Effect of Kinetic, Design, and Operating Parameters on Reactor Gain

The larger the reactor gain, the more likely open loop instability will occur and the ..... The capital costs of the compressor and reactor vessel are...
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Ind. Eng. Chem. Res. 2000, 39, 2384-2391

Effect of Kinetic, Design, and Operating Parameters on Reactor Gain William L. Luyben† Department of Chemical Engineering, Lehigh University, Bethlehem, Pennsylvania 18015

Many industrial reactions are carried out in adiabatic tubular reactors. The desired inlet feed temperature is usually achieved by using the hot reactor effluent to preheat the cold feed in a feed-effluent heat exchanger (FEHE). This positive feedback of energy introduces the potential for open loop instability. Previous papers have explored the control of this type of process. The key parameter in FEHE/reactor systems is the reactor gain, i.e. the change in the reactor exit temperature Tout for a given change in the reactor inlet temperature Tin: KR ) ∆Tout/∆Tin. The larger the reactor gain, the more likely open loop instability will occur and the more difficult the control problem will be. This paper explores the impact of various kinetic, design, and operating parameters on reactor gain. Three reaction systems are studied. The first is the hypothetical reaction A + B f C, where kinetic parameters (activation energy and heat of reaction) can be varied. Then two real industrial reactions are studied: the hydrodealkylation of toluene (HDA process) and the chlorination of propylene (allyl chloride process), where design parameters (inlet reactant feed concentration and reactor length) are varied. Larger reactor gains occur for increases in reactant feed concentration, activation energy, and heat of reaction. Reactor gains can vary non-monotonically for some design parameters when interactions between concentration and temperature effects occur. 1. Introduction Tubular reactors present challenging design and control problems. If the flow patterns are truly plug flow with no back-mixing or radial gradients in temperature and composition, there is no potential for multiple steady states or instability. In a stirred tank reactor process (CSTR), these possibilities exist. Therefore one might conclude that the control of plug-flow tubular reactors (PFR) should be easier than that of CSTRs. However, this is usually not the case. Tubular reactors have unique features that distinguish them from CSTRs. First, the inlet feed temperature to a tubular reactor is a critical parameter. We consider an exothermic, irreversible reaction occurring in an adiabatic tubular reactor. Normally we would like to run at some maximum exit temperature Tmax out from the reactor since this minimizes reactor size in the design stage or maximizes the production rate in the operating stage. But exit temperature typically has an upper bound limited by safety, undesirable side reaction (yield considerations), catalyst degradation, etc. For a given per-pass conversion, a fixed amount of heat is liberated from the exothermic reaction. This energy will result in an adiabatic temperature rise through the reactor, whose magnitude will depend on the total flow rate through the reactor and the heat capacity. The total feed to the reactor typically consists of fresh reactant feed streams plus recycles. The lower the total feed flow rate, the larger the adiabatic temperature rise. Since providing recycle to serve as a thermal sink involves capital and operating cost in the separation section of the plant, we would like to keep the total feed flow rate to the reactor as low as possible. This means that we would like to keep the inlet feed temperature to the reactor as low as possible. † Telephone: 610-758-4256. Fax: 610-758-5297. E-mail: [email protected].

However, unlike CSTRs in which feed temperature is usually unimportant, feed temperature to a tubular reactor is critical. At the design stage, a low inlet temperature requires a very large reactor. At the operating stage, when the size of the reactor is fixed, a low inlet temperature may result in low conversion (the PFR may “quench”). Some PFR systems also exhibit extreme sensitivity to inlet temperature; i.e., a very small change in inlet temperature can result in large changes in exit temperature. This parametric sensitivity problem can be both a steady-state problem and a dynamic problem. The transient “wrong-way” effect or “inverse response” occurs in some packed PFRs due to the difference in propagation speeds through the bed of concentrations and temperature changes. We consider in this paper only the steady-state gain relating Tin and Tout. Another important difference between CSTRs and PFRs involves heat-transfer area considerations in the nonadiabatic case. Heat removal in CSTRs can be achieved in a number of ways that permit the heattransfer area to be a design parameter: jacket cooling, internal coils, evaporative cooling, and circulation through external heat exchangers. Heat removal in PFRs is limited to transfer through tubes. PFR design requires careful consideration of tube diameter to guarantee adequate heat-removal area per reactor volume or catalyst weight. CSTRs and PFRs also differ in the importance of pressure drop. In CSTRs pressure drop is negligible. In PFRs pressure drop is usually quite significant, particularly in heterogeneous systems (solid catalyst packed in tubes). Important trade-offs exist between pressure drop, tube diameter, tube length, reactor size, and recycle flow rate, particularly in gas-phase systems where recycle compression costs can be very important. Because of the need to provide a desired inlet temperature in PFR systems, the cold feed stream needs to heated. Also the hot effluent stream from the reactor

