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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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


PVE

Gas

CF12

Fl.

CF12

Fl.

CF22

Gas

CF22

Ø3

3

3

3

3

3

Ø

Ø

Ø

3

PVEHE

3


PVE

3

3

3PVEHE

3


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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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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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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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.