This document summarizes a class on mathematical modeling of thermal systems. It describes how thermal capacitance relates temperature change to heat flow based on the first law of thermodynamics. Thermal resistance relates temperature difference to heat flow rate. The document provides equations for thermal capacitance and resistance, and develops a mathematical model for a simple thermal system consisting of a tank with heated fluid, relating changes in the system's temperature and heat input over time.

1. ICE401: PROCESS INSTRUMENTATION
AND CONTROL
Class 9: Mathematical Modeling of
Thermal Systems
Dr. S. Meenatchisundaram
Email: meenasundar@gmail.com
Process Instrumentation and Control (ICE 401)
Dr. S.Meenatchisundaram, MIT, Manipal, Aug – Nov 2015

2. Thermal Systems:
Process Instrumentation and Control (ICE 401)
Dr. S.Meenatchisundaram, MIT, Manipal, Aug – Nov 2015
• The basic thermal processes encountered in the process
industries are the mixing of hot and cold fluids, the exchange
of heat through adjoining bodies, and the generation of heat
by combustion or chemical reaction.
• Two laws of thermodynamics are used in the study of thermal
systems. The first governs the way in which heat energy is
produced and determines the amount generated. The second
governs the flow of heat.
• Temperature changes in an isolated body conform to the first
law of thermodynamics. For a given body, heat input raises
the internal energy, and the rate of change of body
temperature will be proportional to the heat flow to the body.

3. Thermal Systems:
Process Instrumentation and Control (ICE 401)
Dr. S.Meenatchisundaram, MIT, Manipal, Aug – Nov 2015
• The constant that relates temperature change and heat flow is
called the thermal capacity of the body:
(7.1)
where
C = thermal capacitance (cal/°C)
dT/dt = the rate of change of temperature (°C/s)
q = heat flow (cal/s)
The thermal capacitance of a body is found by:
C = MS (7.2)
where
M = the mass of the body (gm);
S = the specific heat of the material (cal/gm)(°C)
dT
C q
dt
=

4. Thermal Systems:
Process Instrumentation and Control (ICE 401)
Dr. S.Meenatchisundaram, MIT, Manipal, Aug – Nov 2015
• Thermal capacitance is analogous to electric capacitance.
• For example, as shown in Figure, heat flowing into a body
with thermal capacitance C causes the temperature (T) to rise
above the ambient value To.

5. Thermal Systems:
Process Instrumentation and Control (ICE 401)
Dr. S.Meenatchisundaram, MIT, Manipal, Aug – Nov 2015
• Heat flow and charge flow as well as temperature and voltage
are analogous quantities. Heat transmission takes place by
conduction, convection, or radiation.
Conduction involves transmission through adjoining
bodies.
Convection involves transmission and mixing.
Radiation uses electro-magnetic waves to transfer heat.
• The rate of heat flow through a body is determined by its
thermal resistance.
• This is defined as the change in temperature that results
from a unit change in heat flow rate.

6. Thermal Systems:
Process Instrumentation and Control (ICE 401)
Dr. S.Meenatchisundaram, MIT, Manipal, Aug – Nov 2015
• Thermal resistance is normally a linear function, in which
case,
(7.3)
where
RT = thermal resistance (°C/ cal/s)
T2 - T1 = temperature difference in (°C)
q = the heat flow (cal/s)
• Thermal resistance is analogous to the resistance in an
electrical circuit.
• If the temperature of a body is considered to be uniform
throughout, its thermal behavior can be described by a
linear differential equation.
2 1
T
T T
R
q
−
=

7. Thermal Systems:
Process Instrumentation and Control (ICE 401)
Dr. S.Meenatchisundaram, MIT, Manipal, Aug – Nov 2015
• This assumption is generally true for small bodies of gases
or liquids where perfect mixing takes place.
• For such a system, thermal equilibrium requires that at any
instant the heat added (qi) to the system equals the heat
stored (qs) plus the heat removed (qo). Thus,
0i sq q q= +

8. Thermal Systems:
Process Instrumentation and Control (ICE 401)
Dr. S.Meenatchisundaram, MIT, Manipal, Aug – Nov 2015
• Consider a thermal system shown in figure.

9. Thermal Systems:
Process Instrumentation and Control (ICE 401)
Dr. S.Meenatchisundaram, MIT, Manipal, Aug – Nov 2015
• The following assumptions are made to make the analysis
simple:
− 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 in-flowing liquid,
− θ = Steady state temperature of out-flowing liquid,
− H = Steady state heat input rate from heater.

10. Thermal Systems:
Process Instrumentation and Control (ICE 401)
Dr. S.Meenatchisundaram, MIT, Manipal, Aug – Nov 2015
• 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.
• Change in outflow heat rate is given by
∆H1 = Q Cs ∆θ
Where
Q = Steady state liquid flow rate; Cs = Specific heat of liquid

11. Thermal Systems:
Process Instrumentation and Control (ICE 401)
Dr. S.Meenatchisundaram, MIT, Manipal, Aug – Nov 2015
∆H1 = ∆θ/R
Where, R = 1/QCs which is defined as the Thermal Resistance.
• 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 dθ∆/dt
• Where
− C = MCs which is defined as Thermal Capacitance.

12. Thermal Systems:
Process Instrumentation and Control (ICE 401)
Dr. S.Meenatchisundaram, MIT, Manipal, Aug – Nov 2015
∆H= ∆H1 + ∆H2
• The mathematical model of a thermal system shown in
figure is
• Applying Laplace transform
H= + C
d
R dt
θ θ∆ ∆
∆
( )
H(s)= + Cs ( )
1
= + Cs ( )
1 Cs
= ( )
s
s
R
s
R
R
s
R
θ
θ
θ
θ
∆
∆ ∆
 
∆  
+ 
∆  
( )
=
H(s) 1 Cs
s R
R
θ∆  
⇒  ∆ + 

13. References:
• Modern Control Engineering, 5th Edition, by Katsuhiko Ogata.
• Advanced Control Systems Engineering, Ronald Burns
• Control Systems, Nagoor Kani.
• A course in Electrical, Electronic Measurements and
Instrumentation, A.K. Sawhney.
Process Instrumentation and Control (ICE 401)
Dr. S.Meenatchisundaram, MIT, Manipal, Aug – Nov 2015