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Simulation of isothermal catalytic fixed-bed reactors operated in successive reaction-regeneration cycles
The Chemical Engineering
Journal, 31 (1985) 137 - 144
137
Simulation of Isothermal Catalytic Fixed-bed Reactors Operated in Successive Reaction-Regeneration Cycles .J. M. ARANDES, Departamento
M. J. AZKOITI
and J. BILBAO
de Quimica Te’cnica, Universidad
(Received April 15,1984;
de1 Pais Vasco, Apdo.
ABSTRACT The reaction time and catalyst initialactivity that maximize the apparent production rate of isothermal fixed-bed reactors operated in reaction-regeneration cycles have been computed. The calculated results have been compared with those from experiments on reaction and regeneration carried out for a system where kinetic equations for the main reaction, deactivation and regeneration, and the activity-coke relationshi# are known. The proposed calculation procedure has been employed for several reaction systems with separable kinetics. The effects of the reaction orders in the main and deactivation reactions and of the kinetic constants of both reactions on the optimum operating conditions have been analysed. The effects of space time and dead time between two cycles have also been studied.
as a process variable and only the kinetic equations of the main reaction arid deactivation were employed in the calculations. Weekman [5] studied the relation between the optimum conditions of the cycle and the process variables: namely the space time and the kinetic constants of the main and deactivation reactions. If the reactor works on the basis of cycles of reaction and regeneration, in order to optimize the system we shall need to determine not only the reaction time but also the regeneration time or the level of catalyst activity which must be recovered. To relate the reaction and regeneration times it is necessary to elucidate the following kinetic expressions kinetic equation at zero time: krA.A)O=
Reactors operated with decaying catalyst will have an operating time after which it will be necessary either to replace the deactivated catalyst with fresh or regenerated catalyst, or to regenerate in the same reactor which will then be operated in successive reactionregeneration cycles. In the literature, there are reports describing studies of the replacement of the deactivated catalyst by determining the operating time that maximizes production or minimizes operating costs [ 1 - 31. Miertschin and Jackson [4] analysed the effect of the economic parameters of the process on the initial and final activity of the optimum cycle. In those studies, the operating time was assumed to be much larger than the regeneration time, so that the regeneration time was not considered
f(XA9
T)
(1)
kinetic equation for deactivation: -da/dt
1. INTRODUCTION
0300-9467/85/$3.30
644, 48080 Bilbao (Spain)
in final form February 10,1985)
= f(a, T, pM)
(2)
relationship between activity and coke content during regeneration: C, = f(a)
(3)
kinetic equation for regeneration : t reg = f(Cc, T, PO,)
(4)
Le Goff [6] has analysed cyclic reactionregeneration systems by examining the operating conditions in order to maximize several objective functions, such as reaction time, costs and production. The kinetic equations for the main reaction and for deactivation assumed by Le Goff, with constant concentration of reactants and zero order respectively, are capable of simple analytical resolution. In this paper, the design equations for reactors with decaying catalyst have been @ Elsevier Sequoia/Printed in The Netherlands
138
determined for a complex case in which the catalyst is regenerated in the same reactor and the regeneration time depends on the reaction time. For this purpose eqns. (1) - (4) are used. Two cases are examined: (A) An experimental system: comparison of the experimental yield from each cycle and the calculated values allows us to evaluate the quality of the calculation method used to simulate these cycles. (B) Reaction systems with separable kinetic equations: the simplicity of the equations involved permits analytical solutions to be obtained and the effects of different parameters on the optimum operating conditions can also be studied.
is in series with the main reaction), so that at any given time the mean activity of the bed may be calculated from WIFA .f
0
ij=
aa x d(W/h,)
When deactivation is independent of the partial pressures of reactants and products, the activity will be uniform. It will depend only on the reaction time and may be computed directly from the deactivation equation. (2) The mean value of conversion at the reactor outlet at time t will be t
x, 2. CALCULATION PRODUCTION
(6)
( wIFA,)
dt (7)
OF THE APPARENT
RATE
(3) The apparent In the design and optimization of reactionregeneration systems, maximizing production or minimizing costs will be the decisive criteria. In this work, which concentrates on general design relationships, we have chosen production maximization for detailed study, since it is outside the scope of this paper to study the costs, sales possibilities etc. of a particular reaction. On the other hand, the fundamental design problem is to calculate the relationship between the mean value of the conversion and reaction time, and, when the regeneration time is dependent on the operation time, to determine the relationship between the two. The steps followed in calculating production are as follows. (1) Once the space time and reaction temperature (kinetic constants) are fixed, the variation of activity with time is calculated by integrating eqn. (2) and subsequently substituting in the mass balance equation for plug flow
a.4 = (-f-A)\)oa d( WI&,,)
(5)
The variation of conversion at the reactor outlet with time is obtained by integrating eqn. (5). Deactivation can depend on reactant and/or product concentrations or it can be independent of them. In the first case the activity of the catalyst varies along the reactor (the activity profile will be rising for parallel deactivation and decreasing when deactivation
production
rate is defined
as FM =
(t + to) w/FA,,
(8)
where the time t, between the end of a cycle and the beginning of the next one, the regeneration time treg and the dead time td are related by t, = &eg +
td
(9)
When the regeneration time is negligible compared to the dead time, the time t, necessary to prepare, clean and fill the reactor will be constant, known and independent of the operating time. Introducing t, into eqn. (ES), the apparent production rate may be computed. On the other hand, when the regeneration time and dead time are of similar magnitude, the former will be a function of the reaction time and of the initial activity at the beginning of the cycle. This initial activity will then have to be recovered during the regeneration stage. (4) From eqn. (3) one obtains two expressions that relate the coke content Ccf at the end of the reaction stage with the activity a = f(a,, t), and the coke content C,i at the beginning of the reaction stage with the initial activity aO. By integrating eqn. (4) between C,i and Ccf and substituting into it the expressions deduced above, a relationship between regeneration time, reaction time and final
139
activity is obtained. On introducing this relationship into eqns. (9) and (8) we obtain the apparent production rate.
