160 Ind. Eng. Chem. Fundam. 1980, 19, 160-166

Design, Simulation, and Operation of a Laboratory Recycle Reactor with Controlled Gas Concentrations

A. Lowe

Institut fur Chemische Technologie, TU Braunschweig, D-3300 Braunschweig, West Germany

A relatively simple control system has been designed to start up and operate a laboratory recycle reactor so as to establish and maintain specified gas concentrations. Thereby it becomes possible to perform kinetic investigations using selected values of partial pressures and to uncouple the effects of gas concentrations and catalytic activity upon reaction rates. The design has proved efficient in simulation and in experimental studies of a muttiple reaction system, the simultaneous catalytic methanation and water gas shift reaction. Mainly due to the novel experimental approach, some interesting insights into reaction kinetics could be obtained and, for the first time, selectivity changes became evident.

Introduction The recycle reactor is a well-known tool for kinetic in-

vestigations of heterogeneous gas reactions. Several ex- cellent reviews exist, e.g., Bennett et al. (1972), Weekman (1974), Doraiswamy and Tajbl (1974), which describe the different versions (e.g., spinning basket, internal recycle, external recycle reactor), all of them designed to establish gradientless behavior. The mode of operation exclusively practiced previously is to set inlet concentrations and/or feed rates and to measure some or all concentrations in the exit stream. In this sense the inlet flow rates may be called controlled variables; the resulting concentrations in the recycle reactor then represent observed variables.

It should be possible to reverse the roles of input and output variables by applying control loops to a recycle reactor so as to establish and maintain particular gas concentrations. Such a mode of operation promises several advantages, for example the possibilities (a) to perform experiments a t specified concentration settings, such as prescribed by factorial or other experimental designs, (b) to evaluate reaction rates r(xref,TRf) at reference conditions of concentrations ( x d and of course of temperature (Tref), thus providing a well-defined and important measure of the absolute catalytic activity, (c) to follow the relative catalytic activity aj = rjjxref,Tref,t)/rj(xref,Tref,t = 0) with time, and (d) through this uncoupling of concentration and activity effects upon the reaction rate considerably to fa- cilitate kinetic modeling and parameter estimation for both reactions and deactivation. In particular this latter point has been discussed recently by Levenspiel (1972) for dif- ferent types of catalyst deactivation. A special technique to realize (c) has been reported earlier by Hedden and Lowe (1966).

Presumably the main application of the new approach will be in the field of catalytic gas-solid reactions. How- ever, it may likewise be applied to noncatalytic gas-solid reactions or to adsorption processes to keep gas concen- trations constant. Some experiences with the method in kinetic investigations of the reaction between carbon and carbon dioxide have been gathered (Bilgesu, 1978). The present paper intends to describe the general approach to multiple reactions; some examples are given mainly for illustrative purposes. Subsequent papers will concentrate on applications. Design

Feedback-Feedforward Control. The following de- sign is based on the steady-state balances of a gradient-free

0196-43 13/80/ 1019-01 60$01 .OO/O

recycle reactor. This means that catalyst decay must be slow enough to justify a pseudo-steady-state treatment and, on the other hand, that the control action does not create too large deviations from the steady state. Fulfilling the latter condition should be much easier when the pseudo- steady state has been established and the main task of the control loops is to counteract the slow catalyst decay. More difficulties are to be expected during the start-up period. This point is further discussed in a subsequent section of the paper.

Now consider the set of R stoichiometrically independ- ent reactions

N

i = l C ~ i j A i = 0 j = 1, ..., R (1)

The kinetics of this reaction network, taking place on a solid catalyst, as well as the kinetics of catalyst deactivation and of selectivity changes are to be investigated. Especially for control purposes, the system (1) must account for concentration changes by deactivating processes.

We assume (i) that the stoichiometry of the reaction system has already been established. Only in such cases does a detailed kinetic investigation with a sophisticated method seem justified. In complex reaction systems where the reaction components are not completely known, it is, nevertheless, possible to control the concentrations of some important components. This possibility and some other examples will be outlined below to illustrate the many modifications which are conceivable if the first and the following assumptions are relaxed.

