ME 383Q – R.G. LongoriaModeling of Physical Systems

Department of Mechanical EngineeringThe University of Texas at Austin

Thermofluid effects in dynamic systems

ME 383Q – R.G. LongoriaModeling of Physical Systems

Department of Mechanical EngineeringThe University of Texas at Austin

Use of entropy (from BP)

ME 383Q – R.G. LongoriaModeling of Physical Systems

Department of Mechanical EngineeringThe University of Texas at Austin

From BP: Equilibrium and statesWhen we make a ‘thermodynamic assumption’, we assume a homogeneous substance has

near uniform properties of temperature and pressure.

These are intensive variables.

Extensive variables ‘distribute among the parts’.

Intensive are common over a substance, while extensive sum over the substance.

ME 383Q – R.G. LongoriaModeling of Physical Systems

Department of Mechanical EngineeringThe University of Texas at Austin

Gibbs internal energy

( , , ) Gibbs (internal) energy

entropy, measures dispersal of system energy

volume, measures size

moles, measures amount or matter content

SVN i

i

U U S V N

S

V

N

= =

=

=

=

The state of a simple thermodynamic system can be quantified by,

SVN SVN SVN SVNi

i

i

dU U U US V N

dt S V N

T P µ

∂ ∂ ∂= = + +

∂ ∂ ∂

−

ɺ ɺ ɺ

��� ��� ���≜ ≜ ≜

P

Now we quantify power flow into the thermodynamic system,

P = power

ME 383Q – R.G. LongoriaModeling of Physical Systems

Department of Mechanical EngineeringThe University of Texas at Austin

Thermodynamic CNow we can represent this storage of energy using a multiport-C

element:

ME 383Q – R.G. LongoriaModeling of Physical Systems

Department of Mechanical EngineeringThe University of Texas at Austin

U is a homogeneous 1st order function

du Tds Pdv= −Example: Basic Gibbs equation for pure substance:

ME 383Q – R.G. LongoriaModeling of Physical Systems

Department of Mechanical EngineeringThe University of Texas at Austin

Energy stored

in a solid

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1

2

3

4

5

6

7

8

9

10

T

∆S

C

This is an equation of state; i.e., a constitutive relation for the C.

ME 383Q – R.G. LongoriaModeling of Physical Systems

Department of Mechanical EngineeringThe University of Texas at Austin

For ideal gas…see BP

Pv RT=Where: and V

duc

dT=

(constant volume specific heat)

ME 383Q – R.G. LongoriaModeling of Physical Systems

Department of Mechanical EngineeringThe University of Texas at Austin

See Example 7-2 from BP

1

( )

0 (adiabatic)

A

oo

oo

p mg A P P

AV p

m

S

VP P

V

VT T

V

γ

γ −

= − + −

=

=

=

=

ɺ

ɺ

ɺ

F0

T

S =ɺ

ME 383Q – R.G. LongoriaModeling of Physical Systems

Department of Mechanical EngineeringThe University of Texas at Austin

Electro-thermal-mechanical systemConsider the electro-thermo-mechanical system shown

below. A piston is forced to move by the expansion of

air in the cylinder. The cylinder and piston are made of

steel. The piston is h thick and the cylinder walls are tw

with the inner radius of the cylinder being rc. The height

of the cylinder is Lc. The ambient temperature of Ta is

fixed and known. The heater coil has known electrical

resistance R and the voltage input is AC at 60 Hz.

a. Develop a bond graph model of this system.

b. Develop state equations for this system.

c. Starting with the air at 25 deg C and compressed

enough to balance the piston, the heater is turned on.

Perform a simulation of the system. What are the steady

state values of critical variables?

Need to model the heat generated by the resistor as well as heat transfer. Let’s

look at how to model these elements.

ME 383Q – R.G. LongoriaModeling of Physical Systems

Department of Mechanical EngineeringThe University of Texas at Austin

“Basic conduction R”

Other modes of heat transfer

have same bond graph form,

just the constitutive relation

changes.

ME 383Q – R.G. LongoriaModeling of Physical Systems

Department of Mechanical EngineeringThe University of Texas at Austin

ME 383Q – R.G. LongoriaModeling of Physical Systems

Department of Mechanical EngineeringThe University of Texas at Austin

F

2

( )

exp

exp

s

o

o

V

oo

V

vS f

T R T

S ST T

c

S SP P

c

= =⋅

−=

−=

ɺ

P−

ME 383Q – R.G. LongoriaModeling of Physical Systems

Department of Mechanical EngineeringThe University of Texas at Austin

You now have enough to model a piston compressing air in a cylinder or chamber.

FP−

ME 383Q – R.G. LongoriaModeling of Physical Systems

Department of Mechanical EngineeringThe University of Texas at Austin

KMR

Note: KMR reverse the sign and bond on the

pressure port – equivalent formulation.

ME 383Q – R.G. LongoriaModeling of Physical Systems

Department of Mechanical EngineeringThe University of Texas at Austin

KMR

ME 383Q – R.G. LongoriaModeling of Physical Systems

Department of Mechanical EngineeringThe University of Texas at Austin

ME 383Q – R.G. LongoriaModeling of Physical Systems

Department of Mechanical EngineeringThe University of Texas at Austin

Thermal effects in

a PMDC motor

ME 383Q – R.G. LongoriaModeling of Physical Systems

Department of Mechanical EngineeringThe University of Texas at Austin

You can derive a fairly good estimate of the thermal limitations

on PMDC motors with this basic model. Here, it is critical that

the rotor/windings temperature not exceed a specified limit. In

turn, this limits the torque capacity of the motor.

ME 383Q – R.G. LongoriaModeling of Physical Systems

Department of Mechanical EngineeringThe University of Texas at Austin

Summary• We can incorporate thermal effects into our system models using basic

elements that represent the generation, storage and transfer of thermal

energy/power.

• Up to this point, only systems with a fixed amount of matter (closed) have

been considered.

• In some practical systems, it may be necessary to keep track of how much

matter enters and/or leaves the system, and for those cases we need to track

moles or mass and the conveyance of energy as well. This can be done with

an extension to the methods we’ve already described.

References

[1] J.J. Beaman and H.M. Paynter, Modeling of Physical Systems, notes for ME

383Q, UT-Austin, 1993. Chapter 7.

[2] D.C. Karnopp, et al, System Dynamics, Wiley (any edition). Chapter 12.

ME 383Q – R.G. LongoriaModeling of Physical Systems

Department of Mechanical EngineeringThe University of Texas at Austin

These models don’t tell us anything about the internal energy storage.

ME 383Q – R.G. LongoriaModeling of Physical Systems

Department of Mechanical EngineeringThe University of Texas at Austin

Example: Air-spring suspension (KMR, P12-12)