3. EXPERIMENTAL
SYSTEM
To test the calculation scheme, we simulated a reaction-regeneration system with which we have worked experimentally. For this study we carried out the dimerization of acetaldehyde to crotonaldehyde in an isothermal fixed-bed reactor of 17 mm inside diameter on a silicaalumina catalyst. The details of this reaction have been reported earlier [ 71. The kinetic equations of the system are as follows. The kinetic equation at zero time (200 “C) is 1.108 X 10-4(Pi -P,p,/9.403 bA)O
=
x lo-*)
Ps(l+ 0.8OlPA + 2.782 X 10-3Pi/Ps)
(10) The kinetic equation for deactivation (200 “C) has the form a= -
(-A)
= 0.77exp(-6.08
X 10e2t) +
(-A)0 +
0.22exp(-9.78
X 10p4t)
(11)
The term 0.22exp(-9.78 X 10m4t) is, in practice, constant at 0 < t < 120 min, so that we can write a = 0.77exp(-6.08
X 10m2t) + 0.23
(12)
or -Z
da
= 6.08 X 10-2(a - 0.23)
(13)
The relationship between a and C, is given by C, = 8.42 - 8.03exp[-3.5(1 - 0.39exp[-2.5(1
-a)]
-a)]
-
(14) In eqn. (14) term C, corresponds to the carbonaceous material that is deposited on the catalyst after it is heated at the regeneration temperature under N2 flow to eliminate the lighter components of coke, the fast combustion of which would otherwise cause uncontrollable regeneration [ 81. Finally, we turn to the kinetic equation of regeneration (570 “C). It has been proved [9] that coke burning at this temperature and particle size is influenced by diffusional
effects in the catalyst. The following expression which relates coke conversion and regeneration time has been obtained by solving the conservation equations for gaseous and solid components by orthogonal collocation with a single point 1-x In - 0.4292exp(-0.4634&,,) 0.699 + 0.463&,,,
1 +
1
+ 0.4292exp(-0.4634&s)]
(15)
The operating conditions for the reactionregeneration cycles were: reaction temperature, 200 “C; regeneration temperature, 570 “C; space time, 1 g cat. h mol-‘. The values of initial activity and reaction time are shown in Table 1. To compute the apparent production rate we have assumed a dead time of 6 min. The computed values of regeneration time necessary to return to the initial activities are shown in Table 1. The average conversion was measured by gas chromatography. Table 1 shows the experimental values for the apparent production rate. As an explanatory example of reaction evolution, conversion has been plotted against time for three successive cycles and an initial activity of 0.9 (Fig. 1). Continuous lines were calculated by solving eqn. (5) and points were obtained experimentally by means of gas chromatographic analysis of the reaction products. In Fig. 2, lines of constant apparent production rate are shown in the a,-t plane. It can be seen that the experimental values in Table 1 are close to the calculated apparent production rate. In this figure the existence of a pair of values of ao, t that maximize the apparent production rate FM can be observed. In previous work [lo] we studied the use of different methods to search for the extremum of multivariable functions. In this case a modified “simplex” approach [ 111 produces good results. The optimum at the conditions of Fig. 2 corresponds to a, = 0.80, t = 26 min and FMmax = 0.0109 mol gg’ h-l. 4. SEPARABLE
KINETIC EQUATIONS
The reaction-regeneration system under study has complex kinetic equations and its
140 TABLE 1 Experimental values of T;, ( low3 mol(h g cat.)-’ ae and t (min)
0.9
(10 = t t
reg
fM cl0
) measured in reaction-regeneration
cycles for several values of
85 9.7 9.35
150 9.8 8.58
5.3 9.26
120 5.3 8.83
140 5.3 8.70
50 3.0 10.00
70 3.1 9.40
100 3.1 8.90
120 3.1 8.60
40 1.8 9.70
53 1.8 9.30
70 1.8 9.00
60 1.1 8.70
80 1.1 8.40
100 1.1 8.20
10 7.9 8.60
14 8.4 9.37
20 9.1 10.20
30 9.5 10.50
50 9.7 10.10
60 9.7 9.90
10 4.0 9.50
14 4.3 10.10
20 4.7 10.80
36 5.1 10.60
60 5.2 9.96
90
10 2.1 9.50
14 2.4 10.10
20 2.7 10.60
30 2.9 10.60
10 1.2 9.00
14 1.4 9.60
20 1.6 10.00
10 0.6 8.10
20 0.9 9.10
30 1.0 9.20
0.8
=
t t _rw? rM
(10 = 0.7
t t reg FM
0.6
a0 = t treg r-,
1.9 8.70
120 1.9 8.30
120 1.1 8.10
140 1.1 7.90
90
0.5
arJ =
t treg fM
x,,t 3
2
I
0
30
0
60
t
90
0
(mln)
30
60
90
0
30
60
90
f(rnl”)
t ImInI
Fig. 1. Conversion VS.time for three successive cycles of reaction-regeneration.
