The document discusses thermal systems and their electrical analogies. It describes three methods of heat transfer: conduction through solid media, convection through fluids, and radiation. For conduction, temperature difference is directly proportional to heat flow and inversely proportional to thermal resistance. For convection, a convection coefficient is used instead of thermal conductivity. Radiation heat transfer depends on the temperatures, areas, and properties of the surfaces. Thermal capacitance represents the ability of an object to store thermal energy and is determined by its mass, specific heat, and temperature change over time.

1. Industrial Control
Behzad Samadi
Department of Electrical Engineering
Amirkabir University of Technology
Winter 2009
Tehran, Iran
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2. Thermal Systems
Electrical Analogy
Type of System Electrical Thermal
T-Variable i, current q, heat flow
A-Variable v, voltage T, temperature
Resistance resistor thermal resistor
Storage (A-Type) capacitor thermal capacitor
Storage (T-Type) inductor unknown
Note that q is in units of power and no T-type thermal energy storage
has been identified.
[Ogata, 1997]
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3. Heat Transfer
Conduction: Heat transfer through solid media
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4. Heat Transfer
Conduction: Heat transfer through solid media
T1 − T2 =Rcq
Rc =
d
KA
d =thickness of the wall
A =area of the wall
K =thermal conductivity of the wall’s material
[Macia and Thaler, 2004]
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5. Heat Transfer
Convection: Heat transfer through a fluid (liquid or gas)
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6. Heat Transfer
Convection: Heat transfer through a fluid (liquid or gas)
T1 − T2 =Rhq
Rc =
d
hA
d =thickness
A =area
h =convection coefficient
[Macia and Thaler, 2004]
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7. Heat Transfer
Radiation:
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8. Heat Transfer
Radiation:
q =AεFσ(T4
1 − T4
2 )
A =area of the surface of the object of interest
ε =parameter descriptive of the emission of surfaces 1 and 2
σ =Stefan-Boltzman Constant
F =geometrical shape factor
T1, T2 =absolute temperatures of bodies 1 and 2
[Macia and Thaler, 2004]
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9. Heat Transfer
Radiation:
q =AεFσ(T4
1 − T4
2 )
Rr =
1
AεFσ(T2
1 + T2
2 )(T1 + T2)
⇒ T1 − T2 = Rr q
[Macia and Thaler, 2004]
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10. Thermal Capacitance
q =C
dT
dt
C =mcx
C =thermal capacitance of the object
m =mass of the object
cx =specific heat of material
[Macia and Thaler, 2004]
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11. Macia, N. F. and Thaler, G. J. (2004).
Modeling and Control of Dynamic Systems.
Delmar Learning.
Ogata, K. (1997).
Modern Control Engineering.
Prentice Hall, 3 edition.
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