MASS TRANSFER AND CHEMICAL REACTION
IN A TURBULENT BOUNDARY LAYER W.
R. V I E T H , J .
H. PORTER, AND T. K. SHERWOOD
MasJachusetts Institute of Technology, Cambridge, Mass.
An equation is derived to describe mass transfer to a fluid in fully developed turbulent flow in a pipe. The assumption that the eddy conductivity is proportional to the cube of the distance to the wall provides the basis for a derivation of the well-known Chilton-Colburn analogy. The analysis is extended to the case of simultaneous mass transfer and irreversible first-order chemical reaction. Computer solutions show the result to conform to those based on the penetration and film models for this case.
HE TRANSFER of mass, heat. 01 momentum between a phase Tboundary and a turbulent fluid is encountered in many chemical engineering processes, and the interrelationships between the three have received much study. The best-known "analogies" developed to relate the three phenomena were described in a 1959 review ( 6 ) . Analogies applicable to fully developed turbulent flow usually assume that molecular and eddy transport occur in parallel, and employ ii rate equation involving a transport coefficient which is the sum of the molecular and eddy diffusion coefficients. D and e. T h e latter is assumed to be equal or proportional to the eddy viscosity, obtainable from the slope of the universal velocity profile, relating U T to y + , and the function E = f ( j - ) developed. T h e analogies developed in this way have certain theoretical weaknesses. but have been effective in relating mass, heat, and momentum transport over a \bide range of variables. For large values of U , the Schmidt or Prandtl number, resistance to transport is localized near the phase boundary, where the velocity profile is not well known This makes errors in e a t large y + unimportant, but requires the introduction of essentially empirical expressions for e a t y+ < 30. The use of the universal velocity profile to obtain e leads to expressions in which the transfer coefficient is proportional to the square root of the friction factorfif u is large (7). Experimental data on mass and heat transfer to liquids in pipes show clearly. however, that 1 he transfer coefficient is proportional to f and not This difficulty might be avoided if the
a t the phase boundary and c, a t the center line. The principal assumptions on which the analysis is based may be expressed by
h. = -(D
+
E)
dc dtJ
-
and (2)
c/v = P(y++)3
whence, for steady-state transport, d
dY
)(1
+ aY3) d Y \
= 0
(3)
where
T h e possible effect of any diffusion-induced interfacial velocity is neglected. This employs the normalized distance and concentration c - C" so the boundary conditions ratios, Y = y/yo, and C = c, - co are C = 1 at Y = 0, and C = 0 a t Y = 1. Integration of Equation 3 gives ~
and
V'T
universal velocity profile related u L to y++
(-=YLa ti;)
-
stead of to y +
.
(7). T h e development to be .
described is based on this suggestion, with e assumed proportional to the exponent is that employed by Murphree (5) and, for y + < 5, by Lin, Moulton, and Putnam ( 4 ) to relate E to distance from a pipe wall.
