Heat Transfer:
Transient Heat Transfer
© 2011 Autodesk
Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.
www.autodesk.com/edcommunity
Education Community
Section 6 – Thermal Analysis
Objectives
Module 5: Transient Heat Transfer
Page 2

Understand the basics of transient heat transfer.

Examine lumped mass systems.

Compare linear unsteady heat transfer with nonlinear unsteady heat
transfer.

Study two examples:
 Heat sink assembly temperature rise
 Transient heat transfer through a rod
© 2011 Autodesk
Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.
www.autodesk.com/edcommunity
Education Community
Section 6 – Thermal Analysis
Transient Heat Transfer
Module 5: Transient Heat Transfer
Page 3

If systems can be approximated as lumped mass systems, transient
heat transfer is easy to find.

Lumped mass systems rarely exist in reality, but particularly for large
systems, this can prove a valuable approximation with little loss in
accuracy.

For materials which are anisotropic rather than uniform, lumped
mass systems may not be a good assumption.

The Biot number (Bi) is normally used to find if a lumped mass
system would be applicable: Bi  0 . 1
© 2011 Autodesk
Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.
www.autodesk.com/edcommunity
Education Community
Section 6 – Thermal Analysis
Simplifying Heat Transfer Analysis
Module 5: Transient Heat Transfer
Page 4

Lumped mass systems simplify transient simulation. The following
equation is normally used:
T(t) = Temperature after time t
T( t )  T
Ti  T
e
 bt
Tα = Ambient Temperature
h = Heat transfer coefficient
Where ,
b
As = Surface Area
hA s
 VC
V = Volume of lumped mass
p
Cp = Specific heat of lumped mass
© 2011 Autodesk
Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.
www.autodesk.com/edcommunity
Education Community
Section 6 – Thermal Analysis
Temperature – Time Relationship
Module 5: Transient Heat Transfer
Page 5
T( t )  T
Ti  T
 e
 bt
Where,
b
hA s
 VC
The temperature of a lumped system approaches the
temperature of the environment as time progresses.
p
The above relationships enable us to determine the temperature T(t)
of a body at time t, or alternatively, the time t required for the
temperature to reach a specified value T(t).
© 2011 Autodesk
Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.
www.autodesk.com/edcommunity
Education Community
Section 6 – Thermal Analysis
Linear Unsteady Heat Transfer
Module 5: Transient Heat Transfer
Page 6




Temperature distribution throughout time is required.
The specific heat and the density of material is also required.
Computationally less expensive than nonlinear cases.
Very few cases for heat transfer in real life are linear unsteady
because many thermophysical properties do change with
temperature.
The graph shows asymptotic
rise of temperature with time.
© 2011 Autodesk
Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.
www.autodesk.com/edcommunity
Education Community
Section 6 – Thermal Analysis
Nonlinear Unsteady Heat Transfer
Module 5: Transient Heat Transfer
Page 7

Problems with unsteady temperature dependent conduction or
involving radiation are nonlinear.

Exact solutions do not exist and an iterative scheme is used to solve
such problems, as the number of unknowns is higher than the
number of equations.

Because of nonlinearity, the time steps should be very small.

How often the nonlinear terms are updated can be controlled
through programming.
© 2011 Autodesk
Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.
www.autodesk.com/edcommunity
Education Community
Section 6 – Thermal Analysis
Example: Heat Sink Assembly
Temperature Rise


Module 5: Transient Heat Transfer
Page 8
A heat sink assembly example that
was used in the earlier modules is
used again here. For this module,
we will be looking at the rise in
Fins
temperature with time as the
(Aluminium)
processor starts to dissipate heat
from an initial ambient temperature
of 20⁰C.
A video presentation is available
with this module detailing the
setting up and solving of transient
heat transfer analysis for this
example.
© 2011 Autodesk
20°
B C
Heat Spreader
(Copper)
Microprocessor
(Silicon)
Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.
40
Watts
www.autodesk.com/edcommunity
Education Community
Section 6 – Thermal Analysis
Additional Example: Transient
Heat Transfer Through a Rod


Module 5: Transient Heat Transfer
Page 9
A solution for transient heat transfer of an insulated rod is available
and can be compared to a simulation.
At the initial condition, the rod is hotter in the center and cooler at
the ends, with a parabolic temperature distribution.
Problem Specification
Insulated rod is allowed to cool
t =0
T=T0
T=T0
T(x, t=0) = f(x) and heat flux q=0
T0
t→
T
t
© 2011 Autodesk
T0
∞
 T
2

x
2
x =0
Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.
x =L
Cooling Temperature Profile
www.autodesk.com/edcommunity
Education Community
Section 6 – Thermal Analysis
Summary
Module 5: Transient Heat Transfer
Page 10

Transient heat transfer analysis occurs mostly during the
starting/stopping of machinery, when the objective is to observe
temperature variation with time; thus the additional variable of
“time” has to be solved.

When solving analytically, the approximation of lumped mass
systems can simplify the analysis.

For this approximation to be valid, the dimensionless Biot Number
parameter is used.
© 2011 Autodesk
Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.
www.autodesk.com/edcommunity
Education Community
Section 6 – Thermal Analysis
Summary
Module 5: Transient Heat Transfer
Page 11

In the case of numerical analysis, the number of time steps will
dictate the length of the analysis.

Therefore, provided that the gradient of temperature with time is
low, larger time steps can be taken.

Just like steady state problems, transient heat transfer problems can
be linear or nonlinear, with the latter adding another dimension of
complexity into the mix.
© 2011 Autodesk
Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.
www.autodesk.com/edcommunity
Education Community