236
Ind. Eng. Chem. Process Des. Dev. 1986, 2 5 , 236-241
Levenspiel, 0. "Chemical Reaction Engineering", 2nd ed.: Wiiey: New York, 1972. Mas&& H. A.; Maymo, J. A. J . c8&/.igse, 74, 61. Prasad, K. E. S.; Doraiswamy, L. K. J . Catal. 1974, 32, 384. Ravimohan, A. L.; Chen, W. H.;Seinfeld, J. H. Can. J . Chem. Eng. 1970, 48, 420.
Wei, J. Ind. Eng. Chem. 1966, 58, 38.
Received for review October 22, 1984 Revised manuscript received June 4, 1985 Accepted July 3, 1985
Deactivation Kinetics of Platinum-R henium Re-forming Catalyst Accompanying the Dehydrogenation of Methylcyclohexane Ajlt K. Pal, Madhumlta Bhowmlck, and Rameshwar D. Srlvastava' Department of Chemical Engineering, Indian Institute of Technology, Kanpur -2080 76, India
The kinetics and mechanism of deactivation by coking of Pt-Re-AI,O, catalyst for the dehydrogenation of methylcyclohexane have been examined along with the kinetics of the main reaction. A statistically best rate expression for the main reaction, developed on the basis of the single-site adsorption of methylcyclohexane, was determined from the experimental data. Deactivation occurred in parallel with the main reaction where methylcyclohexane was adsorbed in different ways in the main and the deactivation reactions. The deactivation kinetic equation was governed by the reaction of two adjacent adsorbed methylcyclohexane molecules, resulting in the coke precursor.
Catalysts for re-forming reactions are small crystallites of Pt or Pt-Re supported on alumina. The addition of Re greatly improves the catalyst's resistance to poisoning by coke. The deposition of carbon may be due to side reactions or due to the decomposition of the organic reactant. As a consequence, the kinetic study of the main reaction in itself becomes seriously complicated. Although investigationsto determine the role of rhenium in the platinum-re-forming catalyst have been made for some time, deactivation-kinetic analysis has been slow to develop. It is only recently that some rate data have been obtained (Jossens and Petersen, 1982; Pacheco and Petersen, 1984a, 1984b). A recent bibliographic review on how Re addition brings about the promotional effects can be found in the references (Jothimurugasan et al., 1985). Jossens and Petersen (1982) and Pacheco and Petersen (1984a, 1984b) studied the deactivation kinetics of the Pt-Re re-forming catalyst accompanying the dehydrogenation of methylcyclohexane (MCH) in a single pellet diffusion reactor. Their analysis with the use of a single pellet diffusion reactor required an accurate measurement of the "center plane concentration". Furthermore, in their models, the fouling analyses are restricted to simple power law model relations. Recently, Corella and Asua proposed (1981) and generalized (1982) theoretically the kinetics of deactivation by coking that relates activity directly to the deactivation reaction and obtained the deactivation equation of the form
where $(pi,") is the deactivation function and d = (m + h - l)/m, m and h are the number of active sites involved in the controlling step of the main reaction and deactivation reaction, respectively. This method has been suc-
* Present address: Department of Chemical Engineering, University of Delaware, Newark, D E 19716. 0196-4305/86/1125-0236$01.50/0
cessfully applied in the studies of catalyst deactivation by coke formation in furfural decarbonylation on Pd-Al2O3 (Srivastava and Guha, 1985) as well as in the dehydration of isoamyl alcohol over silica-alumina catalyst (Corella and Asua, 1981). The above proposal is a more fundamental approach to the analysis of catalyst deactivation. As an example of the application of this approach and in keeping with the goal of this study, .it is appropriate to consider the case of MCH dehydrogenation over Pt-Re-A1203 catalyst. The experimental results are analyzed on the basis of LangmuirHinshelwood kinetics for the main reactiion as well as for the deactivation reaction with statistical data interpretation. Experimental Section Catalyst Preparation. Platinum-rhenium catalyst on
