thermal effectiveness factor as taken directly from Figure 1. Figure 2 sho\vs that the 4, = 0 result closely approximates the curves for finite 4 (finite rnass transfer resistance) over a considerable range of Damkoehler numbers: For example, for a catalyst pellet-reaction system for which #, = 10, q may be approximated within 1OyG by a thermal effectiveness factor for -Da > 20, provided y = 20. For larger y the range of thermal control would be larger. T h e dashed lines divide the figure into three regions. I n B the thermal effectiveness factor can be used with a n error of less than lOy0and in region A the isothermal effectivness factor (zero heat transfer resistance) is valid within 10%. Only in region C is it necessary to include both heat transfer and ma\s transfer resistances. Correspondingly three 7 values may be discussed: the isothermal q for region A , the thermal q for region C, and the mixed-control q for region B. As a n example, Satterfield and Sherwood ( 3 ) ,using the data of Prater ( Z ) , have estimated the following values for the dehydrogenation of cyclohexane a t 25 atm. and 450’ C. on a platinum-alumina catalyst y = 40
p
= -0.18
The form of the rate equation is not known, but we can compare the three 11 values only by assuming a first-order rate equation. Then the mixed-control result is immediately available from TYeisz and Hicks graphs ( 5 ) . The isothermal q can be obtained from the conventional Wheeler equation ( 6 ) . T h e thermal 7 is given in Figure 1 for any form of rate equation, probided D a is known. This is obtainable from the knoivn p and Equation 6 for the first-order case. The comparison can be made as a function of 4, which measures the level of the
Table I .
Isothermal, Theirmal, and Mixed-Control Effectiveness Factors EffectivenessFactor Rate L e d IsoMixed(40) P Da thermal Thermal control Slow,+, = 4 -0.18 .2.88 0.56 0.31 0.31 Intermediate, = 10 -0.18 .- 18 0.27 0.14 0.14 Fast, +o = 50 -0.18 -450 0.059 0.033 0 , 0 3 0
reaction rate, and, hence, the relative importance of mass transfer. Thus for a comparatively rapid reaction (high diffusion resistance), 4, might be 50. Then D a = -(0.18)(50)2 = -450. From Figure 1 a t y = 40, 7 (thermal) = 0.033. Table I gives the results for three levels of reaction rate. For this highly endothermic reaction, the isothermal values are in error a t all rate levels. However, the thermal q is a n adequate representation of the mixed-control result. Nomenclature
C,
= concentration of reactant a t pellet surface, gram moles/
cc. D a = Damkoehler number, R,2(-AH)r,p/k,Ta De = effective intraparticle diffusivity, sq. cm./sec. E = activation energy AH = heat of reaction k , = effective intraparticle thermal conductivity k l = first-order reaction rate constant, cc./(g.) (sec.) R = radial distance from center of catalyst pellet R, = gas constant R, = radius of pellet T = absolute temperature T, = temperature at pellet surface T* = dimensionless temperature, T / To x = dimensionless radial distance, R/R, r, = rate of reaction a t pellet surface, gram moles/(g.) (sec.) 6 = Weisz and Hicks parameter (Equation 7 ) y = Arrhenius number, E/R,T, q = effectiveness factor # = Thiele modulus (Equation 6) = density of catalyst pellet, g./cc. p literature Cited (1) Carberry, J. J., A.I.Ch.E. J . 7, 350 (1961). (2) Prater, C. D., Chem. Eng. Sci. 8, 284 (1958). ( 3 ) Satterfield, C. N., Sherwood, T. K., “The Role of Diffusion in Catalysis,” pp, 88-90, Addison-LYesley Publishing Co., Reading, Mass.. 1963. (4) Tinkler. J. D., Metzner, A. B., Ind. Eng. Chem. 53, 663 (1961). (5) TVeisz. P. B., Hicks, J. S., Chem. Eng. Sei. 17, 265 (1962). (6) \\’heeler, A., “Catalysis,” P. H. Emmett, ed., Vol. 2, Reinhold, Ye\+ York, 1955.
J. A. MAYMO R. E. CCNNINGHAM J. M. S M I T H ‘
Y
University of Buenos Aires Buenos A i m , Argentina
RECEIVED for review August 10, 1965 ACCEPTED December 20, 1965 Present address, University of California, Davis, Calif.
