Page 78 - Bird R.B. Transport phenomena

Problems 63

2В.1 Different choice of coordinates for the falling film problem. Rederive the velocity profile
and the average velocity in §2.2, by replacing x by a coordinate x measured away from the
wall; that is, x = 0 is the wall surface, and x = 8 is the liquid-gas interface. Show that the ve-
locity distribution is then given by

2
2
v = (pg8 /fjL)[(x/8) - \(x/8) \ cos /3 (2B.1-1)
z
and then use this to get the average velocity. Show how one can get Eq. 2B.1-1 from Eq. 2.2-18
by making a change of variable.
2B.2 Alternate procedure for solving flow problems. In this chapter we have used the following
procedure: (i) derive an equation for the momentum flux, (ii) integrate this equation, (iii) insert
Newton's law to get a first-order differential equation for the velocity, (iv) integrate the latter
to get the velocity distribution. Another method is: (i) derive an equation for the momentum
flux, (ii) insert Newton's law to get a second-order differential equation for the velocity profile,
(iii) integrate the latter to get the velocity distribution. Apply this second method to the falling
film problem by substituting Eq. 2.2-14 into Eq. 2.2-10 and continuing as directed until the ve-
locity distribution has been obtained and the integration constants evaluated.

2B.3 Laminar flow in a narrow slit (see Fig. 2B.3).

Fluid in

Fluid out Fig. 2B.3 Flow through a slit, with В « W « L

(a) A Newtonian fluid is in laminar flow in a narrow slit formed by two parallel walls a dis-
tance 2B apart. It is understood that В « W, so that "edge effects" are unimportant. Make a
differential momentum balance, and obtain the following expressions for the momentum-flux
and velocity distributions:

(2B.3-1)

V- = (2B.3-2)
In these expressions & = p + pgh = p - pgz.
(b) What is the ratio of the average velocity to the maximum velocity for this flow?
(c) Obtain the slit analog of the Hagen-Poiseuille equation.
(d) Draw a meaningful sketch to show why the above analysis is inapplicable if В = W.
(e) How can the result in (b) be obtained from the results of §2.5?
Answers: (b) (v )/v = I
z Zfmax
3
2 (9>o ~ ® )B W
L P
3 fiL