alumina support was prepared by the impregnation (incipient wetness) technique. The chemicals used were chloroplatinic acid (Johnson Mathey, London), rhenium heptaoxide (Riedel-De Haen, West Germany), and 7-alumina with a BET surface area of about 220 m2/g. After impregnation, the catalyst was dried in air a t 373 K and calcined in an air stream for 5 h a t 723 K. The catalyst composition was alumina with 0.3 wt ?& Pt and 0.3 wt 7'0 Re (Pt-Re-Al,03; metal area = 0.35 m2/g). Individual metal catalysts with 0.3 wt % Pt and 0 . 3 w t % Re on y-alumina were also prepared under the same conditions except for Re-A1203 where calcination was not performed. The catalysts have been recently characterized by the use of proton-induced X-ray emission and Rutherford backscattering spectrometry (together with electron microscopy and chemisorption studies) for the dehydrogention of methylcyclohexane (Jothimurugesan et al., 1985). Kinetic and Deactivation Runs. Catalyst was tested primarily for the initial activity and resistance to deactivation by using a micropulse reactor-chromatograph assembly, Hewlett-Packard Model 5880. With this apparatus, either single-pulse or a train of pulses of known volume of MCH and frequency can be automatically 0 1985 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 1, 1986
passed through the catalyst bed (1cm length, 0.3 cm diameter, containing 0.2-0.3 g of crushed catalyst of 0.1 mm mean diameter), using a suitable carrier gas such as N2or HP. The MCH pulse rate was 1 pL/s. The flow rate of the carrier gas was 30 mL/min and the operating temperature range 220-700 K, R (mol of Hz/mol of MCH) or R' (mol of N,/mol of MCH) = 2-5, and W/FMo (g of catalyst h/gmol) = 5. The activity of Pt-Re-AI,O3 catalyst was checked in the presence of hydrogen, and it remained constant for 24 h. For comparison, the activities of Pt-A1203and Re-A1203 catalysts were also carried out. Rhenium catalyst showed no activity. The activity of the Pt-A1,03 catalyst was nearly the same as that of Pt-Re-A1203 catalyst. The experiments in which the reaction feed was a mixture of MCH and toluene have shown that in the presence of hydrogen, toluene had a retarding effect on the rate of dehydrogenation and the catalyst activity. In contrast, when nitrogen was present, the influence of the addition of 10% toluene to the feed showed negligible inhibition. The micropulse reactor enabled promising operating conditions to be quickly found. Catalysts are then readily tested in a large-scale reactor to obtain kinetic data. Before the bulk of the experimental work was taken in the integral reactor, certain preliminary studies were made to aid in an analysis of the results. The experimental conditions were so chosen that the reaction rate was not influenced by external and internal diffusion (Ross and Walsh, 1961). The fixed-bed reactor with general experimental procedure was similar to that described by Sharma and Srivastava (1981, 1982). The flow rate of MCH was maintained by a metering pump. The MCH was fed to the preheater where it was vaporized and mixed with either Hz or N2 gas depending on the kinetic or deactivation study. The mixed vapor was then to the reactor containing the catalyst bed. The temperature of the catalyst bed was measured by an iron-constantan thermocouple located in a thermowell. The effluent from the reactor was cooled in a series of condensers. The noncondensed gases were vented via a gas flow meter. The liquid condensate was analyzed by employing a gas-liquid chromatograph. A 2.0 m long, 3.2 mm diameter column of 10% SE-30 on Chromosorb W was used to separate the various components. The major products from dehydrogenation of MCH were hydrogen (H,) and toluene. The conversions at zero time, (XM)o, were calculated from the initial kinetic runs, whereas the conversions at any time, t , XM,were obtained from the collected samples of the deactivation runs. Prior to the reaction and after calcination, the catalyst was reduced by passing hydrogen gas at 723 K for 6 h. The catalyst was then presulfided with thiophene vapor for 5 h. During the time of reduction, the pressure inside the reactor was maintained at 1.0 atm.
