Dynamic Comparison of Alternative Tubular Reactor Systems

Ind. Eng. Chem. Res. 2004, 43, 1003-1029

1003

Dynamic Comparison of Alternative Tubular Reactor Systems Phisit Jaisathaporn† and William L. Luyben* Process Modeling and Control Center, Department of Chemical Engineering, Lehigh University, Bethlehem, Pennsylvania 18015

Direct quantitative comparisons of the dynamic controllability of four alternative tubular reactor systems are given in this paper. An exothermic, irreversible gas-phase reaction, A + B f C, is carried out in a packed-bed tubular reactor in a process that includes reaction, separation, feed preheating, and gas recycle. The four alternative tubular reactor designs are (1) a single-stage adiabatic reactor, (2) multistage adiabatic reactors with interstage cooling, (3) multistage adiabatic reactors with cold-shot cooling, and (4) a single-stage cooled reactor. All of the adiabatic systems are open-loop unstable (exhibit limit-cycle behavior) because of the sensitivity of the first reactor, which propagates thermal waves through the system. The cooled reactor system is open-loop stable for base-case parameters. The single-stage adiabatic reactor and the multistage adiabatic reactor system with intermediate cooling are closed-loop stable. However, the optimal seven-bed multistage adiabatic reactor system with cold-shot cooling is closed-loop unstable because of the severe disturbances generated by manipulating cold-shot flows. If the number of beds is reduced to three, the system is closed-loop stable. The cooled reactor is the most controllable. 1. Introduction A previous paper1 provided a quantitative comparison of the steady-state economic optimal design of several alternative tubular reactor systems. This paper studies the dynamic controllability of four systems: (1) a singlestage adiabatic reactor; (2) multistage adiabatic reactors with interstage cooling; (3) multistage adiabatic reactors with cold-shot cooling; (4) a single-stage cooled reactor. An exothermic, irreversible gas-phase reaction, A + B f C, is carried out in packed-bed tubular reactors. The reactors are part of a plantwide system that includes reaction, separation, feed preheating, compression, and gas recycle. Many papers have appeared in the literature that discuss the dynamics and control of tubular reactors in isolation.2-10 Only a handful of papers have dealt with the dynamics and control of tubular reactors in a plantwide environment. Design and control of feedeffluent exchanger/reactor systems were reported by Douglas et al.11 Tyreus and Luyben12 discussed the inverse response, dead time, and open-loop instability of a reactor coupled with a preheater. The chaotic behavior of a similar system was reported by Bilden and Dimian.13 Plantwide systems with single-stage adiabatic tubular reactors have been studied recently in several papers.14-21 Stephens22 investigated a methanol plant consisting of four adiabatic catalyst beds with cold-shot cooling and showed that the exit temperature control failed but the inlet temperature control was successful. Luyben23 discussed the effect of design and kinetic parameters on the control of cooled tubular reactor systems. * To whom correspondence should be addressed. Tel.: 610758-4256. Fax: 610-758-5057. E-mail: [email protected]. † Current address: Department of Chemical and Process Engineering, King Mongkut’s Institute of Technology North Bangkok, 1518 Piboolsongkram Road, Bangkok 10800, Thailand. Tel.: 66-2-913-2500 ext 8230. Fax: 66-2-587-0024. E-mail: [email protected].

We compare the dynamics and control of the four systems considered in the previous paper.1 The four optimal flowsheets of alternative tubular reactor systems are shown in Figure 1. These alternative designs use different reactor configurations, but the separation, recycle, and preheating sections are essentially identical. Steady-state conditions, equipment sizes, and design parameters are given in these figures for the optimal economic steady-state designs. The cold-shot design for three beds is shown, but a seven-bed design is optimal. As we will show, the seven-bed process is uncontrollable. 2. Dynamic Model The reactor is modeled by three partial differential equations: component balances on A and B (eqs 1 and 2) and an energy balance (eq 3 for an adiabatic reactor or eq 4 for a cooled reactor). The overall heat-transfer coefficient U in the cooled reactor in eq 4 is calculated by eq 5 and is a function of the Reynolds number Re (eq 6). Equation 7 is used for pressure drop in the reactor using the fiction factor f given in eq 8. The dynamics of the momentum balance in the reactor are neglected because they are much faster than the composition and temperature dynamics. A constant mass flow through the reactor is assumed. The reaction rate r′C is based on the volume of the reactor and is calculated by eq 9.

