Page 497 - Bird R.B. Transport phenomena

Problems 477

(b) Substitute the result in (a) for T h - T c into Eq. 15.4-12, and integrate the equation thus ob-
tained over the length of the exchanger. Use this result to show that 1

(15B.1-2)

15B.2o Pressure drop in turbulent flow in a slightly converging tube (Fig. 15B.2). Consider the tur-
bulent flow of an incompressible fluid in a circular tube with a diameter that varies linearly
with distance according to the relation

D = D ] + (D 2-D ])y (15B.2-1)

At z = 0, the velocity is v x and may be assumed to be constant over the cross section. The
Reynolds number for the flow is such that / is given approximately by the Blasius formula of
Eq. 6.2-13,
0.0791
f (15B.2-2)
Re 1/4
- m D 2, p, L, and v = /x/p.
Obtain the pressure drop p } p 2 terms of v uD u
(a) Integrate the d-form of the mechanical energy balance to get

(15B.2-3)

and then eliminate v 2 from the equation.
(b) Show that both v and / are functions of D:
0.0791
(15B.2-4)

Of course, D is a function of z according to Eq. 15B.2-1.
(c) Make a change of variable in the integral in Eq. 15B.2-3 and show that

(15B.2-5)

(d) Combine the results of (b) and (c) to get finally

^ - 1 ^ - 1 (15B.2-6)

(e) Show that this result simplifies properly for D^ = D 2.

Diameter D
Diameter D 2

2 Fig. 15B.2o Turbulent flow in a hori-
1
2 = 0 Direction of flow z = L zontal, slightly tapered tube (D } is
(z direction) slightly greater than D 2).

1
A. P. Colburn, lnd. Eng. Chetn., 25, 873 (1933).