The steady-state mass balances for the reaction system (1) occurring in a gradient-free reactor may be represented by

(2) with

(3)

f,.x, - f," = N.r(x,)

f, = f," + sT-r(x,) where

N

i= l rb , ) = m.r,(x,); f," = Cfsi"

and N is a ( L X R ) matrix of the stoichiometric coefficients ui j . Since these N - 1 = L equations contain 2N - 1 var- iables (the elements of f,", x,", and fd"), properly setting the values of N variables will determine the steady state completely and uniquely apart from pathological cases that

0 1980 American Chemical Society

Ind. Eng. Chem. Fundam., Vol. 19, No. 2, 1980 161

We remember that for an approximately ideal mixing characteristic of a recycle reactor a small inlet flow rate f," is highly profitable. Hence we may search for an op- timal f s I O value. With the optimal f s I O the solution of eq 2 and 3 (all the mole fractions being set) must fulfill (at t = 0)

f," = min (6)

Figure 1. Schematic representation of a gradientless reactor with controlled concentrations.

will not be considered in the following. Equation 2 may be split and rearranged to give

f s - x , R - fd" = N ( R p ) - r ( X , ) ( 2 4

fa 'Xs(L-R) - fOs(L-R) = N(L-RP)'N-l(RJl)'(fs'XaR - fsRo) (2b) with

f," - s T . ~ - l (RP )&R " (3a)

1 - s ~ . N - ~ ( R $ ) * x , R f s =

or fsl" - n?"-*(R,R)'fsR"

xs1 - n l T . N - l ( R , R ) . X B R f s = (3b)

It is always possible to select R key species so that the inverse N-l(RR) does exist (Aris, 1965); to some extent the selection can be made arbitrarily (Schubert and Hofmann, 1975).

Ignoring possible complications caused by loop inter- actions or other dynamic influences for the moment, perhaps the most obvious way to control M mole fractions (L 1 M 1 R) is to install R single feedback loops (see Figure 1). Each loop must be able to establish and maintain constant one particular xi variable by proper adjustment of the corresponding flow rate f ," . The mea- sured values of the R manipulated feed rates f j " have to be transferred to an on-line computing device which pro- vides the set-points of the ( M - R ) yet missing inlet flow rates f k o s by solving the system of eq 2b. Probably this can best be accomplished, when xI and f I " of an inert component have been set. (It should be noted that if more than one x r f l " pair belongs to the N design variables, a solution of eq 2 cannot be guaranteed. Such cases have been excluded from the following considerations.) With f , = f s I o / x s I from eq 3b, the relationships needed to com- pute the missing fkoB values from the measured flow rates f j o become linear, are completely uncoupled, and have constant coefficients. For the k components with selected x k values they correspond to stoichiometric invariants (Aris, 1965)

with f s ' S ' X s = S.fS" = w (4)

S = ( - N ( M - R , R ) " - ' ( R , R ) ~ ~ ~ ~ - R ~ - R ) ) (5) Other methods to provide the f k o s values result in slightly more complicated relationships (see below).

Care must be taken, of course, that setting the design variables results in adjustable feed flow rates. This point is best analyzed, if we assume (ii) that L mole fractions and the flow rate of an inert, fsIo, are set. Further we assume (iii) that some preliminary knowledge about the reaction vector r i (x) (at t = 0) is available.

with the constraints f," 1 g 1 0 (7)

gi values may be chosen with reference to the smallest flow rates that can be reliably regulated with the flow valves. The optimization according to eq 6 with the constraints (2), (3), and (7) may be performed by a linear programming technique. However, in not too complicated cases the solution should be found by simple inspection combined with some trial and error.

For example, the following procedure may be applied. (a) Find the products in the reaction system, Le., those components Ai with

Euij.rj > 0

(b) Among the products find by trial and error the com- ponent AI that fulfills

I

T V i j . r j x,l > -

$uk j . r j &k for all k # 1

(c) Solve eq 2 separately with f s 1 0 = gl to obtain all the other feed flow rates. (d) Compare them with the con- straints (7). (e) Calculate f s10 from each constraint gi that is violated, according to

( f ) With the highest value f s l 0 found in this way calculate f s I O from eq 2 and 3b. This is the optimal fsI".

