Page 423 - Bird R.B. Transport phenomena

Problems 405

(d) Obtain the lowest eigenvalue by the method of Stodola and Vianello. Use Eqs. 71a and
12
2
72b on p. 203 of Hildebrand's book, with ф = 2(1 - £ ) for Newtonian flow and Xi = 1 - £ 2
as a simple, but suitable, trial function. Show that this leads quickly to the value 0\ = 3.661.
12D.3. The Graetz-Nusselt problem (asymptotic solution for large z). Note that, in the limit of very
large z, only one term (i = 1) is needed in Eq. 12D.2-2. It is desired to use this result to com-
pute the heat flux at the wall, q 0, at large z and to express the result as
q Q = (a function of system and fluid properties) X (T Ll - T o) (12D.3-1)

where T b is the "bulk fluid temperature" defined in Eq. 10.8-33.
(a) First verify that

1
% = ~ of <J b - T o ) (12D.3-2)
Here © is the same as in Problem 12D.2, and & b = (T b - T Q)/{T X - T o).
(b) Show that for large z, Eq. 12D.3-2 and Eq. 12D.2-2 both give
qo = ^fi(T b-T o) (12D.3-3)

Hence for large z, all one needs to know is the first eigenvalue; the eigenfunctions need not be
12
calculated. This shows how useful the method of Stodola and Vianello is for computing the
limiting value of a heat flux.
12D.4. The Graetz-Nusselt problem (asymptotic solution for small z).
(a) Apply the method of Example 12.2-2 to the solution of the problem discussed in Problem
12D.2. Consider a Newtonian fluid and use the following dimensionless quantities:

Show that the method of combination of variables gives

3
© = \ Г е х р ( - ^ ) ^ (12D.4-2)
Г(з) v
J
in which ri = (Na /9& .
3
U3
(b) Show that the wall flux is
r
( Г
-"
[
*!'-« = I ^ ( Re Pr тГ] ' " °> (12D 4 3)
The quantity (Re Pr D/z) = (4/Tr)(ivC /kz) appears frequently; the grouping Gz = (wC /kz) is
p
p
called the Graetz number. Compare this result with that in Eq. 12D.3-3, with regard to the de-
pendence on the dimensionless groups.
(c) How may the results be written so that they are valid for any generalized Newtonian
model?
12D.5. The Graetz problem for flow between parallel plates. Work through Problems 12D.2,3, and
4 for flow between parallel plates (or flow in a thin rectangular duct).
12D.6. The constant wall heat flux problem for parallel plates. Apply the methods used in §10.8,
Example 12.2-1, and Ex. 12.2-2 to the flow between parallel plates.

12
F. B. Hildebrand, Advanced Calculus for Applications, Prentice-Hall, Englewood Cliffs, NJ. (1963),
§5.5.