Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 3, 1979
399
Analysis of Temperature-Time Data for Deactivating Catalysts S. Krishnaswamy and J. R. Kittrell” Department of Chemical Engineering, University of Massachusetts, Amherst, Massachusetts 0 1003
A simple mathematical model based on a single irreversible primary reaction and nth order, concentration-independent deactivation, is presented. The model predicts the time-temperature relationship in order to maintain constant conversion of reactant. The applicability of the model to hydrocracking and reforming pilot plant data is demonstrated. Effects of process variables such as start-of-run temperature, space velocity, and conversion level of operation on t h e fouling rate are examined.
Introduction In the operation of heterogeneous catalytic reactors, the activity of the catalyst commonly drops progressively with time on stream. If such a reactor is operated a t constant feed rate and temperature, the conversion of reactant will decrease with time. A shut-down period is required when the conversion drops too low, and the catalyst is either discarded or regenerated. Alternately, conversion can be maintained constant by adjusting process variables such as reactant feed rate and reactor temperature. Commercial reactors undergoing catalyst decay are often controlled to produce constant conversion of reactants by gradually increasing the temperature with time-on-stream till the catalyst is no longer of use and has to be replaced. Szepe and Levenspiel (1968) used the Bolza form of the calculus of variations to examine the optimal temperature policy in a batch reactor. They demonstrated that in order to maximize the final conversion for a specified reaction time and final catalyst activity, the effective rate constant should be invariant during the entire process, provided the deactivation reaction is more temperature sensitive than the main reaction. Operating the reactor at the maximum allowable temperature was found to be optimal when the activation energy for deactivation is less than that for the main reaction. The isothermal plug flow problem was solved by Chou et al. (19671, who developed appropriate time-temperature relationships for the case of first-order, concentration-independent decay. For concentrationdependent deactivation, Lee and Crowe (1970) showed that the constant conversion policy is not optimal for unconstrained temperatures. Other aspects of the problem have been considered by Ogunye and Ray (1968, 1971), Pommersheim and Chandra (1974), Earp and Kershenbaum (1975), Therien and Crowe (19741, and Crowe (1976). Even though several commercial processes, such as hydrocracking and desulfurization of gas oil, are operated with the constant conversion, increasing temperature policy, few attempts have been made to apply the theoretical predictions of simple mathematical models, such as the one developed by Chou et al. (1967), to industrial data. Butt and Rohan (1968) have analyzed reactor operation with catalyst deactivation assuming several general cases of nonselective catalyst poisoning. The integration of their model forms is rather complex and, consequently, their solutions for temperature-time profiles are difficult to use. Theoretical Development Assuming the constant conversion mode of operation, a simple temperature-time relationship can be derived for the general case of nth order, concentration-independent deactivation. Szepe and Levenspiel (1970) have discussed 0019-7882/79/1118-0399$01 .OO/O
the applicability of such separable rate forms for deactivation studies. For concentration-independent decay, we have
where k d is the deactivation rate constant, exhibiting an Arrhenius temperature dependence kd
= Ad exp(-Ed/Rn
-
(2)
For an irreversible reaction of the type A B, the following condition must be satisfied for constant conversion
k.a = k o
(3)
where k , the rate constant for the primary reaction, is given by k = AA exp(-EA/RT) (4) and ko corresponds to the value of this rate constant a t the start of the run, Le., when T = To. From eq 3 and 4 the following relationship between catalyst activity and temperature is obtained
Substituting eq 5 and 2 into eq 1 and integrating, we find
wherein the initial catalyst activity was assumed to be unity. Rearrangement of eq 5 and subsequent substitution in eq 6 finally yields the required time-temperature relationship
For the special case of first-order deactivation, this expression becomes identical with the result derived by Chou et al. (1967). When the order of deactivation equals 2 and when the activation energies for reaction and deactivation are 0 1979 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 3, 1979
400
0
700
---
Run 1
A Run 2
I
-
%de
Predl'tlor
L
I
,
I
I
1
I
4""
0
400
200
800
600
1000
T i m e (hrs.)
Figure 1. Temperature-time trajectories for deactivating hydrocracking catalyst, runs 1 and 2. 820
U
;
140
L
a
I
I
E" f
660
580;
580
0
200
100
"
"
100
200
400
300
"
300
400
! IO
T i m e , hours
T h e (hrs)
Figure 2. Temperature-time trajectories for deactivating hydrocracking catalyst, runs 3 and 4.