Some Modifications of the Design. (1) I t should be clear from the foregoing section that a preliminary knowledge of reaction rates as functions of mole fractions is invaluable. If assumption (iii) does not hold, it would also suffice, however, to substitute the right sides of eq 2a by first-order polynomials L(f,"). These may be found, for example, from preliminary runs in which all the f s i" have been set according to a factorial design. When neither an estimate of r(x,) nor equivalent knowledge is available such as the L(f,") polynomials just mentioned, a proper f s I o value must be searched for during the experimental run by adjusting the flow rate of the inert component manually. If a component flow is nearly zero, f s I O has to be increased. Of course, adjusting f s I 0 manually may make it difficult to start up and to stabilize the recycle reactor. However, it also provides a strategy, when the a priori knowledge of the reaction rates proves insufficient or not precise enough after the start of the experimental run, or when in the course of the run selectivity changes cause flow rates of one or more components to decay too strongly.

(2) If less than L mole fractions are set, the computa- tional search for properly fixed values of several f l " vari- ables becomes a nonlinear problem in the general case. It is recommended not to abandon this part of assumption (ii) unless it is absolutely necessary. (It must be violated, for example, if some components can be fed only as a mixture.) Providing for the separate adjustability of all component flow rates does not necessarily mean that they can all be manipulated automatically. Hence it would also suffice to set all the initial mole fractions of a run, even

162

if it is not possible or not desirable to control all of them. Then the initial inlet flow rates may be searched for as outlined above.

Also with more than one f l " value being fixed for the whole run the key to control M x variables (L L M L R) through R primary control loops is to have the f , value available. The simplest way seems to set the x I - f I o pair as mentioned. Other possibilities are (a) to set one x l - f lo pair of a reactive component and compute f, from eq 3b, (b) not to set but to measure x l corresponding to a fixed

f s 1 0 and again use eq 3b, (c) to compute f, from eq 3a, and (d) to measure f , .

Points a-c have some minor flaws or disadvantages with respect to the requirements on the computing device which can be seen from eq 3a and 3b, respectively. Obtaining the flow rate f, of a gaseous mixture of slowly changing composition (point d) ready for on-line processing might be difficult in many cases.

(3) If assumption (i) does not hold, each component the concentration of which is to be controlled requires a par- ticular control loop as mentioned above. Initial manual adjusting of an (inert) component flow rate will be nec- essary to avoid too large or too small or even vanishing values of the automatically manipulated flow rates. Be- cause all or most of the control loops will interact via the chemical processes, tuning of more than a few may become a tedious and difficult task.

Workability of the Control Scheme. The sets of key components that are admissible in terms of stoichiometry will not be equally suited for control purposes. Further- more, the pairing of x I and f l o , tacitly assumed up to this point (see Figure l), is not self-evident. To assess the choice of variables to be measured and manipulated in the primary (feedback) control loops, the relative-gain method (Shinskey, 1979) may be used. It can provide some mea- sure of loop interactions and may be helpful in designing a decoupling scheme if necessary. However, the design of any multivariable control scheme will suffer from the fact that a t most an approximate knowledge of the reaction vector will be available. Remember that a complete and accurate knowledge about the rates is the ultimate goal of the whole procedure. Then, of course, no further measurements and hence no control is needed.

Among the multivariable control schemes, modal control (Rosenbrock, 1962) would appear to be a straightforward extension of the proposed design. First of all, the computing device has already been de- signed to provide linear combinations of measured varia- bles; it should be easy to extend it to do the same for modal control purposes. Secondly, even apart from the control problem, it is highly desirable to measure as many exit concentrations as possible in order to check the mass balance of the recycle reactor. For these reasons the ap- plicability of modal control to the present problem has been studied in some detail.

