Page 681 - Bird R.B. Transport phenomena

§21.4 Enhancement of Mass Transfer by a First-Order Reaction in Turbulent Flow 661

of the wall is proportional to the third power of the distance from the wall. When the in-
tegrals are rewritten in terms of <x = y/'R, we get the dimensionless equation

J
(a> AB /v) + Ko* V V I o <Sb AB /v) + K0 3
(21.4-7)
This equation contains several dimensionless groupings: the Schmidt number Sc = v/%b ,
AB
2
a dimensionless reaction-rate parameter Rx = /с"'К /^, and a dimensionless mass transfer
coefficient Sh = k D/4b known as the Sherwood number (D being the tube diameter).
c AB
In the limit that Rx —> <», the solution to Eq. 21.4-3 under the given boundary condi-
tions is С = exp(—Shcr/2). Substitution of this solution into Eq. 21.4-7 then gives after
straightforward integration
1^^-1=2^/0-2^/! (21.4-8)

in which
1
1
• - / : о Sc" + Ко 3 do- (21.4-9)
fi exp(-Sho-/2)
J,= — — . da (21.4-10)
1
Jo Sc" + Ко r 3
This can be solved 1 to give Sh as a function of Sc, Rx, and K.
The foregoing solution of Eq. 21.4-3 is reasonable when Sc, Rx, and z are sufficiently
large, and is an improvement over the result given by Vieth, Porter and Sherwood. 2
However, in the absence of chemical reaction, Eq. 21.4-3 fails to describe the downstream
increase of С caused by the transfer of species A into the fluid. Thus, the mass-transfer
enhancement by the chemical reaction cannot be assessed realistically from the results of
either Ref. 1 or Ref. 2.
For a better analysis of the enhancement problem, we use Eq. 21.4-1 to get a more
complete differential equation for C:
k
%)
=
°* f Hr { r&AB + ® f f ) - "' C
The assumption that С = 0 at r = 0 is then replaced by the zero-flux condition дС/дг = О
2
there. We represent 4b% in this geometry as I \dv,/dr\ for fully developed flow, by use of
a position-dependent mixing length / as in Eq. 21.3-3. Introducing dimensionless nota-
tions v + = v /v*, z + = zv*/v, r + = Wtlv, and Г = lv*/v based on the friction velocity
2
v* = Vr /p of §5.3, we can then express Eq. 21.4-11 in the dimensionless form
o
SC _ 1 д ( /Эдв + Эй. < ? с \ Г
+ (21.4-12)

r + dr + dr +
in which a Damkohler number Da = k'"v/vl has been introduced.
An excellent model for the mixing length / is available in Eq. 5.4-7, developed by
Hanna, Sandall, and Mazet by modifying the model given by van Driest. 4 This model
3

W. R. Vieth, J. H. Porter, and T. K. Sherwood, Ind. Eng. Chem. Fundam., 2,1-3 (1963).
2
3
О. Т. Hanna, О. С Sandall, and P. R. Mazet, AIChE Journal, 27, 693-697 (1981).
E. R. van Driest, /. Aero. Sci., 23,1007-1011,1036 (1956).
4