I
ROBERT 8. ANDERSON Bureau of Mines,
U. S.
Department of the Interior, Pittsburgh, Pa.
Graphical Differentiation in Ammonia Synthesis Kinetics This method simplifies testing rate equations, and a differentiation method based on log-log plots is often useful
IN
-
CATALYTIC KINETICS a number of reasonable fundamental rate equations are examined, and the best equation is chosen on the basis of its ability to fit the kinetic data and the consistency of sign, magnitude, and temperature dependente of the constants with the postulated reaction mechanism. Where conditions of the differential reactor are not attained, the laboriousness of testing rate equations can be substantially decreased if the differential reaction rate can be determined by graphical differentiation. In the work described here, a simple method of graphical differentiation is examined, using published data for the ammonia synthe i3 on iron catalysts. For catalytic reactions the differential reaction rate, r , may be defined as ( 6 ) 7
= dx/d(V/Q)
(1)
where V i s volume or weight of catalyst; Q is weight, moles, or S.T.P. volume of total feed or certain components of feed per unit time; and x is fraction consumed of the components defining Q. When Vis defined as bulk volume of the catalyst and Q is the S.T.P. gas flow per hour, V/Q = 1/S, where S i s hourly inlet space velocity as usually defined. For simplicity r = V/Q is used. T is not contact time, and its dimensions are the reciprocal of those for the differential reaction rate. I n ammonia synthesis kinetics, three definitions of differential reaction rate a)-" da/d have been used; r = (1 (l/S), where n = 1, 2, or 3, a is the mole fraction of ammonia, and S is the inlet space velocity (2, 4, 8, 9 ) . Recent workers ( 9 ) have used n = 2, which is identical to Equation 1, and this equation is obtained by both contact-time and flow-balance derivations (7). The three definitions lead to only small numerical a) usually differences, as the term (1 has a value between 1.00 and 1.10. For the ammonia synthesis, x can be defined as the fraction of hydrogen, nitrogen, or H P Nz consumed. The same component or components must, of course, be used to define space velocity. I n the present paper, conversion and space velocity are based on HZ Nz. The rate equation
where PA, p B , p R , and ps are partial pressures or concentrations of reactants, products, other components, and T , the temperature, may usually be transformed to 7 = d x / d r = g(x, T ) (3) and tested by integrating
Integration may be difficult, when a volume change occurs or the stoichiometry is not constant. The FischerTropsch synthesis on iron is a complex reaction with variable stoichiometry in which the relative usage of hydrogen and carbon monoxide varies with both conversion and feed gas composition, For this system integration is impractical if not impossible. In any case, if the differential reaction rate, r , can be determined by graphical differentiation, testing of the rate equation in its differential form (Equation 2), is simpler than in Integral form (Equation 4). The kinetics of the ammonia synthesis is commonly expressed by the rate equation of Temkin and Pyzhev (8), =
k l P N 2 (fi3H%/p2NH,)"
-
k2 (pzNH8/p3H*)'
(5)
+
where a is mole fraction of ammonia produced, L is a constant, and P is total pressure in atmospheres-is so complex that it must be evaluated numerically. a ) term in the The exponent of the (1 denominator of Equation 6 has been given the values 1, 2, and 3 by various workers, depending on the definition of the differential reaction rate. According to Equation 1 the exponent should be 2. In this numerical integration the value of a is assumed, and kz is calculated. The value of QI that yields from data a t a given temperature the most nearly constant values of kz is taken as the best choice. When a is small, of the order of 0.01, the integral can be simplified enough so that it can be integrated. By differentiating the rate data to obtain r , the problem is simplified, and both k2 and a may be evaluated directly. Equation 5 may be rearranged to
+
( KP(nhs/a2) - 1 / r = (l/kz)(Ph3/u2)1(7)
where h, n, and a are mole fractions of
+
I
-a
where k l , k z , and CY are constants. Usually the equation is expressed in terms of k?, the rate constant for ammonia de-
composition, where kl = Kk2 and K is the equilibrium constant, K,. Despite the relative simplicity of the reaction and Equation 5, the integral-for ex1Np and a = 0.5 ample, for 3Hz
l
1
20
30
+
+
+
r = d x / d r = f($a.$e,
. . , P R , ~ s ., . . T )
(2)
2
3
4
6 0422 X
8
1
0
/lo8(?l2
Figure L. Graphical differentiation permits linear plots of the Temkin equation Data of Brill a i atmospheric pressure and 321 C.