10.1021/ie990471y CCC: $19.00 © 2000 American Chemical Society Published on Web 06/03/2000

Ind. Eng. Chem. Res., Vol. 39, No. 7, 2000 2385

typically needs to be cooled before sending it to the separation section. This could be achieved by providing upstream of the reactor a steam-heated exchanger or a fired furnace (depending on the required temperature level). Then the hot effluent could be cooled by steam generators and cooling water exchangers. This type of system can be very effectively controlled because there are independent manipulated variables on the heating and cooling phases, and there is no interaction. Obviously this type of system would be very inefficient from a steady-state energy consumption point of view. Logic would dictate that some type of heat exchanger network be used to preheat the cold feed with the hot reactor effluent. Thus most industrial applications of this type of system use the flowsheet sketched in Figure 2. However, this more energy efficient process is more difficult to control. This is an important example of the ever-present trade-offs and conflicts between economic steady-state design and dynamic controllability. The dynamics and control of these commonly encountered feed-effluent heat exchanger/adiabatic exothermic reactor systems have been studied for many years. Pioneering papers are those of Douglas et al.,1 Anderson,2 and Shinnar et al.3 More recent work is reported by Tyreus and Luyben4 and Luyben.5 The potential for chaotic behavior in a similar system has recently been reported by Bilden and Dimian,6 but they examined only open loop, uncontrolled systems. All of these papers have looked at reactors with given reactor gains. No literature reports have been found that explore what parameters determine reactor gain. That is the purpose of this paper. One would certainly expect that the chemistry of the system would have a strong impact on reactor gain. Activation energy and heat of reaction should be key parameters, and these are explored in this paper. It should be noted that we consider only exothermic irreversible reactions. When reactions are reversible, control problems are seldom encountered because reactor gains are typically small. They can even be negative, which eliminates the positive feedback of energy. This occurs because of the decrease in the equilibrium constant with increasing temperature. Per-pass conversion can decrease as inlet temperature is increased. One final introductory note should be made. The startup of these systems requires that some type of heater be provided to initially achieve the temperature required for the reaction to “light off”. Once the reactor exit temperature becomes sufficiently higher than the inlet temperature, the required preheating can be achieved in the FEHE and no heat input is required in the heater. However, since this startup heater or furnace is available, it could potentially be used as an additional manipulated variable. A comparison of these alternative process and control structures is presented by Reyes-DeLeon and Luyben.7

Figure 1. FEHE/reactor process with inlet temperature control.

Figure 2. Block diagrams.

exchanger. Split-ranged valves are used to manipulate the bypass flow rate and the flow rate through the FEHE. A linear transfer-function analysis of this process reveals the basic dynamic problem. We assume that the reactor itself has a very simple first-order lag transfer function. The reactor is open loop stable with a negative pole at s ) -1/τR.

