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Modeling of Thermal Systems
In this lesson, we will learn and study the modeling fluidic systems by
creating equivalent circuits.
Analogies for the thermal and electrical systems are given in the table.
Potential difference generates a current in electrical circuits. Temperature
difference makes heat transmit in thermal systems.
Ht
Heat
(Joule)
Qt
Heat flow
rate
(J/s)
q
Charge
(Coulomb)
i
Current
(Amper)
L
Induc.
(Henry)
1/Ct
Ct:Thermal
capacitance
Rt
Thermal
resistance
T
Temp.
(oC)
1/C
C:Capac.
(Farad)
R
Resistance
(Ohm)
V
Voltage
(Volt)
Ht
Heat
(Joule)
Qt
Heat flow
rate
(J/s)
1/Ct
Ct:Thermal
capacitance
Rt
Thermal
resistance
T
Temp.
(oC)
D.Rowell & D.N.Wormley, System Dynamics:An Introduction, Prentice Hall, 1997
Thermal Capacitance:
Ct=mCp
Ct: Thermal capacitance, Cp: Specific heat, m: Mass
Thermal resistance in heat transfer by conduction :
1
A
Rt 
Rt:Resistance, ρc:Conduction coefficient, A:Cross
CD  c
CD
L
section, L:Length
Thermal resistance in heat transfer by convection :
1
Rt 
Ch: Convection coefficient
ACh
Analogous Electrical Circuits of Thermal Systems
Thermal systems can be analyzed by creating equivalent electrical circuits. Let’s
consider Example 6.1.
A thermal system is given here. We will create the equivalent electrical circuit for the
system by using the analogy.
Ta
Example 6.1
o
Qk1
2
1
TB
o
o Qk2 o
Qk1, Qk2, TB
Inputs:
q k 1 , q k 2 , VB
Thermal Elec.
q n
Qn
Rn
Cn
Rn
Cn
Tn
Vn
Thermal System
Q1a
Ta
o
1
Qk1
o
Q1B
Analogous Circuit
C1
R1B
R1a
q 1a
q 1B
q k 1
Va
In the thermal system, the room #1 is heated with the
heat source Qk1. Ta s the atmospheric temperature.
While the objects in the room #1 are heating depending
on the their thermal capacitance, at the same time
there is a heat transfer from the room #1 to the next
room #B.
In the equivalent circuit, the current supply qk1_dot
can supply the line with the resistance R1a, capacitor
C1 and the line with the resistor R1B.
We remember the analogy that the heat flow, thermal
capacitance, thermal resistance in thermal systems
correspond to the electrical current, capacitor, resistor
in electrical systems, respectively.
R1a corresponds to the thermal resistance between the
room #1 and atmosphere. R1B corresponds to the
thermal resistance between the room #1 and room B.
C1 in the circuit corresponds to the thermal
capacitance of the room #2.
Thermal System
Q1a
Ta
o
1
Qk1
QB2
o
TB
Q1B
We remember from the analogy table that
a temperature in thermal systems
correspond to a voltage in electrical
systems.
Analogous Circuits
q Ba R
Ba
C1
R1B
RB2
R1a
q 1B q
q 1a
+
q k 1
Va
-
The room B has a heat source which the
temperature is controlled. The value of
the temperature in the room B is TB.
There is a heat transfer from the room B
to the room #2 or vice versa.
q B 2
VB
So, a voltage source VB corresponding to
TB is placed into the circuit in the
electrical system.
This voltage source’s current q1b_dot can
supply the lines with the resistance RBa
and the resistance RB2.
Thermal System
Q1a
Q2a
Ta
o
1
Qk1
The room #2 is heated with the heat
source Qk2. There is a heat transfer
between the room #2 and the
atmosphere.
2
o
TB
Q1B
In the equivalent circuit, the current
supply qk2_dot can supply the line
with the resistance R2a, capacitor
C2 and the line with the resistor
RB2. C2 in the circuit corresponds
to the thermal capacitance of the
room #2.
QB2
QBa o Q o
k2
Analogous Electrical Circuits
q Ba R
Ba
C1
R1B
RB2
R1a
q 1B q
q 1a
+
q k 1
Va
C2
-
R2a
q B 2
VB
q k 2
q 2a
The heat sources Qk1, Qk2 and the
controlled temperature TB are the
inputs of the thermal system.
The currents qk1_dot, qk2_dot and
the voltage source VB are the inputs
of the electrical system.
CAD-CAE
Dynamic (Transient) behaviour
Steady-state behaviour
By analyzing the equivalent circuit, transient or steady-state dynamic behavior
at a desired point of a circuit can be calculated.
Thus, the corresponding heat flow and temperatures at a desired point of a
thermal system are found.
Nowadays, thermal systems can be analyzed by the computers.
Before the developments in computer technology, in order to analyze thermal
systems engineers have used the equivalent circuits, which are produced easily
and cheaply.
Engineers have developed thermal systems by doing experiments with the
circuits.