Result and Discussion Kinetic Equation at Operation Time Zero. The kinetics of the main reaction were studied in an integral reactor by using conversion data at zero-time coke content. The experiments were conducted for 12 h with the following operating conditions: R = 3; temperature range, 598-673 K; W/FMo = 5-25. The conversion (xM)o data vs. space time ( w/FMo) for the Pt-Re-Alz03 catalyst at zero coke content are shown in Figure 1. The numerical values of the rate of reaction at zero time (-rMl0were obtained by a second-order polvs. W/FM, followed by ynomial fit of the curves (XM)o analytical differentiation. A nonlinear regression algorithm (Sharma and Srivastava, 1981) utilizing Marquardt's algorithm (Marquardt,
237
W/FMo(g-COt h / g mole)
Figure 1. Con_versiona t time zero against space time at different temperatures; R = 3.
1963)was used to obtain a mathematical fit for various rate expressions. The starting values of the parameters were obtained from linear regression. Once the model satisfied the criterion of parameter values to be physically significant, then the goodness of fit of the model was checked by performing an F test. The dehydrogenation of methylcyclohexane (MCH) to toluene proceeds according to the reaction y 3
y 3
The Langmuir-Hinshelwood isothermal rate equations based on both the single- and dual-site adsorption and surface reaction for the above reaction were considered. These rate equations consist of adsorption for each of the reaction components and hydrogen. The regressions of the rate equations were performed at each temperature. All possible combinations of the various adsorption terms were examined to allow for various negligible adsorption terms. The isothermal rate equations, basic types of which are shown in Table I, thus resulted in 1 2 different mathematical forms which were confronted with the experimental data. The following model SA-2 (derived from,the SA series) described the experimental data most satisfactorily.
The values of the kinetic constants are given in Table 11. An F test was performed on the residual for this model in which six replicates were carried out. The Fobsd value 5.03 lies within the range 0 < Fobsd < F(4,5,0.95) = 6.26, where 4 is the degree of freedom for calculating the pure error variance, 5 is the degree of freedom for estimating the lack of fit variance, and 0.95 is the confidence interval. The following relations between the kinetic constants and the temperature were determined. K = 1664 exp(-6750/T) (4) KT = 1.59 X exp(6000/7') (5) The activation energy for the main reaction is 56.4 kJ/mol
238 Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 1, 1986 Table I. Isothermal Rate Equations (Equation 2 ) model
rate-controlling step
rate equation
Single-Site Mechanism k(pM
SA
adsorption of MCH
ss
surface reaction
(-rM)o
=
PTPH3 -
7)
KMPTPHa + KTPT + KHPH K
Dual-Site Mechanism
S surface reaction
DS
Table 11. Values of t h e Kinetic Constants T, K k, gmol h-' atm-' 598 0.021 f 0.001 623 0.037 f 0.003 648 0.049 f 0.005 673 0.064 f 0.007
Table 111. Deactivation Function Values
Kr, atm-' 3.160 f 1.490 2.423 f 1.549 2.055 A 1.475 1.234 f 1.054
