Computers and Chemical Engineering Vol. 7, No. 5, pp. 645-661, 1983 0098 1354/83 $3.00 + .00 Printed in Great Britain. © 1983 Pergamon Press Ltd.

PROCESS OPTIMIZATION: A COMPARATIVE CASE STUDY

LORENZ T. BIEGLER t and RICHARD R. HUGHES* Chemical Engineering Department and Engineering Experiment Station, University of Wisconsin-

Madison, Madison, Wl 53706, U.S.A.

(Received 1 February 1982)

Abstract--Four recently developed algorithms were tested on a realistic propylene chlo- rination process simulation. All four are based on successive quadratic programming and interface easily with most sequential modular simulation packages. Using SPAD for simu- lation, optimal cases were obtained in as few as 29 simulation-time equivalents. The paper includes model details, reactor kinetics, and algorithm performance.

Scope Our previous papers [1-3] describe four algorithms developed for efficient optimization of sequential-modular simulation models. Each algorithm was tested extensively on a simple flash process to determine optimal tuning parameters and to study algorithmic performance.

All four algorithms are" based on the Successive Quadratic Programming (SQP) algorithm of Powell[4] and use a BFGS update to develop the quadratic matrix. The methods differ in the strategies used for the three basic steps: function evaluation, gradient evaluation, and formulation of the quadratic program.

The first algorithm, Quadratic/Linear Approximation Programming (Q/LAP)[1] requires a converged process flowsheet for each function evaluation. Gradients are calculated by construction of linear models for each flowsheet module. These are then assembled into a large sparse linear system which is perturbed in lieu of the actual process model, to obtain the gradient in decision-variable space. This gradient is the basis for the next quadratic program.

Infeasible Path Optimization of Sequential Modular Simulations (IPOSEQ)[2] converges and optimizes simultaneously. Function evaluations and gradient evaluations are performed by simple (non-iterative) passes through the calculation sequence. Tear equations (the difference between guessed and calculated stream elements) are linearized and included as equality constraints in the quadratic program. The dimensionality of the gradient and the quadratic program is increased to include the decision variables and the tear (guessed recycle stream) variables. Implementation is very simple; for most modular simulators, IPOSEQ merely requires the substitution of an optimization "block" for the recycle convergence algorithm.

The last two algorithms[3] are feasible variants of IPOSEQ. The Complete Feasible Variant (CFV) and Reduced Feasible Variant (RFV) algorithms use the same gradient calculation strategies as IPOSEQ, but require converged flowsheets for each function evaluation. CFV, like IPOSEQ, includes tear equations in the quadratic program while RFV calculates reduced gradients and solves the smaller quadratic program that Q/LAP uses. Convergence is accelerated because the quadratic programming solution yields good starting values for the tear variables, either directly (in CFV) or through the reduced gradients (in RFV).

Here, we present a comparative study of these four algorithms on a realistic problem--the design of a process for direct chlorination of propylene. The simulator used was again SPAD [5], as installed on the Univac 1100/82 at the University of Wisconsin. Special modules were added to handle the reaction kinetics and the HCI scrubber.

Conclusions and significance--The nine-variable model was successfully optimized by each of the four algorithms. From a starting point design which showed a net value added of 1010.13 $/hr, the following results were obtained:

Algorithm STE Optimum ($/hr)

RFV 28.83 1609.97 CFV 28.74 1609.92

IPOSEQ 45.29 1609.84 Q/LAP 61.18 1600.55

*Author to whom correspondence should be addressed. fPresent address: Carnegie-Mellon University, Pittsburgh, PA 15213, U.S.A.

645

646 LORENZ T. B1EGLER and RICHARD R. HUGHES

(STE or Simulation Time Equivalents is the ratio of the CPU time for optimization to that for simulation.)

Comparison of the optimal points shows that there is an optimal "ridge" along the lower reaction temperature bound, in the space of the other three variables which influence the reactor kinetics.

