Regimes of Mass Transfer with Chemical Reaction GlOVANNl ASTARITA
for which the phenomena of m a s transfex Pand of chemical reaction take place simultaneously OCRSSES
are very frequent, and have been studied both theoretically and experimentally for a variety of systems. I n particular, much work has been carried out in the field of chemical absorption (gas-liquid systems) and of heterogeneous catalysis (fluid-solid systems). A completely rigorous analysis of any specific process requires taking into account simultaneously so many mutually interfering phenomena as to be practically impossible. Fortunately enough, if some reasonable simplifyinq hypotheses concerning both the kinematics of motion in the phase where the reaction takes place and the kinetics of the reaction are made, theoretical results can be obtained, which have been satisfactorily substantiated by experimental evidence. I t h unfortunate that, in many caws, the assumptions made are either insufficiently stated, or unnecessarily mtrictive, or, more often, both. This is particularly true when asymptotic solutions are wnsidaed, which by the way are the only ones whose malyric form is simple enough to make them useful for calculational purposes, and whose reliability and sophistication do not exceed the reliability of data which a r c used in practice for design. It is almost always truc that the asymptotic forms of equations which have been obtained on the basis of some set of assumptions wncerning the kinematics of motion and the kinetics of t h e reaction are valid also if the restriction of the above said agsumptions is removed, although the complete non. asymptotic equation is only valid under the specified conditions. It will be shown in this paper that five basic “regimes” (or asymptotic solutions of the pertinent differential equations) of mass transfer with chemical reaction exist, the most wnspicuous featurts of which do not depend on the particular system which is being considered; simple rules are stated by means of which any specific pmcess can be identified as approaching the conditions ofanyoneofthesefiveregimes. Such rules are in the form of inequalities which need to be largely and not only barely fulfilled (as can be expected for any asymptotic equation). Thus, specific pmcurxs may take place in transition conditions; where such is the case, a nonasymptotic equation, say an interpolation formula, is required, which can in principle be obtained only on the basis of well speciiied assump18
INDUSTRIAL AND ENGINEERING CHEMISTRY
tions. Yet, it will be shown that even in the transition region an interpolation between asymptotic solutions can be obtained which is only weakly sensitive to the particular system which is being considered. Coned Model and Parameten Ddlniiion
Consider a two-phase system, where a chemical reaction is taking place within phase 2, which involves a reactant being continuously transferred into phase 2 from phase 1.
Asymptotic solutions of the differential equations of mass transfer with chemical reaction can be derived without any specific assumption concerning the kinematics of motion of the phase in which reaction takes place, and without any assumptions regarding the reaction kinetics. The author identifies five “regimes” for mass transfer and gives simple rules for recognizing which regime is active.
phase concentration within phase 1, and c1‘ the interface concentration, the equation for the maSS transfer flux is: N = kl(C1 - a’) (1)
kl being a known mass transfer coefficient. It will bc
I n phase 1, a phenomenon of simple mass transfer is taking place, which is not directly influenced by the reaction; thus, the nature of phase 1 need not be specified. Of course, the nature of phase 1 influences the velocity distribution within phase 2 in the neighborhood of the interface; but, as will be shown below, asymptotic solutions are such that the velocity distribution within phase 2 is largely immaterial. It will be assumed that the mass transfer phenomenon within phase 1 is well understood, so that, if c1 is the bulk-
further assumed that physical equilibrium prevails a t the interface, so that c1’ and CO‘ are related to each other in a known way, co’ being the interface concentration of the transferring component within phase 2. Therefore, the various possible cases will be examined up to the point where an equation is obtained relating the mass transfer rate N to the value of co’ and of the relevant physical and chemical parameters (kf, 0, D, fa, c-). Such an equation, coupled with Equation 1, allows to eliminate the interface concentrations CI‘ and co’ provided the relevant distribution coefficient co’/cl’ is known. Knowledge of the latter may imply some difficulty in practical cases, as discussed in a number of works (8, 70, 13, 14,37,44). It will initially be assumed that phase 2 is a liquid; consideration of a gaseous phase would be less realistic but substantially equivalent. The case where phase 2 is a solid, such as a porous catalyst, can be regarded as a particular case of the general treatment for a liquid phase (only three regimes, instead of five, are possible when phase 2 is a solid). The interface between phases 1 and 2 will be assumed to be plane, which is not a restrictive assumption in a majority of cases (geometric factor corrections are generally of minor importance). The volume of phase 2 per unit interface area will be designated by 4; if phase 2 wets the packing in a packed tower, 4 is the average thickneap of the wetting layer; when phase 2 is a porous catalyst, 4 is a characteristic dimension of the pellets, say ‘/I of the radius in the case of spherical pellets. In the absence of the chemical reaction, the mass transfer coefficient, as referred to phase 2, would have same value k# such that:
- CO)
No = k&‘
(2)
where k 2 is supposed to be known. A diffusion time fD is defined (4,6, 7,73) such that:
k,O =
dDFD
(3)
The form of Equation 3 is obviously suggested by the penetration theory formulation, but the equation itself is VOL 5 8
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here regarded as a definition of tD) and as such is not related to the penetration theory formulation. I t has recently been shown (35) that the concept of a diffusion time, defined by Equation 3, and interpreted as a n average life of surface elements of phase 2, has a much broader validity than implied by the assumptions on the kinematics of motion underlying the penetration theory formulation. Indeed, if an average life of surface elements is calculated for any velocity distribution within phase 2 at the velocity level prevailing within the concentration boundary layer, and compared with the value of t o obtained from Equation 3, it is seen that they are of the same order of magnitude (values of k2O have been calculated for a variety of kinematic conditions 5, 75, 79, 29, 37, 54). Another parameter, namely a depth of penetration A, can be defined as the distance within phase 2 over which concentrations appreciably different from c, are to be expected (1 is the thickness of the concentration boundary layer). A rigorous definition is obtained through the following equation :
where, if (dc/dx),-o depends on some other variable such as time or position on the interface, its lowest value is to be considered. Whatever the velocity distribution within phase 2, the values of Xo, say of the depth of penetration in the absence of the reaction, and of t D are related to each other by the equation,
as can be shown by simply inserting Equations 3 and 4 into the definition of k2O. When phase 2 is a solid, both tD and Xo are infinity. When phase 2 is a fluid, k2O, tD, and Xo could, in principle, have any value whatsoever, but in practice only a limited range of values is encountered in usual mass-transfer equipment. Orders of magnitude for the relevant parameters, such as deduced by usually accepted empirical or semitheoretical correlations, are listed in Table I. Figure 1 is a sketch of concentration profiles such as are encountered in the absence of the chemical reaction. As can be seen from the values listed in Table I, the figure is not to scale, because in reality Xo is two orders of magnitude smaller than 4. This allows no distinction between the values of co and c,. Slow a n d Fast Reactions
Whenever the reaction rate r essentially depends only on the concentration of the transferring reactant, the AUTHOR Giovanni Astarita is Assistant in the Chair of Indus-
trial Chemistry at the University of Naples. A t the time this paper was written, he was Visiting Associate Professor in the Department of Chemical Engineering, University of Delaware. He haspublished more thanjifty papers in thejields o f mass transfer and fluid mechanics. 20
INDUSTRIAL AND ENGINEERING CHEMISTRY
diffusion-reaction differential equation takes the following form:
where x ( x ) is a function depending on the shape of the velocity distribution in the neighborhood of the interface within phase 2. When t is the time measured along the path of phase 2 within the concentration boundary layer, the order of magnitude of x is unity; according to the penetration theory model, x = 1. A dimensionless concentration y, a dimensionless time 6, and a reaction time t R may be defined as :
(7)
According to the discussion following Equation 3, t D is indeed the natural yardstick required for nondimensionalizing the time t. The definition of tR is believed to be particularly relevant, and will be discussed in some detail. It is evident that, for first-order reactions, tR is the inverse of the kinetic constant, and as such is independent of the concentration level. I n general, tR depends on the concentration level, and in Equation 9 it is defined with reference to the interface conditions. I n contrast with t D , which is a measure of the time availabfe for the diffusion process, tR is a measure of the required time for the reaction to take place appreciably. Inserting Equations 7, 8, and 9 into 6, the following equation is obtained:
The terms within square brackets in Equation 10 are both of the order of unity. Thus, whatever the analytic forms of x ( x ) and r(c), the second term on the right-hand side of Equation 10 can be dropped if: tR
