Design and Control of a Gas-Phase Adiabatic Tubular Reactor

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Ind. Eng. Chem. Res. 2001, 40, 3762-3774

Design and Control of a Gas-Phase Adiabatic Tubular Reactor Process with Liquid Recycle Francisco Reyes Programa de Simulacion Molecular, Instituto Mexicano del Petroleo, Eje Central Lazaro Cardenas 152, 07730 Mexico City, Mexico

William L. Luyben* Department of Chemical Engineering, Lehigh University, Bethlehem, Pennsylvania 18015

Previous studies have explored the design and control of processes with gas-phase adiabatic tubular reactors that feature a gas recycle and a simple separation section consisting of a single ideal separator vessel. The gas recycle leads to high compressor capital and operating costs. This paper extends this work to the case in which a distillation column is required in the separation section and the recycle stream is liquid. The liquid recycle means that there are no compressor costs to counterbalance the reactor costs. However, there are large capital and energy costs associated with the vaporization/condensation of the recycle stream. For the numerical case studied, the liquid recycle process is more expensive than the gas recycle process, and it is more difficult to control. The basic reaction is A + B f C. Three reaction systems are considered: case 1 (irreversible with moderate activation energy), case 2 (irreversible with high activation energy), and case 3 (reversible). The optimum steady-state designs for cases 1 and 3 can be effectively controlled by the same control structure. The optimum steady-state design for case 2 cannot be controlled, and the process has to be redesigned to prevent reactor runaways. The concentration of one of the reactants has to be reduced so that it becomes a limiting reactant, thus providing self-regulation. This self-regulation in the liquid recycle process is not as effective as that in the gas recycle system because of the slower changes in concentrations due to the larger holdups of material in the liquid phase. 1. Introduction

Table 1. Physical Properties of Components A-C

The design and control of tubular reactors have been the subjects of many papers. Almost all of these studies have looked at the tubular reactor in isolation. Few papers have studied the design and control of tubular reactors in a plantwide environment. Luyben1 and Reyes and Luyben2 studied a gas-phase adiabatic tubular reactor system with a gas recycle. The exothermic irreversible reaction was A + B f C. The reactor effluent was sent to a separation section consisting of a simple flash tank in which pure component C was removed in the liquid phase and reactants A and B went overhead in the vapor stream. This was compressed and recycled back to join the gas-phase fresh feed streams before entering the preheat system. The optimum steady-state economic design featured a flowsheet with only a feed-effluent heat exchanger for reactor feed preheating. However, effective dynamic control required the use of both a heat exchanger and a furnace. This paper studies the flowsheet in which the separation section is a distillation column and the recycle stream is in the liquid phase. The fresh feed streams are also liquids. A compressor is no longer needed, but a vaporizer is required. 2. Process Studied The process is shown in Figure 1. Although hypo* To whom correspondence should be addressed. E-mail: [email protected]. Telephone: 610-758-4256. Fax: 610-7585297.

component kmol-1

molecular weight, kg heat of vaporization at 273 K, kg kmol-1 liquid heat capacity, kJ kmol-1 K-1 vapor heat capacity, kJ kmol-1 K-1 Antoine constantsa Aj Bj a

A

B

C

15 16 629 48 30

20 16 629 64 40

35 16 629 112 70

9.2463 8.5532 7.8600 -2000 -2000 -2000

Pjs in bar and T in Kelvin: ln Pjs ) Aj + Bj/T.

thetical components are used in this paper (A + B f C), the chemistry and the flowsheet are typical of a very large number of real industrial processes, e.g., production of isooctane, amines, methanol, ammonia, ethylbenzene, etc. Two liquid-phase fresh feed streams F0A and F0B, containing the reactants, are mixed with the liquid recycle stream D and sent to a steam-heated vaporizer. Vapor is first preheated in a feed-effluent heat exchanger (HX1) and then in a furnace before entering the adiabatic packed tubular reactor. The reactor inlet temperature is Tin, and the reactor outlet temperature is Tout. The hot gas from the reactor preheats the reactor feed in HX1 and the liquid recycle stream in HX2. The stream leaving HX2 is partially condensed and is fed into a distillation column. The relative volatilities are RA ) 4, RB ) 2, and RC ) 1, so a simple phase separator would not give a sufficiently pure product stream. Therefore, the separation section consists of a distillation column. Only one column and one recycle stream are required because the product C is the

10.1021/ie0004141 CCC: $20.00 © 2001 American Chemical Society Published on Web 07/14/2001

