Page 504 - Modelling in Transport Phenomena A Conceptual Approach

484 CHAPTER 11. UNSTEADY MICROSCOPIC BALANCES WITH GEN.

and it is subject to the boundary conditions
at <=O G=O (1 1.2-33)

at [=1 G=O (1 1.2-34)
Equation (11.2-32) is a Sturm-Liouville equation with a weight function of unity.
The solution of Eq. (11.2-32) is

G(<) = Asin(X<) + Bcos(XE) (11.2-35)
where A and B are constants. From Eq. (11.2-33), B = 0. Application of the
boundary condition defined by Eq. (11.2-34) gives
AsinX = 0 (1 1 .%36)

For a nontrivial solution, the eigenvalues are given by
sinX=O =+ An =nn n= 1,2, ... (11.237)
Therefore, the general solution is

(11.238)
n=l
The unknown coefficients Cn can be determined by using the initial condition, Eq.
(11.2-25), with the result

(1 1.239)

(1 1.2-40)

Therefore, the transient solution is given by

(11.241)

Substitution of the steady-state and the transient solutions, Eqs. (11.222) and
(11.241), into Eq. (11.2-18) gives the solution as