Page 580 - Modelling in Transport Phenomena A Conceptual Approach

560 APPENDUC B. SOLUTIONS OF DI.F’FER.ENT!lAL EQUATIONS

TI > To. The governing differential equation together with the initial and boundary
conditions are
aT
-=a- d2T (B.3-28)
at at2
at t = 0 T = To for all 2 (B.3-29)
at x=O T=T1 t>O (B.3-30)

at x=L T=Tl t>O (B.3-31)
?.&e that Le differential equation is linear and homogeneous. On the other
hand, the boundary conditions, although linear, are not homogeneous. The bound-
ary conditions in the x-direction become homogeneous by introducing the dimen-
sionless quantities
e=- Ti -T (B.3-32)
Tl - To
c=z (B.3-33)
z
at
T=- (B.3-34)
L2
In dimensionless form, Eqs. (B.3-28)-(B.3-31) become

de d2e
-=- (B.3-35)
at2
at T=O 8=1 (B .3-36)

at c=O 8=0 (B.3-37)
at c=1 8=0 (B.3-38)
The separation of variables method assumes that the solution can be represented
as a product of two functions of the form

e(T, a = F(7) G(c) (B .3-39)

Substitution of Eq. (B.3-39) into Eq. (B.3-35) and rearrangement gives

(B.3-40)

While the left side of Eq. (B.3-40) is a function of 7 only, the right side is dependent
only on e. This is possible only if both sides of Eq. (B.3-40) are equal to a constant,
2
say - X , i.e.,
1 dF
_---
--- 1 dLG - -A2 (B.3-41)
F dr G q2