With the usual notation, the linearized dynamic system equations may be represented by

X = A-x + B*u (8) where x = x(t) - x, and u = f"(t) - f,". From eq 2 and 3 it follows that

Ind. Eng. Chern. Fundam., Vol. 19, No. 2, 1980

Modal Control.

a r a xs

n.A = (N - X,.S*).-- - f , . I

n.B = I - x,.eT (10) where ar/ex, is a short notation for (ar/ax),. The system matrix A is found to have R distinct eigenvalues and one (L-R)-fold eigenvalue hR+l = - f , with L-R independent

1

eigenvectors (Zurmuhl, 1964). Since pathological cases have been excluded from the

consideration, XR+l has the smallest absolute value. Therefore a t least L-R control loops have to be applied in order to improve the dynamics of the recycle reactor fundamentally.

If we retain the structure of Figure 1 and restrict the number of control loops to the minimum R, it might be advantageous to apply modal control to the R feedback loops. The remaining L-R flow rates have to be manipu- lated according to the feedforward strategy as mentioned above. It can be shown that this strategy will create L-R noncontrollable modes belonging to XR+l.

Nevertheless it remains a potential remedy against loop interactions and can speed up the response essentially as long as the noncontrollable modes, i.e., the noncontrollable components of the canonical variables y = M-x, are not excited. Excitations can be avoided if any transient of the recycle reactor starts with x = 0 and if the estimated matrix A is a matrix polynomial of the actual matrix A+, therefore having the same matrix M of left-hand eigen- vectors. The second condition will be met strictly only in special cases, e.g., when the estimated initial rates are wrong by a common factor and no selectivity change occurs during the run.

Frequently the greatest influence on control system performance will be caused by inevitable time delays due to transport lags between the recycle system and gas an- alyzers in the exit line. It is mainly because of these dead times that the analysis of a specific control scheme should be followed by some simulation studies. Simulation

Control Loops for a Particular Multiple Reaction System. In order to study the applicability of the simple feedback-feedforward control design, a system of R = 2 reactions between N = 6 components, one being an inert, was investigated.

With the stoichiometric matrix

it corresponds to the methanation of synthesis gas reaction 1: CO + 3Hz = H 2 0 + CH, ( 1 2 4 reaction 2: CO + H 2 0 = C 0 2 + H2 (12b) which was investigated experimentally after the simulation studies.

Questions left open by the general design concern the measured and manipulated variables which constitute the feedback loops, the control algorithm, or the setting of controller parameters; they were answered as follows.

Usually those components which can be analyzed in the simplest way continuously will be selected for measure- ment. The data should be available without much delay and with signals suited for on-line processing. When more than R components meet the requirements, those with the smallest concentration settings should be prefered because generally they will react most sensitively on disturbances or drifts. In the particular case of reaction systems ( la) , the components A, and A5 (CO and COz) having the smallest concentrations in the recycle reactor (see the legend to Figure 3) and actually being well suited for continuous measurement were selected to be measured for control purposes. Mainly because it agrees best with our intuition, these variables were paired with the corre-

Ind. Eng. Chem. Fundam., Vol. 19, No. 2, 1980 163

0.124 1.62

and, for comparison, those with the "true" reaction rates r(x) = 0.5.ri(x)

0.005

From the first gain matrix, eq 16, the Ziegler-Nichols method yields the control parameters

sponding feed rates f l " and f 5 " . In order to restrict the number of parameters, three-

mode controllers were ruled out. Following common usage, PI controllers were prefered to PD controllers.

Controller Tuning for the Start-up. All the answers to the open questions mentioned at the beginning of the preceding section, though based on intuition, may be re- garded as general hints for similar situations. There re- main two main problems which are best studied by sim- ulation. The first one is the tuning of the control loops in view of loop interactions and time delays. The second one is the question if the control design based on the steady-state balances will work well even when larger de- viations from the set points will occur, especially in view of the inherently incomplete knowledge of the reaction rates. Larger deviations are to be expected during the start-up or other transients. If such periods prove reliably controllable, no particular control difficulties should occur as soon as the (pseudo-) steady state has been established and the controllers have mainly to counteract slow catalyst decay. Therefore it was felt that simulation of the start-up combined with a search for optimal controller settings would be most interesting and helpful for the operation of the experimental setup.

