Page 422 - Bird R.B. Transport phenomena

404 Chapter 12 Temperature Distributions with More Than One Independent Variable

that viscous dissipation and axial heat conduction effects are negligible. Use the following di-
mensionless variables:

Show that the temperature profiles in this system are

2 3?f) (12D.2-2)

in which X/ and /3 ; are the eigenfunctions and eigenvalues obtained from the solution to the
following equation:

* * * - 0 ( 1 2 D 2 3 )
- -
with boundary conditions X = finite at £ = 0 and X = 0 at £ = 1. Show further that

A, = - ^ (12D.2-4)

(b) Solve Eq. 12D.2-3 for the Newtonian fluid by obtaining a power series solution for X ;. Cal-
culate the lowest eigenvalue by solving an algebraic equation. Check your result against that
given in Table 12D.2.
(c) From the work involved in (b) in computing fi] it can be inferred that the computation of
the higher eigenvalues is quite tedious. For eigenvalues higher than the second or third the
10
Wenzel-Kramers-Brillouin (WKB) method can be used; the higher the eigenvalue, the more
accurate the WKB method is. Read about this method, and verify that for the Newtonian fluid
$ = \{M ~ If (12D.2-5)

A similar formula has been derived for the power law model. 11

Table 12D.2 Eigenvalues pj for the Graetz-Nusselt Problem for Newtonian Fluids"
By Stodola and
i By direct calculation* 7 By WKB method' Vianello method^
3.56
3.67
1 3.67 3.56 3.661 е
2 22.30 22.22
3 56.95 56.88
4 107.6 107.55
a
The (3j here correspond to \k] in W. M. Rohsenow, J. P. Hartnett, and Y. I. Cho, Handbook of
Heat Transfer, McGraw-Hill (New York), Table 5.3 on p. 510.
b
Values taken from K. Yamagata, Memoirs of the Faculty of Engineering, Kyushu University,
Volume VIII, No. 6, Fukuoka, Japan (1940).
c Computed from Eq. 12D.2-5.
d For the particular trial function in part (d) of the problem.

10
J. Heading, An Introduction to Phase-Integral Methods, Wiley, New York (1962); J. R. Sellars,
M. Tribus, and J. S. Klein, Trans. ASME, 78,441-448 (1956).
"I. R. Whiteman and W. B. Drake, Trans. ASME, 80, 728-732 (1958).