Page 648 - Bird R.B. Transport phenomena

628 Chapter 20 Concentration Distributions with More Than One Independent Variable
The boundary conditions are taken to be:

at x < 0 or у = oo,

(20.2-24)
O) A
at у = 0, v = 0
x

O) A = °> (20.2-25)
A0
at у = 0, = v (x) (20.2-26)
o
Here the function v (x) stands for v (x, y) evaluated at у = 0 and describes the distribution of
o
y
mass transfer rate along the surface. This function will be specified later.
Equation 20.2-20 can be integrated, with the boundary condition of Eq. 20.2-26, to give
(20.2-27)

This expression is to be inserted for v into Eqs. 20.2-21 to 23.
y
To capitalize on the analogous form of Eqs. 20.2-21 to 23 and the first six boundary condi-
tions, we define the dimensionless profiles

K T^T " "A^-^IO (20.2-28)
Uv = U T = П =
and the dimensionless physical property ratios

A = j) = 1 Л = ^ = Pr А = ^ - = Sc (20.2-29)
v 7 ш
With these definitions, and the above equation for v yf Eqs. 20.2-21 to 23 all take the form
(
П, Ш + ^ - ± £ n^) f = fx f? (20.2-30)

and the boundary conditions on the dependent variables reduce to the following:
a t x < 0 o r y = oo, П = 1 (20.2-31)
at у = 0, П = 0 (20.2-32)
Thus the dimensionless velocity, temperature, and composition profiles all satisfy the same
equation, but with their individual values of Л.
The form of the boundary conditions on П suggests that a combination of variables be
tried. By analogy with Eq. 4.4-20 we select the combination:

(20.2-33)

Then by treating П and IL^, as functions of r/ (see Problem 20B.3), we obtain the differential
equation
2
/4W Lv»x p , \dU 1 d U
n
\——-/2—T, ii drf I — = — — -
v
\ Усс V ^o / ay A dr\
with the boundary conditions
at y) = oo, П = 1 (20.2-35)
at г) = 0, П = 0 (20.2-36)