∂yA (1 - yA)r′C ∂yA ) -v ∂t ∂z C

(1)

∂yB (1 - yB)r′C ∂yB ) -v ∂t ∂z C

(2)

∂T ∂T ) -cFv - λr′C ∂t ∂z

(3)

Fcatccat Fcatccat

4U ∂T ∂T (T - Tst) (4) ) -cFv - λr′C ∂t ∂z Dtube

10.1021/ie030434d CCC: $27.50 © 2004 American Chemical Society Published on Web 01/20/2004

1004 Ind. Eng. Chem. Res., Vol. 43, No. 4, 2004

Figure 1. (A) Optimized flowsheet of a single-stage adiabatic reactor system. (B) Optimized flowsheet of a multistage adiabatic reactor system with interstage cooling. (C) Optimized flowsheet of a multistage adiabatic reactor system with cold-shot cooling. (D) Optimized flowsheet of a cooled tubular reactor system.

Ind. Eng. Chem. Res., Vol. 43, No. 4, 2004 1005

U ) 0.01545 +

0.6885 × 10-6 Re Dp

Table 1. Process Parameters

(5)

Re ) DpvF/µ

(6)

dP -fFv2 ) dz D × 105

(7)

p

f)

1- 1- 1.75 + 150 3 Re

(

)

r′C ) RFcate-E/RTyAyBP2

(8) (9)

where ) voidage of the bed, v ) gas superficial velocity (m s-1), C ) gas concentration (kmol m-3), c ) heat capacity of the gas (kJ kg-1 K-1), Dtube ) diameter of the reactor tube (m), and DP ) catalyst-pellet diameter (m). See the Nomenclature section for a complete list. Table 1 lists parameters used. The system pressure P is computed from a total material balance on all of the units in the gas loop (eq 10 with average temperature Tav ) 500 K and average molecular weight Mav ) 20 kg kmol-1). The gas volume Vgas is fixed and includes the gas volumes of the FEHE, furnace, reactor, condenser, and flash drum. The diameter of tubes in all heat exchangers is 1 in. Equal volumes on the shell and tube sides are assumed. The volume of the furnace is computed from the furnace heat duty and is 14 ft3 (106 Btu/h)-1.24 The diameter of the flash drum is calculated using an F factor of 0.79 (in SI units) and a height-to-diameter ratio of 2. Table 2 lists equipment gas volumes for the four optimal alternative tubular reactor systems.

dP RTav(MAF0A + MBF0B - MCLC) ) dt VgasMav

R ) 0.190 38 kmol s-1 bar-2 (kg of catalyst)-1 E ) 69 710 kJ kmol-1 λ ) -23 237 kJ kmol-1 DP ) 0.003 m Fcat ) 2000 kg m-3 ) 0.4 cPA ) 30 kJ kmol-1 K-1 cPB ) 40 kJ kmol-1 K-1 cPC ) 70 kJ kmol-1 K-1 cg ) 2 kJ kg-1 K-1 ccat ) 0.5 kJ kg-1 K-1 MA ) 15 kg kmol-1 MB ) 20 kg kmol-1 MC ) 35 kg kmol-1

F0A ) 0.12 kmol s-1 F0B ) 0.12 kmol s-1 T0 ) 313 K LC ) 0.12 kmol s-1 PR ) 50 bar TS ) 313 K ∆THX ) 25 K ∆PHX ) 0.5 bar µ ) 1.8 kg m-1 s-1 γ ) 1.312 UFEHE ) 0.142 kJ s-1 m-2 K-1 UHX ) 0.227 kJ s-1 m-2 K-1

Table 2. Equipment Gas Volume

one-bed adiabatic

equipment FEHE furnace interstage HX reactor condenser flash drum total (m3)

15.38 1.37 15.50 3.80 24.09 60.14

three-bed adiabatic interstage HX 6.75 0.59 2.29 15.10 1.59 3.61 29.93

three-bed adiabatic cold-shot

cooled

7.77 0.69

11.93 0.88

8.69 2.81 11.64 31.60

6.27 1.37 2.87 23.32

multistage adiabatic reactor system. Heat is transferred from the hot gas to saturated water on the shell side generating steam at temperature Tst. The overall heattransfer coefficient UHX is constant and is equal to 0.227 kJ s-1 m-2 K-1. The heat-transfer area per volume AHX/ VHX is 157 m2 m-3.