A recycle reactor may be started as follows. The desired gas concentrations are established in the cold reactor. Then, a t t = 0, the temperature is quickly raised to the reaction temperature T, and the feed flow rates are si- multaneously adjusted to values f"' which have been calculated with an initial estimate ri(xs) according to eq 6, 7, 2, and 3.

For modeling the start-up, the following assumptions have been made. The initial estimates ri(x,) are wrong by factors d, the time lags of the flow meters and of the controllers for the feed rates f k o (see Figure 1) can be regarded as negligible, the time lags between the exit of the recycle reactor and the gas analyzers for the compo- nents AI and A5 can be accounted for by dead times td l , l and td,2, and the rates of both reactions can be represented by mass action kinetics. Reaction parameters as well as operation parameters have been given in the legend to Figure 3. The reaction weight-time t , of the j t h reaction is defined by

Assuming the initial estimates ri(x,) wrong by factors d is formally equivalent to a step-change in the catalytic activities. Therefore these factors may likewise account for rapid deactivation during the start-up. Further changes in the activities are not considered in the present simu- lation of the system dynamics.

Now initial estimates for the control parameters V and T have to be found. Rough estimates may be obtained by neglecting the interaction between the control loops and tuning the single feedback loops according to one of the well-known methods. For that reason the gain matrix (ax,/afSR") was calculated by solving the system of equa- tions

where

a f,"

The results obtained with the initial estimates ri(x,) are

T = (':'I 4i.2) s In the present case the relative gain array can be repre- sented by the so-called coefficient of interaction, K,

it seems to indicate a very weak loop interaction (Star- kermann, 1974). In spite of this low K, value, using the values of eq 18 as starting values in a search for optimal control parameters failed to give physically meaningful vectors x(t) and f"(t). Apparently this failing was caused by the time delays.

The objective function of the search was chosen the integral square error

The search was performed by means of a gradient tech- nique using difference approximations to the first partial derivatives. In the present case the optimal control pa- rameters were found to be

In this and other cases the optimal values were close to regions of instability or infeasibility of the state vector x. For actual control the control action should be somewhat weaker than the optimal one.

Figures 2 and 3 represent the course of x and f" with time, showing that the control system is very efficient. Though not surprising, it is perhaps interesting to note that without time delays the deviations from the steady-state values x, could be held less than 1% during the whole start-up with optimal controller settings. Operation

Based on the simulation studies a recycle reactor was built up and equipped with control loops for the reaction system (12). The main parts of the setup will be described in the light of the schematic representation in Figure 1. Then the performance of the system is intended to be demonstrated by examples showing some possibilities and advantages of the novel technique.

Experimental Setup. The reactor itself is a small U-shaped tube reactor filled with inert catalyst support material and a diluted catalyst sample (supported Ni- catalyst Girdler G-65) and heated with a fluidized sand bath. It is connected to an outer recirculation loop. The

164 Ind. Eng. Chem. Fundam., Vol. 19, No. 2, 1980 'pl- ;60 1.2 ] O o 'It I = L I b o

- ' 0 lo+Oo - t i m e , s

t o s ; -10 - ' O

-c ; a

-10 -10

4 0

Figure 2. Start-up of reaction 12 with optimal controller settings (simulated). Deviations of controlled concentrations from steady- state values (set points). For data, see Figure 3.

. E

N

0 0.-

t 8.8-

, 3 6 1 i: I i=5;

Figure 3. Start-up of reaction 12 with optimal controller settings (simulated). Course of manipulated flow rates with time. Assumed reaction and operation data: lo2 X x,T = (1.0, 18.3, 3.2, 9.9, 1,6); p = 1 atm; T, = 673 K; t,: = 0.586 g s/cm3; t,; = 0.62 g s/cm3; rn =

14 s. 1.0 g; dT = (0.5, 0.5); T, = 1.12 g S/Cm3; 71 = 48.6 S; tdl , l = 7 S; tdl,:, =

whole system including a diaphragm pump contains some 50 cm3. The gaseous components are fed through separate lines by means of mass flow controllers. The exit gas is continuously analyzed for CO and CO, by NDIR instru- ments. Deviations from the set points act through primary two-mode controllers (not shown in Figure 11, the outputs of which direct the set points of the corresponding cas- caded flow controllers. The on-line computation of the set points of the other feed flows is done by means of some operational amplifiers. The inert gas flow (N,) constant as a whole is split up by control action so as to pass a portion of it through a temperature-controlled water trap. Thus the proper H,O feed rate is provided. For super- vision of the other gas concentrations the exit gas can be analyzed at times with gas chromatographs.

