Page 459 - Modelling in Transport Phenomena A Conceptual Approach

10.2. ENERGY TRANSPORT 439

at r=O e=i
at E=1 e=o (10.2-15)
at [=-1 8=0
Since the governing equation as well as the boundary conditions in the E-direction
are homogeneous, this parabolic partial differential equation can be solved by the
method of separation of variables as explained in Section B.6.1 in Appendix B.
The solution can be represented as a product of two functions of the form

(10.2-16)

so that Eq. (10.2-14) reduces to
1 dF 1 d2G
(10.217)

While the left-side of Eq. (10.2-17) is a function of r only, the right-side is depen-
dent only on E. This is possible only if both sides of Eq. (10.2-17) are equal to a
constant, say - x2, i.e.,
1 dF 1 d2G
=
- - - - -A2 (10.2-18)
F dr G g2
The choice of a negative constant is due to the fact that the solution will decay to
zero as time increases. The choice of a positive constant would give a solution that
becomes infinite as time increases.
Equation (10.2-18) results in two ordinary differential equations. The equation
for F is given by
dF
-+A2F=0 (10.2-19)
dr
The solution of Eq. (10.2-19) is

=
e-
~(7) X2r (10.2-20)
On the other hand, the equation for G is
dLG
-+X2G=0 (10.2-21)
g2
and it is subject to the boundary conditions

at [=1 G=O (10.2-22)
at c=-1 G=O (10.2-23)

Note that Eq. (10.2-21) is a Sturm-Liouville equation with a weight function of
unity. The solution of Eq. (10.2-21) is

G(E) = A sin(A[) + B cos(AE) (10.224)