∂T ∂T UHXAHX(Tst - T) ) -v + ∂t ∂z FcgVHX

(14)

(10)

The FEHE is assumed to be a single-pass, countercurrent shell-and-tube design. Three partial differential equations are used for the temperatures of gas on the tube side, gas on the shell side, and the tube metal: eqs 11-13, respectively. The overall heat-transfer coefficients on both the tube and shell sides, Ut and Us, are constant and are equal to 0.284 kJ s-1 m-2 K-1. Equal heat-transfer area per volume is assumed for the shell and tube sides (At/Vt ) As/Vs) and is 157 m2 m-3 based on a 1-in. tube diameter. The density of the tube material Fm is 7700 kg m-3. The heat capacity of the tube cm is 0.5 kJ kg-1 K-1. The thickness of the tube thm is 3.048 × 10-3 m.

∂Tt UtAt(Tm - Tt) ∂Tt ) -vt + ∂t ∂z FtcVt

(11)

∂Ts ∂Ts UsAs(Ts - Tm) ) -vs + ∂t ∂z FscVs

(12)

∂Tm Us(Ts - Tm) - Ut(Tm - Tt) ) ∂t Fmcmthm

(13)

Equation 14 is used for the temperature of gas in the tubes of the intermediate heat exchanger used in the

The system with ordinary differential equations and partial differential equations (PDEs) is programmed in MATLAB. The method of lines is used to solve the PDEs. The partial derivatives in z are computed by using a first-order finite-difference approximation. The number of grid points in z used for the reactor in the single-stage configurations is 51. The number of grid points in z used for each reactor in multistage configurations is 31. The number of grid points in z used for each heat exchanger is 21. MATLAB’s ode23 is used for numerical integration. The computing times (on a PC equipped with the Intel Pentium IV 2.4 GHz processor) for 1-h of closed-loop simulation for the single-stage adiabatic reactor system, the multistage adiabatic reactor system with interstage cooling, the multistage adiabatic reactor system with cold-shot cooling, and the cooled reactor system are 9.4, 54.9, 19.8, and 10.2 min, respectively. Note that we assume a plug-flow tubular reactor in which there are no radial gradients and in which axial diffusion and conduction are assumed negligible. 3. Control Structures Figure 2 shows the control structures used for the four alternative tubular reactor systems. In parts B and C of Figure 2 (the multistage adiabatic systems with


Figure 2. (A) Control structure of a single-stage adiabatic reactor system. (B) Control structure of a multistage adiabatic reactor system with interstage cooling. (C) Control structure of a multistage adiabatic reactor system with cold-shot cooling. (D) Control structure of a cooled tubular reactor system.

interstage cooling and cold-shot cooling), only two reactor sections are shown so as not to clutter the diagram too much. The remaining reactor sections have the same control structures.

All of these systems have some common control loops. The system pressure is controlled by manipulating the fresh feed of A. The concentration controller with ratio control is used to control inlet gas composition by

Ind. Eng. Chem. Res., Vol. 43, No. 4, 2004 1007 Table 3. Controller Tuning for a Single-Stage Adiabatic Reactor System ZN tuning

Table 6. Controller Tuning for a Three-Bed Adiabatic Reactor System with Cold-Shot Cooling

TL tuning

ZN tuning

TL tuning

control loop

KC

τI (min)

KC

τI (min)

control loop

KC

τI (min)

KC

τI (min)

P, F0A Tmix, FB T1, QF

0.83 -2.76 11.27

2.35 0.99 2.56

0.57 -1.90 7.75

6.19 2.62 6.77

P, F0A Tmix, FB T1, QF T2, F2 T3, F3

0.53 -2.70 11.26 -7.21 -7.96

2.18 0.99 2.56 1.01 1.01

0.36 -1.85 7.74 -4.95 -5.47

5.75 2.61 6.77 2.67 2.67

Table 4. Controller Tuning for a Three-Bed Adiabatic Reactor System with Interstage Cooling ZN tuning control loop P, F0A Tmix, FB T1, QF T2, Tst,2 T3, Tst,3

