Class 13 - Mathematical Modeling of Thermal System

System Modeling Coursework

P.R. VENKATESWARANFaculty, Instrumentation and Control Engineering,

Manipal Institute of Technology, ManipalKarnataka 576 104 INDIAPh: 0820 2925154, 2925152

Fax: 0820 2571071Email: [email protected], [email protected]

Web address: http://www.esnips.com/web/SystemModelingClassNotes

Class 13: Modeling of Thermal systems

mailto:[email protected]

mailto:[email protected]

http://www.esnips.com/web/SystemModelingClassNotes

July – December 2008 prv/System Modeling Coursework/MIT-Manipal 2

WARNING!

•

I claim no originality in all these notes. These are the compilation from various sources for the purpose of delivering lectures. I humbly acknowledge the wonderful help provided by the original sources in this compilation.

•

For best results, it is always suggested you read the source material.

•

Of late, this has becoming supplement to what is taught in the class. So BEWARE! You are on two tracks!


Contents

•

Description of a Thermal system•

Model of the Thermal system


Introduction

•

Thermal systems are those that involve the transfer of heat from one substance to another.

•

Thermal system may be analyzed in terms of thermal resistance and thermal capacitance although they may not be represented as lumped parameters.

•

But by making some assumptions, they can be represented as distributed parameters, which make the analysis simple.


A Thermal System


Assumptions for the system

• Fluid in the tank is perfectly mixed so that it is at uniform temperature

• The tank is insulated to eliminate heat loss to the surrounding air.

• There is no heat storage in the insulation.


Definitions for variables of the system

• θi

= Steady state temperature of inflowing liquid,•

θ

= Steady state temperature of out-flowing liquid,

•

H = Steady state heat input rate from heater.


What happens when you move from steady state?

• Let ∆H be a small change in the heat input rate from its steady state value. This change in H will result in the following changes.

– Change in heat output rate by an amount ∆H1

.–

Change in heat storage rate of liquid in the tank by an amount ∆H2

.–

Change in temperature of out-flowing liquid by an amount ∆θ.


Thermal Resistance

•

Change in outflow heat rate is given by∆H1

= Q Cs ∆θ•

WhereQ = Steady state liquid flow rateCs = Specific heat of liquid

• ∆H1

= ∆θ/R–

If R = 1/QCs which is defined as the Thermal Resistance


Thermal Capacitance

•

Change in heat storage rate is given by∆H2

= MCs d∆θ/dt•

Where–

M = mass of the liquid in the tank

–

∆dθ/dt

= rate of rise of temperature in the tank•

∆H2

= C d∆θ/dt–

Where C = MCs which is defined as Thermal Capacitance.


Transfer function for the system

∆H= ∆H1 + ∆H2 •

The mathematical model of a thermal system shown in figure is

•

Applying Laplace transform

H= + C dR dtθ θΔ Δ

Δ

( )H(s)= + Cs ( )

1 = + Cs ( )

1 Cs = ( )

s sR

sR

R sR

θ θ

θ

θ

ΔΔ Δ

⎡ ⎤ Δ⎢ ⎥⎣ ⎦+⎡ ⎤ Δ⎢ ⎥⎣ ⎦


Transfer function of the system

( ) = H(s) 1 Cs

s RR

θΔ ⎡ ⎤⎢ ⎥Δ +⎣ ⎦


Summary

•

Thermal systems are simple systems like level and do not yield to complexities like pneumatic or hydraulic systems.

•

Hence, it is easy and possible to associate analogous situations and derive the transfer function.


References

1.

Advanced Control Systems Engineering, Ronald Burns

2.

Modern Control Engineering, Ogata3.

Control Systems, Nagoor

Kani

4.

A course in Electrical, Electronic Measurements and Instrumentation, A.K. Sawhney

…amongst others


And, before we break…

•

Love has the patience to endure the fault we cannot cure.

Thanks for listening…

Class 13 - Mathematical Modeling of Thermal System

Documents

Transcript of Class 13 - Mathematical Modeling of Thermal System