02-
0.. 0
I,
20
I.
10
1,
60
I,‘.
80
100
1.1,.
120
1LO
time
lmin)
Fig. 2. Lines of constant apparent production rate in the ao-t plane.
solution has to be numerical. We have extended the simulation of reaction-regeneration cycles to reactions with separable kinetic equations in order firstly to solve analytically and so to obtain equations that give the apparent production rate, and secondly to analyse the effect of the operating conditions on the maximum apparent production rate and the optimum values of a,, and t. The kinetic equations proposed are as follows kinetic equation (-rA)o
= hPz
at zero time: rz = 0,l
(16)
141 TABLE 2 Expressions for xAt corresponding to various values of n and d in the kinetic equations of the main and deactivation reactions: (-r_& = kPi, (-da/dt) = kdad, C = k x W/F*,, D = kdt
n=O,d=O
Caoexp(-D) n=O,d=l kd
1 t
0
n=l,d=O 0
d
n=l,d=l
x*t =
-Caoexp(-D)
+
(-Cao)2exp(-2D)
kinetic equation for deactivation: -iii
= kdUd
d=O,l
(17)
relationship between a and C,: C, = b - b exp[-m(1
-a)]
(13)
kinetic equation for regeneration: (-rc’,) = W&o1
(-Cao)3exp(-3D)
+ . ..
3!3kd
2!2kd
kci
du
+
t 1 0
assumed that activity is uniform along the bed. Also, the time constant of deactivation is very much larger than fluid contact time. Now, we shall analyse the effect of the reaction parameters on the maximum apparent production rate and the optimum variables a0 and t. Figure 3 shows how FMdecreases with space time, except at n = 0, because FM
(19)
The general expression for a and C,, eqn. (MS), has been verified in several regeneration studies for silica-alumina catalysts carried out by the authors. Here a value of 3 has been assumed for m. Table 2 shows expressions &t obtained for power-law kinetics and various orders of reaction and deactivation. The parameters that affect the apparent production rate are the space time parameter W/F, ,the kinetic constant k for the main react:on; the kinetic constant of deactivation kd ; the kinetic constant of regeneration k,,,; the partial pressure of O2 in the regeneration Paz; and the dead time td. In simulating several reaction systems we have chosen the following values for each of these parameters: W/FAo, 1 - 9 g cat. h mol-‘; k, 0.05 mol (g cat. h atm”)-l; kd, 0.001 - 0.05 min-‘; kreg, 0.567 min-’ atm-‘; PO,, 0.21 atm; and td, 2 - 10 min. A value of 0.567 min-’ for kreghas been obtained experimentally in the regeneration of a silica-alumina catalyst [9]. By setting po, equal to a constant 0.21 atm along the reactor, we are assuming regeneration under differential conditions. In eqn. (17) it was
2”’
n:l
3.2
(a)
1._1
thin)
120 @I
0
*
1. 2
1. L
' 6
*
1. 6
' 10
'&//FA4(gcat_h/mol)
Fig. 3. (a) Maximum apparent production rate vs. space time; (b) optimum reaction time us. space time.
142
is defined per unit of catalyst weight in eqn. (8). Data shown in Fig. 3 correspond to kd = 0.001 min-’ and td = 6 min. It can be observed that as n increases the drop in the maximum apparent production rate becomes higher. Taking into account the fact that n is the exponent of PA = PAo(l - X) in the kinetic equation, (eqn. (16)) and that PA, is 1, the term (-r,& becomes smaller as the reaction order increases. However, the effect of the deactivation order on the maximum apparent production rate is smaller than the effect of n and the curves of ?, versus W/FAo are in practice parallel for the different values of d. The optimum initial activity of each cycle, except at n = 0, becomes lower as the space time increases, but this fall is negligible and then a0 = 0.963 for y1= 0 and d = 0, a, = 0.962 for it = 0, d = 1, a0 = 0.0960 for n = 1, d = 0 and a0 = 0.960 for n = 1, d = 1. In Fig. 3 the values of optimum operating time have been plotted against space time. It can be seen that, for y1= 1, the optimum reaction time increases as space time increases. The comments above refer to the effect of space time. But, as is noted in Table 2, W/FAo acts as a multiplier on k, so that the kinetic constant of the main reaction will have the same effect as that described above for W/FA,. Since the parameter is actually k X W/FA,, any changes in k and W/F*, that still keep k X W/FAo constant will have no effect on the optimum conditions a, and t, although they will move the maximum apparent production rate, because it is defined in terms of the weight of catalyst. It should also be noted that the kinetic constant k includes the term E,. When PA, # 1, a change in the reaction order implies a change in k, with the effects described earlier. To study the effect of the deactivation constant, the results for W/F*, = 1 g cat. h malli and td = 6 min have been analysed. Figure 4 reveals how the apparent production rate varies with the deactivation constant kd. As kd increases, the apparent production rate decreases and this fall becomes smaller as the values of kd become higher. The increase in deactivation order and the drop in order of the main reaction cause an increase in the apparent production rate, the second cause being the more significant. The initial activity decreases as kd increases, and this increment is higher as tz and/or d
’L
(a)
I
001
,
I
002
I
1
003
,
,
OOL
,
I
005
t kq160 I
120’
(b)
‘0 ’ 001 ’ ’ 002 ’ ’ ’ ’ ’ 003
001
’
005
Fig. 4. Effect of deactivation rate constant on optimum -,n=O;---,n=l.(a) operating conditions: Maximum apparent production rate; (b) optimum reaction time; (c) initial catalyst activity.
