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Chapter 1 Reaction Performance in the Catalyst Unsteady State
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Chapter 1
REACTION PERFORMANCE IN THE CATALYST UNSTEADY STATE UNSTEADY STATE OF THE CATALYST According to Academician Boreskov's definition of catalysis (ref. 1), it is an acceleration of a chemical reaction over certain substances (called catalysts) which repeatedly interact with reagents without entering into the composition of the product. A catalytic process usually involves three successive stages: adsorption of initial components, chemical conversion on the surface and desorption of the reaction product. Each stage consists of consecutive or parallel steps of both physical and chemical interaction of intermediate compounds on the surface as well as with the components of the gas phase. Regardless of its specific action, the nature of catalysis is common and can be defined as follows: the catalyst, being a part of the intermediate compounds, increases the degree of compensation of the energy required for the splitting of old bonds by the energy released in the formation of the new bonds. This results in a decrease in the activation energy of the chemical reaction. That is, in short, an up-to-date point of view to the considerable growth of the chemical transformation rate in catalysed reactions. 1.1
The state of the catalyst is affected by the reaction medium. To every gas phase composition invariable over a lengthy period of time, catalyst surface temperature and initial conditions there corresponds an unique catalyst state, characterized by specific structure, composition and catalytic properties, provided there is no external-diffusion resistance. This catalyst state, of which time-invariant activity and selectivity are characteristic, does not, however, always prove optimal. One can imagine a situation, e.g., periodic variation in some range of gaseous mixture composition when the unsteady catalyst activity and selectivity exceed on average those under the steady state. It is recommended to compare catalyst states at identical concentrations both under stationary and averaged non-stationary conditions. The efficiency of a catalytic process under forced non-steady-state conditions is determined by the
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mechanism of the process, the rate constants of various reaction stages and the dynamic properties of the catalyst surface. Therefore, it makes sense briefly to remind the reader of those properties (ref. 2) and then proceed to the problems concerned with the performance of the processes in the unsteady state of the catalyst surface. The chemical composition of the catalyst, its surface structure and catalytic properties undergo some changes as the result of variation of the reaction medium. Numerous experimental data confirm that the composition, structure and properties of different catalysts, massive and supported metals and alloys, simple and complex oxides and acid-base catalysts really are influenced by the concentrations of the components in the reaction mixture and temperature (ref. 3). The modification of the chemical composition and the change in the catalytic properties in many cases are distinct as, for example, in sulphur dioxide oxidation on vanadium (ref. 4). The metal oxidation state decreases from VS + to V 4 + with increasing concentration of 802 in the reaction mixture and decreasing temperature. These variations are reversible and have small relaxation times.Furthermore, at low temperatures with a reaction mixture enriched with 802 and long period of interaction with the catalyst, a special state of inactive quadrivalent crystalline vanadium appears. Its concentration is in good agreement with the decrease in catalytic activity. These types of variations are characterized by longer relaxation times. A considerable transformation in the chemical composition and catalytic properties under the influence of the reaction medium is also observed for solid catalysts, for example, for oxidation of CO on various manganese oxide catalysts (ref. 5). The surface composition of the catalyst with different initial composition in the steady-state regime corresponds to the one and the same catalyst structure which can only exist in this reaction medium. Similar steady states of manganese oxides could be obtained after 30 hours of catalyst treatment by the reaction mixture at 2]OoC. The same results were obtained for oxidation of CO on copper-containing catalysts (ref. 6). Interaction of these catalysts with the reaction mixture involved a relatively rapid
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change in the oxidation state of the catalyst surface and its underlayer and also significantly slowed the creation or destruction of the space defects (inhomogeneities), which are evidently connected with phase rearrangements of the surface layer. As was shown for conversion of CO on copper-containing oxide catalysts, the oxidation of the catalyst by water vapour affects the catalytic activity. In '~his case, the stationary concentration of oxygen is attained much more slowly as compared with the catalytic reaction itself (ref. 7). The changes in the oxide catalyst properties under the influence of the reaction medium is evidently the most studied aspect of the variety of heterogeneous catalytic processes. The oxidation state of the catalyst is decreased following the reduction of this ratio and this results in a sharp drop in the general reaction rate, while the selectivity toward the incomplete oxidation product is simultaneously increased. Acrolein oxidation to acrylic acid on a vanadium-molybdenum oxide catalyst is the perfect example of the above mentioned change in parameters (ref. 8). The effect of the reaction mixture has also been observed for acid-base catalysts. For example, under the influence of water vapour, the extent of hydration of acidic centres and also the ratio between the Bronsted and Lewis centres can be changed. As reported for metallic catalysts, their surface is easily reconstructed under the influence of the reaction medium and tends to maximum surface energy. Very often the reconstructive adsorption of reactants leads to a change in the structure of the metal surface (ref. 9). Moreover, the composition of the near-surface layer is usually also modified as a result of dissolution of the reaction mixture components. The amount of the components adsorbed is often many times greater than a monolayer coverage. Levchenko et al. (ref. 10) observed a slow change in the rate of ethene oxidation on silver, which they related to oxygen diffusion in the near-surface layer of the catalyst. A similar process was observed in the catalysed oxidation of hydrogen on silver films (ref. 11). All these factors result in a change in the sorption heat of the reactants
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on the metal surface and in the activation energy of elementary steps. Consequently, the total catalytic activity and selectivity of the reaction are changed. The surface composition of solid alloys is also modified under the influence of the reaction medium. The free surface energy is changed due to the chemisorption and, therefore, it is more advantageous for the alloy component with the higher chemisorption energy to be on the surface. As reported in ref. 12 for Pt-Sn alloys, in a vacuum their surface is enriched with tin (compared to the catalyst bulk), whereas in a carbon monoxide atmosphere it is enriched with platinum. Atom shift in metals is usually limited to the thin near-surface layer, but with an increase in temperature the thickness of this layer may grow. We have only considered some examples of a large body of experimental data. These data taken together permit one to conclude that solid catalysts are components of the reaction system and are sensitive to variations of the reaction mixture composition, temperature and other parameters. To each composition and temperature of the reaction mixture, there corresponds a certain catalyst steady state. Some experimental data obtained about )0 years ago demonstrate the existence of hysteresis in the rate of a chemical reaction. Recently, isothermal self-excited oscillative regimes on the catalyst surface at invariable gas phase composition were discovered. This does not invalidate the above deduction, but only indicates that the steady state of the catalyst is determined by the composition and temperature of the gas phase as well as by the changes in the condition of this phase. The influence of the reaction mixture on catalyst properties must be taken into account in the kinetic dependences of the reaction in heterogeneous catalysis. In the vast majority of cases, kinetic equations are usually derived on a tacit assumption that the composition of the solid catalyst and its properties are independent of the reaction mixture composition and of its influence on the catalyst. In reality, this condition is not fulfilled because the chemical composition of
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the catalyst is affected by the reaction medium, which eventually leads to a phase change of the active component and alters the volume composition of the near-surface layer. That is why, besides the effects of temperature and composition, the reaction rate is also changed owing to modification of the catalyst properties. The dependence of the reaction rate, W, on the concentration of reagents must therefore include two functions. The first function, f[c(t); 9(c(t) )], characterizes the true kinetic dependence which describes the reaction mechanism. The second function, y?[c(t); Q(c(t))], is determined by the effect of the reaction mixture composition on the catalyst properties (ref. J) W
=
F
[f (r;
§)
i
Cf (c,
8)J
where t is the time, c is the vector of the reagent concentration, g is the surface-packing vector. Most often, expression (1.1) of the rate of the chemical reaction can be represented as the product of functions f and ~ • These functions can be determined experimentally by varying either the concentration of the reagents at the same catalyst state ( (jJ = constant) or by changing 'fl via catalyst pretreatment by reaction mixtures of various compositions. In the case of heterogeneous catalytic reactions, an important factor is the time and character of the transition to the steady state. It is necessary that this time, Mc (in other words, the relaxation time), be estimated in comparison with the time of the entire catalytic process, Mp, i.e., with the time of the effect of the reaction mediwn on the catalyst. It is useful to introduce a time-scale notion of the change in the gas phase condition, tc,for processes where the gas phase parameters are varied. If the gas phase variation is of periodic character, t c stands for the duration of the period. In the majority of cases Mp » t c. As a result of a change in the reaction mixture parameters, two different types of unsteady catalytic state can occur: (a) the concentrations of the intermediate products of the catalytic cycle are unsteady;
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(b) the state of the catalyst in the near-surface layer is unsteady leading to a variation in the rate constants for the elementary steps. The variation in the surface concentrations occurs as the result of the stepwise character of the catalytic process, that is, it occurs at a rate similar in magnitude to that of the reaction rate. For reactions which occur rapidly enough and, therefore, are of practical interest, the time-scale of the surface concentrations generally ranges from 10- 1 to 102 s. Modifications of the surface structure and near-surface composition affecting the catalytic properties usually include other stages of transition to the steady state than those of the catalytic reaction itself. That is why, in most cases, the transition to the steady state occurs far more slowly compared to the time of the catalytic reaction. The value of Mc for the non-steady-state process on the catalyst surface can, for example, be quantitatively estimated by the following expression
~ [W(t)- W(=J] tit o W(=) - W(Oj
(1.2)
where W(O) and W (00) are the values of the observable conversion rates under the steady-state conditions before and after inducing a jump-like disturbance at the moment of time t = 0; Wet) is the current value of this rate. A catalytic process may be carried out under three types of states: steady, unsteady or quasi-steady, depending on the character of the catalyst state variation and external conditions. The observable rates of chemical transfonnation of the reagents, W, in the steady-state regime depend on the temperature and instantaneous concentrations of all reaction participants as well as on the concentrations of the intennediate substances on the catalyst surface.The rates of fonnation and consumption of the gaseous substances participating in one and the same reaction under steady-state conditions are equal, within the accuracy of the stoichiometric relationship. In the unsteady regime these rates may appear to be different. If the composition, structure and properties of the catalyst in the unsteady regime adjust to the condition of
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the gas phase so rapidly as to fulfil at any time the inequality
W-r[c(6J, rr6J] W where r(o, T) is the reaction rate calculated via the kinetic model of the stationary process, this process on the catalyst surface is regarded as a quasi-steady-state one relative to the gas phase condition. In inequality (1.3), G is some small value corresponding, for example, to the allowed experimental error in the reaction rate measurement. The difference between Wand r in the non-steady-state regime is determined by the dynamic properties of the catalyst, which are connected to the relaxation characteristics of the catalytic cycle and to the effect of the reaction mixture on the catalyst. wnen the catalyst properties are changed owing to a modification in the surface structure and in the near-surface composition, the approximation to the steady state usually comprises stages other than those of the catalytic reaction. So, as a rule, the approximation to the steady state is carried out within a greater time than the characteristic time of the transient regime of the catalytic cycle. Generally, the non-steady-state kinetic model of the reaction is as follows:
dB
db
_
=
_
F, (c, 8,
ri:, T);
(1. 4a)
(1. 4b)
j
==
J (c, 8, s, T);
CjJ '=
If (C, B,
W=
F[jfe,8, z, T); !j(e,B,
i"i, T) ;
( 1 • 4c)
(1.4d)
(1. 4e)
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Eqn. 1.4a describes the dynamic character of the catalytic cycle: 8 is the concentration vector of substances on the catalyst surface, d is the vector of the catalyst characteristics changing under the influence of the reaction medium, T is the temperature. Eqn. 1.4b describes the dynamic character of the catalyst state under the influence of the reaction medium. Expression 1.4e is the vector of the observable rates of chemical conversion detennined by the kinetics of the process shown in eqn. 1.4c and by the influence of the reaction mixture on the catalyst expressed by eqn. 1.4d. Expressions 1.1 and 1.4e may coincide when only one chemical reaction is involved. 1.2
WAYS TO CREATE THE CATALYST UNSTEADY STATE
Two practical methods to maintain un unsteady catalyst surface can be distinguished: (a) periodic activation of the catalyst under conditions considerably different from those of the catalytic process and (b) periodic variation of the reaction mixture parameters, including the temperature. As was stated above, variation of the characteristics of the gas phase may result in the appearance of two types of catalyst surface states. (1) The concentrations of the intermediate products of the catalytic cycle are unsteady. In the case of heterogeneous catalytic reactions on solid catalysts these are the surface concentrations of reactants and products of conversion participating in the elementary steps of the reaction. The variation of the concentrations of the surface substances is the result of the stepwise manner of the catalytic process occuring at a rate similar in magnitude to that of the chemical conversion. For reactions carried out rapidly enough to be of practical interest, the re~axation time of the surface concentrations ranges mostly from 1 to 100 s. (2) The state of the catalyst in the near-surface layer is unsteady, which leads to variations in the rate constants of the elementary steps. These variations are connected with concurrent interactions Which do not take part in the catalytic cycle. As a rule, the activation energies of these interactions are rather large, whereas the rates are small. Therefore, the relaxation time of the concurrent interactions is far greater
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than that of the catalytic cycle. In both cases the unsteady-state character is determined by the divergence of the catalytic properties corresponding to the time-averaged values of the gas phase parameters. In the first case, the efficiency can be increased owing to the oscillation regime which provides the possibility of controlling the concentration of the intermediate species and, under certain conditions, to change the productivity and selectivity. In the second case the efficiency of the process can be increased by creating an optimum (in the mean) unsteady state of the catalyst. Let us assume that the time-scale of the variation of the surface concentrations, Mf, is much less than the time-scale of the stabilization of the catalytic properties, M'fJ' and than the time-scale of variation of the gas phase condition, t c• Also, let the values Jilf and t c be related as Mf « M'fJ ~ t c• If these conditions are valid, the catalytic reaction can be carried out in the catalyst unsteady state. The intermediate products of the cycle will be thus in a quasi-steady state, which means that at any moment of time their concentrations are slightly different from the concentrations corresponding to the composition and temperature of the gas phase at the same instant. Of course, we have to assume further that the variation range of the gas phase condition allows for the existence of only one steady-state regime on the catalyst surface for every given invariable composition of the gas phase. The difference between the steady and unsteady states for various reactions lies in the degree of hydration and oXidation, the content of dissolved components or in other parameters. The unsteady state can also be obtained by varying the condition to which the catalyst is exposed. Let ~ be the parameter determining the catalyst state. It is due to change in the course of the reaction carried out under non-steady-state conditions. During the steady-state process with the parameters of the gas phase corresponding to their mean values at time t c in the unsteady process, oc = ~S. In many cases the catalyst unsteady state can exceed its steady state in both activity and selectivity within a certain interval of variation. Then, with the gas phase condition invariable during
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the entire course of the catalytic function and with W being linearly dependent on the functions f and ~ (1.1), the relative gain in the chemical conversion rate for a simple reaction will take the form tc
~ o
LlW W(d S)
'I [ d (b), § (OJ tit - f
=
where Ll W is the absolute value of the reaction rate gain compared to the steady-state operation with W(d S ) = r; 'IS is a function of the influence of the reaction mixture of the catalyst in the stationary regime '1(C£.s'
r.:
B.s).
In case of complex reactions, the change in the selectivity is the most important practical aspect. For example, for two parallel reactions carried out in the unsteady regime with rates W for the usual and W2 for the concurrent one, the selectivity 1 can be expressed as tc
tc
~ ~tit + 0
o
0
tc
tc
2
0
tc
tC
~ iff dt / ~ '1 dt
~ w, db s=
tc
SVti 0
dt
~ lit dt / ~ <1 o
2
db + 12
/
h
0
where f 1 and f 2 are kinetic functions of the first and the second reactions, respectively. With catalyst state variation the ratio tc
~ liz dt o
o
is changed. If under the catalyst unsteady state characterized
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by the value of parameter
d
the following condition is fulfilled
then there is a gain in selectivity. One can apply the above expressions to estimation of the efficiency of an unsteady process performed with periodic activation. With non-'steady-state operation of the catalyst, the parameter a: tends to approach c(,s • In order to return to the optimum state of the catalyst it is necessary to treat it periodically with a regenerating gaseous mixture other than the reaction mixture used in the process. This can be achieved either by periodically blowing down the reactor with the regenerating mixture or by continuous extraction of a portion of the catalyst to be regenerated in a separate apparatus. The last method seems convenient in the case of a quasi-fluidized catalyst bed or moving large-grain bed. The catalyst unsteady state can be attained by means of periodic variation of the reaction mixture parameters, i.e. composition, temperature, pressure and volume flow rate. Vilien the catalytic properties are changed due to variations in the conditions of the gas phase, the analytical determination of the optimum oscillations for the parameters presents a serious difficulty. To avoid it, it is necessary to know the kinetics of the investigated reaction in various catalytic states and, in addition, the kinetics of the influence of reaction mixtures of various compositions and temperature on the catalyst state. If the mean-integral reaction rate equals IV during a fraction fl of the whole period, t c ' of the composition and/or gas phase temperature variation when the state of the catalyst is close to its optimum, and W2 in the interval of regeneration, then 'the increase in the observable rate compared to the rate under 2(1_ 2 steady conditions, R, is expressed by W~ + W fl) - R. W may be (and always is) a small value, so the gain in the reaction rate is determined by the difference between v 1p and R.
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At present there is a discrepancy between experimental research dealing with processes occurring under unsteady catalyst operation and the theory devoted to the creation and investigation of kinetic models describing the unsteady-state process on the catalyst surface.There have been few bench-scale examinations of the methods to increase the efficiency of a catalytic process under non-steady-state conditions. The authors of experimental studies dealing with the increase in process efficiency obtained at considerable absolute values of chemical conversion tend to pay attention only to the influence of the reaction mixture on the catalytic properties, while theoretical works are mainly concerned with the dynamic character of the catalytic cycle. Thus, the theory allows one to understand the practical findings only in a ve.ry general sense. Furthermore, it has to be admitted that the theoretical process modelling is not necessarily related to particular processes. We shall now examine the important experimental results and then proceed to consider some theoretical aspects of performing catalytic reactions in the catalyst unsteady state. 1 .3
EXPERllIENTAL DATA
Ref. 13 reports the influence of forced oscillations of the reaction mixture composition at the reactor inlet on the oxidation of sulphur dioxide on vanadium. The ratio of the reagents, S02 and O2, was changed in a cyclic manner around a mean value of 0.6. The minimum concentration ratio of the reagents was 0.2 with a peak of 1.0. The dependence of the mean reaction rate averaged over the period, t c' on the length of this period exhibited a maximum in the range t c = 4-5 h and a reagent ratio between 0.3 and 0.4. The temperature of the reaction mixture at the inlet was measured to be 405°C. The reactions were carried out at small conversion extents and were far from equilibrium states. Evaluations of the rates pf the processes in this system showed that the characteristic times of transient regimes in the catalytic cycle were much smaller than the duration of the period at which a noticeable increase in chemical conversion rate was observed. Evidently, the explanation lies in the fact that a considerable part of the vanadium in the low temperature range is in the inactive
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quadrivalent crystalline form V4+. The characteristic times of the transient regimes required to change the concentration (crystallization or dissolution) of this form during the cycle may appear to be rather long and commensurable with the discussed periods of concentration oscillations in the gas phase. Ref. 14 also reports the influence of the cyclic variation of the initial mixture composition for oxidation of 802 in a fixed catalyst bed. This research was carried out in a twin-bed catalytic reactor. 802 mixed with air and concentrated to 12.4% was fed into the first bed of vanadium catalyst. This bed, used for initial conversion of about 90% of the sulphur dioxide into SO)' operated in the steady-state regime. The composition of the mixture fed into the second bed was varied in a cyclic manner: 13 minutes of pure air blow, then 1) min of the mixture from the outlet of the first bed. The temperature of the mixture at the second bed outlet was kept at 406°C. The results showed that under steady-state conditions the conversion extent at the second bed outlet was 95.7%, while in the cyclic regime it reached 98.7%. This value exceeds the equilibrium conversion extent under experimental conditions calculated for the mean values of the temperature and composition. This can be explained by the unfavourable shift of the steady-state reaction from the equilibrium state caused by the large amount of SO) dissolved in the catalyst body. After air or nitrogen blowing, the sulphur anhydrite is desorbed and helps to change the quadrivalent vanadium into its pentoxide. The ratio of the catalyst volume to the volume of the reaction mixture (conventional time of contact) in the experimental examples discussed here was measured as only a few seconds. For this catalyst amount the optimum value of the period lies between 10 and 20 min. It is noteworthy that the effect of blowing air through the catalyst bed at 406°C was evident during 24 h. The catalyst transition (approximation) to the steady-state was evidently close to completion within this considerable interval. The adsorbed hydrogen and the reaction products are known to be mainly present on the catalyst surface as in the process of ammonia synthesis in the steady-state regime on ferrous
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catalysts. Nitrogen is practically absent due to its slow adsorption rate in relation to the other stages of ammonia formation. Feeding of pure nitrogen onto the refined catalyst surface by means of an inert gas flow (for example, helium) provides a noticeable rise in nitrogen adsorption rate and a considerable accumulation of adsorbed surface nitrogen is observed. One can suppose that periodic introduction of H and 2 N2 (separately) into a reactor containing a ferrous catalyst should lead to an increase in ammonia output owing to the increase in the cyclically averaged concentration of adsorbed nitrogen. However, periodic variation of the inlet concentrations of H2 and N2 in this reaction on ferrous and ruthenium catalysts proved efficient at low pressures. In the temperature range 325-425°C and pressure about 1 roPa, the productivity o~ the ferrous catalyst was increased by 30% with the optimum oscillation period being about 30 seconds (ref. 15). An identical productivity increase was detected at a pressure of about 2.5 roPa and at 400°C, the period of oscillation in this case being 7-10 min (ref. 16). Table 1.1 reports values of the productivity, G, and of the extent of conversion of nitrogen into ammonia, x, with a stoichiometric nitrogen-hydrogen feed by means of an helium flow at normal (atmospheric) pressure (ref. 17). Shortening the time between impulses by about 50 times and a temperature drop from 500°C to 250°C had little effect on x, whereas the productivity increased. TABLE
1.1
Dependence of G [sm3 IT C] and x [%J on the time interval between impulses of a mixture containing N2 + 3H2 (in an helium flow)
x 500 400 250
Time interval ( s) 20 180
10
T (Oe) 4.4 4.0 2.1
G
0.62 0.55 0.53
x 4.5
4.2 2.3
G
0.35 0.32 0.30
360
540 ,..
x
G
x
\.r
x
G
4.8 4.5 3.8
0.026 0.022 0.020
4.6 4.5 3.6
0.021 0.020 0.017
4.8 4.4 3.7
0.013 0.011 0.010
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This indicates that the efficiency of the non-steady-state process is evidently determined by the dynamic character of the catalytic cycle, characterized by the value of the time-scale of the transition regime on the surface, Me ~ Mf ~ 10 s. These considerations are in full agreement with the experimental data on relaxation processes on the surface of the ferrous catalyst at normal (atmospheric) pressure (ref. 18). It has been shown here that slow processes of the reaction medium interaction are not observed, at least at temperatures between 220°C and 400°C, and the dynamic properties of the surface are determined by the rates of variation in the intermediate product coating the surface. Table 1.2 reports the extent of conversion of nitrogen into ammonia as a function of the catalyst amount at constant volume of the impulse and different temperatures for the impulse and flow (filtration) stationary regimes. It is clear that the conversion extent under steady-state conditions is a factor of 10 2 lower than that in the unsteady state. The catalyst productivity under these conditions is an order of magnitude greater than that in the steady-state regime.
Extent of nitrogen conversion into ammonia, x [%], in various regimes T (oC)
Catalyst amount in reactor 2 g 0.5 g N2 into H2 flow
500 ~o
250
2 g (flow)
N2 + 3H2
N2 into
N2 + 3H2
N2 +
into He flow
H2 flow
into He flow
+ 3H2
2.0 1.2 0.8
1.6 1.4 1.0
0.07 0.024 0.0043
The results described above are similar to those obtained for experimental synthesis of ammonia on ruthenium. The rate of formation of ammonia is not high because of strong hydrogen
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inhibition. With a cyclical character of the process this resistance can be noticeably reduced, as been demonstrated experimentally: the reduction rate was increased by a factor of 10 2 due to periodic hydrogen feed onto the catalyst body at JOOoC and 1 atm pressure. The period of the impulses was measured as a few minutes (ref. 19). It must be added that the observable rate of chemical conversion on the ruthenium catalyst remained very low. Operation under the non-steady-state appeared efficient for the reactions of oxidative dehydration of ethylbenzene into styrol on activated coals (refs. 20, 21). The initial period of this process for the catalyst oxidized by oxygen from the air is unsteady, provided the oxygen and the ethyl benzene contents are invariable. Coal is highly active for the first 10 min of the oxidative dehydration process, while in the reaction of deep oxidation it remains quite passive. As steady-state conditions of the reaction are approached, the activity of coal in the main conversion decreases, whereas in the concurrent reaction the activity increases. There appeared to be at least two forms of chemisorbed oxygen on the catalyst surface, one of which, being less mobile and fixed, shows activity in the reaction of oxidative dehydration, the other being mobile and slightly fixed is active in the reaction of deep oxidation. The decrease in coal activity is explained by the considerable oxygen flow and the gradual coating of the catalyst surface with product. Application of an oxygen-containing mixture free from ethylbenzene resulted in full regeneration of the catalyst. Tests on oxidative dehydration under non-steady-state conditions at 400°C were carried out in a flow converter with two chambers operated in turn. The reaction mixture was blown through one chamber and hot air through the other. Every 10 min the reactors were alternated. It was possible to increase the styrol output and selectiVity of the reaction by means of cyclical dehydration and regeneration as compared to those obtained under steady-state conditions. At a styrol output of 70-78%, the selectiVity was 92-96%. Better characteristics, to my mind, were the low product coating and high average content of slightly fixed oxygen on the catalyst surface brought by the periodic reaction control.
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An increase in efficiency under non-steady-state conditions was obtained for ethene hydration in an isothermal reactor with a fixed catalyst bed consisting o£ Pt ont-A1203 (ref. 22). The reagent concentrations at the reactor inlet were changed according to a stepwise rectangular function: for ethene, in the region 0-50% (v/v); for hydrogen, 100-50% (v/v). The results were then compared with those obtained under steady-state conditions for a reaction mixture composition at the reactor inlet corresponding to mean composition values of a cycle in the non-steady-state regime. The rate of ethene conversion was strongly dependent on the concentration oscillation frequency, being highest at period of about 10 s. The average observable ethene hydration rate in this case was 50% higher than that under steady-state conditions with identical mean concentrations of H2 and C2H in the gas phase. This increase in conversion 4 rate must be related to the kinetic characteristics of the elementary reaction steps. If it were possible to eliminate the infl\tence of the reactor volume on the variation of the gas phase conditions, the peak performance would surely be reached in the so-called slidip~ regime with an high frequency of the gas phase concentration oscillations. Because of the damping properties of the reaction volume, the incremental constant variations in the composition at the reactor inlet are not evident when they reach the catalyst surface. The influence of the reactor volume is also seen at periods commensurable with the time during which the reaction mixture is present in the contact zone, which is a few seconds. At very high frequencies the reactor operates under conditions corresponding to invariable and mean-integral (over the period) values of the concentrations. Ref. 23 describes the formation of ethyl ester from ethene and acetic acid. Sulphuric acid implanted in the inner surface of silica gel particles served to catalyse the reaction. The efficiency of the catalyst was studied in a fixed-bed reactor under both steady- and non-steady-state conditions. The unsteady state was created by a periodic stepwise feed of acetic acid into the reactor inlet. The ethene flow-rate was not effected in this process. The optimum period of oscillation of the acid concentration was about 30 min. Under invariable conditions the
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catalytic activity reached a constant value after 400 h of operation. It must be noted that the catalytic activity gradually increased to a maximum within the first 100 h and then decreased approximately by half. Under periodic performance of the process the product output largely depended on the period and on the relation between the acetic acid feed time and the time of zero concentration at the reactor inlet. The catalytic activity observed in the first 100 h of non-steady-state operation coincided with the peak value for the steady-state case and remained stable for a longer time. This allowed for a sufficient increase in the reactor productivity compared to the optimum steady-state regime. An investigation of the influence of the inlet concentration oscillations of ethene on the selectivity of its oxidation to ethene oxide on silver was carried out under isothermal conditions (ref. 24). It revealed the following phenomena: (a) transition from steady-state to periodic control leads to a decrease in the extent of conversion; (b) an increase in the period causes a decrease in the conversion extent; (c) with a decrease in conversion extent it is possible to increase the selectivity under non-steady-state conditions. The same reaction was studied under non-isothermal conditions at atmospheric pressure in a 30 mm tube reactor filled with a silver-containing catalyst (ref. 25). The inlet ethene concentration was periodically varied stepwise from 0 to 12% (v/v). A considerable increase in selectivity was obtained at a period of about 30 s and at identical values of the conversion extent related to the value under steady-state conditions. Fig. 1.1 shows experimental data concerning the process selectivity under conditions of practically invariable conversion extent and temperature fields in the reactor. Attention should be paid to the fact that though the productivity of the non-steady-state regime characterized by the output of ethene oxide was somewhat lower, nevertheless, the selectivity remained within 50-60%. Ref. 26 reports the non-linear dependence of the ethene formation rate on the extent of oxygen coverage of the catalyst surface. A maximum in the formation rate is observed at a
19
coverage of 0.5-0.6. This, evidently, can-be explained by a structural modification of the metal surface, causing a change in the type of link between metal and oxygen as the result of the latter's interaction with both the surface and near-surface layers of the catalyst. Penetration of oxygen into the near-surface layers of silver results in their modification, which may also be induced by some other electronegative elements, as reported in ref. 27. The similarity between the deeply adsorbed oxygen and electronegative promoters and the character of the activity and selectivity change after application of the promoter allow one to suppose that the effect of an increase in selectivity in the oxidation of ethene in the cyclic non-steady-state regime is explained by a decrease in activation energy in the stages determining full and partial ethene oxidation. The most noticeable energy drop occurs in the process of partial oxidation. The use of non-steady-state conditions may permit operation with higher concentrations of adsorbed oxygen, which is the reason for the selectivity improvement. We cannot as yet exclude the possibility that the maximum selectivity at a period of 30' s is connected with the dynamic properties of the reactor and not determined by the dynamic properties of the catalyst surface. Periodic oscillations in the alcohol concentration at the inlet of an isothermal tUbe reactor with a fixed bed consisting of Al 20 particles resulted in a selectivit~ increase in the J dehydration of ethanol (ref. 28). A pronounced influence of the frequency and asymmetry of the inlet function of the ethanol output was observed. The optimum set of parameters raised the ethanol output two-fold as compared to that obtained under steady-state operation. Now let us have a look at the process of butadiene hydration on nickel under non-steady-state isothermal conditions (ref. 29). The tests were carried out in a flow reactor at 70°0. Theoretical calculations were made according to an ideal substitution model for isothermal conditions. The concentrations of butadiene, butene, hydrogen and butane as well as these of the three supposed intermediate compounds on the catalyst surface were taken into account. The inlet concentrations of hydrogen and 04H6 were changed stepwise in antiphase; the total
20
........