The results suggest that, if numerical perturbation must be used for gradient calculation, then the feasible variant algorithms, CFV and RFV, are the best SQP methods for optimization of sequential-modular models.

PROCESS EXAMPLE The process chosen for this comparative study is the synthesis of allyl chloride, a valuable intermediate in the manufacture of epoxy resins and glycerine[6]. The flowsheet adopted (Fig. 1) is an elaboration of the example described by Hughes[7]. It uses the reaction scheme and separation sequence from pub- lished process descriptions[6,8], with an added quench loop to reduce carbon deposition in the reactor effluent transfer line and cooler. The plant scale is set by using a fresh propylene feed rate of 100 lb-mole/hr. This results in an allyl chloride prod- uction of about 34 million lbs/yr at a stream factor of 95%.

This process, modeled on a sequential modular simulator, requires the ordering and solution of thousands of unit operations, physical property, and mass and energy balance relationships. The opti- mization of such a system can clearly be termed a large-scale optimization problem.

A block diagram of the simulation of this process on SPAD [5] is given in Fig. 2. For the most part, this uses standard SPAD unit modules, as follows:

CMP--Single-stage polytropic compressor DIS---Shortcut distillation (Fenske-Underwood-

Gilliland) HCB--Process cooler HHB--Process heater MIX--Adiabatic mixer PMP--Process pump SPL--Stream splitter (mechanical) ZWB--Iterative convergence via Wegstein pro-

cedure.

Two special modules were developed: UU1 for the reactor; and UU2 for the scrubber-drier. Details of these models are given in the following process description.

In the example process, propylene feed at 60°F and

O-[] Propylene 171o Feed

60 °

Chlorine Feed

80 °

PREHEATER

674 o ®

Direct Chlorination of Propylene

- Decision Variables ~ ' - ~ - Constraints

[~o - psia'~ Conditions °F J at RFV

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l = d ~ -~ Ally, Chloride 20 ~ N° / ~ 116

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Fig. 1. Process flowsheet.

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Process optimization: a comparative case study

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200 psia is mixed with a recycle stream which is mostly unconverted propylene. The combined gas- eous feed is then preheated and fed directly into the reactor with make-up chlorine at 80°F. The principal (and desired) reaction is substitution by chlorine to produce allyl chloride:

C12 + CH2~-~CH-CH3 -~ CH2=CHCH2CI + HCI.(1)

Other restrictions that yield less desirable by- products are the addition of chlorine, to form 1,2-dichloropropane:

C12 -F- CH2--CHCH3 -~ CH2CHC1CH2CI (2)

and further chlorine substitution in allyl chloride, to yield 1,3-dichloropropene:

C12 + CH2--CHCH2C1 ~ CHC1--CHCH2CI + HC1. (3)

Rate expressions for reactions (1) and (2) are given by Smith[9]; these were derived from the data of Groll & Hearne[10]. Further analysis of these data led to a rate expression for reaction (3). All three reactions are exothermic and have overall second-order, Arrhenius-type kinetics, of the form ( for j = 1, 2, 3):

rj = kjp?Pa2 (4)

where

kj = Aj exp[ - Bj/(Tx + 459.69)]. (5)

Here, ,p? is the partial pressure of propylene for reactions (1) and (2) and of allyl chloride for reaction

(3). Values for the heats of reaction and the constants Aj, Bj are given in Table 1. The table also lists values of the kj's at several reaction temperatures.

The kinetics of this reaction system are series- parallel:

+ CI2

CH2--CHCH 3 ~ CH2=CHCHI + HCI kl

k 2 ~ Cl2 ka~X~+ C12

CH2CHCICH2CI CHCI--CHCH2CI + HCI.