become higher. It can be observed how the optimum operation time becomes higher and the effect of d becomes larger than the effect of n when n and/or d are increased. To visualize the effect of dead time we have studied results obtained at k4 = 0.001 min-’ and W/F*, = 1 g cat. h mol-‘. As the dead time rises, the apparent production rate falls, causing optimum initial activity and reaction time to occur at higher values as is shown in Fig. 5. The optimum reaction time and optimum initial activity increase with dead time, the slope being progressively lower.
143
reactant concentration in the feed and others. As we have specified a particular temperature and also that deactivation is independent of concentration, such a study is outside the scope of this paper. The determination of a relationship between activity and coke content during the regeneration is of particular interest. This topic has hardly been studied experimentally and it may well be that the relationship is not the same as that observed for the reaction stage, because the coke content may develop in a different way with time during reaction and regeneration. Here we have only approached the calculation and posterior optimization of the apparent production rate. However, in the planning of an industrial process, economic criteria will generally prevail. Knowing the economic data for a particular process it is possible to optimize other objective functions. With profit maximization as the objective function, it is necessary to consider the economic influence of production (sales), costs of reactants and catalyst regeneration. In order to optimize the operation of an existing reactor, this objective function can be formulated as
t (mm)
1eo-
160
B = a.4%?AFA,0 - flFA,O - rW&,,
0.96
0.95
0
2
6
8
10
tdcmln) (c)
Fig. 5. Effect of dead time between two cycles on optimum operating conditions. (a) Maximum apparent production rate; (b) optimum reaction time ; (c) initial catalyst activity.
5. DISCUSSION
In this paper we have proposed a procedure for simulating fixed-bed reactors that work with reaction-regeneration cycles. This procedure may be used with systems exhibiting separable or non-separable kinetics. In a more extensive study, variables other than initial activity and reaction time that have an influence on the kinetic equation at zero time and on the deactivation equation could be considered, such as temperature,
(20)
where CY= sale price ($(product mol))‘), /3 = cost of reactants ($(reactant mol))‘), y = cost of regeneration ($(catalyst weight X time))‘) and 8 and ereg are respectively the fractions of reaction and regeneration time in one cycle. In computing the expression defined by eqn. (20) or an analogous expression including investment and fixed costs, the fundamental problem is to compute the relationship between reaction and regeneration times which is the central topic of this paper.
REFERENCES R. B. Smith and J. Dresser, Petrol. Refiner, 36 (1957) 199. P. R. Walton, Chem. Eng. Prog., 57 (1961) 42. H. Kramers and K. R. Westerterp, Elements of Chemical Reactor Design and Operation, Academic Press, New York, 1963. G. N. Miertschin and R. Jackson, Can. J. Chem. Eng., 48 (1970) 702. V. W. Weekman, Ind. Eng. Chem. Process Des. Deu., 7 (1968) 252. P. Le Goff, Znt. Chem. Engng., 23 (1983) 225.
144 7 J. M. Arandes, Ph.D. Thesis, Universidad de1 Pais Vasco, Bilbao, 1982. 8 E. Furimsky, Ind. Eng. Chem. Prod. Res. Dev., 18 (1979) 206. 9 J. Bilbao, A. Romero and J. M. Arandes, Chem. Eng. Ski., 38 (1983) 1356. 10 J. M. Arandes, J. Bilbao and A. Romero, Ing. Quim., 16 (5) (1984) 153. 11 R. Fletcher and M. J. D. Powell, Comput. J., 6 (1963) 163.
APPENDIX A: NOMENCLATURE
A, R, S reactant and products of the main
reaction, respectively initial catalyst activity and average activity of the bed after the reaction stage coke content at zero activity percentage of coke in the catalyst coke content at the beginning and end of the reaction stage respectively order of reaction with respect to activity in the deactivation equation feed rate (mol A h-l) rate constants in the main reaction, deactivation and regeneration respectively parameter in eqn. (18)
a,ao,c catalyst activity,
d
%I
k, h, kr
m
order of reaction with respect to A in the main reaction partial pressure of coke precursor (atm) pM partial pressure of A at inlet and in pA,, pA the reactor (atm) partial pressures of R and S (atm) pry pS P02 oxygen partial pressure (atm) apparent production rate (mol g-’ FM h-l) reaction rate with respect to A and (--A), reaction rate with respect to A at (-A)0 zero time respectively (mol g-’ h-‘) t--c‘,) coke burning rate (g coke g cat.-’ min-‘) temperatures of reactor and regenerT, T, ator respectively (K) t, treg reaction and regeneration times respectively (min) time between the end of a cycle and b, td the beginning of the next one and dead time respectively (min) W catalyst weight (g) conversion of coke in the regeneraX tion, 1 - C, /C,, xA,,~ XA conversion with respect to A and conversion with respect to A at zero time respectively time averaged conversion at the xA reactor exit n
Journal, 31 (1985) 137 - 144
137
Simulation of Isothermal Catalytic Fixed-bed Reactors Operated in Successive Reaction-Regeneration Cycles .J. M. ARANDES, Departamento
M. J. AZKOITI
and J. BILBAO
de Quimica Te’cnica, Universidad
(Received April 15,1984;
de1 Pais Vasco, Apdo.