~ ...., CI) <;j ~
>. +> •.-1 :> 'r-! +> 0
(!)
20
fa
r-I
Q)
02
'H 0 III
§
,.q 0
duration of cycle, t c (8)
Fig. 1.1. Ethene oXidation to ethene oxide under non-isothermal conditions on a silver catalyst. Relative selectivity change (~S) as a function of the period, t , at invariable ethene conversion extent (0.05) and constaRt temperature field in the reactor. ~S = 0 corresponds to steady-state conditions. concentrations of both hydrogen and butadiene were constant. The optimum steady-state regime found by calculation and tested in practice served as a comparison. It had the following parameters: extent of conversion, 50%, selectivity of butene formation, 60%. In the non-steady-state operation of this process, the selectivity of this chemical transformation could be increased by 20% with the conversion extent unchanged. The period of the cycle was varied in the range between 2 and 30 8 and the amplitudes of oscillation in the range from 0.2 to 1.0. As seen in Fig. 1.2, the butene productivity, 2 ,is increased while the duration of the cycle is reduced and the selectivity, S, is increased with increasing oscillation time, t c• This may be explained by the following hydration mechanism. The rate of butane formation is largely dependent on the hydrogen concentration. With a decrease in the oscillation period there is an increase in the amount of butadiene adsorbed leading to an increased rate of butene formation and to an even more marked
21
increase in the rate of butane formation. The mean adsorbed butadiene content is decreased due to slow operational switches, providing thereby an increase in the process selectivity.
0.2S
0,50
0.2L(
. . . ,."ifF-
0.23
-Q,,)
-0----01
----- __ L_
0.22
o.os o.vs
al PI al
0
rl 0
>
'-J
Cf.l A
OJ{I{
:>..
+>
-r-i
A
:>.. +> 'n 0
*.
........
>
0.2t
OJ{2
'n
+> 0
Q)
rl
0
L(
t6
20
2'"
O"{Q
Q)
Ul
duration of cycle, t c ( s) Fig. 1.2. Bench-scale and experimental data (marked with circles) regarding the productivity and selectivity of butene hydration on a nickel catalyst in the periodic regime. 1,2 are the optimum steady-state selectivity, S, and productivity,g • Interesting data are reported in ref. )0 concerning Fischer-Tropsch reactions of paraffin synthesis on a Ru/.r -A1 20 catalyst. 3 This study shows that the distribution of the reaction products according to their molecular weights largely depends on the conditions used to achieve the non-steady-state. Impulses of hydrogen and a CO/H2 mixture were periodically blown down through the inlet of the fixed catalyst bed. With a period of 8-12 min, there was a considerable narrowing of the distribution function. In the steady-state regime the ratio of concentrations of the components C12H26 and C6H14 was 0.74 and it was as low as 0.12 under non-steady-state conditions. Considerable variation of the molecular weight distribution of the products was observed in a Fischer-Tropsch process on promoted ferrous catalysts at a pressure of approximately 0.4 mPa and a temperature of about 250°C (ref. 31). The methane
22
output was significantly increased owing to the cyclic regime of the reaction. Ref. 32 reports the results of experiments with Pt/A1 20 3 catalysed non-steady-state reactions at 505°C. The time-averaged mixture composition of CO, NO and O2 was oxygen-stoichiometric. The concentrations of CO and NO were constant at 1% and 0.1%, respectively, whereas the oxygen concentration was periodically varied at different amplitudes averaged over 0.4%. In this case, forced non-steady-state operation of the process did not produce any effect on the reaction efficiency. The extent of oxidation of CO and of reduction of NO monotonously increased in parallel with the oscillation frequency, tending to the steady-state values corresponding to the mean values of the oxygen concentration. Considerable changes in the rates of carbon monoxide oxidation and NO reduction were observed during variation of the oxygen concentration (during the period, t c) from 0.24 to 0.66% and at a period of about 1 s. Oxidation of CO in the non-steady-state regime on a supported platinum catalyst has also been studied (ref. 33). The reaction mixture was fed through the inlet of a non-gradient isothermal reactor with a periodic variation of the mixture composition: a mixture of carbon monoxide and argon was blown in during the first part of period; during the second part of the period an argon-oxygen mixture was applied. The process was carried out at a temperature of approximately so-c, carbon monoxide concentrations of 0-2% and oxygen concentrations of 0-3%. The maximum cycle duration was about 3 min. A 20-fold increase in the reaction rate as compared to that obtained under steady-state conditions was possible in the non-steady-state regime. The maximum in the reaction rate occurred at a cycle duration of about 1 min. The authors explained the results of their experiments assuming the following reaction mechanism: (1)
CO
+
[z]
(2)
O2
+
2 [Z] -:::= 2 [OzJ
(J)
[COZ]
+
~
[C02]
[ozJ - CO 2
+
2[ZJ
23
(4)
co + [oz] -- CO 2 + [z]
Supposing that carbon dioxide formation is mostly determined by the rate of the third stage, this rate is highest when the concentrations of surface forms, [COl] and [Ol] , are about equal. Then, a considerable increase in the carbon dioxide formation rate under·non-steady-state conditions follows from the fact that concentrations of surface coatings were kept in the vicinity of optimum values. At the same time, in the case of a steady-state operation, the extents of surface coating by the intermediate substances, [cOZ] and [Ol] (as independent tests have shown), are quite different and their product is a small value. Similar experiments were performed by Barshed and Erdogan (ref. 34) in a flow isothermal reactor with periodic variation of the carbon monoxide and oxygen concentrations at the inlet of the fixed catalyst bed. The minimum oscillation frequency was 0.2 Hz. The authors demonstrated that the dependence of the observable rate of carbon monoxide and oxygen interaction on cycle duration has a maximum value if the cycle duration is commensurable with the time-scale of the reaction carried out on the catalyst surface, i.e., with the dynamic properties of the catalytic cycle. Oxidation of carbon monoxide in the non-steady-state regime was also investigated on a supported V 20 catalyst (refs.35,36). 5 The ratio of the partial pressures of CO and 02 (CCo/C 02) was periodically varied according to an incremental constant function at the inlet of the reactor with a fixed catalyst bed. The variation took place around different averaged values. The oxidation rate appeared to depend on the initial mixture composition and on the oscillation period. If the mean values of the CCo/CO ratio are not sufficient, the influence of the amplitude Becomes more and more noticeable. It seems of interest that the dependence of the averaged over the cycle rate of chemical conversion on the period has three regions of resonant frequencies corresponding to t c = 1-2 min, 15-20 min, and 40-45 min. The rate at these frequencies is 1.5-2.5 times higher than in the steady-state regime. To explain the results obtained
24
the authors tried to resort to a well known hypothesis of carbon monoxide oxidation mechanism on a vanadium catalyst. However, they could achieve neither quantitative nor qualitative correspondence with the facts. In the ideal models used in their investigation, which contained stages of sorption and reactions on an homogeneo~s surface, the duration of the period was considerably shorter than that observed experimentally and which allowed for a pronounced increase in the oXidation rate under non-steady-state conditions (ref. 37). Temperature variation in a cyclical manner applied to the same system also proved of little avail in increasing the efficiency of the process (ref. 38). Let us consider some examples of catalytic process at the catalyst unsteady state related to its periodic regeneration in the fluidized bed. First, the OXidation of ortho-xylene on a catalyst containing 3.2 mol.% of V20 0.64 mol.% of Te0 2 and 5, 96.2 mol.% of Ti0 2 (ref. 39). The activity and selectivity of the catalyst in its preliminary oxidized state practically coincide with the corresponding parameters for a steady state. A preliminarily reduced catalyst is characterized by an higher activity at any extent of conversion, x, and its selectivity rises if x is large and slightly decreases when x is small. A change in the extent of conversion under non-steady-state conditions leads to a considerable variation in the oxidation state of the catalyst: at small values of x the catalyst is reduced and its oxidation state increases as the value of x increases. If the catalyst is formed at small values of x and is continuously placed in the zone of higher conversion extents, one expects an increase in phthalic anhydride output as the result of the decrease in the relative rate of its additional oxidation on the reduced catalyst. This was confirmed in a pilot reactor with a pseudo-fluidized catalyst bed. A directed circulation of the catalyst in the reaction zone was created by means of its reverse flow against the interacting mixture and by returning it to the upper part of the apparatus. The mean catalyst residence time in the reaction zone was approximately 20 seconds. A change in x was achieved by variation of the reactor temperature. When the process is carried out in the fluidized bed organised with the help of special devices and if
25
conversion extent lies between 0.8 and 0.95, the output of the partial oxidation products appeared to be 10% lo~er than in the case of the catalyst circulation. This can be explained by the fact that if the catalyst is not directly circulated, it tends approximately to a steady-state which corresponds to a conventional operation in the fixed catalyst bed. Such a state in the organised fluidized bed was possible due to a small value of the time-scale of the transition regime on the catalyst surface. This seems to be the most reasonable explanation at the moment. The process of oxidative dehydration of n-butene on a bismuth-molybdenum catalyst at 480-500°C was carried out in a twin-reactor system with a fluidized bed (ref. 40). The catalyst was circulated between two apparatus, one of which served as the reactor and the other as the regenerator. Fig. 1.3 shows the dependence of selectivity on the butene conversion in various operational regimes. One can see that the highest selectivity at constant butene conversion is reached when butene and oxygen are separately fed into different apparatus: C S is fed into the 4H reactor while 02 is fed into the regenerator.
90
.......
*
.-; 0
::-
80
70
'-' u.l ~
?>
:> 50
-r-i
~,
2
60
.p
'r!
~f
-0-----00-0-.
.p
o
QJ
.-;
QJ
m
60
70
80
go
extent of conversion, x(vol.%)
Fig. 1.3. Dependence of selectivity, S, in the oxidative dehydration of n-butene in various operation modes: 1 = "reactor-regenerator" system without oxygen supply into the reaction zone. 2 = the same system with the oxygen supply; 02/C4HS = 0.2. 3 = fixed bed of the catalyst; 02/C4HS = 1.5. 4 = fluidized bed of the catalyst; 02/C4HS = 1.5.
26
Investigation of the catalyst under non-steady-state conditions (ref. 41) indicated that in the process of butene dehydration without oxygen supply, it (the catalyst) has a large period of continuous activity. At the end of this period, the butadiene output gradually decreases while the selectivity is unaffected. A certain oxygen volume of about 15 monolayers covering the internal surface of the catalyst corresponds to the period of constant catalyst activity irrespective of the conditions. Oxygen treatment of the catalyst regenerated its initial properties. Thus, the oxygen necessary for the reaction to be carried out was introduced together with the catalyst, the latter taking the role of the oxygen-transfer agent. A redox mechanism is sure to take place in this process. According to this mechanism, the adsorbed n-butene interacts with the surface oxygen of the catalyst. Should it (oxygen) b€ absent in the gas phase, it is diffused to the surface from the catalyst volume. The rate of such a supply is rather high if the content of applied oxygen is less than that corresponding to about 15 monolayers. In the course of the reaction, the rate of oxygen thrust onto the catalyst surface starts to set a limit on the dehydration process and the observable rate of the main process is reduced. During the stage of oxidative dehydration one can detect a slow deposition of carbon compounds. The beginning of rapid deposition of such compounds coincides with the time at which an extent of reduction of the catalyst corresponding to the beginning of a noticeable selectivity decrease is attained. Carbon compounds are easily oxidized by oxygen blow and the catalyst is thereby regenerated. The catalyst in the above process may continuously serve as the oxygen-transfer agent only under conditions enabling its reduction corresponding to the removal of approximately three oxygen monolayers. At higher reduction states of the catalyst metallic bismuth and M0 20 are likely to appear. 3 Evidently, one can always expect a change in the efficiency of the catalytic process under forced non-steady-state catalyst conditions, provided these processes are carried out according to different mechanisms: in particular, such redox reactions as full oxidation of hydrogen, carbon monoxide, hydrocarbons and
27
many other organic substances at high temperatures as well as partial oxidation of olefins, alcohols and aromatic compounds. Interaction of an oxidant, one way or another, with the regenerated catalyst introduced into the reaction zone (in presence of the oxidant or not) may often allow one to increase the activity and/or selectivity. This increase may be due to the catalyst's ability to maintain under the non-steady-state regime its optimum state With respect to the energy of oxygen binding with the surface. Another illustration may be the process of naphthalene oxidation to phthalic anhydrite on vanadium (ref. 42). The catalytic activity in this process increases with the degree of oxidation, O. The selectivity of the process does not depend on 0 when it is over 0.3. The process was carried out in a "reactor-regenerator" system and only 25% of the catalyst body was stored in the apparatus. The catalyst circulating factor was equal to 6 inVerse hours. In the pilot reactor at a regeneration temperature of 385°C, the yield of phthalic anhydride reached 87% at 8 ~ 0.35, while without regeneration these values only were 82% and 0.29, respectively. At a temperature of 400°C, the oxidation extent was 0.43, the catalyst productivity increased 1.5 times and the phthalic anhydride yield was constant at 87%.
1.4
OPTn~IZATION
OF THE CATALYST UNSTEADY STATE
Upon consideration of the above-mentioned facts one can conclude that there have been many experimental studies on heterogeneous reaction performance under non-steady-state catalyst conditions. There is no doubt now that the transition to an non-steady-state regime offers a favourable increase in the process efficiency compared with that obtainable in the steady-state. Still, the effect of the efficiency increase, as we have noted, cannot always be explained by purely quantitative factors obtained through mathematical simulation on the basis of independent kinetic research. This presents serious difficulties in non-steadY-state process control and in determination of the optimum conditions for these processes. At the same time there have been a number of theoretical attempts to distinguish classes of chemical reactions according
28
to simulated kinetic schemes ~or which the possibility o~ increasing the catalytic process e~~iciency under periodic variation o~ control parameters has been demonstrated. Therewith arouse the cyclic optimization problem which is related to the traditional theory o~ optimum control. In the same time it has a number o~ speci~ic peculiarities on which we shall touch ~urther. The regimes o~ periodic character allowing ~or a considerable increase in process e~~iciency as compared with the optimum under steady-state conditions are thought to be the main goal of such investigations. Let us take as an example the mechanism of a catalytic process illustrating the efficiency of the forced non-steady-state regime (ref. 43). 1.4.1
An example of an efficient non-steady-state regime
Consider the case where the efficiency of the process can be increased due to periodic oscillations of the gas phase concentrations ~~th invariable rates of the elementary steps. In other words, only non-steady-state of the first type will be taken into account. The efficiency gain in this situation may be caused by the oscillative regime in the reactions o~ heterogeneous catalysis enabling control of the concentration of the chemisorbed particles, leaving the average concentration of the reagents in the gas phase unaffected and, under certain conditions, enabling variation of the productivity and selectivity. Let substances A and B interact according to a one-route mechanism in the presence of a catalyst with periodic variation of the concentration of A in the gas phase. Two possible variants are open for discussion. In the first instance the initial substance A is adsorbed on the catalyst surface and interacts with substance B in the gas phase according to the collision mechanism. In the other case, the initial substance B is adsorbed and the end product is formed as the result of interaction of A (contained in the gas phase) and the adsorbed reagent B via the collision mechanism. An elementary scheme describing the formation of the product C for the first case can be represented as: kf
A +[Z] ~ [Al]
(I)
29
B + [AZ]
1£ .z:
[Z] + C
The second case as
(1)
in which product D is formed can be represented
B + [l] (II)
A + [BZ]
ft
-!.. [Z] + D
where [AZ] and [BZ] are the products of the chemisorption, and [Z] represents the vacant active surface area. Kinetic models corresponding to mechanisms (I) and (II) can be derived
L
d[Al] dt
=
1£( CA rt-[AZ]) - k_f [AZ] - h 2 CB [AZJ (1.5b)
where CA and CB are the concentrations of substances A and B in the gas phase, L is the number of active centres per catalyst surface unit, k i are velocity constants (i = 1, 2, 3, 4, -1,-3). Eqns. 1.5a and 1.5b were derived upon consideration of the conditions of the surface concentration balance: [Z] + [AZ] = and [Z] + [BZ] = 1. Let us divide eqn. 1.5 by a normalization coefficient, kC, to make it dimensionless. Then we obtain
d [AZ] _ r , dt' - K, CA (I-[Al]) - K_ 1 [AZ] -~ CB [AZ]
(1.Ga)
(10Gb)
30
where K1 = ki/kC, i = -1, -3; Ki = k;/k; i = 1, 2, 3, 4; C~ = CAlC; CB = CB/C; t'= tkC/L; a dimensionless period, t c' equals tckc/L. For simplicity, the prime symbols will be omitted. The formation rates of C and D averaged over the dimensionless period will be equal to
~
~
tc
1
~ K2 CB [AZ] db
=-
t
c
(1
.7 a)
0
and
W
D
1 =-
to
tc
~ K" CA [BZ] db 0
Let us assume that the concentration Cn is invariable during the whole period and CA is varied according to
2CA • C = A
{
o
(1.8)
where CA is the period-averaged concentration of substance A. The concentrations of the intermediate substances in the steady cyclic regime are identical both at the beginning and at the end of the cycle:
31
The reaction rates under non-steady-state conditions will be regarded as the rates averaged over the period (see eqn. 1.7). Now let us compare the reaction rates in this case with the values under steady-state conditions determined by the period-averaged values for the control actions. In the steady-state regime under the above-stated conditions and where CA = CA = = constant, the reaction rates according to mechanisms I and II have the forms:
(1.10)
where
k
he
aC f
f !f.,g
k_ f
=
1£1 ILf
!l;])=
it.5 kif IL,J
a2C
k
z
!f,-1
Fig. 1.4a shows the dependence of the difference in the .....reaction rates, .fJc = We - RC and.P D = liD - RD, under both steady and non-steady-state conditions on the oscillation period at K1 = K_ 1 = 3; K2 = K = K = 1; K_ = 0.3; CB = 1; CA = 2. 3 3 4 Values of Wc and WD were derived analytically from eqns. 1.6 with consideration of the conditions described in eqns. 1.8 and 1.9 by means of substitution of the obtained solutions into ~
eqns. 1.7a and 1.7b and their integration.
32
0
.fJc .f};
.p oj
H
'l-i
a)
0.1
0
§
,.q 0
0.2
duration of cycle, t c (s)
tr.l
0.1
O.t
Ps
Pc
Pn
.p
0
0
oj
'"
~
.p
.,;
>
'M .p 0
r-l
H
o:l
'l-i 0
'l-i 0
go
Q
-0. t
-O.t
ro
,.q
§ ,.q 0
0
-0.2-
-0.2.
o.t
fO
duration of cycle, to (s)
100
1000
b)
33
0.1
O. f
">
.
V)
»
a
+>
'ri
P+>
.Pc ..PlJ (1)
+> al
'ri C)
(1)
-O.t
(1)
OJ
1-1
'H
'H
0
0
(1)
§ .q
(1)
a hi)
c)
rl
-0.2
.q
C)
C)
duration of cycle, t c (s) Fig. 1.4. Difference in reaction rates,.pr.(1)'.Pn~2), and selectivities, Ss (3) in the steady- and non-stenay-state regimes vs. the oscillation period, tel of the concentration of A in the gas phase for a one-route (al, two-route (b) and three-route (c) mechanism. At small periods of oscillations the so-called sliding regime is asymptotically realized. The concentrations of the intermediate compounds are practically unaffected during the period OWing to its small time and remain constant, equalling the steady-state values at corresponding average values of the control parameters. If the oscillation periods are extensive, the so-called quasi-steady-state regime is asymptotically realized, where the surface concentrations at every moment of time take up their steady-state values, corresponding to the values of the control parameters at the same moments of time. Taking into account eqn. 1.8 and that t c --- 00 we obtain:
~ f W=-R c
2.
C
I
C A
=
2~
As the steady-state reaction rates, Re and RD (see eqn. 1.10),
34
are lower than the first-order reaction in A, the reaction rates in the quasi-steady-state regime have smaller values than in the steady-state. With increasing magnitude of the period a monotonous transition from the sliding regime to the quasi-steady-state occurs within a comparatively shorter interval of the proper dynamic regime where unsteady properties of the system (eqns. 1.6) are significantly pronounced. The dynamic regime is realized when the value of the oscillation period of the concentration of A is measured by a factor of the characteristic time MAZ of the transient regime of the intermediate substance [A~, , or by a factor of the characteristic time M BZ of the substance [BZ]. For eqns. 1.6 these characteristic times can be estimated as follows:
The reader is reminded that if another type of reaction mechanism is considered, for example, where the steady-state reaction rate has a larger order than first, i.e., in the concentration of A, the observable reaction rate in the cyclic regime is always higher than that under steady-state conditions. It would also monotonously increase with increasing of oscillation of CA period. Dynamic properties in the second reaction mechanism (II) can become more visible at greater periods in comparison with the first reaction mechanism (I) owing to the different characteristic times of the transient regimes on the catalyst surface. Now let us make the problem a little more complicated, with two parallel reactions, A + B--C and A + B-D, on the same catalytic centres, [Z], in conformity with the two-route mechanism, (I) - (II). The corresponding kinetic model with [z] + [AZ] + [BZ] = 1 will have the following dimensionless form:
35
If D is the useful product, the process selectivity is determined by the expression (1.12)
where Wc and WD, as before, are given by the expressions 1.7a and 1.7b. For a steady-state regime with time-invariant concentrations CA and CB' the kinetic characteristics will be:
S=RD jrR D +R C )
Rc = Kf K2
cAeB (
fl = Kj C
i«, + K2 CB ) + ( K_J + K'I CA )( K CA + r, + K2 CB )
B
K -.7 + Kif CA ) / jU ;
f
As is seen in Fig. 1.5, the depenednce of the steady-state rate of formation of product D, RD on the concentration of A is of an extreme character. It is almost linear if the value of CA is small. An increase in CA leads to such an increase in the concentration of the intermediate compounds, [AZ], that the content of BZ is dramatically decreased. The quantity of product D which is produced is proportional to the concentration of BZ. Fig. 1.4b showed the dependence of the differences between the rates for reactions jJ C and P D and between the selectivities Ps = S - S in both the steady- and non-steady-state regimes relative to the oscillation period of the concentration of A (see eqn. 1.8). The parameters are identical to the ones selected for the previous example. Values of W c and W D were
36
derived from the solutions of mechanisms I and II with conditions 1.8 and 1.9 by their substitution into eqn. 1.7 and integration. It is clear from Fig. 1.6 that with increasing period, t c' there is a transition from the sliding regime to the proper dynamic one. With the set of constants determined above, the characteristic times for variation of the concentration of intermediate compound AZ are smaller than that for BZ. That is why, with small values of t c' [AZ] begins to change in time following the oscillations of CA' whereas [BZJ remains practically invariable over the entire period.
---.
~~
"-
I~~
0.8
0.8
~<:r ><:
0.6
0.6
Q)
tf)
H
>. +> 0..-1
+> (Ij 1=1
0 0..-1
~
OJI
OJ!
>
•..-1
+>
+>
0
0
(Ij
OJ
H
Q)
r-I
0.2
0.2
QI
Ul
Fig. 1.5. Dependence of the rates of the chemical reactiQllS and selectivities in the stea~-state(l =_R~, 2 = RD, 3 = S'C R = 1) and non-steady-state(4 = W ,5 = s, C ~t) = 1) regimes on the average concentration ~A fBr the proce~s described by mechanism (I) - (II). The dot (6) designates the rate of formation of D(J ) at optimum control u 1(t)= CA, u 2(t) CB, in addition 1
~A - 2, ~B
= 1.
=
As a result, averaged over the period, (AZ] is decreased resulting in a dearease in the rate of formation of C, WC' and an increase in the average concentration of BZ. As long as (BZ] is little affected in the course of the period, there is a gain in the rate of formation of TI, W D, and a rise in selectiVity.
37
When the duration of the period, t c' becomes commensurable with the characteristic time of [BZ] variation, the absolute values of BZ also begin to change significantly. A further increase in t c leads to a decrease in W D and in selectivity. When t c--= a quasi-steady-state regime is realised with proper performance characteristics as compared to the steady-state operation. The values of RD and S in the steady-state regime at CA = 2 are 0.269 and 0.367. For a cyclic non-steady-state regime with identical values of average concentrations the peak value 0.344 is attained at t c = 1.1. The selectivity maximum (3), 0.464%, is attained at t c = 1.7 (see fig. 1.5). fAZ]
0.50 I=-+-~""----'I""":'-t-~~-------l
a)
duration of cycle, t/t c
[HZ] 0.75r---------------------,
b)
o
0.5"
0.75"
duration of cycle, t/t c Fig. 1.6. Change in the concentrations of surface substances [AZ] (a) and [BZ] (b) in a cyclic non-steady-state process carried out according to mechanism (I) - (II) with the dimensionless period, t c = 0.01 (1); t c = 0.7 (2); t c = 4.3 (3).
38
Thus, in this example the efficiency of the cyclic process is provided by a vast difference in the characteristic times of the surface concentration variations. The maximum period for the main reaction in this case is the result of the cc etition of the adsorbing substances to occupy as much of the catalyst surface as possible. Let us consider a three-route reaction mechanism when one of ~he reagents is completely dissolved in the catalyst volume: parallel reactions, A + B-- C and A + B - D, are carried out on the same active surface sites in accordance with the first and the second reaction mechanisms with addition of the fifth step, for example, dissolution of the adsorbed substance B in the catalyst volume: (1)
A +
[z]
(2)
B
+
[z]
(J)
B
+
[AZ]
( 4)
A
+
[BZ]
(5)
[BZ]
+
--s.,----s., ----
[V]
K{
K.J
K2
f5!-
--KJ
K- s
[AZ] [BZ] C
+
[z]
D
+
[z]
[BV]
+
(III)
[z]
Here [V] and [BV] are the concentrations of the vacant and occupied sites in the catalyst volume. A kinetic model of the mechanism with consideration of the balances [Z] + [AZ] + [BZ] = 1 and [V] + [BV] = 1, may be represented in the following dimensionless form
d[BZ] di
=
K" CB (f-[AZ]-[BZ]J - K_J[BZ] - XII CA [Bl] +
+ Ks[BZ](t-[BV]J + K_s[BV]( f -[AZ] - [BZ])
=
39
where ~ = L*/L and L* is the number of sites in the catalyst volume available for penetration of substance B. The existence of the fifth stage does not affect the kinetic characteristics of the process under steady-state conditions, which have the same appearance (see eqn. 1.13). Reaction rates (eqns. 1.7a and 1.7b) undernon-steady-stateconditions were calculated by means of integration of eqns. 1.14 with consideration of the conditions 1.8 and 1.9 and BVlt=o = BVlt=~. The selectivity, S, of the non-steady-state regime was calculated via expression 1.12. Fig. 1.4c showed the dependences of the differences between process indexes under both steady- and non-steady-state conditions on the oscillation period of the concentration of A of the sort described in (1.8) with the same values of the parameters described above. It was also assumed that K = K_ = 5 5 = 50 and c£ = 10. In Fig. 1.4b and 1.4c it is easily seen that the reaction rate and the selectivity are similar (qualitatively), depending on the cycle duration, with the exception of the case when the function (see Fig. 1.4c) was at its minimum. This is explained by the fact that the presence of the fifth stage leads to a greater process sluggishness as regards variation of [BZ]. As a result, the unsteady character in relation to the reaction rate, R, is observed at large values of the period. If the fifth stage is carried out at low rates, the exchange processes With the catalyst volume can be observed only at large periods of the gas phase oscillations. If the exchange processes are rapid enoU$h the and correspond to large values of the constants, K and K_ 5, 5 second and the third equations in system 1.14 could be replaced by the following:
+
K C [Bl] -
(1-[AZ]J2
d[AZ] d/;
40
For simplicity, we assume K be equal to K_ We can see now 5 5• that the catalyst surface sluggishness relating to [BZ] is increased by approximately 1 + ce times, the value for [AZ] remaining low. An increase in [AZ] only makes the catalyst sluggishness more pronounced. Naturally, the fifth stage does not influence either the steady-state dependences of the process or the quasi-steady-state values of Wc, W D and S, which occur if the period, t c' is long enough. That is why the asymptotic values of the corresponding process rates and selectivity at t - 0 and t _ oo concide (see Fig. 1.4b and 1.4c). The maximum value of the rate, RD, equalling 0.314 (Fig.1.4c) is attained at t c ~ 1.8 and the maximum selectivity, 0.478, at t c ~ 7.9. Some decrease in the maximum rate in comparison with the above examples (Fig. 1.4b) is explained by partial consumption of the intermediate substance [BZ] to [BV] formation in the course of the first semi-period. An increase in the selectivity maximum is provided by a greater separation of the characteristic times of the dynamics of the surface concentration variation. Let us estimate the real period values at which the maxima of the reaction rate, W D, and selectivity, S, are attained in examples I and II. The values of the rate constants for the individual stages shall be of the order that the absolute values of the conversion rates are of practical interest (k = 10 18 m2s~ C - 0.1 mPa, L = 1.4 10 19 m 2 , specific internal surface of the catalyst S = 50 m 2. g -1), then in a steady-state regime with C, = 0.2 mPa and Cn = 0.1 m Pa, one has RD = 0.55 ml/g·s, R = C Jl .u = 0.95 ml/g-s and the normalization factor, L/kC = 14 s. In this case the maxima in the rate (RD = 0.64 ml/g·s) and the selectivity in the second example are reached with periods of 15.4 and 23.8 s. respectively.