Figure 5. Temperature-time trajectories for deactivating hydrocracking catalyst, runs 9, 10, and 11. 1000
t
'
"
0
c
o Run 5 A Run 8
980
c"
960 L
700
'
0
1
I
200
I
I
I
400
600
I
I
800
I
I
940
k
Figure 3. Temperature-time trajectories for deactivating hydrocracking catalyst, runs 5 and 6.
I
identical, eq 7 is not valid. For this special case, an analogous derivation leads to the equation
Application of Model to Reactor Data Figures 1-6 show typical time-temperature data obtained from pilot plant hydrocracking studies, taken a t constant conversion (Stangeland and Kittrell, 1972) but a t a variety of hydrogen partial pressures and space velocities. Included in these profiles are data from eleven runs for hydrocracking of gas oil (runs 1-11) and two runs for catalytic reforming of naphtha (runs A and B). The hydrocracking data were taken with the reactor being controlled to produce constant conversion of gas oil. The reforming data, on the other hand, were obtained with a
0
Runn
b
Run E
-
roo0
T i m e , (hrs.)
0
I
200
I
I
I
400
I
600
I
b d t l Prelirtlon
I
800
I
I 1000
T i m e , ( hrs.)
Figure 6. Temperature-time trajectories for deactivating reforming catalyst.
constant octane mode of operation, and hence the constant conversion model is not directly applicable. However, the reforming process is often modeled as a system of several parallel reaction schemes, and the overall effective rate constant may possibly be viewable as being maintained approximately constant. Furthermore, since the shapes of the temperature-time profiles for reforming data are similar to those for hydrocracking, one might expect that the model could describe both sets of data. Conventionally, the temperature-time curves for hydrocracking are divided into three regimes: fouling due to nitrogen titration of acidic sites of the catalyst, a linear regime due to slow coking, and a regime of accelerated coking characterized by an exponential increase in temperature with time. However, the data sets investigated
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 3, 1979
401
Table I. Deactivation Parameters for Hydrocracking and Reformine run no.
Ad,
Ed,
3
h- '
cal/g-mol
10
1049.0 1045.0 1200.0 1053.0 1211.0 1192.0 1225.0 1109.0 1054.0 1052.0
Hydrocracking 40 592.3 6 715.4 42 145.1 37 581.1 49 833.3 64 988.3 60 015.1 9 681.0 1 2 948.6 1 7 187.6
A B
1400.0 1400.0
Reforming 51 125.8 51 125.8
1 2 3 4 5 6 7 8 9
a
To "R
E , = 30 000 cal/g-mol; A* = 0.52 x
2.24 E 0.43 E 1.54 E 2.08 E 9.00 E 1.35 E 5.84 E 1.36 E 2.63 E 5.07 E
12 00 11 11 12 18 15 00 02 02
1.00 E 11 1.49 E 11 1012
h-1; n = 1 0.01
in this study correspond only to the linear and exponential coking regimes, as typified by Figure 1. The hydrocracking data cover a wide range of temperatures, from 580 to 880 O F , while the reforming data are from 940 to 1000 O F . Since it is apparent that the form of eq 7 does not allow an independent estimation of the order and activation energy of deactivation, a simple first-order deactivation scheme was assumed to obtain estimates of the deactivation parameters from each temperature-time run. The estimation of the constants in the nonlinear model of eq 7 is relatively straightforward and can be achieved by using nonlinear regression techniques or by plotting sums-of-squares contours of the parameters. However, the parameters reported in Table I were obtained using an alternate technique which is less cumbersome than these methods. Equation 7 is of the form
t = C(1 - eAY)
0
0.04
0.08
0.12
Figure 7. Linearized deactivation plot by constant interval method. 1
0.
-
,\"
I c
(9)
0
Figure 8. Linearized deactivation plot by constant interval method.
If plots of t vs. 1 / T are made and the reciprocal temperature axis is subdivided into four equal segments, namely
it can be shown that
where tl, t 2 , t3, and t4 are times corresponding to the reciprocal temperatures l / T l , 1/T2,l/T3, and l / T 4 , respectively. Once an estimate of the constant C has been obtained, eq 9 can be rearranged to the form In (1 - t / C ) = AY (12)
Hence, a plot of the left-hand side of this equation vs. Y will be linear, if the model is a valid representation of the data, and the slope of this line will be a measure of the second constant. Using eq 11,values of the parameter C were determined for the data of Figures 1-6. The expected linear behavior of the In (1- t / C ) vs. Y plots was indeed demonstrated by all the experimental runs investigated, as shown in Figures 7-9. The deactivation parameters estimated in this manner are presented in Table I, assuming n = 1. Figures 10 and 11 are sums-of-squares contour plots for runs 1 and 2, respectively, shown in Figure 1. The estimates of the constants, obtained by using the technique outlined above, fall within the 90% confidence region for run no. 1as shown in Figure 10, demonstrating the validity of the technique. However, since the temperature-time curve for run no. 2 is approximately linear, the sumsof-squares plots of the parameters for this case (Figure 11) show extended contours, suggesting that the magnitudes of the parameters are imprecisely estimated. In fact, if the temperature-time plots are exactly linear, as for run no. 11 in Figure 5, the parameter estimation technique out-
Ind. Eng. Chem. Process Des. Dev., Vol.
