be different, leading to an apparent discrepancy between laboratory and plant data. Most catalyst pellets have very large values of L, which would tend to make these effects small. Also, the differences are largest in the region of large m, or small G , which is usually avoided in design because of the excessive diffusion limitations. However, there may be some situations in which the discrepancies discussed above would be important. A useful approximation to G for small ( m / L ) can be obtained by using the relation that as (m/L)+ 0, a1 -+ m/L, a2 + 3.8, ivhich reduces Equation 8 to
Grv-
tanh m
(;)>‘I-’ +
[
m
1+-
0.018
m2
~3 tanh
(3.8L)
(9)
with (m/’L) restricted to ( m / L ) 2< 0.2. As L + a, Equation 9 reduces to the one-dimensional result (Equation 1). Equation 3 would then be useful for determining easily whether or not the transverse gradients are likely to be of importance. I t is strictly true only for cylindrical cross section, but the eigenvalues, a t , for other geometries have similar properties and so it ivould be expected that Equation 9 could be used in general with, for example, the relations of Wheeler (7) for arbitrary particles. Some estimates of the parameters, L, can be obtained from the data in the above references. Most pore radii are in the range of 10 to 400 A., although very porous materials such as kieselguhr and the “macropores” of compressed pellets as discussed by Wakao and Smith ( 6 ) could have sizes of lo4 A. The pore length could have a length of about 100 micronsfor example, see “micro-particles” of IVakao and Smith (6)which would lead to L 100. Deposited thin films would have a smaller length, but the pore radius would also be much smaller. If TVheeler’s (7) model of pore length being related to over-all particle size is used, ordinary catalyst pellets would have L N IO5+. Thus the extreme range of L for most materials would be about 50 < L < 105.
-
From Figure 1, this leads to the conclusion that transverse bulk diffusion effects are essentially negligible for most practical cases of interest and only for exceptional types of catalyst pellets would they have to be taken into account. Equation 9 provides a simple formula for checking the possible importance.
Nomenclature ai = roots of Bessel function, Equation 9
C = reactant concentration C, = reactant concentration at pore mouth D = pore bulk diffusion coefficient E
= effrctiveness factor
k
= intrinsic surface reaction rate coefficient
L
=
dimensionless total pore length (all length quantities dimensionless with respect to pore radius, R ) m = L d 2 k R / D = Thiele modulus r = dimensionless radial position in pore R = pore radius x = dimensionless axial position in pore J = Bessel function
literature Cited
(1) Carberry, J. J., Chem. Proc. Eng. 44, 306 (June 1963). (2) Hougen, 0. A,, Ind. Eng. Chem. 53, 509 (1961). (3) Satterfield, C. Pi.,Sherwood, T. K., “The Role of Diffusion in Catalysis,” Addison \$’esley, Reading, Mass., 1963. (4) Sneddon, I. N., “Fourier Transforms,” McGraw-Hill, New York, 1951. (5) Thiele, E. W., Znd. Eng. Chem. 31, 916 (1939). (6) Wakao, N., Smith, J. M., Chem. Eng. Sci. 17, 825 (1962). (7) \Vheeler, A., Aduan. Catalysis 3, 250 (1951). K E S S E T H B. BISCHOFF Department of Chemical Engineering, The University of Texas, Austin, Tex. RECEIVED for review February 15, 1965 ACCEPTED August 9, 1965
CONDITIONS FOR AZEOTROPE FORMATION A T CONSTANT TEMPERATURE AND FOR PARTIAL MISCIBILITY I N BINARY LIQUID MIXTURES The conditions are determined in terms of the constants of the Redlich-Kister empirical equation order and tested on experimental data.