The desired product, allyl chloride, is an intermediate in the series reaction. This would normally call for a plug-flow reactor, to keep the allyl chloride away from the inlet with its high chlorine concentration. However, the system is highly exothermic and the rate constants in Table 1 suggest that high tem- peratures favor reaction (1) vs reaction (2). This is confirmed by Smith's analysis[9], based on reactions (1) and (2) alone; his Figs. 5-8 indicates a much better product for stirred-tank reactors. Thus, the reactors actually used promote back-mixing; they are really flame reactors[11], although the residence times are longer--about one-tenth of a second. To prevent excessive loss of allyl chloride via reaction (3), the process uses a high propylene/chlorine ratio, with recycle of unconverted gases. A high ratio also re- duces the rate of co*ke build-up on the reactor inter- nals.

Based on these considerations, we model the reac- tor (in special subroutine UUI), as a gas-phase continuous stirred tank reactor (CSTR). The design

648 LORENZ T. BIEGLER and RICHARD R. HUGHES

Table 1. Reaction constants for propylene chlorination

Reaction

Stoiehiometric Coefficient% vij

Substitution Addition Sec'y. Subst. (j=l) (j=2) (if3)

1. HC~ +1 0 ÷1 2. C¢ 2 - 1 - 1 - 1 3. C3H 6 -1 - 1 0 4. C385C~ +1 0 -1 5. C3H6C~ 2 0 +1 0 6. C3H4C£ 2 0 0 +1

0 -I 0

Heats of Reaction

-AHj (BTU/Ib-mol)

Kinetic Coefficients

4,800 79,200 91,800

f 1b-tool" ] AJlkr 3_a-Sj j 206,000 . . 7

Bj(*R) 13,600 3,430

Rate Constants, ~(Ib-mol/~r- ft3-atm 2)

at 600*F 0.55 0.46 800*F 4.23 0.77

IO00*F 18.55 1.12

Conversion to SI Units:

K = ( * R ) / 1 . 8

J /mo le = 2 .326 (BTU/ lb -mo l )

4 .6 x 108

21,300

0.86 20.95

212.24

variables are the propylene/chlorine ratio in the feed (q), the reactor temperature (Tx), and the reactor volume (Vx). (Another possible variable is the reactor pressure (Px), but, in the present model, this is set arbitrarily to make the pressure-molal flow product of the effluent equal that of the combined feed.)

The reactor calculation begins by setting the combined-feed component flows, F~,--adjusting the make-up chlorine to match the specified propylene/ chlorine ratio, q = F i / F 3. The effluent flows, Ei, can then be defined from the stoichiometric coefficients (v~) (Table 1) and the reaction extents (Xj):

E i = F i + ~ v ~ ; i=1 ,6 . (6) J

The reaction extents are obtained from the kinetic Eqs. (4) and (5) with usual CSTR and ideal-gas partial pressure assumptions:

Xj = Vxrj = V,,kj(Px/14.7)2E~,EJE2r; j = 1,3 (7)

where

ET = ~ ei = FT-- x~ (8)

and (for the present results)

Px = PFFr /ET • (9)

In the reactor model, the simultaneous solution of Eqs. (6)-(9) is done by iteration. For this calculation, Eq. (7) are transformed, with the use of (6), to:

6=_x, - v ) , P ~ ( e , - x , - X g ( F : - X , - X~ - X ~ ) / E ~

(10)

X 2 = ( k J k O X ~ (11)

k3X,(e, + X,) )(3 = k3Xl + k l ( F 3 - X~ - - )(2)" (12)

Now X~ is adjusted iteratively by the secant method until 6 < e( ~ 10-4). For each assumed X~, the equa- tion order is (11), (12), (8), (9), (10). After closure, Eqs. (6) are used to fix the Ei values. Finally, by heat balance, the necessary feed preheat temperature is calculated.