402
18, No. 3, 1979
l.C
irreversible reactions, the policy of operating a t the maximum allowable temperature is optimal when Ed is less than E A (Szepe and Levenspiel, 1968; Chou et al., 1967). However, since both hydrocracking and reforming involve a complex network of reactions, optimal policies different from those for single irreversible reactions are possible. More importantly, since the order of deactivation is unknown, the reported Ed values were calculated assuming first-order kinetics for deactivation. Furthermore, the effects of process variables such as hydrogen partial pressure and coke precursor concentration in the feed, as well as the presence of diffusional limitations, can have a significant impact on the magnitudes of Ed and Ad.
0.5
-
I
r
Effect of Process Variables on Fouling R a t e In commercial operation of deactivating systems, it is sometimes found that the temperature-time profile is approximately linear throughout a large fraction of the run. For this case, eq 7 or 8 can be approximated by the linear form
0.1
0
0.02
0.06
0.04
T = To + (FR)t
Figure 9. Linearized deactivation plot by constant interval method.
eo
where the fouling rate FR is given by
0 Eaflmals -909/oConlldencs Contour
sol \
\\
\ \
401
-
\\
Run No. 1
\
I
30
I
550
500
450
I
600
C
Figure 10. Contours of sums of squares of deviations, eq 12, for run 1. 7.5
0
"
-
Eatlmale 90 yo Conlldcncs
6.5
Note that this linear form is independent of the order of deactivation. Also, since the zero-time intercept contains no information regarding the deactivation parameters, it becomes apparent that good estimates of both Ed and Ad cannot be obtained from this linearized form of the temperature-time profile. However, it is possible to get an estimate of the magnitude of the deactivation rate constant at the start of the run by measuring the slope of the line, and hence the fouling rate. The fouling rate FR, given by eq 14, is seen to be a strong function of To. However, since conversion and space velocity are related to Tothrough the primary reaction kinetics, the fouling rate has a secondary dependence on these two process variables. For any order of the primary reaction, for the constant conversion mode of operation, we have
h
( k o / S V ) = fx(x,CAo)= constant
0
4
5.5
c
(13)
(15)
Hence
\ \
-1= l n ( f,SV x )
-RIEA
TO 4.51
3000
'
I
I
3200
3400
I
3600
3800
4000
C
Figure 11. Contours of sums of squares of deviations, eq 12, for run
Since To2 is a much weaker function than the exponential temperature function in eq 14, we can, using eq 16, rewrite the expression for the fouling rate as
n
L.
lined earlier will break down, since the denominator of eq 11 will become identically equal to zero. However, for all the data sets that exhibit curvature in the temperaturetime plots, the deactivation parameters can be accurately estimated by this technique. The solid lines in Figures 1-6 are the predictions of the model and it is apparent that the fit of the model is adequate to represent both hydrocracking and reforming data. It is noteworthy that the same model describes the linear as well as the exponential regimes of the temperature-time curves, thereby making it unnecessary to postulate two different mechanisms for the two regimes. The values of Ed listed in Table I vary over a wide range, with four runs having Ed values less than EA. For single,
FR = K(fXSV)Ed/EA
(17)
where
K=
AdRTo2 EAAAEdIE~
Taking logarithms on both sides, we have In FR = In K
Ed +In (f,SV) EA
(18)
Hence, eq 18 predicts that a plot of In FR vs. In (f,SV)will yield a straight line with the slope being a measure of the ratio of the activation energies for deactivation and reaction.
10"
.
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 3, 1979
Table 11. Apparent Orders of Deactivation for Hydrocracking and Reforming
[ P I
I
.. I
I
order of
run no.
..
10.~ 1
403
1 2 3 4 5 6 7 8 9 10
X:SO%
o
1.0
deactivation,b
cal/g-mola
n
Hydrocracking
4
llo7
Ed,
4
X 170
rn
6x:eo% 10.0
O/O
4 0 592.3 6 715.4 4 2 145.1 37 581.1 4 9 833.3 64 9 8 8 . 2 6 0 015.1 9 681.1 1 2 948.6 17 1 8 7 . 6
1.31 2.44 1.26 1.41 1.00 0.50 0.67 2.34 2.24 2.10
Reforming
L
A and B 0.1
a
n = 1.
