Page 282 - Bird R.B. Transport phenomena

266 Chapter 9 Thermal Conductivity and the Mechanisms of Energy Transport

§9.1 FOURIER'S LAW OF HEAT CONDUCTION
(MOLECULAR ENERGY TRANSPORT)
Consider a slab of solid material of area A located between two large parallel plates a
distance У apart. We imagine that initially (for time t < 0) the solid material is at a tem-
perature T throughout. At t — 0 the lower plate is suddenly brought to a slightly higher
o
temperature T and maintained at that temperature. As time proceeds, the temperature
x
profile in the slab changes, and ultimately a linear steady-state temperature distribution
is attained (as shown in Fig. 9.1-1). When this steady-state condition has been reached, a
constant rate of heat flow Q through the slab is required to maintain the temperature dif-
ference AT = Т г - T . It is found then that for sufficiently small values of AT the follow-
o
ing relation holds:

(9.1-1)
Y
That is, the rate of heat flow per unit area is proportional to the temperature decrease
over the distance У. The constant of proportionality к is the thermal conductivity of the
slab. Equation 9.1-1 is also valid if a liquid or gas is placed between the two plates, pro-
vided that suitable precautions are taken to eliminate convection and radiation.
In subsequent chapters it is better to work with the above equation in differential
form. That is, we use the limiting form of Eq. 9.1-1 as the slab thickness approaches zero.
The local rate of heat flow per unit area (heat flux) in the positive у direction is desig-
nated by q . In this notation Eq. 9.1-1 becomes
XJ
q =-kj- (9.1-2)
y
This equation, which serves to define k, is the one-dimensional form of Fourier's law of
2
heat conduction}' It states that the heat flux by conduction is proportional to the tempera-

Solid initially at
t<0 temperature T o

Lower plate
. . . suddenly raised
to temperature T }

Tiy.t) Small t

Fig. 9.1-1. Development of the
steady-state temperature pro-
Large t file for a solid slab between two
parallel plates. See Fig. 1.1-1 for
the analogous situation for mo-
mentum transport.