KC 0.37 -2.82 11.24 5.19 5.80

τI (min) 2.40 0.99 2.56 1.06 1.05

TL tuning KC

ZN tuning

τI (min)

0.25 -1.94 7.73 3.57 3.99

6.32 2.62 6.77 2.79 2.78

Table 5. Controller Tuning for a Seven-Bed Adiabatic Reactor System with Cold-Shot Cooling ZN tuning

Table 7. Controller Tuning for a Cooled Reactor System

TL tuning

control loop

KC

τI (min)

KC

τI (min)

P, F0A Tmix, FB T1, QF T2, F2 T3, F3 T4, F4 T5, F5 T6, F6 T7, F7

0.44 -2.65 11.28 -13.02 -12.47 -13.20 -13.77 -14.36 -15.00

2.09 0.99 2.56 1.01 1.05 1.06 1.06 1.06 1.06

0.25 -1.94 7.75 -8.95 -8.57 -9.08 -9.47 -9.87 -10.31

6.32 2.62 6.77 2.66 2.77 2.79 2.79 2.79 2.80

manipulating the fresh feed of B. Bypassing around the FEHE is used to control the gas mixture temperature Tmix. The reactor inlet temperature T1 is controlled by

TL tuning

control loop

KC

τI (min)

KC

τI (min)

P, F0A Tmix, FB T1, QF Tpeak, Tst

0.40 -5.24 11.22 2.67

2.22 1.00 2.57 2.76

0.27 -3.60 7.71 1.84

5.86 2.63 6.78 7.28

manipulating the furnace heat input. The setpoints of these two controllers are the same, and the controller output signals are split-ranged so that bypassing and furnace heat input cannot occur simultaneously. The liquid level in the flash drum is controlled by manipulating the liquid product. A temperature controller manipulates heat removal in the condenser to hold the drum temperature constant. The rotation speed of the compressor is controlled, which fixes the recycle gas flow rate. The multistage adiabatic reactor system with interstage cooling has a controller on the inlet temperature of the second reactor, as shown in Figure 2B. The temperature is controlled by manipulating the steam pressure in the steam-generating heat exchanger. The level in the steam generator is controlled by manipulating the boiler feedwater (BFW) makeup flow.

Figure 3. Open-loop response of a single-stage adiabatic reactor system.


Figure 4. (A) Results for +2.8 K T1 (single-stage adiabatic). (B) Results for -3.7 K T1 (single-stage adiabatic).

Figure 2C shows the control structure of the multistage adiabatic reactor system with cold-shot cooling for the two-reactor case. The inlet temperature of the second reactor T2 is controlled by manipulating the coldshot flow rate F2. Note that changing the cold-shot flow F2 produces an opposite change in the gas flow fed to the first reactor F1 because the total recycle flow rate

FR is constant. For the multibed systems, the inlet temperature Tn of each bed is controlled by manipulating the cold-shot flow at the inlet of that bed Fn. Changing any of these flows affects the flow to the first reactor. The effect of this interaction on controllability is important and will be discussed in the next section. It represents a basic problem with cold-shot cooling


Figure 5. (A) Results for +22% FR (single-stage adiabatic). (B) Results for -13% FR (single-stage adiabatic).

when the total gas recycle flow rate is fixed, which is the case in most practical systems because of compressor limitations. Figure 2D shows the control structure of the cooled reactor system. Five temperature transmitters at different axial locations are used in the reactor. One of the transmitters is located where the steady-state peak

temperature occurs. Two other transmitters are placed upstream and two downstream of this peak location. The distance between each transmitter is 10% of the length of the reactor. A high selector receives the five temperature transmitter signals and sends the highest signal to a peak temperature controller. The controller controls the peak temperature by manipulating the