which compares with the literature value of 71.1 kJ/mol (Jossens and Petersen, 1982). The lower value and activation energy obtained in the present study can be attributed to the increased toluene inhibition at reaction temperatures in excess of 598 K. A similar inhibition effect has been reported by Jossens and Petersen (1982). Conversion of methylcyclohexane to toluene proceeds through several steps and is accompanied by a considerable number of side reactions. Toluene is generally formed via a series of consecutive reactions involving dehydrogenation of methylcyclohexane to methylcyclohexene followed by the dehydrogenationof methylcyclohexadiene. This f m d y dehydrogenatedto toluene. A detailed reaction mechanism for the main reaction is given elsewhere (Jothimurugesan et al., 1985). In the present study, a reaction mechanism describing the surface reaction phenomena based on the initial rate data is given in eq 9. Kinetics of Deactivation. The results of the experimentations in the micropulse reactor showed that the addition of 10% toluene in the MCH feed has no appreciable effect on the rate of deactivation reaction. These results indicated that the deactivation reaction was parallel; that is, coke was formed from the MCH feed and not from the product toluene. Accordingly, the deactivation reaction may be described as
$' (PMplr)
t, h
0.00 1.00 2.00 3.00 4.00 5.00 0.00 1.0 2.00 3.00 4.00 5.00 0.00 1.00 2.00 3.00 4.00 5.00 0.00 1.00 2.00 3.00 4.00 5.00
p M ,atm
pT,atm
h=1
h=2
Temperature = 598 K 0.5525 0.0607 0.0977 0.0557 0.0949 0.5690 0.5872 0.0501 0.0925 0.6073 0.0439 0.0916 0.0370 0.0907 0.6296 0.6546 0.0294 0.0896
0.0977 0.1049 0.1135 0.1264 0.1425 0.1623
Temperature = 623 K 0.0795 0.1749 0.4917 0.0739 0.1557 0.5098 0.5299 0.0677 0.1413 0.0605 0.1356 0.5535 0.0523 0.1296 0.5802 0.6119 0.0425 0.1209 Temperature = 648 K 0.0906 0.3999 0.4554 0.4733 0.0851 0.2972 0.4944 0.0786 0.2369 0.0711 0.1941 0.5188 0.5477 0.0623 0.1644 0.0514 0.1431 0.5831 Temperature = 673 K 0.0996 0.7497 0.4263 0.4440 0.0941 0.4408 0.0878 0.3111 0.4646 0.0802 0.2411 0.4894 0.0710 0.1961 0.5192 0.0594 0.1662 0.5569
0.1749 0.1844 0.1969 0.2285 0.2692 0.3063 0.3999 0.4228 0.04503 0.4647 0.4798 0.5034 0.7497 0.7882 0.8232 0.8713 0.9098 0.9836
and the reaction rate at zero time, (-rM)io,with both rates measured at the same conversion or partial pressure of the reaction mixture utilizing the relation (Corella and Asua, 1982)
M-T+H M
-
coke
In order to determine the kinetic equation for the deactivation reaction, the deactivation rate data were taken from the experiments carried out in an integral reactor for 5.0 h with the following operating conditions: R' = 3; temperature range, 598-673 K; &'IFMo = 5-25. These data are represented in Figure 2. A typical plot of XMvs. t (operation time) is shown in Figure 3. From the XM vs. W/FM, curves at different operation times, Figure 2, the reaction rate, which is the slope of the curve, was calculated for a particular space time (W/FM,) = 20, at different operation times: (-rM)O,(-rM)l, (-r& ..., etc. Then the activity at any time t was determined as the ratio between the reaction rate at any time t , (-rM)j,
(7)
where j = 0-5. The values of the activities at different times are shown in Figure 4. The deactivation rates (-daldt) were evaluated again by the usual second-order polynomial fitting as well as analytical differentiation. The deactivation function values were then calculated from eq 1and are listed in Table 111. The possible isothermal rate equations for the deactivation reaction (eq 6) were derived based on single-site and dual-site mechanisms by the same method as described by Corella and Asua (1982) and are represented in Table IV. All these derivations were based on the assumption that the controlling step was the formation of the coke
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 1, 1986
I
Tamp.=598K
239
o 598 K A 623 K 648 K 673 K
X
Tamp. z 623 K
uO
2
1
5
4
3 t. h
Figure 3. Conversion against process time a t different temperatures; WIF,, = 20, R’ = 3.
