Page 467 - Modelling in Transport Phenomena A Conceptual Approach

10.2. ENERGY TRANSPORT 447

reduces Eqs. (10.2-7), (10.2-62), (10.2-63) and (10.2-64) to
ae
_- a2e (10.2-69)
--
67 at2

at 7=o e=i (10.2-70)
at t=o ae (10.2-71)
--
at -O
ae
at --- (10.2-72)
at - BiH e
The use of the method of separation of variables in which the solution is sought in
the form
e(r, E) = F(7) G(t) (10.2-73)
reduces the differential equation, Eq. (10.2-69) to

1 dF - 1 d2G (10.2-74)
F dr G dt2 = - X2
Equation (10.2-74) results in two ordinary differential equations:

dF + X2F = 0 j F(r) = e-A2T (10.2-75)
dr
d2G
+
- X2G = 0 + G(<) = Asin(Xt) + B cos(Xt) (10.2-76)
dt2
Therefore, the solution becomes

6 = e-A2T [Asin(X<) +Bcos(X<)] (10.2-77)
The application of Eq. (10.2-71) indicates that A = 0. Application of the boundary
condition defined by Eq. (10.2-72) gives

B X e- sin X = BiH B e- cos X (10.2-78)
Solving for X yields
A, tan A, = BiH (10.279)
The first five roots of Eq. (10.2-79) axe given as a function of BiH in Table 10.3.
The general solution is the summation of all possible solutions, i.e.,

M
(10.2-80)
n=l