CONCENTRIATION DEPENDENCE OF T H E B I N A R Y DIFFUSION COEFFICIENT UPT O
the present time, no general framework has been available for the quantitative explanation of the variation of the binary mutual diffusion coefficient with composition. This is particularly true for the case of liquid phase binary diffusion, where a comparative wealth of data (4) has indicated some rather noticeable, yet not fully explainable, qualitative trends. T h e composition dependence of the binary diffusion coefficient tends to be linear in nearly ideal systems, while nonideal systems generally show minima in their diffusivities a t some point between the infinitely dilute extremes. (Associated systems which tend to exhibit maxima seem to be the only exception to this rule of thumb.) T h e obvious interrelationship between the diffusional behavior and the thermodynamic properties of binary mixtures,
coupled with phenomenological arguments based on the theory of irreversible processes, has led some to attempt a quantitative explanation based upon “fundamental” coefficients defined on the basis of chemical potential gradients as driving forces for the diffusion process. Since no kinetic model of the diffusion process is invoked from this viewpoint, very little absolute success can be expected. I n this paper, a modification of the absolute rate theory is developed based upon the “hole” model of Eyring (3) to yield an expression which predicts the concentration dependence of the binary mutual diffusion coefficient in terms of the two infinitely dilute coefficients and a thermodynamic factor. Starting with Eyring’s basic model (3, 5 ) , a free energy barrier is postulated for the transport of a molecule of diffusing VOL. 5
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281
A modification of the absolute rate theory is used to develop an expression for the variation of the binary mutual diffusion coefficient with composition. This relationship, in terms of the diffusion coefficient a t infinite dilution and a thermodynamic factor, predicts the behavior of all liquid and solid mixtures except for associated systems.
species between two successive equilibrium positions. Such barriers exist for each of the two diffusing species in a binary system and, indeed, persist even in a completely homogeneous (on a macroscopic scale) mixture. When a macroscopic chemical potential gradient is imposed on such a n equilibrium state, the energy barriers for each species are distorted. Referring to Figure 1, this distortion may, to a first approximation, be represented by
I
where i. refers to the direction of a jump, a is the distance between equilibrium positions, and b p , / d x is the macroscopic gradient of chemical potential. According to the theory of absolute rates ( 3 ) , the specific rates (frequencies) for the diffusion of species i are vi& = k _ T e - A G t + / R T
h
(2)
T h e net diffusional velocity of species i in the direction of the gradient may then be calculated from = a(v,+
- vi-)
(4)
Considering diffusion in a n i - j binary mixture, the diffusional velocity of species i may be represented by a similar expression with the result that
I n view of the Gibbs-Duhem relation:
CiVpi
+c,V~j
= 0
t
;-.I
I
X
Figure 1. Distortion of equilibrium free energy barrier by chemical potential gradient
in which AG,, is the net activation energy for the diffusion process as a whole. Now, Equation 8 may be inverted:
(3)
If Equations 1 and 2 are combined with Equation 3 with the restriction that 2 R T >> a dpt/dx (this restriction corresponds to the domain of validity of linear phenomenological rate laws), the final result, generalized to three dimensions, is
- v v L- - e - ~ G L o / R TV P % hN
/
(6)
Equation 5 may be reduced to
and compared with the defining equation for the binary friction coefficient (7) : VPi
FijCj(Vj
- Vi)
(11)
Thus
F I.1. =
C,AGtj/RT
ca2
(12)
Evaluation of the friction coefficient a t the two composition extremes gives
A reasonable mixing rule [employed by Eyring and coworkers (2, 5 ) in predicting the viscosity of liquid mixtures] is AGij = x,AGPj
+ xiAGi,
(14)
Combination of Equations 12, 13, and 14 gives
Now, according to Eyring (3) there is some net rate process in terms of which the diffusion process may be described, and in this over-all process there is some configuration that can be identified as the activated state. The existence of the over-all activated state is confirmed by experiment, so it is reasonable to rewrite Equation 7 as
Now the relationship between the friction coefficient and the binary mutual diffusion coefficient is (7)
Use of Equation 16 in Equation 15 yields the desired result,
This relationship has been tested by Vignes (6) and shown to be valid in all binary liquid and solid systems measured to date 282
l&EC FUNDAMENTALS