ABSTRACT The reaction time and catalyst initialactivity that maximize the apparent production rate of isothermal fixed-bed reactors operated in reaction-regeneration cycles have been computed. The calculated results have been compared with those from experiments on reaction and regeneration carried out for a system where kinetic equations for the main reaction, deactivation and regeneration, and the activity-coke relationshi# are known. The proposed calculation procedure has been employed for several reaction systems with separable kinetics. The effects of the reaction orders in the main and deactivation reactions and of the kinetic constants of both reactions on the optimum operating conditions have been analysed. The effects of space time and dead time between two cycles have also been studied.
as a process variable and only the kinetic equations of the main reaction arid deactivation were employed in the calculations. Weekman [5] studied the relation between the optimum conditions of the cycle and the process variables: namely the space time and the kinetic constants of the main and deactivation reactions. If the reactor works on the basis of cycles of reaction and regeneration, in order to optimize the system we shall need to determine not only the reaction time but also the regeneration time or the level of catalyst activity which must be recovered. To relate the reaction and regeneration times it is necessary to elucidate the following kinetic expressions kinetic equation at zero time: krA.A)O=
Reactors operated with decaying catalyst will have an operating time after which it will be necessary either to replace the deactivated catalyst with fresh or regenerated catalyst, or to regenerate in the same reactor which will then be operated in successive reactionregeneration cycles. In the literature, there are reports describing studies of the replacement of the deactivated catalyst by determining the operating time that maximizes production or minimizes operating costs [ 1 - 31. Miertschin and Jackson [4] analysed the effect of the economic parameters of the process on the initial and final activity of the optimum cycle. In those studies, the operating time was assumed to be much larger than the regeneration time, so that the regeneration time was not considered
f(XA9
T)
(1)
kinetic equation for deactivation: -da/dt
1. INTRODUCTION
0300-9467/85/$3.30
644, 48080 Bilbao (Spain)
in final form February 10,1985)
= f(a, T, pM)
(2)
relationship between activity and coke content during regeneration: C, = f(a)
(3)
kinetic equation for regeneration : t reg = f(Cc, T, PO,)
(4)
Le Goff [6] has analysed cyclic reactionregeneration systems by examining the operating conditions in order to maximize several objective functions, such as reaction time, costs and production. The kinetic equations for the main reaction and for deactivation assumed by Le Goff, with constant concentration of reactants and zero order respectively, are capable of simple analytical resolution. In this paper, the design equations for reactors with decaying catalyst have been @ Elsevier Sequoia/Printed in The Netherlands
138
determined for a complex case in which the catalyst is regenerated in the same reactor and the regeneration time depends on the reaction time. For this purpose eqns. (1) - (4) are used. Two cases are examined: (A) An experimental system: comparison of the experimental yield from each cycle and the calculated values allows us to evaluate the quality of the calculation method used to simulate these cycles. (B) Reaction systems with separable kinetic equations: the simplicity of the equations involved permits analytical solutions to be obtained and the effects of different parameters on the optimum operating conditions can also be studied.
is in series with the main reaction), so that at any given time the mean activity of the bed may be calculated from WIFA .f
0
ij=
aa x d(W/h,)
When deactivation is independent of the partial pressures of reactants and products, the activity will be uniform. It will depend only on the reaction time and may be computed directly from the deactivation equation. (2) The mean value of conversion at the reactor outlet at time t will be t
x, 2. CALCULATION PRODUCTION
(6)
( wIFA,)
dt (7)
OF THE APPARENT
RATE
(3) The apparent In the design and optimization of reactionregeneration systems, maximizing production or minimizing costs will be the decisive criteria. In this work, which concentrates on general design relationships, we have chosen production maximization for detailed study, since it is outside the scope of this paper to study the costs, sales possibilities etc. of a particular reaction. On the other hand, the fundamental design problem is to calculate the relationship between the mean value of the conversion and reaction time, and, when the regeneration time is dependent on the operation time, to determine the relationship between the two. The steps followed in calculating production are as follows. (1) Once the space time and reaction temperature (kinetic constants) are fixed, the variation of activity with time is calculated by integrating eqn. (2) and subsequently substituting in the mass balance equation for plug flow
a.4 = (-f-A)\)oa d( WI&,,)
(5)
The variation of conversion at the reactor outlet with time is obtained by integrating eqn. (5). Deactivation can depend on reactant and/or product concentrations or it can be independent of them. In the first case the activity of the catalyst varies along the reactor (the activity profile will be rising for parallel deactivation and decreasing when deactivation
production
rate is defined
as FM =
(t + to) w/FA,,
(8)
where the time t, between the end of a cycle and the beginning of the next one, the regeneration time treg and the dead time td are related by t, = &eg +
td
(9)
When the regeneration time is negligible compared to the dead time, the time t, necessary to prepare, clean and fill the reactor will be constant, known and independent of the operating time. Introducing t, into eqn. (ES), the apparent production rate may be computed. On the other hand, when the regeneration time and dead time are of similar magnitude, the former will be a function of the reaction time and of the initial activity at the beginning of the cycle. This initial activity will then have to be recovered during the regeneration stage. (4) From eqn. (3) one obtains two expressions that relate the coke content Ccf at the end of the reaction stage with the activity a = f(a,, t), and the coke content C,i at the beginning of the reaction stage with the initial activity aO. By integrating eqn. (4) between C,i and Ccf and substituting into it the expressions deduced above, a relationship between regeneration time, reaction time and final
139
activity is obtained. On introducing this relationship into eqns. (9) and (8) we obtain the apparent production rate.