41
In the third example the possibility of dissolution of the adsorbed substance in the catalyst volume is taken into consideration. For example, we know that oxygen can readily be dissolved in both oxide and metallic catalysts. The same is true of hydrogen in metallic catalysts. The process of dissolution provides a resistance to transient regimes in catalytic systems and the dissolved substance chemically interacts with the catalyst. The catalyst energy properties may be modified and, consequently, there may be a change in the activation energies of the various catalytic reaction steps. In this part of the chapter the modifications of the catalyst properties under non-steady-state conditions, which could also lead to a large increase in efficiency, are not considered. Introduction of the dissolution stage in the present case only served to show the vast difference in sluggishness of intermediate compounds of the catalyst surface. As a result, with periods of 25.2 and 110.6 seconds, respectively, the peaks in the rate, W D, and the selectivity, S, are reached. k, C and L have identical values to those in the previous example. These considerations and quantitative examples allow one to express a general idea. Let two parallel reactions be carried out on different catalytic centres and each reaction occur according to the two-step mechanism. Then, the rate of each reaction is changed in a monotonous manner with increasing period of oscillation of the reagent concentration. In addition, considerable variations in the reaction rate in the dynamic regime occur as soon as the characteristic time of the given reaction route dynamics gets shorter. In the present case the characteristic time is determined by the time of the transient regime measured for the intermediate substance of this reaction route. The selectivity of a reaction route, the dynamics of which may be determined by a more complex dependence, should, evidently, have extreme values in the following cases: (1) Both reactions have an observable order less than first relative to the controlled concentration. The characteristic time of the dynamics of the reaction leading to the end product is greater than that of the concurrent reaction route. This
42
example of a simple reaction mechanism has already been discussed. (2) Both reactions have orders higher than first relative to the controlled concentration. The characteristic time of the dynamics of the main reaction route is smaller than that of the concurrent one. This case corresponds to one described in ref. 44 where, in addition, some simplifying assumptions were made. The reaction rates under the forced non-steady-state conditions in both cases are of monotonic behaviour in the course of period variation, for example, as reported in (ref. 45). The maxima in the reaction rates may appear when there is a competition between the surface substances participating in different reaction routes for the active centres. This phenomenon occurs in the above examples I - II and III. The scheme of stages I - II may probably be applied to many catalytic reactions. Isomerization, partial oxidation, fermentation reactions and other catalysed reactions can be carried out according to this (or a similar) scheme. Of course, the quantitative character is meant rather than the scheme's total quantitative correspondence. To make these schemes applicable to processes of partial oxidation one can interpret them as follows. If A is an hydrocarbon and B is oxygen, then adsorption of the hydrocarbon leads to the formation of oxidation products. The collision reaction mechanism leads to partial oxidation of olefin (en H 2n ) carried out in conformity with the following stages (1)
( 2)
A +
[zJ ~ [AZJ
== [JZ]
°2
+
[AZJ
0) °2
+
[z] ~ [02ZJ AO
( 4)
A +
(5)
(1.5n - 1)02
(6)
2 [ozJ...f\- [02z J
[ °2 zJ +
[JzJ +
[ozJ
+
J'L.
[zJ
nC02
+
nH20
+
[z]
43
where A is olefin, [JZ] is an intermediate compound easily transformed on the catalyst surface during the fifth reaction stage, AO is the olefin oxide and the sign --fI- designates a rapid irreversible reaction. The kinetic model for this reaction mechanism coincides with the model discussed above within the accuracy of the stoichiometric coefficients. The kinetic model for the mechanism (1)
A
+
[z]
(2)
B
+
[B]
(3)
A +
--
[AZ] [BZ]
[BZ] -
D +
[Z]
in which a portion of the active surface is occupied by the adsorbed and inert (in the sense of interaction) substance, A, is also identical to the above equations, where the concentration of B is constant. Considerations of the qualitative character discussed here may be applied to a great many multi-route catalytic reaction mechanisms including non-linear ones. For example, in reactions which are carried out according to the two-stage schemes
[z]
[AZ]
(1)
A +
(2)
[AZ] -
(3)
B2
+ [A,Z] -
( 4)
B
+
(5)
A +
[BZ] -
(1)
A
[Z] ~ [A 1Z]
( 2)
[AZ]
2
~
[A,Z] C + [z]
(IV)
2[ZJ ~ 2[BZ]
D + [z]
and +
+
[BZ] -
D +
2[ZJ
(V)
44
0)
B2
+
2[Z]:;:'= 2[BZJ (V)
an extremal dependence of the rate of formation of D on the period of oscillation of the concentration of A in the cyclic non-steady-state regime is observed. This all can only take place in case of large sluggishness of the reaction route toward substance D formation.
1.4.2
Statement of the optimization problem and ways to solve it
Fig. 1.7 shows calculational results for example I-II where one can distinguish three regions of the cyclic process. In the first region (the so-called sliding regime) the duration of the oscillation period, t c' is so short that it is less than the time-scale of the transition process on the catalyst surface: t c « Mc• In Fig. 1.7 this region corresponds to plane SOA and to some small value in the vicinity of t > O. Due to the linear c dependence of the rates of the catalytic steps I - II on the concentration CA at short duration of the cycle, the situation is a little different from that in the steady-state reaction, corresponding to point SS. As a general case, profile 1, of course, may show some other pattern of behaviour, provided the dependences of the rates of the reaction stages on the control are non-linear. The region of small values of t c discussed here is called the region of sliding regimes (ref. 46). In another extreme case (profile 3) the duration of the period of variation of the concentration of the gas phase is far greater than the characteristic time for the surface transient regime, i.e., t c » Mc' The dynamic properties of the system here do not have any effect on the process characteristics and the observable rates. of chemical conversion are determined by the kinetic models describing the steady-state process, which is dependent on the gas phase concentration and temperature at each moment of time. This is the so-called quasi-steady-state proces&
45
s 55
2
a
Fig. 1.7. Change of selectivity, S, of the catalytic process I - II depending on duration of the period, t , and amplitude, a, of oscillation of the concentration of comBonent A in the gas phase with fixed average values of the concentrations of A and B. 1 = the sliding regims; 2 = steady-state regimes; 3 = quasi-steady-state cyclic regimes. A geometric approach (ref. 47) has often proved illustrative and useful for the analysis of extreme cases. Many process indices for the steady, quasi-steady and sliding regimes can be constructed. A quasi-steady-state process cannot be efficient lest there are no limitations on some average process characteristics. If this is the case, one may consider steady-state operation at u = constant as the optimum, which provides the maximum value of some criterion, J. An efficiency gain in the sliding regime compared to the steady catalyst state can be achieved only if" the reaction stage rates are non-linearly dependent on the concentration of the gas phase or if criterion J is non-linearly dependent on some other process parameter. If t c ~ Mc , then as has already been discussed, the dynamic properties of the system considerably affect the non-steady-state process characteristics. Therefore, for purely extreme cases, the question concerning the efficiency of the non-steady-state process still requires. an answer.
46
Now we have come closer to the mathematical statement of the problem of finding the optimum cyclic regime with the best performance characteristics relative to any criterion of estimation. We are, certainly, in the first place interested in improved cycle-averaged indices, like, for example, the selectivity of the process. One can divide the problems of the cyclic optimization into two classes: (1) Problems concerned with the search for optimum periodic control providing such a cyclic regime in which the optimized process characteristics, for example, productivity or selectivity, reach their extreme values. (2) The same problems but with addition of limits of the process parameters within the period, i.e., the so-called limitations of the mean value, e.g., of the control parameters. In this kind of problem, for example, mean values of the concentration or temperature of the gas phase in the course of the cycle may be set. All problems of periodic control discussed in the literature related to this field belong to these classes. After having had a good look at the example of an efficient cyclic process in the previous part of this chapter, it is easier now to formulate a general statement of the problem of the cyclic optimization for processes described by a system of common differential equations. Let us take as an example the process described by such a system of equations as
The conditions pertaining to cyclic character are (1 .1Ga)
or
tc
~ h [B(t), u(t>] dt = o
0
(1.1Gb)
47
where ll,(tJ =. crt) = [Ut (t ), Uz (b), ... , um, (b) J is the reagent concentrations vector in the gas phase which serves as the control or, u (b) EVE R rn. the incremental constant functions determined for [0, t c]; e(t) = [e 1 ( t ) , e 2 ( t ) , ••• ,e n ( t )] is the vector of the concentration of the intermediate substance on the catalyst surface or B(t) ERn variable states continuous for [0, tcl; t c is the period. The process characteristics are designated as tc
!I =J... \' 5fJ,,[Brt),u(bY]dt, If., te J
1£=1,2, ... ,e
f10
o
where ~~ is the process characteristics at any moment of time. The efficiency function to be maximized is
J
=
9 (y> o
max
==?
(1.18)
iI(6),t c
y= [!Jt' 'Iz' ... , Ve J The average process characteristic, y, and also the function, J, can represent the observable rates of chemical transformation or/and the selectivity. As a rule, there is a limitation on the minimum and maximum values of the control, u(t), like u. m i Tb ~
if
{. Uj "uJ (b) "
ma:x
, J.
=
1" 2 ... ,
77U
In the second class of problems, the mean-integral limitations in addition are set as
fli ( V-)
=
0,
i.
- f», 2 , .. r
e1
tt < t
Thus, the problems of cyclic optimization (see eqns. 1.15-1.20) involve determination of ur:», li*(t), t 0* such that
48
It follows that the absence of information about the duration of the period as well as about the initial and final states of the system and the presence of mean-integral limitations which occur during the period, all add to this class of prob~ems. It may be on interest to juxtapose the solution of the problem of the non-steady-state cyclic ~rocess optimization and the solution of the problem of the steady-state process optimization. Let us write the relationship for the steady-state case which follows from eqns. 1.15-1.20 under conditions that the state vector, e(t), and the control vector are constant: /(8,u)=0
g(ij)
=
0
J = 9o(ijJ ~ mq-x u.
u-E:V , e-E-RTIFor many problems the set V can degenerate to a point. It should be noted that·if there is an optimum, J = J un, in the steady-state regime and J = JS the following condition (1
.25)
is always fulfilled. The term "efficient" will be applied to the non-steady-state process where Jun is greater than JS (ref. 48). There is a direct way to estimate the efficiency of the periodic regime, e.g., through solution of the cyclic optimization problem. However, this can be difficult in many cases. That is why it is thought to be of interest to evaluate the efficiency of the non-steady-state process (or at least obtain an answer if the inequality, Jun > JS is really valid) without resorting to a search for the optimum cyclic regime described in eqns 1.15-1.20.
49
Various methods including analytical ones which efficiency of the non-steady-state process is to be evaluated are discussed in the literature. Let us briefly go through these methods. The solution of the problem of cyclic regime optimization, in analogy to the problems of optimum control, should satisfy necessary optimum conditions. The well known Pontryagin's principle of maximum (refs. 49, 50) for the said problem can be formulated by introduction of Hamiltonian function H [Brt), u(tJ, lj/(tJ,
J:]
v
=
'(t)j(tJ
+
i'ij(t)
(1.26)
Here iji (t) are the conjugated variables, sP (t) is the vector function consisting of functions of the mean indexes from expression 1.17 and A is a vector of real numbers. The principle of the maximum is: let [e*(t), ii*(t)], 0< t~tc be the solution of the cyclic optimization problem (see eqns. 1.15-1.20). Then, there exist a continuous vector function, yr(tJ: [0, cc] R '", which is differentiated at the left on [0, t c]' .it E Rtf and real numbers, do"'" Q t : so that the following conditiona for all t E [0, t c ] are fulfilled
max H[1ktJ, ire V
»to. 7jdtJ, iJ
=
[B "(t),
U
*(t), 7jr(t),
J:]
e
.x' = ?= d i 'l J V) L ~D
(1.28)
50
Expression 1.28 describes conditions for the conjugated variables and expression 1.29 describes the transversality condition. The principle ofm~imum allows only for analytical solution of the problems posed by the optimization of the catalyst unsteady state as a rare case. Sometimes, one can have vector, 9*, which is the solution of problem 1.21-1.24, which is not in accord with the necessary conditions of the optimum regime. This means that Jun> JS (ref. 51). Most often the required optimum conditions only allow one to give a qualitative description of the optimum solution or/and to build up numerical algorithms of the optimization. It is expedient to make use of methods based on the analysis of extreme cases and to formulate some sufficient conditions for the efficiency of the periodic regimes. It is sufficient to analyse the behaviour of the system at both very large and very small values of the period (compared to the characteristic time of the system) which, as nreviously discussed, correspond to the quasi-steady-state and sliding regimes. In the quasi-steady-state regime the system will satisfy to eqn. 1.21 due to the extensiveness of the cycle, provided 0< t < t c• For steady-states assumed as unique, the value of the control, u(t), defines the state
where Bs t is the solution of eqns. 1.21. If we substitute expression 1.30 into sub-integral function 1.17, the non-linear programming problem in the mean (ref. 48) can be obtained. This problem can eaRily be reduced to a common problem of non-linear programming. Notice that any vector which is the index of the averaged values attained in a quasi-steady-state re~ime can be realized by means of incrementAl constant control consisting of not more than 1 + 1 increments, where 1 is the dimensionality of the process index vector. The main theorem for the forced cyclic regimes having dynamics approximating to these of the quasi-steady-state operation is formulated as follows: for any quasi-steady-state process operated by a set of admissible values, V, the resulting efficiency vector is the locking element of the convex
51
curve Ss(V), i.e.,it belongs to clcoSs(V), where Ss may designate, for example, the reaction rate in the steady-state regime. As long as the vector index of any quasi-steady-state periodic control belongs to clcoSs(V), it follows from Caratheodory's theorem in his convex set theory that the reaction control is of incremental constant character. Another extreme case of the cyclic regime is a sliding regime (refs. 52, 53). One has to bear in mind two peculiarities of this regime: (a) the period of oscillation is considerably shorter than the characteristic time of the system's transient process and (b) optimum control can always be realized with the help of n + 1 + 1 switches between constant values, where n is the dimensionality of the state vector and 1 is the dimensionality of the vector index. Under certain circumstances, there is a more restricted limitation to the number of switches. Consequently, one can believe that the state of the variables is constant and satisfies the system of differential equations 1.15 in the mean
t:
tc
~ Ii [ 8.st .tut)J dt
=
0,
i = /, 2, ... >
7U
o The time-invariant composition of the intermediate compounds on the catalyst surface, 9s1 ' in the sliding regime is subjected to quick control-varied action of the gas phase composition. That is why the process performAnce indexes may appear significantly different from those of the steady-state regime, because the control a!fpcts the system in a non-linear manner. It is possible that the solution of the steady-state problem can satisfy the principle of maximum, since it only serves as a necessary condition of the optimum, though in reality, transition to an non-steady-state operation seems expedient. The sliding regime efficiency test has, therefore, more convincing power than the test of the principle of maximum.
52
To analyse the forced non-steady-state regimes under conditions when the dynamic properties of the catalyst surface play an important role t one should use the X-criterion (refs. 51, 54, 55) based on the analysis of the efficiency function with small sinusoidal variations of the control about its steady-state value. The eXistence of optimum steady-state control is assumed to be the inner point of the set of admissible controls. In this case, the first variation of the quality criterion (1.18) vanishes and the second variation of the efficiency function in the vicinity of optimum steady-state control is investigated. One can derive the process variables under steady-state conditions with u(t) = s t = constant from the system of equations (1.21). If there is only one solution, the value of criterion (1.24) is unambiguously determined.
u
At small sinusoidal oscillations of the controlling parameters art)
=
st + Re ;X cos wt - Jm
U
~ SiTU
wt
(1.32)
where ;t E em, lI;d c m <6, [, is a small positive number, m is the dimensionality of the control and W = 2%/tc • The quality criterion is determined as accurately as small values of the fifth order by the following expression
where the asterisk designates the conjugate and n.» TTl" the Hermitian matrix; ITriw) is evaluated as follows. The Hamiltonian function of the system's steady-state conditions is introduced
H Of, iI, where
v . J) =
1jJ
7j/'[ + I.'
is derived by setting
ESc
= O. Then,
nr/ w ) = G *r/w)pf.i(jw) +Q'r;(/w) +R
( 1 .34)
where
C(/w) =(/w/-A)-IB ; A =
Ie (if,
P=j
e,
B
u. (
(1);
P
=
H eo
(15, ii , 1Jf),
=
iZ)
H s« ( 8, U, 1Jf)
53
It is assumed that u does not lie on the boundary of the determination region of (1.19) and that matrix A's proper numbers do not have zero real figures. One can apply the so-called X -test to various frequencies, thus obtaining data on its controlling efficiency. The use of this criterion allows one to establish an effective phase shift between the components of the reaction control. With this method taken as the foundation in ref. 56, an approach was suggested which sometimes gives the possibility of obtaining an analytical' expression for the corresponding value of the efficiency improvement in various types of periodic reaction control. A specific control test (ref. 57) represents a particular case of the x -criterion. Its application is thought to be expedient in the cases when the optimum steady-state control is of a specific type and permits a considerable reduction in mathematical computation. Let Uo be the vector of constant optimum controls of a specific type and lie within the boundaries of the region of admissible values. Then the following theorem is valid: if at least one of the two following conditions satisfy some k value of 0, 1, ~, ~tc., the optimum cyclic regime is efficient: (1) ~ = 0 for m = 0.1, ••• , 2k - 1 and there exists such ~ that (- t) Ii S T If2/r, S > D
(2)
~
=0
for m
= 0.1,
, 2k
-
And R2k + 1 I: 0, where
aH(ao,U,7j/ -0-0 ,
ou
It is clear that in this case the calculAtion of ~1' Ro' R2 up to their first non-zero member (base) is preferable than direct calculation of the matrix, octca); The small-value parameter technique (refs. 58, 59) and the conditions of non-steady-state optimum control (ref. 60) also allow one to come to a certain
54
conclusion about the efficiency of the transition to the non-steady-state regime. An important note must be made concerning research based on the construction of class sets of the solutions to periodic control problems and also the relationship between these classes in the form of necessary and sufficient conditions (refs. 48, 50). One can discard the existence of a large class of cyclic optimization problems which offer no advantage over the optimum steady-state regime.
The estimation of the optimum conditions for non-steady-state regimes so important for general efficiency evaluation of the non-steady-state process has proved of little use ;n respect of determination of the optimum control law as well as for construction of numerical algorithms. The most promising method seems to be that of direct calculation. One can distinguish three main calculational problems, posed by the solution of the optimum non-steady-state regime problem: (1) computation of the periodic regime with a prescribed value of the duration of the cycle and form of control action; (2) determination of the optimum form of the control action within the prescribed duration of the period; (3) determination of the optimum frequency of the control action. It is advisable to solve problems 2 and 3 simultaneously to ensure a considerable increase in accuracy. The use of numerical methods in solution of the cyclic optimization problem has two peculiarities, which are accounted for as already discussed by the existence of the periodic boundary conditions: neither the initial nor the final state of the system, nor the optimum duration of the period are known. The second peculiarity arises during consideration of various integral limitations on the mean process indices. At present, there is no general method of solving cyclic optimization problems. All algorithms used for that purpose are based on classic understanding of the function variation and of the modified maximum principle. The gradient method, which is based upon variational calculus, is believed to be the most general and reasonable. The method was discussed previously
55
(ref. 61) and is as follows: fixing the period of duration, t , c one numerically determines the corresponding optimum control. Another period duration is then asssigned and its corresponding optimum control is again found. After that the values of the efficiency functions are compared and the optimum period is calculated directly. This approach requires much computer time. Ref. 62 reports another numerical algorithm which is free from the use of the cyclic conditions described in eqn. 1.16. The optimum control was determined over a large section of the time-scale with arbitrary initial conditions. The optimum control and variable states found were similar to periodic functions at the right-hand side of the time interval. One can try to apply the conjugated gradients method to the search for the optimum cyclic process. To obtain sliding and quasi-steady-state regimes one can make use of the known methods of non-linear programming. However, one must be aware of the fact that, in general, the solution of the boundary periodic problem represents a serious difficulty. Its requirements in every particular case of the cyclic regime computation (if it is ever possible) are immense. Before commencing algorithmic calculation of the optimum cyclic regime, it is of practical use to do a preliminary analysis of the problem, which may permit prediction of the form of the optimum control which may be, for example, constant or incremental constant. Kinetic models of heterogeneous catalytic reactions are, as a rule, linearly dependent on the gas concentration, i.e., differentiation of expression 1.26 on u(t) will show that ~ = HI)" [8(6), 1jF(6J, l]. In case of this reaction, the controls which satisfy the necessary conditions of the optimum (eqn. 1.27) take up either their boundary values (eqn. 1.19) as a function of the above mentioned ~ or can be derived from the condition described in ref. 63.
Hu =D;
!fJ = 1,2, ...
This condition describes the so-called specific control and it can be fulfilled in accordance with all components of the
56
vector, u(t). If system 1.15 is bilinear,which is often the case with linear mechanisms of catalysed reactions, one can see that specific control determined by the said condition with limitations (eqn. 1.20) and with m = n always has a steady-state character: u(t) Steady-state reaction control may not satisfy this condition if the kinetic equations are non-linear with respect to the surface concentrations. Therefore, this control cannot be optimal.
= u.
we Rhall further consider an example of the efficient numerical algorithm for optimum regime calculation where the character of the optimum control and its period are determined simultaneously (refs. 64, 65). To generalize its formulae, let new variable states be introduced 8.{I;);J'[u, {t)]=u.~-n (t) ~ ~ ~-TU
:». i~n+1,TU+2"",TU+m j-'~-7b' (1.36)
where Pi-ru are constants. Note that eqns. 1.35 and 1.36 are identical to the mean-integral limitations set in 1.20b. Let the new variable, 't = t/t c' also be introduced. Then expressions 1.15 - 1.17 and 1.35 - 1.36 shall take the following forms:
(1.38) f
IIlit
=
~
tc
= "
2, ... ,
e
o
Let l5 J be the change in efficiency function with variation of the control, oliJ-c) and of the period, o'te . I t is possible to show that with the use of linearization of expressions 1.18 and 1.37 - 1.39 in the form
57
O.l·
+ -j-~-
otc
0&'{, (0) "" 00.· (t);
c
i =- /,2, ... , ru +
~
!'u=1,2, ...
06
i
~ 1,2, ... ,
n
+ n&
7Tl;1
,t
and with the use of the earlier introduced Hamiltonian function
that the change in efficiency function in the linear approximation can be determined by the following expression f
s: =
f.. J /=1
\'
o
iJH A
ou:J
tfU , da: + J
(1.40)
In this expression oBi represents the variations of state variables corresponding to the cyclic conditions of the process and to variations, ~Ui and obc ; IJillv is the variation of the process indices; Y'"i are conjugated variables which can be
58
determined as follows
rYi (7:)
=
oH
of}· , z
i
=
f, n.«
TTL t
(1. 42)
Thus, to make 0" J positive, which corresponds to an increase in the efficiency function, it is sufficient for the following expression to be valid
and variations small enough for the linear character in expression 1.40 to be observed. It follows from eqns. 1.33, 1.41 and 1.42 that (with mf
i n + 1, n + m) , .li(·); constant. Hence, L !Fili =D and instead eqn, 1.44 we can write: i~n-+t
''1n,( ote ) ~ siU"
[
S t, r, Ii d~
]
(1. 45)
a
Let us now formulate the algorithm for simultaneous search of the optimum control and period. Step one. Arbitrary control parameters, uj, satisfying eqns. 1.19 and 1.20 and an arbitrary period, t~, are assigned. Step two. With the assigned control parameter, u~, and period, t~, where k is the iteration number, the system of eqns. 1.37 is solved at i = 1, 2, ••• , n and with the cyclic conditions
59
described in eqn. 1.38 taken into account. To fulfil these conditions, the Newton-Raphson method can be applied (ref. 61). After calculation of the functions 9~(~) (i = 1, 2, ••• , n), the value of the efficiency function, J, is found with the help of expressions 1.18 and 1.39. Then, in time-reserved order, the conjugated system (eqn. 1.41) with i = 1, 2, ••• , n + 2 is integrated beginning with the arbitrary initial conditions. Complete solution of !f/i!f,r satisfying the conditions of eqn, 1.42 is reached via both the given particular and fundamental solutions obtained earlier by means of the Newton-Raphson method. o
)
Step three. Arbitrary values of assigned.
1.)/,
i
n + 1, n + mare
Step four. New controlling functions are calculated according to the formula
(1. 46)
e; > 0
where
(ft+t)( )
Uj
.
>
is a vector of a small positive numbers. If
u/ max.
then
uJ
Iv+!} (7: = U
J
max-.
It+f()
,J.f u/
7:
min-
J
then
mirv
1';,+1
u; (7:) = Uri . In other words, if the control determined by expression 1.46 at some moment of time is found to be beyond the limit, it takes up its boundary value. Step five. Fulfilment of conditions 1.20 is checked. In case it is violated, 1JI"i (i = n + 1, n + m1) is changed so as to redress this violation. So, we can represent it as solution of m incremental linear algebraic equations in relation to lJ'i' Step six. A new value of the period is calculated using the formula 1
te/t+1 =: t/'+
0T
~ ~ lJI"i Fe
d-t:
Q
where
(;r
>o.
Step two is then repeated. Values of
6T
and a
60
e
fragment of vector (eqn. 1.46) must be small enough to ensure accuracy of expression 1.40. The iteration process continues until no growth of the efficiency function is observed.
1.4.3
Some examples of optimum performance of processes
Some studies have considered on a theoretical basis the problems of finding optimum controlling parameters, making use of a selection of catalysed reaction mechanisms as models. These studies help to distinguish classes of chemical reactions allowing for an efficiency increase under non-steady-state conditions. Let us probe deeper into the non-steady-state process on catalyst surface. With periodic variation of the gas phase condition, the period-averaged reaction rate, t c' is described by an expression similar to eqn. 1.17 Cc
W=.i. t \'j W [8ft), e
TfO, crt)] dt
o
where B(t), as before, is the vector of the concentration of the intermediate substance on the catalyst surface; T(t) is the temperature; c(t) is the gas concentration vector. All parameters of the gas phase, c and T, (or a part of the parameters)may serve as controls, u(t). In the case where there are two pArallel reactions andW 1 is the average rate of the end product formation and 2 is the rate of formation of the side product, the selectivity of the non-steady-state process, as it has already been shown in this chapter (see eqn. 1.12), will be determined by the following expression:
W
It is problematic whether it is possible to attain an higher
61
rate, W, or selectivity, s, in a periodic non-steady-state regime at u = u(t) than the corresponding values of the rate, R1, and selectivity, s, in the steady-state regime with u = ~
f
fi
S
u(tJd6 = con.stanc . Let the controlling parameter oscillate c 0 with a larger period than all the characteristic times of the initial system. The behaviour of the intermediate substances on the catalyst surface may be then regarded as quasi-steady. In this case, instead of expression 1.47,one can write the following: = t
(;c
vv, '>3:c ~ u, [ tto, crt) dt o
--'
From eqn. 1.49 W1
> R1 (u av ) if
where u av tc
corresponds to the average temperature, Tav = (;c
and to the average concentration, C
all
=
f
t
c
~ ~ T(b)dt 0
j crt) cit.
\'
Since the
o
mean period selectivity involves integrals, Sis greater than S if the following inequality is fulfilled:
Now let the controlling parameters oscillate with a much smaller period than those of all the characteristic times of the initial system. Under these circumstances, the catalyst operates in a sliding regime where the concentrations of all intermediate substances, 9(t), on its surface remain unchanged during the whole period and are equal to 9 av = constant. Values 9 av are derived from expressions 1.31. Since the characteristics of the sliding regime will differ from the corresponding characteristics of the steady-state regime only in the case of a non-linear dependence of f 1 on n, periodic control of the gas phase composition or pressure at low values of the period of oscillation is often unable to alter the process efficiency
62
indices. Still, the sliding regime may become efficient when it is temperature-controlled. Let us now look back at the example discussed in section 1.4.1. This example shows the possibility in some classes of reaction mechanisms with intermediate values of the period of variation of the gas phase condition to accomplish an higher observable rate of conversion and selectivity than in the quasi-steady-state or sliding regimes. Let us try to evaluate the process efficiency with the help of the X -criterion, taking as an example the qualitative characteristics of the non-steadystate process described by the system of eqns. 1.11, namely, cyclically-averaged side-product formation rates, W (see eqn. c 1.17), end product formation rate, WD (see eqn. 1.18b),and selectivity, (see eqn. 1.12). The system of equations 1.11 is bilinear. Hence, according to eqn, 1• .34, P = 0 and R = o. Consequently: roJ
s
flriw)=/lr;(iw)+G*(/w)/l= [n 11 n f2
n n 21
]
22
In addition, n 12 and n 21 are complex conjugates. In accordance with eqn. 1.32 and without any loss in general character, oscillations of the control for this problem will be described by
where A1 and A2 are small enough oscillation amplitudes of the concentration of A, U1, and of B, U2; ff is the phase shift between these oscillations; 1 , are the mean values of the concentrations CA and CB during the cycle.
u u2
According to eqns , 1• .32 - 1.33 and expressions for
nr/co) and
u(t):
If 02 J is expressed as a function of the oscillation frequency, then
63
1J2J =- At <:£, + Ai <:£2 + AfA 2 0(;.3 sin (J + wAfA z (p-W 2)2 +Jl2
2
co 2
+ w (A ;!!. a 5 +Ajo(;6 +A 1A z
d 7 siTL
QI(
0) + w
r./U-w2) 2 +Jl 2C0 2
cos 0 +
5A
1 A z 0(;8 cos
If
where di ' i = 1.8 are rather unwieldy composites of kinetic 2 jt 2 . cons t ants an d ~ u 1, -u 2; jU = f 8t t f 82 82/ 81 JoS a product of proper numbers of system 1.11; I = k 1u1 + k 2 + kJU2 + k4U2 + + k + k 6u1• If CB = u 2 = constant and CA = u1 + A1coswt is the 5 only controlling parameter, then a +8w 2 6' }(w) -At {jU_(2)2 +cY 2 W 2
_
2
where a = x
-ZRIJ
[k f { k_ 3
2
{(k 3 cB + k _,3 )( !£1 S4 + !£-/-j-!£2 CB ) x
-t- k~ CA)
+
kL((Jf,,~ + tc., +!f,z
C B)]
k/ ~ [/1,,,/c.q
CA CB +
Rn is determined by expression 1.13. The period-averaged rate of the reaction in the non-steadYstate regime is:
WD
=;
R
D
-j-
6'](w)
I t is higher than in the steady-state regime if tf 2 Jtco) > o.
