Page 360 - Modelling in Transport Phenomena A Conceptual Approach

340 CHAPTER 9. STEADY MICROSCOPIC BALANCES WITH GEN.

Elimination of (Po - PL) between Eqs. (9.1-100) and (9.1-105) gives

(9.1- 106)

Since Dh = 2R(1 - K), the Reynolds number based on the hydraulic equivalent
diameter is
(9.1-107)

so that Eq. (9.1-106) becomes

9.1.4.2 Investigation of the limiting cases

1
R Case (i) K t
When the ratio of the radius of the inner pipe to that of the outer pipe is close to
unity, Le., n --f 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=.IrR(l+/C) (9.1-109)
B = R(1- K) (9.1-110)

Therefore, the product WB3 is equal to
WB3
WB3 = ~R4(1- ~~)(1- K)~ + xR4 = (9.1-1 11)
(1 - K2) (1 - K)2

so that Eq. (9.1-99) becomes

(9.1.112)

Substitution of + = 1 - K into Eq. (9.1-112) gives

(Po - PL) WB3 lim [
+2 - 2+ + 2
&= 1 (9.1-113)
2-+
8 PL +-to $2 + +W-+)