Multiobjective Optimization of Industrial FCC Units Using Elitist

The elitist nondominated sorting genetic algorithm (NSGA-II) is used to solve a three-objective function optimization problem. The objective functions...
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Ind. Eng. Chem. Res. 2002, 41, 4765-4776

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Multiobjective Optimization of Industrial FCC Units Using Elitist Nondominated Sorting Genetic Algorithm Rahul B. Kasat, D. Kunzru, D. N. Saraf, and Santosh K. Gupta* Department of Chemical Engineering, Indian Institute of Technology, Kanpur 208016, India

This study provides insights into the optimal operation of the fluidized-bed catalytic cracking unit (FCCU). A five-lump model is used to characterize the feed and the products. The model is tuned using industrial data. The elitist nondominated sorting genetic algorithm (NSGA-II) is used to solve a three-objective function optimization problem. The objective functions used are maximization of the gasoline yield, minimization of the air flow rate, and minimization of the percent CO in the flue gas using a fixed feed (gas oil) flow rate. The decision variables and several important state variables corresponding to the optimal conditions of operation are obtained. The optimal solutions correspond to the unstable, saddle-kind, middle steady states. The procedure used is quite general and can be applied to other industrial FCCUs. The optimal results obtained here provide physical insights that can help one in obtaining and interpreting such solutions. Introduction Increase in the demand for gasoline, LPG, and diesel over the last several decades have led to major improvements in refinery operations. The fluidized-bed catalytic cracking unit (FCCU) is an important conversion unit in most integrated refineries. Several studies have been reported in the open literature that deal with various aspects of FCCUs. These include their modeling, simulation, kinetics, multiplicity of steady states, chaotic behavior, on-line optimization, and control. Avidan and Shinnar1 reviewed the developments and commercialization of catalytic cracking in detail. Different workers have discussed the kinetics2-4 in the reactor and the regenerator and have modeled5,6 these units separately, whereas a few7-12 have developed an integrated model for the reactor-regenerator system. In the case of the reactor, various models are available that describe the feed and the products in terms of different kinetic “lumps”. These include describing the products in terms of two,13 three2, four,14 five,5,6 ten,3 or twelve15 lumps while considering the feed as a single lump (except the last two studies). The advantage of tenand twelve-lump kinetic schemes over the others is that the apparent rate constants are independent of the feed composition, but the main problem is that a relatively large number of kinetic parameters need to be evaluated using experimental or industrial data. These are not easily available. Moreover, the feed also needs a more detailed characterization. Several models5,6,16,17 are available in the literature for the regenerator. Several studies7-12,18,19 have been reported on the integrated models that include both the reactor and the regenerator. A detailed review on these several models of the FCCU is provided by Arbel et al.7 It is clear that a considerable amount of effort has been put into the modeling of this unit. Several approaches have been taken, extending from relatively empirical ones,6,20,21 where the hydrodynamics is not too detailed, to detailed models from the group of Elnashaie,10-12 who * To whom correspondence should be addressed. E-mail: [email protected]. Phone: 91 512 598722. Fax: 91-512590104.

focus on the complex, two-phase nature of the fluidized beds in both the reactor and regenerator. The strengths and shortcomings of these models are reviewed critically by Elnashaie and Elshishini.11 A review of the several models existing at present, suggests that one can possibly use any reasonable model, even if empirical, for simulation purposes as well as for optimization, provided we use experimental or (scarce) industrial data to tune the parameters usually associated with the models. The model discrepancies and approximations made (e.g., use of average values of the specific heats for the gas, etc.), are usually made up for by the tuning process, though it restricts the range of applicability of the model somewhat. In the past decade, several interesting studies10-12,20-25 have been reported dealing with the multiplicity of steady states in FCCUs. Indeed, the detailed model of Elnashaie et al.12,25 has been used to study bifurcation as well as the chaotic behavior24 of FCCUs under dynamic conditions. These studies use three kinetic lumps (Weekman et al.2) with the two-phase model of Kunii and Levenspiel17 or Davidson et al.26 Several studies have also been carried out on the optimization of FCCUs.27-34 Most of them use some type of a profit-function as the objective. Some of the commonly used decision variables in these studies are the regenerator temperature, reactor temperature, catalyst circulation rate and the air supply rate. These studies use a single objective function. In the past several years, more detailed optimization studies using multiple objective functions and constraints have been reported in the chemical engineering literature using a variety of mathematical algorithms. These are reviewed by Bhaskar et al.35 Besides being more realistic (several noncommensurate objective functions need not be clubbed together, by scalarization/parametrization, into a single profit-function), these studies have the added advantage that they do not miss out the optimal solutions when a duality gap is encountered due to the nonconvexity of the objective function.35,36 In such multiobjective studies, we often obtain a Pareto set of nondominating (equally good) solutions, and a decisionmaker needs to use his intuition or additional information to decide upon the

10.1021/ie020087s CCC: $22.00 © 2002 American Chemical Society Published on Web 08/24/2002

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Figure 1. Schematic diagram of the fluid catalytic cracking unit (FCCU).

Figure 2. Five-lump kinetic scheme19 used in this work.

preferred solution. We use a recent adaptation of genetic algorithm (referred to herein as NSGA-II), as developed by Deb and co-workers37 to optimize industrial FCCUs using more than one objective function. This is the first attempt in FCCU optimization where more than a single objective function has been used. This is also the first study in chemical engineering in which we are using NSGA-II.37,38 Model of the FCCU The FCCU consists of two major units, a reactor/riser and a regenerator, as shown in Figure 1. The description of the process is available in Avidan and Shinnar.1 In the present study, the five-lump kinetic scheme of Ancheyta et al.4 has been slightly modified and used with the empirical reactor models of Arbel et al.7 and Krishna and Parkin6 for the riser and the regenerator, respectively. This has been done because of the simplicity of these models. Indeed, these workers claim that their reactor models are being used in industry for simulation purposes. Dave19 modified the original scheme of Ancheyta et al.4 by assuming that gasoline and LPG also convert to coke. This modified kinetic scheme, used in the present study, is shown in Figure 2. Several of the rate constants and the heats of reaction have been compiled from earlier sources,7,39,40 and four rate constants have been obtained by Dave19 by tuning some industrial data available to us. The complete set of model equations as well as the values of the associated parameters are given in Appendix 1.

In the modeling study of Dave,19 the reactor/riser has been modeled as a steady-state plug flow reactor with the assumption that gas oil cracking follows secondorder kinetics, whereas gasoline and LPG cracking reactions follow first-order kinetics. The riser unit is assumed to be working under pseudo-steady-state conditions. This leads to a set of ordinary differential equations and is simple, as well as quite adequate, given the significant difference in the characteristic times of the riser and regenerator.9 We have used an empirical model primarily because of its simplicity, even though a more comprehensive, two-phase flow model having three kinetic lumps2 and incorporating the fact41 that the solid and gas flows are coupled through the chemisorption process, has been proposed by Elnashaie and co-workers10-12 and could have been used as well. In fact, Arandes et al.9 have also used an empirical model similar to ours very recently for modeling their industrial data. The catalytic deactivation function for the cracking of vacuum gas oil is that provided by Yingxun.42 Dave19 assumed the temperature drop around the stripper to be about 10 °C and adjusted the hydrogen-to-carbon ratio in the catalyst at the entry point of the regenerator, to match industrial data on the regenerator temperature. The regenerator is modeled using the threeregion model proposed by Krishna and Parkin6 with some modifications. The first region is the transfer line carrying spent catalyst to the regenerator. The second region is the dense bed in which the bulk of the combustion of coke takes place, and the third zone is the dilute phase above the dense bed. Simulation studies indicate that less than about 4% removal of carbon on the spent catalyst occurs in the transfer line, so modeling of the latter is not done. The gas is assumed to be in plug flow throughout the regenerator bed. The catalyst is assumed to be well mixed in the dense bed and in plug flow in the dilute phase. It is also assumed that coke combustion is kinetically controlled and the resistance to mass transfer from the gas to the catalyst phase is negligible. This model also hypothesizes a wellmixed emulsion phase. In contrast, Elnashaie and coworkers10-12 propose a two-phase model for the regenerator. However, Errazu et al.5 show that the two-phase model, with realistic parameters, gives results very similar to those of a stirred tank reactor. The model of Krishna and Parkin6 uses such empirical correlations and is used in the present study because of its simplicity, as well as because of its applicability to industrial units. The main reactions taking place in the regenerator are k10

C + 1/2O2 98 CO k11

C + O2 98 CO2

(1a) (1b)

k12c

CO + 1/2O2 98 CO2 (heterogeneous CO combustion) (1c) k12h

CO + 1/2O2 98 CO2 (homogeneous CO combustion) (1d) The balance equations for O2, CO, CO2, and for H2O (reaction not listed in eq 1) are given in Appendix 1, along with other necessary information. Table 1 also

Ind. Eng. Chem. Res., Vol. 41, No. 19, 2002 4767 Table 1. Design Data for the FCCU Used19 in This Study parameter

value

riser length (m) riser diameter (m) regenerator length (m) regenerator diameter (m) inventory of catalyst in regenerator (kg) feed rate (kg/s) riser pressure (kPa) regenerator pressure (kPa)

37.0 0.685 19.4 4.5 34 000 29.0 253.85 267.23

gives the details of the industrial FCCU used in this work. These are very slightly modified from those corresponding to an existing industrial unit,19 for proprietary reasons. The Runge-Kutta-Gill method is used to solve the several ordinary differential equations, because the equations are not too stiff.43 The equations for the riser and the regenerator are coupled because of the recycling of the catalyst, and we have a set of equations similar to two-point boundary value problems. The procedure of solving such equations is similar.43 Values of the coke on regenerated catalyst, Crgc, and the temperature, Trgn, of the regenerated catalyst, are assumed. The equations are solved to give new values (iterates) for these. Convergence is obtained using the Newton-Raphson method.43 Use of techniques more sophisticated than the Newton-Raphson method was not found necessary. A tolerance of 1.5 °C is used for the convergence of Trgn, whereas a tolerance of 1 × 10-4 kg of coke/kg of catalyst is used in the case of Crgc. For any prescribed set of operating variables (air flow rate, catalyst flow rate, air preheat temperature and feed preheat temperature), we solve the model equations as described and compute the output variables (yields of gasoline and other products, coke on regenerated catalyst, temperature of the regenerated catalyst, percentage of CO in the flue gas, etc.). These are either used as the objective functions themselves or may be used to evaluate them. The model equations form part of the optimization algorithm. NSGA-II uses random numbers to generate sets of operating conditions/decision variables in the form of coded binary chromosomes. The model equations are solved for each chromosome and the corresponding objective functions evaluated. The chromosomes are then changed using operations (crossover and mutation) that are similar to those in genetics to evolve toward optimal solutions. It was found that sometimes the model does not converge, particularly when the decision/ input variables generated by the optimization algorithm are practically infeasible. In these situations, new chromosomes replace the infeasible ones. It was observed that the computed temperature at the base of the riser was always higher than the industrial values. Similarly, the model-predicted values of the dense bed temperature, Trgn, differed slightly from industrial values. To minimize these two deviations, empirical heat loss terms were introduced for the riser base (dotted box shown in Figure 1) as well as for the dense bed of the regenerator. The appropriate equations are also included in Appendix 1. A similar loss term was incorporated by Elnashaie et al.11 and Han et al.8 The model is tuned using three sets of operating, steady-state data taken on an industrial FCCU (details of the unit are given in Table 1). This tuned model is then tested on three additional sets of industrial data. Good agreement is observed for all three cases. Table 2

Table 2. Comparison of Model Results with Industrial Data

feed flow ratea (kg/s) catalyst flow ratea (kg/s) feed preheat temperaturea (K) riser pressurea (atm) regenerator pressurea (atm) air flow ratea (kmol/s) air preheat temperaturea (K) riser top temperatureb (K) regenerator temperatureb (K) flue gas temperatureb (K) dry gas yieldb (%) LPG yieldb (%) gasoline yieldb (%) gas oil conversionb (%) coke yieldb (%) a

data from industry

results from the tuned model

29.11 208.33 625.1 2.546 2.68 0.5707 493.9 768.8 937.5 948.2 3.4 12.4 34.0 54.4 4.6

29.11 208.33 625.1 2.546 2.68 0.5707 493.9 770.14 936.88 948.4 3.69 13.05 36.13 57.4 4.52

Fixed variables. b Predicted variables.

shows the comparison for one set of data. This confirms the validity of our simple though empirical model. Optimization The elitist nondominated sorting genetic algorithm37,38 (referred to, here, as NSGA-II) is used in this study to obtain Pareto-optimal solutions. This is a robust algorithm and incorporates the concept of elitism to make it more powerful than the earlier algorithm, NSGA-I.38 Details of NSGA-II are provided in Appendix II (it is assumed that the reader is aware of the preliminary details35,38 on GA and NSGA-I). It is well established35 that GA and its adaptations (like NSGA) are very robust and give global optima, and several studies are available that demonstrate that real-life problems are better solved using GA than other techniques such as sequential quadratic programming (SQP). In fact, SQP has to be used with the -constraint method and applied repeatedly for different values of  to obtain the Pareto, whereas NSGA gives the Pareto in a single application of the algorithm. Formulation Several objective functions can be considered in any optimization study of FCCU to maximize its profitability and satisfy real-life constraints. In this unit, an increase in the yields of gasoline, LPG, or diesel always leads to an increase in the profit. So, it is clear that the maximization of these yields should be taken as the objective functions. Interestingly, an increase in the gasoline yield usually leads to an increase in the yield of LPG, and of dry gas, and so the use of gasoline yield as one of the objective functions is sufficient. The fivelumped model4,19 used here is not detailed enough to predict the yield of diesel, and so the latter is not included. In contrast, an increase in the yield of gasoline has an adverse effect on (increase in) the coke formation. This coke decreases the activity of the catalyst and needs to be burnt off in the regenerator, requiring higher amounts of feed air. Burning of the coke results in the formation of CO and CO2. The former needs to be converted to CO2 in the dilute phase before the gases enter the cyclones. This after-burning of CO can produce very high temperatures. There are usually two options available: One is to carry out full combustion in the regenerator, so that the CO emitted in the flue gas is very low. The other is to carry out only partial combus-

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tion in the regenerator and allow emission of higher amounts of CO from this unit. The flue gas is then sent to a CO reboiler where it is converted to CO2 before being emitted finally to the atmosphere. The full combustion mode of the FCCU requires large amounts of air supply. This automatically increases the operating costs. In addition, full combustion leads to much higher amounts of exothermic heat produced, which could create problems. There is also a need to burn off the maximum amount of the coke in the regenerator so as to keep the catalyst activity in the riser at high levels. On the basis of this discussion, we could select three objective functions for the study. The first is to maximize the gasoline yield (profitability), the second is to minimize the CO in the flue gas (pollution), and the third is to minimize the air flow rate (operating costs). If we were to study only a single objective function, e.g., the profit, one would have to assign appropriate weightage factors to several important objectives while incorporating them into a single economic objective function. This would be quite difficult to do if we wish to assign a cost to the emission of CO. Obviously, use of multiobjective functions, as in the present study, is a superior and more detailed option than just minimizing the cost or maximizing the profit. The algorithm provides a set of equally good solutions to a plant engineer from which he can select the preferred solution, using specified pollution norms or his intuition. Consideration of all three objective functions simultaneously at the start of any multi-objective study is difficult to attempt and understand, and several simpler, two-objective function problems are first studied to build insights. Only one such two-objective function problem is described here for the sake of brevity. This is given by

Problem No. 1: Max I1 (Tfeed, Tair, Fcat, Fair) ) gasoline yield

(2a)

Min I2 (Tfeed, Tair, Fcat, Fair) ) % CO in the flue gas (2b) 700 K e Trgne 950 K

(2c)

Crgc e 1%

(2d)

The following bounds are used on the four decision variables

575 e Tfeed e 670 K

(3a)

450 e Tair e 525 K

(3b)

115 e Fcat e 290 kg/s

(3c)

11 e Fair e 46 kg/s

(3d)

The feed preheat temperature, Tfeed, is selected as a decision variable because it plays a major role in controlling the heat balance in the FCCU. The lower limit of Tfeed is selected so as to supply sufficient heat required for the cracking and vaporization of the feed. Too low a value of Tfeed results in a decrease in the riser temperature, which, ultimately, decreases the yields of gasoline, LPG, etc. The upper limit7 of Tfeed is decided so as to prevent coking of the heating coils in the preheater. The riser top and the regenerator tempera-

tures are functions of the catalyst circulation rate, Fcat, and the air flow rate, Fair. The lower and upper bounds of the catalyst flow rate, Fcat, are taken such that the catalyst-to-feed flow ratio, Fcat/Ffeed (keeping Fair/Ffeed constant), lies between7 4 and 10, respectively. Similarly, the bounds on the air flow rate, Fair, are taken such that the air-to-feed flow ratio, Fair/Ffeed (keeping Fcat/Ffeed constant), lies between7 0.4 and 1.6. Below and above these ratios, trivial steady states7 exist that have no relevance in industrial operations. Many workers7,10-12,20-25 have studied the multiplicity of steady states of industrial FCCUs. It is well-known that commercial FCCUs operate at the unstable middle steady state at which point the maximum gasoline yield is obtained. It is also a well-known fact that because of the very high capacity of the system, the drift from the operating unstable middle steady state is relatively slow. Elnashaie and co-workers10-12,23-25 have studied this in great detail and have studied the effect of different parameters. The upper and lower bounds of Fair and Fcat are selected in this study so as to obtain optimal solutions in this range of unstable steady states. The lower and upper bounds of the air preheat temperature, Tair, are determined by the amount of heat that air can pick up in actual operations. Too low a value of Tair results in a cooling of the regenerator, and too high a value has the opposite effect. The first end-point constraint (eq 2c) on Trgn is incorporated so as to maintain reasonable temperatures in the dense bed. Too low a value of Trgn decreases the temperature of the regenerated catalyst, which causes a decrease in the riser temperature, which, in turn, leads to a decrease in the yields of gasoline, LPG, etc. Too high a value of Trgn reduces the catalyst activity rapidly and also causes damage to the catalyst in the regenerator. The upper and lower bounds on the Trgn indirectly forces the operation toward the middle steady state.7 The second end-point constraint (eq 2d) is on the coke on regenerated catalyst. Too high a coke content on the regenerated catalyst results in a decrease in the catalyst activity, which automatically decreases the yields of gasoline, LPG, etc. In the present study, the flow rate of the gas oil feed is kept constant at 29 kg/s. Generally, the feed rate is taken at a fixed value while industrial FCCUs are optimized. Two additional two-objective function optimization problems studied by us are

Problem Nos. 2 and 3: Max I1 (Tfeed, Tair, Fcat, Fair) ) gasoline yield Min I2 (Tfeed, Tair, Fcat, Fair) ) Fair

(4a) (4b)

subject to (s.t.): CO in flue gas e 8% (Problem 2) or

CO in flue gas e 1000 ppm or 0.1% (Problem 3) (4c) In addition, eqs 2c,d and 3a-d are incorporated. The above optimization problems are solved to obtain optimal solutions for the FCCU in the partial combustion mode (Problem 2) or the full combustion mode (Problem 3).

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Figure 3. Pareto set, decision variables and state variables for the optimal solutions for Problem 1.

To encompass Problems 1-3 into a single interesting multi-objective optimization problem, we solved the following three-objective function optimization problem:

Problem No. 4: Max I1 (Tfeed, Tair, Fcat, Fair) ) gasoline yield Min I2 (Tfeed, Tair, Fcat, Fair) ) Fair

(5a) (5b)

Min I3 (Tfeed, Tair, Fcat, Fair) ) % CO in the flue gas (5c) The bounds and constraints (eqs 2c,d, 3 a-d) are the same as in Problems 1-3. Most workers in the field have used the penaltyfunction approach while solving optimization problems with end-point constraints.35 This works when the process is not too complex. In the case of the FCCU, the model is quite complex and a different approach is

being used. Because GA generates solutions randomly, the moment a solution is generated, it is checked to see if it is able to satisfy all the end-point constraints. If it does not, we do not accept it as a member of the population and generate another chromosome till these conditions are satisfied. This enables all the members in a population to satisfy all the constraints. A similar procedure is followed when daughter chromosomes are obtained after crossover and mutation. This technique was found to be superior to the conventional penalty function approach35 (penalty proportional to the deviation) for the present problem. It enables the generation of wider variety (diversity) of solutions. However, it is a little slow in the beginning. Results and Discussion A computer code was written in MS Fortran Power Station to obtain the results of the multi-objective optimization problems. The code was tested (using a

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Figure 4. Surface plot for the three-objective function optimization problem (Problem 4).

single chromosome) for simulation of a few runs for which industrial data were available to us and compared with results obtained by Dave.19 The initial guesses of Trgn and Crgc were 900 K and 0.002 kg of coke/kg of catalyst, respectively. The computational time required for solving Problem 1 is 12 h on a Pentium 4 (1.7 GHz) machine. The Pareto optimal solutions and the associated variables for Problem 1 are shown in Figure 3. It can be observed from Figure 3b-e that the catalyst flow rate, the air flow rate, and the feed preheat temperature increase with the increase in the gasoline yield. An increase in the feed preheat temperature results in an expected increase in the temperature at the base of the riser (Figure 3f). Similarly, an increase in the catalyst flow rate results in an increase in the cracking reactions. This leads to an increase in the total conversion (Figure 3g) as well as in the individual yields of gasoline, dry gas, and LPG (Figure 3a,j,k). Simultaneously, this leads to a small increase in the amount of coke formed (Figure 3i) in the riser. This results in an increased requirement of air (Figure 3b) to satisfy the constraint on the coke on the regenerated catalyst (eq 2d). The air preheat temperature (Figure 3e) is almost at its lower bound. This is necessary to keep the temperature of the regenerated catalyst below 950 K (eq 2c). It is observed that the optimal solutions are associated with extremely low values of the CO concentration in the flue gas (almost at ppm levels). The optimal solutions in Figure 3 correspond to complete combustion of CO to CO2 in the dilute phase of the regenerator, leading, simultaneously, to a large difference in the temperatures of the dense and dilute phases (Figure 3h). It is worth mentioning that solutions with larger values of CO (of about 8%; partial combustion) are not picked up by the optimization algorithm in Problem 1 because they are inferior to those in the complete-combustion zone (because CO is minimized). One could have obtained solutions in the partialcombustion zone by reducing the upper bound on Fair. The results of Problems 2 and 3 are not reported here for the sake of brevity. It is obvious that lower air flow rates are required in Problem 2 (partial-combustion zone) compared to Problem 3 (complete-combustion zone).

Table 3. Details of Two Sets of Chromosomes in Problem 4 set 1 % CO in the flue gas gasoline yield (%) air flow rate (kg/s) air preheat temperature (K) feed preheat temperature (K) catalyst flow rate (kg/s)

0.896 42.62 20.92 456.45 666.75 285.04

0.897 35.62 19.42 471.99 596.73 202.93

set 2 4.268 41.49 18.56 453.01 665.45 261.77

5.212 32.31 17.33 458.72 584.84 192.83

With the physical insights now developed, we proceed to solve the more difficult Problem 4 (eq 5), involving three-objective functions. The population size was increased to 100 for this problem, and the number of generations was taken as 90. The computer time was about 48 h. Figure 4 shows the 3-dimensional plot characterizing the nondominating Pareto optimal points for Problem 4. This surface has been obtained by curvefitting the several Pareto optimal points actually obtained. Interestingly, it is found that both the completecombustion (low CO, high air flows) and the partialcombustion (high CO, low air flows) zones are obtained in the present problem, simultaneously. It can be observed from Figure 4 (and detailed tabulated information; not being provided here) that every point on the Pareto surface is equally good (nondominating) because any point may be worse than another in terms of one or two of the three objectives but is superior in terms of least one other objective function. If we look at the surface plot closely, we see several maxima and a few adjacent minima. Table 3 shows details of two sets of such maxima-minima combinations. Set I in this table shows that the air flow rate is lower (better) when the gasoline yield is lower (worse), for almost the same amount of CO in the flue gas; i.e., the peak is superior in terms of the gasoline yield, but the minima is better in terms of the other objective function (air flow rate). A small decrease in the air flow rate (from 20.92 to 19.42 kg/s) is associated with a sharp decrease in the catalyst flow rate (from 285.04 to 202.93 kg/s), which is responsible for the sharp decrease in the gasoline yield (from 42.62 to 35.62%). Set II in Table 3 shows similar results. This is because of the high sensitivity of the model to the catalyst flow rate in certain regions of the parameter

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Figure 5. Pareto set, decision variables and state variables for selected optimal solutions (peaks) of Problem 4.

space. Similar sensitivities were reported by Arbel et al.7 It is difficult to ascertain, at this stage, if this high sensitivity is real or just a weakness of the model being used here. Some regions of the ridge between nearby peaks are shown by rectangles in Figure 4. There are no optimal points in these regions, at least from among the 100 chromosomes obtained in this study. It must also be mentioned here that the computed Pareto solutions occur only in the inner zone in Figure 4 (where all the maxima and minima lie), and that no solutions exist in the adjacent flat surfaces in Figure 4 (these are artifacts generated by the graphical package). In other words, in Figure 4, the Pareto points essentially form a series of several hillocks of decreasing heights (with a few nearby minima), starting from the low CO-high air flow end (region A) and curving gently toward the high COlow air flow end (region B). If we assign some extra importance to the gasoline yield (a decisionmaker’s post-Pareto job) than used for

generating Figure 4, the several hills in this plot would become a set of several preferred solutions. The details corresponding to several of these selected peaks in Figure 4 (some indicated by +), have been plotted as simpler 2D plots in Figure 5. It is observed that for the optimal solutions shown in Figure 5a,b, the catalyst flow rate and the feed preheat temperature lie near their upper bounds. This is because high (∼40%) values of the gasoline yield require high values of both these variables. The constraint on Crgc is taken care of by the air flow rate. The constraint on Trgn can be satisfied either by adjusting the air flow rate or by adjusting the air preheat temperature. Minimization of the air flow rate leaves Tair as the only variable to satisfy the constraint on Trgn. As the percent CO in the flue gas decreases (moving from the partial-combustion zone to the complete-combustion zone), the air flow rate is found to increase. This leads to an increase in the oxygen available at the end of the dense bed and its subsequent use in the afterburning of CO. This simultaneously

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results in an increase in the temperature rise in the dilute phase of the regenerator (Figure 5i), as well as a decrease in the coke on the regenerated catalyst (Figure 5l). Figure 5 can be used to advantage by a decisionmaker to select an appropriate “preferred solution” (point of operation), based on his or her intuition, something that is often not quantifiable. It is well established20-23 that FCCUs have multiple steady states and that the maximum gasoline yield occurs at the unstable, saddle type, middle steady state. The optimal solutions obtained here are, indeed, these middle steady states. This has been ensured by using appropriate bounds on the decision variables as well as constraints on Trgn, as observed for similar situations by Arbel et al.7 We also found that varying the mutation and crossover probabilities, as well as the seed number, does not affect the 3-D Pareto set. Also, we did not study the affect of the feed composition on the Pareto set, because the rate constants depend on the composition for the five-lumped model assumed in this work, and there are insufficient industrial data available to us to tune these constants for other feeds. Elshishini and Elnashaie44 have, however, studied the effect of feed composition on the multiplicity of steady states, using their three-lump model and some data available to them. A rough idea of the effect of feed composition could possibly be had from their results on four charged stocks. Conclusions

Acknowledgment Partial financial support from the Department of Science and Technology, Government of India, New Delhi [through grant III-5(13)/2001-ET], is gratefully acknowledged. We thank all the reviewers for their several valuable suggestions. Appendix 1 Parameters are given in Tables A1 and A3. Model equations19 follow. Reactor.

dh

Ei Cφ i ) 5-7 RT 2 Ei Cφ i ) 8, 9 ri ) ko,i exp RT 3 Ffeed/Fv ) Ffeed/Fv + Frgc/Fc Fv )

9

Rijri ∑ i)1

) ArisHris(1 - )Fc

( )

Ei ri ) ko,i exp C 2φ RT 1

j ) 1, 2, ..., 5 i ) 1-4

(A1)

(A3) (A4) (A5)

PrisMWg RT

(A6)

5

MWg )

xjMWj ∑ j)1

(A7)

φ ) (1 + 51Cc)-2.78 dT

)

dh

ArisHrisFc(1 - )

(A8)

9

∑ri(-∆Hi)

(A9)

FrgcCpc + FfeedCpfvi)1

T(h)0) ) {[FrgcCpc(Trgn - 10.0) + FfeedCpflTfeed ∆HevpFfeed] - 0.019[FrgcCpc(Trgn - 10.0) + FfeedCpflTfeed - ∆HevpFfeed]}/(FrgcCpc + FfeedCpfv) (A10) Regenerator. Parameters are given in Tables A2 and A3. Dense Bed.

dfO2

An empirical model is tuned using some data on an industrial FCCU. The procedure is quite general and any other FCCU can be similarly modeled and tuned using associated industrial data. An adaptation of NSGA is used to obtain Pareto optimal solutions for this unit, using three objective functions with constraints. The optimal results incorporate the full-combustion (of CO) as well as the partial-combustion modes. The procedures developed are quite general and can be applied to other FCCUs. In fact, several other interesting multiobjective optimization problems can also be formulated and solved, e.g., maximizing the feed flow rate as well as the production of products, etc.

dFj

( ) ( )

ri ) ko,i exp -

dz

) -Argn

(

)

r12 r10 + r11 + 2 2

1 fO2(0) ) 0.21Fair - fH2O (A11) 2 dfCO ) -Argn(r12 - r10) dz dfCO2 ) Argn(r11 + r12) dz

fCO(0) ) 0

(A12)

fCO2(0) ) 0

(A13)

Crgc fO2 r10 ) (1 - )Fck10 P MWc ftot rgn

(A14)

Crgc fO2 P r11 ) (1 - )Fck11 MWc ftot rgn

(A15)

fO2fCO Prgn2 (A16) ftot2

r12 ) (xpt(1 - )Fck12c + k12h)

CH fH2O ) Frgc(Csc - Crgc) MWH2

(A17)

fN2 ) 0.79Fair

(A18)

dTrgn )0 (A19) dz Trgn(z)0) ) Tbase + {[fCO(Zbed)HCO + fCO2(Zbed)HCO2 + fH2OHH2O + FairCpair(Tair - Tbase) + FscCpc(Tsc Tbase)] - 0.11[fCO(Zbed)HCO + fCO2(Zbed)HCO2 + fH2OHH2O + FairCpair(Tair - Tbase) + FscCpc(Tsc Tbase)]}/{[FrgcCpc + fCO2(Zbed)CpCO2 + fCO(Zbed)CpCO +

(A2)

fO2(Zbed)CpO2 + fH2OCpH2O + fN2CpN2]} (A20)

Ind. Eng. Chem. Res., Vol. 41, No. 19, 2002 4773 Table A1. Kinetic and Thermodynamic Parameters19 Used for Reactor Modeling rate constants

frequency factora,19,39

activation energy38 (kJ/kmol)

heat of reaction7 (kJ/kmol)

k1 k2 k3 k4 k5 k6 k7 k8 k9

14054.59b 2293.00b 390.95b 29.90b 65.40c 0.00c 0.00c 0.32c 0.19c

57540 52500 49560 31920 73500 45360 66780 39900 31500b

45000 159315 159315 159315 42420 42420 42420 2100b 2100b

Table A3. Thermodynamic and Other Parameters19 Used for Simulation

a m6 (kg catalyst)-1 (kmol gas oil)-1 s-1 for k to k and m3 (kg 1 4 catalyst)-1 s-1 for k5 to k9. b From ref 19. c From refs 19 and 39.

Table A2. Kinetic Parameters7 Used for Regenerator Modeling parameter

frequency factor

activation energy (E/R, K)

βca kc (atm-1 s-1) k12c (kmol of CO) (kg cat)-1 s-1 m-3 k12h (kmol of CO) m-3 atm-2 s-1

2512 1.0694 × 108 117 5.07 × 1014

6795 18 890 13 890 35 555

a β ≡ (k /k ) ) β exp[(-E /RT)]. k ≡ k c 10 11 c0 β c 10 + k11 ) kc0 exp[(Ec/RT)].

{with fi(Zbed) being the value of fi at z ) Zbed}

0.305u1 + 1 0.305u1 + 2

(A21)

u1 )

Fair 0.3048FgArgn

(A22)

(ri as in eqs A14-A16, but with appropriate parameters/ variables)

Prgn RTrgn

(A23)

Argn dTdil ) [H (r - r12) + HCO2(r11 + r12)] dz Cptotftot CO 10 (A32)

Zbed ) 0.3048(TDH)

(A24)

TDH ) TDH20 + 0.1(D - 20)

(A25)

log10 (TDH20) ) log10 (20.5) + 0.07(u1 - 3)

{with fi(Zbed) being the value of fi at z ) Zbed}. Dilute Phase.

(

)

r12 r10 + r11 + 2 2 fO2(0), same as at the end of dense bed (A28)

) -Argn

dfCO ) -Argn(r12 - r10) dz fCO(0), same as at the end of dense bed (A29) dfCO2 dz

a R ) for the jth lump corresponding to the reaction from lump ij i to lump j (in mass terms).

Cptot ) CpN2fN2 + CpO2fO2 + CpCOfCO + CpCO2fCO2 + CpH2OfH2O + CpCFent

(

) Argn(r11 + r12) fCO2(0), same as at the end of dense bed (A30)

dfC ) -Argn(r10 + r11) dz initial condition from eq A40 (A31)

)

ftot

(A26)

Crgc ) {[FscCsc(1 - CH) - {fCO(Zbed) + fCO2(Zbed})MWc]}/{[Frgc(1 - CH)]} (A27)

dz

value 1.003 3.430 3.390 30.530 32.280 36.932 30.850 47.400 350.0 1.078 × 105 3.933 × 105 2.42 × 105 1089.0 0.15 350 170 65 30 12 2.06 5.39 11.67 29.17 2.62 5.67 14.17 2.17 5.42 2.0 × 10-4 0.10

)

Fg )

dfO2

parameter Cpc, kJ kg-1 K-1 Cpfl, kJ kg-1 K-1 Cpfv, kJ kg-1 K-1 CpN2, kJ kg-1 K-1 CpO2, kJ kg-1 K-1 CpH2O, kJ kg-1 K-1 CpCO, kJ kg-1 K-1 CpCO2, kJ kg-1 K-1 ∆Hevp, kJ kg-1 HCO, kJ kmol-1 HCO2, kJ kmol-1 HH2O, kJ kmol-1 Fc, kg m-3 CH, (kg of H2) (kg of coke)-1 MWgas oil MWgasoline MWLPG MWdry gas MWcoke Rgas oil,gasolinea Rgas oil,LPG Rgas oil,dry gas Rgas oil,coke Rgasoline,LPG Rgasoline,dry gas Rgasoline,coke RLPG,dry gas RLPG,coke Dp, ft xpt

Fent ) 0.4535WA

(A33) (A34)

W ) FfYu1

(A35)

log10 Y ) log10 60 + 0.69 log10 X - 0.445(log10 X)2 (A36) X)

u12 gDpFp2

(see eq A22)

dil ) 1 Fdil )

Fdil Fc

Fent 0.3048Argnu1

(1 - CH) 12 Zdil ) Zrgn - Zbed

fC(0) ) FentCrgc

gasoline yield (%) ≡

F1(MW1) 5

Fi(MWi) ∑ i)1

(A37) (A38) (A39) (A40) (A41)

× 100 (A42)

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Ind. Eng. Chem. Res., Vol. 41, No. 19, 2002

(g) Repeat for all chromosomes in P. P′ constitutes the first front or sub-box (of size eNp) of nondominated chromosomes. Assign it Rank ) 1. (h) Create subsequent fronts in (lower) sub-boxes of P′ using the chromosomes remaining in P. Compare these members only with members present in the current sub-box. Assign these Ranks ) 2, 3, .... Finally, we have all Np chromosomes in P′, boxed into one or more fronts. This sequential procedure is superior to that used in NSGA-I38 where any chromosome is compared to all other Np - 1 chromosomes. 3. Evaluate the crowding distance, Ii,dist, for the ith chromosome in any front: (a) Rearrange all chromosomes in front j in ascending order of the values of any one of their fitness functions,37,38 Fi. (b) Find the largest cuboid (rectangle for 2 fitness functions) enclosing i that just touches its nearest neighbors in the F-space. (c) Ii,dist ≡ 1/2 (sum of all sides of this cuboid). (d) Assign large values of Ii,dist to solutions at the boundaries. This helps maintain diversity in the Pareto set. This procedure is superior to the sharing operation of NSGAI.38 4. Copy the best of the Np chromosomes of P′ in a new box, P′′ (Best Parents): (a) Select any pair, i and j, from P′ (randomly, irrespective of fronts). (b) Identify the better of these two chromosomes. i is better than j if

Ii,rank * Ij,rank: Ii,rank < Ij,rank Ii,rank ) Ij,rank: Ii,dist > Ij,dist

Figure A1. Flowchart for NSGA-II.

Appendix II Elitist Nondominated Sorting Genetic Algorithm, NSGA-II37,38 1. Generate box, P, of Np parent chromosomes38 (see flowchart in Figure A1). 2. Classify these chromosomes into fronts based on nondomination:38 (a) Create new (empty) box, P′, of size, Np. (b) Transfer ith chromosome from P to P′, starting with the first. (c) Compare chromosome i with each member, e.g., j, in P′, one at a time. (d) If i dominates over j, remove j from P′ and put back in P. (e) If i is dominated by j, remove i from P′ and put back in P. (f) If i and j are nondominating, keep both i and j in P′. Continue for all j.

(c) Copy (without removing from P′) the better chromosome in a new box, P′′. (d) Repeat till P′′ has Np members. (e) Copy all of P′′ in a new box, D, of size Np. Not all of P′ need be in P′′ or D. 5. Carry out crossover and mutation38 of chromosomes in D. This gives a box of Np daughter chromosomes. 6. Copy all the Np best parents (P′′) and all the Np daughters (D) in box PD (elitism). Box PD has 2Np chromosomes. 7. Reclassify these 2Np chromosomes into fronts (box PD′) using only nondomination. 8. Take the best Np from box PD′ and put into box P′′′. 9. This completes one generation. Stop if criteria are met. 10. Copy P′′′ into starting box, P. Go to Step 2 above. Nomenclature A ) cross-sectional area of regenerator, ft2 Argn ) cross-sectional area of regenerator, m2 Aris ) cross-sectional area of riser, m2 Cc ) coke on catalyst at any location, (kg of coke)(kg of catalyst)-1 Ci ) concentration of ith lump, kmol m-3 CH ) weight fraction of H2 in coke, (kg of H2)(kg of coke)-1 (Appendix 1) Crgc ) coke on regenerated catalyst, (kg of coke)(kg of cat)-1 Csc ) coke on spent catalyst, (kg of coke)(kg of cat)-1

Ind. Eng. Chem. Res., Vol. 41, No. 19, 2002 4775 Cpc ) heat capacity of catalyst, kJ kg-1 K-1 Cpfl ) heat capacity of liquid feed, kJ kg-1 K-1 Cpfv ) heat capacity of vapor feed, kJ kg-1 K-1 Cpi ) mean heat capacity of i [H2O, N2, O2], kJ kg-1 K-1 Cptot ) heat capacity of (total) mixture, kJ kg-1 K-1 D ) diameter of regenerator, ft Dp ) average diameter of catalyst particle, ft Ec, Eβ ) activation energies, kJ kmol-1 Ei ) activation energy of ith reaction, kJ kmol-1 fi ) molar flow rate of i [CO, CO2, H2O, N2, O2, carbon] in the regenerator, kmol s-1 ftot ) total gas flow rate at any location in the regenerator, kmol s-1 Fair ) flow rate of air feed to the regenerator, kmol s-1 Fent ) entrained catalyst flow rate, kg s-1 Ffeed ) feed flow rate of oil, kg s-1 Fj ) molar flow rate of jth lump, kmol s-1 Frgc ) flow rate of regenerated catalyst, kg s-1 Fsc ) flow rate of spent catalyst, kg s-1 g ) gravitational acceleration, 32.2 ft s-2 h ) dimensionless height of riser (≡z/Hris) ∆Hevp ) heat of vaporization of gas oil feed, kJ kg-1 Hi ) heat of formation of i, kJ kmol-1 ∆Hi ) heat of ith reaction, kJ kmol-1 Hris ) height of riser, m k0,i ) frequency factor for ith reaction (Appendix 1) kc ) overall rate of combustion of coke ki ) reaction rate constant for ith reaction (Appendix 1) MWc ) molecular weight of coke, kg kmol-1 MWg ) average molecular weight of gas phase, kg kmol-1 MWj ) molecular weight of jth lump, j ) 1, 2, ..., 5, kg kmol-1 MWH ) molecular weight of H2, kg kmol-1 Nchr ) total chromosome length (40) Nga ) number of generations (50 for Problems 1-3; 90 for Problem 4) Np ) population size (50 for Problems 1-3; 100 for Problem 4) pc ) crossover probability (0.95) pm ) mutation probability (0.05) Prgn ) pressure in regenerator, atm Pris ) pressure in riser, atm ri ) rate of the ith reaction, i ) 1-9 (riser); i ) 10-12 (regenerator), kmol (kg catalyst)-1 s-1 or kmol m-3s-1 R ) universal gas constant, J K-1 kmol-1 Tair ) temperature of air fed to the regenerator, K Tbase ) base temperature for heat balance calculations, K (assumed, 866.6 K) Tdil ) temperature of dilute phase at any location, K Tfeed ) temperature of gas oil feed, K Trgn ) temperature (uniform) of dense bed, K Tris ) temperature of riser at any location, K Tris,top ) temperature at top of riser, K Tsc ) temperature of spent catalyst [)Tris,top - ∆Tst], K ∆Tst ) temperature drop in stripper (assumed 10 K) u ) velocity of gas in the riser or the regenerator, m s-1 u1 ) superficial linear velocity, ft s-1 W ) catalyst entrainment flux, lb (ft2 regenerator area)-1 s-1 xj ) mole fraction of jth lump, j ) 1, 2, ..., 5 xpt ) relative (catalytic) CO combustion rate (eq A16) Y ) (kg catalyst entrained in dilute phase) (kg fluidizing vapor)-1 z ) height from the entrance of the regenerator, m Zbed ) height of the dense bed, m Zdil ) height of the dilute phase, m Zrgn ) total height of the regenerator, m Greek Letters Rij ) stoichiometric coefficient of jth species in ith reaction, based on mass

βc ) CO/CO2 ratio at catalyst surface in regenerator (Appendix 1)  ) void fraction in riser or regenerator at any location dil ) void fraction in the dilute phase at any location Fc ) density of solid catalyst (not including void fraction), kg m-3 Fden ) density of catalyst in the dense bed, kg m-3 Fdil ) density of catalyst in the dilute phase, kg m-3 Ff ) density of fluidization vapor, lb ft-3 Fg ) density of gas phase in the regenerator, kmol m-3 Fp ) density of catalyst particle (solid), lb ft-3 Fv ) density of vapor at any location, kg m-3 φ ) activity of the catalyst Subscripts i, j ) ith or jth lump (1, gas oil; 2, gasoline; 3, LPG; 4, dry gas; 5, coke)

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Received for review January 29, 2002 Revised manuscript received May 17, 2002 Accepted July 23, 2002 IE020087S