Trade-Offs Between Design and Control in Chemical Reactor Systems

8/13/2019 Trade-Offs Between Design and Control in Chemical Reactor Systems

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Trade-offs between design and control in chemicalreactor systemsWilliam L. LuybenDepartment o f Chemical Engineering, lacocca Hall , Lehigh University I l l , Bethlehem, PA18015, USA(Received 5 Janu ary 1993)

This paper presents two basic ideas. First, some simple but insightful examples are given of direct conflictsbetween steady-state economic design and dynamic process control in jacket-cooled continuous stirred-tankreactor systems. Second, a design methodology is proposed that provides a quantitative approach toresolving these conflicts. The essential element of the method is to specify the ratio of the maximum heat-removal rate to the normal design heat-removal rate. It is shown that this is equivalent to setting the ratio ofthe heat-transfer temperature difference (reactor temperature minus jacket temperature) at design con-ditions to the maximum possible temperature difference (reactor temperature minus coolant supply temper-ature). Several typical kinetic systems are simulated to illustrate the significant improvement in temperaturecontrol and disturbance rejection that is achievable when controllability is specifically incorporated into thedesign of the reactor. The cost of this improved controllability is increased capital investment in somekinetic systems, i.e., larger reactors or additional heat-transfer area. With other reaction systems, the costcan be reflected in both higher capital investment and lower yield unless the process is modified to improvecontrollability.(Keywords: trade-offs; reactor control)

The literature contains a large number of papers thatdiscuss the design and control of chemical reactors. Text-books such as Folger and Froment and Bischoff2 presentthe fundamentals, but the emphasis is primarily on thesteady-state aspects. These texts cover some topics indynamics, but most of the discussion is about stabilityand multiple steady-states.The many papers on reactor dynamics also concen-trate primarily on questions of stability, starting with theclassic paper of Aris and Amundson3, continuing withthe multiple steady-state work of Vejtassa and Schmitz4,the book by Perlmutte? and several recent papers (forexample, Balakotaiah and Lus$).There have been only a limited number of papers inthe literature that explore the interaction between steady-state design and dynamic controllability in reactionsystems. There have been several papers that haveattempted to provide a general methodology for assess-ing the impact of steady-state decisions on dynamic oper-ability in chemical processes. Lenoff and Morari studiedthe design of integrated distillation systems using amulti-objective function approach. Morari analysed thedynamic resiliency of plants based on concepts of IMCdesign. Palazoglu et ~1.~ and Palazoglu and Khamba-nonda studied the dynamic operability of chemical pro-cesses from the process design standpoint. All of thesestudies have been quite theoretical and mathematical intheir approach, and the techniques have found little0959-1524/93/010017-250 1993 Butterworth-Heinemann Ltd

application in industry. However, they have attempted tohandle a universal problem for a large class of chemicalprocesses.No such grandiose claims are made for this paper. Thetreatment will be narrowly limited to jacketed, perfectlymixed chemical reactors. However, since this type ofreactor system is so widely used in industry, the resultsgained from this study should have wide application inthe design and control of many practical industrial reac-tors. The reactions and the reactors considered in thispaper are quite simple. However, they are rich enough toprovide some very important insights into the trade-offsbetween steady-state design and control. The consider-ation of design and control trade-offs is often overlookedby both reactor designers and control engineers.In the next section, we illustrate the problem by consi-dering the simplest possible reaction case: one irrever-sible reaction, equal-size reactors and given (and equal)temperatures in all reactors. This will serve to show thata process with a higher capital cost (one large CSTR)provides much better temperature control than a lessexpensive process (multiple CSTRs in series). We thenconsider the problem of selecting the best reactor tem-perature while considering both steady-state economicsand dynamic controllability. We next consider a morecomplex kinetic system (consecutive reactions) in whichthe conflict between design and control impacts on bothcapital cost and, more importantly, product yield. The

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Trade-off s i n chemi cal react or systems: W . L. Luy beninsights gained from these specific case studies are thengeneralized and a design methodology presented.

Alternative processes with a single reactionLet us consider an irreversible, exothermic first-orderreaction A+B occurring in the liquid phase in one ormore stirred-tank reactors in series. All reactors have thesame volume and operate at the same temperature,which is given. A height-to-diameter ratio of two wasused.

rr~, = I,&,, (1)where G = rate of consumption of reactant A (lb-molh-) in the nth reactor stage, V, = holdup of nth reactor(lb-mol), k, = specific reaction rate (h-) in the nth reac-tor stage and z,, = concentration of reactant A in the nthreactor (mole fraction A).In the numerical case studied later via dynamic simula-tion, the steady-state operating temperature of all reac-tors was 140F. At this temperature the specific reactionrate was 0.5 h-. An activation energy of 30 000 Btu lb-mol- was used. Constant reactor holdups wereassumed. Feed enters the first reactor at a flow rateF = 100 lb-mol h- with temperature T,, = 70F andconcentration z, = 1 (mole fraction A). Molecularweight was 50 lb lb-mole- and liquid density was 50 lbft-j. Perfect mixing in the reactor was assumed. The heatof reaction was - 30 000 Btu lb-mole- unless otherwisenoted.The reactor is cooled by flowing cooling water at a rateF, (ft3 h-) through a jacket that surrounds the verticalwalls of the reactor. Perfect mixing in the jacket wasassumed. A four-inch jacket clearance was used. Inletcooling water had a temperature of 70F. The overallheat-transfer coefficient U was assumed to be constant(300 Btu h- F- ft- for this section).