The reactor effluent is mixed with a quench stream and the combined stream further cooled to 50°F. The stream is then fractionated in a 14-plate distillation

Process optimization:

column to separate synthesis gases from the heavier chlorinated hydrocarbons. The vapor overhead is scrubbed and dried to remove the HCI, and a fraction vented to prevent build-up of impurities. The remain- der is then recompressed before mixing with the feeds. The bottoms from the column is split to form the quench and product streams. The quench is pumped and cooled to 20°F before mixing with the reactor effluent, and the product is sent to a 14-plate finishing column to separate allyl chloride from the heavier dichlorides.

The HCI scrubber is a gas-liquid countercurrent absorption tower. Water enters the top at 80°F and leaves as boiling 32wt% hydrochloric acid[12]. In addition to absorbing 99.9% of the HCI, the scrubber also absorbs the following compounds (if present):

Max % wt, dry basis in aqueous 32% HCI

Chlorine 0.1 Propylene 0.1 Allyl chloride 0.3 1,2-dichloropropane 0.7

These values were estimated by Hughes[7] from handbook solubility data. The scrubbed gas is then dried with caustic or sulfuric acid to remove any water vaporized in the scrubber.

The special scrubber-drier unit subroutine UU2 calculates a scrubber mass balance based on the above solubilities. This yields the component flows for the scrubber outlet gas and the hydrochloric acid, on a dry basis. Then an energy balance assuming effluent acid at its boiling point yields an outlet temperature for the gas. No calculations are made for the drier; at the low gas temperatures expected, water vapor carryover is too small for any significant heat effect.

In this preliminary optimization study, it was desirable to keep the computation time for simulation as short as possible. Accordingly, simplified physical property correlations were used, with vapor-liquid equilibrium data fitted to Antoine equations. Vapor pressure data for 1,3-dichloropropene were estimated by comparing boiling points and vapor pressures of chlorinated hydrocarbon hom*ologues. Although these physical property correlations may yield inaccu- rate flash and enthalpy calculations, the effect of error is minimized by the choice of design variables-- temperatures, pressures and split fractions--and by the use of an objective function which is affected mainly by reactor kinetics.

OPTIMIZATION PROBLEM AND PROCESS DESIGN RESULTS

To optimize a process design, the preferred objec- tive is usually the venture profit or venture worth. However, for this process, the product and feed values are high and the plant is fairly simple and relatively inexpensive. Thus the objective function can be approximated as the net value added, i.e. the sales return less the feed costs. This is fortunate, since SPAD has only limited capabilities for estimating capital and utility costs.

a comparative case study 649

The prices used for raw materials and products are given in Table 2. The last column (net value added) is calculated by adjusting for stoichiometric feed requirements and by-product HC1. With these values the following objective function was established, and used in all the calculations.

Max ff ---22.17(AC) + 12.48(DCP) + 10.06(DCP-) (13)

If the product rates are in lb moles/hr, ~b has units of S/hr.

(Note that the coefficient of DCP= does not agree with Table 2. This is a numerical error which was found after the optimization calculations were com- pleted. It does not really invalidate the comparative study, so there has been no attempt to repeat the calculation).

The nine decision variables and three constraints for the optimization are listed in Table 3. This table also summarizes the optimization results. Tables 4 and 5 provide additional details for the best design found--the RFV optimal point. The temperatures and pressures for this design are also shown in Fig. 1.

In Table 3, the Q/LAP results are somewhat disap- pointing, although the best point obtained is a marked improvement over the starting point. The best points for the other three methods--IPOSEQ, CFV, and RFV--are within a range of 13 ¢/hr of each other. However, the design variable values differ considerably, especially those for the propylene/ chlorine ratio(r/), the reactor volume (Vx) and the compressor outlet pressure (Pc) (which directly con- trols the reactor pressure). The optimum apparently lies on a "ridge" in variable space; along this ridge, different values for these three variables result in the same product yields, and, thus, the same objective value. This can also be seen by straightforward in- spection of the kinetic expessions. The other reactor variable, the temperature (Tx), seeks its lower bound of 800°F. The present CSTR model finds a better yield structure at low temperatures, and the cost of the high recycle required is not recognized in the objective functions.

