Acknowledgment
{
One of the authors held a National Science Foundation Fellowship during this work. The numerical calculations were carried out at the Princeton University Computation Center, which is supported in part by the National Science Foundation.
T
4 7
w
plant parameter 3.14159 phase angle dead time = plant parameter = = = =
literature Cited Nomenclature
Cohen, G., Coon, G., Trans. A.S.M.E. 75,827 (1953). Eisenberg, L., ZSA Trans. 6 , 225-33 (1967). Farrington, G., “Fundamentals of Automatic Control,” pp. 196203, Chapman and Hall, London, 1951. McAvoy, T., Ph.D. dissertation, Princeton University, Princeton, N. J.. 1964. McAvoy, T., Johnson, E. F., Znd. Eng. Chem. Process Design Develop. 6. 440 11967). Ream, N.‘, Trahs. Soc. Znst. Technol. 6 , 19 (1954).
= derivative compensating constant G = plant transfer function G, = controller transfer function G, = feedforward controller transfer function K , = proportional gain M = magnitude s = Laplace transform variable t = time Td = derivative constant Ti = integral constant
c
RECEIVED for review September 28, 1967 ACCEPTED July 19, 1968
T. J. McAVOYl
GREEKLETTERS = = y = 8, = 8? = e,, = (Y
p
E. F. JOHNSON
damping factor frequency variable controlled variable reference variable upset variable
Department of Chemical Engineering Princeton University Princeton, A’. J . 1 Present address, Department of Chemical Engineering, University of Massachusetts, Amherst, Mass. 01003.
EFFECTIVENESS FACTORS FOR REVERSIBLE REACTIONS The generalized graphical methods presented in previous papers for calculating catalyst effectiveness X. Illustrative calculations demonstrate use of factors are extended to a simple reversible reaction A the method.
e
ROBERTS and Satterfield (1965,
1966) have presented generalized graphical methods for calculating catalyst effectiveness factors for cases in which the kinetics follows a Langmuir-Hinshelwood (Hougen-Watson) type of rate equation and is irreversible. The present paper extends the method to a simple reversible reaction.
A e X
PA - P X ( P A , e / p X , e ) l
1
+ KAPA+ K X P X -I- C K i p i
(2)
I
The index i denotes any species present that can be adsorbed onto the surface of the catalyst but does not undergo reaction. As in the previous papers, it is assumed that the catalyst is infinite in two directions and of thickness L in the third and is exposed to the gas stream on one face and sealed on the other, that the effective diffusivities of all species are constant but not necessarily equal, that the catalyst is isothermal, and that the ideal gas laws are applicable. The simple first-order, reversible reaction in flat plate or spherical geometry has been treated by a number of authors: Boreskov (1947), Boreskov and Slinko (1952), Carberry (1962), Smith and Amundsen (1951), Wagner (1943), and Wicke and Brotz (1949). Schneider and Mitschka (1966a) recently calculated effectiveness factors for the same reversible reaction following the more general rate expression of Equation 2, but their results are couched in terms of parameters difficult to use for calculations and difficult to identify with physical meaning. Their analysis has been extended (1966b) to the rate expression 664
l&EC FUNDAMENTALS
(3)
(1)
T h e reaction rate is assumed to obey the relationship:
r =
derivable by assuming a second-order rather than first-order surface process. We follow our previous practice of presenting the effectiveness factor in terms of the modulus aL,containing only observable or calculable quantities. As before, G L is defined as:
Details of the mathematical procedure and numerical calculations are given by Kao (1967). Two new parameters are required over those used for the irreversible reactions. One is P A , ~ / P A ,= ~ C, the ratio of the equilibrium partial pressure of A to that at the outside surface of the catalyst. The other parameter is defined by:
where
(5)
The generalized definitions of K and w as used in previous papers reduce to Equations 5 and 6, respectively, for the present case. Their values are independent of whether or not the reverse reaction is considered. For irreversible reaction, B in Equation 4 reduces to K p A having the same definition as previously used. An irreversible zero-order reaction corresponds to K P A , ~approaching infinity, a first-order reaction to KpA,s approaching zero, and irreversible higher order reaction to negative values of K P A , ~ . The minimum possible value is -1,
present graphs of q os. aL,for values of C of 0.3, 0.5, 0.7, or 0.9, and for a family of curves of B. Figure 5 is a cross plot on 7 Q L coordinates of families of curves of C, for values of B = -0.98, 0, or 50. A graph of q os. aL for C of 0.1 (Kao, 1967) differs little from Figure 3 of Roberts and Satterfield (1965), and is not reproduced. The limiting case of C = 0 represents irreversible reaction.
