Page 498 - Modelling in Transport Phenomena A Conceptual Approach

478 CHAPTER 11. UNSTEADY MICROSCOPIC BALANCES WTH GEN.

Therefore, the transient solution is
m
(1 1.1-45)
n=l
The unknown coefficients Cn can be determined by using the initial condition given
by Eq. (11.1-32). The result is
00
1 - t2 = Cn Jo(Xnt) (11.1-46)
n=l
Since the eigenfunctions are orthogonal to each other with respect to the weight
function w(<) = 5, multiplication of Eq. (11.1-46) by <Jo(Xn<) and integration
from 5 = 0 to = 1 gives

Note that the integral on the right-side of Eq. (11.1-47) is zero when n # m and
nonzero when n = m. Therefore, when n = m the summation drops out and Q.
(11.1-47) reduces to the form

(11.1-48)

Evaluation of the integrals gives

(1 1.1-49)

The transient solution takes the form
00
(1 1.1-50)

Substitution of the steady-state and the transient solutions, Eqs. (11.1-29) and
(11.1-50), into Eq. (11.1-24) gives the solution as

The volumetric flow rate can be determined by integrating the velocity distri-
bution over the cross-sectional area of the tube, i.e.,

(11.1-52)