Page 679 - Bird R.B. Transport phenomena

§21.4 Enhancement of Mass Transfer by a First-Order Reaction in Turbulent Flow 659

§213 SEMI-EMPIRICAL EXPRESSIONS FOR
THE TURBULENT MASS FLUX
In the preceding section we showed that the time-smoothing of the equation of conti-
nuity of A gives rise to a turbulent mass flux, with components f Ai = v\c . To solve
A
mass transport problems in turbulent flow, it may be useful to postulate a relation be-
tween f Ai and the time-smoothed concentration gradient. A number of empirical expres-
sions can be found in the literature, but we present here only the two most popular ones.

Eddy Diffusivity

By analogy with Fick's first law of diffusion, we may write

jit) _ _Q\(0 д. СУЛ ^ Л\
I Ay ~ ^AB j VZl.J-ly
dy
as the defining equation for the turbulent diffusivity ЯЬ , also called the eddy diffusivity. As
АВ
is the case with the eddy viscosity and the eddy thermal conductivity, the eddy diffusiv-
ity is not a physical property characteristic of the fluid, but depends on position, direc-
tion, and the nature of the flow field.
{
U)
The eddy diffusivity 4t and the eddy kinematic viscosity v {t) = fi /p have the same
AB
dimensions—namely, length squared divided by time. Their ratio

is a dimensionless quantity, known as the turbulent Schmidt number. As is the case with
the turbulent Prandtl number, the turbulent Schmidt number is of the order of unity
(see the discussion in §13.3). Thus the eddy diffusivity may be estimated by replacing it
by the turbulent kinematic viscosity, about which a fair amount is known. This is done
in §21.4, which follows.

The Mixing-Length Expression of Prandtl and Taylor

According to the mixing-length theory of Prandtl, momentum, energy, and mass are all
transported by the same mechanism. Hence by analogy with Eqs. 5.4-4 and 13.3-3 we
may write
dc
jit) — _ / 2 t A
I Ax, - l dy Ту (21.3-3)
2
where / is the Prandtl mixing length introduced in Chapter 5. The quantity I \dv /dy\ ap-
x
pearing here corresponds to ЯЬ% of Eq. 21.3-1, and to the expressions for v {t) and a u) im-
plied by Eqs. 5.4-4 and 13.3-3. Thus, the mixing-length theory satisfies the Reynolds
analogy v U) = a u) = Э&, or Pr (0 = Sc (/) = 1.

§21.4 ENHANCEMENT OF MASS TRANSFER BY A FIRST-ORDER
REACTION IN TURBULENT FLOW 1
We now examine the effect of the chemical reaction term in the turbulent diffusion equa-
tion. Specifically we study the effect of the reaction on the rate of mass transfer at the
wall for steadily driven turbulent flow in a tube, where the wall (of material A) is slightly

1
О. Т. Hanna, О. С Sandall, and C. L. Wilson, Ind. Eng. Chem. Research, 26, 2286-2290 (1987). An
analogous problem dealing with falling films is given by О. С Sandall, O. T. Hanna, and F. J. Valeri,
Chem. Eng. Communications, 16,135-147 (1982).