W / F M (9-cot ~ h lg mole)
o,6
c
Time, h Temp.=6CBK
0
I
I
I
1
2
3
I L
I
5
t,h
Figure 4. Activity against process time a t different temperatures; R‘= 3. Table IV. Isothermal Deactivation Rate Equations Assuming Step 2 as Rate-Controlling
model
step 1 M + 1 M*l Step 2 nM(g) + hM*l+ Step 3 (p,lh) e (P21h)* (Pllh)’ ‘ .coke m h n rate equations
DE-3
1
W/FMo(g-cat. h l g mole)
Figure 2. (a, left) Conversion against space time for a different process time at temperatures 598 and 623 K; R’ = 3. (b, right) Conversion against space time for a different process time a t temperatures 648 and 673 K; 8’ = 3.
precursor (step 2). To evaluate the parameters of the deactivation rate equation, the value of K T for the main reaction has been used. The same linear and nonlinear regressions as used for the main reaction were also employed. The model which described the experimental results most satisfactorily is (model DE-3)
_ -da dt
k,jKM*2pM*2
(1
+ K T p T + K M * p M * ) ’ a2
(8)
DE-4
1
2 2
0
1
dt
-
-*
d KM
MZ
f
KTpl
+
a’
KM*PM)’
kdKM*2PM
=
dt
*
(1 + KTPT +
a2
KM*PM)’
This model was based on the assumption that two adjacent adsorbed methylcyclohexane molecules are involved in the formation of the coke precursor P,lz,which ultimately leads to coke through several equilibrium steps. The mechanism is
240
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 1, 1986 Mi
M*i
e
+M"I r.d.s.
TI
+
P,ie
3H2(g)
e==
Ppiz
Table V. Kinetic Constants for the Deactivation Reaction T,K k i . h-' KM*,atm-' 598 0.214 f 0.008 6.077 f 0.944 623 0.485 f 0.007 4.505 f 0.127 648 1.054 f 0.009 4.295 f 0.575 673 2.599 f 0.106 3.069 f 0.148
d
P3iz***coke (9)
where the formation of the coke precursor is the rate-determining step, and MCH is adsorbed in a different way from the main reaction. The same phenomenon has been reported by Jodra et al. (1976) for the dehydrogenation of benzyl alcohol over a Cu-Cr,O,/asbestos catalyst, where benzyl alcohol was adsorbed in a different way in the main reaction and the deactivation reaction. The parameter values corresponding to eq 8 are listed in Table V. The relations between the constants of the deactivation rate equation and temperature are
kd = 5.64 X 10" exp(-17500/T) KM* = 0.014 exp(3625/T)
(10) (11)
The activation energy for deactivation was calulated as 146.6 kJ/mol, in agreement with the value of 163.2 kJ/mol obtained by Jossens and Petersen (1982). This value is higher than that for the main reaction, indicating that the deactivation reaction is more sensitive to temperature than the main reaction. This higher activation energy also indicates that the deactivation reaction is very slow. The present model describes that the coke precursor is formed from the adsorbed reactant where the adsorption for the coking reaction occurs in a manner different from the main reaction. This differs from the model proposed by Pacheco and Petersen (1984b) where the coke precursor was considered to be formed from the adsorbed reactant, but the adsorption for coking was similar to the main reaction. In the proposed mechanism, it is conceivable that the adsorbed sulfur on the Pt-Re surface impedes the ultimate transformation of Pill, P,l,, ..., "coke". This is in general agreement with the observations that sulfur pretreatment in naphtha re-forming lowers the aging rate of the industrial Pt-Re-Al,03 catalysts. In a parallel study (Bhowmick, 1984), the kinetics and mechanism of deactivation of the Pt-Al,03 catalyst have been investigated for the same reaction. Bhowmick obtained the same deactivation model as presented by eq 8 with a value of KM*= 21.6 atm-l a t 598 K. The present value of KM*= 6.1 atm-' is considerably lower than that of the Pt-A1203 catalyst, indicating a weak adsorption of MCH. This is mainly because of the addition of Re since this diluent metal does not form strong bonds with MCH, thereby reducing the chance of adsorbed molecules forming multiple bonds with several adjacent sites. Platinum is probably unique in its formation of stable, strongly bound polymeric residues, because of its very low activity as a hydrogenolysis catalyst. It is interesting to note that in the experiments conducted in the micropulse reactor, air addition is seen to lower the toluene yield, raise the catalyst bed temperature substantially, and produce no apparent deactivation of the catalyst over the 2-day period of the experiments. The idea would seem to warrant further evaluation in the longer term, to permit trade-offs between hydrocarbon and hydrogen losses, hydrogen recovery, and catalyst utilization, and more to quantify a more straight-forward reactor design.