In the quasi-steady-state regime when w
= 0:
64
In the sliding regime
tim
uJ-=
0 2 J(W) =0
The latter inequality is valid i f the following conditions are fulfilled (1) a>O,
8>0
In the first case 13JU 2 - at + 2 aJU > 0, and in the third case the function 02J(W) will always reach its peak at the frequency:
Fig. 1.8 shows the dependences of the second variations, J1 - W C ' J 2 -Wn and J 3 - S, on the period, tc' and on the oscillation amplitudes, A1 and A2, with K1 = K_1 = 3; K2 = K) =
K4 = 1; K_ 3 = 0.3; CA = u1 = 2; CB = U2 = 1 and various phase shifts, 0, in the case of joint and asynchronous oscillations of u 1(t) and u 2(t). The effect would be expected to be most pronounced when the characteristic times of the transient regimes of the intermediate compounds, [AZ] and [BZJ, on the catalyst surface are considerably different. The above data, certainly, indicate that the largest efficiency gain can be reached in the case of joint oscillations of the controlling parameters. For example, if the period t c = 1 and A1 = A2 = -2 ff=0.83x 410, and tf2J1 =0.1357-10-,meanwhJ.leatA 2=O -2 tf2J1 = 0.238-10 -4 and at A = 0 and 11. = 10 -2 , ann A1 = 10, 1 2 02J1 = 0.0144-10- 4 •
65
It can be concluded from expression 1.50 that the second variation of the quality criterion has a sinusoidal dependence on the phase shift when the period is fixed. The maximum is reached when tg 0 = -Jm n f2 IRe n,2 • Fig. 1.9 illustrates the dependence of the optimum phase shift, 00, of the concentration oscillation and the maximum of the function, J 1, i.e., the maximum of the rate of formation of substance D as a function of the period, t c' The optimum phase shift, 00, for function J 2 increases monotonically from -0.5% to 0 in parallel with the increase in the period. For function J the phase shift 3, monotonically increases from 0.5 to x.
1\ 0.02
a)
100 -0.02
-o.a«
:5
r
0.1 fOO
-0.2
-0-'" -0.6
tc
b)
66
It
oJ., • io« 5"
0.2
2
0.1
c)
f
~
fOO
tc
-0.1
Fig. 1.8. Dependences of the period of second variations of the rate of formation of substance D,02J 1 (a), of substance C,02J 2 (b), and the selectivity, 02J:3 (c), at (1) A1 = 10- 2, A = 0; 2 (2) A1 = 0, A2 = 10- 2; (3) A1 = A2 = 10- 2, d = 0; (4) A1 = A2 = = 10- 2, If = 0.83 ; (5) A1 = A2 = 10-2, If = 1.33
O.Of
duration of cycle
Fig. 1.9. Dependence of the optimum (for lf2 J 1 ) phase shift, 00, on the period of oscillation of the gas phase components A and B, t c •
67
Let us strictly formulate the problem of the optimum control determination for system (1.11) with cyclic conditions as described in eqn. 1.16a : 9 i ( 0 ) = 9 i ( t c ) ' i = 1,2; in additio~ 9 '" [AZ], 9 =. [BZ]. The efficiency function, J 1-=* max must be t 2 maximized with the following limitations tc
:c
~
~ u., (n~ u. m a x [ul(t)-Ui]dt=O; U.min., z ~ z '
o
where u 1 '" CA' u 2 == CB• According to the required optimum conditions and if CB = constant and value of the period is fixed, either incremental constant control with boundary-set values of concentration or special, in other terms, steady-state control can be regarded as the optimum. Thus, it appears that the solution of the problem shown in Fig. 1.5 corresponds to the optimum: curve RD represents the rate of formation of product D in the steady-state regime with the concentration of component A in the gas phase equalling CA. Curve WD represents the rate of formation of D under the optimum non-steady-state conditions with the same values of the constants. u 1m i n and u:l: max, correspond to 0 and 4, respectively, and u 2 ( t ) = 1. If u1
In the case when both the concentrations of substances A and B are the controls, the problem is solved analytically via the known algorithm and the optimum control profile and the optimum period value should be simultaneously derived. With 0 ~ u 1 <; 4, o ~ u 2 ~ 2, 1 = 2, = 1 (values of constants unchanged) an incremental constant control must be regarded as optimal. With the period, t c' about 1.01 and the phase shift,5= 0.77x , it completely satisfies the optimum condition. The optimum values of the period and the phase shift accurately coincided with the optimum values obtained with the use of the x-criterion. With and = 1, in the steady-state regime, J 1 = 0.296, while 1 =
u
u
2
u2
u2
68
with u 2 = constant and under the optimum control, u 1(t), J = 0.344 and with the optimum control, u 1(t) and u 2(t), J = 0.424. I shall now illustrate both the method of limited cases and the geometric approach. For example, the initial component, A, is converted via a catalytic process into products C and D (ref& 44, 47) in accord with the following mechanism: A +
[z]
A +
[A*] -
~ A 11-
C + [Z]
(VI)
D+[ZJ
A -
The process is assumed to be carried out in the region where the adsorption rate of A does not depend on the concentration of the adsorbed substance, [A*]. Further, the first two reactions of mechanism VI represent elementary steps, and the third reflects the complex observable reaction of second order. We shall consider the situation Where all intermediate substances resulting in the formation of product D have small times of transition in comparison with those resulting in the formation of substance [A*J. Let the concentration of A, CA, be the control parameter u, and concentration of the adsorbed component, [A*] , be the control parameter e. Then one can derive the following equation for the material balance
Expressions designating the rates of formation of products C and D take the forms tc
~ W = -f ~ c tc
'It2 uBdt
a
to
~ W
IJ
= -
1~
tc
a
11,% u
"
2
dt
69
The limitations of the variation of concentration should obey . al ~. t y, u rru.n. '" ./ /' . the a.nequ u ( t ) '" u max• Prof~le 1 in Fig;. 1.10 shows the re,:ationship between the rates of formation of C 'W ' and 1 of D ,W2, in the steady-state regime when:
In the quasi-steady-state regime with a large period of variation of the concentrations and with the region of variation of the concentration of substance A within the range u min<, 0 and u ma.x; the relationship between the rates of formation of C and D lies in the region enclosed by profiles 2 and 3. If D is the reaction end product, a selectivity increase will be provided in the quasi-steady-state regime owing to a stronger non-linear character of th~ dependence of the rate of formation of D (W2 ) compared to C (W1). As the period is decreased (especially in the sliding regime), the selectivity relative to product D increases. The reason may be that the rate of formation of product C is decreased and at the extremum (as in the sliding regime) it would tend to the rate corresponding to that in the steady-state regime with the concentration of A(u) determined by the mean-integral value of the period. Behaviour of this kind can be explained by the inertial properties of the catalyst surface, provided by the adsorptive and catalytic steps of the reaction mechanism VI. Owing to the lack of inertia of the third step, the rate of formation of the desired product, D, will still be described by a square-law dependence on the concentration of substance A. The rate of the first reaction in the sliding regime is totally identical to that in the steady-state regime and the rate, W 2 ' is identical to that in the quasi-steady-state regime. Profile 3 in Fig. 1.10 displays the relationship between the rates, W 2 ' in the sliding 1 and W regime. It is of interest that a regime may appear almost quasi-steady-state if the values of the period of concentration variation in the gas phase are approximately 100 times as large as the characteristic time of the transient regime of the system. On the other hand, a regime may become almost sliding if
70
the oscillation period appears to be equal or smaller than this characteristic time.
2 1
Fig. 1.10. Selectivity of the process described by mechanism VI in the steady-state (1), quasi-steady-state (2) and. sliding ()) regimes. (k 1 = k_ 1 = k 2 = 10; k) = 5; u m== 1; u mm= 0). Mechanism VI showed the formation of the reaction product, C, to be the result of the interaction of the intermediate substance, [A*] , with molecules of the initial components, A, from the gas phase via the collision mechanism. Scheme VII shows the formation of product C directly from product [A*] (1)
A + [ZJ~[A*J
(2)
[A*J~ C
0)
A -- D
+
[z]
(VII)
Such a difference in reaction mechanisms opens up a possibility to attain a selectivity increase toward product C with the same parameters, which prov.ide, according to mechanism VI, a selectivity decrease (ref. 44). In the first place it can be referred to the value of the observable period of formation of
71
the side product, D. The smaller the order of reaction 3 in mechanism VII (A-D), the larger is the gain in selectivity. An increase in the selectivity relative to product, P2' can also be attained for reactions which are carried out according to the scheme (ref. 44): (1)
A + [zJ~ [A*]
( 2)
A + [A*J~ c
0) A ( 4)
+
[zJ (VIII)
+ [Zj::::[A**J
A + [A**J --- D +
[Z]
It is possible only if the cyclic regime is actually sliding in reaction to [A*] and quasi-steady-state relative to [A**] •
1.4.4
A non-isothermal process on a catalyst surface
Let two parallel reactions be carried out according to the following common mechanism
~
(1)
A + [Z]
(2)
B
+
[AZ]
0)
B
+
[z] -"'3 [Z]
[AZ]
~ [Z]
+ +
C
(IX)
D
Since the concentration of the intermediate substance [AZ] on the catalyst surface in the sliding regime is unchanged, the selectivity of the process (relative to the formation of the end product, C) under conditions of fast periodic temperature variation is I.e
~k
$=
o
2(t)dt
72
If the activation energy of formation of the main product, C, is higher than that for formation of product D, then s ) s and the non-steady-state process is efficient (s is the selectivity in the steady-state regime with the same mean values of the temperature). Let us further consider the example of a catalytic process carried out according to the following adsorption mechanism (ref. 66)
+
(2)
B
0)
[AZ]
(X)
[Z]::= [BZ]
+
+
[BZ] -- 2[ZJ
AB
and the example of a two-route mechanism formed by steps of mechanism X and the collisional step (4)
B
+ [AZ] -- [Z]
+
AB
1-3
(XI)
Mechanism X includes adsorption steps for various reagents on the surface of the catalyst. As the result of this adsorption the surface substances are formed and the interaction between these is also included in mechanism X. Step XI represents, as a rule, a part of the mechanism of oxidation of simple molecules on metals or a primitive mechanism providing plurality of steady states of the catalyst surface. Mechanisms X-XI are typically used to describe the oxidation reactions of carbon monoxide on metals. The values
characterize the contribution (selectivity) of each reaction route to the total rate of the product formation (W1 is the rate of the ith step, i = 3,4).
73
The partial pressures of the gaseous reagents, CAzand CB' and the surface temperature, T, serve as the control parameters. These parameters have different effects on the kinetic dependences of some stages. An increase in the partial pressures causes an increase in the rates of reactions 1 and 2 and in the rate of the collisional step 4. There are often large differences in the temperature dependences of the reactions in the detailed mechanism. Let us assume that the activation energies of the adsorptive reactions corresponding to those described in mechanism X are small enough to state that E1 ~ 0, E2 ~ O. Also let the activation energy in mechanism XI be small, so that E ~ O. The desorption activation 4 energies are rather high: E_ 1 = 120 ~ 280 kJ/mol; E_ 2 = 80 + +160 kJ/mol. The activation energy, Es' of interaction between substances [AZ] and [BZ] is intermediate between the activation energies of adsorption and desorption, equalling 40 + 80 kJ/mol. Some typical dependences of the steady-state reaction rate on the temperature at various pressures are shown in Fig. 1.11. As stated before, a quasi-steady-state regime may be desired particularly in the region of variation of a controlling parameter (in our case in the region of temperature variation, T1 < T ~ T2) where the curves of the kinetic dependences on this parameter are concaVe. For example, in the case described by profile 2 (Fig. 1.11) the ratio, W/r, of the mean rates in the quasi-steady-state and in the average steady-state regime in the region of temperature variation, T = 530 ~ 570 oK, is about 25. When some processes are performed via mechanism X or a similar mechanism, a considerable increase in the efficiency can be achieved at intermediate values of the temperature variation frequency. It follows that : (a) the rate of adsorption of the gases in only weakly related to the temperature; the process of adsorption is also rapid at low temperatures; (b) the rate of desorption can become sufficiently high only at elevated temperatures; (c) at low temperatures (T 1), a special composition of the catalyst surface is provided with a considerable content of the intermediate substances [AZ] and [BZ]. These substances react at high temperature (T 2) which
74
2.'(
.
p:::
2.0
'.6
Q)
..p
al
H
s::0
'r-!
..p
1.2
0.8 0.'(
0
al Q)
H
sao
520
500
580
(jQO
820
temperature t T (K)
Fig. 1.11. Dependence of the role of the steady-state reaction carried out via mechanism X on temperature. k. = k? exp (-E/rT) (i = 1,2, 3); k~ = 1.5 8- 1 , k 2 = 1; k~ = 2.23:105;~k~1 = 3'10 12; k~2 = 3.10 11 8- 1 ; E1 = E2 0; E3 = 40 kJ/mol; E1 = E2 = = 30 kJ/mol; Cn = 5; CA = 5 (1), 4 (2), 3 (3). 2
[BZ] 0.8
[BZ]
a)
0.8
0.6 0.4 0.2
[AZ]
0.2
0.4
0.6
0.8 [AZ]
Fig. 1.12. Phase situation in the system where mechanism X is realized with periodic temperature control. CA = Cn = 5. Sections of the phases trajectories marked 1, ~ correspond to temperatures, T1, T2 (a) T1 = 350 oK, T2 = 580 oK; (b) T1 = 350 oK, T = 530 oK. The values of the parameters are identical to those 2 shown in Fig. 1.11.
75
still should not be very high otherwise the desorption rate might be increased. Fig. 1.12 shows phase trajectories (profiles) corresponding to non-steady-state control by means of the temperature variation. At first, the catalyst is treated with the reaction mixture at low temperature, T1; adsorption takes place and thus the system moves into the region of middle-range coatings characterized by higher rates of reaction. If the temperature is fixed, the highest rates are observed at surface compound concentrations, [AZ] ,.. [BZ] ~ 0.5. The temperature is further increased to T2 and the adsorbed substances interact yielding the reaction product. The concentration of intermediate substances on the catalyst surface is reduced. After several periodic variations of the temperature a periodic "two-stroke" regime of reaction control is established: (1) Adsorption: the system operates at temperature T1 for some time and the surface is enriched with substances [AZ] and [BZ] • (2) Reaction: the temperature is increased to T2 and the reaction product is liberated at an high rate. The number of vacant centres on the surface is increased. In addition, the parameters of the "two-stroke" regime (the times of every cycle and temperatures, T1 and T2) should be chosen in such a way that the phase trajectories are always in the region of surface compositions at which high rates are characteristic. Otherwise the system can drift into the region of coating having low reaction rates after a few "strokes" (Fig.1.12b).
Selectivity on the catalytic process in an isothermal reactor with ideal mixing The selectivity in this case can be detailed for the example of the phenomenologic description touched upon in ref. 67. The essence of the description was discussed in ref. 3. Let us take the case of two reactions carried out in parallel:
~B
A~
C
76
Let an non-steady-state cyclic regime be created by means of periodic variation of the inlet concentration, Co (b) = Co (/; +tc J. The ratio, V =S/S, should be determined, i.e., the average selectivity in the periodic regime created to the selectivity in the steady-state regime when the inlet concentration, cos ' is similar to the average periodic inlet concentration:
tc c .:'. o be
\' co(t)dt
J o
Let us assume that the steady-state (quasi-steady-state) rate of the main reaction, r 1, is a convex-shaped function of the concentration, C, and the rate of the side reaction, r 2, is a concave-shaped function of C. This actually means that
where Cs = ). C1 + (1- J.) c2 is the average concentration, corresponding to that in the cyclic regime and also in the sliding regime; 0 < ). < 1; C1 < C2• Let at least one of these inequalities be strictly fulfilled. With the properties determined above, ri(C) (i = 1,2), the following inequality is always valid
where W1 and W 2 are the cyclically-averaged rates of formation of products Band C in a regime close to the quasi-steady-state, i.e., at long duration of the cycle. It can be derived from this inequality that under the given conditions the selectivity in the non-steady-state regime is always higher than in the steady-state regime. The value of the maximum selectivity is equal to
77
The initial amplitude of variation of the concentration of component A in a reactor operating in the quasi-steady-state cyclic regime (with very low frequencies of the inlet concentration variation) is smaller than the amplitude of variation of the inlet concentration. Also, the difference in amplitudes is proportional to the reactor volume and to the extent of conversion. Consequently, the selectivity is expected to increase in the non-steady-state regime if the reactor has a small volume, the extent of conversion is low and the inlet concentration is varied at low frequencies. At high frequencies the reactor operation will correspond to that under steady-state conditions with average values of the inlet concentration. The phenomenological mathematical description of the observable rates of chemical conversion for the example under consideration where the dynamic properties of the catalyst surface are essential has the following form (ref. 67) dWf 1 db
M
dW2.
M2 dt
=
r.tc) - W + Il 1
f
=
r2
drt(c)
fdt
(ct : W + IT drz(c) C
2.
2.
eU
where M1 and M2 are the time-scales of the transient regimes on the catalyst surface; are the corresponding times of the , l 2 preact. The material balance in the reactor is
n n
where Mp is the mean residence time of the reaction mixture in the reactor; ~o is the conventional contact time of the mixture and the catalyst. Let 1f = hfe m, '2 = kg C TTl, and the inlet concentration be varied according to C = CDS cot: + a sin wt . In Fig. 1.13 the o numerical solution of the system of eqn. 1.51-1.52 is shown at m = 2, n = 1. Similar results were naturally obtained at m > n. The increase in selectiVity in the non-steady-state regime continues to grow with increasing residence time of the mixture
78
in the reactor. The extremal properties of the mentioned dependences is explained by the fact that in the vicinity of the dimensionless frequency, GO' = Mp/t c' the amplitude of oscillation of the concentration of A in the reactor is larger than in the quasi-steady-state or sliding (GV=oo) regime. This causes a slight increase in selectivity in the region of the natural "resonant'" frequency, approximately 1/Mp' The lower the natural frequency of the system, the lower is the frequency of the external influence required to attain the maximum selectivity. The impact of the preliminary properties of the surface is clear from Fig. 1.1Jb. Parameter A alters the absolute value, V, without a change in the position of the maxima. One can see that the dynamic properties of the catalyst surface are able either to increase or decrease the selectivity gain depending on the type of kinetic mode. The same can be said about the case of two reactions, the main and the side, for which Fig. 1.14 shows calculational results. a)
..
1
';:".
ro
+>
.r-!
::-
0r-l
+> o
Q)
r-I
Q)
Ul
'.03
0
'r-!
+> a;l 1-1
frequency, co'
frequency, co'
Fig. 1.13. A reactor with ideal m~x~ng in which two parallel reactions are carried out. Influence of the frequency of variation of the inlet concentration, .»: , on the ratio, V , of selectivities in the cyclic and stead~-state regimes. (a) A = 0; M = J.J (1); 2.0 (2); 1.0 (3); 0.5 (4)~ 0 (5). (b) M = 3.3; A = = 0 (1); 0.5 (2); 1.0 0); 2.0 (4). (5) bl = 0, A = O. co: = = M/t c ; M1 = M2; M = M1 / Mp ; IT, = ilz ; A = n, IMp'
79 ';:,.
t.O'I
til
Q)
'M
+>
'M
::-
'.03
'M
+>
o
Q)
r-f
1.02
Q)
m ~
0
un
0 'M
+> al H
frequency, W'
Fig. 1.14. Influence of forced oscillations of the inlet concentration, ur , on -the ratio, P, of selectivities in the nonsteady and steady-state regimes. A = 0, M = 0.01 (1); O.OJ (2); 0.05 (3) (see Fig. 1.13).
Refs. 68 and 69 dealt with the example of the non-steady-state process of carbon monoxide oxidation on platinum in a reactor with ideal mixing. The kinetic pattern of the process corresponded to mechanism X where A2 "" 02' [AZ] == [OZ] , B == Icol, [BZ] == [COZ], AB == CO. In accordance with this mechanism, a molecule of CO from the gas phase is adsorbed on one active centre of the catalyst while every molecule of oxygen is adsorbed in a dissoiative fashion on two centres (ref. 70). Carbon dioxide formation occurs according to a step mechanism. It is assumed that CO 2 is not adsorbed on the catalyst surface. The peculiarities of the steady-state regimes via this mechanism are well reported in the chemical literature. One should remember that there are interacting intermediate substances on the surface and with cartain values of the rate constants of the reaction stages there can be a plurality of steady-state rates of CO 2 formation at similar concentrations of the reacting components above the catalyst surface or at the reactor inlet. With the supposition that the reaction is carried out in an isothermal reactor with
80
ideal mixing operating at constant pressure, the dynamic character of the system behaviour was studied. As is seen in Fig. 1.15 which shows the results of calculation of the steady-state regimes, the rate of reaction in the region of low CO concentrations increases with increasing carbon monoxide content, but decreases at higher CO concentrations. At intermediate values of the CO concentration three steady states of the system can exist. Two of them are stable and one is unstable. The maximum and the minimum in the oxidation rate correspond to the steady states. Let the carbon monoxide concentration in the mixture be varied in accordance with the sinusoidal law
cco (t:') =
C + a sin. cat' co,o
where Ceo , 0 is the time-averaged CO concentration in the mixture; a is the amplitude of oscillations, w is the frequency. I t is assumed that CO2 (6') = t - Ceo (t '), I f Ceo , 0 = 0.5 is chosen, the steady state of the system corresponds to the lower steady profile of the rate (see Fig. 1.15) at which the normalized rate, r, is 0.0254. In this case it is possible to attain a considerable increase in the reaction rate through transition to a cyclic variation of the mixture concentration. This will happen when the amplitude and the frequency of the forced oscillations are of a kind that for a portion of the period of oscillation the concentration, eeo,o will correspond to the upper profile of the reaction rate. It is clear from Fig. 1.16 that with invariable values of the oscillation amplitudes and an initial CO concentration in the region of dimensionless frequencies, W~0.45, a resonant behaviour of the system is observed at the time-averaged reaction rate curve crosses the maximum, corresponding to the mean normalized rate in the non-steady-state regime, = 0.262. This value of the rate is ten times larger than the corresponding value of the rate in the steady-state regime and twice as large as the value of the rate in the quasi-steady-state, cyclic regime (w~ 0) at Q = 0.4. This behaviour is explained by the dynamic interactions inside the system, which are connected with the forced transition of CO coatings of the catalyst surface from lower to higher values. Vllien the extent of coating is high, the time-averaged values for
W
81
0.7 0.6
po:
as
~
Cll
+>
o.«
s::0
0.3
+>
0.2
al H
'M
0
al
Cll
H
o. 1
r.o C
co
Fig. 1.15. Steady-state (normalized) CO oxidation rate, r, on Pt as function of the initial concentration, Cro o (fraction), in the mixture at the inlet of the reactor with id~al mixing LK1 = 35; K_ 1 = 0.5; K2 = 20; K_ 2 = 50; K) = 6; CC02 = OJ (r is normalized by division of the absolute rate by its maximR~ value obtained with stoichiometric amounts of CO and 02 and their complete conversion into CO 2 j. the rate often tend to the steady-state values, whereas at low coatings they tend to quasi-steady-state values. Notice, that for this example, an extreme dependence of the observable rate of CO oxidation on the value of the amplitude of oscillation is observed, provided the oscillation frequency is fixed. Ref. 69 considers mechanism XI where the possibility of interaction of gaseous carbon monoxide with the adsorbed oxygen is taken into account CO
+
[OZ] -
CO 2
+
[Z]
A new mathematical simulation of this model did not bring about the desired quality. It was found only that the resonance frequency was strongly dependent on the ratio of the reactor volume capacities to the surface area of the catalyst.
82
0.30 , . . - - - - - - - - - - - - - - , 0.25
0.20
0.15'
0.10
17.03
____ L
0.001
0.01
_
2.
1.0
0.1
frequency,
10
UJ
Fig. 1.16. Non-steady-state (cyclic, normalized) rate of CO oxidation on Pt, W, (3) as a function of the (dimensionless) frequency of oscillations, w (Q ; 0.4; CCO,O ; 0.5); (1) quasi-steady-state (cyclic) regime; (2) steady-state regime where rlC co,a ; 0.5.
REFERENCES 1 G.K. Boreskov, Zh. Vses. Khim. Ova., 22, 5(1977) 495-505. 2 Yu.Sh. Matros, Unsteady Processes in Catalytic Reactors, Elsevier, 1985. 3 G.K. Boreskov, Kinet. Katal. (Engl. Transl.), 21, 1(1980) 5-16. 4 B.S. Balzhinimaev, S.V. Kozyrev, G.K. Boreskov (editor), A.A.lvanov, N.P. Belyaeva and V.I. Zaikovsky in Heterogeneous Catalysis, Proceedings of the 3rd All-Union Conference on Mechanism of Catalytic Reactions, Institute of Catalysis, Novosibirsk, 1982, pp. 177-180. 5 V.A. Sadykov, P.G. Zyrulnicov, V.V. Popovsky and S.F. Tihov, Kinet. Catal. (Engl. Trans.), 22, 5(1981) 1219-1226. 6 V.A. Sadykov, S.F. Tihov and V.V. Popovsky, Kinet. Catal. (Engl. Transl.), 27, 1(1986) 147-155.
83
7 E. Fiolitakis, U. Hoffman and H. Hofmann, Chem. Eng. Sci., 35, 5(1980) 1021-1030. 8 G.Ya. Popova, T.V. Andrushkevish and G.A. Metalkova, React. Kinet. Catal. Lett., 12, 4(1979) 469-473. 9 K.A. Dadayan, G.K. Boreskov, V.I. Savchenko, N.N. Bulgakov and M.D. Smolikov, Dokl. Akad. Nauk SSSR, 239, 2(1978) 356-359. 10 L.P. Levchenko, N.V. Kulkova and M.M. Tyomkin, Kinet. Catal. (Engl. Transl.), 14, 6(1976) 1542-1547. 11 V.Sh. Gruver, A.V. Hasin and G.K. Boreskov, Kinet. Catal. (Engl. Transl.), 12, 1(1971) 156-159. 12 Verbeek H. and W.M.H. Sachtler, J. Catal., 42, 2(1976) 257-267. 13 M.P. Unni, R.R. HUdgins and P.L. Silveston, Can. J. Chem. Eng., 51, 6(1973) 623-629. 14 V.P. Briggs, R.R. HUdgins and P.L. Silveston, Chem. Eng. Sci., 32, 9(1977) 1087-1092. 15 Wilson H.D. and R.G. Rinker, Chem. Eng. Sci., 37, 3(1982) 343-355. 16 A.K.Jain, R.R. HUdgins and P.L. Silveston, Can. J. Chem. Eng., 61(1983) 824-832. 17 A.H. Abdel Ghani, A.V. Krylova and G.P. Klimova, Investigation of the Reaction for Ammonia Synthesis in Impulse Regime, Moscow Chemico-Technological Institute, Minvus SSSR, 1983, pp. 1-6; VINITI Dep. Card Number: 2341-83 (in Russian). 18 E.N. Shapatina, V.L. Kuchaev and M.I. Tyomkin, Kinet. Catal. (Engl. Transl.), 26, 3(1985) 660-667. 19 G. Rambeau and R.G. Amariglio, Applied Catal., 1, 5(1981) 291-302. 20 T.G. Alhasov, A.E. Lisovsky, Yu.A. Ismailov and A.I. Kozharov, Kinet. Catal. (Engl. Transl.), 19, 3(1978) 611-614. 21 A.E. Lisovsky, A.I. Kozharov, Yu.A. Izmailov and T.G.Alhasov, Kinet. Catal. (Engl. Transl.), 19, 4(1978) 950-954. 22 H. Helmrich, A. Renken and K. Schugerl, Chem.-Ing.-Tech., 46, 15(1974) 647. 23 E.J. Leupold and A. Renken, Germ. Chem. Eng., 1(1978) 218-222. 24 D. Coustien, Diss. Dokt., Naturwiss., Fach: Math. and
84
Naturwiss., Techn. Univ., Hannover, 1979, Paragraph 203. 25 A. Renken, Int. Chern. Eng., 24, 2(1984) 202-213. 26 A.V. Hasin, in Theoretical Problems of Catalysis, Institute of Catalysis, SO AN SSSR, Novosibirsk, 1977, pp. 135-146. 27 L.P. Levchenko, N.V. Kulkova and M.I. Tyomkin, Kinet. Catal. (Engl. Transl.), 17, 6(1976) 1542-1547. 28 M. Leuthe, Diss. Dokt. Naturwiss., Fach: Math. und Naturwiss. Techn. Univ., Hannover, 1974, Paragraph 163. 29 A.S. AI-Tai und J.S. Kershenbaum, Am. Chern. Soc. Symp. Ser., 65(1978) 512-525. 30 F.M. Dautzenberg, J.N. Helle, R.A. Sauten and H. Verbeek, J. Catal., 50, 1(1977) 8-14. 31 S.L. Feimer, P.L. Silveston and R.R. HUdgins, Can.J. Chern. Eng., 62(1984) 241-248. 32 L.L. Hedegus, C.C. Chang, D.J. McEwen and E.M. Sloan, Ind. Eng. Chem. Fundam., 19, 4(1980) 367-372. 33 M.P. Cultip, AIChE J., 25, 3(1979) 502-508. 34 Yo. Barshad, Gulari Erdogan, AIChe J., 31, 4(1985) 649-658. 35 N.K. Abdul-Kareem, A.K. Jain, P.L. Silveston and R.R. Hudgins, Chem. Eng. Sci., 35, 1-2(1980) 273-282. 36 N.K. Abdul-Kareem, P.L. Silveston and R.R. HUdgins, Chem. Eng. Sci., 35, 10(1980) 2077-2084. 37 J.L. Feimer, A.K. Jain, R.R. HUdgins and P.L. Silveston, Chem. Eng. Sci., 35, 12(1982) 1797-1805. 38 N.K. Abdul-Kareem, R.R.Hudgins and P.L. Silveston, Chern. Eng. Sci., 35, 10(1980) 2085-2088. 39 O.M. Ilyinich, G.K. Boreskov and A.A. Ivanov, Dokl. Akad. Nauk SSSR, 268, 5(1983) 1171-1173. 40 A.L. Zeilingold, N.A. Komarovsky, E.E. Chehov, M.E. Basner, G.A.Stepanov, A.N. Bushin and G.N. Abaev in Oxydative Dehydration of Hydrocarbons, "Ely" Publishers, Baku, 1970, pp. 72-78 (in Russian). 41 A.L. Zeilingold, N.A. Komarovsky, E.E. Chezov, G.A. Stepanov, L.S. Tuktanova, Yu.D. Pankratyev and M.G. Slinko, Kinet. Catal. (Eng. Transl.), 13, 5(1972 1181-1189. 42 Z.P. Ilyina, O.Ya. Polotnyuk, V.I. Timoshenko, L.I. Oks and M.G. Slinko in 7th All-Union Conference "Khimreactor-7" , Vol. 4, Baku, 1980, pp. 29-33. 43 I.A. Zolotarsky and Yu.Sh.Matros, React. Kinet. Cutal. Lett., 20, 3-4(1982) 321-326.
