Page 264 - Modelling in Transport Phenomena A Conceptual Approach

244 CHAPTER 8. STEADY MICROSCOPIC BALANCES WITHOUT GEN.

8.1.2.1 Investigation of the limiting case

Once the solution to a given problem is obtained, it is always advisable to in-
vestigate the limiting cases if possible, and compare the results with the known
solutions. If the results match, this does not necessarily mean that the solution is
correct, however, the chances of it being correct are fairly high.
In this case, when the ratio of the radius of the inner pipe to that of the outer
pipe is close to unity, i.e., ~d --+ 1, a concentric annulus may be considered to be a
thin-plane slit and its curvature can be neglected. Approximation of a concentric
annulus as a parallel plate requires the width, W, and the length, L, of the plate
to be defined as
w = aR(l+ K) (8.1-34)

B = R (1 - K) (8.1-35)
Therefore, the product WB is equal to

WB
WB = rR2(1 - K~) + aR2 = - (8.1-36)
1 - 62
so that Eq. (8.1-30) becomes

&=- WBV 1 (8.1-37)

Substitution of $ = 1 - K into Eq. (8.1-37) gives

(8.1-38)

The Taylor series expansion of the term ln(1- $) is

(8.1-39)

Using Eq. (8.1-39) in Eq. (8.1-38) and carrying out the divisions yields
+... -2(&-z--- 3+ ...)] (8.1-40)
3

&=- WBV (8.1-41)

Note that Eq. (8.1-41) is equivalent to Eq. (8.1-15).