Ed =
51 125.8
1.04
50 000 cal/g-mol.
Work Function (SV.X),h;'
Figure 12. Dependence of fouling rate on catalyst work function.
A simpler correlation between the fouling rate and the process variables x and SV can be obtained by rewriting eq 18 in the form In FR = In K
Ed fx Ed +In - + - In ( x S V ) EA
EA
(19)
where (xSV) is referred to as the "work function". For a zero-order main reaction, f,= (CAOx)and a plot of In FR vs. In (xSV)will be exactly linear. In many runs in support of commercialization, the effect of space velocity on fouling rate is measured by simultaneously adjusting Tosuch that conversion is constant between runs. For first-order kinetics f,= -In (1- x) and, at constant conversion, the plots would still be linear, according to eq 19. This is demonstrated by the plots of Figure 12, which were generated from eq 14 using the parameter values of run no. 7 , listed in Table I. I t is also apparent from Figure 12 that the intercept of the lines (which depends upon the square of To)is indeed not very sensitive to changes in the conversion level of operation, thereby yielding an excellent linear correlation between fouling rate and the work function. It is expected that this sensitivity will increase slightly with reaction order. Since a valid estimate of Ed can be obtained from this plot, it is then possible to use eq 7 to predict the apparent order of deactivation. Assuming a value of 50000 cal/g-mol for Ed, the values of n that correspond to the apparent deactivation activation energies listed in Table I were calculated using the relationship These values of n, listed in Table 11, were found to range from 0.5 to 2.5. Of course, effects of other variables, such as hydrogen partial pressure, feed coke precursor concentrations, and diffusional limitations, have not been accounted for and the calculated values of n reflect only apparent orders of deactivation. Conclusions A mathematical model based on nth order, concentration-dependent deactivation has been developed which predicts temperature-time relationships required for constant conversion of reactant. The model has been tested and verified by using pilot plant hydrocracking and reforming data. It has been shown that linear as well as
exponential regimes of the temperature-time curve are well described by the model. A linear correlation is shown to exist between fouling rate and the work function of the catalyst. Acknowledgment The authors are indebted to Chevron Research Company for permission to publish the data presented in the paper. They also wish to express their appreciation to Dr. R. L. Laurence of the University of Massachusetts for his helpful suggestions and to Mr. Bruce Savatsky for help with the data analyses.
Nomenclature a = fractional catalyst activity A = constant defined by eq 10 A A = preexponential factor for primary reaction, h-' Ad = preexponential factor for deactivation, h-l C = constant defined by eq 10 E A = activation energy for primary reaction, cal/g-mol Ed = activation energy for primary reaction, cal/g-mol F R = fouling rate, "F/h K = reaction rate constant, h-' K O = initial reaction rate constant, h-' k d = deactivation rate constant, h-' K = constant defined by eq 17 n = order of deactivation R = gas constant S V = space velocity, h-' t = process time, h T = catalyst temperature, OR TO= initial catalyst temperature, O R x = fractional conversion Y = ( 1 / T ) - (l/To), (OR)-' Literature Cited Butt, J. E., Rohan, D. M.,Chem. Eng. Sci., 23, 489 (1968). Chou, A., Ray, W.H., Aris, R., Trans. Inst. Chem. Eng.. 45, T 153 (1967). Crowe, C. M., Cbem. Eng. Sci., 31, 959 (1976). Earp. R. G.,Kershenbaum, L. S., Chem. Eng. Sci., 30, 35 (1975). Lee, S.I., Crowe. C. M.,Cbem. Eng. Sci., 25, 743 (1970). Ogunye, A. F., Ray, W.H., Trans. Inst. Chem. Eng., 46, T 225 (1968). Ogunye, A. F., Ray, W.H., AICbE J . , 17, 43, 365 (1971). Pommersheim, J. M., Chandra, K., AICbE J . , 21, 1029 (1975). Stangeland, B. E., Kittrell, J. R., Ind. Eng. Cbem. Process Des. Dev., 11, 15 (1972). Szepe, S., Levenspiel, O., Chem. Eng. Sci., 23, 881 (1968). Szepe, S.,Levenspiel. O., Cbem. React. Eng., Proc. Eur. Symp., 4tb, 1968 ( 1970). Therien, N.,Crowe, C. M., Can. J . Cbem. Eng., 52, 822 (1974).
Received for revieu January 30, 1978 Accepted April 2, 1979
