Ind. Eng. Chem. Fundam. 1980, 19, 126-128
126
OPP85
GPPPG
o b ' o - o%&lrT%NE in ilAP0"R Figure 3. Pressure as a f u n c t i o n o f CzHs calculated
GGP5
'
1bGqO
molar fraction
in gaseous phase a t different temperatures.
used. A serious comparison would have to be done on real experimental results of these authors. It is also possible that the error of 1%in calculated values of usGand u,L is underestimated (Donohue, 1977). In their paper, Lee and Kohn (1969) did not say whether n-dodecane is degassed before introducing ethane. Their method to purge the cell free of air by a stream of ethane is not very satisfactory because ethane is slightly soluble a t atmospheric pressure. The accuracy of the P,T,x results obtained by use of the static method described here is very sensitive to data treatment. In this study, we have used a special equation of state to determine physical properties of mixtures in order to deduce x from z. A better equation of state would provide better determination of x 2 from the knowledge of
nlT, n2T(listed in Table 11),and V (0.0415 dm3). Estimated error on is dm3, and this could result in a maximum error of 0.2 bar in P. Conclusion The apparatus for measuring vapor-liquid equilibria was tested on the system ethane-n-dodecane. The agreement with the P,T,x data of Lee and Kohn proves the validity of this rapid method. Total pressures for one mixture at five temperatures are obtained in 12 h. The installation is suitable for the study of mixtures of components either both liquid or gas or one liquid and on8 gas a t normal conditions. The apparatus proposed in this paper can be used extensively for studies in relation to absorption or compression heat pumps. Nomenclature n? = total number of moles of component i introduced in the still niG = number of moles of component i in the vapor phase niL = number of moles of component i in the liquid phase = total volume inside the cell and tubing up to valve VS2, dm3 VL, @ = volumes of liquid and vapor phases, dm3 uk,usG = saturated molar volumes of liquid and gas phases, dm3 mol-' x i = mole fraction of component i in the liquid phase zi = mole fraction of component i in the still yi = mole fraction of component i in the vapor phase T = temperature, K Literature Cited Donohue, M. D., Prausnitz, J. M., Research Report 26, "Statistical Thermodynamics of Solutions in Natural Gas and Petroleum Reflning", Gas Process Association, Tulsa, Okia., 1977. Donohue, M. D., Prausnitz, J. M., AIChE J., 24, 849 (1978). Kohn, J. P., Kurata, F., Pet. Process., 11, 57 (1956). Lee, K. H., Kohn, J. P., J . Chem. Eng. Data, 14, 292 (1969).
Received for review M a r c h 22, 1979 Accepted October 18,1979
COMMUNICATIONS Use of Active Site Balances for Catalyst Deactivation Models
Catalyst deactivation models, which have traditionally been used as empirical forms, are derived from utilization of the concepts of active site balances and the law of mass action. This approach has both pedagogic value and practical utility in the interpretation of catalyst deactivation phenomena.
