TRANSIENT HEAT
CONDUCTION
T
he temperature of a body, in general, varies with time as well
as position. In rectangular coordinates, this variation is expressed as
T
(
x
,
y
,
z
,
t
), where (
x
,
y
,
z
) indicates variation in the
x
,
y
, and
z
directions,
respectively, and
t
indicates variation with time. In the preceding chapter, we
considered heat conduction under
steady
conditions, for which the tempera
ture of a body at any point does not change with time. This certainly simpli
fied the analysis, especially when the temperature varied in one direction only,
and we were able to obtain analytical solutions. In this chapter, we consider
the variation of temperature with
time
as well as
position
in one and multi
dimensional systems.
We start this chapter with the analysis of
lumped systems
in which the tem
perature of a solid varies with time but remains uniform throughout the solid
at any time. Then we consider the variation of temperature with time as well
as position for onedimensional heat conduction problems such as those asso
ciated with a large plane wall, a long cylinder, a sphere, and a semiinfinite
medium using
transient temperature charts
and analytical solutions. Finally,
we consider transient heat conduction in multidimensional systems by uti
lizing the
product solution.
209
CHAPTER
4
CONTENTS
4–1
Lumped Systems Analysis
210
4–2
Transient Heat Conduction
in Large Plane Walls, Long
Cylinders, and Spheres
with Spatial Effects
216
4–3
Transient Heat Conduction
in SemiInfinite Solids
228
4–4
Transient Heat Conduction in
Multidimensional Systems
231
Topic of Special Interest:
Refrigeration and
Freezing of Foods
239
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Page 209
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4–1
LUMPED SYSTEM ANALYSIS
In heat transfer analysis, some bodies are observed to behave like a “lump”
whose interior temperature remains essentially uniform at all times during a
heat transfer process. The temperature of such bodies can be taken to be a
function of time only,
T
(
t
). Heat transfer analysis that utilizes this idealization
is known as
lumped system analysis,
which provides great simplification
in certain classes of heat transfer problems without much sacrifice from
accuracy.
Consider a small hot copper ball coming out of an oven (Fig. 4–1). Mea
surements indicate that the temperature of the copper ball changes with time,
but it does not change much with position at any given time. Thus the tem
perature of the ball remains uniform at all times, and we can talk about the
temperature of the ball with no reference to a specific location.
Now let us go to the other extreme and consider a large roast in an oven. If
you have done any roasting, you must have noticed that the temperature dis
tribution within the roast is not even close to being uniform. You can easily
verify this by taking the roast out before it is completely done and cutting it in
half. You will see that the outer parts of the roast are well done while the cen
ter part is barely warm. Thus, lumped system analysis is not applicable in this
case. Before presenting a criterion about applicability of lumped system
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 Spring '10
 Ghaz
 Heat Transfer, TI

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