This neglect of costs also permits the choice, for the separation system, of a low column pressure (Ps), high recovery factors (fLx, fHx) and high recycle quench (fe)" These all result in slight improvements in product recovery, even though they mean a relatively-large, low-temperature, high-reflux col- umn. The allyl chloride content of the recycle, small as it is, also drives the vent fraction (f~) to its lower bound. An implied assumption of the objective, Eq. (13), is that the remaining components in the vent have a net added value of zero, i.e. are priced the same as the feed, so the allyl chloride content alone provides the incentive to reduce the vent. This is not really an adequate decision criterion for the vent, which really exists to prevent build-up of undesirable trace impurities. To handle this properly, the im- purities should be included in the feeds, and the reactor and separation units should allow for this presence. However, without such improvements, the only solution is to fix a .reasonable lower bound.

Table 3 lists three problem constraints. The first tests the need for a recycle compressor. If this con-

650 LORENZ T. BIEGLER and RICHARD R. HUGHES

Table 2. Raw material and product prices

Price at Plant Limit* Net Value Added

Compound As Quoted $/ib-mol $/ib-mo]

HC£ $35/ ton (36 ° B~ ) 0.96 Note**

C£ 2 $145/ ton 5.15

C3H 6 21 ¢ / i b 8.84

A l l y l c h l o r i d e (AC) 46 e / l b 35.20 22.17

Dlchloropropane (DCP) I 1.35/gal if 126.47 12.48

Dichloropropene (DCPffi)~ ~ 50% unsaturated {22.13 4.91

*For C£ 2 and C.H. price is cost delivered to plant. For products price J o' p is net to plant, excluding marketing costs. For dichlorides, marketing costs have been taken as 50% of quoted market price of $2.70/gai.

**Stolchiometrlc by-product credits added to C 3 chloride values.

straint were active, and remained active when the test value (now 10psi) was lowered to zero, the com- pressor could be eliminated. However, this is phys- ically unlikely and would probably mean there is an error elsewhere in the simulation. The second con- straint enforces high purity in the chlorinated hydro- carbon product stream. A third constraint is needed to guard against a simulation problem in the reactor block. Here we force the reactor inlet temperature to be no less than 90°F; certain combinations of the reactor design variables may calculate a temperature that is unrealistically low due to the high heats of reaction. Constraints (1) and (3) remained inactive throughout all of the optimization calculations. The second constraint was active at least sometime during the optimization on Q/LAP, IPOSEQ and CFV. A fourth constraint that could have been added would have assured that the production of dichloropropene was greater than that of dichloropropane as required by the product price in Table 2. This was not in the original model forumlation and was not added be- cause the requirement was satisfied in all of the optimization calculations.

In retrospect, constraints (1) and (3) were really unnecessary. More significantly, some of the vari- ables should not have been included in the opti- mization. The simple objective function of Eq. (13) does not properly evaluate the effects of the column variables and the split functions. Thus the process conclusions with respect to these variables are not realistic; a large low-pressure, low-temperature sepa- ration column is probably not justified. However, as an optimization study, the present model is quite valuable. It is apparent that all four of the opti- mization methods adjusted to nine variables in a manner consistent with the chosen objective function, and the process characteristics. In the remainder of this article, we discuss the relative performance of the methods on the nine-variable model.

OPTIMIZATION PERFORMANCE Q /LAP (Quadratic/Linear Mpproximation Pro- gramming)

The progress of the Q/LAP optimization is shown in Fig. 3, and significant base points along the path are presented in Table 6. An objective function im- provement of $590.42/hr was obtained after nine new base points and 313.2 sec of CPU time. For this calcu- lation, the convergence tolerance was 10 _3 and the perturbation factor 10 -2. The fifth variable was scaled to make the corresponding objective function deriva- tive fall between 1 and 10; for the other variables, the scale factors were adjusted to yield derivatives be- tween 10 and 100[1].

