Page 677 - Bird R.B. Transport phenomena

2
Chapter 1

Concentration Distributions

in Turbulent Flow

§21.1 Concentration fluctuations and the time-smoothed concentration
§21.2 Time-smoothing of the equation of continuity of A

§21.3 Semi-empirical expressions for the turbulent mass flux
§21.4° Enhancement of mass transfer by a first-order reaction in turbulent flow
§21.5* Turbulent mixing and turbulent flow with second-order reaction

In preceding chapters we have derived the equations for diffusion in a fluid or solid, and
we have shown how one can obtain expressions for the concentration distribution, pro-
vided no fluid turbulence is involved. Next we turn our attention to mass transport in
turbulent flow.
The discussion here is quite similar to that in Chapter 13, and much of that material
can be taken over by analogy. Specifically, §§13.4, 13.5, and 13.6 can be taken over di-
rectly by replacing heat transfer quantities by mass transfer quantities. In fact, the prob-
lems discussed in those sections have been tested more meaningfully in mass transfer,
since the range of experimentally accessible Schmidt numbers is considerably greater
than that for Prandtl numbers.
We restrict ourselves here to isothermal binary systems, and make the assumption
of constant mass density and diffusivity. Therefore the partial differential equation de-
scribing diffusion in a flowing fluid (Eq. 19.1-16) is of the same form as that for heat con-
duction in a flowing fluid (Eq. 11.2-9), except for the inclusion of the chemical reaction
term in the former.

§21.1 CONCENTRATION FLUCTUATIONS AND
THE TIME-SMOOTHED CONCENTRATION

The discussion in §13.1 about temperature fluctuations and time-smoothing can be taken
over by analogy for the molar concentration c A. In a turbulent stream, c A will be a rapidly
oscillating function that can be written as the sum of a time-smoothed value c A and a tur-
bulent concentration fluctuation c' A
(21.1-1)
c A = c A + c' A
which isjmalogous to Eq. 13.1-1 for the temperature. By virtue of the definition of c A we
see that c' A = 0. However, quantities such as v' xc' A, v\f A, and vx A are not zero, because the
local fluctuations in concentration and velocity are not independent of one another.
The time-smoothed concentration profiles c A{x, y, z, t) are those measured, for exam-
ple, by the withdrawal of samples from the fluid stream at various points and various

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