Page 640 - Bird R.B. Transport phenomena

620 Chapter 20 Concentration Distributions with More Than One Independent Variable

2)
and a) described by
A
2
P.D.E.: - ^ - + (v • VOJ^) = 3) У ^ 2 ) - к"'(off (20.1-48)
ЛВ
I.C.atf = 0: ^ 2 ) = 0 (20.1-49)
B.C. at surfaces: w? = "AO (*, y. z) (20.1-50)

We now proceed to solve these two auxiliary problems by means of Laplace transform.
Taking the Laplace transform of the equations for o) gives
])
A
P.D.E. + I.C.: (p + k"W A ]) - o> {x, y, z) + (v • VSJJ*) = ЯЬ Ч п " (20.1-51)
2
АВ
Al
А
B.C. at surfaces: п А ]) = 0 (20.1-52)
Now, the function g in Eq. 20.1-43 is the solution for a) with k"' replaced by zero. Corre-
])
A
spondingly the Laplace transform g satisfies Eqs. 20.1-51 and 52 with p + k'" replaced by p:
uTty, x, У, z) = g(p + k'", x, y, z) (20.1-53)
Hence by taking the inverse Laplace transform we get
k"'t) (20.1-54)
which is the first part of the solution.
Next, taking the Laplace transform of Eqs. 20.1-48 to 50 gives
2)
2
P.D.E. + I.C.: (p + k"W? + (v • V^ ) = °b V uf (20.1-55)
AB
B.C. at surfaces: w? = i a) (x, y, z) (20.1-56)
A0
The Laplace transform / satisfies the same two equations with k"' replaced by zero. That is, if
we now use s for the transform variable in lieu of p, we have
2
P.D.E. + I.C.: s/ + (v • V/) = 9) V ~f (20.1-57)
AB
B.C. at surfaces: / = ]• u) (x, y, z) (20.1-58)
AQ
We see that the function sf satisfies the same boundary condition as puff and that the differ-
ential equations for sf and puff are identical when s = p + k". Hence
V^A\ = ^\S- ^ (20.1-59)
P
or

^ (р, x, у, z) = V -^j f{p + k"', x, y, z) (20.1-60)
2 )
lr
Taking the inverse transform then gives

off = I* exp(-k"'t') ^-f(x, y, z, t'W (20.1-61)
Jo at
{
as the second part of the solution. Addition of the two parts of the solution, u> and off, then
A
gives Eq. 20.1-43 directly.
Equation 20.1-43 provides a means for predicting concentration profiles in reacting sys-
tems from calculations or experiments on nonreacting systems at the same flow conditions.
Several extensions of this treatment are available, including multicomponent systems, 9 turbu-
8 9
7 9
lent flow, ' and more general boundary conditions. "
9
Y.-H. Pao, A1AA Journal, 2,1550-1559 (1964); Chem. Eng. Sci., 19,694-696 (1964); ibid., 20, 665-669
(1965).