Steady-state designsGiven the fresh feed flow rate (F = 100 lb-mol h-) andfresh feed composition (zO = 1; pure reactant), thesteady-state design procedure was:(1) Specify the conversion x.(2) Calculate the concentration of the product streamleaving the last (n = N) reactor stage zW

z&.= z,(l - x) (2)(3) Calculate the required reactor size for N = 1, 2 and3. (a) One CSTR (N = 1)

FX = k(1 - X)Z = z,(l - x)

18 J. Proc. Cont . 1993, Vol 3, No 1

(3)(4)

(b) Two CSTR (N = 2)VI = v* = fll - Jl - 4

kJi x22 = zO(l - x)

Fzo = F + V,k(c) Three CSTR (N = 3)

V = v* = v3 = F[l - (1 - x)]k(1 - x)~z3 = z,(l - x)

fzo = F + V,kFZI = F + V2k

(5)(6)

(7)

(8)(9)

(10)

(11)(4) Calculate the diameter, length and heat-transfer areaof each reactor, and determine its capital cost:

D, = (2v/x)3 (12)L, = 20, (13)Cost = 19 16.9 (D,).j 6 (L,).802 (14)A H?l = 27~:(D,)~ (15)

(5) Calculate the heat-removal rate Q, the jacket tem-perature TJnand the cooling water flow rate F,, for eachreactor stage.

Ql = -(zO - zl )Fh - c,M F(T, - TO) (16)Q = -(z,_ - z,)Fh (17)

where h = heat of reaction (Btu lb-mol-), cp = heatcapacity reaction liquid = 0.75 (Btu lb- OF-), Tl =reactor temperature = 140 (OF), TO = feed temperature= 70 (F)

T,, = T,, -Hll

FJn= QnPJCJVJ, TJ,)(18)(19)

where F, = flow rate of COOhg Water (ft3 h-l), pJ =density of cooling water = 62.3 (lb ftm3), J = heat capa-city of coolant = 1 (Btu lb- OF-), TJ, = inlet tempera-ture of cooling water = 70 (F).

Figure I gives flow-sheet conditions for three altema-tive processes for the k = 0.5 and 95% conversion case.Table 1 gives more steady-state design results for a


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I I T I TRf.eee Fle.eee w3.2es

Figure 1 One-, two- and three-CSTR processes

number of cases. Conversion was varied from 99.5% to75% and specific reaction rate was varied from 0.1 to12.5 h-. These results shows some interesting trendsthat give us some insights about the controllability ofthese various design cases.

(1)

(2)

(3)

(4)

The heat removal rate is much higher in the firststage than in the later stages. This occurs because theconcentration of reactant is the highest at this point.This makes the control of the first stage the mostdifficult, as will be demonstrated later.As conversion is increased, the required reactor hold-ups increase. This increases the size of the reactors,giving larger heat-transfer areas. Therefore jackettemperatures are higher and cooling-water flow ratesare lower. As will be demonstrated later, these effectsimprove the controllability of the system. So reactorsystems with high conversions are easier to controlthan reactor systems with low conversions.When considering multi-stage reactors, as conver-sion is decreased, the heat-removal rates areincreased in the later stages, as compared to the firststage.As reaction rates increase, reactor sizes decrease.This decreases heat-transfer area, lowers jacket tem-peratures and increases cooling water flow rate. Forexample in a three-CSTR process with k = 2.5 and95% conversion, the jacket temperature in the firstreactor is 70.13F and the cooling water flow rate is24 300 gpm. As you might expect intuitively and aswill be demonstrated later, these effects degrade thecontrollability of the process. So reactor systemswith small specific reaction rates are easier to controlthan reactor systems with large specific reaction rates(as the same level of conversion).

Note that for very high specific reaction rates, a multi-stage reactor system is not feasible for the numerical casestudied. This is because the reactor size, and thereforeheat transfer area, is so small that the required jackettemperature is lower than the temperature of the avail-able cooling water. Refrigeration would have to be used.

Trade-offs in chemical reactor systems: W. L. LuybenThe capital costs of some of the alternative processesare given below:

Case 1: k = 0.5 and 95% conversionCost of one-CSTR Process = $427 300Cost of two-CSTR Process = $296 600Cost of three-CSTR Process = $286 700Case 2: k = 0.5 and 99% conversionCost of one-CSTR Process = $1 194 000Cost of two-CSTR Process = $536 700Cost of three-CSTR Process = $458 200These numbers show that the optimum steady-statedesign is the multiple-stage reactor system. The higher

the conversion, the larger the economic incentive to go tomultiple stages. At 95% conversion, the capital cost of athree-CSTR process is 67% of a one-CSTR process. At99% conversion, this percentage is only 38%. Thus, ifonly steady-state economics are considered, the designsof choice in these numerical cases would be two or threeCSTRs in series.

Dy namics and controlNow let us look at the dynamic aspects of these alterna-tive designs. We will use both linear analysis and non-linear rigorous simulations to study the controllability ofthese reactor systems. The model used was developed byLuyben and Devia and Luyben13. Three differentialequations describe each stage: reactor componentbalance, reactor energy balance and jacket energybalance (see Appendix). Two 1 min first-order lags in thetemperature measurement were used. Proportional-integral (PI) temperature controllers were used. It isrecognized that considerable improvement in tempera-ture control could have been obtained by using deriva-tive action14. Since our principal objective is to assess theinherent controllability of the alternative processes, onlyPI controllers were used so that the comparisons wouldbe easier.

