 2 u 2
0
Start Presentation November 22, 2012
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Volume and Entropy Storage
• Let us consider once more the situation discussed in the previous lecture.
0 Sf 0 C th C 0
S/V
1 C th C 0
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.
I
Start Presentation November 22, 2012
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The Internal Energy of Matter I
• As we have already seen, there are three different (though inseparable) storages of matter: Mass Volume Heat • 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
.
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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 :
Chemical potential Flow of internal energy
· · · S
i
m
i · N
·
i Heat flow Mass flow Molar mass flow Volume flow
• This is the
Gibbs equation
.
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The Internal Energy of Matter III
• The internal energy is proportional to the the total mass
n
.
• By normalizing with intensive.
n
, all extensive variables can be made
n i = N i n
• Therefore:
d dt (n·u) = T · d dt (n·s) - p · d dt (n·v) +
S
i
m
i · (n· n dt i )
d dt (n·u) - T · d dt (n·s) + p · d dt (n·v) -
S
i
m
i · d dt (n· n i ) = 0
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M athematical M odeling of Physical S ystems
The Internal Energy of Matter IV
d dt (n·u) - T · d dt (n·s) + p · d dt (n·v) -
S
i
m
i · d dt (n· n i ) = 0
n · + [ du dt - T · ds dt + p · dv dt dn dt · [ u - T · s + p · v -
S
i -
S
i
m
i
m
i · dn i dt · n i ] ] = 0
This equation must be valid independently of the amount
n
, therefore:
Finally, here is an explanation, why it was okay to compute with funny derivatives.
du dt T · ds dt + p · dv dt -
S
i
m
i · dn i dt = 0 u - T · s + p · v -
S
i
m
i · n i = 0 Flow of internal energy Internal energy
Start Presentation November 22, 2012
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M athematical M odeling of Physical S ystems
The Internal Energy of Matter V
U = T · S - p · V +
S
i
m
i ·N i
· · · S
i
S
i
m
i
m
i · N
·
· N
·
i i
· · S
i
m
i
·
· N i
S m
i
·
· N i = 0
This is the
Gibbs-Duhem equation
.
November 22, 2012
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November 22, 2012
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The Capacitive Field of Matter
T S
·
C
S T
·
GY
V p
·
T
·
S
GY
n i
m
i
· CF
n i
m
i
·
GY
V p
·
C C
m
i
n
i p q
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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
.
dU
T
S
p
V
g
M dt
November 22, 2012
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Bus-Bond and Bus-0-Junction
• The three outer legs of the CF-element can be grouped together.
0 November 22, 2012 C 0 Ø C
CF
C 0
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3
CF
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November 22, 2012
Once Again Heat Conduction
CF
2
CF
1 Ø 1 1 1 D 1x mGS 2 1 1 1 0 2 1 2 1 mGS Ø 2 D 2x 3 3
CF
1 HE
CF
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Volume Pressure Exchange
CF
1 Ø 1 D 1x
CF
1 GS 3 p q 1 1 q 2 D p q D p 0 D p q 2 PVE 3 p 2 q GS
CF
2
CF
2 Ø 2 D 2x 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 moved in the process.
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1 1
General Exchange Element I
2 1 1 Sw 1 0 mGS 1 GS 0 GS 1 1 0 mGS 2 2 Sw 2
The three flows are coupled through RS elements.
This is a switching element used to encode the direction of positive flow.
November 22, 2012
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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
.
r 1 , S 1 r 2 , S 2
CF
1
CF
2 November 22, 2012 Ø 3 RF 3
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Multi-phase Systems
• We may also wish to study phenomena such as
evaporation
and
condensation
.
Ø 3 3 HE, PVE, Evaporation, Condensation 3 Ø 3
CF
gas
CF
liq November 22, 2012
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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 November 22, 2012
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November 22, 2012
Condensation On Cold Surfaces
• Here, a boundary layer must be introduced.
Boundary layer
CF
gas
CF
gas Ø 3 Heat conduction (HE) Volume work (PVE) Condensation and Evaporation HE PVE RF 3 Ø Rand schicht HE Heat conduction (HE) Volume work (PVE) Condensation and Evaporation gas s s Ø liq 3
CF
surface Ø 3
CF
liq 3 HE PVE RF 3 Ø 3
CF
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M athematical M odeling of Physical S ystems
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.
CF 1 {M 1 } {x 1 } MI {M 2 } {x 2 } CF 2 Start Presentation November 22, 2012
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M athematical M odeling of Physical S ystems
Entropy of Mixing
• The mixing entropy is taken out of the Gibbs potential.
It was assumed here that the fluids to be mixed are at the same temperature and pressure.
CF
11
CF
21 1 2 1 2 1 2 1 1 1 1 1 1 1 RS mix 1 mix 2 1 2 1 2 D S id mix 1
CF
12 x 11 M 11 M 21 MI x 21 HE PVE
CF
22 RS D S id mix 2 2 Start Presentation November 22, 2012
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M athematical M odeling of Physical S ystems
CF
11
CF
21 November 22, 2012 2 2 2 1 1 1 1 1 1 1 RS RS mRS mix 1 1 D S 1 0 1 2 1 1 2 RS RS mRS 0 mix 2 2 D S 2
CF
12 MI HE PVE
CF
22
© Prof. Dr. François E. Cellier
It is also possible that the fluids to be mixed are initially at different temperature or pressure values.
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Convection in Multi-element Systems
CF 11 Ø 3 PVE HE 3 RF PVE HE 3 CF 21 PVE HE 3 Ø 3 RF PVE HE HE PVE vertical exchange (mixture) Ø 3 CF 12 horizontal exchange (transport) Ø 3 PVE HE 3 CF 22 3 Ø 3 RF PVE HE 3 CF 13 HE PVE PVE HE 3 Ø CF 23 Start Presentation November 22, 2012
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M athematical M odeling of Physical S ystems
Two-element, Two-phase, Two-compartment Convective System
Gas CF 21 Ø 3 PVE HE PVE HE + Ø 3 3 3 Gas CF 11 HE PVE RF RF PVE HE Gas CF 22 3 3 Gas CF 12 3 Ø + PVE HE 3 Ø PVE HE HE Condensation/ Evaporation PVE PVE HE Ø 3 PVE HE Fl.
CF 21 HE Condensation/ Evaporation PVE Ø 3 {x 21 , D S E 21 , D V E 21 } {M 21 ,T 21 , p 21 } 3 3 Fl.
CF 11 MI 1 phase boundary HE PVE RF RF PVE HE 3 Fl.
CF 12 3 HE Condensation/ Evaporation PVE Ø MI 2 PVE HE HE Condensation/ Evaporation PVE PVE HE 3 3 {x 22 , D S E 22 , D V E 22 } {M 22 ,T 22 , p 22 } Ø Fl.
CF 22 November 22, 2012
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Concentration Exchange
• It may happen that neighboring compartments are not completely homogeneous.
In that case, also the concentrations must be exchanged.
CF i CF i+1
...
3 November 22, 2012 Ø 3 HE PVE
CE
3 Ø 3
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References I
• Cellier, F.E. (1991),
Continuous
Springer-Verlag, New York, Chapter 9 .
System Modeling
• 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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M athematical M odeling of Physical S ystems
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.
November 22, 2012
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