Bondgraph modeling of thermo-fluid systems

ME270 Fall 2007

Stephen Moore

Professor Granda

Introduction

• Study of thermofluid bondgraphs

• Series of three thermofluid bondgraph example models– Heat transfer- Conduction– Incompressible flow– Compressible flow

• To gain knowledge of bondgraph modeling of thermofluid systems

Heat transfer• Resistance is thermal • T- temperature • - heat flow • - entropy flow • Pseudo bonds

– T * ≠ Power

Note: Refer to Figure 12.1, “System Dynamics”

Q

S

QT1 T2

RT1 T2

1

S 2

S

S

Heat transfer

• Related equations• H- heat conduction

coefficient • R is a function of the

average to maintain linearity

)21(

2

)21(1

)21(1/1

2

)21(2/2

2211

)21(

2

TTH

TaveR

S

T

TTTave

T

TTHTQS

T

TTHTQS

STQST

TTHQ

Heat transfer

• Results– Differential equations in Matlab are

developed from momentum and displacement- I and C elements

– Simulink used to display results

Heat transfer• Simulink model

T1 = 373K, T2 = 273KhGW = 0.037 W/mK

hAl = 237 W/mK

Glass Wool Aluminum

Tank emptying

• Incompressible, one-dimensional flow

• Model gives estimate of the time it takes to empty a tank

Tank emptyingAT

ρ

p1 p2

A2

pl=0

h

l

AT>>A2

0

V

pp

Rb

CQ

I

SpQ

1 1p1

Note: Refer to Figure 12.9,

“System Dynamics”

Tank emptying

• -Volumetric flow rate out of the tank

• -Rate of pressure momentum in the pipe

Tpp

p

A

gVp

L

A

Ap

pL

AV

2

222

2

2

V

pp

• Rb- Bernoulli resistance of pipe– Indicates a loss of kinetic energy

as the fluid leaves the system– Difficult to accurately determine

without experimental data• C - capacitance of the tank• I – inertia of the flow

Tank emptying

• System parameters– Water at ambient

conditions (μ, λ, ρ)– Tank diameter- 10 m– Tank depth- 10 m– Outlet pipe diameter-

0.5 m– Length- 1 m

• Resistance-

5625 N*s/m^5

Resistance was determined by P3/Q3

(R~ P3/Q3)

• Capacitance-

.008 m^4*s^2/kg

• Inertia-

4000 kg/m*s

Tank emptying

Tank emptying

Tank emptying

Air cylinder

• Models compressible flow

• Capacitive fields

• Resistive fields

Air cylinderF(t)xdot

P1,T1,m1,V1

P2,T2 m2,V2

mp,Ap

Ar

P1

C

C

R0

1

00

0

0

0

Sf

Sf

Se:F

T1

P2

T2

P2

TF: Ap

(Ap-Ar):TF

P1

I:mp

Note: Refer to Figure 12.17, “System Dynamics”

Air cylinder

• The single R element with 4 bonds requires 16 values

• Two C elements 4 bonds each require 18 values

• The values are approximate values

Air cylinder

• The working fluid:– Air at 25oC and 100 KPa– Cp - 1005 N-m/Kg K– Cv - 718 N-m/Kg K– Volume - 0.012272 m3

– Mass – 0.014253 Kg– Lower chamber is empty– Upper chamber is full

• Geometry:– Cylindrical chamber– 0.25 m diameter– 0.25 m height– Mass cylinder is 3.4 kg

• Applied force– 25 N upward

Air cylinder• Results

– Volume in upper and lower chambers• Expect upper chamber to decrease volume and lower chamber to

increase volume with time

Air cylinder• Results

– Pressures in upper and lower chambers• Expect pressure in the upper chamber to increase while the lower

chamber decreases

Air cylinder• Results

– Mass flow in the chambers• Expect mass flow out of the upper chamber and into the lower

chamber

Air cylinder

• The model worked, however, the results obtained are incorrect

• The values of the R-field and C-field are based on rough approximations

• More work is required to adequately model the air cylinder

Conclusion

• Thermofluid bondgraphs are significantly different than typical bondgraphs

• Care must be taken to ensure the correct parameters are chosen for C, I and R elements, especially for R-fields, C-fields and I-fields

• Expect most thermofluid bondgraphs to represent non-linear systems

• CampG and Matlab obtains the differential equations easily.