corresponding to very strong product adsorption effects. The 00 and maximum and minimum values of B are likewise -1 and B = 0 corresponds to a simple first-order reversible reaction. T h e ratio of PA,e/PA,o = C will vary between zero (for irreversible reaction) and 1 (for reaction a t equilibrium and, therefore, corresponding to a zero rate). The absolute value of B is always less than that O f & A , $ . Figures 1 through 4
+
-
I O
F
-b (L"
0 IO
YI
a(Y2 c
-
2 e 9! W
0 01
0001
001
01
10
IO0
6, Figure 1. Effectiveness factor as a function of modulus @L for a reversible firstorder surface reaction C = 0.3
0001
001
10
01
100
P L
Figure 2. Effectiveness factor as a function of modulus order surface reaction
aL for
a reversible first-
C = 0.5
P, Figure 3. Effectiveness factor as a function of modulus @, order surface reaction C = 0.7
for a reversible first-
VOL. 7
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NOVEMBER
1968
665
6L Figure 4. Effectiveness factor as a function of modulus order surface reaction C = 0.9
CP,
for a reversible first-
E=-098.C:0 I E = -098,C.O 3 E;-098,C2051 8=-098,C:07 ' 8.- 098,C;O9.
9
t-
0.0I
t
I
I
I I I I I I
0.001
0.01
I .o
0.1
10.0
@L
Figure 5.
7 vs.
CP,
relationship for representative combinations of 6 and C
As pointed out in previous papers, neglecting to allow for reverse reaction results in a calculated effectiveness factor higher than the true value. The examples below illustrate this point and demonstrate the use of the graphs. The parameters used here for characterizing the degree of reversibility of the reaction cannot be directly related to those used by Schneider and Mitschka-namely, M , and B. Their F' corresponds to our w and their B' to our Kw. For the irreversible case, their B reduces to our KpA,,. Illustrative Examples Case I. Gupta and Douglas (1967) report studies on isobutylene hydration to tert-butyl alcohol on Dowex 50W cation exchange resin. Water is present in large excess and the reaction rate is first order with respect to isobutylene concentration. T h e reverse reaction may be assumed to be first order with respect to tert-butyl alcohol. They report that the equilibrium constant a t 100' C . corresponds to a minimum of 94y0 conversion of isobutylene and they assumed reaction was essentially irreversible under their experimental conditions. For a simple first-order reaction, Kpa,*+ 0, B + 0 ,
C = P a , e / P a , , = 0.06/0.94 = 0.064 666
I&EC FUNDAMENTALS
Calculate the effectiveness factor for a set of their experimental conditions corresponding to the greatest degree of diffusional limitations (highest temperature and largest particles). At 95' C. on 48- to 65-mesh resin (radius in the swollen state = 0.0213 cm.), the rate was about 55 X gram mole/(hr.) (gram resin) a t a surface concentration of 0.0172 X gram mole of isobutylene per cc. of swollen resin. Deft a t 95' C. was 1.6 X 10-5 sq. cm. per second. The rate was specified per gram of anhydrous resin, whereas we wish it on the basis of cubic centimeters of swollen resinvolume. The degree of swelling was not specified, but for illustrative purposes we may assume about 1 cc. of swollen resin per gram of anhydrous resin. For simple isothermal reactions mathematical relationships derived for flat plate geometry may be applied with little error to reaction in a sphere by taking L = R/3 where R is the radius of the sphere. Substituting into Equation 1,
a,
= =
(0.0213/3)2 (55 X 1.6 X 3600
1 1.72 X
2.8
From Figure 5, interpolating between the curves for C = 0 and C = 0.1, and taking B = 0, q = 0.34. Comparison of the
curves shows that neglecting to allow for the reverse reaction results in a calculated value of q high by only about 5% if equilibrium corresponds to 94(r, conversion. The degree of error incurred is independent of the value of 7 in this asymptotic region. I n this case the reaction is taken to be simple first order in each direction; hence B = 0, and it is unnecessary to know the effective diffusivity of the product. This, however, will be needed for more complex kinetic expressions. Case 11. Consider a gas-phase reaction in which the combination of catalyst dimension, L, effective diffusivity, D e f f , observed reaction rate, ( - I / V , ) ( d n j d t ) , and reactant concentration a t the outside surface, results in a value of @L = 1.0. We suppose that the reaction is retarded by product adsorption to such a degree that K = -0.7 and we takepA,s= 1. If we assume that the reaction is irreversible, from Figure 3 of Roberts and Satterfield ( 1 9 6 5 ) , the effectiveness factor, 9, corresponding to @ = 1.O and Kp,,, of -0.7 would be 0.50. However, we now discover that the reaction is in fact reversible and t h a t p A , ,is, let us say, 0.5. Then,
c
=
PA,B/PA,S
=
+0.5
c
D or D e f f
= pA,e/p Ass = effective diffusion coefficient of reactant A, sq.
