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
Ind. Eng. Chem. Fundam., Vol. 19, No. 1, 1980 129
the third virial coefficient, can only describe the low density gas phase behavior of a fluid. An equation for pressure of the above form which can describe both the gas and liquid phase behavior of a fluid can be written as follows m
P=
zajpJ-1 j=1
However, eq 2 is of infinite order in density. An equation of state which can approximate a n infinite series i n density using pressure as t h e dependent variable and yet requires solution of only a cubic expression for density (given pressure) is a ratio of polynomials a1 + a2p + a3p2 + a4p3 P= (3) + a7p2 + asp3 a5 + Equation 3 represents the most general form of a density-cubic or, alternatively, volume-cubic mathematical equation, where al through a8 are parameters which can be temperature and composition dependent. When multiplied out, eq 3 can be shown to yield a cubic in density, as follows (pa5 - al) + (Pa6 - a2)p + (Pa, - a3)p2 + (Pas - a4)p3 = 0 (4) In terms of the compressibility factor Z (= PIpRT), eq 3 becomes
z=
(C,/P)
+ cz + C3P + C4P2
+ a7p2 + asp3
+
( 5)
a5 where Ci= a i / R T , i = 1, 2, 3, 4, R is the universal gas constant, and T is the absolute temperature. Before eq 5 can be subjected to the thermodynamic ideal gas limit that as p -,0, Z 1, C1 is required to be zero to prevent the divergence of Z as p 0. Letting C1 = 0, we have
- -
Equation 9 represents the most general form for a cubic equation of state in density which satisfies the requirement that in the limit as p -,0, Z -,1. The five coefficients d l , d z , d3, d4, and d5 must be density independent but can be temperature dependent (for a given pure fluid) and composition dependent (for a mixture). At this juncture we can easily show that Professor Martin's general volume-cubic equation is only a special case of eq 9. If we let
d3 ='(y
+ P); d4 = yP; d5 = 0
we have
or
Z=1-
RT
(1 + PpMl
+ yp)
+ (1 + Pp)(l + y p )
In terms of the specific volume, V , eq 1 2 becomes
" RT
Z=1-
( V + P)(V + y) In terms of P, eq 13 becomes
p = -R-T
4T)
+
+
RT (13) ( V + P)(V + y) 6(T)
v
Now in the limit as p
-+
0 (7)
-
To satisfy the thermodynamic requirement as p -,0,Z 1, C2 must be equal to a5. Letting C2 = u5 and dividing the numerator and denominator on the right-hand side of eq 6 by Cz we have n
1
z=
L'3
n
L'4
+ -c2p + -p2 c2
(12)
(14)
(V + @)(V+ y) V ( V + P)(V + y) which is Professor Martin's equation (eq 1 in his paper), where P is the pressure, V is the specific volume, T is the absolute temperature, R is the universal gas constant, a and 6 are functions of temperature, and P and y are constants. Since Professor Martin's equation was shown to be a general volume-cubic equation of state from which previous cubic equations of state can be obtained as special cases, it is obvious that all previous cubic equations of state also can be obtained from eq 9 as special cases. On the other hand, there is no previously reported cubic equation of state from which eq 9 can be obtained. Literature Cited
Now letting C3/C2 = d l , C 4 / C 2= d 2 , a6/C2 = d3, a,/C2 = d4, and a8/C2= d5, we have 1 + dlp + d2p2 z = 1 + d3p + d4p2 + d5p3 (9)
Martin, J. J. Ind. Eng. Chem. Fundam., 18, 81 (1979).
School of Chemical Engineering and Materials Science The University of Oklahoma Norman, Oklahoma 73019
K. Hemanth Kumar Kenneth E. Starling*