GR(s) )

Tin(s)

)

KR τ Rs + 1

(1)

We assume that the dynamics of the heat exchanger are negligible compared to the dynamics of the reactor, so the reactor inlet temperature Tin is related to the flow rate of the bypass stream FB and the reactor exit temperature Tout by the algebraic equation

Tin(s) ) K1Tout(s) + K2FB(s)

(2)

Note that K2 is negative since an increase in bypass flow rate decreases Tin. This means that the temperature controller Gc(s) must increase the bypass flow rate when the temperature increases, so the controller gain must be negative. To avoid confusion, we will use positive controller gains and a positive K2 in the analysis that follows. Figure 2 gives a block diagram of the individual components in the open loop system. Combining eqs 1 and 2 gives

KR T + K2FB Tin ) K1 τRs + 1 in

2. Coupled FEHE/Reactor Process Figure 1 shows a feed-effluent heat exchanger coupled with an adiabatic exothermic reactor. The heat of reaction produces a reactor effluent temperature Tout that is higher than the temperature of the feed stream to the reactor Tin. Therefore heat can be recovered from the hot stream leaving the reactor. The control objective is to maintain the reactor inlet temperature by manipulating the bypass flow of cold material around the heat

Tout(s)

[

Tin 1 -

[

(3)

]

K1KR ) K2FB τRs + 1

]

(4)

K2(τRs + 1) F ) τRs + 1 - K1KR B K2 τR (τRs + 1) (s - 1) FB(s) (5) K1KR - 1 K1KR - 1

Tin(s) )

[(

)

/(

)

]

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Ind. Eng. Chem. Res., Vol. 39, No. 7, 2000

GCP(s) )

Tin(s) FB(s)

)

[(

)

K2 (τ s + 1) K1 KR - 1 R τR (s - 1) (6) K1KR - 1

(

/

)

]

Thus the coupled open loop system has a transfer function GCP(s) relating the controlled variable Tin(s) to the manipulated variable FB(s) that is open loop unstable if the product of the gains K1KR is greater than 1. The heat exchanger gain K1 depends on the heat-transfer area and the approach temperature differential on the hot end of the process (the temperature difference between the entering hot stream and the exiting cold stream), but it cannot be greater than unity. Equation 6 clearly shows the importance of the reactor gain. Typical values reported by Tyreus and Luyben4 range from 2 to 6, and we illustrate this in the processes studied below. At this point it might be useful to demonstrate quantitatively the impact of reactor gain on the controllability of the FEHE/reactor process. We use the example used by Luyben5 and eq 6 with the parameter values τR ) K1 ) K2 ) 1 and assume three first-order measurement lags with τm ) 0.1. Table 1 gives the dynamic properties of the process when two values of reactor gain (KR ) 2 and KR ) 4) are used. The controller tuning procedure is to find the values of controller gain Kc and reset time τI that give the largest resonant frequency ωr with a reasonable maximum peak in the closed loop log modulus Lc curve. Figure 3 gives Lc curves for three values of reset and different gains. The system is conditionally stable since there are both a maximum gain Kmax and a minimum gain Kmin for a given reset time. As Figure 3 illustrates, if the gain is too low, there is a large low-frequency peak. If the gain is too high, there is a large high-frequency peak. Thus, there is an optimum gain that minimizes the peak height. When reactor gain is KR ) 2, the optimum settings are τI ) 1.5Pu ) 1.5(0.397) and Kc ) 3Kmin ) 3(0.440), giving a maximum peak height of +3dB at a resonant frequency of 9 rad. When reactor gain is KR ) 4, the optimum setting are τI ) 1.5Pu ) 1.5(0.446) and Kc ) 1.75Kmin ) 1.75(1.32), giving a maximum peak height of +8dB at a resonant frequency of 9 rad. Since this peak height is larger than that obtained with the smaller reactor gain, the closed loop performance is predicted to be more underdamped. The time-domain unit step responses for the two reactor gain cases are shown in Figure 4. The degradation in dynamic controllability as reactor gain increases is clearly demonstrated by this example. Note that the closed loop time constants for the two cases are about the same, as predicted by the similar resonant frequencies.

Table 1. Dynamic Parameters for Two Values of Reactor Gain KR Pu (h) Ku τI/Pu Kmax Kmin Kc/Kmin)opt Lmax (dB) c ωr

2 0.397 6.56 1 4.75 0.350 3 3 1.5

1.5 5.32 0.440 3 3 9

2 5.63 0.507 3 4 10

4 0.446 5.27 1 3.86 1.06 1.75 7 8

1.5 4.33 1.32 1.75 8 9

2 4.56 1.52 2 9 10

Figure 3. Closed loop log modulus.

3. Hypothetical Process Studied We consider first a process that has been adapted from that presented by Jones and Wilson.8 An exothermic, gas-phase irreversible reaction A + B f C occurs in an adiabatic tubular reactor with a reaction rate R (kmol‚s-1‚(kg of cat.)-1).

R ) kPAPB

(7)

where Pj ) the partial pressure of component j and the

Figure 4. Closed loop setpoint responses for KR ) 2 and KR ) 4.

specific reaction rate k is temperature dependent (kmol‚s-1‚(kg of cat.)-1)‚bar-2).

k ) Re-E/RT

(8)

Ind. Eng. Chem. Res., Vol. 39, No. 7, 2000 2387 Table 2. Parameter ValuessBase Case -23 237

heat of reaction λ (kJ/(kmol of C)) activation energy E (kJ/(kmol)) k at 500 K (kmol‚s-1‚bar-2‚(kg of cat.)-1) (kJ/kmol) heat capacities (kJ‚kmol-1‚K-1) cpA cpB cpC molecular weights (kg/kmol) MWA MWB MWC

69 710 3.309 × 10-8 30 40 70 15 20 35

contains a specified composition (yRC) of component C. The fresh feed contains only pure reactants A and B, and following Jones and Wilson,8 we assume the fresh feed contains precisely stoichiometric amounts of the two reactants. In reality, this is never true because of flow measurement inaccuracy. Any practical control scheme must be able to keep track of the inventories of reactants in the system. Some type of feedback control must be used to adjust the quantities of the two reactants fed to the system. Since these reactants cannot leave the process, they must be completely reacted, and this means that every molecule of A fed requires exactly one molecule of B. This issue is discussed by Tyreus and Luyben.9 In this paper we assume that perfect stoichiometric amounts of reactants are fed. The desired production rate is set at 0.12 kmol/s of product C, which means that the fresh feed flow rate is 0.24 kmol/s of an equimolal mixture of reactants. The value of the recycle flow rate is determined from steadystate economics discussed in the next section. 4. Optimum Steady-State Design

Figure 5. Nomenclature.

where E is the activation energy (kJ/kmol). Parameter values used in the base case are given in Table 2, and Figure 5 shows the nomenclature used. The heat of reaction λ and the activation energy E are varied in the study. The catalyst bulk density is 2000 kg/m3, and the porosity is 0.5. The reactor pressure is 50 bar. The steady-state changes in the molar flow rate of component C (FC(W)) and temperature (T(W)) down the length of the PFR are calculated by numerical integration of two ordinary differential equations. The independent variable is reactor catalyst weight W, with the limits of integration between W ) 0 and W ) Wcat..

dFC/dW ) R

(9)

with the boundary value FC(0) ) FCo.

dT λR )dW cPAFA + cPBFB + cPCFC

(10)

with the boundary value T(0) ) Tin and where cPj ) molar heat capacity of component j. The molar flow rates of components A and B are calculated at each axial position from the molar flow rate of C and the total molar flow rates (FoA ) FoB) fed to the inlet of the reactor, which depend on the fresh feed flow rate and the recycle flow rate.

FA(W) ) FB(W) ) FoA - (FC(W) - FCo)

(11)

where FCo is the molar flow rate of component C entering the reactor, which depends on the flow rate and composition of the recycle gas.

FCo ) FRyRC

(12)

A simplified separation section is assumed in which the liquid from the separator drum contains no reactant components A or B, and the gas from the separator

The optimum steady-state design of the system involves an economic trade-off between the capital cost of the reactor vessel and catalyst and the capital cost and energy cost of the compressor. The exit temperature of the reactor is fixed at its maximum value Tout ) 500 K. The reactor inlet temperature Tin is calculated from an energy balance around the reactor and varies with the recycle flow rate; i.e., the higher the recycle flow rate FR, the higher Tin can be and still give the 500 K exit temperature. The design procedure is the following: 1. Specify the desired production rate, the concentration yRC in the recycle gas and the kinetic parameters E and λ. 2. Pick a value of recycle flow rate FR (to be optimized). 3. Calculate the required inlet reactor temperature from an energy balance. 4. Calculate the inlet molar flow rates to the reactor. 5. Integrate down the length of the reactor until the temperature is 500 K. This also corresponds to producing the desired amount of product (0.12 kmol/s). This gives reactor size and the amount of catalyst Wcat. required for the selected recycle flow rate. 6. Evaluate the total annual cost (TAC) of the process for the recycle flow rate and reactor size. See the Appendix. 7. Vary the recycle flow rate FR and repeat steps 3-6 until the minimum TAC is obtained. Figure 6 shows typical results of this optimization procedure. These results are for the base-case conditions with an activation energy E ) 69 710 kJ/kmol and heat of reaction λ ) -23 237 kJ/kmol. Reactor size and cost decrease as the recycle flow rate increases, but compressor cost and energy cost increase. Therefore an optimum recycle flow rate exists that corresponds to an optimum reactor size. Table 3 gives the optimum design parameters for the base case. The effect of increasing yRC (lower reactant concentrations in the reactor) is to increase the total annual cost. The optimum recycle flow rate is increased slightly. The size of the reactor increases. Figure 7 gives results for the case where the activation energy is increased by 20%. The value of the specific

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Ind. Eng. Chem. Res., Vol. 39, No. 7, 2000

Figure 7. Optimization +20% E.

Figure 6. Optimization base case. Table 3. Optimum DesignssBase Case and Worst Case λ (kJ/(kmol of C)) E (kJ/kmol) FR (kmol/s) yRC (mol fraction of component C in recycle) Wcat. (103 kg) Tin (K) Tout (K) FC,spec (kmol of C/s) TAC (106 $/year) reactor gain KR

base case

worst case

-23 237 69 710 2.4 0

-23 237(1.9) 69 710(1.9) 5.7 0

35.8 469.8 500 0.12 2.34 2.92

51.3 474.5 500 0.12 3.96 6.34

reaction rate at 500 K is kept constant at 3.3088 × 10-8 kmol‚s-1‚(kg of cat.)-1‚bar-2 as E is changed by backcalculating a new preexponential factor R. The effect of increasing E on the optimum design is to increase the optimum values of TAC and the recycle flow rate. This occurs because the larger increase in reaction rates with temperature requires that the reactor inlet temperature be higher for the same exit temperature. For the same production rate, this requires a higher recycle flow rate. These optimum designs are obtained over a range of values of the parameters E, λ, and yRC. Then reactor gains are calculated as described below.

5. Reactor Gain Once the steady-state design is obtained, the reactor gain is easily obtained by considering the reactor in isolation and making a small change (1 K) in inlet temperature. Integrating down the length of the reactor from W ) 0 to W ) Wcat. for that particular case gives a new exit temperature. Then the reactor gain KR is calculated.

KR ) ∆Tout/∆Tin

(13)

Figure 8 shows the effect of the parameters E, λ, and yRC on reactor gain. Each case is an optimum steadystate economic design for the selected parameter values. Each of these parameters has a significant impact on reactor gain. As less reactant is available (increasing yRC), reactor gain decreases. Higher activation energies and higher heats of reaction lead to higher reactor gains. The base-case reactor gain is KR ) 2.92 when yRC ) 0. As a worst-case condition, if both the activation energy and the heat of reaction are increased by 90%, the resulting reactor gain is KR ) 6.34. The optimum design of this case is given in Table 3. A large recycle flow and a large reactor are required. It is interesting to note that for this worst case, the 1 K increase in Tin (from 474.5 to 475.5 K) produces a change in Tout from 500 to 506.3 K. The corresponding

Ind. Eng. Chem. Res., Vol. 39, No. 7, 2000 2389 Table 4. Parameter Values for HDA Process reactor effluent

flow rate (lb-mol/h) temp (°F) composn (mole fraction) H2 CH4 benzene toluene diphenyl reactor vol (ft3) a

reactor inlet

Luyben et al.

this papera

4382 1150

4382 1263

4382 1263

0.4291 0.4800 0.0053 0.0856 0

0.3644 0.5463 0.0685 0.0193 0.0015 4066

0.3653 0.5456 0.0675 0.0208 0.0018 4066

Using reduced heat capacities.

Table 5. Heat Capacity Data (HDA) cp at 1500 °F (ideal gas; Btu/(lb-mol‚°F)) H2 CH4 benzene toluene diphenyl

Figure 8. Effect of yRC, E, and heat of reaction on KR.

change in the production rate of C in the reactor is from 0.12 to 0.145 kmol/s. Thus a 1 K change in reactor inlet temperature results in a 20% increase in the rate of production. This high sensitivity illustrates the need for very good inlet temperature control. 6. HDA Process In this section we calculate reactor gains for the adiabatic tubular reactor in the HDA (hydrodealkylation) process, which has been extensively studied by Douglas.10 There are two reactions: the first major reaction is irreversible

toluene + H2 f benzene + CH4 and the second minor reaction is reversible.

2benzene h diphenyl + H2 Detailed kinetic parameters and flow sheet design values are given in Douglas10 and in Luyben et al.11 The first reaction rate (lb-mol‚min-1‚ft-3) is given by

r1 ) 3.6858 × 106PTPH0.5e-25 616/T

(14)

The second reaction rate is given by

r2 ) 5.987 × 104PB2e-25 616/T - 2.553 × 105PDPBe-25 616/T (15) where T is in kelvin and the partial pressures of the components are in psia. The reactor operates adiabatically at 500 psia, 1150 °F inlet, and 1263 °F outlet. The heat of reaction of the first reaction is 21 500 Btu/lbmol and essentially zero for the second. Table 4 gives the reactor inlet and exit conditions and values of reactor design parameters. The steady-state composition and temperature profiles down the length of the reactor are found by integrating the ordinary differential equations from the five component balances and one energy balance. The boundary conditions at the inlet are volume V ) 0 and T ) 1150 °F. The total volume is 4066 ft3. Note that there is a large excess of hydrogen in the reactor. Toluene is the limiting reactant. It enters at a concentration of about 8 mol % and

a

7.34 16.1 47.7 59.2 94.6a

Estimated.

leaves at a concentration of about 2 mol % under the base-case conditions. Average heat capacities taken from the literature12 were initially assumed for all components. See Table 5. However, the resulting reactor exit temperature predicted was 1224 °F instead of the 1263 °F reported by Douglas10 and Luyben et al.11 The values of the heat capacities had to be reduced by about 25% to attain the higher exit temperature. An alternative would be to adjust the heat of reaction. Figure 9 gives results of reactor gains for the HDA reactor over a range of inlet toluene concentrations. The concentrations of hydrogen and methane were adjusted for each toluene feed concentration to keep the same H2/CH4 ratio and the same benzene concentration (0.0053 mole fraction) in the feed stream. Different magnitude changes in inlet temperature were used to calculate the reactor gain, and the effect of this parameter is also shown in Figure 9. Very small ∆T’s give small changes in Tout and result in unreliable gain calculation due to numerical accuracy limitations. At low toluene feed concentrations, reactor gain is small because there is little reactant present to fuel additional reaction. At high toluene feed concentrations, reactor gain is also small because the adiabatic temperature has become so high that essentially all of the toluene has been consumed and there is none left for additional reaction when inlet temperature is increased. 7. Allyl Chloride Process As our second real reaction process, we consider the chlorination of propylene. Complete kinetic and physical property data are provided by Smith.13 There are two parallel reactions, one giving allyl chloride and one giving 1,2-dichloropropane.

Cl2 + C3H6 f CH2dCH-CH2Cl + HCl with reaction rate R1

Cl2 + C3H6 f CH2Cl-CHCl-CH3 with reaction rate R2, where the reaction rates (lb-mol‚ h-1‚ft-3) are given by

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Ind. Eng. Chem. Res., Vol. 39, No. 7, 2000 Table 6. Heat Capacity Data (Allyl Chloride) cp (Btu/(lb-mol‚°F)) C3H6 Cl2 HCl allyl chloride dichloropropane

25.3 8.6 7.2 28.0 30.7

Figure 10. Effect of reactor length (allyl chloride).

left for further reaction when the reactor inlet temperature is increased. Therefore reactor gains decrease. For a very long reactor, in which the limiting reactant is completely converted, the reactor gain becomes one. 8. Conclusion

Figure 9. Effect of feed toluene concentration (HDA).