Numerous presentations have been made of deactivation models, both to represent experimental deactivation data and for use in theoretical reactor studies. To date, these models have often been depicted as useful empirical representations of the deactivation behavior of the catalyst. For example, Anderson and Whitehouse (1961) have presented activity-poison relationships in an algebraic form and Froment and Bischoff (1961) have used these forms in integral reactor analysis. Levenspiel (1972) has presented the differential form of the deactivation models, and Butt (1972) has described several applications of these models. 00 19-7874/80/1019-0 126$01 .OO/O
In the present note, we show how these models can be derived using the concept of active site balances and power-law deactivation orders. Although this methodology does not necessarily reduce the empiricism, such an approach has utility in providing a simple, hypothetical basis for observed deactivation behavior. Furthermore, the approach provides a consistent basis for extension of empirical models on batch-solids reactors to reactor types involving input and output of catalyst (e.g., fluid beds), thereby having pedagogic value as well as potential practical utility. A useful starting point for describing the catalyst
0 1980 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 19, No. 1, 1980
127
Table I. Differential Deactivation Models case
primary reaction kinetics
deactivation model
parallel deactivation A - + R ; A + PJ aa - - k D a d C R m _
series deactivation A+R+P$
at
side-by-side deactivation (poisoning) A- R;P+P$
-'A
= kaCA"
-aa_ - - k D a d C p m
concentration independent deactivation
-'A
= kaCA"
aa -- -kDad
deactivatioh phenomenon has been advocated by Levenspiel (1972) and Butt (1972), wherein the rate of the primary reaction is expressed in separable form r = fl(r)fdC)f3(4 (1) For a case wherein the total available concentration of active sites is being diminished by deactivation, this approach has both appeal and utility, for example, through consideration of the loss of total sites on the kinetic numerator term in the Hougen-Watson expression (Carberry, 1976). For reversible adsorption of a poison, however, the use of an adsorption term in the denominator of the Hougen-Watson model has also been contemplated (Hougen and Watson, 1947; Prater and Lago, 1956). For such cases, the separable form is not obtained. Furthermore, Butt et al. (1978) have identified cases wherein the rate separable form is incorrect. If the separable rate form for the primary reaction is used, a similar form can be hypothesized for the rate of catalyst decay r D = f4(r)fs(c)f6(a) (2) At this point, we may view the reactor operation involving deactivating catalysts as exhibiting two independent reactions, the primary reaction and the deactivation reaction. The description of such a reactor will require two independent mass balances, on species germane to each of these independent reactions. The mass balances on the reactant, describing the primary reaction, are widely utilized, leading to the well-known design equations for plug flow or stirred tank reactor geometries. For the mass balance pertaining to the deactivation reaction, it is convenient to provide a balance on the discrete active sites in the reactor. Considering the total number of active sites per unit mass of fresh catalyst to be So and the number of active sites per unit mass of. partially deactivated catalyst at time t to be S in a batch-solids reactor, a conservation equation for active sites over an increment in time, At, becomes wsIt+,t - wslt = r D k w (3) where rD is the rate of deactivation, which is expressed as the number of sites lost per unit time per unit mass of catalyst. If the catalyst is deactivated uniformly over the entire reactor (e.g., CSTR or PFR with concentration independent deactivation), W is the total mass of catalyst in the reactor. If not, W is the mass of catalyst in a small volume element of the reactor. Dividing by WAt and taking the limit, we obtain
2
=
rD;
S(0) = s o
(4)
Assuming the rate of active site decay (reaction), rD, can
at
at
be expressed by the law of mass action, the parallel deactivation case of Table I becomes
Equation 5 can be normalized by dividing through by Sod to provide aa = -
-Sod-lktD a d c a m
at
(6)
where a is the normalized catalyst activity a =
s/s,;
a(0) = 1
(7)
The decay rate constant can be redefined to incorporate the constant, SOd-l
St-'kb Hence the deactivation model becomes kD
E
(8)
A similar approach can be taken with any of the Levenspiel (1972) models, also shown in Table I. Furthermore, with appropriate reactor assumptions, consistent conservation equations for the active sites may readily be written for other reactor geometries. For example, with first-order, concentration-independent deactivation in a CSTR with continuous addition of fresh catalyst and removal of spent catalyst, the site balance becomes WSIt+At - WSIt = mSoAt - mSaVgAt- k,Sav,WAt (10) where Sa is the average activity over the time increment At. Fur&ermore, if the catalyst addition and removal rates, m, are maintained to achieve a constant conversion with time, the left-hand side becomes identically zero, leading to
