Math Modeloing of Thermal Systems1

7/23/2019 Math Modeloing of Thermal Systems1

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Mathematical Modeling of Thermal Systems

Contents

Introduction

The Energy Balance

Examples Involving only Thermal Resistance and Capacitance

Example: Two thermal resistances in series

Example: Heating a Building with One Room

Example: Heating a Building with One Room, but with Variable

External Temperature.

Example: Heating a Building with Two Rooms

Examples Involving Fluid Flow

Example: Cooling a Block of Metal in a Tank with Fluid Flow.

Aside: Modeling a Fluid Flow with and Electrical Analog

Solving the Model

IntroductionThis page discusses how the system elements can be included in larger

systems, and how a system model can be developed. The actual solution of such

models is discussedelsewhere.

The Energy BalanceTo develop a mathematical model of a thermal system we use the concept of an

energy balance. The energy balance equation simply states that at any given

location, or node, in a system, the heat into that node is equal to the heat out of the

node plus any heat that is stored (heat is stored as increased temperature in thermal

capacitances).

Heat in = Heat out + Heat stored

To better understand how this works in practice it is useful to consider several

examples.

Examples Involving only Thermal Resistance and CapacitanceExample: Two thermal resistances in series

Consider a situation in which we have an internal temperature, i, and an ambient

temperature, awith two resistances between them. An example of such a situation is

your body. There is a (nearly) constant internal temperature, there is a thermal

resistance between your core and your skin (at s), and there is a thermal resistance

between the skin and ambient. We will call the resistance between the internal
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temperature and the skin temperature R is, and the temperature between skin and

ambient Rsa.

a) Draw a thermal model of the system showing all relevant quantities.

b) Draw an electrical equivalent

c) Develop a mathematical model (i.e., an energy balance).

d) Solve for the temperature of the skin if i, =37C, a=9C, Ris=0.75/W; for a patchof skin and Rsa= 2.25/W for that same patch.

Solution:

a) In this case there are no thermal capacitances or heat sources, just two know

temperatures ( i, and a), one unknown temperature (s), and two resistances ( Risand

Rsa.)

b) Temperatures are drawn as voltage sources. Ambient temperature is taken to be

zero (i.e., a ground "temperature), all other temperatures are measured with respect to

this temperature).

c) There is only one unknown temperature (at s), so we need only one energy

balance (and, since there is no capacitance, we don't need the heat stored term).


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Note: the first equation included a, but the second does not, since ais our reference temperature and is taken to be zero.

d) Solving for s

Note: you may recognize this result as the voltage divider equation from electrical circuits.

We can now solve numerically (we use 28C for the internal temperature since it is

28C above ambient (37-9=28)

This says that sis 21C above ambient. Since the ambient temperature is 9C, the

actual skin temperature is 30C.

Note: If Rsais lowered, for example by the wind blowing, the skin gets cooler, and it feels like it is colder. This is the mechanism

responsible for the "wind chill" effect.

Example: Heating a Building with One Room

Consider a building with a single room. The resistance of the walls between the

room and the ambient is Rra, and the thermal capacitance of the room is Cr, the heat

into the room is qi, the temperature of the room is r, and the external temperature is a

constant, a.



c) Develop a mathematical model (i.e., a differential equation).

Solution:

a) We draw a thermal capacitance to represent the room (and note its

temperarature). We also draw a resistance between the capacitance and ambient.


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b) To draw the electrical system we need a circuit with a node for the ambient

temperature, and a node for the temperature of the room. Heat (a current source)

goes into the room. Energy is stored (as an increased temperature) in the thermal

capacitance, and heat flows from the room to ambient through the resistor.

c) We only need to develop a single energy balance equation, and that is for the

temperature of the thermal capacitance (since there is only one unknown

temperature). The heat into the room is qi, heat leaves the room through a resistor and

energy is stored (as increased temperature) in the capacitor.

by convention we take the ambient temperature to be zero, so we end up with a first

order differential equation for this system.

Example: Heating a Building with One Room, but with Variable External

Temperature.Consider the room from the previous example. Repeat parts a, b, and c if the

temperature outside is no longer constant but varies. Call the external temperature

e(t) (this will be the temperature relative to the ambient temperature). We will also

change the name of the resistance of the walls to Rre to denote the fact that the

external temperature is no longer the ambient temperature.