Tabl e 2 gives results from the linear model for twocases where k = 0.5 (conversion = 95% and 99O/,). Inthe one-CSTR process the open-loop eigenvalues arenegative, indicating open-loop stability. Bit in the two-CSTR process the first reactor has two positive open-loop eigenvalues. For the three-CSTR process the firstand second reactors have positive open-loop eigenvaluesfor the 95% conversion case. Thus the mathematicalanalysis gives us our first quantitative indication that themore reactors we have in a series of CSTRs, the moredynamically inferior the process will be. And the lowerthe conversion, the poorer the dynamics.

Next, the linear model was used to calculate prelimi-nary controller settings. The ultimate gain and frequencywere calculated for each reactor. The controller gain thatgave +2 dB maximum closed-loop log modulus wascalculated for a proportional controller. This gain and areset time that was twice the Ziegler-Nichols value wereused as preliminary settings. The larger reset value was

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Trade-of fs in chemical reactor systems: W. L. LuybenTable 1 Steady-state designs of CSTR processes

0.100.975 0.100.95 0.100.75 0.500.995 0.500.95199000 39000 19000 3000 39 800 7800 380050.2 29.16 76.82 12.4 29.4 17.06 13.4215800 5344 3309 967 5417 1828 11310.005 0.025 0.05 0.25 0.005 0.025 0.052722 2662 2588 1988 2722 2662 2588139.43 138.34 137.39 133.15 138.32 135.14 132.3878.46 77.95 76.82 62.97 79.72 81.77 82.9913 142 5325 347220.3 15.02 13.022588 1417 10660.0707 0.1581 0.22360.005 0.025 0.052525 2263 2067197 399 521136.75 134.68 133.54139.75 139.06 138.3775.7 70.01 65.085.66 11.57 15.24

FE4b50.500.251238750131.12134.6240.5123.22

2628 106511.87 8.78885 4850.0707 0.15810.005 0.0252525 2263197 399130.49 124.43139.26 137.2583.53 83.185.69 11.88

6947.62??2360:os2067521121.1135.2480.9215.97

4847 242014.55 11.551331 8380.171 0.29240.0293 0.08550.005 0.0252224 1860425 62172.7 182134.43 132.6138.93 137.53139.82 139.2869.1 59.512.3 18.42.1 5.2

171410.29g840:13570.051632698257131.83136.5138.7152.821.07.5

587:i60.630.39690.25E441131.33132.85135.4927.622.313.5

969 484 3438.51 6.75 6.02455 286 2280.171 0.2924 0.36840.0293 0.0855 0.13570.005 0.025 0.052224 1860 1632425 621 69872.7 182 257123.71 118.36 116.1136.89 132.78 129.78139.47 137.89 136.2382.9 77.0 70.812.7 19.8 23.42.1 5.4 7.8

z95 ;:;I5 ;::5 12.5 12.5 12.5 12.50.995 0.975 0.95 0.757960 156017.17 9.971853 6250.005 0.0252722 2662135.10 125.8083.67 95.46

E53j370.052588117.72108.49

120 1592 312 1524.24 10.04 5.83 4.59113 634 214 1320.25 0.005 0.025 0.051988 2722 2662 258881.42 125.68 98.49 74.85348.24 97.83 186.96 106610516.943030.07070.0052525197112.19137.83119.755.81

4265.14A?5810:0252263F49131.97184.8812.89

1394.451250.22360.05206752184.74126.07280.618.58

*

1944.981560.1710.02930.005222442572.7130.89130.89138.44198.914.02.1

96.83.95?29240:08550.025186062118276.72118.88133.82553.9::i

68.63.5277.90.36840.13570.05163269825770.13110.12128.9924 33034.88.7

8::

7%2i80.251988119.9679.59

5.031590.500.251238750114.05124.2756.2127.65117.54.211120.638

441114.66119.09126.8338.028.515.5

*Design not feasibleF,s are in gpm, Qs are in 10 Btu h-

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Table 2 Linear model open-loop resultsk = 0. 5Conversion1 CSTR:

aah

2 CSTR:First stage&Ih.

Second stageczlL

3 CSTR:First stageaun,

Second stage&I11.

Third stageaxn.

0.95- 1.57-0.0197 f i 0.337- 18.3

- 0.5970.1881.32- 22.5

-3.51-0.161 f 0.315i- 19.3

+ 0.567 -0.661- 0.018 0.2062.81 1.234- 24.9 - 22.4

- 3.330.0799 f 0.372i-21.2- 3.49- 0.170 f 0.302i- 19.2

-4.77-0.30 f 0.2663- 20.2-4.10- 0.502-0.157- 18.7

0.99

- 1.21-0.325- 0.0793- 16.3

- 1.440.204 f i 0.394- 19.8

- 2.94-0.400-0.111- 17.7

used so that the integral action would not degrade thefeedback dynamics too much, but still gradually drivethe temperature back to its setpoint. Table 3 gives resultsof these calculations.Simulation resul tsThe settings were tested on the rigorous nonlinear modelof the process, and controller gains and resets wereempirically adjusted to give reasonable closed-loopdamping coefficients with as tight control of temperatureas possible.

F igure 2u gives the response of a one-CSTR processfor step changes in feed rate from 100 to 150 lb-mol h- and from 100 to 50 lb-mol h- for the system with k =0.5 and 95% conversion. When feed rate is increased, thetemperature in the reactor initially decreases. This is dueto the sensible heat effect of the colder feed (70F versus140F). After about 5 min, the temperature starts toincrease. This is because the concentration of reactanthas increased enough to increase the rate of reaction. Themaximum temperature deviation is only O.O6F, but ittakes over 5 h to drive back close to the setpoint becauseof the slow change in reactor concentration and the largereset time used in the controller tuning.F igure 26 gives the response of the two-CSTR processfor the same disturbances. Now the maximum tempera-ture deviation in the first reactor is about 0.6F. This isten times larger than the deviation experienced in theone-CSTR process. Thus we have another indication

Trade-of fs in chemical reactor systems: W. L. Luyb enOne CSTR _ 55 Conversion

~Timetlcm)

Figure 2 Response to 50% step changes in feed flow rate: a, temper-ature in one CSTR, b, temperature in first reactor in two-CSTR process

that the control of the two-CSTR process will not be asgood as the one-CSTR process.In order to put these alternative reactor systems to afairly severe test, the heat of reaction was assumed toincrease at time equal zero. This is a somewhat artificialdisturbance, but it serves the purpose of exercising thecontrol system and testing the rangeability and resilienceof these different processes. It could correspond physi-cally in a real system to a sudden change in catalystactivity or to the initiation of a side reaction that has ahigher heat of reaction.In F igure 3 the heat of reaction is increased from30 000 to 45 000 Btu lb-mol- at time equal zero. Nowthe better controllability and rangeability of the one-CSTR process compared to the two-CSTR process arestriking. A single CSTR handles this disturbance withonly a l.YF temperature deviation. In a two-reactorsystem, this disturbance causes a 10F jump in the tem-perature in the first reactor.

F igure 4 shows that the one-CSTR process can handleincreases in the heat of reaction to over 60 000 Btu lb-mol-. However, the two-CSTR process can barely han-

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Trade-of fs in chemical reactor systems: W. L. LuybenTable 3 Controller tuning resultsk = 0.5, x = 0.95 1 CSTR process 2 CSTR processFirst Second First 3 CSTR processSecond ThirdKUo,, (rad h )GN (h)P controllerK +*WEmpirical KEmpirical r,

16.1 4.21 15.9 2.71 6.26 16.921.4 21.9 22.1 21.6 22.4 22.60.245 0.239 0.237 0.484 0.467 0.4633.98 0.20 4.08 0.161 1.49 4.418.51 2.19 9.41 2.16 7.95 9.883.13 1.21 4.74 0.956 2.05 4.00.452 0.392 0.242 0.784 0.412 0.25

Gains are dimensionless; temperature transmitter span was 25F and valve span was 4 times steady-state flow rate50% Ructbtt Heat Increase; solid4 CSTR; dashed=TRl; d0tUd=TRZ Reaction Heat lnrxevcd to 47,ooO; solid=TRl; dzskdSR2

Tii (hours) Time&ours)Figure 3 One-CSTR and two-CSTR processes: responses to 50%increase in heat of reaction

Reaction Heat Incmsc; solid=1 CSm dashed=TRl; dotted=TR2 Three csm solid= So; dahed=100155 t

,--* i150 1CSTR f&&t~_ ;; . . Y ,;j j , .._...*.,.._...__.,..: . ._: ::

Figure 4 One-CSTR process (heat of reaction increased to 60 000);two-CSTR process (heat of reaction increased to 46 500)Figure 6 Three-CSTR process response to 50% step changes in feedflow rate

dle an increase in the heat of reaction to 46 500 Btu lb- evaluated. Figure 6 shows that the responses to changesmo1-. As shown in Figure 5, the two-CSTR process in feed flow rate exhibit larger temperature deviationsexperiences a runaway when the heat of reaction is (1.2F). Figure 7 compares the temperatures in the firstincreased to only 47 000 Btu lb-mol-. The dynamic reactor for the one-, two- and three-CSTR processes forsuperiority of the one-CSTR system is dramatically an increase in the heat of reaction to 38 000 Btu lb-demonstrated in these results. .I mol-. The one-CSTR process shows a peak temperatureThings get even worse when the three-CSTR process is deviation of only about 1F. The two-CSTR process

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Figure 5 Two-CSTR process: heat of reaction increased to 47 000


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Trade-offs in chemical reactor systems: W. L. Luyb en2 CS7R; solid=46,50@ More AH1 - d&ted+S,XQ dmed=6O_KtORe~~~dcat eat 38.OCQ olid=1 CSTR; &s41eds2 CSTR; dotted=3 CSTR

:

Tim (lmm) The (licw)Figure 7 One-, two- and three-CSTR processes: heat of reactionincreased to 38 000 Figure 8 Two-CSTR process with and without increased heat transferarea: heat of reaction increased

gives a 4F peak temperature error. The three-CSTRprocess can barely handle this disturbance, giving a peaktemperature deviation of almost 55F.These results clearly demonstrate that the process thatis the most economical from a steady-state point of viewis not the best from a dynamic point of view.

Improving controllability in thejirst reactorSince a series of CSTRs is more economical from asteady-state point of view but performs worse from adynamic point of view, an improvement in the dynamicswould be desirable. The dynamic problems occur in thefirst stage because this is where the reactant concen-trations are the highest.

difference between the reactor and the coolant, whichshould improve the dynamics of the system. We select acoolant temperature of 132.38F (the same as in the one-CSTR process). In order to transfer the required amountof heat, the total heat-transfer area is 904 ft2, but thejacket area is only 364 ft. Therefore an external heatexchanger is added that has 540 ft2 of area. Assuming l-inch OD tubes 8 feet long installed with 1. l-inch squarepitch gives a heat exchanger with 258 tubes, a shell dia-meter of 1.66 ft and a net shell volume of 127 ft3. Coolingwater flows through both the jacket and the external heatexchanger. The cooling water flow rate is 66.3 gpm. Thusless cooling water is used with the larger area reactorsince the exit temperature of the cooling water is higherand the heat pickup is the same.Increasing heat-transfer area in the first reactor isobviously what has to be done. There are several ways toaccomplish this, but we will discuss two of the mostpractical approaches: using an external heat exchangerand using two reactors in parallel. A third approach thatcan be used in some cases is the use of internal coolingcoils. The use of coils is somewhat restricted because theyoften interfere with mixing. The use of auto-refrigeration(evaporative cooling) with an external condenser can

provide the necessary increase in heat-transfer area insome reaction systems if the vapour-liquid equilibrium isappropriate, i.e., gives reasonable pressures at therequired reaction temperature.

Figure 8 compares the responses of the first reactor in atwo-CSTR process with and without the external heatexchanger for an increase in the heat of reaction. Thesolid line is the temperature in the first reactor withoutadditional heat-transfer area when the heat of reaction isincreased to 46 500 Btu lb-mol-. The reactor, barelyrides through this disturbance. The maximum tempera-ture deviation is over 12F.

External heat exchanger. If the reaction liquid can bepumped through an external heat exchanger, a largeamount of heat-transfer area can be achieved. To illus-trate the improvement, let us consider a case withk = 0.5 h- and 95% conversion. Using a two-CSTRsystem (see Figure 1) requires two reactors, each having695 lb-mol of holdup and 364 ft of heat-transfer area. Inthe first reactor the heat-transfer rate is 2.067 x lo6 Btuh- and the jacket temperature is 121F. The flow rate ofthe cooling water is 80.9 gpm.

The dashed and dotted lines show how the tempera-ture in the first reactor changes when an external heatexchanger is added. The dashed line is for a 46 000 Btulb-mol- heat of reaction. The temperature deviation isonly 6F. The dotted line shows that the system can nowhandle much larger changes in the heat of reaction (up toabout 60 000 Btu lb-mol-, a 100% increase). Clearly theadditional heat exchanger area has made this systemmuch more controllable. Capital investment would beincreased because of the cost of the heat exchanger, butthe improvement in controllability may be well worth theadditional cost. Product quality and yield should beimproved because temperatures will be held much closerto their optimum setpoints. Safety and pollution risksfrom runaway reactions should be reduced.

Now suppose- we want to have a lower temperature Using two reactors in parallel. Keane suggested that it

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Trade-of fs in chemical reactor systems: W. L. Luyb en9 CSTR UITH FIRST TUD IN PRRRLLEL

TUO CSTR PROCESSF=l-

Figure 9 Parallel first-stage reactors

may be practical in some systems to improve the dyna-mics of series CSTR systems by operating dual reactorsat the first stage, i.e., run two reactors in parallel. Usingtwo parallel tanks instead of one tank at the first stageprovides more heat-transfer area for the same total hold-up.To illustrate the benefits, let us consider the samesystem as discussed above. Total fresh feed of 100 lb-molh- is now split into two equal streams and each is fedinto tank with holdup V. Since both tanks are operatingat the same temperature and have the same volume andthe same feed stream, the concentrations and heat-transfer rates will be the same. Then the outlet streamsfrom both reactors are fed into another CSTR. Figure 9compares this first-stage-parallel process with a conven-tional two-stage series process. In each case, equal reac-tor sizes are assumed; but reactor size is different in theparallel and series processes.In order to calculate the volume I/ of each of thereactor vessels, we use the equations given below:

z* = z,(l - x) (20)Fz, = Fzz + VkzZ (21)Fz, = Fz, + 2 Vkz, (22)

Combining Equations (21) and (22) gives a quadraticequation which can be solved for K(2k* z,/F)V * + (3kz2)V + (z2 - z,)F = 0 (23)For the numerical case considered, z, = 0.1711 and

V = 484 lb-mol. The first-stage concentration is lowerthan in the series structure, and the reactor size issmaller. However, there are now three vessels in this two-stage process, so the total reactor volume is larger. Thecapital cost of this parallel three-reactor, two-stage pro-cess is $356 000. The conventional series two-CSTR pro-cess cost is $296 600.The cooling jacket temperature required in the firststage is now 127.1F, compared to 12l.lF in the conven-

155

1x

145

140

135

130

125

iarSeries - solid=46,500; Parallel _ da&d=i6JO@ doned=52,000

I0.5 1 1.5 2 2.5 3

Sties - solid=46JCQ PamIle _ dashcd=46,500: dotted=52,000350 b :\ : ; i j:3@J _...... y..j ..; ;., . ._......._............ ./ _

i, j:so _ _... fSS _i; . .. .: .._.... ___ . . _: :.

0 10 0.5 1 1.5 2 2.5 3

Time (hours)

Figure 10 Series and parallel reaction systems responses to increasesin heat of reactiontional series process. This indicates that this processshould have better dynamics. Figure 10 confirms this forheat of reaction changes. The solid curves are the con-ventional series two-CSTR process. Increasing the heatof reaction from 30 000 to 46 500 Btu lb-mol- yields atemperature deviation in the first reactor that is over12F. The dashed curves are the parallel system. Thetemperature deviation is reduced to 7F. This is not asgood as the external heat exchanger system, as we wouldexpect because of the lower jacket temperature (127.1versus 132.38F) and the smaller heat-transfer area in thefirst reactor stage (574 versus 904 ft).The dotted curves in Figures IOa and 106 show howthe temperature and coolant flow rate in the first reactorchange for an increase in the heat of reaction up to52 000 Btu lb-mol-. The cooling water flow saturatesfor about 10 min, but the system rides through thisdisturbance. As shown in the previous section, the exter-nal heat exchanger process could handle a larger distur-bance (60 000 Btu lb-mol-).Thus, the use of parallel reactors in the first stage togain additional heat-transfer area does improve the con-trollability of the process.

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Trade-of fs in chemical reactor sys tems: W. L. Luyb en

Selection of reactor temperature consideringcontrollabilityIn this section the interaction between steady-statedesign and dynamic controllability in continuous stirred-tank reactors will be considered in the situation wherereactor temperature is a design parameter. A simple irre-versible, exothermic reaction is assumed to occur in aperfectly mixed vessel with an external cooling jacket.The optimum reactor temperature from the stand-point of steady-state economics would be the highestpossible temperature since this minimizes reactor holdupfor a given conversion, which results in the smallestcapital cost. The temperature limitation may be due tometallurgical constraints, product thermal degradation,safety, undesirable side reactions or other factors.We will demonstrate in this section that the besttemperature from the standpoint of controllability is northe highest possible temperature. This is the result of thereduction in cooling-jacket heat-transfer area that occursas the size of the reactor is reduced. The temperaturedifference between the reactor and the jacket becomesbigger, giving a reactor that is more difficult to control, isless resilient, has less rangeability and is more likely to be

(6) Calculate the diameter DR and length LR of the reac-tor, the heat-transfer area AH and the installed capitalcost.DR = (2 V&C)~ (28)LR = 2DR (2Cost = 1916.9 (D, w (LR)o~802 (30)AH = 2x (DR) (31)

(7) Calculate the heat-removal rate Q, the jacket tem-perature 7 and the cooling water flow rate F,.Q= -(z,-z)F I -cpMF (TR- T,,) (32)

where h = heat of reaction (Btu lb-mol- of A reacted),To = feed temperature = 70 (F), M = molecularweight = 50 (lb lb-mol-)

open-loop unstable.The numerical values of parameters used in the simu- T, = T, - Hlation study are the same as were used in the previoussection. Several different values of conversion, feed flowrate and heat-transfer coefficient were studied. The speci-fic reaction rate at a base temperature of 140F (kl& wasassumed to be 0.5 h-.

Steady-state designGiven a fresh feed flow rate F and fresh feed composition(z, = 1, pure reactant), the following procedure wasused to explore the effects of operating at various reactortemperatures.(1) Specify the conversion x, overall heat-transfer coeffi-cient U and klN.(2) Calculate the concentration of the reactant in thereactor z.

z=z,(l -x) (24)(3) Select a reactor temperature TR.(4) Calculate the specific reaction rate at this tempera-ture.

k = k, e-ER(T~+-) (25)where

k, = k,,, eElmRand E = activation energy = 30 000 Btu lb-mall andR = 1.99 Btu lb-mol- R-(5) Calculate the required reactor holdup.

F, = QPJCJPJ - TJA

(33)

(8) Repeat steps 4 through 7 for a range of reactortemperatures.Detailed steady-state design results for two levels ofconversion are given in Table 4, and the significantresults are illustrated in F igures I 1 through 14. FigureZla shows how reactor holdup VR varies with reactortemperature for three different feed flow rates (25,50 and100 lb-mol h-l). Figure I I represents the base case inwhich the conversion is 95% (z = 0.05), the overall heat-transfer coefficient U is 100 Btu h- ftm2 F- and klm s0.5 h-. As reactor temperature is increased, the size ofthe reactor (and its cost) decreases.In Figur e 22a he conversion is increased from 95% to99%, and reactor volumes increase by a factor of 5, asexpected. In Figur e Z3a,conversion is decreased to 75%,and reactor volumes decrease by a factor of 19 from thebase case. In Figur e 14a, heat-transfer coefficient U isincreased, but this has no effect on reactor volume. It willbe shown that it does have a significant effect on control-lability.These steady-state design calculations all indicate thatthe reactor should be operated at the highest possibletemperature in order to minimize reactor size.However, as reactor temperature is increased and reac-tor volume decreases, the heat-transfer area AH alsodecreases. The rate of heat transfer decreases only veryslightly as reactor temperature is increased because mostof the heat to be transferred is from the heat of reactionand sensible heat effects are quite small. This means thatthe heat-transfer differential temperature AT between the

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Trade-of fs in chemical reactor systems: W: L. LuybenTable 4 Steady-state designs of CSTR processes(U = 100, k,, = 0.5)A. Conversion = 95% (z = 0.05)TR 100 120 140V, 22 867 9038 3800DE 24.41 17.91 13.42AH 3744 2016 1132 659.3 397.30 2737 2662 2587 2512 2437T, 92.69 106.80 117.144 241.4 144.8 109.8AT 7.31 13.2 22.86rGo 83.1325 90.53942 93.81228Qm./Q 2.311 2.232 2.021cost 1306 733 427B. Conversion = 99% (z = 0.01)TR 120 140 160V, 47091 19 800 8804DR 31.05 23.27 17.76AH 6059 3401 1981Q 2782 2707 2632TJ 115.41 132.04 146.714 122.6 87.3 68.7AT 4.59 7.96 13.29FR;o

872605.6 807516.26 729823.17

Q&Q 3.137 2.983 2.773cost 2048 1194 721

160 1801690 79010.24 7.95

121.89 118.6596.9 100.338.11 61.3592.86 88.204427 3647

2003876.27246.92362104.30137.895.7080.692945

1.762 1.497 1.247258 161 103

180 2004118 201713.78 10.871194 7422557 2482158.58 166.5457.8 51.521.42 33.46125.91 124.476457 5603

fy = 4pB (37)Combining gives:

2.525 2.257 T _ u&T, + I;;wJ,o449 288 J - rc,p, + UAHQs are in 10 Btu h-; F s are in gpm; capital cost is in $looO, temper-atures are in F.

reactor and the cooling jacket must increase. F igur es 1 b,12b, 23b and 24b all show this trend: the higher thereactor temperature, the higher the AT. As these figuresshow, the AT increases as we increase feed flow rate Fand decreases as we increase overall heat-transfer coeffi-cient U or conversion.