Ind. Eng. Chem. Res., Vol. 40, No. 17, 2001 3763

Figure 1. Optimized flowsheet for case 1. Table 2. Kinetic Parameters of Reaction Systems Case 1

heat of reaction, kJ kmol-1 specific reaction rate, kmol s-1 bar-2 kgcat-1

-23 237 0.3882e-69710/8.314T

Case 2 heat of reaction, kJ kmol-1 -44 150 specific reaction rate, 1.3908 × 106e-132450/8.314T -1 -2 -1 kmol s bar kgcat Case 3 heat of reaction, kJ kmol-1 -14 000 specific reaction rate 61.224e-94000/8.314T of the forward reaction, kmol s-1 bar-2 kgcat-1 specific reaction rate of 225000e-108000/8.314T the reverse reaction, kmol s-1 bar-2 kgcat-1

heaviest component and can be removed from the bottom of the column. If C were an intermediate boiling component between the reactants, two columns and two recycles would be required. Vapor-liquid equilibrium and physical property data are given in Table 1. Ideal vapor-liquid equilibrium is assumed. Three different kinetic systems are studied: case 1 [irreversible reaction with a moderate activation energy (69 710 kJ kmol-1)], case 2 [irreversible reaction with a high activation energy (132 450 kJ kmol-1)], and case 3 [reversible reaction (KEQ ) 0.0079 at 500 K)]. Table 2 gives the kinetic parameters used. In the following sections, the optimum design and the dynamic control of each one of these cases are considered. Table 3 gives the optimum steady-state design conditions for each case. The following assumptions and specifications are used in all cases: 1. The product flow rate (stream L) from the base of the column is fixed at 0.12 kmol s-1. 2. The product purity xC,L is fixed at 0.98 mole fraction C.

Table 3. Optimum Steady-State Design Conditions case 1 case 2, opt case 2a case 2b case 3 Wcat, kg yA,in, mole fraction yC,in, mole fraction yA,out, mole fraction yC,out, mole fraction Tin, K Tmix, K xA,V, mole fraction xC,V, mole fraction xC,D, mole fraction TV, K Tout, K TH,out, K D, kmol s-1 FF, kmol s-1 FB, kmol s-1 AX1, m2 AX2, m2 AV, m2 AC, m2 AR, m2 QX1, MW QX2, MW QF, MW QV, MW QC, MW QR, MW MV, kmol MC, kmol MB, kmol NT NF RR V/L Pcol, bar P, bar TAC, 106$/yr

80114 0.445 0.168 0.395 0.273 453 441 0.235 0.355 0.201 396 500 455 1.18 1.30 0.735 671 685 534 3092 174 2.62 7.77 0.66 18.95 25.82 6.17 854 968 379 11 9 0.35 4.1 12.7 35 7.11

122038 0.385 0.206 0.364 0.247 466 453 0.190 0.407 0.221 401 500 448 3.34 3.46 1.50 2297 1790 1336 7177 194 7.81 20.89 1.95 47.42 59.51 6.88 2155 2222 436 11 10 0.10 4.5 12.1 35 12.85

245917 0.073 0.2 0.050 0.230 477 464 0.031 0.344 0.210 413 500 449 4.65 4.77 1.66 3910 2542 1814 12222 251 11.31 32.36 2.83 64.37 79.21 8.91 2936 3015 468 13 12 0.08 5.5 9.1 35 20.19

284952 0.727 0.2 0.721 0.229 473 456 0.435 0.478 0.211 386 500 431 4.72 4.84 1.64 4064 2045 1946 9015 259 13.31 21.68 3.33 69.05 84.20 9.19 2976 3129 500 5 4 0.09 6.1 15.5 35 21.52

45783 0.527 0.01 0.509 0.048 520 491 0.353 0.027 0.011 378 534 421 2.97 3.09 0.724 3968 1068 1307 12148 851 12.74 11.03 3.19 46.37 78.73 30.19 1951 3053 1,656 20 15 0.66 20.6 14.4 35 13.29

3. The operating pressure in the adiabatic reactor is assumed to be 35 bar, and the pressure drop is neglected.

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Ind. Eng. Chem. Res., Vol. 40, No. 17, 2001

4. The minimum approach temperature differentials for the heat exchangers are fixed at 10 K in HX1 and 25 K in HX2. 5. The vapor leaving the vaporizer is at its dewpoint temperature. 6. The reflux drum temperature in the distillation column is fixed at 316 K so that cooling water can be used. 7. Distillation columns were designed by setting the total number of trays equal to twice the minimum number and estimating the optimum feed tray from the Fenske equation. 8. The mass heat capacity of all components is assumed to be constant and equal (2 kJ kg-1 K-1). This means that the product of the mass flow rate and the mass heat capacities is a constant for any stream. Therefore, despite the fact that molar flow rates of individual components are varying with distance down the reactor, the sum of the product of the component molar flow rates and the corresponding molar heat capacities is also constant. 3. Case 1: Irreversible, Moderate Activation Energy 3.1. Optimum Steady-State Design. The optimum steady-state design involves an economic tradeoff between (1) the capital costs of the reactor vessel, catalyst, heat exchangers, furnace, distillation column, and vaporizer and (2) the operating costs of providing energy to the furnace, vaporizer, and reboiler and removing energy in the column condenser. The exit temperature of the reactor is fixed at its assumed maximum design value Tout ) 500 K, as limited by safety, catalyst degradation, undesirable side reaction, etc. The highest reactor exit temperature is used at the design stage because it minimizes the reactor size. There are four design degrees of freedom (the method developed by Luyben3can be used to find this number). The following design optimization variables are chosen: (1) reactor inlet temperature Tin, (2) mole fraction of C in the reactor inlet yC,in, (3) ratio of mole fractions of A to B in the reactor inlet yA,in/yB,in, (4) ratio of furnace duty to total preheat duty QF/QTOT. This set of independent variables is chosen because the steady-state evaluation of the tubular reactor performance requires the integration of ordinary differential equations starting from given inlet conditions. All other variables are now dependent variables and can be calculated. The following stepwise steady-state design procedure was used. 3.1.1. Reactor. The total flow rate of material into the reactor Fin can be calculated from an overall reactor energy balance because the product flow rate L and the C yj,incpj product purity xC,L are fixed and the term Fin∑j)A is constant throughout the reactor (see assumption 8 above)

Fin )

-λLxC,L C

[Tout - Tin]

(1)

∑yj,incpj

j)A

where λ is the heat of reaction per mole of C and cpj is the molar heat capacity of component j in the vapor phase. The catalyst weight required to reach the specified outlet temperature and all other exit conditions are

obtained by integration of the differential component and energy balance equations with respect to catalyst weight.

dFC ) RC dw

(2)

-λRC dT ) dw FAcpA + FBcpB + FCcpC

(3)

RC ) kPAPB

(4)

where Fj is the molar flow rate of component j at any axial position, RC is the rate of production of C per kilogram of catalyst, Pj is the partial pressure of component j, and k is the temperature-dependent specific reaction rate (see Table 2). At the reactor inlet (w ) 0) the molar flow rates are known (Fj,in ) Finyj,in) and the temperature is known (Tin). At any axial position down the length of the reactor, the molar flow rates are FA ) FA,in - FC + FC,in and FB ) FB,in - FC + FC,in. The partial pressure of component j is calculated from the total pressure (P ) 35 bar) and the molar flow rates

Pj )

PFj FA + FB + FC

(5)

The two ordinary differential equations given above are numerically integrated from the known inlet conditions (T ) Tin at w ) 0) to T ) Tout. This gives the total weight of the catalyst Wcat and all of the reactor exit conditions. 3.1.2. Furnace. The vapor leaving the vaporizer is at its dewpoint temperature TV, which can be calculated from the known pressure and composition. Then the furnace inlet temperature Tmix and its heat duty QF are calculated from the following equation because the ratio QF/QTOT has been set as one of the design optimization variables.

Tin - Tmix QF ) QTOT Tin - TV

(6)

C

QF ) Fin

∑yj,incpj(Tin - Tmix)

(7)

j)A

It should be noted that the capital cost of a furnace is included in all designs because a furnace is required for startup. The furnace is assumed to be capable of providing 20% of the total preheating duty in all cases, except for those in which higher QF/QTOT ratios are specified. 3.1.3. Heat Exchanger HX1. The cold and hot streams in HX1 are both gas phase. The exit temperature of the cold stream leaving the heat exchanger TC,out is calculated from the known temperature of the entering hot stream (Tout) and the given minimum approach temperature (10 K): TC,out ) Tout - 10. The bypass flow rate FB is calculated from an energy balance around the mix point before the furnace.

FB )

Fin(TC,out - Tmix) TC,out - TV

(8)


The temperature of the hot stream leaving the heat exchanger TH,out is calculated from the overall energy balance

TH,out ) Tout - (Tmix - TV)

(9)

because the product of the flow rates and heat capacities are the same in both the hot and cold streams. The heattransfer area is calculated from the known log mean temperature differential driving force and an overall heat-transfer coefficient of 0.142 kW m-2 K-1 (25 Btu h-1 ft-2 R-1). 3.1.4. Heat Exchanger HX2. To design HX2, we need the flow rate, composition, and temperature of the distillate recycle stream from the column D. Because we know the flow rates and compositions of the column feed (reactor exit) and column bottoms, the distillate flow rate and composition can be calculated. From assumption 6 above, we know the distillate temperature. The stream entering the cold side of HX2 is pumped to a high enough pressure so that it stays in the liquid phase throughout the heat exchanger. The hot-side stream enters as a superheated vapor but is partially condensed. In the desuperheater section, the temperature drops from TH,out to a dewpoint temperatue defined by the reactor outlet composition and pressure. The rate of heat transfer in the desuperheater section is obtained from these two temperatures and the value of the reactor exit flow rate. The minimum pinch temperature difference occurs at the boundary between the desuperheating section and the condensing section. A minimum ∆T of 25 K is used, which gives the temperature of the cold-side stream at the boundary TCP. Knowing the coldside inlet temperature (316 K; see assumption 6 above) and TCP permits the calculation of the heat-transfer rate in the condensing section. Then the temperature and fraction of vapor of the hot stream leaving HX2 can be calculated. The temperature of the heated liquid recycle stream leaving the desuperheating section TR can be calculated from the known heat-transfer rate and TCP. Because all temperatures and heat-transfer rates are known for both sections, the area required in each section can be calculated. The overall heat-transfer coefficient in both sections is assumed to be 0.284 kW m-2 K-1 (50 Btu h-1 ft-2 R-1). 3.1.5. Vaporizer. To design the vaporizer, we need to know the two liquid fresh feed stream flow rates, temperatures, and compositions. Each fresh feed is assumed to be pure reactant A or B and enters at a temperature of 305 K. The flow rates are calculated from the overall component balances, accounting for the consumption of reactants to produce C (LxC,L) and the losses of reactants in the product stream [LxA,L ) 0 and LxB,L ) (0.12)(0.02)].

F0A ) L(xC,L + xA,L)

(10)

F0B ) L(xC,L + xB,L)

(11)

Now the vaporizer heat duty can be calculated from an energy balance. The overall heat-transfer coefficient is assumed to be 1.988 kW m-2 K-1, and a temperature differential of 18 K is used. A value of TV of about 400 K, depending on the composition, permits the use of 5 bar steam. 3.1.6. Column. With the four design optimization variables specified and with the calculations completed

up to this point, the flow rates and compositions of all the streams entering and leaving the column are known. Then the design of the column requires the selection of the number of trays, the feed tray location, and the operating pressure. The pressure is calculated from the known distillate composition and temperature. The total number of trays is set at twice the minimum number, as calculated by the Fenske equation, which is also used to estimate the feed tray location. A rigorous WangHenke distillation rating program is then used to calculate the reflux ratio (and reboiler energy consumption) required to achieve the specified bottoms purity. The column reboiler is sized by assuming an overall heat-transfer coefficient of 1.988 kW m-2 K-1 and a temperature differential of 18 K. The condenser is sized by assuming an overall heat-transfer coefficient of 0.8517 kW m-2 K-1 (150 Btu h-1 ft-2 R-1) and an inlet cooling water temperature of 305 K. The exit cooling water temperature is set either at 323 K or at a temperature that gives a minimum differential temperature difference of 5 K. The column diameter is calculated assuming an F factor ) VmaxFV1/2 ) 0.79, where Vmax is the superficial vapor velocity (m s-1) and FV is the vapor density (kg m-3). The column height results from assuming a tray spacing of 0.61 m and 20% additional height. Because the column feed is almost completely vapor, the diameter of the rectifying section is larger than that of the stripping section. 3.2. Optimization Procedure and Results. Once the plant sizing is completed and energy consumption is determined, the next step is to calculate capital and operating costs. The cost correlations used are taken from Douglas,4 assuming a capital cost of the catalyst of $100/kg. The objective function in the optimization is the total annual cost (TAC), which includes the annual capital costs of all of the equipment and the energy costs of the vaporizer, column, and furnace. The optimum steady-state design is found by varying the four design optimization variables. Figure 2 illustrates the effects of yC,in and yA,in/yB,in for QF/QTOT ) 0 with Tin optimized at each point. The optimum value of yC,in is about 0.15. Note that this is different from that found in the perfect separator case,1-3 in which yC,in ) 0 is the optimum. The optimum value of yA,in/yB,in is about 1. The recycle flow rate has a direct influence on the capital and energy costs of the vaporizer. The column costs decrease as the concentration of the more volatile component (A) increases because the separation is easier. Column costs also decrease as the concentration of the least volatile component (C) increases because the lower purity distillate requires a lower reflux ratio. Reactor costs are the largest, followed by vaporizer costs and column costs. In Figure 3 the optimum is found in terms of the QF/ QTOT ratio when the values of the remaining three design optimization variables (yC,in, yA,in/yB,in, and Tin) that minimize the TAC at each QF/QTOT value are found using the MATLAB function “fmins”. The optimum ratio is 0.2, but the impact of QF/QTOT is actually small. TAC, column costs, and reactor costs vary only slightly. The main effect is a tradeoff between vaporizer costs and furnace costs. Remember that the capital cost of the furnace is fixed for QF/QTOT ratios of less than 0.2, which explains the slight change in the slope of the furnace costs curve at a QF/QTOT ratio of 0.2. The optimum design has a TAC of $7 110 000/year. The optimized flowsheet is shown in Figure 1. It is

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Figure 2. Case 1: effects of yC,in and yA,in/yB,in for QF/QTOT ) 0 and Tin optimized at each point.

Figure 3. Optimization in case 1 in terms of QF/QTOT.

interesting to compare the cost of this liquid recycle system with the cost of the vapor recycle system2 in which an ideal separator is assumed. The TAC of the vapor recycle process is less than half ($3 000 000/year) that of the liquid recycle. This is a result of the high energy and capital costs of the column and the vaporizer. 3.3. Dynamics and Control. The dynamic model of the entire process is similar to that presented in previous papers1,2 with the distillation column dynamic

model added. Simulations were performed using FORTRAN for computational speed. A control system is developed for the process by using the plantwide design procedure discussed in work by Luyben et al.5 Conventional proportional-integral (PI) controllers are used for temperature, pressure, flow, and composition loops. Level controllers are proportional only. Parts A and B of Figure 4 show the two control structures CS1a and CS1b, respectively. Their main difference consists of the way pressure is controlled. In


Figure 4. (A) Control structure CS1a. (B) Control structure CS1b.

CS1a we manipulate the column feed flow rate; in CS1b the rate of vaporization is manipulated. Tuning is done using relay-feedback tests and the Tyreus-Luyben tuning rules because these settings are more robust than those of Ziegler-Nichols. Temperature transmitters are modeled as a cascade of three 6 s lags, composition transmitters as four 1 min lags, and the pressure transmitter as two 30 s lags. The tuning of the FEHE mixpoint is done by keeping the furnace controller in automatic.2 The vaporizer holdup in kilomoles is set to correspond to the amount of liquid collected over a 10 min period at the total recycle flow rate. The same applies for the column condenser and reboiler holdups, where computations are based on the liquid flows leaving each vessel. The holdups on the column trays are obtained from the tray geometry, the steady-state liquid flow rates, and liquid density, using the Francis

weir formula. Perfect control of all flows and of column pressure is assumed. Note that the bottoms composition is controlled by manipulating the bottoms flow rate, while the bottoms level is controlled by the reboiler heat input. This R-B structure is used because of the high boilup ratio V/L. From Figure 5 we can see that the performances of both control structures are about the same for a step change of +5 K in the setpoint of the reactor inlet temperature controller. There is, however, a very noticeable difference for setpoint changes in the recycle or column feed flow controllers, as shown in Figure 6. The flow rate FR,TOT in control structure CS1a and FF in CS1b are changed by -20% with respect to their nominal values. In CS1b this produces a sudden pressure increase that takes a fairly long time to compensate for because of the large thermal inertia of the vaporizer.

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Figure 5. Case 1: responses to a +5KTin variation under CS1a and CS1b.

Figure 6. Case 1: responses to -20% recycle flow rate or column feed flow rate variations under CS1a and CS1b, respectively.

As a result of this pressure increase, the reaction rate increases and yields a 566 K temperature spike at the reactor outlet, even for the relatively moderate activation energy assumed in this case study. This undesirable feature is not observed with CS1a. Dual composition control on the column was explored, but it was found that product quality control was not as good and settling times were increased. The setpoint to the vaporizer xB,V composition controller can be adjusted to change the production rate. Its final impact was found to be very modest, however, with very long response times because of the large holdup of the vaporizer. The column reflux flow rate can also be used for throughput manipulation, as can be seen in Figure 7. The changes in Figure 7 are step changes in the reflux flow rate, which result in large variability in the product purity. Using ramp changes in the reflux flow rate lessens the impact, but product quality control is poor. The PI composition controller cannot eliminate error in the face of ramp disturbances. A more effective strategy would be to use some feed-forward control between the reflux and reboiler heat input. 4. Case 2: Irreversible, High Activation Energy 4.1. Steady-State Design and Optimization. The design optimization variables in this case are the same as those in case 1, so the identical steady-state design

Figure 7. Case 1: responses to column reflux variations under CS1a.

and optimization procedure is followed. The only differences are the values of the activation energy and the heat of reaction. See Table 2. The preexponential factor was modified to give the same specific reaction rate at 500 K. Optimization results are shown in Figure 8 where the QF/QTOT ratio is changed. For every QF/QTOT value, the three remaining optimization variables that minimize the TAC have been determined. As found in case 1, the economic impact of the furnace is small. There is a tradeoff between furnace costs and vaporizer costs, and there is little change in reaction or separation costs. A QF/QTOT ratio of 0.2 minimizes the TAC of the plant at $12 850 000/year. Flowsheet information is given in Table 3. Note that the optimum economic steady-state design for the high activation energy case has a higher TAC than that for the moderate activation energy case. This occurs because the high-temperature sensitivity requires a small temperature change through the reactor, which means a large recycle flow rate. 4.2. Dynamics and Control. An important difference from case 1 is made evident in Figure 9: the optimum steady-state design is not controllable. The high activation energy makes the reactor extremely sensitive to any changes in the reactor inlet temperature. We feel that this process, as designed, is uncontrollable no matter what control structure or controller algorithm is used. The inherent problem is dynamic sensitivity of the reactor, which makes the response independent of the type of controller used. Control becomes more difficult as the reaction rates increase and particularly as the activation energy increases. As pointed out by Shinnar et al.,7 the dynamic runaway problem is an inherent feature of the reactor and the “...response is independent of the controller employed (nonlinear, IMC and so on)....” Our approach to achieving a dynamically controllable process is to modify the basic process design. Following Luyben,6 we design a system with one of the reactants having a low concentration. This “limiting reactant” provides a strong degree of self-regulation to the process, which helps to compensate for the extreme temperature sensitivity. Figure 10 shows the economic impact of designing the process at different reactant concentration ratios. The QF/QTOT ratio is fixed at 0.2, and three values of yC,in are used. For every point, the reactor inlet temperature


Figure 8. Optimization in case 2 in terms of QF/QTOT.

Figure 9. Case 2: responses to a +1.25KTin variation under CS1a.

is varied to minimize the corresponding TAC. It is clear that designing for reactant ratios away from unity has a large economic penalty. A plant designed for yA,in/yB,in ratios of 0.1 or 10 has a TAC that is more than 50% greater than that when the ratio is 1. Dynamic simulations showed that a yA,in/yB,in ratio of greater than 10 or less than 0.1 had to be used to prevent large-temperature excursions in the reactor. It is interesting to note that in the gas recycle process studied by Luyben3 a reactant ratio of about 0.2 provided sufficient self-regulation. This indicates that the liquid recycle system has more difficult dynamics than the gas recycle process. One reason for this could be the slower composition changes in the liquid recycle system because of the much larger amounts of reactant liquid holdups in the column and in the vaporizer. Two suboptimum design cases are studied. In case 2a the limiting reactant is A, and in case 2b the limiting reactant is B. The values of steady-state design parameters are given in Table 3 for the two cases. Most of the

parameters are similar in the two cases, with the exception of the number of trays in the column. Because the relative volatility of A is greater than that of B, the separation is easier when there is more A present in the feed to the column. Therefore, fewer trays are required in case 2b. Case 2a: Limiting Reactant A. Figure 11 shows control structure CS2a, which is similar to CS1a with the exception that the fresh feed F0A is flow controlled. The composition of the limiting reactant A in the recycle stream is not controlled. This “floating composition” scheme permits the product of the reactant compositions at the inlet of the reactor yA,inyB,in to decrease as reactant A is consumed. The overall reaction rate depends on this concentration term (yA,inyB,in) and on the specific reaction rate k, which is temperature dependent. As the temperature increases, k increases rapidly, but the consumption of A reduces the yA,inyB,in product and provides composition self-regulation, which helps to stabilize this very temperature-sensitive system. Figure 12 shows the responses of the system for (8 K step changes in the reactor inlet temperature. Significant changes occur in reactor outlet temperature Tout that take several minutes to return to steady-state levels. Reactant composition changes take place, but these changes occur over a 30-120 min time period because of the large liquid holdups in the recycle path: the column, the reflux drum, and the vaporizer. Product quality (xC,L) is adversely affected for decreases in Tin because the production rate of C decreases and the column sees a significant change in the feed composition. Figure 12 also shows that the system cannot handle a 10 K decrease in the reactor inlet temperature. The reactor temperatures drop so low that the reactant concentrations cannot build up enough to provide the reaction rate required at the fixed fresh feed rate. The system fills up with A as the fresh feed of B and the product flow rate L drop to zero.

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Figure 10. Optimization of case 2 in terms of yA,in/yB,in at QF/QTOT ) 0.2.

Figure 11. Control structure CS2a.

Figure 13 shows the responses to (20% changes in the flow rate FR,TOT. The system handles these large disturbances fairly well in terms of temperature, but product quality control (xC,L) is poor for increases in flow rate. This occurs because the feed flow rate to the columns changes very rapidly. Figure 14 shows the responses to (50% variations in F0A. The CS2a control structure can handle this very large change in the production rate. The reactor exit temperature reaches a new steady state. For the 50% increase, the reactor exit temperature increases about 10 K, and for the 50% decrease, it drops about 10 K.

The product purity is well controlled. The system takes a long time (>10 h) to come to the new steady state. Case 2b: Limiting Reactant B. In this design the limiting reactant is component B, so the piping shown in Figure 11 is modified to exchange the entry locations of F0B and F0A. The control structure is changed to flow control F0B, and F0A is used to hold the total recycle flow rate FR,TOT constant. The resulting control structure is labeled CS2b. Figure 15 shows how this system responds to (8 K step changes in the reactor inlet temperature. When Tin is increased, the changes in the reactor exit temperature


Figure 12. Case 2a: responses to Tin variations under CS2a. Figure 15. Case 2b: responses to Tin variations under CS2b.

separation more difficult, which explains the increase in C going overhead in the recycle stream. Another factor is the fewer trays in the case 2b design (due to the easier A/C separation). This lessens the column’s ability to handle changes in the fractionation load. A comparison of the dynamic performances of the two designs indicates that case 2a is somewhat better (operating with A as the limiting reactor). 5. Case 3: Reversible Reaction

Figure 13. Case 2a: responses to FR,TOT variations under CS2a.