Ki
= adsorption equilibrium constant for species
cm./sec.
K
k L
P r
= = =
GREEKLETTERS w
=
+L
=
defined by Equation 6 defined by Equation 3
SUBSCRIPTS
A e 2
S
X
and
= =
i
in rate expression, atm.-' defined by Equation 5 reaction rate constant, (g. mole)(atm.)/(cc.) (sec.) thickness of catalyst slab, cm. partial pressure, atm. reaction rate, g. moles reactant/(sec.) (cc. cat.) of reaction, g. moles/(sec.)(cc.
reactant a t equilibrium any species present that can be adsorbed but does not undergo reaction = conditions at outside surface of catalyst particle = product = = =
literature Cited
From Figure 2, the effectiveness factor is calculated to be 0 . 3 5 . Allowing for reverse reaction does not change the value of K , but changes the values of both B and C. Ifp,,, were 0.9, then
C = +0.9 -0.7 (1 - 0.9) = -0.19 1 (-0.7)(0.9)
B=-
+
From Figure 4, 7 = 0.09 as opposed to 0.50 if reverse reaction were neglected. Calculations such as this indicate that when reaction is occurring near equilibrium (relatively large values of C ) neglect of the reverse reaction can result in erroneously high effectiveness factors being calculated. A compensating factor, however, is that under such conditions the observed reaction rate will tend to be small, resulting in a low value of aL. Nomenclature
B 6,
= =
defined by Equation 4 concentration of A at outside surface of catalyst pellet, g. moles/cc.
Boreskov, G. K., Khim. Prom. 1947, p. 256. Boreskov, G. K., Slinko, M. G., Zh. Fir. Khim. 26,235 (1952). Carberry, J. J., A.Z.Ch.E. J . 8, 557 (1962). Gupta, V. P., Douglas, W. J. M., A.Z.Ch.E. J . 13,883 (1967). Kao, H. S-P., S.M. thesis, Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, 1967. Roberts, G. W., Satterfield, C. N., IND.ENG.CHEM.FUNDAMENTALS 4, 288 (1965). Roberts, G. I$'., Satterfield, C . K., IND.ENG.CHEM. FUKDAMENTALS 5 , 317 (1966). Schneider, P., Mitschka, P., Chem. Eng.Sci. 21, 455 (1966a). Schneider, P., Mitschka, P., Collection Czech. Chem. Commun. 31, 3677 (1966b) (inEnglish). Smith, N. L., Amundsen, N. R., Znd. Eng. Chem. 43, 2156 (1951). Wagner, C., Z. Phys. Chem. 193, l(1943). Wicke, E.,Brotz, W., Chem.-Zng7.-Tech. 21, 219 (1949). H E N R Y SO-PING K A O Department of Chemical Engineering Princeton University Princeton, N. J .
CHARLES N. SATTERFIELD Department of Chemical Engineering Massachusetts Institute of Technology Cambridge, M a s s . RECEIVED for review h'ovember 6, 1967 .4CCEPTZD June 11, 1968
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