Page 683 - Bird R.B. Transport phenomena

§21.5 Turbulent Mixing and Turbulent Flow with Second-Order Reaction 663

functions of z" for various values of the Damkohler number Da. These results lead to the
following conclusions:
1. In the absence of reaction (that is, when Da = 0), the Sherwood number falls off
rapidly with increasing distance into the mass-transfer region. This behavior is
consistent with the results of Sleicher and Tribus 8 for a corresponding heat trans-
fer problem, and confirms that the convection term of Eq. 21.4-11 is essential for
this system. This term was neglected in References 2 and 3 by regarding the con-
centration profiles as "fully developed."
2. In the presence of a pseudo-first-order homogeneous reaction of the solute (that is,
when Da > 0), the Sherwood number falls off downstream less rapidly, and ulti-
mately attains a constant asymptote that depends on the Damkohler number.
Thus, the enhancement factor, defined as Sh (with reaction)/Sh (without reaction),
can increase considerably with increasing distance into the mass-transfer region.

§21.5 TURBULENT MIXING AND TURBULENT
FLOW WITH SECOND-ORDER REACTION
We now consider processes occurring within turbulent fluid systems, with particular ref-
erence to the two mixers shown in Fig. 12.5-1. In Fig. 12.5-1 (a) is shown a steady state sys-
tem, in which two input streams enter a system of fixed geometry at constant rates, and
in Fig. 12.5-1 (b) an unsteady state system, in which two initially stationary, segregated,
miscible fluids are mixed by turning an impeller at a constant angular velocity, starting at
time t = 0. One stream [in (a)] or one initial region [in (b)] contains solute A in solvent S,
and the other contains solute В in solvent S. All solutions are sufficiently dilute that the
solutes do not appreciably affect the viscosity, density, or species diffusivities. Then the
behavior of the solute (A or B) in either system [(a) or (b)] is described by the non-time-
smoothed diffusion equations

2
2
^ = ® V c A + R A ^ = % V c B + R B (21.5-1, 2)
AS
5
with suitable initial and boundary conditions.
For these systems, we may write that at z = 0 [in (a)] or t = 0 [in (b)]
c = c a n d c = 0 (21.5-3,4)
A A{) B
over the A inlet port [in (a)] or the initial region [in (b)], and
c = с во and c = 0 (21.5-5, 6)
B A
over the В inlet port [in (a)] or the initial region [in (b)]. In addition, we consider all con-
fining surfaces to be inert and impenetrable. 1

No Reaction Occurring

For this situation, the terms R A and R are identically zero. We now define a single new
B
independent variable

c
B0

s
С A. Sleicher and M. Tribus, Trans. ASME, 79, 789-797 (1957).
1
In system {a), these boundary conditions are only approximations. The indicated values of c and
A
:, are regarded as asymptotic values for z << 0.
: