Page 61 - Bird R.B. Transport phenomena

46 Chapter 2 Shell Momentum Balances and Velocity Distributions in Laminar Flow

The double integral in the denominator of the first line is the cross-sectional area of the
film. The double integral in the numerator is the volume flow rate through a differential
element of the cross section, v dx dy, integrated over the entire cross section.
z
(iii) The mass rate of flow w is obtained from the average velocity or by integration of
the velocity distribution
2
f w f 8 p ?W8 3 cos В
w =\ pv dxdy = pW8(v.) = ^ - ^ r (2.2-21)
z
Jo Jo ^ M
(iv) The film thickness 8 may be given in terms of the average velocity or the mass
rate of flow as follows:

5 = . I^Hh: = .з/ 2 "7 д (2-2-22)
/р gWcos /3
(v) The force per unit area in the z direction on a surface element perpendicular
to the x direction is +T evaluated at x = 8. This is the force exerted by the fluid (re-
XZ
gion of lesser x) on the wall (region of greater x). The z-component of the force F of the
fluid on the solid surface is obtained by integrating the shear stress over the fluid-solid
interface:
fL fW fL fW/ fo \
Я = (т J x=6 )dy dz=\ [-V-r Vydz
h Jo Jo Jo \ dx =s/
( p%8 cos p\ x
= (LW)(-/i) ~ „ = pgSLWcos P (2.2-23)

This is the z-component of the weight of the fluid in the entire film—as we would have
expected.
Experimental observations of falling films show that there are actually three "flow
1
regimes/' and that these may be classified according to the Reynolds number, Re, for the
flow. For falling films the Reynolds number is defined by Re = 48(v )p//A. The three flow
z
regime are then:
laminar flow with negligible rippling Re < 20
laminar flow with pronounced rippling 20 < Re < 1500
turbulent flow Re > 1500

The analysis we have given above is valid only for the first regime, since the analysis
was restricted by the postulates made at the outset. Ripples appear on the surface of the
fluid at all Reynolds numbers. For Reynolds numbers less than about 20, the ripples are
very long and grow rather slowly as they travel down the surface of the liquid; as a re-
sult the formulas derived above are useful up to about Re = 20 for plates of moderate
length. Above that value of Re, the ripple growth increases very rapidly, although the
flow remains laminar. At about Re = 1500 the flow becomes irregular and chaotic, and
the flow is said to be turbulent. ' 2 3 At this point it is not clear why the value of the

This dimensionless group is named for Osborne Reynolds (1842-1912), professor of engineering at
]
the University of Manchester. He studied the laminar-turbulent transition, turbulent heat transfer, and
theory of lubrication. We shall see in the next chapter that the Reynolds number is the ratio of the inertial
forces to the viscous forces.
2
G. D. Fulford, Adv. Chem. Engr., 5,151-236 (1964); S. Whitaker, Ind. Eng. Chem. Fund., 3,132-142
(1964); V. G. Levich, Physicochemical Hydrodynamics, Prentice-Hall, Englewood Cliffs, N.J. (1962), §135.
3
H.-C. Chang, Ann. Rev. Fluid Mech., 26,103-136 (1994); S.-H. Hwang and H.-C. Chang, Phys. Fluids,
30,1259-1268 (1987).