Factorial Experiments. In the first instance some experiments were performed without concentration con- trol, in the course of which the several feed rates were varied according to factorial designs. These experiments provided some ideas about the feasible region of concen- trations and about the order of magnitude of the reaction rates. Then based on simulation, on increasing 4. and on occasional improvements by trial and error, suitable or even nearly optimal controller settings could be arrived at in the following experiments. An example is given in Figure 4. Responses of concentrations and feed rates to

r 0.30 1:: h b

72

i n

' 'h 1 . 5 1 : c k , , , ,

0 100 260 360 1.3

+tim(.s

Figure 4. Response of concentrations (a) and flow rates (b) to a step change in the concentration settings: X, recorded; -, calculated. Reaction and operation data: lo2 X qT = (0.28,7.39, 1.53,9.69, 1.08); fsNp = 6.648 X = 6.396 X mol/s; p = 1 atm; TI = 548 K; tr< = 0.045 g s/cm3; tr$ = 0.094 g s/cm3; m = 0.4 g; T, = 0.21 g s/cm3; 71 = 26.8 s; td1,l =

mol/s; lo2 X xST = (0.20, 5.35, 1,89, 10.4, 1.00), f N

7 9; td1,2 = 14 S.

a step change in the concentration settings and a simul- taneous minor adjustment of the Nz feed rate have been recorded with a twelve-point intermittent recorder and are indicated by crosses in the figure. After having established reliable operation during steady states and transients (including the start-up), factorial designs were performed with the concentrations being the factors.

Most of the work concerned with methanation kinetics has more or less completely ignored the simultaneous oc- currence of the water-gas shift reaction (12b), a notable exception being a paper by Saletore and Thomson (1977). Obviously this is not justified at least under the present reaction conditions. This may be seen from the reaction weight-times t , which are felt to present a well suited measure of the absolute catalytic activity. These times are comparable in magnitude (see the legend to Figure 4).

The systematic approach of factorial designs enabled us to measure the rates of both the reactions and represent them by significant main effects and factor interactions. The significant effects have been used for simulating the transients in Figure 4 (full lines) with obviously satisfactory success. Conservative controller settings have been used in experiment and simulation; optimal settings computed according to eq 19 actually result a t least in stronger os- cillations similar to those in Figures 2 and 3 and lead frequently to unstable behavior.

The possibility to establish selected distances from equilibrium for reaction 12b proved especially helpful for measuring the rate of this reaction accurately. Further- more, a t particular concentration levels a net consumption of carbon dioxide being in contrast to the driving force could be observed. I t appears that a kinetic reaction set must contain at least one macroscopic reaction step more than the stoichiometric set (12). We have tried to model the kinetics of the reaction system assuming that this additional step is the direct hydrogenation of C02 to CHI. The results are not presented here because they are ca- pable of further improvements and a discussion is beyond the scope of this paper.

Uncoupling the Effects of Concentrations and Ac- tivity. Even with highly purified gases and at only mod- erate temperatures methanation catalysts will initially deteriorate more or less rapidly and then continue to deactivate slowly for long times on stream. Such behavior

Ind. Eng. Chern. Fundam., Vol. 19, No. 2, 1980 165

experimental designs with the concentrations being the variables. Presumably even more important is the un- coupling of concentration and activity effects upon the rates. Above all, the (changing) catalytic activity, defined by aj = rj(xRf)/r/(xref), can be obserued and needs no longer be only estimated.

To control concentrations in a recycle reactor during start-up and operation is, in principle, a problem of optimal and/or adaptive control. The present design, however, is as simple as possible, based on steady-state balances and using some analogue units for on-line data processing. The design is realizable with common and commercially available instruments.