OR
a binary liquid mixture the total pressure is given by the
F equation
rI
=
7l(l
-
x)plo
+
xp2O
(1)
T h e Redlich-Kister empirical equations of the third order (9) can be used to express the variation of the activity coefficients with concentration:
loge
71 = x’{
(B
+ 3 C + 5 D ) - (4 C + 16 0).+ 12 Dx2} (2)
loge 7 2 = (1
- x)’{ ( B + C -P
D)
- (4 C + 8 D)x + 12 Dx’) (3)
136
I&EC FUNDAMENTALS
of the third
Condition for Azeotrope Formation at Constant Temperature
I t can be shown that the condition for azeotrope formation a t constant temperature is the existence of a single root in the composition range 0 < x < 1 of the equation (71p1O
- 7 2 p z O ) [l
+f(x)I = 0
(4)
where
- (12 C + 96 D)x3 + + 18 C + 58 D ) x ~- (2 B + 6 C f
f ( x ) = 48 Dx4
(2 B
10 0). (5)
and that such a root exists if
,B
+ DI > IC + log,pzo/ploi
(6)
Prediction of Azeotrope Occurrence at Constant Temperature Occurrence of Areotrobe Detd. Predicted 2 B 4- DI IC loge p2 o/pl‘i exptly. from ( 6 ) 0.443 0.770 NO No Acetone 1.172 0.803 Yes Yes Nitromethane 1.735 0.771 Yes Yes 2-Propanol Acetone 0,930 0.508 Yes Yes Benzene 0.355 0.663 N O KO
Table I.
Component
7 Benzene Benzene Benzene Carbon tetracLloride n-Heptane
+
Table II.
-
(B D) 0.906 Acetonitrile Benzene 1.439 Benzene Ethanol I-Butanola LVater 1.10 1.737 Nitromethane Carbon tetrachloride 1.91 Ethanola Decane LVater 1.92 Ethyl acetatea Approximate calues of 13, C, and D calculated using Hala’s method ( 8 ) . 2
(The condition for constant pressure presents many more problems because p i ” , pZo, 71, and y2 are all temperaturedependent.) T h e validity of Equation 6 is seen by a comparison of experimental and predicted azeotrope occurrence (Table I).
I t can be shown that a condition for the existence of a region
of immiscibility is that ihe equation
- (12 C + 96 1 1 ) f ~ ~(2 B + 18 C + 58 D)x’ (2BfGC+lOD)x+l
= O
(7)
should have two roots in the composition range 0 < x < 1. This is so if one or more of the following conditions are obeyed:
2B
B - D > 2
(8)
B+d3
(9)
+ d 4 C i + 5 D > 9.6
+ 4D)
( B 4-
(10 )
These conditions are tested on experimental data (Table 11). Furthermore since whatever the values of B, C, and D , Equation 7 has a t most two roots in the interval 0 < x < l, a binary liquid mixture cannot have a multiplicity of regions of immiscibility. The usefulness of the results is limited by the necessity of having experimental liquid-vapor equilibrium data in order to calculate the empirical constants; and so one cannot pre-
(2 B C
1.150 2,548 4.61 2,598 2.37 5.14
(5) (2) (6) (7)
++5 d/Zs D)
2.645 6.250 12.84 6.224 5.15 13.58
Occurrence of Partial Miscibilitv Predicted Detd. from exptly. ( 8 , 9, 70) No NO XO
NO
Ref. ( 4) (3)
Yes
Yes No No Yes
( 70) (5) (7) (8)
NO
No Yes
dict the existence of an azeotrope or a region of partial miscibility. Nomenclature
rI
total pressure mole fraction of component 2 y1, 7 2 activity coefficients pio,pzo vapor pressure of components a t ambient temperature B, C, D = constants of Redlich-Kister equation X
Condition for Partial Miiscibility
48 Dx4
(6)
Prediction of Partial Miscibility
Component
7
Ref.
= = = =
Literature Cited
(1) Brown, I., Australian J . Sci. Res. A5, 530 (1952). (2) Brown, I., Fock, W., Smith, F., Australian J . Chem. 9, 364 (1956). (3) Brown, I., Smith, F., Zbid., 7, 264 (1954). (4) Zbid., 8, 62 (1955). (5’1 Zbid.. D. 501. (6) Zbid.: io, 423 (1957). (7) Ellis, S.R. M., Spurr, K. J., Brit. Chem. Eng. 6 , 92 (1961). (8) Hala, E., Pick, J., Fried, V., Vilim, O., “Vapour-Liquid Equilibrium.” D. 93. Pewamon Press. London. 1958. (9) Redlich, O., Kister, A.UT.,Znd. Eng: Chem. 40, 345 (1948). (10) Smith, T. E., Bonner, R. F., Zbid., 41,2867 (1949).
D. JAQUES D. A . LEE Chance Technical College Smethwick StaJordshire, England
RECEIVED for review April 26, 1965 ACCEPTED November 29, 1965
VOL. 5
NO. 1
FEBRUARY 1966
137
