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Chapter 2
HEAT CONDUCTION
EQUATION
Mehmet Kanoglu
University of Gaziantep
Copyright © 2011 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Heat and Mass Transfer: Fundamentals & Applications
Fourth Edition
Yunus A. Cengel, Afshin J. Ghajar
McGraw-Hill, 2011

Objectives
• Understand multidimensionality and time dependence of heat transfer, and
the conditions under which a heat transfer problem can be approximated as
being one-dimensional.
• Obtain the differential equation of heat conduction in various coordinate
systems, and simplify it for steady one-dimensional case.
• Identify the thermal conditions on surfaces, and express them mathematically
as boundary and initial conditions.
• Solve one-dimensional heat conduction problems and obtain the temperature
distributions within a medium and the heat flux.
• Analyze one-dimensional heat conduction in solids that involve heat
generation.
• Evaluate heat conduction in solids with temperature-dependent thermal
conductivity.
2

INTRODUCTION
• Although heat transfer and temperature are closely related, they are of a different
nature.
• Temperature has only magnitude. It is a scalar quantity.
• Heat transfer has direction as well as magnitude. It is a vector quantity.
• We work with a coordinate system and indicate direction with plus or minus signs.
3

• The driving force for any form of heat transfer is the temperature difference.
• The larger the temperature difference, the larger the rate of heat transfer.
• Three prime coordinate systems:
• rectangular T(x, y, z, t)
• cylindrical T(r, , z, t)
• spherical T(r, , , t).
4

5
• Steady implies no change
with time at any point within
the medium
• Transient implies variation
with time or time
dependence
• In the special case of
variation with time but not
with position, the
temperature of the medium
changes uniformly with
time. Such heat transfer
systems are called lumped
systems.
Steady versus Transient Heat Transfer

Multidimensional Heat Transfer
• Heat transfer problems are also classified as being:
• one-dimensional
• two dimensional
• three-dimensional
• In the most general case, heat transfer through a medium is three-
dimensional. However, some problems can be classified as two- or one-
dimensional depending on the relative magnitudes of heat transfer rates in
different directions and the level of accuracy desired.
• One-dimensional if the temperature in the medium varies in one direction
only and thus heat is transferred in one direction, and the variation of
temperature and thus heat transfer in other directions are negligible or zero.
• Two-dimensional if the temperature in a medium, in some cases, varies
mainly in two primary directions, and the variation of temperature in the third
direction (and thus heat transfer in that direction) is negligible.
6

• The rate of heat conduction through a medium in a specified direction (say, in
the x-direction) is expressed by Fourier’s law of heat conduction for one-
dimensional heat conduction as:
8
Heat is conducted in the direction
of decreasing temperature, and
thus the temperature gradient is
negative when heat is conducted
in the positive x -direction.

• The heat flux vector at a point P on the
surface of the figure must be
perpendicular to the surface, and it
must point in the direction of decreasing
temperature
• If n is the normal of the isothermal
surface at point P, the rate of heat
conduction at that point can be
expressed by Fourier’s law as
9

Heat
Generation
• Examples:
• electrical energy being converted to heat at a rate of I2R,
• fuel elements of nuclear reactors,
• exothermic chemical reactions.
• Heat generation is a volumetric phenomenon.
• The rate of heat generation units : W/m3 or Btu/h·ft3.
• The rate of heat generation in a medium may vary with time as well as
position within the medium.
10

Heat Generation in a Hair Dryer
• The resistance wire of a 1200-W
hair dryer is 80 cm long and has a
diameter of D = 0.3
• Determine the rate of heat
generation in the wire per unit
volume,
• and the heat flux on the outer
surface of the wire as a result of
this heat generation
11

12
ONE-DIMENSIONAL HEAT CONDUCTION
EQUATION
Consider heat conduction through a large plane wall such as the wall of a
house, the glass of a single pane window, the metal plate at the bottom of
a pressing iron, a cast-iron steam pipe, a cylindrical nuclear fuel element,
an electrical resistance wire, the wall of a spherical container, or a
spherical metal ball that is being quenched or tempered.
Heat conduction in these and many other geometries can be
approximated as being one-dimensional since heat conduction through
these geometries is dominant in one direction and negligible in other
directions.
Next we develop the one-dimensional heat conduction equation in
rectangular, cylindrical, and spherical coordinates.

