Page 401 - Bird R.B. Transport phenomena

§12.2 Steady Heat Conduction in Laminar, Incompressible Flow 383

EXAMPLE 12.2-1 Solve Eq. 10.8-19 with the boundary conditions given in Eqs. 10.8-20,21, and 22.
Laminar Tube Flow
with Constant Heat SOLUTION
Flux at the Wall The complete solution for the temperature is postulated to be of the following form:

0(£ О = <3U& 0 ~ 0 (& 0 (12.2-4)
d
in which j ^ , 0 is the asymptotic solution given in Eq. 10.3-31, and 0 rf(£, О is a function that
e
will be damped out exponentially with time. By substituting the expression for 0(f, 0 in Eq.
12.2-4 into Eq. 10.8-19, it may be shown that the function 0 (f, О must satisfy Eq. 10.8-19 and
rf
also the following boundary conditions:

В.С1: atf = O, -jF = 0 (12.2-5)

B.C. 2: atf = l, ^f = ° (12.2-6)
B.C. 3: at £ = 0, 0 d = ©„(£, 0) (12.2-7)

We anticipate that a solution to the equation for 0 rf(f, £) will be factorable,
(12.2-8)

Then Eq. 10.8-19 can be separated into two ordinary differential equations

(12.2-9)

(12.2-10)

2
in which -c is the separation constant. Since the boundary conditions on X are dX/dg = 0 at
13
£ = 0, 1, we have a Sturm-Liouville problem. Therefore we know there will be an infinite
number of eigenvalues c k and eigenfunctions X k, and that the final solution must be of the
form:

О = в«(£ 0 ~ (12.2-11)
where

(12.2-12)

The problem is thus reduced to finding the eigenfunctions Х (£) by solving Eq. 12.2-10, and
к
then getting the eigenvalues c k by applying the boundary condition at £ = 1. This has been
done for к up to 7 for this problem. 14

13
M. D. Greenberg, Advanced Engineering Mathematics, Prentice-Hall, Upper Saddle River, N.J.,
Second Edition (1998), §17.7.
14 R. Siegel, E. M. Sparrow, and T. M. Hallman, Appl. Sci. Research, A7, 386-392 (1958).