NUMERICAL METHODS
IN HEAT CONDUCTION
S
o far we have mostly considered relatively simple heat conduction prob
lems involving
simple geometries
with simple boundary conditions be
cause only such simple problems can be solved
analytically.
But many
problems encountered in practice involve
complicated geometries
with com
plex boundary conditions or variable properties and cannot be solved ana
lytically. In such cases, sufficiently accurate approximate solutions can be
obtained by computers using a
numerical method.
Analytical solution methods
such as those presented in Chapter 2 are based
on solving the governing differential equation together with the boundary con
ditions. They result in solution functions for the temperature at
every point
in
the medium. Numerical methods, on the other hand, are based on replacing
the differential equation by a set of
n
algebraic equations for the unknown
temperatures at
n
selected points in the medium, and the simultaneous solu
tion of these equations results in the temperature values at those
discrete
points.
There are several ways of obtaining the numerical formulation of a heat
conduction problem, such as the
finite difference
method, the
finite element
method, the
boundary element
method, and the
energy balance
(or control
volume) method. Each method has its own advantages and disadvantages, and
each is used in practice. In this chapter we will use primarily the
energy bal
ance
approach since it is based on the familiar energy balances on control vol
umes instead of heavy mathematical formulations, and thus it gives a better
physical feel for the problem. Besides, it results in the same set of algebraic
equations as the finite difference method. In this chapter, the numerical for
mulation and solution of heat conduction problems are demonstrated for both
steady and transient cases in various geometries.
265
CHAPTER
5
CONTENTS
5–1
Why Numerical Methods
266
5–2
Finite Difference Formulation of
Differential Equations
269
5–3
OneDimensional
Steady Heat Conduction
272
5–4
TwoDimensional
Steady Heat Conduction
282
5–5
Transient Heat Conduction
291
Topic of Special Interest:
Controlling the
Numerical Error
309
cen58933_ch05.qxd
9/4/2002
11:41 AM
Page 265
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5–1
WHY NUMERICAL METHODS?
The ready availability of
highspeed computers
and easytouse
powerful soft
ware packages
has had a major impact on engineering education and practice
in recent years. Engineers in the past had to rely on
analytical skills
to solve
significant engineering problems, and thus they had to undergo a rigorous
training in mathematics. Today’s engineers, on the other hand, have access to
a tremendous amount of
computation power
under their fingertips, and they
mostly need to understand the physical nature of the problem and interpret the
results. But they also need to understand how calculations are performed by
the computers to develop an awareness of the processes involved and the lim
itations, while avoiding any possible pitfalls.
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 Spring '08
 mamoli
 Differential Equations, Numerical Analysis, Heat Transfer, Partial differential equation, finite difference

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