85
44 J.E. Bailey and F. Horn, Ber. Bunsenges. Phys. Chern., 74, 7(1970) 611-617. 45 A.K. Jain, R.R. Hudgins and P.L. Silveston, Can. J. Chern. Eng., 61, 1(1983) 46-49. 46 J.E. Bailey, Chern. Eng. Comm., 1(1973) 111-124. 47 J.E. Bailey and F. Horn, Ber. Bunsenges. Phys. Chern., 73, 3(1969) 274-279. 48 A.M. Zyrlin, Variation Methods of Chemical Apparatus Calculation, Mashinostroenie, Moskow, 1978 (in Russian). 49 A.M. Zyrlin, V.S. Balakirev and E.G. Dundukov, Variation Methods of Optimization of Controllable Objects, Energia, Moskow, 1976 (in Russian). 50 E.G. Gilbert, SIAM. J. Control Opt., 15, 5(1977) 717-746. 51 R. Schladlich, U. Hoffman and H. Hoffman, Chem. Eng. Sci., 38, 9(1983) 1375-1384. 52 J.E. Bailey, Optimal Periodic Process in the Limit of Very Fast and Very Slow Cycling, Automatica, Pergamon Press (G.B.), 1972, pp. 451-454. 53 S. Rinaldi, I.E.E.E. Trans. Aut. Control, AC-15(1970) 671-673. 54 S. Bittani, G. Fronsa and G. Guardabassi, I.E.E.E. Trans. Aut. Control, AC-18(1973) 33-38. 55 N. Watanabe, K. Onogi and M. Matsubara, Chern. Eng. Sci., 36 (1980) 809-818. 56 R. Sincic and J.E. Bailey, Chern. Eng. Sci., 35(1980) 1153-1161. 57 N. Wattanabe, Y. Nishibura and M. Matsubara, I.E.E.E. Trans. Aut. Control, AC-21 , 4(1976) 609-610. 58 V.A. Novikov, Prikl. Mat. and Mekh., 43(1979) 566-569. 59 N.N. Dorogov and A.M. Zyrlin, in Dynamic Regims in Chemistry and Chemical Technology, Institute of Catalysis SO AN SSSR, Novosibirsk, 1979, pp. 76-82 (in Russian). 60 M.S. Dukelsky and A.M. Zyrlin, Avtom. Telemekh., 9(1977) 5-11. 61 T.J.M. Horn and R.C. Lin, Ind. Eng. Chern. Process Des. Dev., 6, 1(1967) 21-30. 62 F.A. Farhadpour and L.G. Gibilaro, Chem. Eng. Sci., 36, 1 (1981) 143-147. 63 A. Braison and Ho-Yu-Shi, Application Theory of Optimum Controls, Mir, Moscow, 1972.
86
64 S.M. Bogdashev, J.A. Zolotarsky and Yu. Sh. Matros, Froc. 8th. Int. Congr. Chern. Eng., Chern. Equipm. Design. and Automation, Praha (Chisa-84), 1984, pp. 12)-124. 65 I.A. Zolotarsky, S.M. Bogdashev and Yu. Sh. Matros, Autom. Telekh. (1987) in press •. 66 V.I. Bykov and G.S. Yablonsky, Dokl. Akad. Nauk SSSR, 251 (1980) 616-619. 67 V.A. Chumachenko, A.I. Oruzheinikov and Yu. Sh. Matros, Kinet. Catal. (Engl. Transl.), 24, 6(1983) 1479-1486. 68 D.T. Lynch, Can. J. Chem. Eng., A7, 2(1983) 183-188. 69 D.T. Lynch, Can. J. Chern. Eng., 62, 5(1984) 691-698. 70 T. Eugel, G. Erte, Adv. Catal., 28(1979) 1-78.
Chapter 1
REACTION PERFORMANCE IN THE CATALYST UNSTEADY STATE UNSTEADY STATE OF THE CATALYST According to Academician Boreskov's definition of catalysis (ref. 1), it is an acceleration of a chemical reaction over certain substances (called catalysts) which repeatedly interact with reagents without entering into the composition of the product. A catalytic process usually involves three successive stages: adsorption of initial components, chemical conversion on the surface and desorption of the reaction product. Each stage consists of consecutive or parallel steps of both physical and chemical interaction of intermediate compounds on the surface as well as with the components of the gas phase. Regardless of its specific action, the nature of catalysis is common and can be defined as follows: the catalyst, being a part of the intermediate compounds, increases the degree of compensation of the energy required for the splitting of old bonds by the energy released in the formation of the new bonds. This results in a decrease in the activation energy of the chemical reaction. That is, in short, an up-to-date point of view to the considerable growth of the chemical transformation rate in catalysed reactions. 1.1
The state of the catalyst is affected by the reaction medium. To every gas phase composition invariable over a lengthy period of time, catalyst surface temperature and initial conditions there corresponds an unique catalyst state, characterized by specific structure, composition and catalytic properties, provided there is no external-diffusion resistance. This catalyst state, of which time-invariant activity and selectivity are characteristic, does not, however, always prove optimal. One can imagine a situation, e.g., periodic variation in some range of gaseous mixture composition when the unsteady catalyst activity and selectivity exceed on average those under the steady state. It is recommended to compare catalyst states at identical concentrations both under stationary and averaged non-stationary conditions. The efficiency of a catalytic process under forced non-steady-state conditions is determined by the
2
mechanism of the process, the rate constants of various reaction stages and the dynamic properties of the catalyst surface. Therefore, it makes sense briefly to remind the reader of those properties (ref. 2) and then proceed to the problems concerned with the performance of the processes in the unsteady state of the catalyst surface. The chemical composition of the catalyst, its surface structure and catalytic properties undergo some changes as the result of variation of the reaction medium. Numerous experimental data confirm that the composition, structure and properties of different catalysts, massive and supported metals and alloys, simple and complex oxides and acid-base catalysts really are influenced by the concentrations of the components in the reaction mixture and temperature (ref. 3). The modification of the chemical composition and the change in the catalytic properties in many cases are distinct as, for example, in sulphur dioxide oxidation on vanadium (ref. 4). The metal oxidation state decreases from VS + to V 4 + with increasing concentration of 802 in the reaction mixture and decreasing temperature. These variations are reversible and have small relaxation times.Furthermore, at low temperatures with a reaction mixture enriched with 802 and long period of interaction with the catalyst, a special state of inactive quadrivalent crystalline vanadium appears. Its concentration is in good agreement with the decrease in catalytic activity. These types of variations are characterized by longer relaxation times. A considerable transformation in the chemical composition and catalytic properties under the influence of the reaction medium is also observed for solid catalysts, for example, for oxidation of CO on various manganese oxide catalysts (ref. 5). The surface composition of the catalyst with different initial composition in the steady-state regime corresponds to the one and the same catalyst structure which can only exist in this reaction medium. Similar steady states of manganese oxides could be obtained after 30 hours of catalyst treatment by the reaction mixture at 2]OoC. The same results were obtained for oxidation of CO on copper-containing catalysts (ref. 6). Interaction of these catalysts with the reaction mixture involved a relatively rapid
3
change in the oxidation state of the catalyst surface and its underlayer and also significantly slowed the creation or destruction of the space defects (inhomogeneities), which are evidently connected with phase rearrangements of the surface layer. As was shown for conversion of CO on copper-containing oxide catalysts, the oxidation of the catalyst by water vapour affects the catalytic activity. In '~his case, the stationary concentration of oxygen is attained much more slowly as compared with the catalytic reaction itself (ref. 7). The changes in the oxide catalyst properties under the influence of the reaction medium is evidently the most studied aspect of the variety of heterogeneous catalytic processes. The oxidation state of the catalyst is decreased following the reduction of this ratio and this results in a sharp drop in the general reaction rate, while the selectivity toward the incomplete oxidation product is simultaneously increased. Acrolein oxidation to acrylic acid on a vanadium-molybdenum oxide catalyst is the perfect example of the above mentioned change in parameters (ref. 8). The effect of the reaction mixture has also been observed for acid-base catalysts. For example, under the influence of water vapour, the extent of hydration of acidic centres and also the ratio between the Bronsted and Lewis centres can be changed. As reported for metallic catalysts, their surface is easily reconstructed under the influence of the reaction medium and tends to maximum surface energy. Very often the reconstructive adsorption of reactants leads to a change in the structure of the metal surface (ref. 9). Moreover, the composition of the near-surface layer is usually also modified as a result of dissolution of the reaction mixture components. The amount of the components adsorbed is often many times greater than a monolayer coverage. Levchenko et al. (ref. 10) observed a slow change in the rate of ethene oxidation on silver, which they related to oxygen diffusion in the near-surface layer of the catalyst. A similar process was observed in the catalysed oxidation of hydrogen on silver films (ref. 11). All these factors result in a change in the sorption heat of the reactants
4
on the metal surface and in the activation energy of elementary steps. Consequently, the total catalytic activity and selectivity of the reaction are changed. The surface composition of solid alloys is also modified under the influence of the reaction medium. The free surface energy is changed due to the chemisorption and, therefore, it is more advantageous for the alloy component with the higher chemisorption energy to be on the surface. As reported in ref. 12 for Pt-Sn alloys, in a vacuum their surface is enriched with tin (compared to the catalyst bulk), whereas in a carbon monoxide atmosphere it is enriched with platinum. Atom shift in metals is usually limited to the thin near-surface layer, but with an increase in temperature the thickness of this layer may grow. We have only considered some examples of a large body of experimental data. These data taken together permit one to conclude that solid catalysts are components of the reaction system and are sensitive to variations of the reaction mixture composition, temperature and other parameters. To each composition and temperature of the reaction mixture, there corresponds a certain catalyst steady state. Some experimental data obtained about )0 years ago demonstrate the existence of hysteresis in the rate of a chemical reaction. Recently, isothermal self-excited oscillative regimes on the catalyst surface at invariable gas phase composition were discovered. This does not invalidate the above deduction, but only indicates that the steady state of the catalyst is determined by the composition and temperature of the gas phase as well as by the changes in the condition of this phase. The influence of the reaction mixture on catalyst properties must be taken into account in the kinetic dependences of the reaction in heterogeneous catalysis. In the vast majority of cases, kinetic equations are usually derived on a tacit assumption that the composition of the solid catalyst and its properties are independent of the reaction mixture composition and of its influence on the catalyst. In reality, this condition is not fulfilled because the chemical composition of
5
the catalyst is affected by the reaction medium, which eventually leads to a phase change of the active component and alters the volume composition of the near-surface layer. That is why, besides the effects of temperature and composition, the reaction rate is also changed owing to modification of the catalyst properties. The dependence of the reaction rate, W, on the concentration of reagents must therefore include two functions. The first function, f[c(t); 9(c(t) )], characterizes the true kinetic dependence which describes the reaction mechanism. The second function, y?[c(t); Q(c(t))], is determined by the effect of the reaction mixture composition on the catalyst properties (ref. J) W
=
F
[f (r;
§)
i
Cf (c,
8)J
where t is the time, c is the vector of the reagent concentration, g is the surface-packing vector. Most often, expression (1.1) of the rate of the chemical reaction can be represented as the product of functions f and ~ • These functions can be determined experimentally by varying either the concentration of the reagents at the same catalyst state ( (jJ = constant) or by changing 'fl via catalyst pretreatment by reaction mixtures of various compositions. In the case of heterogeneous catalytic reactions, an important factor is the time and character of the transition to the steady state. It is necessary that this time, Mc (in other words, the relaxation time), be estimated in comparison with the time of the entire catalytic process, Mp, i.e., with the time of the effect of the reaction mediwn on the catalyst. It is useful to introduce a time-scale notion of the change in the gas phase condition, tc,for processes where the gas phase parameters are varied. If the gas phase variation is of periodic character, t c stands for the duration of the period. In the majority of cases Mp » t c. As a result of a change in the reaction mixture parameters, two different types of unsteady catalytic state can occur: (a) the concentrations of the intermediate products of the catalytic cycle are unsteady;
6
(b) the state of the catalyst in the near-surface layer is unsteady leading to a variation in the rate constants for the elementary steps. The variation in the surface concentrations occurs as the result of the stepwise character of the catalytic process, that is, it occurs at a rate similar in magnitude to that of the reaction rate. For reactions which occur rapidly enough and, therefore, are of practical interest, the time-scale of the surface concentrations generally ranges from 10- 1 to 102 s. Modifications of the surface structure and near-surface composition affecting the catalytic properties usually include other stages of transition to the steady state than those of the catalytic reaction itself. That is why, in most cases, the transition to the steady state occurs far more slowly compared to the time of the catalytic reaction. The value of Mc for the non-steady-state process on the catalyst surface can, for example, be quantitatively estimated by the following expression
~ [W(t)- W(=J] tit o W(=) - W(Oj
(1.2)
where W(O) and W (00) are the values of the observable conversion rates under the steady-state conditions before and after inducing a jump-like disturbance at the moment of time t = 0; Wet) is the current value of this rate. A catalytic process may be carried out under three types of states: steady, unsteady or quasi-steady, depending on the character of the catalyst state variation and external conditions. The observable rates of chemical transfonnation of the reagents, W, in the steady-state regime depend on the temperature and instantaneous concentrations of all reaction participants as well as on the concentrations of the intennediate substances on the catalyst surface.The rates of fonnation and consumption of the gaseous substances participating in one and the same reaction under steady-state conditions are equal, within the accuracy of the stoichiometric relationship. In the unsteady regime these rates may appear to be different. If the composition, structure and properties of the catalyst in the unsteady regime adjust to the condition of
7
the gas phase so rapidly as to fulfil at any time the inequality
W-r[c(6J, rr6J] W where r(o, T) is the reaction rate calculated via the kinetic model of the stationary process, this process on the catalyst surface is regarded as a quasi-steady-state one relative to the gas phase condition. In inequality (1.3), G is some small value corresponding, for example, to the allowed experimental error in the reaction rate measurement. The difference between Wand r in the non-steady-state regime is determined by the dynamic properties of the catalyst, which are connected to the relaxation characteristics of the catalytic cycle and to the effect of the reaction mixture on the catalyst. wnen the catalyst properties are changed owing to a modification in the surface structure and in the near-surface composition, the approximation to the steady state usually comprises stages other than those of the catalytic reaction. So, as a rule, the approximation to the steady state is carried out within a greater time than the characteristic time of the transient regime of the catalytic cycle. Generally, the non-steady-state kinetic model of the reaction is as follows:
dB
db
_
=
_
F, (c, 8,
ri:, T);
(1. 4a)
(1. 4b)
j
==
J (c, 8, s, T);
CjJ '=
If (C, B,
W=
F[jfe,8, z, T); !j(e,B,
i"i, T) ;
( 1 • 4c)
(1.4d)
(1. 4e)
8
Eqn. 1.4a describes the dynamic character of the catalytic cycle: 8 is the concentration vector of substances on the catalyst surface, d is the vector of the catalyst characteristics changing under the influence of the reaction medium, T is the temperature. Eqn. 1.4b describes the dynamic character of the catalyst state under the influence of the reaction medium. Expression 1.4e is the vector of the observable rates of chemical conversion detennined by the kinetics of the process shown in eqn. 1.4c and by the influence of the reaction mixture on the catalyst expressed by eqn. 1.4d. Expressions 1.1 and 1.4e may coincide when only one chemical reaction is involved. 1.2
WAYS TO CREATE THE CATALYST UNSTEADY STATE
Two practical methods to maintain un unsteady catalyst surface can be distinguished: (a) periodic activation of the catalyst under conditions considerably different from those of the catalytic process and (b) periodic variation of the reaction mixture parameters, including the temperature. As was stated above, variation of the characteristics of the gas phase may result in the appearance of two types of catalyst surface states. (1) The concentrations of the intermediate products of the catalytic cycle are unsteady. In the case of heterogeneous catalytic reactions on solid catalysts these are the surface concentrations of reactants and products of conversion participating in the elementary steps of the reaction. The variation of the concentrations of the surface substances is the result of the stepwise manner of the catalytic process occuring at a rate similar in magnitude to that of the chemical conversion. For reactions carried out rapidly enough to be of practical interest, the re~axation time of the surface concentrations ranges mostly from 1 to 100 s. (2) The state of the catalyst in the near-surface layer is unsteady, which leads to variations in the rate constants of the elementary steps. These variations are connected with concurrent interactions Which do not take part in the catalytic cycle. As a rule, the activation energies of these interactions are rather large, whereas the rates are small. Therefore, the relaxation time of the concurrent interactions is far greater
9
than that of the catalytic cycle. In both cases the unsteady-state character is determined by the divergence of the catalytic properties corresponding to the time-averaged values of the gas phase parameters. In the first case, the efficiency can be increased owing to the oscillation regime which provides the possibility of controlling the concentration of the intermediate species and, under certain conditions, to change the productivity and selectivity. In the second case the efficiency of the process can be increased by creating an optimum (in the mean) unsteady state of the catalyst. Let us assume that the time-scale of the variation of the surface concentrations, Mf, is much less than the time-scale of the stabilization of the catalytic properties, M'fJ' and than the time-scale of variation of the gas phase condition, t c• Also, let the values Jilf and t c be related as Mf « M'fJ ~ t c• If these conditions are valid, the catalytic reaction can be carried out in the catalyst unsteady state. The intermediate products of the cycle will be thus in a quasi-steady state, which means that at any moment of time their concentrations are slightly different from the concentrations corresponding to the composition and temperature of the gas phase at the same instant. Of course, we have to assume further that the variation range of the gas phase condition allows for the existence of only one steady-state regime on the catalyst surface for every given invariable composition of the gas phase. The difference between the steady and unsteady states for various reactions lies in the degree of hydration and oXidation, the content of dissolved components or in other parameters. The unsteady state can also be obtained by varying the condition to which the catalyst is exposed. Let ~ be the parameter determining the catalyst state. It is due to change in the course of the reaction carried out under non-steady-state conditions. During the steady-state process with the parameters of the gas phase corresponding to their mean values at time t c in the unsteady process, oc = ~S. In many cases the catalyst unsteady state can exceed its steady state in both activity and selectivity within a certain interval of variation. Then, with the gas phase condition invariable during
10
the entire course of the catalytic function and with W being linearly dependent on the functions f and ~ (1.1), the relative gain in the chemical conversion rate for a simple reaction will take the form tc
~ o
LlW W(d S)
'I [ d (b), § (OJ tit - f
=
where Ll W is the absolute value of the reaction rate gain compared to the steady-state operation with W(d S ) = r; 'IS is a function of the influence of the reaction mixture of the catalyst in the stationary regime '1(C£.s'
r.:
B.s).
In case of complex reactions, the change in the selectivity is the most important practical aspect. For example, for two parallel reactions carried out in the unsteady regime with rates W for the usual and W2 for the concurrent one, the selectivity 1 can be expressed as tc
tc
~ ~tit + 0
o
0
tc
tc
2
0
tc
tC
~ iff dt / ~ '1 dt
~ w, db s=
tc
SVti 0
dt
~ lit dt / ~ <1 o
2
db + 12
/
h
0
where f 1 and f 2 are kinetic functions of the first and the second reactions, respectively. With catalyst state variation the ratio tc
~ liz dt o
o
is changed. If under the catalyst unsteady state characterized
11
by the value of parameter
d
the following condition is fulfilled
then there is a gain in selectivity. One can apply the above expressions to estimation of the efficiency of an unsteady process performed with periodic activation. With non-'steady-state operation of the catalyst, the parameter a: tends to approach c(,s • In order to return to the optimum state of the catalyst it is necessary to treat it periodically with a regenerating gaseous mixture other than the reaction mixture used in the process. This can be achieved either by periodically blowing down the reactor with the regenerating mixture or by continuous extraction of a portion of the catalyst to be regenerated in a separate apparatus. The last method seems convenient in the case of a quasi-fluidized catalyst bed or moving large-grain bed. The catalyst unsteady state can be attained by means of periodic variation of the reaction mixture parameters, i.e. composition, temperature, pressure and volume flow rate. Vilien the catalytic properties are changed due to variations in the conditions of the gas phase, the analytical determination of the optimum oscillations for the parameters presents a serious difficulty. To avoid it, it is necessary to know the kinetics of the investigated reaction in various catalytic states and, in addition, the kinetics of the influence of reaction mixtures of various compositions and temperature on the catalyst state. If the mean-integral reaction rate equals IV during a fraction fl of the whole period, t c ' of the composition and/or gas phase temperature variation when the state of the catalyst is close to its optimum, and W2 in the interval of regeneration, then 'the increase in the observable rate compared to the rate under 2(1_ 2 steady conditions, R, is expressed by W~ + W fl) - R. W may be (and always is) a small value, so the gain in the reaction rate is determined by the difference between v 1p and R.
12
At present there is a discrepancy between experimental research dealing with processes occurring under unsteady catalyst operation and the theory devoted to the creation and investigation of kinetic models describing the unsteady-state process on the catalyst surface.There have been few bench-scale examinations of the methods to increase the efficiency of a catalytic process under non-steady-state conditions. The authors of experimental studies dealing with the increase in process efficiency obtained at considerable absolute values of chemical conversion tend to pay attention only to the influence of the reaction mixture on the catalytic properties, while theoretical works are mainly concerned with the dynamic character of the catalytic cycle. Thus, the theory allows one to understand the practical findings only in a ve.ry general sense. Furthermore, it has to be admitted that the theoretical process modelling is not necessarily related to particular processes. We shall now examine the important experimental results and then proceed to consider some theoretical aspects of performing catalytic reactions in the catalyst unsteady state. 1 .3
EXPERllIENTAL DATA
Ref. 13 reports the influence of forced oscillations of the reaction mixture composition at the reactor inlet on the oxidation of sulphur dioxide on vanadium. The ratio of the reagents, S02 and O2, was changed in a cyclic manner around a mean value of 0.6. The minimum concentration ratio of the reagents was 0.2 with a peak of 1.0. The dependence of the mean reaction rate averaged over the period, t c' on the length of this period exhibited a maximum in the range t c = 4-5 h and a reagent ratio between 0.3 and 0.4. The temperature of the reaction mixture at the inlet was measured to be 405°C. The reactions were carried out at small conversion extents and were far from equilibrium states. Evaluations of the rates pf the processes in this system showed that the characteristic times of transient regimes in the catalytic cycle were much smaller than the duration of the period at which a noticeable increase in chemical conversion rate was observed. Evidently, the explanation lies in the fact that a considerable part of the vanadium in the low temperature range is in the inactive
13
quadrivalent crystalline form V4+. The characteristic times of the transient regimes required to change the concentration (crystallization or dissolution) of this form during the cycle may appear to be rather long and commensurable with the discussed periods of concentration oscillations in the gas phase. Ref. 14 also reports the influence of the cyclic variation of the initial mixture composition for oxidation of 802 in a fixed catalyst bed. This research was carried out in a twin-bed catalytic reactor. 802 mixed with air and concentrated to 12.4% was fed into the first bed of vanadium catalyst. This bed, used for initial conversion of about 90% of the sulphur dioxide into SO)' operated in the steady-state regime. The composition of the mixture fed into the second bed was varied in a cyclic manner: 13 minutes of pure air blow, then 1) min of the mixture from the outlet of the first bed. The temperature of the mixture at the second bed outlet was kept at 406°C. The results showed that under steady-state conditions the conversion extent at the second bed outlet was 95.7%, while in the cyclic regime it reached 98.7%. This value exceeds the equilibrium conversion extent under experimental conditions calculated for the mean values of the temperature and composition. This can be explained by the unfavourable shift of the steady-state reaction from the equilibrium state caused by the large amount of SO) dissolved in the catalyst body. After air or nitrogen blowing, the sulphur anhydrite is desorbed and helps to change the quadrivalent vanadium into its pentoxide. The ratio of the catalyst volume to the volume of the reaction mixture (conventional time of contact) in the experimental examples discussed here was measured as only a few seconds. For this catalyst amount the optimum value of the period lies between 10 and 20 min. It is noteworthy that the effect of blowing air through the catalyst bed at 406°C was evident during 24 h. The catalyst transition (approximation) to the steady-state was evidently close to completion within this considerable interval. The adsorbed hydrogen and the reaction products are known to be mainly present on the catalyst surface as in the process of ammonia synthesis in the steady-state regime on ferrous
14
catalysts. Nitrogen is practically absent due to its slow adsorption rate in relation to the other stages of ammonia formation. Feeding of pure nitrogen onto the refined catalyst surface by means of an inert gas flow (for example, helium) provides a noticeable rise in nitrogen adsorption rate and a considerable accumulation of adsorbed surface nitrogen is observed. One can suppose that periodic introduction of H and 2 N2 (separately) into a reactor containing a ferrous catalyst should lead to an increase in ammonia output owing to the increase in the cyclically averaged concentration of adsorbed nitrogen. However, periodic variation of the inlet concentrations of H2 and N2 in this reaction on ferrous and ruthenium catalysts proved efficient at low pressures. In the temperature range 325-425°C and pressure about 1 roPa, the productivity o~ the ferrous catalyst was increased by 30% with the optimum oscillation period being about 30 seconds (ref. 15). An identical productivity increase was detected at a pressure of about 2.5 roPa and at 400°C, the period of oscillation in this case being 7-10 min (ref. 16). Table 1.1 reports values of the productivity, G, and of the extent of conversion of nitrogen into ammonia, x, with a stoichiometric nitrogen-hydrogen feed by means of an helium flow at normal (atmospheric) pressure (ref. 17). Shortening the time between impulses by about 50 times and a temperature drop from 500°C to 250°C had little effect on x, whereas the productivity increased. TABLE
1.1
Dependence of G [sm3 IT C] and x [%J on the time interval between impulses of a mixture containing N2 + 3H2 (in an helium flow)
x 500 400 250
Time interval ( s) 20 180
10
T (Oe) 4.4 4.0 2.1
G
0.62 0.55 0.53
x 4.5
4.2 2.3
G
0.35 0.32 0.30
360
540 ,..
x
G
x
\.r
x
G
4.8 4.5 3.8
0.026 0.022 0.020
4.6 4.5 3.6
0.021 0.020 0.017
4.8 4.4 3.7
0.013 0.011 0.010
15
This indicates that the efficiency of the non-steady-state process is evidently determined by the dynamic character of the catalytic cycle, characterized by the value of the time-scale of the transition regime on the surface, Me ~ Mf ~ 10 s. These considerations are in full agreement with the experimental data on relaxation processes on the surface of the ferrous catalyst at normal (atmospheric) pressure (ref. 18). It has been shown here that slow processes of the reaction medium interaction are not observed, at least at temperatures between 220°C and 400°C, and the dynamic properties of the surface are determined by the rates of variation in the intermediate product coating the surface. Table 1.2 reports the extent of conversion of nitrogen into ammonia as a function of the catalyst amount at constant volume of the impulse and different temperatures for the impulse and flow (filtration) stationary regimes. It is clear that the conversion extent under steady-state conditions is a factor of 10 2 lower than that in the unsteady state. The catalyst productivity under these conditions is an order of magnitude greater than that in the steady-state regime.
Extent of nitrogen conversion into ammonia, x [%], in various regimes T (oC)
Catalyst amount in reactor 2 g 0.5 g N2 into H2 flow
500 ~o
250
2 g (flow)
N2 + 3H2
N2 into
N2 + 3H2
N2 +
into He flow
H2 flow
into He flow
+ 3H2
2.0 1.2 0.8
1.6 1.4 1.0
0.07 0.024 0.0043
The results described above are similar to those obtained for experimental synthesis of ammonia on ruthenium. The rate of formation of ammonia is not high because of strong hydrogen
16
inhibition. With a cyclical character of the process this resistance can be noticeably reduced, as been demonstrated experimentally: the reduction rate was increased by a factor of 10 2 due to periodic hydrogen feed onto the catalyst body at JOOoC and 1 atm pressure. The period of the impulses was measured as a few minutes (ref. 19). It must be added that the observable rate of chemical conversion on the ruthenium catalyst remained very low. Operation under the non-steady-state appeared efficient for the reactions of oxidative dehydration of ethylbenzene into styrol on activated coals (refs. 20, 21). The initial period of this process for the catalyst oxidized by oxygen from the air is unsteady, provided the oxygen and the ethyl benzene contents are invariable. Coal is highly active for the first 10 min of the oxidative dehydration process, while in the reaction of deep oxidation it remains quite passive. As steady-state conditions of the reaction are approached, the activity of coal in the main conversion decreases, whereas in the concurrent reaction the activity increases. There appeared to be at least two forms of chemisorbed oxygen on the catalyst surface, one of which, being less mobile and fixed, shows activity in the reaction of oxidative dehydration, the other being mobile and slightly fixed is active in the reaction of deep oxidation. The decrease in coal activity is explained by the considerable oxygen flow and the gradual coating of the catalyst surface with product. Application of an oxygen-containing mixture free from ethylbenzene resulted in full regeneration of the catalyst. Tests on oxidative dehydration under non-steady-state conditions at 400°C were carried out in a flow converter with two chambers operated in turn. The reaction mixture was blown through one chamber and hot air through the other. Every 10 min the reactors were alternated. It was possible to increase the styrol output and selectiVity of the reaction by means of cyclical dehydration and regeneration as compared to those obtained under steady-state conditions. At a styrol output of 70-78%, the selectiVity was 92-96%. Better characteristics, to my mind, were the low product coating and high average content of slightly fixed oxygen on the catalyst surface brought by the periodic reaction control.