13
(2-6)
Heat Conduction
Equation in a Large
Plane Wall

15
Heat
Conduction
Equation in a
Long Cylinder

17
Heat Conduction Equation
in a Sphere

18
Combined One-Dimensional Heat Conduction
Equation
An examination of the one-dimensional transient heat conduction
equations for the plane wall, cylinder, and sphere reveals that all
three equations can be expressed in a compact form as
n = 0 for a plane wall
n = 1 for a cylinder
n = 2 for a sphere
In the case of a plane wall, it is customary to replace the variable
r by x.
This equation can be simplified for steady-state or no heat
generation cases as described before.

Heat Conduction Through the Bottom of a Pan
• Consider a steel pan placed on top of an
electric range to cook spaghetti. The
bottom section of the pan is 0.4 cm thick
and has a diameter of 18 cm. The
electric heating unit on the range top
consumes 800 W of power during
cooking, and 80 percent of the heat
generated in the heating element is
transferred uniformly to the pan.
Assuming constant thermal conductivity,
obtain the differential equation that
describes the variation of the
temperature in the bottom section of
the pan during steady operation.
19

Heat Conduction in a Resistance Heater
• A 2-kW resistance heater wire with
thermal conductivity k = 15 W/m⋅K,
diameter D = 0.4 cm, and length
L = 50 cm is used to boil water by
immersing it in water. Assuming the
variation of the thermal conductivity
of the wire with temperature to be
negligible, obtain the differential
equation that describes the variation
of the temperature in the wire during
steady operation.
20

21
GENERAL HEAT CONDUCTION EQUATION
In the last section we considered one-dimensional heat conduction
and assumed heat conduction in other directions to be negligible.
Most heat transfer problems encountered in practice can be
approximated as being one-dimensional, and we mostly deal with
such problems in this text.
However, this is not always the case, and sometimes we need to
consider heat transfer in other directions as well.
In such cases heat conduction is said to be multidimensional, and
in this section we develop the governing differential equation in
such systems in rectangular, cylindrical, and spherical coordinate
systems.

25
Cylindrical Coordinates
Relations between the coordinates of a point in rectangular
and cylindrical coordinate systems:

26
Spherical Coordinates
Relations between the coordinates of a point in rectangular
and spherical coordinate systems:

27
BOUNDARY AND INITIAL CONDITIONS
The description of a heat transfer problem in a medium is not complete without a full
description of the thermal conditions at the bounding surfaces of the medium.
Boundary conditions: The mathematical expressions of the thermal conditions at the
boundaries.
The temperature at any
point on the wall at a
specified time depends
on the condition of the
geometry at the
beginning of the heat
conduction process.
Such a condition, which
is usually specified at
time t = 0, is called the
initial condition, which
is a mathematical
expression for the
temperature distribution
of the medium initially.

28
• Specified Temperature Boundary Condition
• Specified Heat Flux Boundary Condition
• Convection Boundary Condition
• Radiation Boundary Condition
• Interface Boundary Conditions
• Generalized Boundary Conditions
Boundary Conditions

29
1 Specified Temperature Boundary Condition
The temperature of an exposed surface
can usually be measured directly and
easily.
Therefore, one of the easiest ways to
specify the thermal conditions on a surface
is to specify the temperature.
For one-dimensional heat transfer through
a plane wall of thickness L, for example,
the specified temperature boundary
conditions can be expressed as
where T1 and T2 are the specified
temperatures at surfaces at x = 0 and
x = L, respectively.
The specified temperatures can be
constant, which is the case for steady
heat conduction, or may vary with time.

30
2 Specified Heat Flux Boundary Condition
For a plate of thickness L subjected to heat
flux of 50 W/m2 into the medium from both
sides, for example, the specified heat flux
boundary conditions can be expressed as
The heat flux in the positive x-direction anywhere in the
medium, including the boundaries, can be expressed by

31
Special Case: Insulated Boundary
A well-insulated surface can be modeled
as a surface with a specified heat flux of
zero. Then the boundary condition on a
perfectly insulated surface (at x = 0, for
example) can be expressed as
On an insulated surface, the first
derivative of temperature with respect
to the space variable (the temperature
gradient) in the direction normal to the
insulated surface is zero.

32
Another Special Case: Thermal Symmetry
Some heat transfer problems possess thermal
symmetry as a result of the symmetry in imposed
thermal conditions.
For example, the two surfaces of a large hot plate
of thickness L suspended vertically in air is
subjected to the same thermal conditions, and thus
the temperature distribution in one half of the plate
is the same as that in the other half.
That is, the heat transfer problem in this plate
possesses thermal symmetry about the center
plane at x = L/2.
Therefore, the center plane can be viewed as an
insulated surface, and the thermal condition at this
plane of symmetry can be expressed as
which resembles the insulation or zero heat
flux boundary condition.

