Page 678 - Bird R.B. Transport phenomena

658 Chapter 21 Concentration Distributions in Turbulent Flow

times. In tube flow with mass transfer at the wall, one expects that the time-smoothed
concentration c A will vary only slightly with position in the turbulent core, where the
transport by turbulent eddies predominates. In the slowly moving region near the bound-
ary surface, on the other hand, the concentration c A will be expected to change within a
small distance from its turbulent-core value to the wall value. The steep concentration
gradient is then associated with the slow molecular diffusion process in the viscous sub-
layer in contrast to the rapid eddy transport in the turbulent core.

§21.2 TIME-SMOOTHING OF THE EQUATION
OF CONTINUITY OF A

We begin with the equation of continuity for species A, which we presume is disappearing
1
by an j?th-order chemical reaction. Equation 19.1-16 then gives, in rectangular coordinates,

Here k'" is the reaction rate coefficient for the Hth-order chemical reaction, and is pre-
sumed to be independent of position. In subsequent equations we shall consider n = 1
and n = 2 to emphasize the difference between reactions of first and higher order.
When c A is replaced by c A + c , and by v t + v\, we obtain after time-averaging
v
A
s
д r r T r
^А _(д_-- + A - - + sL~c \ - (—~ ~ + -^L~ ~ + A ~
=
V A
'
dt \dx VXCA dy Vy A dz z A ) \dx VX CA dy ^ dz Vz
(21.2-2)
Comparison of this equation with Eq. 21.2-1 indicates that the time-smoothed equation
differs in the appearance of some extra terms, marked here with dashed underlines. The
terms containing v\c A describe the turbulent mass transport and we designate them by
f , the ith component of the turbulent molar flux vector. We have now met the third of
Ai
the turbulent fluxes, and we may summarize their components thus:
turbulent molar flux (vector) ]% = v\c A (21.2-3)
turbulent momentum flux (tensor) т]- = pv'iV- (21.2-4)
]
turbulent heat flux (vector) q) n = pC vjT (21.2-5)
p
All of these are defined as fluxes with respect to the mass average velocity.
It is interesting to note that there is an essential difference between the behaviors of
chemical reactions of different orders. The first-order reaction has the same form in the
time-smoothed equation as in the original equation. The second-order reaction, on the
2
other hand, contributes on time-smoothing an extra term — /C'"Q, this being the manifes-
2
tation of the interaction between the chemical kinetics and the turbulent fluctuations.
We now summarize all three of the time-smoothed equations of change for turbu-
lent flow of an isothermal, binary fluid mixture with constant p, 4b , and fi\
AB
continuity (V • v) = 0 (21.2-6)
(n
(
motion p j£ = Vp - [V • (т '° + T )] + pg (21.2-7)
-
С
continuity of A -=£ = -(V • (Jjj 0 4- J^)) - \ Л, А % (21.2-8)
L/t I fc~> \C A i С А )
Here f A = ~4b c and it is understood that the operator D/Dt is to be written with the
AB Af
time-smoothed velocity v in it.
1
S. Corrsin, Physics of Fluids, 1,42-47 (1958).