3. EXPERIMENTAL
SYSTEM
To test the calculation scheme, we simulated a reaction-regeneration system with which we have worked experimentally. For this study we carried out the dimerization of acetaldehyde to crotonaldehyde in an isothermal fixed-bed reactor of 17 mm inside diameter on a silicaalumina catalyst. The details of this reaction have been reported earlier [ 71. The kinetic equations of the system are as follows. The kinetic equation at zero time (200 “C) is 1.108 X 10-4(Pi -P,p,/9.403 bA)O
=
x lo-*)
Ps(l+ 0.8OlPA + 2.782 X 10-3Pi/Ps)
(10) The kinetic equation for deactivation (200 “C) has the form a= -
(-A)
= 0.77exp(-6.08
X 10e2t) +
(-A)0 +
0.22exp(-9.78
X 10p4t)
(11)
The term 0.22exp(-9.78 X 10m4t) is, in practice, constant at 0 < t < 120 min, so that we can write a = 0.77exp(-6.08
X 10m2t) + 0.23
(12)
or -Z
da
= 6.08 X 10-2(a - 0.23)
(13)
The relationship between a and C, is given by C, = 8.42 - 8.03exp[-3.5(1 - 0.39exp[-2.5(1
-a)]
-a)]
-
(14) In eqn. (14) term C, corresponds to the carbonaceous material that is deposited on the catalyst after it is heated at the regeneration temperature under N2 flow to eliminate the lighter components of coke, the fast combustion of which would otherwise cause uncontrollable regeneration [ 81. Finally, we turn to the kinetic equation of regeneration (570 “C). It has been proved [9] that coke burning at this temperature and particle size is influenced by diffusional
effects in the catalyst. The following expression which relates coke conversion and regeneration time has been obtained by solving the conservation equations for gaseous and solid components by orthogonal collocation with a single point 1-x In - 0.4292exp(-0.4634&,,) 0.699 + 0.463&,,,
1 +
1
+ 0.4292exp(-0.4634&s)]
(15)
The operating conditions for the reactionregeneration cycles were: reaction temperature, 200 “C; regeneration temperature, 570 “C; space time, 1 g cat. h mol-‘. The values of initial activity and reaction time are shown in Table 1. To compute the apparent production rate we have assumed a dead time of 6 min. The computed values of regeneration time necessary to return to the initial activities are shown in Table 1. The average conversion was measured by gas chromatography. Table 1 shows the experimental values for the apparent production rate. As an explanatory example of reaction evolution, conversion has been plotted against time for three successive cycles and an initial activity of 0.9 (Fig. 1). Continuous lines were calculated by solving eqn. (5) and points were obtained experimentally by means of gas chromatographic analysis of the reaction products. In Fig. 2, lines of constant apparent production rate are shown in the a,-t plane. It can be seen that the experimental values in Table 1 are close to the calculated apparent production rate. In this figure the existence of a pair of values of ao, t that maximize the apparent production rate FM can be observed. In previous work [lo] we studied the use of different methods to search for the extremum of multivariable functions. In this case a modified “simplex” approach [ 111 produces good results. The optimum at the conditions of Fig. 2 corresponds to a, = 0.80, t = 26 min and FMmax = 0.0109 mol gg’ h-l. 4. SEPARABLE
KINETIC EQUATIONS
The reaction-regeneration system under study has complex kinetic equations and its
140 TABLE 1 Experimental values of T;, ( low3 mol(h g cat.)-’ ae and t (min)
0.9
(10 = t t
reg
fM cl0
) measured in reaction-regeneration
cycles for several values of
85 9.7 9.35
150 9.8 8.58
5.3 9.26
120 5.3 8.83
140 5.3 8.70
50 3.0 10.00
70 3.1 9.40
100 3.1 8.90
120 3.1 8.60
40 1.8 9.70
53 1.8 9.30
70 1.8 9.00
60 1.1 8.70
80 1.1 8.40
100 1.1 8.20
10 7.9 8.60
14 8.4 9.37
20 9.1 10.20
30 9.5 10.50
50 9.7 10.10
60 9.7 9.90
10 4.0 9.50
14 4.3 10.10
20 4.7 10.80
36 5.1 10.60
60 5.2 9.96
90
10 2.1 9.50
14 2.4 10.10
20 2.7 10.60
30 2.9 10.60
10 1.2 9.00
14 1.4 9.60
20 1.6 10.00
10 0.6 8.10
20 0.9 9.10
30 1.0 9.20
0.8
=
t t _rw? rM
(10 = 0.7
t t reg FM
0.6
a0 = t treg r-,
1.9 8.70
120 1.9 8.30
120 1.1 8.10
140 1.1 7.90
90
0.5
arJ =
t treg fM
x,,t 3
2
I
0
30
0
60
t
90
0
(mln)
30
60
90
0
30
60
90
f(rnl”)
t ImInI
Fig. 1. Conversion VS.time for three successive cycles of reaction-regeneration.