R1 ) 206 000PCl2PC3e-27 200/(RT)

(16)

R2 ) 11.7PCl2PC3e-6860/(RT)

(17)

where component partial pressures Pj are in atmospheres and temperature is in °R. The heats of reaction are λ1 ) -48 000 Btu/lb-mol and λ2 ) -79 200 Btu/lbmol. The adiabatic tubular reactor is fed 0.85 lb-mol/h of a 80/20 molar ratio of propylene to chlorine. The inlet temperature is 392 °F, and pressure is constant at 29.4 psia. The reactor inside diameter is 2 in. Heat capacity data are given in Table 6. Figure 10 gives results for different reactor lengths. For small reactor lengths, little reaction occurs and the chlorine concentration changes little from the inlet feed (20 mol %). There is little temperature rise from the inlet 392 °F. Reactor gain values are low. As reactor length increases, there is more reaction, higher exit temperatures, and lower exit chlorine concentrations. Reactor gain increases, reaching a maximum of 5.5 at a reactor length of 15 ft. Note that there is still quite a bit of chlorine left under these conditions (7 mol %). As reactor length continues to increase, the exit chlorine concentration becomes so low that there is little

The effects of several kinetic, operating and design parameters have been explored. Large concentrations of reactants, large heats of reaction, and large activation energies result in large reactor gains, which imply more difficult dynamic control problems. Whenever a FEHE/reactor process is designed, it is important to include a calculation of the reactor gain as part of the design analysis. Large gain values should be avoided. Since the kinetic parameters are given for a specific reaction system, the designer can only change reactant concentrations and per-pass conversion to reduce reactor gains. Either low conversion or very high conversion result in low reactor gains. If the economics of yield and conversion strongly favor conditions that give high reactor gains, the feed preheating system must be very carefully designed in order to avoid control problems. The use of both heatexchanger bypassing (to avoid runaways) and furnace heat input (to avoid quenching) may be required to handle dynamic disturbances. Nomenclature A ) reactant component B ) reactant component C ) product component cPj ) heat capacity of component j (kJ‚kmol-1‚K-1) DR ) reactor diameter (m) E ) activation energy (kJ‚kmol-1) F ) flow rate in reactor (kmol/s) FC ) production rate (kmol of C/s) Fj ) flow rate of component j (kmol/s) Fo ) fresh feed flow rate (kmol/s) FoA ) fresh feed flow rate of reactant A (kmol/s)

Ind. Eng. Chem. Res., Vol. 39, No. 7, 2000 2391 FoB ) fresh feed flow rate of reactant B (kmol/s) FR ) recycle flow rate (kmol/s) GC ) feedback controller transfer function GCP ) transfer function of coupled exchanger/reactor process GR ) reactor transfer function k ) specific reaction rate K1 ) heat exchanger gain K2 ) heat exchanger gain Kmax ) maximum controller gain for given reset time Kmin ) minimum controller gain for given reset time KR ) reactor gain Ku ) ultimate gain L ) liquid flow rate leaving separator drum (kmol/s) LR ) length of reactor (m) MWj ) molecular weight of component j (kg/kmol) P ) total pressure (bar) Pj ) partial pressure of component j (bar, psia, or atm) Pu ) ultimate period (h) P1/P2 ) suction and discharge pressures of compressor (bar) R ) perfect gas law constant (bar‚m3‚kmol-1‚K-1) RC ) rate of production of C (kmol of C/s) R1 ) reaction rate of allyl chloride (lb-mol‚h-1‚ft3) R2 ) reaction rate of dichloropropane (lb-mol‚h-1‚ft3) T ) temperature in reactor (K or °R) TAC ) total annual cost ($/year) T1 ) compressor suction temperature (K) Tin ) reactor inlet temperature (K) Wcomp ) compressor work (kJ/kmol) Wcat. ) weight of catalyst (kg) yRC ) composition or recycle gas (mole fraction of component C) Greek Letters R ) preexponential factor λ ) heat of reaction (kJ/(mol of C produced)) γ ) ratio of heat capacities τI ) reset time constant τm ) temperature measurement lag τR ) reactor time constant ωr ) resonant frequency (rad)