17
An increase in efficiency under non-steady-state conditions was obtained for ethene hydration in an isothermal reactor with a fixed catalyst bed consisting o£ Pt ont-A1203 (ref. 22). The reagent concentrations at the reactor inlet were changed according to a stepwise rectangular function: for ethene, in the region 0-50% (v/v); for hydrogen, 100-50% (v/v). The results were then compared with those obtained under steady-state conditions for a reaction mixture composition at the reactor inlet corresponding to mean composition values of a cycle in the non-steady-state regime. The rate of ethene conversion was strongly dependent on the concentration oscillation frequency, being highest at period of about 10 s. The average observable ethene hydration rate in this case was 50% higher than that under steady-state conditions with identical mean concentrations of H2 and C2H in the gas phase. This increase in conversion 4 rate must be related to the kinetic characteristics of the elementary reaction steps. If it were possible to eliminate the infl\tence of the reactor volume on the variation of the gas phase conditions, the peak performance would surely be reached in the so-called slidip~ regime with an high frequency of the gas phase concentration oscillations. Because of the damping properties of the reaction volume, the incremental constant variations in the composition at the reactor inlet are not evident when they reach the catalyst surface. The influence of the reactor volume is also seen at periods commensurable with the time during which the reaction mixture is present in the contact zone, which is a few seconds. At very high frequencies the reactor operates under conditions corresponding to invariable and mean-integral (over the period) values of the concentrations. Ref. 23 describes the formation of ethyl ester from ethene and acetic acid. Sulphuric acid implanted in the inner surface of silica gel particles served to catalyse the reaction. The efficiency of the catalyst was studied in a fixed-bed reactor under both steady- and non-steady-state conditions. The unsteady state was created by a periodic stepwise feed of acetic acid into the reactor inlet. The ethene flow-rate was not effected in this process. The optimum period of oscillation of the acid concentration was about 30 min. Under invariable conditions the
18
catalytic activity reached a constant value after 400 h of operation. It must be noted that the catalytic activity gradually increased to a maximum within the first 100 h and then decreased approximately by half. Under periodic performance of the process the product output largely depended on the period and on the relation between the acetic acid feed time and the time of zero concentration at the reactor inlet. The catalytic activity observed in the first 100 h of non-steady-state operation coincided with the peak value for the steady-state case and remained stable for a longer time. This allowed for a sufficient increase in the reactor productivity compared to the optimum steady-state regime. An investigation of the influence of the inlet concentration oscillations of ethene on the selectivity of its oxidation to ethene oxide on silver was carried out under isothermal conditions (ref. 24). It revealed the following phenomena: (a) transition from steady-state to periodic control leads to a decrease in the extent of conversion; (b) an increase in the period causes a decrease in the conversion extent; (c) with a decrease in conversion extent it is possible to increase the selectivity under non-steady-state conditions. The same reaction was studied under non-isothermal conditions at atmospheric pressure in a 30 mm tube reactor filled with a silver-containing catalyst (ref. 25). The inlet ethene concentration was periodically varied stepwise from 0 to 12% (v/v). A considerable increase in selectivity was obtained at a period of about 30 s and at identical values of the conversion extent related to the value under steady-state conditions. Fig. 1.1 shows experimental data concerning the process selectivity under conditions of practically invariable conversion extent and temperature fields in the reactor. Attention should be paid to the fact that though the productivity of the non-steady-state regime characterized by the output of ethene oxide was somewhat lower, nevertheless, the selectivity remained within 50-60%. Ref. 26 reports the non-linear dependence of the ethene formation rate on the extent of oxygen coverage of the catalyst surface. A maximum in the formation rate is observed at a
19
coverage of 0.5-0.6. This, evidently, can-be explained by a structural modification of the metal surface, causing a change in the type of link between metal and oxygen as the result of the latter's interaction with both the surface and near-surface layers of the catalyst. Penetration of oxygen into the near-surface layers of silver results in their modification, which may also be induced by some other electronegative elements, as reported in ref. 27. The similarity between the deeply adsorbed oxygen and electronegative promoters and the character of the activity and selectivity change after application of the promoter allow one to suppose that the effect of an increase in selectivity in the oxidation of ethene in the cyclic non-steady-state regime is explained by a decrease in activation energy in the stages determining full and partial ethene oxidation. The most noticeable energy drop occurs in the process of partial oxidation. The use of non-steady-state conditions may permit operation with higher concentrations of adsorbed oxygen, which is the reason for the selectivity improvement. We cannot as yet exclude the possibility that the maximum selectivity at a period of 30' s is connected with the dynamic properties of the reactor and not determined by the dynamic properties of the catalyst surface. Periodic oscillations in the alcohol concentration at the inlet of an isothermal tUbe reactor with a fixed bed consisting of Al 20 particles resulted in a selectivit~ increase in the J dehydration of ethanol (ref. 28). A pronounced influence of the frequency and asymmetry of the inlet function of the ethanol output was observed. The optimum set of parameters raised the ethanol output two-fold as compared to that obtained under steady-state operation. Now let us have a look at the process of butadiene hydration on nickel under non-steady-state isothermal conditions (ref. 29). The tests were carried out in a flow reactor at 70°0. Theoretical calculations were made according to an ideal substitution model for isothermal conditions. The concentrations of butadiene, butene, hydrogen and butane as well as these of the three supposed intermediate compounds on the catalyst surface were taken into account. The inlet concentrations of hydrogen and 04H6 were changed stepwise in antiphase; the total
20
........
~ ...., CI) <;j ~
>. +> •.-1 :> 'r-! +> 0
(!)
20
fa
r-I
Q)
02
'H 0 III
§
,.q 0
duration of cycle, t c (8)
Fig. 1.1. Ethene oXidation to ethene oxide under non-isothermal conditions on a silver catalyst. Relative selectivity change (~S) as a function of the period, t , at invariable ethene conversion extent (0.05) and constaRt temperature field in the reactor. ~S = 0 corresponds to steady-state conditions. concentrations of both hydrogen and butadiene were constant. The optimum steady-state regime found by calculation and tested in practice served as a comparison. It had the following parameters: extent of conversion, 50%, selectivity of butene formation, 60%. In the non-steady-state operation of this process, the selectivity of this chemical transformation could be increased by 20% with the conversion extent unchanged. The period of the cycle was varied in the range between 2 and 30 8 and the amplitudes of oscillation in the range from 0.2 to 1.0. As seen in Fig. 1.2, the butene productivity, 2 ,is increased while the duration of the cycle is reduced and the selectivity, S, is increased with increasing oscillation time, t c• This may be explained by the following hydration mechanism. The rate of butane formation is largely dependent on the hydrogen concentration. With a decrease in the oscillation period there is an increase in the amount of butadiene adsorbed leading to an increased rate of butene formation and to an even more marked
21
increase in the rate of butane formation. The mean adsorbed butadiene content is decreased due to slow operational switches, providing thereby an increase in the process selectivity.
0.2S
0,50
0.2L(
. . . ,."ifF-
0.23
-Q,,)
-0----01
----- __ L_
0.22
o.os o.vs
al PI al
0
rl 0
>
'-J
Cf.l A
OJ{I{
:>..
+>
-r-i
A
:>.. +> 'n 0
*.
........
>
0.2t
OJ{2
'n
+> 0
Q)
rl
0
L(
t6
20
2'"
O"{Q
Q)
Ul
duration of cycle, t c ( s) Fig. 1.2. Bench-scale and experimental data (marked with circles) regarding the productivity and selectivity of butene hydration on a nickel catalyst in the periodic regime. 1,2 are the optimum steady-state selectivity, S, and productivity,g • Interesting data are reported in ref. )0 concerning Fischer-Tropsch reactions of paraffin synthesis on a Ru/.r -A1 20 catalyst. 3 This study shows that the distribution of the reaction products according to their molecular weights largely depends on the conditions used to achieve the non-steady-state. Impulses of hydrogen and a CO/H2 mixture were periodically blown down through the inlet of the fixed catalyst bed. With a period of 8-12 min, there was a considerable narrowing of the distribution function. In the steady-state regime the ratio of concentrations of the components C12H26 and C6H14 was 0.74 and it was as low as 0.12 under non-steady-state conditions. Considerable variation of the molecular weight distribution of the products was observed in a Fischer-Tropsch process on promoted ferrous catalysts at a pressure of approximately 0.4 mPa and a temperature of about 250°C (ref. 31). The methane
22
output was significantly increased owing to the cyclic regime of the reaction. Ref. 32 reports the results of experiments with Pt/A1 20 3 catalysed non-steady-state reactions at 505°C. The time-averaged mixture composition of CO, NO and O2 was oxygen-stoichiometric. The concentrations of CO and NO were constant at 1% and 0.1%, respectively, whereas the oxygen concentration was periodically varied at different amplitudes averaged over 0.4%. In this case, forced non-steady-state operation of the process did not produce any effect on the reaction efficiency. The extent of oxidation of CO and of reduction of NO monotonously increased in parallel with the oscillation frequency, tending to the steady-state values corresponding to the mean values of the oxygen concentration. Considerable changes in the rates of carbon monoxide oxidation and NO reduction were observed during variation of the oxygen concentration (during the period, t c) from 0.24 to 0.66% and at a period of about 1 s. Oxidation of CO in the non-steady-state regime on a supported platinum catalyst has also been studied (ref. 33). The reaction mixture was fed through the inlet of a non-gradient isothermal reactor with a periodic variation of the mixture composition: a mixture of carbon monoxide and argon was blown in during the first part of period; during the second part of the period an argon-oxygen mixture was applied. The process was carried out at a temperature of approximately so-c, carbon monoxide concentrations of 0-2% and oxygen concentrations of 0-3%. The maximum cycle duration was about 3 min. A 20-fold increase in the reaction rate as compared to that obtained under steady-state conditions was possible in the non-steady-state regime. The maximum in the reaction rate occurred at a cycle duration of about 1 min. The authors explained the results of their experiments assuming the following reaction mechanism: (1)
CO
+
[z]
(2)
O2
+
2 [Z] -:::= 2 [OzJ
(J)
[COZ]
+
~
[C02]
[ozJ - CO 2
+
2[ZJ
23
(4)
co + [oz] -- CO 2 + [z]
Supposing that carbon dioxide formation is mostly determined by the rate of the third stage, this rate is highest when the concentrations of surface forms, [COl] and [Ol] , are about equal. Then, a considerable increase in the carbon dioxide formation rate under·non-steady-state conditions follows from the fact that concentrations of surface coatings were kept in the vicinity of optimum values. At the same time, in the case of a steady-state operation, the extents of surface coating by the intermediate substances, [cOZ] and [Ol] (as independent tests have shown), are quite different and their product is a small value. Similar experiments were performed by Barshed and Erdogan (ref. 34) in a flow isothermal reactor with periodic variation of the carbon monoxide and oxygen concentrations at the inlet of the fixed catalyst bed. The minimum oscillation frequency was 0.2 Hz. The authors demonstrated that the dependence of the observable rate of carbon monoxide and oxygen interaction on cycle duration has a maximum value if the cycle duration is commensurable with the time-scale of the reaction carried out on the catalyst surface, i.e., with the dynamic properties of the catalytic cycle. Oxidation of carbon monoxide in the non-steady-state regime was also investigated on a supported V 20 catalyst (refs.35,36). 5 The ratio of the partial pressures of CO and 02 (CCo/C 02) was periodically varied according to an incremental constant function at the inlet of the reactor with a fixed catalyst bed. The variation took place around different averaged values. The oxidation rate appeared to depend on the initial mixture composition and on the oscillation period. If the mean values of the CCo/CO ratio are not sufficient, the influence of the amplitude Becomes more and more noticeable. It seems of interest that the dependence of the averaged over the cycle rate of chemical conversion on the period has three regions of resonant frequencies corresponding to t c = 1-2 min, 15-20 min, and 40-45 min. The rate at these frequencies is 1.5-2.5 times higher than in the steady-state regime. To explain the results obtained
24
the authors tried to resort to a well known hypothesis of carbon monoxide oxidation mechanism on a vanadium catalyst. However, they could achieve neither quantitative nor qualitative correspondence with the facts. In the ideal models used in their investigation, which contained stages of sorption and reactions on an homogeneo~s surface, the duration of the period was considerably shorter than that observed experimentally and which allowed for a pronounced increase in the oXidation rate under non-steady-state conditions (ref. 37). Temperature variation in a cyclical manner applied to the same system also proved of little avail in increasing the efficiency of the process (ref. 38). Let us consider some examples of catalytic process at the catalyst unsteady state related to its periodic regeneration in the fluidized bed. First, the OXidation of ortho-xylene on a catalyst containing 3.2 mol.% of V20 0.64 mol.% of Te0 2 and 5, 96.2 mol.% of Ti0 2 (ref. 39). The activity and selectivity of the catalyst in its preliminary oxidized state practically coincide with the corresponding parameters for a steady state. A preliminarily reduced catalyst is characterized by an higher activity at any extent of conversion, x, and its selectivity rises if x is large and slightly decreases when x is small. A change in the extent of conversion under non-steady-state conditions leads to a considerable variation in the oxidation state of the catalyst: at small values of x the catalyst is reduced and its oxidation state increases as the value of x increases. If the catalyst is formed at small values of x and is continuously placed in the zone of higher conversion extents, one expects an increase in phthalic anhydride output as the result of the decrease in the relative rate of its additional oxidation on the reduced catalyst. This was confirmed in a pilot reactor with a pseudo-fluidized catalyst bed. A directed circulation of the catalyst in the reaction zone was created by means of its reverse flow against the interacting mixture and by returning it to the upper part of the apparatus. The mean catalyst residence time in the reaction zone was approximately 20 seconds. A change in x was achieved by variation of the reactor temperature. When the process is carried out in the fluidized bed organised with the help of special devices and if
25
conversion extent lies between 0.8 and 0.95, the output of the partial oxidation products appeared to be 10% lo~er than in the case of the catalyst circulation. This can be explained by the fact that if the catalyst is not directly circulated, it tends approximately to a steady-state which corresponds to a conventional operation in the fixed catalyst bed. Such a state in the organised fluidized bed was possible due to a small value of the time-scale of the transition regime on the catalyst surface. This seems to be the most reasonable explanation at the moment. The process of oxidative dehydration of n-butene on a bismuth-molybdenum catalyst at 480-500°C was carried out in a twin-reactor system with a fluidized bed (ref. 40). The catalyst was circulated between two apparatus, one of which served as the reactor and the other as the regenerator. Fig. 1.3 shows the dependence of selectivity on the butene conversion in various operational regimes. One can see that the highest selectivity at constant butene conversion is reached when butene and oxygen are separately fed into different apparatus: C S is fed into the 4H reactor while 02 is fed into the regenerator.
90
.......
*
.-; 0
::-
80
70
'-' u.l ~
?>
:> 50
-r-i
~,
2
60
.p
'r!
~f
-0-----00-0-.
.p
o
QJ
.-;
QJ
m
60
70
80
go
extent of conversion, x(vol.%)
Fig. 1.3. Dependence of selectivity, S, in the oxidative dehydration of n-butene in various operation modes: 1 = "reactor-regenerator" system without oxygen supply into the reaction zone. 2 = the same system with the oxygen supply; 02/C4HS = 0.2. 3 = fixed bed of the catalyst; 02/C4HS = 1.5. 4 = fluidized bed of the catalyst; 02/C4HS = 1.5.
26
Investigation of the catalyst under non-steady-state conditions (ref. 41) indicated that in the process of butene dehydration without oxygen supply, it (the catalyst) has a large period of continuous activity. At the end of this period, the butadiene output gradually decreases while the selectivity is unaffected. A certain oxygen volume of about 15 monolayers covering the internal surface of the catalyst corresponds to the period of constant catalyst activity irrespective of the conditions. Oxygen treatment of the catalyst regenerated its initial properties. Thus, the oxygen necessary for the reaction to be carried out was introduced together with the catalyst, the latter taking the role of the oxygen-transfer agent. A redox mechanism is sure to take place in this process. According to this mechanism, the adsorbed n-butene interacts with the surface oxygen of the catalyst. Should it (oxygen) b€ absent in the gas phase, it is diffused to the surface from the catalyst volume. The rate of such a supply is rather high if the content of applied oxygen is less than that corresponding to about 15 monolayers. In the course of the reaction, the rate of oxygen thrust onto the catalyst surface starts to set a limit on the dehydration process and the observable rate of the main process is reduced. During the stage of oxidative dehydration one can detect a slow deposition of carbon compounds. The beginning of rapid deposition of such compounds coincides with the time at which an extent of reduction of the catalyst corresponding to the beginning of a noticeable selectivity decrease is attained. Carbon compounds are easily oxidized by oxygen blow and the catalyst is thereby regenerated. The catalyst in the above process may continuously serve as the oxygen-transfer agent only under conditions enabling its reduction corresponding to the removal of approximately three oxygen monolayers. At higher reduction states of the catalyst metallic bismuth and M0 20 are likely to appear. 3 Evidently, one can always expect a change in the efficiency of the catalytic process under forced non-steady-state catalyst conditions, provided these processes are carried out according to different mechanisms: in particular, such redox reactions as full oxidation of hydrogen, carbon monoxide, hydrocarbons and
27
many other organic substances at high temperatures as well as partial oxidation of olefins, alcohols and aromatic compounds. Interaction of an oxidant, one way or another, with the regenerated catalyst introduced into the reaction zone (in presence of the oxidant or not) may often allow one to increase the activity and/or selectivity. This increase may be due to the catalyst's ability to maintain under the non-steady-state regime its optimum state With respect to the energy of oxygen binding with the surface. Another illustration may be the process of naphthalene oxidation to phthalic anhydrite on vanadium (ref. 42). The catalytic activity in this process increases with the degree of oxidation, O. The selectivity of the process does not depend on 0 when it is over 0.3. The process was carried out in a "reactor-regenerator" system and only 25% of the catalyst body was stored in the apparatus. The catalyst circulating factor was equal to 6 inVerse hours. In the pilot reactor at a regeneration temperature of 385°C, the yield of phthalic anhydride reached 87% at 8 ~ 0.35, while without regeneration these values only were 82% and 0.29, respectively. At a temperature of 400°C, the oxidation extent was 0.43, the catalyst productivity increased 1.5 times and the phthalic anhydride yield was constant at 87%.
1.4
OPTn~IZATION
OF THE CATALYST UNSTEADY STATE
Upon consideration of the above-mentioned facts one can conclude that there have been many experimental studies on heterogeneous reaction performance under non-steady-state catalyst conditions. There is no doubt now that the transition to an non-steady-state regime offers a favourable increase in the process efficiency compared with that obtainable in the steady-state. Still, the effect of the efficiency increase, as we have noted, cannot always be explained by purely quantitative factors obtained through mathematical simulation on the basis of independent kinetic research. This presents serious difficulties in non-steadY-state process control and in determination of the optimum conditions for these processes. At the same time there have been a number of theoretical attempts to distinguish classes of chemical reactions according
28
to simulated kinetic schemes ~or which the possibility o~ increasing the catalytic process e~~iciency under periodic variation o~ control parameters has been demonstrated. Therewith arouse the cyclic optimization problem which is related to the traditional theory o~ optimum control. In the same time it has a number o~ speci~ic peculiarities on which we shall touch ~urther. The regimes o~ periodic character allowing ~or a considerable increase in process e~~iciency as compared with the optimum under steady-state conditions are thought to be the main goal of such investigations. Let us take as an example the mechanism of a catalytic process illustrating the efficiency of the forced non-steady-state regime (ref. 43). 1.4.1
An example of an efficient non-steady-state regime
Consider the case where the efficiency of the process can be increased due to periodic oscillations of the gas phase concentrations ~~th invariable rates of the elementary steps. In other words, only non-steady-state of the first type will be taken into account. The efficiency gain in this situation may be caused by the oscillative regime in the reactions o~ heterogeneous catalysis enabling control of the concentration of the chemisorbed particles, leaving the average concentration of the reagents in the gas phase unaffected and, under certain conditions, enabling variation of the productivity and selectivity. Let substances A and B interact according to a one-route mechanism in the presence of a catalyst with periodic variation of the concentration of A in the gas phase. Two possible variants are open for discussion. In the first instance the initial substance A is adsorbed on the catalyst surface and interacts with substance B in the gas phase according to the collision mechanism. In the other case, the initial substance B is adsorbed and the end product is formed as the result of interaction of A (contained in the gas phase) and the adsorbed reagent B via the collision mechanism. An elementary scheme describing the formation of the product C for the first case can be represented as: kf
A +[Z] ~ [Al]
(I)
29
B + [AZ]
1£ .z:
[Z] + C
The second case as
(1)
in which product D is formed can be represented
B + [l] (II)
A + [BZ]
ft
-!.. [Z] + D
where [AZ] and [BZ] are the products of the chemisorption, and [Z] represents the vacant active surface area. Kinetic models corresponding to mechanisms (I) and (II) can be derived
L
d[Al] dt
=
1£( CA rt-[AZ]) - k_f [AZ] - h 2 CB [AZJ (1.5b)
where CA and CB are the concentrations of substances A and B in the gas phase, L is the number of active centres per catalyst surface unit, k i are velocity constants (i = 1, 2, 3, 4, -1,-3). Eqns. 1.5a and 1.5b were derived upon consideration of the conditions of the surface concentration balance: [Z] + [AZ] = and [Z] + [BZ] = 1. Let us divide eqn. 1.5 by a normalization coefficient, kC, to make it dimensionless. Then we obtain
d [AZ] _ r , dt' - K, CA (I-[Al]) - K_ 1 [AZ] -~ CB [AZ]
(1.Ga)
(10Gb)
30
where K1 = ki/kC, i = -1, -3; Ki = k;/k; i = 1, 2, 3, 4; C~ = CAlC; CB = CB/C; t'= tkC/L; a dimensionless period, t c' equals tckc/L. For simplicity, the prime symbols will be omitted. The formation rates of C and D averaged over the dimensionless period will be equal to
~
~
tc
1
~ K2 CB [AZ] db
=-
t
c
(1
.7 a)
0
and
W
D
1 =-
to
tc
~ K" CA [BZ] db 0
Let us assume that the concentration Cn is invariable during the whole period and CA is varied according to
2CA • C = A
{
o
(1.8)
where CA is the period-averaged concentration of substance A. The concentrations of the intermediate substances in the steady cyclic regime are identical both at the beginning and at the end of the cycle:
31
The reaction rates under non-steady-state conditions will be regarded as the rates averaged over the period (see eqn. 1.7). Now let us compare the reaction rates in this case with the values under steady-state conditions determined by the period-averaged values for the control actions. In the steady-state regime under the above-stated conditions and where CA = CA = = constant, the reaction rates according to mechanisms I and II have the forms:
(1.10)
where
k
he
aC f
f !f.,g
k_ f
=
1£1 ILf
!l;])=
it.5 kif IL,J
a2C
k
z
!f,-1
Fig. 1.4a shows the dependence of the difference in the .....reaction rates, .fJc = We - RC and.P D = liD - RD, under both steady and non-steady-state conditions on the oscillation period at K1 = K_ 1 = 3; K2 = K = K = 1; K_ = 0.3; CB = 1; CA = 2. 3 3 4 Values of Wc and WD were derived analytically from eqns. 1.6 with consideration of the conditions described in eqns. 1.8 and 1.9 by means of substitution of the obtained solutions into ~
eqns. 1.7a and 1.7b and their integration.
32
0
.fJc .f};
.p oj
H
'l-i
a)
0.1
0
§
,.q 0
0.2
duration of cycle, t c (s)
tr.l
0.1
O.t
Ps
Pc
Pn
.p
0
0
oj
'"
~
.p
.,;
>
'M .p 0
r-l
H
o:l
'l-i 0
'l-i 0
go
Q
-0. t
-O.t
ro
,.q
§ ,.q 0
0
-0.2-
-0.2.
o.t
fO
duration of cycle, to (s)
100
1000
b)
33
0.1
O. f
">
.
V)
»
a
+>
'ri
P+>
.Pc ..PlJ (1)
+> al
'ri C)
(1)
-O.t
(1)
OJ
1-1
'H
'H
0
0
(1)
§ .q
(1)
a hi)
c)
rl
-0.2
.q
C)
C)
duration of cycle, t c (s) Fig. 1.4. Difference in reaction rates,.pr.(1)'.Pn~2), and selectivities, Ss (3) in the steady- and non-stenay-state regimes vs. the oscillation period, tel of the concentration of A in the gas phase for a one-route (al, two-route (b) and three-route (c) mechanism. At small periods of oscillations the so-called sliding regime is asymptotically realized. The concentrations of the intermediate compounds are practically unaffected during the period OWing to its small time and remain constant, equalling the steady-state values at corresponding average values of the control parameters. If the oscillation periods are extensive, the so-called quasi-steady-state regime is asymptotically realized, where the surface concentrations at every moment of time take up their steady-state values, corresponding to the values of the control parameters at the same moments of time. Taking into account eqn. 1.8 and that t c --- 00 we obtain:
~ f W=-R c
2.
C
I
C A
=
2~
As the steady-state reaction rates, Re and RD (see eqn. 1.10),
34
are lower than the first-order reaction in A, the reaction rates in the quasi-steady-state regime have smaller values than in the steady-state. With increasing magnitude of the period a monotonous transition from the sliding regime to the quasi-steady-state occurs within a comparatively shorter interval of the proper dynamic regime where unsteady properties of the system (eqns. 1.6) are significantly pronounced. The dynamic regime is realized when the value of the oscillation period of the concentration of A is measured by a factor of the characteristic time MAZ of the transient regime of the intermediate substance [A~, , or by a factor of the characteristic time M BZ of the substance [BZ]. For eqns. 1.6 these characteristic times can be estimated as follows:
The reader is reminded that if another type of reaction mechanism is considered, for example, where the steady-state reaction rate has a larger order than first, i.e., in the concentration of A, the observable reaction rate in the cyclic regime is always higher than that under steady-state conditions. It would also monotonously increase with increasing of oscillation of CA period. Dynamic properties in the second reaction mechanism (II) can become more visible at greater periods in comparison with the first reaction mechanism (I) owing to the different characteristic times of the transient regimes on the catalyst surface. Now let us make the problem a little more complicated, with two parallel reactions, A + B--C and A + B-D, on the same catalytic centres, [Z], in conformity with the two-route mechanism, (I) - (II). The corresponding kinetic model with [z] + [AZ] + [BZ] = 1 will have the following dimensionless form:
35
If D is the useful product, the process selectivity is determined by the expression (1.12)
where Wc and WD, as before, are given by the expressions 1.7a and 1.7b. For a steady-state regime with time-invariant concentrations CA and CB' the kinetic characteristics will be:
S=RD jrR D +R C )
Rc = Kf K2
cAeB (
fl = Kj C
i«, + K2 CB ) + ( K_J + K'I CA )( K CA + r, + K2 CB )
B
K -.7 + Kif CA ) / jU ;
f
As is seen in Fig. 1.5, the depenednce of the steady-state rate of formation of product D, RD on the concentration of A is of an extreme character. It is almost linear if the value of CA is small. An increase in CA leads to such an increase in the concentration of the intermediate compounds, [AZ], that the content of BZ is dramatically decreased. The quantity of product D which is produced is proportional to the concentration of BZ. Fig. 1.4b showed the dependence of the differences between the rates for reactions jJ C and P D and between the selectivities Ps = S - S in both the steady- and non-steady-state regimes relative to the oscillation period of the concentration of A (see eqn. 1.8). The parameters are identical to the ones selected for the previous example. Values of W c and W D were
36
derived from the solutions of mechanisms I and II with conditions 1.8 and 1.9 by their substitution into eqn. 1.7 and integration. It is clear from Fig. 1.6 that with increasing period, t c' there is a transition from the sliding regime to the proper dynamic one. With the set of constants determined above, the characteristic times for variation of the concentration of intermediate compound AZ are smaller than that for BZ. That is why, with small values of t c' [AZ] begins to change in time following the oscillations of CA' whereas [BZJ remains practically invariable over the entire period.
---.
~~
"-
I~~
0.8
0.8
~<:r ><:
0.6
0.6
Q)
tf)
H
>. +> 0..-1
+> (Ij 1=1
0 0..-1
~
OJI
OJ!
>
•..-1
+>
+>
0
0
(Ij
OJ
H
Q)
r-I
0.2
0.2
QI
Ul
Fig. 1.5. Dependence of the rates of the chemical reactiQllS and selectivities in the stea~-state(l =_R~, 2 = RD, 3 = S'C R = 1) and non-steady-state(4 = W ,5 = s, C ~t) = 1) regimes on the average concentration ~A fBr the proce~s described by mechanism (I) - (II). The dot (6) designates the rate of formation of D(J ) at optimum control u 1(t)= CA, u 2(t) CB, in addition 1
~A - 2, ~B
= 1.
=
As a result, averaged over the period, (AZ] is decreased resulting in a dearease in the rate of formation of C, WC' and an increase in the average concentration of BZ. As long as (BZ] is little affected in the course of the period, there is a gain in the rate of formation of TI, W D, and a rise in selectiVity.
37
When the duration of the period, t c' becomes commensurable with the characteristic time of [BZ] variation, the absolute values of BZ also begin to change significantly. A further increase in t c leads to a decrease in W D and in selectivity. When t c--= a quasi-steady-state regime is realised with proper performance characteristics as compared to the steady-state operation. The values of RD and S in the steady-state regime at CA = 2 are 0.269 and 0.367. For a cyclic non-steady-state regime with identical values of average concentrations the peak value 0.344 is attained at t c = 1.1. The selectivity maximum (3), 0.464%, is attained at t c = 1.7 (see fig. 1.5). fAZ]
0.50 I=-+-~""----'I""":'-t-~~-------l
a)
duration of cycle, t/t c
[HZ] 0.75r---------------------,
b)
o
0.5"
0.75"
duration of cycle, t/t c Fig. 1.6. Change in the concentrations of surface substances [AZ] (a) and [BZ] (b) in a cyclic non-steady-state process carried out according to mechanism (I) - (II) with the dimensionless period, t c = 0.01 (1); t c = 0.7 (2); t c = 4.3 (3).