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Solution:

The solution is much like that for the previous example. Exceptions are noted below.

a) The image is as before with the external temperature replaced by e(t).

b) To draw the electrical system we need a circuit with a node for the external

temperature and a node for the temperature of the room. Though perhaps not obvious

at first we still need a node for the ambient temperature since all of our temperatures

are measured relative to this, and our capacitors must always have one node

connected to this reference temperature. Heat flows from the room to the external

temperature through the resistor.

c) We still only need to develop a single energy balance equation, and that is for the

temperature of the thermal capacitance (since there is only one unknown

temperature). The heat into the room is qi, heat leaves the room through a resistor and

energy is stored (as increased temperature) in the capacitor.

(the ambient temperature is taken to be zero in this equation). In this case we end up

with a system with two inputs (qiand e).

Example: Heating a Building with Two Rooms


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Consider a building that consists of two adjacent rooms, labeled 1 and 2. The

resistance of the walls room 1 and ambient is R1a, between room 2 and ambient is

R2aand between room 1 and room 2 is R12. The capacitance of rooms 1 and 2 are

C1and C2, with temperatures 1and 2, respectively. A heater in in room 1 generates

a heat qin. The temperature external temperature is a constant, a.



c) Develop a mathematical model (i.e., a differential equation).

In this case there are two unknown temperatures, 1and 2, so we need two energy

balance equations. In both cases we will take ato be zero, so it will not arise in the

equations.

Room 1: Heat in = Heat out + Heat Stored Room 2: Heat in = Heat out + Heat Stored

In this case there are two parts to the "Heat

Out"term, the heat flowing through R1a and the heat

through R12.

In this case we take heat flow throughR12 to (from 1 to 2) to be an input.

We could also take this energy balance

to have no heat in, and write the heat flow

from2 to 1 as a second "Heat out" term. (note the

change of subscripts in the subtracted terms)


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The two first order energy balance equations (for room 1 and room 2) could be

combined into a single second order differential equation and solved. Details about

developing the second order equation arehere.

This describes different ways in which physical systems can be represented

mathematically. Generally system equations are derived as a set of differential

equations. For example consider the system shown below (along with free body

diagrams and system differential equations).

System:

Free Body

Diagrams:

Equations:

Free Body 1 (at x1) Free Body 2 (at x2)
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This representation completely represents the system, but is cumbersome to

use. It requires two equations with multiple symbols and subscripts. It doesn't

generalize well to a wide variety of systems. In this document we first review the

representation of systems with differential equations, then we develop several

systems that are generally easier to work with.

Keep in mind that this document describes how the various forms (e.g., transfer

function, state space...) but does not discuss their solution. That discussion is

elsewhere, and separated by discipline:

Examples Involving Fluid FlowSo far we have not considered fluid flow in any of the examples; let us do so now.

Example: Cooling a Block of Metal in a Tank with Fluid Flow.

Consider a block of metal (capacitance=Cm, temperature=m). It is placed in a well

mixed tank (at termperature t, with capacitance Ct). Fluid flows into the tank at

temperature inwith mass flow rate Gin, and specific heat cp. The fluid flows out at the

same rate There is a thermal resistance to between the metal block and the fluid of

the tank, Rmt, and between the tank and the ambient Rta. Write an energy balance for

this system.

Note: the resistance between the tank and the metal block, Rmt, is not explicitly shown.

Solution:

Since there are two unknown temperatures, we need two energy balance equations.

Metal Block: Heat in = Heat out +

Heat StoredTank: Heat in = Heat out + Heat Stored


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In this case there is not heat in, and

heat outis to the tank through Rmt.

In this case we have heat in from the fluid flow

and from the metal block.We have heat out to ambient through Rta.

Aside: Modeling a Fluid Flow with and Electrical Analog

To model this system with an electrical analog, we can represent the fluid flow as a

voltage source at in, with a resistance equal to 1/(Gincp). If you sum currents at the

nodes tand myou can show that this circuit is equivalent to the thermal system

above.

Solving the ModelThus far we have only developed the differential equations that represent a

system. To solve the system, the model must be put into a more usefulmathematical representation such as transfer function or state space. Details about

developing the mathematical representation arehere.

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