2
1
1
3
4
~ N L E TSPACE VELOC~TI
1 6
8
1 1
20
0
hr
Figure 2. Graphical differentiation using log-log plots Ammonia synthesis with 3Hz atm. and 400' C.
VOL. 52, NO. 1
+ 1N2
a t 100
JANUARY 1960
89
hydrogen, nitrogen, and ammonia, respec tively Graphical differentiation on double logarithmic plots was found satisfactory for determining r . Such plots of x as a function of 7 were approximately linear over about half the range of conversion reported; and, by addition of a constant increment 1 to the values of x , the plots become linear within the possible experimental error of the measurements over virtually the entire range of conversion. Further adjustment by adding an increment to i- is possible, but was not needed. For the general case the equation is
.
= k(r+m)j
x + l
71 Figure 3. Linear plots o f Temkin equation permit direct determination of both constants Data
of
Emmett
and
Kummer
(8)
where k is a constant, and r = dx/dr = jk(r
+ rn)j-l j ( X
=
+ l)/(T + m)
(9)
Equation 9 can be readily evaluated from the slope of the double logarithmic plots, j , and the experimental values. The method has moderate accuracy only for regions in which the log-log plot is almost exactly linear. For testing kinetic equations the differential reaction rate should be evaluated over as wide a range of conversion as possible; therefore, the increments should be adjusted to give the longest linear plot possible. For synthesis at atmospheric pressure, a is very small, and h = h, and n = n,. Equation 7 may be expressed in terms of the efficiency factor, 7, where 7 = ais, and a,: he, and ne are mole fractions of ammonia, hydrogen, and nitrogen 1 S 2 feed at equilibrium. For 3H2 Equation 7 becomes
+
(a2,/a2
(I/$
-
-
l)/r = ( 1 / k 2 ) ( 0 . 7 5 3 ~ 2 , /u~22) ,1 - a
(IO)
l)/r = (1/k:)(O.422/a2,q2)1 -
a
(11)
Atmospheric pressure data of Brill (3) for two iron catalysts at 321' C. were used. Double logarithmic plots of a (= 2x at the low conversions obtained) as a function of S were linear, and no adjustment was necessary. Plots of Equation 11 are shown in Figure 1. The least squares method of Ergun (5) for determining the best common slope for a family of parallel lines was used to obtain the data in Table I. The common slope was 0.409 (a = 0.591),
Table l.
I
3
08
06
3
but for computing the values of kz a slope of 0.4 was used, so that the rate constants Lvould be comparable with those reported by Brill with o( = 0.6. I n ammonia kinetics constant terms are often included in kz, and these terms are not defined by Brill. Thus, the ratios of rate constants obrained by each method \\'ere used to test the values of the rate constant. The agreement is good. The best fit of atmospheric pressure data to the Temkin equation was obtained \virh cy = 0.6 (2, 3). The data of Emmett and Kummer (4) for experiments a t 50 to 100 atm. also were used to test the differential method. Values of mole fractions of ammonia and outlet space velocity were taken from the smoothed curves, and these were N2, x, changed to conversion of HS and inlet space velocity. On the double logarithmic plots x had to be adjusted to obtain a satisfactory linear plot over more than 95% of the rangeof conversion, as shown in Figure 2. To obtain the rate in terms of ammonia Droduced. the values of r determined in Figure 2 ivere multiplied by 2. Plots of rate Equation 7 are presented in Figure 3 and values of constant kz in Table 11. As the differential reaction rate was defined in the same units for calculations bv both methods, the values of kz may be compared directly for the same values of cy. In
+
Ammonia Synthesis a t Atmospheric Pressure
(Iron catalysts at 321' C. with 3Hz
4- 1N2 gas) Integration of Brill
Differential Method k2
a
Catalyst
Each test
1.