The design has proved efficient in simulation and ex- periment, as demonstrated by means of the simultaneous methanation and water-gas shift reactions. Though of a preliminary nature, some interesting new insights into the reaction and deactivation kinetics of this system could be obtained mainly due to the new approach.

Application will be facilitated if the stoichiometry of the reaction system being under consideration is known and some order-of-magnitude knowledge of the reaction rates is available, although these are not absolute prerequisites. It should be obvious that such a more or less sophisticated method will usually not be applied in the very beginning of kinetic investigations of a particular reaction system. Acknowledgment

The experimental work performed by R. Krokoszinski is greatly appreciated. Financial support has been given by the Federal State Niedersachsen and by the Max- Buchner-Forschungsstiftung. Nomenclature A = reaction component A = system matrix, see eq 9 a = catalytic activity B = control matrix, see eq 10 eT = (1, 1, ..., 1) f = molar flow rate f." = inlet flow rate manipulated in a feedback control loop fko = inlet flow rate manipulated in a feedforward control loop f l o = fixed inlet flow rate I = unit matrix g = smallest adjustable flow rate L = N - 1 M = matrix of left-hand eigenvectors m = catalyst mass N = number of reaction components N = matrix of stoichiometric coefficients ui j n = number of moles in the loop nT = row vector of N p = total pressure R = number of reactions in a reaction system R G = gas constant r = r,.m, reaction rate rm = specific reaction rate based on catalyst mass S = matrix of coefficients in the invariants

T = temperature T = diagonal matrix of reset times t = time t , = reaction weight-time, see eq 13 V = diagonal matrix of controller gains u = volume flow rate w = vector of invariants 1c = mole fraction y = canonical variable zi = weight factor Greek Letters X = eigenvalue of A v = stoichiometric coefficient

sT = (Xiuil, **., XiViR)

approximate time onstream, mln 1 2 2LO 480 720

1

I '* 1.

run number

Figure 5. Center points replicates of a factorial design showing activity and selectivity changes in the reaction system (12) a t 275 "C: 0, methanation; 0, shift reaction.

0 4 \ 100 200 300 LOO

00- 1 t i m e onst rearn,mln

0

Figure 6. Continuously observed reaction rates a t fixed concen- trations of the main components showing activity and selectivity changes in the presence of H,S: -, methanation; - - -, shift reaction. has recently been observed again by Dalla Betta e t al. (1975) for the methanation reaction alone. The operation with controlled gas concentrations allows one immediately to obtain a definite measure of activity for each reaction of the system, ai = rj(xref)/rji(xref). In the course of the factorial experiments, replicates of the center point were taken as r(xref) (Figure 5). Not only is the catalyst decay measured quantitatively, but also a selectivity change in the reaction system could be clearly observed for the first time. The activity curves in Figure 5 can be used to relate the measured reaction rates of the plan to a definite level of activity. This will uncouple concentration and activity effects provided reaction and deactivation kinetics can be treated as separable (SzBpe and Levenspiel, 1971). It may further be concluded from Figure 5 that deactivation is concentration-independent in the concentration range of the plan.

A last illustrative example is given in Figure 6. It shows the course of a run with some 300 ppm of H2S in the feed. Over the whole range of reaction rates, the concentrations of the six main components could be kept constant without readjusting the initial controller settings. The H2S con- centration was not controlled but measured at times in the exit stream. In principle, since the deactivation rates a as a function of activity and H2S concentration can be obtained by differentiation, a kinetic evaluation of such activity curves would already be possible. The additional control of the poison level, now in progress, will enable an integral evaluation and make the whole problem of kinetic modeling more amenable. The hitherto unknown selec- tivity change by H2S poisoning should be a challenge for future mechanistic considerations. Summary and Conclusions

Control of some or all gas concentrations in a laboratory recycle reactor opens new possibilities for detailed kinetic studies of catalytic gas-solid or other heterogeneous gas reactions. It becomes possible, for example, to perform

166 Ind. Eng. Chem. Fundam. 1980, 19, 166-175

T I = n / f , mean residence time in the loop T, = r n / u o , weight-time Subscripts dl = delay e = equilibrium I = inert i , I , k = component i, I , k j = reaction j ref = reference s = steady state Superscripts i = initial O = inlet s = set point T = transpose L i t e r a t u r e Ci ted Aris, R., "Introduction to the Analysis of Chemical Reactors", Chapter 2.7,

Prentice-Hall, Englewood Cliffs, N.J., 1965.