Conclusion The rate equation for the dehydrogenation of methylcyclohexane to toluene along with the kinetics and mechanism of deactivation of the Pt-Re-Al,O, catalyst have been determined. Sulfur plays an important role with the bimetallic catalysts. The initial rate equation, deactivation rate equation, and the relation between initial rate and rate at any time have been treated simultaneously to describe the kinetics and the deactivation of the catalytic system completely. The equations obtained from this mechanistic approach are useful for the reactor design. Nomenclature a =
catalyst activity, integral reactor
a. = catalyst activity at time j
d = deactivation order
= feed rate of methylcyclohexane, mol/h h = number of active sites involved in the controlling step of deactivation reaction KM, KT, KH = equilibrium adsorption constants for methylcyclohexane, toluene, and hydrogen, respectively, in the main reaction, atm-' KM* = equilibrium adsorption constant for methylcyclohexane to yield the coke precursor, atm-l k = rate constant of the main reaction, gmol/(g of catalyst h atm) kd = rate constant of the deactivation reaction, h-' K = thermodynamic equilibrium constant, atm3 1 = active site m = number of active sites involved in the controlling step of the main reaction M, T, H = methylcyclohexane, toluene, and hydrogen, respectively n = number of molecules of M in the gas phase which react with adsorbed M to give the coke precursor Pl12,P1lh = coke precursor P212,P312, P& P31h= different forms of coke in the coking sequence pM, p T , pH = partial pressures of methylcyclohexane,toluene, and hydrogen, respectively, atm ei= partial pressure of i, atm R = mole ratio of hydrogen to MCH R' = mole ratio of nitrogen to MCH -rM = rate of reaction of M, gmol/(g of catalyst h) ( - T ~ )=~ rate of reaction of M at zero time T = absolute temperature, K t = time, h W = weight of catalyst in reactor, g XM= conversion of methylcyclohexane at any time (XM)o= conversion of methylcyclohexane at zero time FM,,
Greek Letters
$(p,,T) = deactivation function, general $(pM,T)= deactivation function, MCH system Registry No. Methylcyclohexane, 108-87-2; platinum, 744006-4; rhenium, 7440-15-5.
Literature Cited Bhowmick, M. M.S. Thesis, Indian Institute of Technology, Kanpur, 1984. Coreila, J.; Asua, J. M. Can. J . Chem. Eng. 1981, 59, 506. Coreiia, J.; Asua, J. M. Ind. Eng. Chem. Process Des. Dev. 1982, 27, 55. Jodra, L. G.; Romero, A,; Coreila. J. An. Quim. 1976, 72, 823. Jothimurugesan, K.;Bhatia, S.; Srivastava, R. D. Ind. Eng. Chem. Fundam. 1985, 24, 433. Jothimurugesan, K.; Nayak, A. K.; Mehta. G. K.; Rai, K. N.; Bhati, S.; Srivastava, R. D. AIChE J . , in press. Jossens. L. W.; Petersen. E. E. J. Catal. 1982, 7 6 , 265.
Ind. Eng. Chem. Process Des. Dev. 1986, 25,241-248 Marqwrdt, D. W. J. SOC.Ind. Appl. Math. 1883, 1 1 , 431. Pacheco, M. A.; Petersen. E. E. J. Cafal. 1884a, 86, 75. Pacheco, M. A.; Petersen, E. E. J. Catal. 1884b, 88, 400. Ross, R. A,; Walsh. 0. 0. J. Appl. Chem. IS81. 1 1 , 469. Sharma, R. K.; Srivastava, R. D. AIChE J. 1881, 2 7 , 41. Sharma, R. K.; Srivastava, R. D. AIChE J. 1882, 28, 855.
24 1
Srivastava, R. D.; Guha, A. K. J. Catal. 1885, 91, 254.
Received for reuiew December 17, 1984 Reuised manuscript received June 2 , 1985 Accepted July 3, 1985
Controller Design for Nonlinear Process Systems via Variable Transformations Babatunde A. Ogunnalke’ Chemical Engineering Department, University of Lagos, Lagos, Nigeria
Departing from the traditional approach of local linearization followed by linear controller design for the thus “linearlzed” system, it is shown in this paper how transformations may be found which transform the nonlinear system into one that is exactly linear. The new approach is based on the hypothesis that a system which is nonlinear in its original variables Is linear in some transformation of these original variables. Constructive methods for finding the appropriate transformations are presented, and practical guidelines to facilltate design are included. The design of a controller for the linear transformed system may then be carried out with a great deal of facility (the resulting controller will of course be nonlinear when recast in terms of the original system Variables). The specific problem of ievei control in process flow systems having various geometric configurations is used to illustrate the potentials of this approach. An example simulation is used to demonstrate the design and performance of these controllers.
1. Introduction It is well-known that the great majority of important chemical processes exhibit nonlinear dynamic behaviors. Thus, most of the practical process control problems will involve nonlinear systems. This fact notwithstanding, the bulk of existing control theory involves the design of linear controllers for linear systems (cf.: Ray, 1981; Stephanopoulos, 1984). Only a modest collection of results is available which may be directly applied to nonlinear systems. The traditional and easiest approach to the controller design problem for nonlinear systems involves linearizing the modeling equation around a steady state and applying linear control theory results. It is obvious that the controller performance in this case will deteriorate as the process moves further away from the steady state around which the model was linearized. Apart from the ”local linearization” approach, there are a few other “special-purpose” design procedures (cf.: Ray, 1981) which may be applied directly to nonlinear systems. However, as noted by Ray (1981), these usually have limited applicability and are often based on accumulated experience with a special type of nonlinear system. When the controller design problem for nonlinear systems is approached in a more general fashion, even fewer results are available. These techniques are usually quite complicated (e.g.: Sommer, 1980; Watanabe and Himmelblau, 1982) and often require the use of sophisticated mathematical tools which most practitioners are not familiar with (cf. for example: Sommer, 1980; Marino, 1984; Hoo and Kantor, 1984). *Currently Summer Visiting Professor at the University of Wisconsin-Madison, Department of Chemical Engineering, Madison, Wisconsin 53706, until January 15, 1986. 0196-4305/86/1125-0241$01.50/0
It is obvious that because of the very nature of nonlinear systems, controller design from a general viewpoint will be inherently more difficult than for linear systems. Nevertheless, the objective of this paper is to present a technique which attempts to strike a balance between the generality of treatment on the one hand and the usual accompanying complexity of actual design and implementation on the other. The approach is based on the hypothesis that if the proper transformation can be found, a system which is nonlinear in its original variables can be made into a linear system in a new set of variables. Controller design for the transformed system may then be based on the powerful body of results available for linear systems. When the controller is implemented in the original system variables, it will of course be nonlinear. Although the principle behind this approach is not new (for example, this is a frequently used strategy for solving certain nonlinear differential equations), the usual obstacle consists in developing constructive methods for finding the appropriate transformations which are general enough without sacrificing simplicity and transparency. Such a method is developed in this paper. The nonlinear control problem is first treated in a general fashion, and the general principles are applied to the problem of level control in process flow systems with cylindrical, conical, and spherical configurations. A simulation example is then presented to illustrate the design procedure and the controller performance. 2. A Special Linear System Consider the system described by
dz -=a+bv dt
(1)
where z is the system state variable, u ( t ) is the system input 0 1985 American Chemical Society