Figure 14. Case 2a: responses to F0A variations under CS2a.

are larger in this case than in case 2b. When A is in excess, the mass flow rate through the reactor is smaller, so there is less thermal sink to absorb the changes in heat generation. Product quality control is essentially the same in the two cases (compare Figures 12 and 15). In case 2a the inlet temperature could be decreased by 8 K and the system could handle it, but it could not handle a 10 K decrease, which resulted in the system filling up with A and shutting down. In this case 2b, a decrease of 8 K cannot be handled. The system fills up with B and C and shuts down. The increase in the B concentration of the feed to the column makes the

The cases studied above have an irreversible, exothermic reaction, so dynamic controllability can be difficult because of high-temperature sensitivity. In this section we consider a reversible exothermic reaction. The decrease in the equilibrium constant with increasing temperature provides a significant degree of selfregulation to the process, so dynamic control is easier. However, the reversible reaction makes the steadystate optimization more difficult because the selection of the operating temperature is a compromise between increasing temperature to increase reaction rates and decreasing temperature to increase equilibrium constants. Figure 16 illustrates the effect of flow rate through the reactor and inlet temperatures on the production rate for two different reactor sizes. There is a peak in the curves, which indicates that there is an optimum inlet temperature. Figure 16 also shows that the higher the flow rate (the higher the recycle flow rate) and the larger the reactor, the higher the production rate. Thus, there are tradeoffs between reactor size and recycle flow rate for a given production rate. 5.1. Steady-State Design and Optimization. The equations describing the system are the same as those in the irreversible case with the exception of the reaction rate term, which becomes

RC ) kFPAPB - kBPC

(12)

The optimum reactor exit temperature with reversible reactions is not necessarily at the maximum. We assume that there is no maximum temperature limitation and include the exit temperature Tout as the additional design optimization variable. The five variables are (1) mole fraction of C in the reactor inlet yC,in, (2) ratio of mole fractions of A to B in the reactor inlet yA,in/yB,in, (3) ratio of furnace duty to total preheat duty QF/QTOT, (4) reactor inlet temperature Tin, and (5) reactor exit temperature Tout.

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Figure 16. Case 3: influence of the reactor inlet temperature on the product flow rate at several values of the reactor feed flowstream when yA,in ) yB,in; yC,in ) 0.02; QF/QTOT ) 0.1.

Figure 17. Optimization in case 3 at QF/QTOT ) 0.1 as a function of yC,in.

The optimization procedure was similar to that used in the previous cases, except that the concentration of C at the inlet of the reactor was selected to be yC,in ) 0.01. This concentration depends directly on the purity of the recycle stream from the top of the distillation column. A high-purity distillate is favored because the reversible reaction is sensitive to the concentration of C in the reactor. The higher this concentration, the lower the production rate due to the equilibrium constraint. The optimization procedure tended to drive the

yC,in variable to quite low values, as shown in Figure 17. Because the distillate is a recycle stream and not a product, we decided to limit the purity of the distillate stream from the column so that the dynamics of the column would not be too nonlinear. In addition, the TAC curve starts to flatten out for yC,in values below 0.01. Table 3 gives the optimum design conditions for this case. Figure 17 also shows that the optimum yA,in/yB,in ratio is not exactly unity. A slight excess of A is favored. The


Figure 18. Case 3: responses to Tin variations under CS1a.

Figure 20. Case 3: responses to column reflux ramped variations under CS1a.

on product quality is severe. In Figure 20 ramp changes in reflux of (25% are made over a 3 h period. The limitations of the PI controller are clearly seen. If the process is redesigned for Tin values below (or above) the optimum steady state, we can achieve effective throughput manipulation without sacrificing product quality. This can be understood from Figure 16, which shows that for design reactor inlet temperature values below the optimum, positive or negative changes in Tin will make the product flow rate vary in a positive or negative direction, respectively, while for design Tin values above the optimum, a reversed situation occurs: positive changes in Tin will decrease the product flow rate, and negative changes will increase it. 6. Conclusion Figure 19. Case 3: responses to FR,TOT variations under CS1a.

separation costs are lower because the volatility of A is higher than that of B. 5.2. Dynamics and Control. The control structure CS1a in Figure 4A is used for the optimum reversible reaction design conditions. The reversible reaction system is much easier to control than the irreversible reaction case because of the self-regulation with respect to temperature. The use of a limiting reactant is unnecessary. The control of product quality is very good. Results for the reactor inlet temperature disturbances are presented in Figure 18. Because the design reactor inlet temperature is close to the peak in the production curve (see Figure 16), both positive and negative changes in Tin lower the production rate. Thus, the reactor inlet temperature does not provide an effective production rate handle. Figure 19 illustrates the effect of recycle flow-rate variations. Recall that for irreversible reactions an increase in recycle flow rate yielded a decrease in the production rate because of the decrease in the reactor temperature. Reversible reactions are less sensitive to decreases in temperature because the shift in the equilibrium constant tends to counterbalance the decrease in the specific reaction rates. Thus, as shown in Figure 19, throughput increases as the recycle flow rate increases. Thus, the recycle flow rate provides an effective production rate handle. Another potential production rate handle is the reflux flow rate in the column. However, as discussed in the previous section, the impact of changing reflux flow rate