02-
0.. 0
I,
20
I.
10
1,
60
I,‘.
80
100
1.1,.
120
1LO
time
lmin)
Fig. 2. Lines of constant apparent production rate in the ao-t plane.
solution has to be numerical. We have extended the simulation of reaction-regeneration cycles to reactions with separable kinetic equations in order firstly to solve analytically and so to obtain equations that give the apparent production rate, and secondly to analyse the effect of the operating conditions on the maximum apparent production rate and the optimum values of a,, and t. The kinetic equations proposed are as follows kinetic equation (-rA)o
= hPz
at zero time: rz = 0,l
(16)
141 TABLE 2 Expressions for xAt corresponding to various values of n and d in the kinetic equations of the main and deactivation reactions: (-r_& = kPi, (-da/dt) = kdad, C = k x W/F*,, D = kdt
n=O,d=O
Caoexp(-D) n=O,d=l kd
1 t
0
n=l,d=O 0
d
n=l,d=l
x*t =
-Caoexp(-D)
+
(-Cao)2exp(-2D)
kinetic equation for deactivation: -iii
= kdUd
d=O,l
(17)
relationship between a and C,: C, = b - b exp[-m(1
-a)]
(13)
kinetic equation for regeneration: (-rc’,) = W&o1
(-Cao)3exp(-3D)
+ . ..
3!3kd
2!2kd
kci
du
+
t 1 0
assumed that activity is uniform along the bed. Also, the time constant of deactivation is very much larger than fluid contact time. Now, we shall analyse the effect of the reaction parameters on the maximum apparent production rate and the optimum variables a0 and t. Figure 3 shows how FMdecreases with space time, except at n = 0, because FM
(19)
The general expression for a and C,, eqn. (MS), has been verified in several regeneration studies for silica-alumina catalysts carried out by the authors. Here a value of 3 has been assumed for m. Table 2 shows expressions &t obtained for power-law kinetics and various orders of reaction and deactivation. The parameters that affect the apparent production rate are the space time parameter W/F, ,the kinetic constant k for the main react:on; the kinetic constant of deactivation kd ; the kinetic constant of regeneration k,,,; the partial pressure of O2 in the regeneration Paz; and the dead time td. In simulating several reaction systems we have chosen the following values for each of these parameters: W/FAo, 1 - 9 g cat. h mol-‘; k, 0.05 mol (g cat. h atm”)-l; kd, 0.001 - 0.05 min-‘; kreg, 0.567 min-’ atm-‘; PO,, 0.21 atm; and td, 2 - 10 min. A value of 0.567 min-’ for kreghas been obtained experimentally in the regeneration of a silica-alumina catalyst [9]. By setting po, equal to a constant 0.21 atm along the reactor, we are assuming regeneration under differential conditions. In eqn. (17) it was
2”’
n:l
3.2
(a)
1._1
thin)
120 @I
0
*
1. 2
1. L
' 6
*
1. 6
' 10
'&//FA4(gcat_h/mol)
Fig. 3. (a) Maximum apparent production rate vs. space time; (b) optimum reaction time us. space time.
142
is defined per unit of catalyst weight in eqn. (8). Data shown in Fig. 3 correspond to kd = 0.001 min-’ and td = 6 min. It can be observed that as n increases the drop in the maximum apparent production rate becomes higher. Taking into account the fact that n is the exponent of PA = PAo(l - X) in the kinetic equation, (eqn. (16)) and that PA, is 1, the term (-r,& becomes smaller as the reaction order increases. However, the effect of the deactivation order on the maximum apparent production rate is smaller than the effect of n and the curves of ?, versus W/FAo are in practice parallel for the different values of d. The optimum initial activity of each cycle, except at n = 0, becomes lower as the space time increases, but this fall is negligible and then a0 = 0.963 for y1= 0 and d = 0, a, = 0.962 for it = 0, d = 1, a0 = 0.0960 for n = 1, d = 0 and a0 = 0.960 for n = 1, d = 1. In Fig. 3 the values of optimum operating time have been plotted against space time. It can be seen that, for y1= 1, the optimum reaction time increases as space time increases. The comments above refer to the effect of space time. But, as is noted in Table 2, W/FAo acts as a multiplier on k, so that the kinetic constant of the main reaction will have the same effect as that described above for W/FA,. Since the parameter is actually k X W/FA,, any changes in k and W/F*, that still keep k X W/FAo constant will have no effect on the optimum conditions a, and t, although they will move the maximum apparent production rate, because it is defined in terms of the weight of catalyst. It should also be noted that the kinetic constant k includes the term E,. When PA, # 1, a change in the reaction order implies a change in k, with the effects described earlier. To study the effect of the deactivation constant, the results for W/F*, = 1 g cat. h malli and td = 6 min have been analysed. Figure 4 reveals how the apparent production rate varies with the deactivation constant kd. As kd increases, the apparent production rate decreases and this fall becomes smaller as the values of kd become higher. The increase in deactivation order and the drop in order of the main reaction cause an increase in the apparent production rate, the second cause being the more significant. The initial activity decreases as kd increases, and this increment is higher as tz and/or d
’L
(a)
I
001
,
I
002
I
1
003
,
,
OOL
,
I
005
t kq160 I
120’
(b)
‘0 ’ 001 ’ ’ 002 ’ ’ ’ ’ ’ 003
001
’
005
Fig. 4. Effect of deactivation rate constant on optimum -,n=O;---,n=l.(a) operating conditions: Maximum apparent production rate; (b) optimum reaction time; (c) initial catalyst activity.