>> tD
(11)
The resulting equation is the same one which governs the diffusion process in the absence of the chemical reaction. Thus, a slow-reaction condition is defined as one where the reaction rate is so low that the shape of the concentration profiles and hence the value of the mass transfer coefficient are independent of the reaction rate. Under slow reaction conditions, the two phenomena of diffusion and of chemical reaction may be analyzed separately, inasmuch as there is no direct mutual interference among them. Conversely, if the following inequality is fulfilled : tR
> k2O
Under these conditions, the reaction rate per unit interface area, N , is given by:
N
>> +/kzO
(21)
TABLE I. PARAMETER VALUES F R O M ACCEPTED CORRELATIONS tD
& / D
(22)
When phase 2 is a liquid, the contemporary fulfillment of conditidns 12 and 22 requires a grossly unrealistic value of +/kz0tD, so that it is in practice impossible. When phase 2 is a solid, condition 12 is implicitly satisfied, so that Equation 22 is the only condition required for the kinetic regime equations to apply. A discussion relative to zero-order reactions, showing the equivalence of the couples of conditions 11 and 21 or 12 and 22 has been published by Astarita and Marrucci ( 7 7). Figure 2 shows how, although arising from different mechanisms, the concentration distributions for the kinetic regime are, both for slow- and for fast-reaction conditions, substantially equivalent. Equation 20 is generally assumed as a definition of the reference value of the reaction rate for solid porous catalysts, with respect to which effectiveness factors are defined. Thus, by definition, the effectiveness factor in the kinetic regime is unity. Condition 22 may indeed be written in terms of the usual definition of the Thiele modulus (46): Th =
+/do% 1, 7 = $/Th (30) where the,correction factor 9 is a function of a dimensionless parameter including the energy of activation AE, the heat of reaction - AH, and the heat conductivity of the solid K:
(34) The relative insensitivity of the interpolation formula to the hypotheses on the kinematics of motion made in its derivation shows that, although a detailed knowledge of the kinematics is required in order to predict the value of kzO,it is not required in order to evaluate the influence of the chemical reaction on the mass transfer rate, even outside of the asymptotic regions. Instantaneous Reaction Regime
Equation 29 holds when the above said parameter has a zero value, which can easily be shown to coincide with the isothermal case. The validity of Equation 29 can be
It has been so far assumed that the reaction rate r is uniquely determined by the value of the concentration of the transferring reactant, c. Although a number of VOL. 5 8
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tion theory model (37); "8 according to a boundary layer formulation (26, 36). I t is generally accepted that the film theory formulation overestimates the influence of molecular diffusivities on mass transfer rates ; thus, the actual uncertainty over the value of m is only on the range ' / 2 i 2/3. Moreover, it is seen that the value of m coincides with the value of the exponent in the equation for kzo: kzo
Figure 5. Instantaneous reaction
problems may be handled through such an assumption, this is not generally true, and indeed the concentration distribution of the transferring component and the reaction rate may depend, in a rather complicated way, on the simultaneous phenomena of diffusion of other reactants and/or reaction products within phase 2. The extreme case is found when the reaction between the transferring component and some reactant B initially present within phase 2 is instantaneous, so that at no point within phase 2 can the two reactants coexist at nonnegligible concentration levels. If b is the concentration of B, with b o its bulk-phase value, one may envisage a concentration distribution such as sketched in Figure 5, with the reaction taking place as fast as diffusion brings the two reactants together. If the diffusivity D' of B equals the diffusivity of the transferring component, the reaction rate per unit interface area is given by:
N
=
kzo [ c o t
+ :]
= kzocO'
+
(g)m3 1
(36)
The value of the exponent m is 1.0 according to the film theory model (6, 44) ; '/s according to the penetra24
D"
(37)
which is known to be ' / z for the case of a zero velocity gradient as assumed in the penetration theory formulation, and "3 for the case of a constant velocity gradient as assumed in boundary layer formulations (5, 19, 29, 31). Thus, Equation 36 may be considered as generally applicable, with m being the same value as would be used in Equation 37. For the case of chemical absorption, m = is indeed a very satisfactory value, as substantiated by experimental evidence obtained under conditions where the DID ' ratio is substantially different from unity (8, 9 ) . Equation 36 can be shown to be valid provided that the following condition is fulfilled (6, 16, 27, 28, 34, 37, 39, 44, 52):
which can be written as: tR