This performance was quite efficient, but the algo- rithm failed to find a declared optimum. After reach- ing the last point and resetting the Hessian matrix, a line search failure terminated the algorithm. The final point also violated the purity constraint by 0.18~. In search of better results, several changes of scale and perturbation sizes were tried, but these were in vain.

IPOSEQ (Infeasible Path) Here, the convergence block (ZWB in Fig. 2) was

replaced with an optimization block, and both opti- mization and convergence advanced with each pass of the calculation loop[2]. The tear variables are included with the design variables, and tear equations are present as equality constraints. Because the col- umn and outlet compressor pressure determined the stream pressures for both loops, the pressure tear variable was not included in the optimization prob- lem.

Several scale factors were tried for this algorithm. The most successful run stopped at an optimum of $1609.84/hr in 231.816CPU seconds and 39 iter- ations. A perturbation factor of 0.01 was used and the Kuhn-Tucker error at the optimum was $.012/hr.

Process optimization: a comparative case study 651

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Process optimization: a comparative case study

Table 5. Propylene chlorination---optimal design----equipment summary

653

Columns

Separator Finishing Column

No. of trays -actual 14 14 -theoretical i0 i0

Reflux ratio (L/D) .037 .406 Diameter, ft 6.0 2.0 Max sup. gas tel. -ft/see 2.38 2.81 At gas density -ib/ft 3 0.21 0.19 Pressure, psia -top 20 14.7

-bott 25 19.7 Temp, °F -top -51.7 114.0

-bott 36.3 109.9

Heater

Rx Freheater S.C. Reboiler F.C. Reboiler

Duty, HBTU/hr 19,365 2,904 1,108 Proe. strm-*F -in 135.1 36.3 109.9

-out 674.1 36.3 109.9

C o o l e r s

Quench Effl. Cooler S.C. Condenser F.C. Condenser

Duty, MBTU/hr 126 32,951 1,014 265 Proc. strm-*F -in 38 661 -51.7 114.0

-out 20 50 -72.0 109.2

Rotating ~[aehines

Compressor quench Pump

Pressure , p s f a - i n 18 27 - o u t 88 .8 100

Temp, *F - i n 44.2 36 .3 - o u t 171.2 38.2

Inlet flow 4743 CFM 39.6 GPH Inlet density-lb/ft 3 0.1497 65.49 Brake Horsepower 1335 5.2

The scale factors were chosen so that the initial objective function derivatives had absolute values between 10 and 100.

The IPOSEQ stepwise performance is not really comparable to the other algorithms; each iteration requires much less CPU time, and the objective function does not increase monotonically, since the convergence requirements also affect the path. Thus, in Fig. 4, we show both the progress of the objective function and the reduction of the tear equation residuals. Table 7 presents design variable and in- equality constraint values for iterations with significant changes. Note, however, that only the last two sets given in the table represent feasible designs. Until iteration 24, the residual errors in the equalities are excessive.

CFV (Converged Feasible Variant) This algorithm required only 147.118 CPU sec and

20 iterations. It terminated at an optimum of 1609.925/hr with a Kuhn-Tucker error of 3.10 -3 S/hr. Again, the scales were chosen so that the initial objective function derivatives had absolute values between 10 and 100. A perturbation factor of 10 -2 with a convergence tolerance of 10- 3 was chosen. The

progress of this optimization can be seen in Fig. 3. Design variable movement is shown in Table 8.

As seen in the table, movement for the first six base points is fairly rapid, and comes close to the opti- mum. The remaining points are fairly close together and show very little objective function improvement. At each iteration, the quadratic programming step is now searching for a point which more closely satisfies the tear equations. Since the convergence block has already satisfied these equations within a certain tolerance, the work of the optimization algorithm is in vain. Better termination procedures are needed for the algorithm.

RFV (Reduced Feasible Variant) The RFV algorithm required 12 base points and

147.605 CPU sec. The algorithm stopped after a line search failure at a base point with an objective function of 1609.97 $/hr and a Kuhn-Tucker error of $0.82/hr. The perturbation factor and convergence tolerance were 10 -2 and 10 -3 , respectively and, the scale factors were chosen so that the initial objective function derivatives had absolute values between 10 and 100. RFV made very rapid progress for the first five base points. Later, however, only small steps

654 LORENZ T. BIEGLER and RJCHARD R. HUGHES

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Process optimization: a comparative case study 659

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were taken until the algorithm terminated with a line search failure (Fig. 3). Table 9 lists the changes in design variables over the course of this optimization.

Compared to CFV, the Reduced Feasible Variant method seems to take larger steps, but requires more time per new base point because of more function evaluations per line search. Also, since tear variables and their bounds are included in the CFV opti- mization step, smaller steps are taken per base point. Though this reduces movement of the algorithm, it also prevents extrapolation to a region where the quadratic approximation is invalid. The RFV algo- rithm, like CFV, also calculates good starting guesses for tear variables, but care must sometimes be taken if these can become negative.

COMPARISON AND EVALUATION Table 10 summarizes the results obtained with the

four algorithms on this problem. The two feasible variant algorithms had the best performance. Their CPU times and objective values are almost identical, although CFV required more base points than RFV. A simulation at the best point required 5.119 CPU sec. Thus, the feasible variant algorithms,

CFV and RFV, required less than 29 STE's (Simu- lation Time Equivalents) for a complete opti- mization. IPOSEQ, the infeasible path algorithm, required 45.3 STE's because it used more iterations and gradient calculations.

The performance of the Q/LAP algorithm is some- what disappointing; it required more CPU time than the others and terminated prematurely. The excessive CPU time can be explained by the bookkeeping required in the modeling and condensation steps and the large number of flowsheet evaluations per line search. The latter problem, as well as the line search failure, are due to errors in the gradients from the modeling step. Combination of linear approxi- mations for a sequence of highly nonlinear modules may result in a poor approximation of the combined result. Also, the error introduced by convergence calculations makes the choice of an appropriate perturbation factor difficult.

Consider the initial, unscaled objective function derivatives compared in Table 11. Here, a per- turbation factor of l0 -2 and convergence tolerance of l0 3 is used to calculate the Q/LAP and RFV gradients. (By definition, the reduced gradient for

Table 10. Summary of propylene chlorination

Algorithm Best Point CPU time (see) Iterations Total STE's*

Q/LAP 1600.55 313.200 10 61.18

IPOSEQ 1609.84 231.816 39 45.29

RFV 1609.97 147.605 12 28.83

CFV 1609.92 147.118 20 28.74

it Based on the simulation time at the best point, - 5.119 sec.

Table 11. Unsealed gradient comparison at the starting point

Unscaled Derivative Values

Variable q/LAP RFV (and reduced CFV)

1 ~ -27.99 -27.6

2 T -0.4323 -0.432 X

3 V 2.350 2.356 X

4 P -.06539 -.06395 S

5 f -.1272 - . 08191 q

6 f -10731.9 -8207.46 V

7 P -2524.8 0.4737 C

8 £LK -1885.69 511.05

9 fHK -1944.27 972.47

Process optimization: a comparative case study 661

CFV is equivalent to the RFV gradients.) Note that the first six gradient elements compare reasonably well in sign and magnitude. The last three elements differ greatly in magnitude and are of opposite signs. The difference in the derivatives for the component recoveries (elements 8 and 9) is due to the column nonlinearities introduced when perturbing a heavy or light key recovery from 0.98, say, to (1.01)(0.98) = 0.9898. The effect of this perturbat ion is less severe on the tear equation responses than on the output streams of the separator. Thus, the effect of these design variables is smoothed when using feasible variant algorithms.

This explanation also applies to element 7, the derivative of the compressor outlet pressure, but we should also consider the pressure perturbations in Q/LAP. In the feasible variant algorithms, the pres- sure of the tear stream is not perturbed because it is fixed within the loop by the design variables. With Q/LAP, pressure must be perturbed in order to define thermodynamically the inlet and outlet streams for each module. The pressure relationships within mod- ules are either nonlinear, as in the reactor block, or of the unit response-no response type. Consider a simple mixer block where the outlet pressure is the lowest inlet pressure. Now, if two or more low pressure streams have pressures close to each other, perturbing one causes the wrong stream to have the lower pressure and leads to erroneous model coefficients and gradients. Because the compressor pressure also determines the reactor pressure after passing through a mixer block, the error in the derivative for this variable is compounded by the pressure effect on the nonlinear reactor kinetics.

Acknowledgements--The authors express appreciation for the support of the Paul A. Gorman Fellowship from the International Paper Company Foundation (for L.T.B.) and of the Engineering Experiment Station of the University of Wisconsin-Madison.

NOMENCLATURE Aj premultiplier for rate constant kj; lb-mol/(hr-ft3-atm 2) Bj Arrhenius temperature factor for rate constant, kj, °R Ei flow of component i in reactor effluent, lb-mol/hr Er total flow of reactor effluent, lb-mol/hr

Ep flow of reactant in reactor effluent, lb-mol/hr j = 1,2 - reactant is propylene q

= 3 - reactant is allyl chloride] F i flow of component i in combined reactor feed, lb-

mol/hr F r total flow of combined reactor feed, lb-mol/hr

fLx fractional recovery of light key (C12) to sepn. column tops

fnK fractional recovery of heavy key (AC) in sepn. column bottoms

fq fraction of sepn. column bottoms split to quench f~, fraction of dry gas split to vent

j* key component for reaction j kj rate constant for reaction j, lb-mol/(hr-ft3-atm 2) Pc compressor outlet pressure, psia P~ reactor feed pressure, psia P~ compressor inlet pressure, psia P~ separation column top pressure, psia P~ reactor pressure, psia

Pcl2 partial pressure of CI 2 in reacting mixture, atm pj, partial pressure of reactant in mixture, atm

[ j = 1,2- reactant is propylene 1 = 3 - reactant is allyl chlorideJ

rj reaction rate for reaction j, lb-mol/(hr-ft 3) Tp reactor preheater outlet temperature, °F T~ temperature of reacting mixture, °F V~ reactor volume, ft 3 Xj extent of reaction j, lb-mol/hr 6 residual error (Eq. (10)), lb-mol/hr e closure test value for 6, lb-mol/hr r/ propylene/chlorine ratio in combined reactor feed v~ stoichiometric coefficient for component i in reaction

j (neg. for reactants, pos. for products) tk objective function = net value added, $/hr

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3. L. T. Biegler & R. R. Hughes, Feasible path opti- mization with sequential modular simulators." Comput. Engng (in press).

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8. A. W. Fairbairn, H. A. Cheney, & A. J. Cherniavsky, Commercial scale manufacture of allyl chloride and allyl alcohol from propylene. Chem. Engng Prog. 43(b), 280 (June, 1947).

9. J. M. Smith, Chemical Engineering Kinetics, 2nd Edn, pp. 212-231. McGraw-Hill, New York (1970).

10. H. P. Groll & G. Hearne, Halogenation of hydro- carbons. Ind, Engng Chem. 31(12), 1530 (1939).

11. H. F. Rase, Chemical Reactor Design for Process Plants. Vol. I, pp. 464-471. Wiley, New York (1977).

12. C. F. Oldershaw, L. Simenson, T. Brown & F. Radcliffe, Absorption and purification of hydrogen chloride from chlorinated hydrocarbons. Chem. Engng Prog. 43(7), 371 (1947).