38
Thus, in this example the efficiency of the cyclic process is provided by a vast difference in the characteristic times of the surface concentration variations. The maximum period for the main reaction in this case is the result of the cc etition of the adsorbing substances to occupy as much of the catalyst surface as possible. Let us consider a three-route reaction mechanism when one of ~he reagents is completely dissolved in the catalyst volume: parallel reactions, A + B-- C and A + B - D, are carried out on the same active surface sites in accordance with the first and the second reaction mechanisms with addition of the fifth step, for example, dissolution of the adsorbed substance B in the catalyst volume: (1)
A +
[z]
(2)
B
+
[z]
(J)
B
+
[AZ]
( 4)
A
+
[BZ]
(5)
[BZ]
+
--s.,----s., ----
[V]
K{
K.J
K2
f5!-
--KJ
K- s
[AZ] [BZ] C
+
[z]
D
+
[z]
[BV]
+
(III)
[z]
Here [V] and [BV] are the concentrations of the vacant and occupied sites in the catalyst volume. A kinetic model of the mechanism with consideration of the balances [Z] + [AZ] + [BZ] = 1 and [V] + [BV] = 1, may be represented in the following dimensionless form
d[BZ] di
=
K" CB (f-[AZ]-[BZ]J - K_J[BZ] - XII CA [Bl] +
+ Ks[BZ](t-[BV]J + K_s[BV]( f -[AZ] - [BZ])
=
39
where ~ = L*/L and L* is the number of sites in the catalyst volume available for penetration of substance B. The existence of the fifth stage does not affect the kinetic characteristics of the process under steady-state conditions, which have the same appearance (see eqn. 1.13). Reaction rates (eqns. 1.7a and 1.7b) undernon-steady-stateconditions were calculated by means of integration of eqns. 1.14 with consideration of the conditions 1.8 and 1.9 and BVlt=o = BVlt=~. The selectivity, S, of the non-steady-state regime was calculated via expression 1.12. Fig. 1.4c showed the dependences of the differences between process indexes under both steady- and non-steady-state conditions on the oscillation period of the concentration of A of the sort described in (1.8) with the same values of the parameters described above. It was also assumed that K = K_ = 5 5 = 50 and c£ = 10. In Fig. 1.4b and 1.4c it is easily seen that the reaction rate and the selectivity are similar (qualitatively), depending on the cycle duration, with the exception of the case when the function (see Fig. 1.4c) was at its minimum. This is explained by the fact that the presence of the fifth stage leads to a greater process sluggishness as regards variation of [BZ]. As a result, the unsteady character in relation to the reaction rate, R, is observed at large values of the period. If the fifth stage is carried out at low rates, the exchange processes With the catalyst volume can be observed only at large periods of the gas phase oscillations. If the exchange processes are rapid enoU$h the and correspond to large values of the constants, K and K_ 5, 5 second and the third equations in system 1.14 could be replaced by the following:
+
K C [Bl] -
(1-[AZ]J2
d[AZ] d/;
40
For simplicity, we assume K be equal to K_ We can see now 5 5• that the catalyst surface sluggishness relating to [BZ] is increased by approximately 1 + ce times, the value for [AZ] remaining low. An increase in [AZ] only makes the catalyst sluggishness more pronounced. Naturally, the fifth stage does not influence either the steady-state dependences of the process or the quasi-steady-state values of Wc, W D and S, which occur if the period, t c' is long enough. That is why the asymptotic values of the corresponding process rates and selectivity at t - 0 and t _ oo concide (see Fig. 1.4b and 1.4c). The maximum value of the rate, RD, equalling 0.314 (Fig.1.4c) is attained at t c ~ 1.8 and the maximum selectivity, 0.478, at t c ~ 7.9. Some decrease in the maximum rate in comparison with the above examples (Fig. 1.4b) is explained by partial consumption of the intermediate substance [BZ] to [BV] formation in the course of the first semi-period. An increase in the selectivity maximum is provided by a greater separation of the characteristic times of the dynamics of the surface concentration variation. Let us estimate the real period values at which the maxima of the reaction rate, W D, and selectivity, S, are attained in examples I and II. The values of the rate constants for the individual stages shall be of the order that the absolute values of the conversion rates are of practical interest (k = 10 18 m2s~ C - 0.1 mPa, L = 1.4 10 19 m 2 , specific internal surface of the catalyst S = 50 m 2. g -1), then in a steady-state regime with C, = 0.2 mPa and Cn = 0.1 m Pa, one has RD = 0.55 ml/g·s, R = C Jl .u = 0.95 ml/g-s and the normalization factor, L/kC = 14 s. In this case the maxima in the rate (RD = 0.64 ml/g·s) and the selectivity in the second example are reached with periods of 15.4 and 23.8 s. respectively.
41
In the third example the possibility of dissolution of the adsorbed substance in the catalyst volume is taken into consideration. For example, we know that oxygen can readily be dissolved in both oxide and metallic catalysts. The same is true of hydrogen in metallic catalysts. The process of dissolution provides a resistance to transient regimes in catalytic systems and the dissolved substance chemically interacts with the catalyst. The catalyst energy properties may be modified and, consequently, there may be a change in the activation energies of the various catalytic reaction steps. In this part of the chapter the modifications of the catalyst properties under non-steady-state conditions, which could also lead to a large increase in efficiency, are not considered. Introduction of the dissolution stage in the present case only served to show the vast difference in sluggishness of intermediate compounds of the catalyst surface. As a result, with periods of 25.2 and 110.6 seconds, respectively, the peaks in the rate, W D, and the selectivity, S, are reached. k, C and L have identical values to those in the previous example. These considerations and quantitative examples allow one to express a general idea. Let two parallel reactions be carried out on different catalytic centres and each reaction occur according to the two-step mechanism. Then, the rate of each reaction is changed in a monotonous manner with increasing period of oscillation of the reagent concentration. In addition, considerable variations in the reaction rate in the dynamic regime occur as soon as the characteristic time of the given reaction route dynamics gets shorter. In the present case the characteristic time is determined by the time of the transient regime measured for the intermediate substance of this reaction route. The selectivity of a reaction route, the dynamics of which may be determined by a more complex dependence, should, evidently, have extreme values in the following cases: (1) Both reactions have an observable order less than first relative to the controlled concentration. The characteristic time of the dynamics of the reaction leading to the end product is greater than that of the concurrent reaction route. This
42
example of a simple reaction mechanism has already been discussed. (2) Both reactions have orders higher than first relative to the controlled concentration. The characteristic time of the dynamics of the main reaction route is smaller than that of the concurrent one. This case corresponds to one described in ref. 44 where, in addition, some simplifying assumptions were made. The reaction rates under the forced non-steady-state conditions in both cases are of monotonic behaviour in the course of period variation, for example, as reported in (ref. 45). The maxima in the reaction rates may appear when there is a competition between the surface substances participating in different reaction routes for the active centres. This phenomenon occurs in the above examples I - II and III. The scheme of stages I - II may probably be applied to many catalytic reactions. Isomerization, partial oxidation, fermentation reactions and other catalysed reactions can be carried out according to this (or a similar) scheme. Of course, the quantitative character is meant rather than the scheme's total quantitative correspondence. To make these schemes applicable to processes of partial oxidation one can interpret them as follows. If A is an hydrocarbon and B is oxygen, then adsorption of the hydrocarbon leads to the formation of oxidation products. The collision reaction mechanism leads to partial oxidation of olefin (en H 2n ) carried out in conformity with the following stages (1)
( 2)
A +
[zJ ~ [AZJ
== [JZ]
°2
+
[AZJ
0) °2
+
[z] ~ [02ZJ AO
( 4)
A +
(5)
(1.5n - 1)02
(6)
2 [ozJ...f\- [02z J
[ °2 zJ +
[JzJ +
[ozJ
+
J'L.
[zJ
nC02
+
nH20
+
[z]
43
where A is olefin, [JZ] is an intermediate compound easily transformed on the catalyst surface during the fifth reaction stage, AO is the olefin oxide and the sign --fI- designates a rapid irreversible reaction. The kinetic model for this reaction mechanism coincides with the model discussed above within the accuracy of the stoichiometric coefficients. The kinetic model for the mechanism (1)
A
+
[z]
(2)
B
+
[B]
(3)
A +
--
[AZ] [BZ]
[BZ] -
D +
[Z]
in which a portion of the active surface is occupied by the adsorbed and inert (in the sense of interaction) substance, A, is also identical to the above equations, where the concentration of B is constant. Considerations of the qualitative character discussed here may be applied to a great many multi-route catalytic reaction mechanisms including non-linear ones. For example, in reactions which are carried out according to the two-stage schemes
[z]
[AZ]
(1)
A +
(2)
[AZ] -
(3)
B2
+ [A,Z] -
( 4)
B
+
(5)
A +
[BZ] -
(1)
A
[Z] ~ [A 1Z]
( 2)
[AZ]
2
~
[A,Z] C + [z]
(IV)
2[ZJ ~ 2[BZ]
D + [z]
and +
+
[BZ] -
D +
2[ZJ
(V)
44
0)
B2
+
2[Z]:;:'= 2[BZJ (V)
an extremal dependence of the rate of formation of D on the period of oscillation of the concentration of A in the cyclic non-steady-state regime is observed. This all can only take place in case of large sluggishness of the reaction route toward substance D formation.
1.4.2
Statement of the optimization problem and ways to solve it
Fig. 1.7 shows calculational results for example I-II where one can distinguish three regions of the cyclic process. In the first region (the so-called sliding regime) the duration of the oscillation period, t c' is so short that it is less than the time-scale of the transition process on the catalyst surface: t c « Mc• In Fig. 1.7 this region corresponds to plane SOA and to some small value in the vicinity of t > O. Due to the linear c dependence of the rates of the catalytic steps I - II on the concentration CA at short duration of the cycle, the situation is a little different from that in the steady-state reaction, corresponding to point SS. As a general case, profile 1, of course, may show some other pattern of behaviour, provided the dependences of the rates of the reaction stages on the control are non-linear. The region of small values of t c discussed here is called the region of sliding regimes (ref. 46). In another extreme case (profile 3) the duration of the period of variation of the concentration of the gas phase is far greater than the characteristic time for the surface transient regime, i.e., t c » Mc' The dynamic properties of the system here do not have any effect on the process characteristics and the observable rates. of chemical conversion are determined by the kinetic models describing the steady-state process, which is dependent on the gas phase concentration and temperature at each moment of time. This is the so-called quasi-steady-state proces&
45
s 55
2
a
Fig. 1.7. Change of selectivity, S, of the catalytic process I - II depending on duration of the period, t , and amplitude, a, of oscillation of the concentration of comBonent A in the gas phase with fixed average values of the concentrations of A and B. 1 = the sliding regims; 2 = steady-state regimes; 3 = quasi-steady-state cyclic regimes. A geometric approach (ref. 47) has often proved illustrative and useful for the analysis of extreme cases. Many process indices for the steady, quasi-steady and sliding regimes can be constructed. A quasi-steady-state process cannot be efficient lest there are no limitations on some average process characteristics. If this is the case, one may consider steady-state operation at u = constant as the optimum, which provides the maximum value of some criterion, J. An efficiency gain in the sliding regime compared to the steady catalyst state can be achieved only if" the reaction stage rates are non-linearly dependent on the concentration of the gas phase or if criterion J is non-linearly dependent on some other process parameter. If t c ~ Mc , then as has already been discussed, the dynamic properties of the system considerably affect the non-steady-state process characteristics. Therefore, for purely extreme cases, the question concerning the efficiency of the non-steady-state process still requires. an answer.
46
Now we have come closer to the mathematical statement of the problem of finding the optimum cyclic regime with the best performance characteristics relative to any criterion of estimation. We are, certainly, in the first place interested in improved cycle-averaged indices, like, for example, the selectivity of the process. One can divide the problems of the cyclic optimization into two classes: (1) Problems concerned with the search for optimum periodic control providing such a cyclic regime in which the optimized process characteristics, for example, productivity or selectivity, reach their extreme values. (2) The same problems but with addition of limits of the process parameters within the period, i.e., the so-called limitations of the mean value, e.g., of the control parameters. In this kind of problem, for example, mean values of the concentration or temperature of the gas phase in the course of the cycle may be set. All problems of periodic control discussed in the literature related to this field belong to these classes. After having had a good look at the example of an efficient cyclic process in the previous part of this chapter, it is easier now to formulate a general statement of the problem of the cyclic optimization for processes described by a system of common differential equations. Let us take as an example the process described by such a system of equations as
The conditions pertaining to cyclic character are (1 .1Ga)
or
tc
~ h [B(t), u(t>] dt = o
0
(1.1Gb)
47
where ll,(tJ =. crt) = [Ut (t ), Uz (b), ... , um, (b) J is the reagent concentrations vector in the gas phase which serves as the control or, u (b) EVE R rn. the incremental constant functions determined for [0, t c]; e(t) = [e 1 ( t ) , e 2 ( t ) , ••• ,e n ( t )] is the vector of the concentration of the intermediate substance on the catalyst surface or B(t) ERn variable states continuous for [0, tcl; t c is the period. The process characteristics are designated as tc
!I =J... \' 5fJ,,[Brt),u(bY]dt, If., te J
1£=1,2, ... ,e
f10
o
where ~~ is the process characteristics at any moment of time. The efficiency function to be maximized is
J
=
9 (y> o
max
==?
(1.18)
iI(6),t c
y= [!Jt' 'Iz' ... , Ve J The average process characteristic, y, and also the function, J, can represent the observable rates of chemical transformation or/and the selectivity. As a rule, there is a limitation on the minimum and maximum values of the control, u(t), like u. m i Tb ~
if
{. Uj "uJ (b) "
ma:x
, J.
=
1" 2 ... ,
77U
In the second class of problems, the mean-integral limitations in addition are set as
fli ( V-)
=
0,
i.
- f», 2 , .. r
e1
tt < t
Thus, the problems of cyclic optimization (see eqns. 1.15-1.20) involve determination of ur:», li*(t), t 0* such that
48
It follows that the absence of information about the duration of the period as well as about the initial and final states of the system and the presence of mean-integral limitations which occur during the period, all add to this class of prob~ems. It may be on interest to juxtapose the solution of the problem of the non-steady-state cyclic ~rocess optimization and the solution of the problem of the steady-state process optimization. Let us write the relationship for the steady-state case which follows from eqns. 1.15-1.20 under conditions that the state vector, e(t), and the control vector are constant: /(8,u)=0
g(ij)
=
0
J = 9o(ijJ ~ mq-x u.
u-E:V , e-E-RTIFor many problems the set V can degenerate to a point. It should be noted that·if there is an optimum, J = J un, in the steady-state regime and J = JS the following condition (1
.25)
is always fulfilled. The term "efficient" will be applied to the non-steady-state process where Jun is greater than JS (ref. 48). There is a direct way to estimate the efficiency of the periodic regime, e.g., through solution of the cyclic optimization problem. However, this can be difficult in many cases. That is why it is thought to be of interest to evaluate the efficiency of the non-steady-state process (or at least obtain an answer if the inequality, Jun > JS is really valid) without resorting to a search for the optimum cyclic regime described in eqns 1.15-1.20.
49
Various methods including analytical ones which efficiency of the non-steady-state process is to be evaluated are discussed in the literature. Let us briefly go through these methods. The solution of the problem of cyclic regime optimization, in analogy to the problems of optimum control, should satisfy necessary optimum conditions. The well known Pontryagin's principle of maximum (refs. 49, 50) for the said problem can be formulated by introduction of Hamiltonian function H [Brt), u(tJ, lj/(tJ,
J:]
v
=
'(t)j(tJ
+
i'ij(t)
(1.26)
Here iji (t) are the conjugated variables, sP (t) is the vector function consisting of functions of the mean indexes from expression 1.17 and A is a vector of real numbers. The principle of the maximum is: let [e*(t), ii*(t)], 0< t~tc be the solution of the cyclic optimization problem (see eqns. 1.15-1.20). Then, there exist a continuous vector function, yr(tJ: [0, cc] R '", which is differentiated at the left on [0, t c]' .it E Rtf and real numbers, do"'" Q t : so that the following conditiona for all t E [0, t c ] are fulfilled
max H[1ktJ, ire V
»to. 7jdtJ, iJ
=
[B "(t),
U
*(t), 7jr(t),
J:]
e
.x' = ?= d i 'l J V) L ~D
(1.28)
50
Expression 1.28 describes conditions for the conjugated variables and expression 1.29 describes the transversality condition. The principle ofm~imum allows only for analytical solution of the problems posed by the optimization of the catalyst unsteady state as a rare case. Sometimes, one can have vector, 9*, which is the solution of problem 1.21-1.24, which is not in accord with the necessary conditions of the optimum regime. This means that Jun> JS (ref. 51). Most often the required optimum conditions only allow one to give a qualitative description of the optimum solution or/and to build up numerical algorithms of the optimization. It is expedient to make use of methods based on the analysis of extreme cases and to formulate some sufficient conditions for the efficiency of the periodic regimes. It is sufficient to analyse the behaviour of the system at both very large and very small values of the period (compared to the characteristic time of the system) which, as nreviously discussed, correspond to the quasi-steady-state and sliding regimes. In the quasi-steady-state regime the system will satisfy to eqn. 1.21 due to the extensiveness of the cycle, provided 0< t < t c• For steady-states assumed as unique, the value of the control, u(t), defines the state
where Bs t is the solution of eqns. 1.21. If we substitute expression 1.30 into sub-integral function 1.17, the non-linear programming problem in the mean (ref. 48) can be obtained. This problem can eaRily be reduced to a common problem of non-linear programming. Notice that any vector which is the index of the averaged values attained in a quasi-steady-state re~ime can be realized by means of incrementAl constant control consisting of not more than 1 + 1 increments, where 1 is the dimensionality of the process index vector. The main theorem for the forced cyclic regimes having dynamics approximating to these of the quasi-steady-state operation is formulated as follows: for any quasi-steady-state process operated by a set of admissible values, V, the resulting efficiency vector is the locking element of the convex
51
curve Ss(V), i.e.,it belongs to clcoSs(V), where Ss may designate, for example, the reaction rate in the steady-state regime. As long as the vector index of any quasi-steady-state periodic control belongs to clcoSs(V), it follows from Caratheodory's theorem in his convex set theory that the reaction control is of incremental constant character. Another extreme case of the cyclic regime is a sliding regime (refs. 52, 53). One has to bear in mind two peculiarities of this regime: (a) the period of oscillation is considerably shorter than the characteristic time of the system's transient process and (b) optimum control can always be realized with the help of n + 1 + 1 switches between constant values, where n is the dimensionality of the state vector and 1 is the dimensionality of the vector index. Under certain circumstances, there is a more restricted limitation to the number of switches. Consequently, one can believe that the state of the variables is constant and satisfies the system of differential equations 1.15 in the mean
t:
tc
~ Ii [ 8.st .tut)J dt
=
0,
i = /, 2, ... >
7U
o The time-invariant composition of the intermediate compounds on the catalyst surface, 9s1 ' in the sliding regime is subjected to quick control-varied action of the gas phase composition. That is why the process performAnce indexes may appear significantly different from those of the steady-state regime, because the control a!fpcts the system in a non-linear manner. It is possible that the solution of the steady-state problem can satisfy the principle of maximum, since it only serves as a necessary condition of the optimum, though in reality, transition to an non-steady-state operation seems expedient. The sliding regime efficiency test has, therefore, more convincing power than the test of the principle of maximum.
52
To analyse the forced non-steady-state regimes under conditions when the dynamic properties of the catalyst surface play an important role t one should use the X-criterion (refs. 51, 54, 55) based on the analysis of the efficiency function with small sinusoidal variations of the control about its steady-state value. The eXistence of optimum steady-state control is assumed to be the inner point of the set of admissible controls. In this case, the first variation of the quality criterion (1.18) vanishes and the second variation of the efficiency function in the vicinity of optimum steady-state control is investigated. One can derive the process variables under steady-state conditions with u(t) = s t = constant from the system of equations (1.21). If there is only one solution, the value of criterion (1.24) is unambiguously determined.
u
At small sinusoidal oscillations of the controlling parameters art)
=
st + Re ;X cos wt - Jm
U
~ SiTU
wt
(1.32)
where ;t E em, lI;d c m <6, [, is a small positive number, m is the dimensionality of the control and W = 2%/tc • The quality criterion is determined as accurately as small values of the fifth order by the following expression
where the asterisk designates the conjugate and n.» TTl" the Hermitian matrix; ITriw) is evaluated as follows. The Hamiltonian function of the system's steady-state conditions is introduced
H Of, iI, where
v . J) =
1jJ
7j/'[ + I.'
is derived by setting
ESc
= O. Then,
nr/ w ) = G *r/w)pf.i(jw) +Q'r;(/w) +R
( 1 .34)
where
C(/w) =(/w/-A)-IB ; A =
Ie (if,
P=j
e,
B
u. (
(1);
P
=
H eo
(15, ii , 1Jf),
=
iZ)
H s« ( 8, U, 1Jf)
53
It is assumed that u does not lie on the boundary of the determination region of (1.19) and that matrix A's proper numbers do not have zero real figures. One can apply the so-called X -test to various frequencies, thus obtaining data on its controlling efficiency. The use of this criterion allows one to establish an effective phase shift between the components of the reaction control. With this method taken as the foundation in ref. 56, an approach was suggested which sometimes gives the possibility of obtaining an analytical' expression for the corresponding value of the efficiency improvement in various types of periodic reaction control. A specific control test (ref. 57) represents a particular case of the x -criterion. Its application is thought to be expedient in the cases when the optimum steady-state control is of a specific type and permits a considerable reduction in mathematical computation. Let Uo be the vector of constant optimum controls of a specific type and lie within the boundaries of the region of admissible values. Then the following theorem is valid: if at least one of the two following conditions satisfy some k value of 0, 1, ~, ~tc., the optimum cyclic regime is efficient: (1) ~ = 0 for m = 0.1, ••• , 2k - 1 and there exists such ~ that (- t) Ii S T If2/r, S > D
(2)
~
=0
for m
= 0.1,
, 2k
-
And R2k + 1 I: 0, where
aH(ao,U,7j/ -0-0 ,
ou
It is clear that in this case the calculAtion of ~1' Ro' R2 up to their first non-zero member (base) is preferable than direct calculation of the matrix, octca); The small-value parameter technique (refs. 58, 59) and the conditions of non-steady-state optimum control (ref. 60) also allow one to come to a certain
54
conclusion about the efficiency of the transition to the non-steady-state regime. An important note must be made concerning research based on the construction of class sets of the solutions to periodic control problems and also the relationship between these classes in the form of necessary and sufficient conditions (refs. 48, 50). One can discard the existence of a large class of cyclic optimization problems which offer no advantage over the optimum steady-state regime.
The estimation of the optimum conditions for non-steady-state regimes so important for general efficiency evaluation of the non-steady-state process has proved of little use ;n respect of determination of the optimum control law as well as for construction of numerical algorithms. The most promising method seems to be that of direct calculation. One can distinguish three main calculational problems, posed by the solution of the optimum non-steady-state regime problem: (1) computation of the periodic regime with a prescribed value of the duration of the cycle and form of control action; (2) determination of the optimum form of the control action within the prescribed duration of the period; (3) determination of the optimum frequency of the control action. It is advisable to solve problems 2 and 3 simultaneously to ensure a considerable increase in accuracy. The use of numerical methods in solution of the cyclic optimization problem has two peculiarities, which are accounted for as already discussed by the existence of the periodic boundary conditions: neither the initial nor the final state of the system, nor the optimum duration of the period are known. The second peculiarity arises during consideration of various integral limitations on the mean process indices. At present, there is no general method of solving cyclic optimization problems. All algorithms used for that purpose are based on classic understanding of the function variation and of the modified maximum principle. The gradient method, which is based upon variational calculus, is believed to be the most general and reasonable. The method was discussed previously
55
(ref. 61) and is as follows: fixing the period of duration, t , c one numerically determines the corresponding optimum control. Another period duration is then asssigned and its corresponding optimum control is again found. After that the values of the efficiency functions are compared and the optimum period is calculated directly. This approach requires much computer time. Ref. 62 reports another numerical algorithm which is free from the use of the cyclic conditions described in eqn. 1.16. The optimum control was determined over a large section of the time-scale with arbitrary initial conditions. The optimum control and variable states found were similar to periodic functions at the right-hand side of the time interval. One can try to apply the conjugated gradients method to the search for the optimum cyclic process. To obtain sliding and quasi-steady-state regimes one can make use of the known methods of non-linear programming. However, one must be aware of the fact that, in general, the solution of the boundary periodic problem represents a serious difficulty. Its requirements in every particular case of the cyclic regime computation (if it is ever possible) are immense. Before commencing algorithmic calculation of the optimum cyclic regime, it is of practical use to do a preliminary analysis of the problem, which may permit prediction of the form of the optimum control which may be, for example, constant or incremental constant. Kinetic models of heterogeneous catalytic reactions are, as a rule, linearly dependent on the gas concentration, i.e., differentiation of expression 1.26 on u(t) will show that ~ = HI)" [8(6), 1jF(6J, l]. In case of this reaction, the controls which satisfy the necessary conditions of the optimum (eqn. 1.27) take up either their boundary values (eqn. 1.19) as a function of the above mentioned ~ or can be derived from the condition described in ref. 63.
Hu =D;
!fJ = 1,2, ...
This condition describes the so-called specific control and it can be fulfilled in accordance with all components of the
56
vector, u(t). If system 1.15 is bilinear,which is often the case with linear mechanisms of catalysed reactions, one can see that specific control determined by the said condition with limitations (eqn. 1.20) and with m = n always has a steady-state character: u(t) Steady-state reaction control may not satisfy this condition if the kinetic equations are non-linear with respect to the surface concentrations. Therefore, this control cannot be optimal.
= u.
we Rhall further consider an example of the efficient numerical algorithm for optimum regime calculation where the character of the optimum control and its period are determined simultaneously (refs. 64, 65). To generalize its formulae, let new variable states be introduced 8.{I;);J'[u, {t)]=u.~-n (t) ~ ~ ~-TU
:». i~n+1,TU+2"",TU+m j-'~-7b' (1.36)
where Pi-ru are constants. Note that eqns. 1.35 and 1.36 are identical to the mean-integral limitations set in 1.20b. Let the new variable, 't = t/t c' also be introduced. Then expressions 1.15 - 1.17 and 1.35 - 1.36 shall take the following forms:
(1.38) f
IIlit
=
~
tc
= "
2, ... ,
e
o
Let l5 J be the change in efficiency function with variation of the control, oliJ-c) and of the period, o'te . I t is possible to show that with the use of linearization of expressions 1.18 and 1.37 - 1.39 in the form
57
O.l·
+ -j-~-
otc
0&'{, (0) "" 00.· (t);
c
i =- /,2, ... , ru +
~
!'u=1,2, ...
06
i
~ 1,2, ... ,
n
+ n&
7Tl;1
,t
and with the use of the earlier introduced Hamiltonian function
that the change in efficiency function in the linear approximation can be determined by the following expression f
s: =
f.. J /=1
\'
o
iJH A
ou:J
tfU , da: + J
(1.40)
In this expression oBi represents the variations of state variables corresponding to the cyclic conditions of the process and to variations, ~Ui and obc ; IJillv is the variation of the process indices; Y'"i are conjugated variables which can be
58
determined as follows
rYi (7:)
=
oH
of}· , z
i
=
f, n.«
TTL t
(1. 42)
Thus, to make 0" J positive, which corresponds to an increase in the efficiency function, it is sufficient for the following expression to be valid
and variations small enough for the linear character in expression 1.40 to be observed. It follows from eqns. 1.33, 1.41 and 1.42 that (with mf
i n + 1, n + m) , .li(·); constant. Hence, L !Fili =D and instead eqn, 1.44 we can write: i~n-+t
''1n,( ote ) ~ siU"
[
S t, r, Ii d~
]
(1. 45)
a
Let us now formulate the algorithm for simultaneous search of the optimum control and period. Step one. Arbitrary control parameters, uj, satisfying eqns. 1.19 and 1.20 and an arbitrary period, t~, are assigned. Step two. With the assigned control parameter, u~, and period, t~, where k is the iteration number, the system of eqns. 1.37 is solved at i = 1, 2, ••• , n and with the cyclic conditions
59
described in eqn. 1.38 taken into account. To fulfil these conditions, the Newton-Raphson method can be applied (ref. 61). After calculation of the functions 9~(~) (i = 1, 2, ••• , n), the value of the efficiency function, J, is found with the help of expressions 1.18 and 1.39. Then, in time-reserved order, the conjugated system (eqn. 1.41) with i = 1, 2, ••• , n + 2 is integrated beginning with the arbitrary initial conditions. Complete solution of !f/i!f,r satisfying the conditions of eqn, 1.42 is reached via both the given particular and fundamental solutions obtained earlier by means of the Newton-Raphson method. o
)
Step three. Arbitrary values of assigned.
1.)/,
i
n + 1, n + mare
Step four. New controlling functions are calculated according to the formula
(1. 46)
e; > 0
where
(ft+t)( )
Uj
.
>
is a vector of a small positive numbers. If
u/ max.
then
uJ
Iv+!} (7: = U
J
max-.
It+f()
,J.f u/
7:
min-
J
then
mirv
1';,+1
u; (7:) = Uri . In other words, if the control determined by expression 1.46 at some moment of time is found to be beyond the limit, it takes up its boundary value. Step five. Fulfilment of conditions 1.20 is checked. In case it is violated, 1JI"i (i = n + 1, n + m1) is changed so as to redress this violation. So, we can represent it as solution of m incremental linear algebraic equations in relation to lJ'i' Step six. A new value of the period is calculated using the formula 1
te/t+1 =: t/'+
0T
~ ~ lJI"i Fe
d-t:
Q
where
(;r
>o.
Step two is then repeated. Values of
6T
and a
60
e
fragment of vector (eqn. 1.46) must be small enough to ensure accuracy of expression 1.40. The iteration process continues until no growth of the efficiency function is observed.
