Page 334 - Bird R.B. Transport phenomena

318 Chapter 10 Shell Energy Balances and Temperature Distributions in Solids and Laminar Flow

This equation describes the balance among thejyiscous force, the pressure force, the
gravity force, and the buoyant force -~pgf3(T - T) (all per unit volume). Into this we
now substitute the temperature distribution given in Eq. 10.9-4 to get the differential
equation 2

-
^ d v 7 (dp 1 ш * т у (m9 9)
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which is to be solved with the boundary conditions
B.C. 1: at у = -В, v = 0 (10.9-10)
z
B.C. 2: at у = +B, v = 0 (10.9-11)
z
The solution is

v z =

We now require that the net mass flow in the z direction be zero, that is,

pv dy = 0 (10.9-13)
J -в z
Substitution of v z from Eq. 10.9-12 and p from Eqs. 10.9-6 and 4 into this integral leads to
the conclusion that
dp
-^ = -pg (10.9-14)

when terms containing the square of the small quantity AT are neglected. Equation 10.9-
14 states that the pressure gradient in the system is due solely to the weight of the fluid,
and the usual hydrostatic pressure distribution prevails. Therefore the second term on
the right side of Eq. 10.9-12 drops out and the final expression for the velocity distribu-
tion is

(10.9-15)
D
\ /A
The average velocity in the upward-moving stream is
D2
(10.9-16)

The motion of the fluid is thus a direct result of the buoyant force term in Eq. 10.9-8, as-
sociated with the temperature gradient in the system. The velocity distribution of Eq.
10.9-15 is shown in Fig. 10.9-1. It is this sort of velocity distribution that occurs in the air
space in a double-pane window or in double-wall panels in buildings. It is also this kind
of flow that occurs in the operation of a Clusius-Dickel column used for separating iso-
topes or organic liquid mixtures by the combined effects of thermal diffusion and free
convection. 1

1 Thermal diffusion is the diffusion resulting from a temperature gradient. For a lucid discussion
of the Clusius-Dickel column see K. E. Grew and T. L. Ibbs, Thermal Diffusion in Gases, Cambridge
University Press (1952), pp. 94-106.