AppendixsEconomics and Sizing The optimization of this process involves finding the values of reactor size (Wcat. of catalyst), recycle flow rate (FR), and reactor inlet temperature (Tin). The objective function to be minimized is the total annual cost (TAC, 106$/year), which is the sum of the annual capital cost (reactor, catalyst, and compressor capital investment divided by a payback period of 3 years) and energy cost of driving the compressor.

TAC ) energy +

reactor + compressor + catalyst payback period (A1)

Compressor energy cost is calculated as a function of the recycle flow rate from the equation for reversible adiabatic compression of an ideal gas (kJ/kmol).

Wcomp )

[( )

γRT1 P2 γ - 1 P1

(γ-1)/γ

]

-1

(A2)

The ratio of heat capacities γ is assumed to be 1.312. Compressor suction temperature is 313 K. Suction pressure is 45 bar, and discharge pressure is 50 bar. The annual energy cost of compression, assuming $0.07/ kW-h and 75% efficiency, is

Energy(106$/year) ) 0.227FR

(A3)

where FR is in kilomoles per second. The capital costs of the compressor and reactor vessel are

compressor cost(106$) ) 0.345(FR)0.82 reactor vessel cost(106$) ) 0.035(DR)1.066(LR)0.802

(A4) (A5)

with reactor diameter DR and length LR in meters. An aspect ratio of LR/DR ) 10 is assumed for the reactor vessel to give reasonable pressure drop, and its cost is twice that of a plain pressure vessel. The cost of the catalyst is assumed to be $100/kg. Literature Cited (1) Douglas, J. M.; Orcutt, J. C.; Berthiaume, P. W. Design and control of feed-effluent, exchanger-reactor systems. Ind. Eng. Chem. Fundam. 1962, 1, 253. (2) Anderson, J. S. A practical problem in dynamic heat transfer. Chem. Eng. 1966, 97. (3) Silverstein, J. L.; Shinnar, R. Effect of design on the stability and control of fixed-bed catalytic reactors with heat feedback. 1. Concepts. Ind. Eng. Chem. Process Des. Dev. 1982, 21, 241. (4) Tyreus, B. D.; Luyben, W. L. Unusual dynamics of a reactor/ preheater process with deadtime, inverse response and openloop instability. J. Process Control 1992, 3 (4), 241. (5) Luyben, W. L. External and internal openloop unstable processes. Ind. Eng. Chem. Res. 1998, 37, 2713. (6) Bilden, C. S.; Dimian, A. C. Stability and multiplicity approach to the design of heat-integrated PFR. AIChE J. 1998, 44 (12), 2703. (7) Reyes-DeLeon, F.; Luyben, W. L. Steady-state and dynamic effects of design alternatives in heat-exchanger/furnace/reactor processes. Ind. Eng Chem. Res., submitted for publication. (8) Jones, W. E.; Wilson, J. A. An introduction to process flexibility; Part 2: Recycle loop with reactor. Chem. Eng. Educ. 1998, 32, 224. (9) Tyreus, B. D.; Luyben, W. L. Dynamics and control of recycle systems. 4. Ternary systems with one or two recycle streams. Ind. Eng. Chem. Res. 1993, 32, 1154. (10) Douglas, J. M. Conceptual Design of Chemical Processes; McGraw-Hill: New York, 1988. (11) Luyben, W. L.; Tyreus, B. D.; Luyben, M. L. Plantwide Process Control; McGraw-Hill: New York, 1999. (12) Luyben, W. L.; Wenzel, L. A. Chemical Process Analysis; Prentice-Hall: Englewood Cliffs, NJ, 1988. (13) Smith, J. M. Chemical Engineering Kinetics, 3rd ed.; Ed., McGraw-Hill: New York, 1981; p 229.

Received for review June 28, 1999 Revised manuscript received April 6, 2000 Accepted April 10, 2000 IE990471Y