where 7, = W / m ,the catalyst space time. This equation, also presented by Levenspiel (1972), is to be solved simultaneously with the mass balance on the reactant, describing the primary reaction (e.g., first order)
Empirical equations which provide catalyst activitypoison relationships have also been proposed by Anderson and Whitehouse (1961) and used by Froment and Bischoff (1961) to describe activity decay via a fouling, or coking,
128
Ind. Eng. Chem. Fundam. 1980, 79, 128-129
mechanism. The empirical forms suggested by Anderson and Whitehouse (1961) relate catalyst activity to the concentration of poison on the catalyst. Although their equations are written at a point in a catalyst bed, we choose to drop such subscripts for simplicity of presentation. The proposed relationships thus become a = 1 - aCp (for aCp 5 1) (13) a = exp(-aCp)
(14)
+ aCp)-l
(15)
a = (1 a*= (1 -
(for aCp 5 1)
(16)
where CY is a constant and Cp is the concentration of poison or coke on the catalyst. Froment and Bischoff (1961) utilized eq 14 to remove the catalyst activity functionality from the continuity equation for coke on the catalyst
to obtain
Further insight into eq 13 through 16 can be obtained by differentiating and combining the result with eq 17. From such an approach, eq 13 becomes
respect to activity of 1, 2, 3, and zero, respectively. Obviously, the order with respect to the reactant A is determined by the assumed form of continuity equation for coke. Butt (1972) comments that eq 13 through 16 lead to modes of reactor performance that qualitatively look very similar to observed deactivation behavior and that Voorhies-type time correlations also typically model such data adequately. Using the alternative formulation of active site balances proposed herein not only provides the advantage of a mass action law interpretation of the empirical eq 13 through 16 and the attendant extension to other poison continuity equations such as eq 17, but also allows the consistent extension of the empirical experience represented by these equations to other reactor geometries, such as shown by eq 10. Nomenclature a = normalized catalyst activity, dimensionless C, CA, Cp = chemical concentrations, g-mol/L d , m , n = dimensionless reaction orders k D = deactivation rate constant m = catalyst addition and removal rate, g/s r D = deactivation rate, sites/g-s S = number of active sites per unit weight of catalyst, sites/g So = number of active sites per unit weight of fresh catalyst, sites/g t = time, h T = absolute temperature, K W = weight of catalyst, g I
Greek Letters a = empirical constant T = space time of reactant, s
eq 14 becomes
aa a CP at = -a exp(-aCp)- at = -akD$CA eq 15 becomes
and eq 16 becomes
Hence, with the assumed continuity equation for catalyst coke (eq 17) eq 13 through 16 become interpretable through active site balances and notably become concentration-dependent deactivation equations of order with
T , = space time of catalyst, s Literature Cited
Anderson, R. B., Whitehouse, A. M., Ind. Eng. Chem., 53, 1011 (1961). Butt, J. B., Adv. Chem. Ser., No. 108, 259 (1972). Butt, J. B., Wachter, C. K., Bilimorla, R. M., Chem. fng. Scl., 33, 1321 (1978). Carberry, J. J., "Chemical and Catalytic Reaction Engineering", McGraw-Hili, New York, N.Y., 1976. Froment, G. F., Bischoff, K. B., Chem. Eng. Sci., 18, 189 (1961). Hougen, 0. A., Watson, K. M., "Chemical Process Principles", Part 111, p 934, Wiley, New Yo&, N.Y., 1947. Levenspiel, O., "Chemical Reactlon Engineering", Wky, New York, N.Y., 1972. Prater, C. D., Lago, R:M., A&. Catal., 8, 293 (1956).
Department of Chemical Engineering University of Massachusetts Amherst, Massachusetts 01003
Edward K. Reiff, Jr.* J. R. Kittrell
Received for review May 7, 1979 Accepted October 18, 1979
CORRESPONDENCE 'Comments on: "Cubic Equations of State-Which?'' Sir: In a recent article (1979), Professor Joseph Martin presented a particular equation as the most general form of a volume-cubic equation of state. In fact, the equation presented by Martin is not the most general form of a volume-cubic equation of state, as discussed below. A direct density (or volume) expansion for pressure which is cubic in density is given by the following ex-
pression
P = a,
+ a2p + a$? + a4p3
(1) where P is the absolute pressure, p is the molar density and al, a2,a3,a4 are parameters which can be temperature dependent. It is known that an expansion of the above form, which is similar to the virial equation of state up to
0019-7874/80/1019-0128$01.00/00 1980 American Chemical Society
