Page 412 - Bird R.B. Transport phenomena

394 Chapter 12 Temperature Distributions with More Than One Independent Variable

Fig. 12.4-3. Forced-convection heat transfer from a
sphere in creeping flow. The shaded region shows
the thermal boundary layer (defined by П < 0.99
т
or у < 1.55 ) for Pe = RePr « 200.
T
Separation point
(6 = 0)
= r-R

Stagnation point
(б» = 77")
t t t t t t t f t
Fluid approaching with velocity v x
and temperature T^

also the region where most of the heat transfer occurs. The result for Q is good within about
5% for RePr > 100; this limits its use primarily to fluids with Pr > 100, since creeping flow is
obtained only at Re of the order of 1 or less. 12
Results of the same form as Eq. 12.4-34 are obtained for creeping flow in other geome-
tries, including packed beds. 813
It should be emphasized that the asymptotic solutions are particularly important: they
are relatively easy to obtain, and for many applications they are sufficiently accurate. We will
see in Chapter 14 that some of the standard heat transfer correlations are based on asymptotic
solutions of the type discussed here.

QUESTIONS FOR DISCUSSION
1. How does Eq. 12.1-2 have to be modified if there is a heat source within the solid?
2. Show how Eq. 12.1-10 is obtained from Eq. 12.1-8. What is the viscous flow analog of this
equation?
3. What kinds of heat conduction problems can be solved by Laplace transform and which can-
not?
4. In Example 12.1-3 the heat flux and the temperature both satisfy the "heat conduction equa-
tion." Is this always true?
5. Draw a carefully labeled sketch of the results in Eqs. 12.1-38 and 40 showing what is meant by
the statement that the "temperature oscillations lag behind the heat flux oscillations by тг/4."
6. Verify that Eq. 12.1-40 satisfies the boundary conditions. Does it have to satisfy an initial con-
dition? If so, what is it?
7. In Ex. 12.2-1, would the method of separation of variables work if applied directly to the func-
tion 0(& 0 rather than to 0 rf(£ f)?
8o In Example 12.2-2, how does the wall temperature depend on the downstream coordinate z?
9. By means of a carefully labeled diagram, show what is meant by the two cases A < 1 and A ^
1 in §12.4. Which case applies to dilute polyatomic gases? Organic liquids? Molten metals?
10. Summarize the situations in which the four mathematical methods in §12.1 are applicable.

12
A review of analyses for a wide range of Pe = RePr is given by S. K. Friedlander, AIChE Journal, 7,
347-348 (1961).
13
J. P. S0rensen and W. E. Stewart, Chem. Eng. Sci., 29, 833-837 (1974).