Note the quite large AT, that must be used for the lowconversion case (75%) shown in Figure 23. We willreturn to this important point later when we discuss thepossibility of using multiple CSTRs in series in order toreduce total reactor volume.As we will demonstrate in the next section, thisincrease in AT indicates that the control of the reactor ismore difficult. Accepting this on faith for the moment,we can say that anything that increases AT gives a lesscontrollable reactor. Therefore reactors with low conver-sions, high feed rates and low overall heat-transfer coeffi-cients will be difficult to control. Similar findings werereported by Handogo and LuyberP.

However, as reactor temperature is increased stillfurther, the decrease in reactor heat-transfer area beginsto rapidly increase AT at design conditions. When cool-ing water is increased by a factor of four in these hightemperature designs, the jacket temperature cannot bedecreased enough to really change AT (and Q) thatmuch. At low temperatures, maximum heat removal islimited by inlet cooling water temperature. At high tem-peratures, it is limited by heat-transfer area.aximum heat-removal rate

A vital issue in the design of reactors is the ability of the The non-monotonic curve means that for a given feedcooling system to handle momentary or sustained heat- rate and a specified ratio Qm,/Q, there are two possibleremoval rates that are larger than the nominal design designs. The first would have a low reactor temperatureheat-removal rate Q. This maximum heat-removal rate and a large reactor volume. The second would have a(Q_) may have to be only slightly higher than the high reactor temperature and a small reactor volume.normal if disturbances are small and the uncertainties in Naturally the latter is the design of choice since it offerskinetic and thermodynamic properties are also small. the same controllability at a lower capital cost. ForHowever, this is seldom the situation, particularly in new example, Fi gure lc shows that for a feed rate of 100 lb-processes. So the ratio of QA to Q may have to be quite mol h- and a desired ratio of 2, we could design alarge in some kinetic systems. As we will show later, reactor at about 85F with a holdup of 50 000 lb-mol; or

specifying this ratio sets the design of the reactor. It alsoestablishes the maximum feed rate that can be achievedin a single CSTR with only jacket cooling for a giveninlet coolant temperature.To calculate the maximum steady-state heat-removalcapacity, we assume that the cooling-water flow rate atdesign is 25% of its maximum. Thus, when the controlvalve is wide open, there will be four times the designflow rate of coolant. We also assume that the reactortemperature is held at its specified value. The steady-stateheat-transfer rate under these conditions can be calcu-lated from the following equations.

Qmx = u-40~ - G) (35)

(38)

Then Qmax an be found from Equation (35).Fi gure llc shows how the ratio Q-/Q varies withreactor temperature for three feed flow rates for the basecase conditions. A very interesting phenomenon can beseen from these results: the curve is not monotonic. At

low reactor temperatures the reactor is large, the heat-transfer area is large and AT is small. But the jackettemperature is only slightly above the inlet cooling watertemperature (70F). This means that the cooling waterflow rate at design conditions is quite large, and even if itis increased to four times the design flow rate, the jackettemperature cannot be lowered below the inlet coolingwater temperature. Therefore, as we move to higher tem-peratures we see an increase in the Q-/Q ratio becausethe jacket temperature at design conditions is increasing.

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CSTJt, Feed Flow: solid=25; dashaId& dotted=1001% i:ij iiii.:iii::i~iiii::iii::iii:iii:i:i:iiii~~~~~, ij ; :jj ;j ::::y.j iii:;::y ::j ,j: ii .:: :,I_ _ : ..I _-3; .,.._,,...... 4 .,.....,...,,.. , . .I............ .> -_ 2 . :~ ..? ,.. ,.. ..j .I..... *. .._

102 \ )80 100 120 140 160 180 200Reactor Tempmturedeg.F)

CSTR. Feed Fl ow: solid=25: dashed=ti dated=100 CSTR, Feed Flow: solid=2 dashed=50. dotted=100100 6 / 1 / ; j90 _............ j_ . . . . . . . .,. ..; ,.,..,,......: .,......,,..,. j .:._

,:gc _............. . _..I..,,..,...,...,,......,...~.,....,.,.....,..,..,,.. i .+ ,.._

080 100 120 140 160 180 200ReactorTempezahue (deg. F)

Trade-offs in chemical reactor systems: W. L. LuybenCSTR,eed Flow: soli d=2 dashed=50; dotted=100

1.4 : k,,_.. .,.. ...I L. . .i . :: ~ _.