This paper has studied adiabatic tubular reactor systems with liquid recycles and distillation columns used in the separation section. Irreversible and reversible reaction cases have been explored. Both steady-state economics and dynamic controllability have been considered in the designs. For the numerical case studied, which is typical of many real chemical systems, the liquid recycle system is more expensive because of the high cost of the distillation column and the need to vaporize the recycle. The liquid recycle process is also more difficult to control because the large holdup in the recycle loop produces slow composition changes. For irreversible reactions, the activation energy is shown to slightly affect the steady-state design but to drastically impact the dynamic controllability. Steadystate economic designs are shown to be very difficult to control because of the severe temperature sensitivity with high activation energies. Changes in the design conditions and changes in the control structure can be used to produce a more easily controlled process. For reversible reactions, the steady-state design is more difficult because of the additional degrees of freedom, but the dynamic controllability is much better because of the inherent self-regulation of exothermic reversible reactions as they encounter chemical equilibrium constraints. Nomenclature A ) reactant component AC ) column condenser area, m2

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AR ) column reboiler area, m2 AV ) vaporizer area, m2 AX1 ) area of HX1, m2 AX2 ) area of HX2, m2 B ) reactant component C ) product component cpj ) vapor heat capacity of species j, kJ kmol-1 K-1 D ) recycle flow rate, kmol s-1 Fin ) reactor feed stream flow rate, kmol s-1 FB ) bypassed flow rate at HX1, kmol s-1 FF ) column feed stream flow rate, kmol s-1 FR,TOT ) flow rate of recycle plus fresh feed, kmol s-1 F0A ) fresh feed flow rate of reactant A, kmol s-1 F0B ) fresh feed flow rate of reactant B, kmol s-1 k ) specific reaction rate of irreversible reaction, kmol s-1 bar-2 kgcat-1 kB ) specific rate of backward reversible reaction, kmol s-1 bar-1 kgcat-1 kF ) specific rate of forward reversible reaction, kmol s-1 bar-2 kgcat-1 L ) product flow rate from column bottoms, kmol s-1 MB ) column base holdup, kmol MC ) reflux drum holdup, kmol MV ) vaporizer liquid holdup, kmol NF ) column feed tray location, from bottom up NT ) column total number of trays P ) reactor pressure, bar Pcol ) column pressure, bar Pj ) partial pressure of component j, bar QC ) rate of heat removal at the condenser, kW QF ) furnace heat duty, kW QR ) reboiler heat duty, kW QTOT ) total preheat duty of the reactor feed, kW QV ) vaporizer heat duty, kW QX1 ) rate of heat transfer at exchanger HX1, kW QX2 ) rate of heat transfer at exchanger HX2, kW RR ) column reflux ratio RC ) rate of production of C, kmol s-1 kgcat-1 Tin ) reactor inlet temperature, K Tmix ) furnace inlet temperature, K Tout ) reactor outlet temperature, K TH,out ) hot-side exit temperature of HX1, K TV ) temperature of stream from vaporizer, K TAC) total annual cost, 106$/year

Vmax ) maximum superficial vapor velocity, m s-1 V/L ) boilup ratio w ) catalyst weight, kg Wcat ) total weight of the catalyst, kg xj,D ) composition of the j component in distillate stream, mole fraction xj,L ) composition of the j component in bottoms stream, mole fraction xj,V ) composition of the j component in liquid in vaporizer, mole fraction yj,in ) composition of the j component in reactor feed stream, mole fraction yj,out ) composition of the j component in reactor exit stream, mole fraction Greek Letters Rj ) relative volatility of species j λ ) heat of reaction, kJ kmol-1 FV ) vapor density, kg m-3

Literature Cited (1) Luyben, W. L. Design and control of gas-phase reactor/ recycle processes with reversible exothermic reactions. Ind. Eng. Chem. Res. 2000, 39, 1529. (2) Reyes, F.; Luyben, W. L. Steady-state and dynamic effects of design alternatives in heat-exchanger/furnace/reactor processes. Ind. Eng. Chem. Res. 2000, 39, 3335. (3) Luyben, W. L. Design and control degrees of freedom. Ind. Eng. Chem. Res. 1996, 35, 2204. (4) Douglas, J. M. Conceptual Design of Chemical Processes; McGraw-Hill: New York, 1988. (5) Luyben, W. L.; Tyreus, B. D.; Luyben, M. L. Plantwide Process Control; McGraw-Hill Professional Publishing: New York, 1999. (6) Luyben, W. L. Impact of reaction activation energy on plantwide control structures in adiabatic tubular reactor systems. Ind. Eng. Chem. Res. 2000, 39, 2345. (7) Shinnar, R.; Doyle, F. J.; Budman, H. M.; Morari, M. Design considerations for tubular reactors with highly exothermic reactions. AIChE J. 1992, 38, 1729.

Received for review April 20, 2000 Revised manuscript received May 29, 2001 Accepted May 29, 2001 IE0004141

Design and Control of a Gas-Phase Adiabatic Tubular Reactor

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