Page 284 - Bird R.B. Transport phenomena

268 Chapter 9 Thermal Conductivity and the Mechanisms of Energy Transport

Another possible generalization of Eq. 9.1-6 is to include a term containing the time
derivative of q multiplied by a time constant, by analogy with the Maxwell model of lin-
ear viscoelasticity in Eq. 8.4-3. There seems to be little experimental evidence that such a
generalization is warranted. 4
The reader will have noticed that Eq. 9.1-2 for heat conduction and Eq. 1.1-2 for vis-
cous flow are quite similar. In both equations the flux is proportional to the negative of
the gradient of a macroscopic variable, and the coefficient of proportionality is a physical
property characteristic of the material and dependent on the temperature and pressure.
For the situations in which there is three-dimensional transport, we find that Eq. 9.1-6 for
heat conduction and Eq. 1.2-7 for viscous flow differ in appearance. This difference
arises because energy is a scalar, whereas momentum is a vector, and the heat flux q is a
vector with three components, whereas the momentum flux т is a second-order tensor
with nine components. We can anticipate that the transport of energy and momentum
will in general not be mathematically analogous except in certain geometrically simple
situations.
In addition to the thermal conductivity k, defined by Eq. 9.1-2, a quantity known as
the thermal diffusivity a is widely used. It is defined as

а = Л~ (9.1-8)
C
P P
Here C p is the heat capacity at constant pressure; the circumflex (л) over the symbol indi-
cates a quantity "per unit mass." Occasionally we will need to use the symbol C p in
which the tilde (~) over the symbol stands for a quantity "per mole."
The thermal diffusivity a has the same dimensions as the kinematic viscosity v—
2
namely, (length) /time. When the assumption of constant physical properties is made,
the quantities v and a occur in similar ways in the equations of change for momentum
and energy transport. Their ratio v/a indicates the relative ease of momentum and en-
ergy transport in flow systems. This dimensionless ratio

(9.1-9)

5
is called the Prandtl number. Another dimensionless group that we will encounter in
6
subsequent chapters is the Peclet number, Pe = RePr.
The units that are commonly used for thermal conductivity and related quantities
are given in Table 9.1-1. Other units, as well as the interrelations among the various sys-
tems, may be found in Appendix F.
Thermal conductivity can vary all the way from about 0.01 W/m • К for gases to
about 1000 W/m • К for pure metals. Some experimental values of the thermal con-

4
The linear theory of thermoviscoelasticity does predict relaxation effects in heat conduction,
as discussed by R. M. Christensen, Theory of Viscoelasticity, Academic Press, 2nd edition (1982). The
effect has also been found from a kinetic theory treatment of the energy equation by R. B. Bird and
C. F. Curtiss, /. Non-Newtonian Fluid Mechanics, 79, 255-259 (1998).
0
This dimensionless group, named for Ludwig Prandtl, involves only the physical properties of
the fluid.
6 Jean-Claude-Eugene Peclet (pronounced "Pay-clay" with the second syllable accented)
(1793-1857) authored several books including one on heat conduction.