HEAT CONDUCTION

1.1 Introduction

Heat conduction is one of the three basic modes of thermal energy transport (convection and

radiation being the other two) and is involved in virtually all process heat-transfer operations. In

commercial heat exchange equipment, for example, heat is conducted through a solid wall (often

a tube wall) that separates two fluids having different temperatures. Furthermore, the concept of

thermal resistance, which follows fromthe fundamental equations of heat conduction, is widely used

in the analysis of problems arising in the design and operation of industrial equipment. In addition,

many routine process engineering problems can be solved with acceptable accuracy using simple

solutions of the heat conduction equation for rectangular, cylindrical, and spherical geometries.

This chapter provides an introduction to the macroscopic theory of heat conduction and its engineering

applications. The key concept of thermal resistance, used throughout the text, is developed

here, and its utility in analyzing and solving problems of practical interest is illustrated.

1.2 Fourier’s Law of Heat Conduction

The mathematical theory of heat conduction was developed early in the nineteenth century by

Joseph Fourier [1]. The theory was based on the results of experiments similar to that illustrated

in Figure 1.1 in which one side of a rectangular solid is held at temperature T1, while the opposite

side is held at a lower temperature, T2. The other four sides are insulated so that heat can flow

only in the x-direction. For a given material, it is found that the rate, qx , at which heat (thermal

energy) is transferred from the hot side to the cold side is proportional to the cross-sectional area,

A, across which the heat flows; the temperature difference, T1 −T2; and inversely proportional to

the thickness, B, of the material. That is:

qx ∝ A(T1 − T2)

B

Writing this relationship as an equality, we have:

qx = kA(T1 − T2)

B

(1.1)

T2

qx

x

Insulated

Insulated

Insulated

B

qx

T1

Figure 1.1 One-dimensional heat conduction in a solid.

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HEAT CONDUCTION 1/3

The constant of proportionality, k, is called the thermal conductivity. Equation (1.1) is also applicable

to heat conduction in liquids and gases. However, when temperature differences exist in fluids, convection

currents tend to be set up, so that heat is generally not transferred solely by the mechanism

of conduction.

The thermal conductivity is a property of the material and, as such, it is not really a constant, but

rather it depends on the thermodynamic state of the material, i.e., on the temperature and pressure

of the material. However, for solids, liquids, and low-pressure gases, the pressure dependence is

usually negligible. The temperature dependence also tends to be fairly weak, so that it is often

acceptable to treat k as a constant, particularly if the temperature difference is moderate. When the

temperature dependence must be taken into account, a linear function is often adequate, particularly

for solids. In this case,

k = a + bT (1.2)

where a and b are constants.

Thermal conductivities of a number of materials are given in Appendices 1.A–1.E. Many other

values may be found in various handbooks and compendiums of physical property data. Process

simulation software is also an excellent source of physical property data. Methods for estimating

thermal conductivities of fluids when data are unavailable can be found in the authoritative book

by Poling et al. [2].

The form of Fourier’s law given by Equation (1.1) is valid only when the thermal conductivity

can be assumed constant. A more general result can be obtained by writing the equation for an

element of differential thickness. Thus, let the thickness be x and let T =T2 −T1. Substituting

in Equation (1.1) gives:

qx = −kAT

x

(1.3)

Now in the limit as x approaches zero,

T

x

→ dT

dx

and Equation (1.3) becomes:

qx = −kA

dT

dx

(1.4)

Equation (1.4) is not subject to the restriction of constant k. Furthermore, when k is constant, it can

be integrated to yield Equation (1.1). Hence, Equation (1.4) is the general one-dimensional form of

Fourier’s law. The negative sign is necessary because heat flows in the positive x-direction when

the temperature decreases in the x-direction. Thus, according to the standard sign convention that

qx is positive when the heat flow is in the positive x-direction, qx must be positive when dT/dx is

negative.

It is often convenient to divide Equation (1.4) by the area to give:

ˆqx ≡ qx/A = −k

dT

dx

(1.5)

where ˆqx is the heat flux. It has units of J/s ·m2 =W/m2 or Btu/h · ft2. Thus, the units of k are

W/m·K or Btu/h · ft · ◦F.

Equations (1.1), (1.4), and (1.5) are restricted to the situation in which heat flows in the x-direction

only. In the general case in which heat flows in all three coordinate directions, the total heat flux is

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1/4 HEAT CONDUCTION

obtained by adding vectorially the fluxes in the coordinate directions. Thus,

→ˆ

q = ˆqx

→i

+ ˆqy

→j

+ ˆqz

→k

(1.6)

where

→ˆ

q is the heat flux vector and

→i

,

→j

,

→k

are unit vectors in the x-, y-, z-directions, respectively.

Each of the component fluxes is given by a one-dimensional Fourier expression as follows:

ˆqx = −k ∂T

∂x

ˆqy = −k ∂T

∂y

ˆqz = −k ∂T

∂z

(1.7)

Partial derivatives are used here since the temperaturenowvaries in all three directions. Substituting

the above expressions for the fluxes into Equation (1.6) gives:

→ˆ

q = −k

∂T

∂x

→i

+ ∂T

∂y

→j

+ ∂T

∂z

→k

(1.8)

The vector in parenthesis is the temperature gradient vector, and is denoted by

→

∇T. Hence,

→ˆ

q = −k

→

∇T (1.9)

Equation (1.9) is the three-dimensional form of Fourier’s law. It is valid for homogeneous, isotropic

materials for which the thermal conductivity is the same in all directions.

Equation (1.9) states that the heat flux vector is proportional to the negative of the temperature

gradient vector. Since the gradient direction is the direction of greatest temperature increase, the

negative gradient direction is the direction of greatest temperature decrease. Hence, Fourier’s law

states that heat flows in the direction of greatest temperature decrease.