Page 362 - Bird R.B. Transport phenomena

344 Chapter 11 The Equations of Change for Nonisothermal Systems

and recognizing that viscous heating is unimportant in this flow. Assume that the tempera-
ture dependence of viscosity may be expressed by an equation of the form /x = Ae , with A
B/T
and В being empirical constants; this is suggested by the Eyring theory given in §1.5.
We first solve the energy equation to get the temperature profile, and then use the latter
to find the dependence of viscosity on position. Then the equation of motion can be solved to
get the velocity profile.

SOLUTION We postulate that T = T(x) and that v = 6 ^ (x). Then the energy equation simplifies to
2 z
2
d T = 0 (11.4-15)
dx 2
This can be integrated between the known terminal temperatures to give
T - T o _
x (11.4-16)
- S
T s T o
The dependence of viscosity on temperature may be written as
/л(Т)
(11.4-17)

in which В is a constant, to be determined from experimental data for viscosity versus tem-
perature. To get the dependence of viscosity on position, we combine the last two equations
to get

(11.4-18)

The second expression is a good approximation if the temperature does not change greatly
through the film. When this equation is combined with Eq. 11.4-17, written for T = T , we
5
then get
(11.4-19)
Mo/
This is the same as the expression used in Example 2.2-2, if we set a equal to —
Therefore we may take over the result from Example 2.2-2 and write the velocity profile as
Pg cos j3 (x/8)
v = (11.4-20)
7
This completes the analysis of the problem begun in Example 2.2-2, by providing the appro-
priate value of the constant a.

EXAMPLE 11.4-4 A system with two concentric porous spherical shells of radii KR and R is shown in Fig. 11.4-
1. The inner surface of the outer shell is at temperature T and the outer surface of the inner
u
Transpiration Cooling 2 shell is at a lower temperature T . Dry air at T is blown outward radially from the inner shell
K
K
into the intervening space and then through the outer shell. Develop an expression for the re-
quired rate of heat removal from the inner sphere as a function of the mass rate of flow of the
gas. Assume steady laminar flow and low gas velocity.
In this example the equations of continuity and energy are solved to get the temperature
distribution. The equation of motion gives information about the pressure distribution in the
system.

- M. Jakob, Heat Transfer, Vol. 2, Wiley, New York (1957), pp. 394^15.