with the exception of associated mixtures. In systems of this nature, a mixture rule, similar to Equation 14 but accounting for the degree of association, is required. T h e result given here is similar to that given by Eyring ( 3 ) . T h e major difference here is that the present development avoids the problem of reference frames by considering the diKerence in the net diffusional velocities of the two species as the over all process. Rationally, the present approach appears to be more satisfying. However, in view of the somewhat heuristic nature of the theory, there seems to be little point in belaboring minor differences from the classical approach. O n the other hand, in the light of the remarkably complete agreement of the present result with experimental data, as demonstrated by Vignes 1(6), the utility of the viewpoint of the modified theory with respect to the diffusional behavior of liquids and solids can hardly be minimized. I t is felt that with further refinement and interpretation, the modified absolute rate theory holds considerable hope toirard a better understanding of the liquid state. Also, the diffusional behavior of multicomponent liquid and solid systems seems to be approachable within the framework of the theory, and this will be considered in a later paper. Nomenclature
a
= distance between equilibrium positions
c
=
c,
= = = = =
D,,
F,, AG,* AG,,
total molar concentration molar concentration of species i binary mutual diffusion coefficient friction coefficient free energy barrier for diffusion of species i free energy barrier for diffusion of species i in a homogeneous mixture
AGtJ = net activation energy for the diffusion process h = Planck constant k = Boltzman constant N = Avogadro’s number R = gas constant T = absolute temperature = diffusional velocity of species i in the x direction u,, = velocity vector of species i v, = molar volume of pure species i it x = distance coordinate = mole fraction of species i x,
GREEKLETTERS 7 % = activity coefficient of species i = chemical potential of species i pz = frequency of jumps of species i ut* SUPERSCRIPT = infinite dilution of indicated species 0 literature Cited
(1) Bearman, R. J., J . Phys. Chem. 65, 1961 (1961). ( 2 ) Gjaldbaek, J. C., Anderson, E. K., Acta Chem. Scand. 8, 1398 (1934). (3) Glasstone, S. K., Laidler, K. J., Eyring, H., “Theory of Rate Processes,” McGraw-Hill, New York, 1941. (4) Johnson, P. A,, Babb, .A. L., Chem. Reos. 56,1387 (1956). ( 5 ) Stearn, A. E., Irish, E. M., Eyring, H., J . Phys. Chem. 44, 981 (1940). (6) Vignes, A., IND.ENG.CHEM.FUNDAMENTALS 5 , 189 (1966). HARRY T. CULLINAN, J R . State Universitj of ,“1Tew York at BuJalo Buffalo, N . Y .
RECEIVED for review October 6, 1965 ACCEPTED January 17, 1966
DISTRIBUTION OF RESIDENCE T I M E S IN A CASCADE OF MIXED VESSELS W I T H BACKMIXING Retallick has indicated how to find the residence time distribution for a cascade of n ideal mixers with backmixing ratio cy, the ultimate purpose being to derive a from experimental data. For large n when his method i s cumbersome, a i s obtained more easily from the relative variance, using Van der Laan’s formula. This formula i s lnere derived from a set of recurrent relations, constituting a partial difference equation relating moment of distribution to stage number and order of moment. The moments are obtained therefrom as power series in x = a / ( l +a). This treatment i s also generalized to cover the case (Bell and Babb) of injection and observation at intermediate points.
(6) derives the residence time distribution of a cascade of n ideal mixers with backmixing by multiplying the residence time distribiutions for n, n 2, n 4, etc., stages by the probabilities, JV, that the liquid backmixes 0, 1, 2 , etc., times and therefore passes through n,, n 2, n 4 , etc., stages, and makes a summation of these products. A slight error is stated to be involved, since the fact has been ignored that in respect to entrainment th,e end stages behave differently from the others. If it can be assumed that the model is an adequate description of practical behavior, it is not necessary to compute the full theoretical distribution curve to find the entrainment ratio, a = q / Q , from the experimental curve. For the model under consideration the relative variance (square of standard deviation, u, divided by square of mean, p ) has been derived by Van der Laan ( 5 ): RETALLICK
+ + + +
u2/,u2 = [ n ( l
- 2:)- 2x(1 - x ” ) ] / n 2 ( 1 -
with x given by x =
d(Q+ d
(I)
This formula is exact and, in particular, the deviating behavior of the end stages is correctly accounted for. Since x = a/(l a),Equation 1 can be rewritten as:
+
a2/pz =
(I
+ 2 a ) / n - 2 a ( 1 + a ) [ -~ { a / ( +~ c ~ ) ] ~ ] / n Z (2)
A derivation, hitherto unpublished, is given below. For n 1, Equation 1 approaches
>>
a2/p2 =
(1
+ 2a)/n
(3)
The procedure for calculating a for given n from the observed u 2 / p 2by means of Equation 2 is somewhat tedious. If one uses 2 a ) / n will be too low by Equation 3 instead, the value of ( 1 not more than 2 a ( l a)/n2-i.e., the calculated a is also low a ) / n , of its value. This but by not more than a fraction, ( 1 is considered adequate for practical use. If, on the other hand, one wishes to calculate n for given a, an analogous reasoning shows that for CY 5 0.4 (Retallick’s highest value) the result is off by not more than 0.3 stage. Since for n 1 one may disregard the afore-mentioned
+
+
+
>>
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