FeaOa-A12Od-&0
0.558
2.
FeaOa Ratios of kn's
0.633
Common slope
01
= 0.6
k2 01
= 0.6
53.1
400
13.8 3.85
105 3.81
0.591
90
INDUSTRIAL AND ENGINEERING CHEMISTRY
20
40
I
60 Ph3/oZ
8 0 I00
I
20 0
I
I
400
tests of the Temkin equation by integration a selected value of cy was assumed LO be applicable to a wide range of experimental conditions. In the present work the best average value of 01 for fitting all data under consideration is chosen. Using the best common slope, o( = 0.618, values of ky were computed. For comparison with the rate constants of Emmett and Kummer, the values of k , were also computed from straisht lines passed through the mean values of the ordinates and abscissas of Figure 3, with slopes corresponding to a = 0.5 and 0.667. The values of k2 by the differential and integral methods for the same value of CY for individual experiments are in satisfactory agreement. If the Temkin equation exactly expressed thc kinetic data, the values of X 2 should be equal at a given temperature and should not vary with feed composition or pressure. The values of k2 obtained by the two methods show the same trends, and have about the same variations.
Discussion The preceding sections demonstrdte the advantages of examining rate data in differential form Graphical differentiation of moderate accuracy permits rapid testing of possible rate equations and in some c a m more direct evaluation-e.g., here both constants of the Temkin equation were obtained. Graphical differentiation on double logarithmic plots uses the method of straightening log-log plots by adding constant increments. This is particularly useful for ammonia kinetics, as linear plots covering a wide range of conversion were obtained. The best results are obtained for data that plot approximately linearly on log-log paper. The firstorder equation, x = 1 - exp(---r), for example, has considerable curvature
0 R A P HIC A L DI F F E RE N T I AT10N Values o f Constants for Temkin Equation for Ammonia Synthesis a t 50 to 100 Atm. Graphical Differentiation Emmett-KummerQ Integration Feed Ratio, Pressure, k2 kl k2 k2 k2 a Atm. a = 0.618 a = 0.5 HdN2 a = 0.667 a = 0.5 a = 0.667
Table II. Temp.,
c.
400
100 66.6 100 66.6 100 50 100 66.6
450 450 450
3 3 3 3
0.539 0.558 0.744 0.588 0.763 0.592 0.540 0.590 0.618
1
1 1/3 1/3
From common slope a Average of values reported.
300 395 3400 4390 2810 3720 2810 2890
177 23 1 2155 2765 1860 2420 1970 2010
1,065 1,430 10,170 13,330 7,640 10,490 6,590 6,900
835 1,270 11,470 14,340 9,580 11,670 6,590 7,665
120 185 2020 2470 2020 2340 1640 1870
____
c
Table 111.
Graphical Differentiation of First-Order Equation x = 1
- exp (-7)
dx/d.r T
X
Actual
0.00
0.02 0.05 0.08 0. 0 0.14 0.18 0.25 0.30 0.35 0.40 0.50 0.60 0.70 0.80
0.0198 0.0488 0.0769 0.0952 0.1307 0.165 0.221 0.259 0.295 0.330 0.393 0.451 0.503 0.551 0.632
0.980 0.951 0.923 0.905 0.869 0.835 0.779 0.741 0.705 0.670 0.607 0.549 0.497 0.449 0.368
(0.956)a 0.953 0.928 0.919 (0.903)
1.o
a
Graphical, Value of 2 below 0.02 0.06 (0.962) 0.916 0.856 0.818 0.776 0.740 (0.716)
Acknowledgment
(0.922) 0.846 0.760 0.719 0.686 0.660 0.612 0.576 (0.544)
(0,802) 0.743 0.697 0.662 0.604 0.560 0.523 0.493 (0.448)
Parentheses indicate end points of linear plot.
