Page 600 - Bird R.B. Transport phenomena

580 Chapter 18 Concentration Distributions in Solids and in Laminar Flow

(c) Verify (lengthy!) that the solution

c = * exp[-(u /2Q) )(s - 2)] (18C.1-3)
A o AB
47Г J) AB S
satisfies the differential equation above.
(d) Show further that the following boundary conditions are also satisfied by Eq. 18C.1-3:

B.C. 2: ass-* 0, -4TTS ^ B -Г^-> W A (18C.1-5)
2
dS
B.C.3: atr = O, Ц^ = О (18С.1-6)
оГ
Explain the physical meaning of each of these boundary conditions.
(e) Show how data on c {r, z) for given v and ЯЬ may be plotted, when the preceding solu-
0
A
АВ
tion applies, to give a straight line with slope у /2ЯЬ and intercept In %b .
о
АВ
AB
18C2. Diffusion and reaction in a partially impregnated catalyst. Consider a catalytic sphere like
that in §18.7, except that the active ingredient of the catalyst is present only in the annular re-
gion between r = KR and r = R:
In region I (0 < r < KR), k"a = 0 (18C.2-1)
In region II (KR<T< R), k"a = constant > 0 (18C.2-2)
Such a situation may arise when the active ingredient is put on the particles after pelleting, as
is done for many commercial catalysts.
(a) Integrate Eq. 18.7-6 separately for the active and inactive regions. Then apply the appro-
priate boundary conditions to evaluate the integration constants, and solve for the concentra-
tion profile in each region. Give qualitative sketches to illustrate the forms of the profiles.
(b) Evaluate W , the total molar rate of conversion of Л in a single particle.
AR
18C.3. Absorption rate in a falling film. The result in Eq. 18.5-18 may be obtained by an alternative
procedure.
(a) According to an overall mass balance on the film, the total moles of A transferred per unit
time across the gas-liquid interface must be the same as the total molar rate of flow of A
across the plane z = L. The latter rate is calculated as follows:

W A = lim(W5z; max )(i f * c \ = dx) = V\fo max f" c \ dx (18C.3-1)
L
A z
A z=L
8->oo \O JQ ) JQ
Explain this procedure carefully.
(b) Insert the solution for c A in Eq. 18.5-15 into the result of (a) to obtain:

(18C.3-2)

In the second line, the new variable и = x/V4® L/v max has been introduced.
AB
(c) Change the order of integration in the double integral, to get

(18C.3-3)

Explain by means of a carefully drawn sketch how the limits are chosen for the integrals The
integrals may now be done analytically to get Eq. 18.5-18.