1.280 100 120 140 160 180 ux)Reactor Temperahue (deg. FJ

-&- 1 _

-0.3580 100 120 140 160Reactor Temperature (deg. F)

180 200

Figure 11 Effect of reactor temperature for different feed rates (II = 100, 95% conversion): a, reactor hold-up; b, reactor temperature differential;c, ratio of Q,,JQ; d, open-loop eigenvalues

we could design a reactor at 142F with a.holdup of 3500lb-mol.However, there is no guarantee that the specified ratiocan be achieved at the specified feed flow rate. Figure Ilcshows that the highest ratio obtainable for a feed rate of100 lb-mol h- under the base-case conditions is 2.3. Ifwe desired a ratio greater than this, the process wouldhave to use two (or more) reactors in parall el. For exam-ple, to get a ratio of 2.5 we could use two parallel reac-tors, each feeding 50 lb-mol h-.As can be seen in F igur es 12c, 13~ and 14c, the maximain the Q,,JQ curves decrease significantly as conversionand U are decreased. The 75% conversion case shown inFigure 13~ is particularly important. The maximum ratiois only 1.35 with a feed flow rate of 100 lb-mol h-l, andonly increases to 1.8 with feed reduced to 25 lb-mol h-.This means that to achieve a ratio of 2 in this lowerconversion case we would have to use more than fourreactors in parallel. This implies that the use of multiplereactors in series may not be at all practical because thefirst stages will have lower conversions and may requiremultiple reactors.

Dynamics and controlIt is useful to look at the effect of changing the reactortemperature on the open-loop stability of the reactor.F igur es ll d, 12d, 13d and 14d show how the real part ofone of the open-loop eigenvalues of the reactor changeswith reactor temperature for the various conditions.Each reactor has a different temperature, size, heat-transfer area and jacket temperature under design con-ditions.

The CSTR process is third order with state variables z,TR and T,, so there are three open-loop eigenvalues. Theone corresponding to the cooling jacket is located far outto the left on the negative real axis. The two correspond-ing to the process fluid in the reactor are negative at lowtemperatures for all values of conversion and heat-transfer coefficient. However, for the 75% conversioncase (Figure 23d), the open-loop eigenvalues becomepositive for reactor temperatures greater than about94F. This open-loop instability occurs because the reac-tors get smaller as the temperature is increased.The cost of increasing controllability in this simpleirreversible reaction system is higher capital investment.

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Trade-of fs in chemical reactor systems: W. L. LuybenCSTR, Feed Flow: sotid=25; dashed=SO; dotted=100

3.6CSTR. Feed Fl ow: solid=25; duhed=50. dottcd=C0

2.6 _. U=OO Con&b kMO=O 5_i L. .., .:. .._ . . - . . _...,,. -*..L,2.4 _,.. i... ,. ... . 1.::,:...... _

..10280 100 120 140 160 180 200 2.2180 LOO 120 140 160 180 200

CSTR. Feed Fl ow: solid=25: dashcd=Xk dotted=100 CSTR. Feed Flow: solid=25 dashed= dotted=100

35b : : ; i ;30 -.......... .; j. .,..........,,. ..,.....,., j ;.&.._: :::G 25 .. -...

i ,...i ,.i ,,.. .._.,...: . ,._8i:i

80 100 120 140 160 180 200Reauor Temperahue (deg. F) Reactor~wnpnahnc dce.F)

Reactor Tempwture (deg. F)

0

_0,15 _.. i. _

-0.3580 100 120 Ma 160 180 200

Figure 12 Effect of reactor temperature for different feed rates (U = 100, 99% conversion): a, reactor hold-up; b, reactor temperature differential;c, ratio of QJQ; d, open-loop eigenvaluesTable 5 Designs with different ratios(F = 100, U = 100, conversion = 95%, k,, = 0.5)

Ratio Qm,./Q Capital cost ($) Ts (F) V, (lb-mol) T, (F)1.5 161 700 179.73 798 118.761.6 193 000 172.11 1060 121.071.7 230 600 164.63 1411 121.951.8 276 600 157.15 1890 121.641.9 334 600 149.53 2565 210.232.0 409 200 141.66 3546 117.772.2 663 300 123.59 7701 108.98

Note: highest possible ratio is 2.31This is illustrated in Tabl e 5 for the base-case conditions.As the desired ratio of Qmax o Q is increased from 1.5 to2, the capital cost of the reactor increases from $161 700to $409 200, more than a factor of 2. This is becausereactor holdup increases from 798.3 lb-mol to 3545 lb-mol. However, the improvement in controllability, flexi-bility and dynamic robustness may be well worth theincreased capital investment. The costs of having onereactor run-away can be much more costly to the plantthan the relatively small incremental investment cost ofbuilding a larger reactor. Reactor runaways mean losses

Table 6 Simulation case-study parameters(C/ = 100, conversion = 95%, k,, = 0.5)AT 10 20 30TR 110.40 134.98 161.97V, 13997 4697 1564DR 20.73 14.40 9.98AH 2699 1303 626.3Q 2698 2606 2505z 100.40 114.98 121.974 177.6 115.8 96.4z, IX 87.45195 93.39421 43482.55Ratio Qm../Q 2.296 2.080 1.736cost 962 488 246Empirical settingsK 16 5.0 3.1r1 0.51 0.48 0.88Qs are in 10 Btu h-; 4s are in gpm; capital cost is in $1000; temper-atures are in F

of reactants and product, lost production, environmentalpollution and safety hazards. In addition to preventingsevere conditions like reactor runaways, the larger reac-tor should permit tighter temperature control which mayresult in improved product quality and yield in some

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Trade-of fs in chemical reactor systems: W. L. LuybenCslR. FeedFlow: aoltd=Zk dsshed=so;dottcd=100

10 __...,;_,_.....,..._.... : ,__........,. . .... . ._................ ........_ ..)I .,,.,...... ... *...,a :.-.:. ._+ .................... ..... F i,:...... ._..._..._...........,._...._.......... . . . ._....................,..._...:.A.._...,.,,._._,......_,.....,...,..,...,............_......._,.........;..,.....

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Trade-Offs Between Design and Control in Chemical Reactor Systems