on a log-log plot, and no long linear plots could be obtained by adding constant increments. Nevertheless, reasonably accurate derivatives were obtained from the short linear portions (Table 111). T h e greatest accuracy is usually obtained near the middle of the linear plot, and except for the end points of the linear portion the graphical derivatives are usually within 3% of actual values. Log-log plots of kinetic data for the Fischer-Tropsch synthesis on nitrided iron catalysts are approximately linear (7). Smoothed data were differentiated by the present graphical method and by a numerical central difference method involving fitting a four-constant poly-
Table IV. 1H2
+ 1CO ~-
nominal to the reference point and three equal intervals above and below it by least squares. T h e derivatives obtained by the two methods in Table IV usually agree within about 2y0 except for the end points of the linear portion of the log-log plots. T h e derivatives are essentially independent of the adjustment constant (Tables I11 and IV). T h e present method involves graphical determination only of the slope of the linear portion of the log-log plot. This value is a constant multiplier for the group of data and does not influence accuracy within the group. The slope of the log-log plot can also be determined numerically as
(lOOO/S) 0.25 0.5 0.75
Conversion, Graphical
0.7%
literature Cited
(1) Anderson, R. B., Toor, H. L., J . Phys. Chem., in press. (21 Bokhoven, C., van Heerden, C., Weetrik, R., Zwietering, P., “Catalysis,” vol. 111, ed. by P. H. Emmett, Chap. 7, pp. 325-33, Reinhold, New York, 1955. (3) Brill, R., J . Chem. Phys. 19,1047 (1951). (4) Emmett, P. H., Kummer, J. T., TND. ENG.CHEM.35, 529 (1943). (5) Ergun, S., Ibid., 48, 2063 (19561. (6) Hougen, 0. A., Watson, K. M., Z6id., 35, 529 (1943). (7) Karn, F. S., Seligman, B., Shultz, J. F., Anderson, R. B., J . Phys. Chem. 62, 1039 (1958). (8) Temkin, M., Pyzhev, V . , Acta Physicoc/zim. U.S.S.R. 12, 327 (1940). ( 9 , Uchida, H., Kuraishi, M., Bull. Chem. SOC. Japan 28, 106 (1955). RECEIVED for review May 14, 1959 ACCEPTED September 11, 1959 Catalyst Club of Philadelphia, March 1959.
+ 1co
2H2
(245)b 228 216 206 199 (189)
(235) 213 199 181 167 (157)
-
+ 1CO
r
Numeri- Conversions Graphical cal‘ X 1 = 0.02 E = 0.12
E=O 0.085 (269)b 0.068 0.151 239 0.130 0.208 219 225 0.189 1.o 0.263 208 209 0.245 1.25 0.315 199 199 0.295 1.50 0.362 191 193 0.345 2.0 0.460 182 188 0.443 2.5 0.550 174 175 0.530 3.0 0.635 (167) 150 0.613 Derivatives determined by central difference method. X
T h e author acknowledges valuable discussions with A. M. Whitehouse in the development of the differentiation method. Many of the calculations were made by L. J. Ruzzi.
Differentiation o f Kinetic Data for Fischer-Tropsch Synthesis (Nitrided iron catalyst at 21.4 atm. and 240’ C.) Feed Gas
r 7
a
where subscripts i and f denote the values of points near the beginning and end of the linear portion, respectively.
0.12
r
Numeri- Conversion, Graphical cala X 1 = 0.02 z = 0.12 241 227 211 204 200 185 171 146
0.070 0.134 0.198 0.258 0.314 0.368 0.468 0.555
(256)b 242 231 222 (215)
(240) 220 206 186 (171)
Numericai5 255 245 233 22 1 213 186 149
Parentheses indicate end points of linear plot.
VOL. 52, NO. 1
JANUARY 1960
91
