Page 601 - Bird R.B. Transport phenomena

Problems 581

18C.4. Estimation of the required length of an isothermal reactor (Fig. 18.3-1). Let a be the area of
catalyst surface per unit volume of a packed-bed catalytic reactor and S be the cross-sectional
area of the reactor. Suppose that the rate of mass flow through the reactor is w (in lb, /hr, for
;!
example).
(a) Show that a steady-state mass balance on substance A over a length dl of the reactor leads to
dco A0 _ SaN A M A

(b) Use the result of (a) and Eq. 18.3-9, with the assumptions of constant 8 and %b , to obtain
AB
an expression for the reactor length L needed to convert an inlet stream of composition х (0)
л
to an outlet stream of composition x (L).
A
{Hint: Equation (P) of Table 17.8-1 may be useful.)
Answer: (h)L = l ™" 11 l ' ^ л о
AB/ -4,(0) [M A x A0 + M B (1 - x A0 )] ln(l - 2*AO)
] 2
18C.5. Steady-state evaporation. In a study of the evaporation of a mixture of methanol (1) and ace-
tone (2) through air (3), the concentration profiles of the three species in the tube were mea-
sured 13 after attainment of steady state. In this situation, species 3 is not moving, and species
1 and 2 are diffusing upward, with the molar fluxes N 2l and N , measured in the experi-
z2
ments. The interfacial concentrations of these two species, x w and x , were also measured. In
20
addition, the three binary diffusion coefficients were known. The interface was located at z =
0 and the upper end of the diffusion tube was at z = L
(a) Show that the Maxwell-Stefan equation for species 3 can be solved to get

x — x^e
3
in which A = v + ^ , with v = М Ь/сЯЬр and £ = z/L.
U3 223 a(iy а у
(b) Next verify that the equation for species 2 can be solved to get

B e
*2 = *20** + ^ (1 - e 0 + - ^ = <t* ~ e 0 (18C.5-2)
D A — D
where В = v ul + ^212 a n d С = ^ 21 2 ~ ^223-
(c) Compare the above equations with the published results.
(d) How well do Eqs. 18C.5-1 and 2 fit the experimental data?

I8D.I0 Effectiveness factors for long cylinders. Derive the expression for rj A for long cylinders anal-
ogous to Eq. 18.7-16. Neglect the diffusion through the ends of the cylinders.

Answer: ri A = Л1 (2Л) 0 Bessel functions"
.., where / and Д are "modified
.
]
0
18D.2. Gas absorption in a falling film with chemical reaction. Rework the problem discussed in
§18.5 and described in Fig. 18.5-1, when gas A reacts with liquid В by a first-order irreversible
chemical reaction in the liquid phase, with rate constant k"'. Specifically, find the expression
for the total absorption rate analogous to that given in Eq. 18.5-18. Show that the result for ab-
sorption with reaction properly simplifies to that for absorption without reaction.
r
u
Answer: W = Wc v ^^ U + u) erfV^ + J%e~ \ in which и = k'; L/v
A AO max max

13 R. Carty and T. Schrodt, Ind. Eng. Chem., 14, 276-278 (1975).