1.4.3
Some examples of optimum performance of processes
Some studies have considered on a theoretical basis the problems of finding optimum controlling parameters, making use of a selection of catalysed reaction mechanisms as models. These studies help to distinguish classes of chemical reactions allowing for an efficiency increase under non-steady-state conditions. Let us probe deeper into the non-steady-state process on catalyst surface. With periodic variation of the gas phase condition, the period-averaged reaction rate, t c' is described by an expression similar to eqn. 1.17 Cc
W=.i. t \'j W [8ft), e
TfO, crt)] dt
o
where B(t), as before, is the vector of the concentration of the intermediate substance on the catalyst surface; T(t) is the temperature; c(t) is the gas concentration vector. All parameters of the gas phase, c and T, (or a part of the parameters)may serve as controls, u(t). In the case where there are two pArallel reactions andW 1 is the average rate of the end product formation and 2 is the rate of formation of the side product, the selectivity of the non-steady-state process, as it has already been shown in this chapter (see eqn. 1.12), will be determined by the following expression:
W
It is problematic whether it is possible to attain an higher
61
rate, W, or selectivity, s, in a periodic non-steady-state regime at u = u(t) than the corresponding values of the rate, R1, and selectivity, s, in the steady-state regime with u = ~
f
fi
S
u(tJd6 = con.stanc . Let the controlling parameter oscillate c 0 with a larger period than all the characteristic times of the initial system. The behaviour of the intermediate substances on the catalyst surface may be then regarded as quasi-steady. In this case, instead of expression 1.47,one can write the following: = t
(;c
vv, '>3:c ~ u, [ tto, crt) dt o
--'
From eqn. 1.49 W1
> R1 (u av ) if
where u av tc
corresponds to the average temperature, Tav = (;c
and to the average concentration, C
all
=
f
t
c
~ ~ T(b)dt 0
j crt) cit.
\'
Since the
o
mean period selectivity involves integrals, Sis greater than S if the following inequality is fulfilled:
Now let the controlling parameters oscillate with a much smaller period than those of all the characteristic times of the initial system. Under these circumstances, the catalyst operates in a sliding regime where the concentrations of all intermediate substances, 9(t), on its surface remain unchanged during the whole period and are equal to 9 av = constant. Values 9 av are derived from expressions 1.31. Since the characteristics of the sliding regime will differ from the corresponding characteristics of the steady-state regime only in the case of a non-linear dependence of f 1 on n, periodic control of the gas phase composition or pressure at low values of the period of oscillation is often unable to alter the process efficiency
62
indices. Still, the sliding regime may become efficient when it is temperature-controlled. Let us now look back at the example discussed in section 1.4.1. This example shows the possibility in some classes of reaction mechanisms with intermediate values of the period of variation of the gas phase condition to accomplish an higher observable rate of conversion and selectivity than in the quasi-steady-state or sliding regimes. Let us try to evaluate the process efficiency with the help of the X -criterion, taking as an example the qualitative characteristics of the non-steadystate process described by the system of eqns. 1.11, namely, cyclically-averaged side-product formation rates, W (see eqn. c 1.17), end product formation rate, WD (see eqn. 1.18b),and selectivity, (see eqn. 1.12). The system of equations 1.11 is bilinear. Hence, according to eqn, 1• .34, P = 0 and R = o. Consequently: roJ
s
flriw)=/lr;(iw)+G*(/w)/l= [n 11 n f2
n n 21
]
22
In addition, n 12 and n 21 are complex conjugates. In accordance with eqn. 1.32 and without any loss in general character, oscillations of the control for this problem will be described by
where A1 and A2 are small enough oscillation amplitudes of the concentration of A, U1, and of B, U2; ff is the phase shift between these oscillations; 1 , are the mean values of the concentrations CA and CB during the cycle.
u u2
According to eqns , 1• .32 - 1.33 and expressions for
nr/co) and
u(t):
If 02 J is expressed as a function of the oscillation frequency, then
63
1J2J =- At <:£, + Ai <:£2 + AfA 2 0(;.3 sin (J + wAfA z (p-W 2)2 +Jl2
2
co 2
+ w (A ;!!. a 5 +Ajo(;6 +A 1A z
d 7 siTL
QI(
0) + w
r./U-w2) 2 +Jl 2C0 2
cos 0 +
5A
1 A z 0(;8 cos
If
where di ' i = 1.8 are rather unwieldy composites of kinetic 2 jt 2 . cons t ants an d ~ u 1, -u 2; jU = f 8t t f 82 82/ 81 JoS a product of proper numbers of system 1.11; I = k 1u1 + k 2 + kJU2 + k4U2 + + k + k 6u1• If CB = u 2 = constant and CA = u1 + A1coswt is the 5 only controlling parameter, then a +8w 2 6' }(w) -At {jU_(2)2 +cY 2 W 2
_
2
where a = x
-ZRIJ
[k f { k_ 3
2
{(k 3 cB + k _,3 )( !£1 S4 + !£-/-j-!£2 CB ) x
-t- k~ CA)
+
kL((Jf,,~ + tc., +!f,z
C B)]
k/ ~ [/1,,,/c.q
CA CB +
Rn is determined by expression 1.13. The period-averaged rate of the reaction in the non-steadYstate regime is:
WD
=;
R
D
-j-
6'](w)
I t is higher than in the steady-state regime if tf 2 Jtco) > o.
In the quasi-steady-state regime when w
= 0:
64
In the sliding regime
tim
uJ-=
0 2 J(W) =0
The latter inequality is valid i f the following conditions are fulfilled (1) a>O,
8>0
In the first case 13JU 2 - at + 2 aJU > 0, and in the third case the function 02J(W) will always reach its peak at the frequency:
Fig. 1.8 shows the dependences of the second variations, J1 - W C ' J 2 -Wn and J 3 - S, on the period, tc' and on the oscillation amplitudes, A1 and A2, with K1 = K_1 = 3; K2 = K) =
K4 = 1; K_ 3 = 0.3; CA = u1 = 2; CB = U2 = 1 and various phase shifts, 0, in the case of joint and asynchronous oscillations of u 1(t) and u 2(t). The effect would be expected to be most pronounced when the characteristic times of the transient regimes of the intermediate compounds, [AZ] and [BZJ, on the catalyst surface are considerably different. The above data, certainly, indicate that the largest efficiency gain can be reached in the case of joint oscillations of the controlling parameters. For example, if the period t c = 1 and A1 = A2 = -2 ff=0.83x 410, and tf2J1 =0.1357-10-,meanwhJ.leatA 2=O -2 tf2J1 = 0.238-10 -4 and at A = 0 and 11. = 10 -2 , ann A1 = 10, 1 2 02J1 = 0.0144-10- 4 •
65
It can be concluded from expression 1.50 that the second variation of the quality criterion has a sinusoidal dependence on the phase shift when the period is fixed. The maximum is reached when tg 0 = -Jm n f2 IRe n,2 • Fig. 1.9 illustrates the dependence of the optimum phase shift, 00, of the concentration oscillation and the maximum of the function, J 1, i.e., the maximum of the rate of formation of substance D as a function of the period, t c' The optimum phase shift, 00, for function J 2 increases monotonically from -0.5% to 0 in parallel with the increase in the period. For function J the phase shift 3, monotonically increases from 0.5 to x.
1\ 0.02
a)
100 -0.02
-o.a«
:5
r
0.1 fOO
-0.2
-0-'" -0.6
tc
b)
66
It
oJ., • io« 5"
0.2
2
0.1
c)
f
~
fOO
tc
-0.1
Fig. 1.8. Dependences of the period of second variations of the rate of formation of substance D,02J 1 (a), of substance C,02J 2 (b), and the selectivity, 02J:3 (c), at (1) A1 = 10- 2, A = 0; 2 (2) A1 = 0, A2 = 10- 2; (3) A1 = A2 = 10- 2, d = 0; (4) A1 = A2 = = 10- 2, If = 0.83 ; (5) A1 = A2 = 10-2, If = 1.33
O.Of
duration of cycle
Fig. 1.9. Dependence of the optimum (for lf2 J 1 ) phase shift, 00, on the period of oscillation of the gas phase components A and B, t c •
67
Let us strictly formulate the problem of the optimum control determination for system (1.11) with cyclic conditions as described in eqn. 1.16a : 9 i ( 0 ) = 9 i ( t c ) ' i = 1,2; in additio~ 9 '" [AZ], 9 =. [BZ]. The efficiency function, J 1-=* max must be t 2 maximized with the following limitations tc
:c
~
~ u., (n~ u. m a x [ul(t)-Ui]dt=O; U.min., z ~ z '
o
where u 1 '" CA' u 2 == CB• According to the required optimum conditions and if CB = constant and value of the period is fixed, either incremental constant control with boundary-set values of concentration or special, in other terms, steady-state control can be regarded as the optimum. Thus, it appears that the solution of the problem shown in Fig. 1.5 corresponds to the optimum: curve RD represents the rate of formation of product D in the steady-state regime with the concentration of component A in the gas phase equalling CA. Curve WD represents the rate of formation of D under the optimum non-steady-state conditions with the same values of the constants. u 1m i n and u:l: max, correspond to 0 and 4, respectively, and u 2 ( t ) = 1. If u1
In the case when both the concentrations of substances A and B are the controls, the problem is solved analytically via the known algorithm and the optimum control profile and the optimum period value should be simultaneously derived. With 0 ~ u 1 <; 4, o ~ u 2 ~ 2, 1 = 2, = 1 (values of constants unchanged) an incremental constant control must be regarded as optimal. With the period, t c' about 1.01 and the phase shift,5= 0.77x , it completely satisfies the optimum condition. The optimum values of the period and the phase shift accurately coincided with the optimum values obtained with the use of the x-criterion. With and = 1, in the steady-state regime, J 1 = 0.296, while 1 =
u
u
2
u2
u2
68
with u 2 = constant and under the optimum control, u 1(t), J = 0.344 and with the optimum control, u 1(t) and u 2(t), J = 0.424. I shall now illustrate both the method of limited cases and the geometric approach. For example, the initial component, A, is converted via a catalytic process into products C and D (ref& 44, 47) in accord with the following mechanism: A +
[z]
A +
[A*] -
~ A 11-
C + [Z]
(VI)
D+[ZJ
A -
The process is assumed to be carried out in the region where the adsorption rate of A does not depend on the concentration of the adsorbed substance, [A*]. Further, the first two reactions of mechanism VI represent elementary steps, and the third reflects the complex observable reaction of second order. We shall consider the situation Where all intermediate substances resulting in the formation of product D have small times of transition in comparison with those resulting in the formation of substance [A*J. Let the concentration of A, CA, be the control parameter u, and concentration of the adsorbed component, [A*] , be the control parameter e. Then one can derive the following equation for the material balance
Expressions designating the rates of formation of products C and D take the forms tc
~ W = -f ~ c tc
'It2 uBdt
a
to
~ W
IJ
= -
1~
tc
a
11,% u
"
2
dt
69
The limitations of the variation of concentration should obey . al ~. t y, u rru.n. '" ./ /' . the a.nequ u ( t ) '" u max• Prof~le 1 in Fig;. 1.10 shows the re,:ationship between the rates of formation of C 'W ' and 1 of D ,W2, in the steady-state regime when:
In the quasi-steady-state regime with a large period of variation of the concentrations and with the region of variation of the concentration of substance A within the range u min<, 0 and u ma.x; the relationship between the rates of formation of C and D lies in the region enclosed by profiles 2 and 3. If D is the reaction end product, a selectivity increase will be provided in the quasi-steady-state regime owing to a stronger non-linear character of th~ dependence of the rate of formation of D (W2 ) compared to C (W1). As the period is decreased (especially in the sliding regime), the selectivity relative to product D increases. The reason may be that the rate of formation of product C is decreased and at the extremum (as in the sliding regime) it would tend to the rate corresponding to that in the steady-state regime with the concentration of A(u) determined by the mean-integral value of the period. Behaviour of this kind can be explained by the inertial properties of the catalyst surface, provided by the adsorptive and catalytic steps of the reaction mechanism VI. Owing to the lack of inertia of the third step, the rate of formation of the desired product, D, will still be described by a square-law dependence on the concentration of substance A. The rate of the first reaction in the sliding regime is totally identical to that in the steady-state regime and the rate, W 2 ' is identical to that in the quasi-steady-state regime. Profile 3 in Fig. 1.10 displays the relationship between the rates, W 2 ' in the sliding 1 and W regime. It is of interest that a regime may appear almost quasi-steady-state if the values of the period of concentration variation in the gas phase are approximately 100 times as large as the characteristic time of the transient regime of the system. On the other hand, a regime may become almost sliding if
70
the oscillation period appears to be equal or smaller than this characteristic time.
2 1
Fig. 1.10. Selectivity of the process described by mechanism VI in the steady-state (1), quasi-steady-state (2) and. sliding ()) regimes. (k 1 = k_ 1 = k 2 = 10; k) = 5; u m== 1; u mm= 0). Mechanism VI showed the formation of the reaction product, C, to be the result of the interaction of the intermediate substance, [A*] , with molecules of the initial components, A, from the gas phase via the collision mechanism. Scheme VII shows the formation of product C directly from product [A*] (1)
A + [ZJ~[A*J
(2)
[A*J~ C
0)
A -- D
+
[z]
(VII)
Such a difference in reaction mechanisms opens up a possibility to attain a selectivity increase toward product C with the same parameters, which prov.ide, according to mechanism VI, a selectivity decrease (ref. 44). In the first place it can be referred to the value of the observable period of formation of
71
the side product, D. The smaller the order of reaction 3 in mechanism VII (A-D), the larger is the gain in selectivity. An increase in the selectivity relative to product, P2' can also be attained for reactions which are carried out according to the scheme (ref. 44): (1)
A + [zJ~ [A*]
( 2)
A + [A*J~ c
0) A ( 4)
+
[zJ (VIII)
+ [Zj::::[A**J
A + [A**J --- D +
[Z]
It is possible only if the cyclic regime is actually sliding in reaction to [A*] and quasi-steady-state relative to [A**] •
1.4.4
A non-isothermal process on a catalyst surface
Let two parallel reactions be carried out according to the following common mechanism
~
(1)
A + [Z]
(2)
B
+
[AZ]
0)
B
+
[z] -"'3 [Z]
[AZ]
~ [Z]
+ +
C
(IX)
D
Since the concentration of the intermediate substance [AZ] on the catalyst surface in the sliding regime is unchanged, the selectivity of the process (relative to the formation of the end product, C) under conditions of fast periodic temperature variation is I.e
~k
$=
o
2(t)dt
72
If the activation energy of formation of the main product, C, is higher than that for formation of product D, then s ) s and the non-steady-state process is efficient (s is the selectivity in the steady-state regime with the same mean values of the temperature). Let us further consider the example of a catalytic process carried out according to the following adsorption mechanism (ref. 66)
+
(2)
B
0)
[AZ]
(X)
[Z]::= [BZ]
+
+
[BZ] -- 2[ZJ
AB
and the example of a two-route mechanism formed by steps of mechanism X and the collisional step (4)
B
+ [AZ] -- [Z]
+
AB
1-3
(XI)
Mechanism X includes adsorption steps for various reagents on the surface of the catalyst. As the result of this adsorption the surface substances are formed and the interaction between these is also included in mechanism X. Step XI represents, as a rule, a part of the mechanism of oxidation of simple molecules on metals or a primitive mechanism providing plurality of steady states of the catalyst surface. Mechanisms X-XI are typically used to describe the oxidation reactions of carbon monoxide on metals. The values
characterize the contribution (selectivity) of each reaction route to the total rate of the product formation (W1 is the rate of the ith step, i = 3,4).
73
The partial pressures of the gaseous reagents, CAzand CB' and the surface temperature, T, serve as the control parameters. These parameters have different effects on the kinetic dependences of some stages. An increase in the partial pressures causes an increase in the rates of reactions 1 and 2 and in the rate of the collisional step 4. There are often large differences in the temperature dependences of the reactions in the detailed mechanism. Let us assume that the activation energies of the adsorptive reactions corresponding to those described in mechanism X are small enough to state that E1 ~ 0, E2 ~ O. Also let the activation energy in mechanism XI be small, so that E ~ O. The desorption activation 4 energies are rather high: E_ 1 = 120 ~ 280 kJ/mol; E_ 2 = 80 + +160 kJ/mol. The activation energy, Es' of interaction between substances [AZ] and [BZ] is intermediate between the activation energies of adsorption and desorption, equalling 40 + 80 kJ/mol. Some typical dependences of the steady-state reaction rate on the temperature at various pressures are shown in Fig. 1.11. As stated before, a quasi-steady-state regime may be desired particularly in the region of variation of a controlling parameter (in our case in the region of temperature variation, T1 < T ~ T2) where the curves of the kinetic dependences on this parameter are concaVe. For example, in the case described by profile 2 (Fig. 1.11) the ratio, W/r, of the mean rates in the quasi-steady-state and in the average steady-state regime in the region of temperature variation, T = 530 ~ 570 oK, is about 25. When some processes are performed via mechanism X or a similar mechanism, a considerable increase in the efficiency can be achieved at intermediate values of the temperature variation frequency. It follows that : (a) the rate of adsorption of the gases in only weakly related to the temperature; the process of adsorption is also rapid at low temperatures; (b) the rate of desorption can become sufficiently high only at elevated temperatures; (c) at low temperatures (T 1), a special composition of the catalyst surface is provided with a considerable content of the intermediate substances [AZ] and [BZ]. These substances react at high temperature (T 2) which
74
2.'(
.
p:::
2.0
'.6
Q)
..p
al
H
s::0
'r-!
..p
1.2
0.8 0.'(
0
al Q)
H
sao
520
500
580
(jQO
820
temperature t T (K)
Fig. 1.11. Dependence of the role of the steady-state reaction carried out via mechanism X on temperature. k. = k? exp (-E/rT) (i = 1,2, 3); k~ = 1.5 8- 1 , k 2 = 1; k~ = 2.23:105;~k~1 = 3'10 12; k~2 = 3.10 11 8- 1 ; E1 = E2 0; E3 = 40 kJ/mol; E1 = E2 = = 30 kJ/mol; Cn = 5; CA = 5 (1), 4 (2), 3 (3). 2
[BZ] 0.8
[BZ]
a)
0.8
0.6 0.4 0.2
[AZ]
0.2
0.4
0.6
0.8 [AZ]
Fig. 1.12. Phase situation in the system where mechanism X is realized with periodic temperature control. CA = Cn = 5. Sections of the phases trajectories marked 1, ~ correspond to temperatures, T1, T2 (a) T1 = 350 oK, T2 = 580 oK; (b) T1 = 350 oK, T = 530 oK. The values of the parameters are identical to those 2 shown in Fig. 1.11.
75
still should not be very high otherwise the desorption rate might be increased. Fig. 1.12 shows phase trajectories (profiles) corresponding to non-steady-state control by means of the temperature variation. At first, the catalyst is treated with the reaction mixture at low temperature, T1; adsorption takes place and thus the system moves into the region of middle-range coatings characterized by higher rates of reaction. If the temperature is fixed, the highest rates are observed at surface compound concentrations, [AZ] ,.. [BZ] ~ 0.5. The temperature is further increased to T2 and the adsorbed substances interact yielding the reaction product. The concentration of intermediate substances on the catalyst surface is reduced. After several periodic variations of the temperature a periodic "two-stroke" regime of reaction control is established: (1) Adsorption: the system operates at temperature T1 for some time and the surface is enriched with substances [AZ] and [BZ] • (2) Reaction: the temperature is increased to T2 and the reaction product is liberated at an high rate. The number of vacant centres on the surface is increased. In addition, the parameters of the "two-stroke" regime (the times of every cycle and temperatures, T1 and T2) should be chosen in such a way that the phase trajectories are always in the region of surface compositions at which high rates are characteristic. Otherwise the system can drift into the region of coating having low reaction rates after a few "strokes" (Fig.1.12b).
Selectivity on the catalytic process in an isothermal reactor with ideal mixing The selectivity in this case can be detailed for the example of the phenomenologic description touched upon in ref. 67. The essence of the description was discussed in ref. 3. Let us take the case of two reactions carried out in parallel:
~B
A~
C
76
Let an non-steady-state cyclic regime be created by means of periodic variation of the inlet concentration, Co (b) = Co (/; +tc J. The ratio, V =S/S, should be determined, i.e., the average selectivity in the periodic regime created to the selectivity in the steady-state regime when the inlet concentration, cos ' is similar to the average periodic inlet concentration:
tc c .:'. o be
\' co(t)dt
J o
Let us assume that the steady-state (quasi-steady-state) rate of the main reaction, r 1, is a convex-shaped function of the concentration, C, and the rate of the side reaction, r 2, is a concave-shaped function of C. This actually means that
where Cs = ). C1 + (1- J.) c2 is the average concentration, corresponding to that in the cyclic regime and also in the sliding regime; 0 < ). < 1; C1 < C2• Let at least one of these inequalities be strictly fulfilled. With the properties determined above, ri(C) (i = 1,2), the following inequality is always valid
where W1 and W 2 are the cyclically-averaged rates of formation of products Band C in a regime close to the quasi-steady-state, i.e., at long duration of the cycle. It can be derived from this inequality that under the given conditions the selectivity in the non-steady-state regime is always higher than in the steady-state regime. The value of the maximum selectivity is equal to
77
The initial amplitude of variation of the concentration of component A in a reactor operating in the quasi-steady-state cyclic regime (with very low frequencies of the inlet concentration variation) is smaller than the amplitude of variation of the inlet concentration. Also, the difference in amplitudes is proportional to the reactor volume and to the extent of conversion. Consequently, the selectivity is expected to increase in the non-steady-state regime if the reactor has a small volume, the extent of conversion is low and the inlet concentration is varied at low frequencies. At high frequencies the reactor operation will correspond to that under steady-state conditions with average values of the inlet concentration. The phenomenological mathematical description of the observable rates of chemical conversion for the example under consideration where the dynamic properties of the catalyst surface are essential has the following form (ref. 67) dWf 1 db
M
dW2.
M2 dt
=
r.tc) - W + Il 1
f
=
r2
drt(c)
fdt
(ct : W + IT drz(c) C
2.
2.
eU
where M1 and M2 are the time-scales of the transient regimes on the catalyst surface; are the corresponding times of the , l 2 preact. The material balance in the reactor is
n n
where Mp is the mean residence time of the reaction mixture in the reactor; ~o is the conventional contact time of the mixture and the catalyst. Let 1f = hfe m, '2 = kg C TTl, and the inlet concentration be varied according to C = CDS cot: + a sin wt . In Fig. 1.13 the o numerical solution of the system of eqn. 1.51-1.52 is shown at m = 2, n = 1. Similar results were naturally obtained at m > n. The increase in selectiVity in the non-steady-state regime continues to grow with increasing residence time of the mixture
78
in the reactor. The extremal properties of the mentioned dependences is explained by the fact that in the vicinity of the dimensionless frequency, GO' = Mp/t c' the amplitude of oscillation of the concentration of A in the reactor is larger than in the quasi-steady-state or sliding (GV=oo) regime. This causes a slight increase in selectivity in the region of the natural "resonant'" frequency, approximately 1/Mp' The lower the natural frequency of the system, the lower is the frequency of the external influence required to attain the maximum selectivity. The impact of the preliminary properties of the surface is clear from Fig. 1.1Jb. Parameter A alters the absolute value, V, without a change in the position of the maxima. One can see that the dynamic properties of the catalyst surface are able either to increase or decrease the selectivity gain depending on the type of kinetic mode. The same can be said about the case of two reactions, the main and the side, for which Fig. 1.14 shows calculational results. a)
..
1
';:".
ro
+>
.r-!
::-
0r-l
+> o
Q)
r-I
Q)
Ul
'.03
0
'r-!
+> a;l 1-1
frequency, co'
frequency, co'
Fig. 1.13. A reactor with ideal m~x~ng in which two parallel reactions are carried out. Influence of the frequency of variation of the inlet concentration, .»: , on the ratio, V , of selectivities in the cyclic and stead~-state regimes. (a) A = 0; M = J.J (1); 2.0 (2); 1.0 (3); 0.5 (4)~ 0 (5). (b) M = 3.3; A = = 0 (1); 0.5 (2); 1.0 0); 2.0 (4). (5) bl = 0, A = O. co: = = M/t c ; M1 = M2; M = M1 / Mp ; IT, = ilz ; A = n, IMp'
79 ';:,.
t.O'I
til
Q)
'M
+>
'M
::-
'.03
'M
+>
o
Q)
r-f
1.02
Q)
m ~
0
un
0 'M
+> al H
frequency, W'
Fig. 1.14. Influence of forced oscillations of the inlet concentration, ur , on -the ratio, P, of selectivities in the nonsteady and steady-state regimes. A = 0, M = 0.01 (1); O.OJ (2); 0.05 (3) (see Fig. 1.13).
Refs. 68 and 69 dealt with the example of the non-steady-state process of carbon monoxide oxidation on platinum in a reactor with ideal mixing. The kinetic pattern of the process corresponded to mechanism X where A2 "" 02' [AZ] == [OZ] , B == Icol, [BZ] == [COZ], AB == CO. In accordance with this mechanism, a molecule of CO from the gas phase is adsorbed on one active centre of the catalyst while every molecule of oxygen is adsorbed in a dissoiative fashion on two centres (ref. 70). Carbon dioxide formation occurs according to a step mechanism. It is assumed that CO 2 is not adsorbed on the catalyst surface. The peculiarities of the steady-state regimes via this mechanism are well reported in the chemical literature. One should remember that there are interacting intermediate substances on the surface and with cartain values of the rate constants of the reaction stages there can be a plurality of steady-state rates of CO 2 formation at similar concentrations of the reacting components above the catalyst surface or at the reactor inlet. With the supposition that the reaction is carried out in an isothermal reactor with
80
ideal mixing operating at constant pressure, the dynamic character of the system behaviour was studied. As is seen in Fig. 1.15 which shows the results of calculation of the steady-state regimes, the rate of reaction in the region of low CO concentrations increases with increasing carbon monoxide content, but decreases at higher CO concentrations. At intermediate values of the CO concentration three steady states of the system can exist. Two of them are stable and one is unstable. The maximum and the minimum in the oxidation rate correspond to the steady states. Let the carbon monoxide concentration in the mixture be varied in accordance with the sinusoidal law
cco (t:') =
C + a sin. cat' co,o
where Ceo , 0 is the time-averaged CO concentration in the mixture; a is the amplitude of oscillations, w is the frequency. I t is assumed that CO2 (6') = t - Ceo (t '), I f Ceo , 0 = 0.5 is chosen, the steady state of the system corresponds to the lower steady profile of the rate (see Fig. 1.15) at which the normalized rate, r, is 0.0254. In this case it is possible to attain a considerable increase in the reaction rate through transition to a cyclic variation of the mixture concentration. This will happen when the amplitude and the frequency of the forced oscillations are of a kind that for a portion of the period of oscillation the concentration, eeo,o will correspond to the upper profile of the reaction rate. It is clear from Fig. 1.16 that with invariable values of the oscillation amplitudes and an initial CO concentration in the region of dimensionless frequencies, W~0.45, a resonant behaviour of the system is observed at the time-averaged reaction rate curve crosses the maximum, corresponding to the mean normalized rate in the non-steady-state regime, = 0.262. This value of the rate is ten times larger than the corresponding value of the rate in the steady-state regime and twice as large as the value of the rate in the quasi-steady-state, cyclic regime (w~ 0) at Q = 0.4. This behaviour is explained by the dynamic interactions inside the system, which are connected with the forced transition of CO coatings of the catalyst surface from lower to higher values. Vllien the extent of coating is high, the time-averaged values for
W
81
0.7 0.6
po:
as
~
Cll
+>
o.«
s::0
0.3
+>
0.2
al H
'M
0
al
Cll
H
o. 1
r.o C
co
Fig. 1.15. Steady-state (normalized) CO oxidation rate, r, on Pt as function of the initial concentration, Cro o (fraction), in the mixture at the inlet of the reactor with id~al mixing LK1 = 35; K_ 1 = 0.5; K2 = 20; K_ 2 = 50; K) = 6; CC02 = OJ (r is normalized by division of the absolute rate by its maximR~ value obtained with stoichiometric amounts of CO and 02 and their complete conversion into CO 2 j. the rate often tend to the steady-state values, whereas at low coatings they tend to quasi-steady-state values. Notice, that for this example, an extreme dependence of the observable rate of CO oxidation on the value of the amplitude of oscillation is observed, provided the oscillation frequency is fixed. Ref. 69 considers mechanism XI where the possibility of interaction of gaseous carbon monoxide with the adsorbed oxygen is taken into account CO
+
[OZ] -
CO 2
+
[Z]
A new mathematical simulation of this model did not bring about the desired quality. It was found only that the resonance frequency was strongly dependent on the ratio of the reactor volume capacities to the surface area of the catalyst.
82
0.30 , . . - - - - - - - - - - - - - - , 0.25
0.20
0.15'
0.10
17.03
____ L
0.001
0.01
_
2.
1.0
0.1
frequency,
10
UJ
Fig. 1.16. Non-steady-state (cyclic, normalized) rate of CO oxidation on Pt, W, (3) as a function of the (dimensionless) frequency of oscillations, w (Q ; 0.4; CCO,O ; 0.5); (1) quasi-steady-state (cyclic) regime; (2) steady-state regime where rlC co,a ; 0.5.
REFERENCES 1 G.K. Boreskov, Zh. Vses. Khim. Ova., 22, 5(1977) 495-505. 2 Yu.Sh. Matros, Unsteady Processes in Catalytic Reactors, Elsevier, 1985. 3 G.K. Boreskov, Kinet. Katal. (Engl. Transl.), 21, 1(1980) 5-16. 4 B.S. Balzhinimaev, S.V. Kozyrev, G.K. Boreskov (editor), A.A.lvanov, N.P. Belyaeva and V.I. Zaikovsky in Heterogeneous Catalysis, Proceedings of the 3rd All-Union Conference on Mechanism of Catalytic Reactions, Institute of Catalysis, Novosibirsk, 1982, pp. 177-180. 5 V.A. Sadykov, P.G. Zyrulnicov, V.V. Popovsky and S.F. Tihov, Kinet. Catal. (Engl. Trans.), 22, 5(1981) 1219-1226. 6 V.A. Sadykov, S.F. Tihov and V.V. Popovsky, Kinet. Catal. (Engl. Transl.), 27, 1(1986) 147-155.
83
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