Universität Stuttgart Modeling Multi-Element Systems Using Bond Graphs 1.Introduction 2.Mixture...

Universität Stuttgart

Modeling Multi-Element SystemsModeling Multi-Element SystemsUsing Bond GraphsUsing Bond Graphs

1. Introduction2. Mixture Properties3. Transport Phenomena4. Model of a Pressure Cooker5. Conclusions

University of Arizona

18. Oktober 2001Jürgen Greifeneder, François Cellier

Modeling Multi-Element SystemsUsing Bond Graphs

Modeling Multi-ElementModeling Multi-Element SystemsSystemsUsing Bond GraphsUsing Bond Graphs

Jürgen GreifenederFrançois E. Cellier

18.10.2001






1. Introduction• Review• Basics of Multi-Element Systems

2. Mixture Properties3. Transport Phenomena4. Model of a Pressure Cooker5. Conclusions

Contents






Introduction

• Describing a thermodynamical problem necessitates 3 variables.

• Separation in storage and dissipative elements. Storage elements calculate the potentials and therefore need to know about the matter, they are representing. Dissipative elements calculate flows and do not care, which matter they are dealing with (network theory).

• Elements do not know about each other.

• No quasi-stationary or flow-equilibrium assumptions were made.

• Contrary to earlier efforts in this field, this work delt with real, rather than pseudo bond graphs.






pq

TS

g

M

.

.

0

0 0

CFCF

C C

C Ø3

CFCF

Icon:

The C-field (storage element)






CFCF1CFCF2

3 3CD

CFCF1CFCF2

3 3DVAVolume work:

Conduction:

Ø Ø3RF

3

33

CFCF1 CFCF2

, s1 1 , s2 2Convection:

Basic dissipative Elements






Traditional Thermodynamics

One Temperature, one pressure and n partial mass’ => n+2 equations.

pq

0

0 0

CFCF

C C

CC

TS

.

0 00 ....g

1M1

.g2M

2

.g3M3

.

g n

M n.....

1 2 3 n

n

Ø

?

CFCF






Multi-Element Mono-Phase Systems

CF1Ø

3

3

CF2

3Ø

2 2DVACD

CF3

Ø

DVA

CD

2

2

DVACD

2

2






Ideal and Non-Ideal Mixtures

• In the process of mixing, additionally entropy will be created, which must be distributed among the participating components

• Distribution is a function of the molar fractions

• CF-Elements are not supposed to know about each other only necessary information will be provided

CF1 CF2MIMI

{M1}

{x1

{M2}

{x2






Ideal and Non-Ideal Mixtures

• In the process of mixing, additionally entropy will be created, which must be distributed among the participating components

• Distribution is a function of the molar fractions

• CF-Elements are not supposed to know about each other only necessary information will be provided

CF1CF2MIMI

{M1, V1, S1}

{x1s1Exv1

Ex

{M2, V2, S2}

{x2s2Exv2

Ex






Entropy of Mixing

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

CDDVA.

.






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 CDDVA

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

An François: eigentlich müßte ich hier erwähnen, daß die drei (m)RS-Elemente einem RF-Element entsprechen (vor allem, weil dies auf der nächsten Folie verwendet wird). Allerdings habe ich die interne RF-Struktur nie verwendet und sehe dies auch nicht als erforderlich an.

Daher: Was tun? Die nächste Folie rauslassen?

An François: eigentlich müßte ich hier erwähnen, daß die drei (m)RS-Elemente einem RF-Element entsprechen (vor allem, weil dies auf der nächsten Folie verwendet wird). Allerdings habe ich die interne RF-Struktur nie verwendet und sehe dies auch nicht als erforderlich an.

Daher: Was tun? Die nächste Folie rauslassen?






CFCF12

CFCF22

MIMI CDDVA

Ø Ø3RF

3

CFCF11

Ø Ø3RF

3

CFCF21

3

3

3

3

3CD

DVA

3

CDDVA

Øoutside

3

3






Convection in Multi-Element Systems

CF12

CF13

CF11

Ø

3

3

3

3

3

Ø

Ø

3

DVACD

3

3

CDDVA CF22

CF23

CF21

3

3

3

3

3

Ø

Ø

Ø

3

DVACD

3

3DVACD

3

CDDVA

3

DVACD

3RF

DVACD

3

3 3RFDVACD

3 3RFDVACD

horizontalExchange(transport)

verticalExchange(mixture)






Two-Element, Two-Phase, Two-Compartment Convective System

Gas

CF11

Fl.

CF11

Fl.

CF21

Gas

CF21

Ø

3

3

3

3

3

Ø

Ø

Ø

3

DVACD

3

CDCondensation/Evaporation

DVA

3

3

3

3


DVA

Gas

CF12

Fl.

CF12

Fl.

CF22

Gas

CF22

Ø3

3

3

3

3

3

Ø

Ø

Ø

3

DVACD

3


DVA

3

3

3DVACD

3


DVA

3

DVACD

3RF

DVACD

3

3 3

3 3

CDDVARF

CDDVARF

3 3RF

DVACD

phase-boundary

3

DVACD

3

3

DVACD

3

3

DVACD

3

3

DVACD

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}

An François:

Wie genau muß ich auf diese Abbildung eingehen?

An François:

Wie genau muß ich auf diese Abbildung eingehen?






Equilibration of Concentrations

CFCFii

3

Ø 3 3CDDVAKA

3

Ø

CFCFi+1i+1

33... ...






Model of a Pressure Cooker

water

air

steam

SE: 393 K

CD (t)

KV

DVA

DVA

CD

CD

CD DVA






water

air

steam

KV

DVA

DVA

CD

CD

CD DVA

KV

Air in boundary

layer

Steam in boundary

layer

CD

RF: p

RF: p

CD

CDDVA

CD

CD

SE: 293 K

CD (t)

CD (t)

SE: 393 K

CD (t)

CD (t)

CD (t)

Model of a Pressure Cooker






Temperature Graphs






Pressure Graphs






Humidity Graphs

OHsat

partialsteam

p

p2






Mass Fraction Graphs

airsteam

steam

mm

m






Conclusion

• The elements introduced suffice to model most thermodynamical problems

• Modeling each matter separately as a storage element and connecting them by means of dissipative elements (RF-concept) simplyfies the modeling task, offers insight into physical functioning of multi-element systems and leads to mathematical models that can be simulated in a numerically stable and highly accurate fashion.

• Models are still limited to systems without chemical reactions






Danke

Ende

Thank you!Thank you!RemerRemerciement!ciement!

Jürgen GreifenederFrançois E. Cellier

Universität Stuttgart Modeling Multi-Element Systems Using Bond Graphs 1.Introduction 2.Mixture...

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