33
3 Convection Boundary Condition
For one-dimensional heat transfer in the x-direction
in a plate of thickness L, the convection boundary
conditions on both surfaces:

34
4 Radiation Boundary Condition
For one-dimensional heat transfer in the
x-direction in a plate of thickness L, the
radiation boundary conditions on both
surfaces can be expressed as
Radiation boundary condition on a surface:

35
5 Interface Boundary Conditions
The boundary conditions at an interface
are based on the requirements that
(1) two bodies in contact must have the
same temperature at the area of contact
and
(2) an interface (which is a surface)
cannot store any energy, and thus the
heat flux on the two sides of an interface
must be the same.
The boundary conditions at the interface
of two bodies A and B in perfect contact at
x = x0 can be expressed as

36
6 Generalized Boundary Conditions
In general, however, a surface may involve convection,
radiation, and specified heat flux simultaneously.
The boundary condition in such cases is again obtained
from a surface energy balance, expressed as

37
SOLUTION OF STEADY ONE-DIMENSIONAL
HEAT CONDUCTION PROBLEMS
In this section we will solve a wide range of heat
conduction problems in rectangular, cylindrical,
and spherical geometries.
We will limit our attention to problems that result
in ordinary differential equations such as the
steady one-dimensional heat conduction
problems. We will also assume constant thermal
conductivity.
The solution procedure for solving heat
conduction problems can be summarized as
(1) formulate the problem by obtaining the
applicable differential equation in its simplest
form and specifying the boundary conditions,
(2) Obtain the general solution of the differential
equation, and
(3) apply the boundary conditions and determine
the arbitrary constants in the general solution.

46
HEAT GENERATION IN A SOLID
Many practical heat transfer applications
involve the conversion of some form of energy
into thermal energy in the medium.
Such mediums are said to involve internal heat
generation, which manifests itself as a rise in
temperature throughout the medium.
Some examples of heat generation are
- resistance heating in wires,
- exothermic chemical reactions in a solid, and
- nuclear reactions in nuclear fuel rods
where electrical, chemical, and nuclear
energies are converted to heat, respectively.
Heat generation in an electrical wire of outer
radius ro and length L can be expressed as

47
The quantities of major interest in a medium with
heat generation are the surface temperature Ts
and the maximum temperature Tmax that occurs
in the medium in steady operation.

49
VARIABLE THERMAL CONDUCTIVITY, k(T)
• the thermal conductivity of a material, in general, varies with temperature.
• However, this variation is mild for many materials in the range of practical
interest and can be disregarded. (Use average value)
• When the variation of thermal conductivity with
temperature k(T) is known, the average value of the
thermal conductivity:
Then the rate of steady heat transfer through a plane wall, cylindrical
layer, or spherical layer

50
The variation in thermal conductivity of a material with temperature in the
temperature range of interest can often be approximated as a linear
function
where β is called the temperature coefficient of thermal conductivity.

53
Example (2): Variation of Temperature in a Resistance Heater
A long homogeneous resistance wire of radius r0=0.5 cm and
thermal conductivity k=13.5 W/m.°C is being used to boil water at
atmospheric pressure by the passage of electric current. Heat is
generated in the wire uniformly as a result of resistance heating at a
rate of egen=4.3 x 107 W/m3. If the outer surface temperature of the
wire is measured to be Ts= 108C, obtain a relation for the
temperature distribution, and determine the temperature at the
centerline of the wire when steady operating conditions are reached.

54
Example (3): Centerline Temperature of a Resistance Heater
A 2-kW resistance heater wire whose thermal conductivity is k=15 W/m.K has a
diameter of D=4mm and a length of L=0.5m, and is used to boil water. If the
outer surface temperature of the resistance wire is Ts=105 ºC, determine the
temperature at the center of the wire.

55
Example (4): Heat Conduction in a Solar Heated wall.
Consider a large plane wall of thickness L=0.06m and thermal conductivity k=1.2
W/m.K in space. The wall is covered with white porcelain tiles that have an
emissivity of ε=0.85 and a solar absorptivity of α=0.26. The inner surface of the wall
is maintained at T1=300K at all times, while the outer surface is exposed to solar
radiation that is incident at a rate of qsolar=800W/m2. The outer surface is also
losing heat by radiation to deep space at 0K. Determine the temperature of the outer
surface of the wall and the rate of heat transfer through the wall when steady
operating conditions are reached. What would your response be if no solar radiation
was incident on the surface?

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