Page 503 - Modelling in Transport Phenomena A Conceptual Approach

11.2. UNSTEADY CONDUCTION WTH HEAT GENERATION 483

at <=1 O,=O (11.2-21)

The solution of Eq. (11.2-19) is

(11.2-22)

The use of Eq. (11.2-22) in Eq. (11.2-18) gives

52
e(r,r) = F(t - s2> - @t(r,<) (1 1 2-23)

Substitution of Eq. (11.2-23) into Eqs. (11.2-13)-(11.2-16) leads to the following
governing equation for the transient problem together with the initial and the
boundary conditions
ae, - d2et
ar at2 (1 1.2-24)
(11.2-25)

at [=O Bt=0 (11.2-26)
at [=l &=O (11.2-27)
which can be solved by the method of separation of variables.
Representing the solution as a product of two functions of the form
@t(T, 5) = F(7) G(5) (11.2-28)

reduces Eq. (11.2-24) to

1dF - GE dt (‘g) (11.2-29)
1 d
F dr
While the left side of Eq. (11.2-29) is a function of r only, the right side is dependent
only on 5. This is possible if both sides of Eq. (11.2-29) are equal to a constant,
say - x2, Le.,
1 dF - 1 d2G
=
- - - - - -A2 (1 1.2-30)
F dr G e2
Equation (11.2-30) results in two ordinary differential equations. The equation for
F is given by
dF
+
- A2F = 0 + F(r) = e-x27 (1 1.2-31)
dr
On the other hand, the equation for G is

(1 1.2-32)