become higher. It can be observed how the optimum operation time becomes higher and the effect of d becomes larger than the effect of n when n and/or d are increased. To visualize the effect of dead time we have studied results obtained at k4 = 0.001 min-’ and W/F*, = 1 g cat. h mol-‘. As the dead time rises, the apparent production rate falls, causing optimum initial activity and reaction time to occur at higher values as is shown in Fig. 5. The optimum reaction time and optimum initial activity increase with dead time, the slope being progressively lower.
143
reactant concentration in the feed and others. As we have specified a particular temperature and also that deactivation is independent of concentration, such a study is outside the scope of this paper. The determination of a relationship between activity and coke content during the regeneration is of particular interest. This topic has hardly been studied experimentally and it may well be that the relationship is not the same as that observed for the reaction stage, because the coke content may develop in a different way with time during reaction and regeneration. Here we have only approached the calculation and posterior optimization of the apparent production rate. However, in the planning of an industrial process, economic criteria will generally prevail. Knowing the economic data for a particular process it is possible to optimize other objective functions. With profit maximization as the objective function, it is necessary to consider the economic influence of production (sales), costs of reactants and catalyst regeneration. In order to optimize the operation of an existing reactor, this objective function can be formulated as
t (mm)
1eo-
160
B = a.4%?AFA,0 - flFA,O - rW&,,
0.96
0.95
0
2
6
8
10
tdcmln) (c)
Fig. 5. Effect of dead time between two cycles on optimum operating conditions. (a) Maximum apparent production rate; (b) optimum reaction time ; (c) initial catalyst activity.
5. DISCUSSION
In this paper we have proposed a procedure for simulating fixed-bed reactors that work with reaction-regeneration cycles. This procedure may be used with systems exhibiting separable or non-separable kinetics. In a more extensive study, variables other than initial activity and reaction time that have an influence on the kinetic equation at zero time and on the deactivation equation could be considered, such as temperature,
(20)
where CY= sale price ($(product mol))‘), /3 = cost of reactants ($(reactant mol))‘), y = cost of regeneration ($(catalyst weight X time))‘) and 8 and ereg are respectively the fractions of reaction and regeneration time in one cycle. In computing the expression defined by eqn. (20) or an analogous expression including investment and fixed costs, the fundamental problem is to compute the relationship between reaction and regeneration times which is the central topic of this paper.
REFERENCES R. B. Smith and J. Dresser, Petrol. Refiner, 36 (1957) 199. P. R. Walton, Chem. Eng. Prog., 57 (1961) 42. H. Kramers and K. R. Westerterp, Elements of Chemical Reactor Design and Operation, Academic Press, New York, 1963. G. N. Miertschin and R. Jackson, Can. J. Chem. Eng., 48 (1970) 702. V. W. Weekman, Ind. Eng. Chem. Process Des. Deu., 7 (1968) 252. P. Le Goff, Znt. Chem. Engng., 23 (1983) 225.
144 7 J. M. Arandes, Ph.D. Thesis, Universidad de1 Pais Vasco, Bilbao, 1982. 8 E. Furimsky, Ind. Eng. Chem. Prod. Res. Dev., 18 (1979) 206. 9 J. Bilbao, A. Romero and J. M. Arandes, Chem. Eng. Ski., 38 (1983) 1356. 10 J. M. Arandes, J. Bilbao and A. Romero, Ing. Quim., 16 (5) (1984) 153. 11 R. Fletcher and M. J. D. Powell, Comput. J., 6 (1963) 163.
APPENDIX A: NOMENCLATURE
A, R, S reactant and products of the main
reaction, respectively initial catalyst activity and average activity of the bed after the reaction stage coke content at zero activity percentage of coke in the catalyst coke content at the beginning and end of the reaction stage respectively order of reaction with respect to activity in the deactivation equation feed rate (mol A h-l) rate constants in the main reaction, deactivation and regeneration respectively parameter in eqn. (18)
a,ao,c catalyst activity,
d
%I
k, h, kr
m
order of reaction with respect to A in the main reaction partial pressure of coke precursor (atm) pM partial pressure of A at inlet and in pA,, pA the reactor (atm) partial pressures of R and S (atm) pry pS P02 oxygen partial pressure (atm) apparent production rate (mol g-’ FM h-l) reaction rate with respect to A and (--A), reaction rate with respect to A at (-A)0 zero time respectively (mol g-’ h-‘) t--c‘,) coke burning rate (g coke g cat.-’ min-‘) temperatures of reactor and regenerT, T, ator respectively (K) t, treg reaction and regeneration times respectively (min) time between the end of a cycle and b, td the beginning of the next one and dead time respectively (min) W catalyst weight (g) conversion of coke in the regeneraX tion, 1 - C, /C,, xA,,~ XA conversion with respect to A and conversion with respect to A at zero time respectively time averaged conversion at the xA reactor exit n