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Schubert, E., Hofmann, H., Chern. Ing. Tech., 47, 191 (1975). Shinsky, F. G., "Process Control Systems", Chapter 8, McGraw-Hill, New York,

1979. Starkermann, T., "Mehrgrossenregelsysteme I", in "Theoretische und exper-

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70 (1977).

Received for review March 26, 1979 Accepted December 27, 1979

Cross-Flow Electrofilter for Nonaqueous Slurries

Chang H. Lee, Dlmltri Gldaspow,' and Darsh T. Wasan

Department of Chemical Engineering, I//inois Institute of Technology, Chicago, Illinois 606 16

An improved method of removal of fine particles suspended in nonaqueous media by the application of a high-voltage (1000 to 10000 V/cm) electric field was developed. The technique is a modification of ordinary cross-flow filtration with a porous tube and of forced flow electrophoresis. The electrofilter was tested with a synthetic nonaqueous slurry as well as samples of diluted H-coal process liquids obtained from a coal liquefaction pilot plant at various electric field strengths, driving pressures, and feed rates. Models of clear boundary layers for flat and tubular cross-flow electrofilters were developed as a function of inlet Peclet number, electrophoretic velocity of particles, and rate of filtration.

I. Exper imen ta l B a c k g r o u n d

One of the major bottlenecks in development of coal liquefaction technology is the removal of fine solid particles after the liquefaction step. Rotary drum pressure precoat filtration has been the most widely employed separation procedure. However, its unfavorable economics have spurred research in alternate techniques.

Settli'ng of solids by the addition of a precipitating liq- uid, such as o-xylene, is such an alternate separation technique, as discussed in the patent literature (Snell, 1974; Sze and Snell, 1974) and elsewhere (Gorin et al., 1977). However, to promote settling, large volumes of promoter liquids are necessary. Settling may also not work well for final removal of fine particles. In view of these difficulties a new separation method is being proposed (Gidaspow et al., 1978).

The new separation method is based on some concepts practiced in electrostatic precipitation of aerosols as well as in forced flow electrophoresis of aqueous colloidal suspensions (Moulik et al., 1967; Moulik, 1971; Henry, 1977). In forced flow electrophoresis of aqueous suspen- sions or cross-flow electrofiltration (as called by Henry and Jacques, 1977) only low voltages are applied, to prevent the decomposition of water. In electrostatic precipitation of dust, high voltages are applied, but the method of re- moval of the clarified gas is different from that in elec-

0196-4313/80/1019-0166$01.00/0

trofiltration. The particles accumulate on the electrode which is periodically cleaned by a mechanical vibration. Thus the method proposed here differs from all other techniques described in the literature. Concept of Cross Flow Elec t ro f i l t r a t i on

Figure 1 illustrates the concept of cross-flow electrofil- tration. The filter is a porous metal plate or tube, as in the case of ordinary cross flow filtration (Henry, 1972). The top is a nonporous metal plate or wire, in the case of cylindrical geometry.

If particles are positively charged, they will migrate toward the negative plate. If they are negatively charged, the polarity is reversed. The advantages over conventional filters are the following. (1) The applied potential permits an almost continuous operation since the small colloidal particles migrate away from the filter. A clear filtrate is continuously withdrawn. (2) The filter can be washed by reversing the flow without taking apart the filter. This can have obvious advantages when operating at high temper- atures and at high pressures. (3) The filter may also be washed by reversing the electric field and by letting the concentrated particles drop down by gravity. A screw-type mechanism may remove such particles resulting in a nearly uninterrupted operation. (4) The filter can operate at high energy efficiencies, because of (a) small pressure drop across the filter due to the absence of a filter cake and large pore opening in the porous tube or plate and (b) small

0 1980 American Chemical Society