steam pressure (or, equivalently, the steam temperature Tst). The liquid in the shell side of the reactor is controlled by manipulating the liquid BFW flow. Perfect control is assumed for the composition/ratio controller, level controllers, gas condenser temperature controller, and speed controller. Our study is limited to base-level regulatory control structures using conventional proportional-integral (PI) controllers. The application of more advanced control techniques is not considered in this paper. 4. Controller Tuning and Disturbances Ziegler-Nichols (ZN) and Tyreus-Luyben (TL) PI tunings are used. Ultimate gain and frequency are obtained by performing relay-feedback tests. Temperature control loops have three 20-s lags. The pressure control loop has two 30-s lags. There is a 120-s lag in the furnace heat input. The pressure transmitter span is 10 bar. Temperature transmitter spans are 100 K. The furnace is capable of increasing the inlet gas temperature by 30 K. Tables 3-7 give the controller tuning parameters used in the various systems. The open-loop behaviors of the alternative systems are investigated by using (20% step changes in the recycle flow rate FR. In the open-loop tests, the pressure controller, the temperature controllers, and the composition controller are all on manual. In the closed-loop tests, all controllers are on automatic. The disturbances in the adiabatic reactor systems are setpoint changes in the inlet temperature and changes in the recycle flow rate FR. The cooled reactor system has the same disturbances with an additional disturbance in the peak temperature setpoint change. All closed-loop disturbances are assumed to enter the system as 5-min ramps from the initial steady-state value to a new value. The temperature setpoint change or recycle flow-rate change is made such that the production rate is increased or decreased by 25%. Because each reactor has a different gain, ∆Tout/∆Tin, setpoint changes in the inlet temperatures of multiple adiabatic reactors are made such that the reactor exit temperature of each reactor reaches the same temperature when the new steady-state conditions are attained. 5. Results 5.1. Single-Stage Adiabatic Reactor System. 5.1.1. Open-Loop Response. The open-loop responses of a single-stage adiabatic tubular reactor system to (20% step changes in the recycle flow rate FR are shown in Figure 3. The solid lines are increases in the recycle flow, and the dashed lines are decreases. The results show that the system produces limit-cycle behavior, alternating between high and low temperatures. This type of dynamic response is called “open-loop unstable” behavior in this paper. When the recycle flow rate is increased, the temperatures throughout the reactor decrease (see Tout in Figure 3) because a higher flow rate through the reactor provides more thermal sink for the heat being generated

Figure 6. Open-loop response of a three-bed adiabatic reactor system with interstage cooling (+20% FR).

by reaction, so the adiabatic temperature rise is smaller. The reaction rate decreases, and the pressure starts to increase. This gradual increase in pressure eventually starts to increase the reaction rate (at about 40 min). The higher reaction rate increases the reactor temperatures, and the reactor exit temperature goes above 500 K at about 50 min. The pressure then decreases because of the high reaction rate. The system finally exhibits limit-cycle behavior with a period of about 12 min and with very large changes in temperature and pressure. A decrease in the recycle flow rate causes an increase in the reactor temperature and results in a similar oscillating (limit-cycle) behavior. The fundamental reason for the open-loop instability is the positive feedback between the adiabatic reactor (which has a large reactor gain, ∆Tout/∆Tin) and the FEHE. The limit-cycle behavior results from the nonlinearity of the system. 5.1.2. Closed-Loop Response. Figure 4A shows the response of the closed-loop system to a change in the setpoint of the reactor inlet T1 temperature controller. The setpoint is ramped up by 2.8 K over 5 min. The TL tuning constants (dashed lines) give oscillatory responses, but ZN tuning (solid lines) gives tight control and no oscillation. Oscillations occur with loose tuning because the high reactor gain and the positive feedback of heat require a minimum value of the controller gain


Figure 7. Results for increases in inlet temperatures (adiabatic with interstage cooling).

to stabilize the system. As shown earlier, the system with zero controller gains (open loop) is unstable and oscillates. The 2.8 K increase in the inlet temperature increases the production rate by 25% as a result of the increase in the reactor temperatures. Figure 4B shows the response to a ramped decrease in the reactor inlet temperature of 3.7 K over 5 min. The system with ZN tuning has tighter control and is less oscillatory than the system with TL tuning. The production rate is decreased by 25% as a result of the reactor temperature decrease. Figure 5A shows the response to a 22% ramped increase in the recycle flow rate FR. The production rate

is decreased by 25% as a result of the reactor temperature decrease. Figure 5B shows the response to a 13% ramped decrease in recycle. The production rate is increased by 25% as a result of the reactor temperature increase. The system oscillates with TL tuning, but ZN tuning gives tight control. These results show that aggressive tuning (ZN) is required to stabilize this open-loop unstable system. The production rate can be effectively changed by changing either the reactor inlet temperature or recycle flow. 5.2. Multistage Adiabatic Reactor System with Interstage Cooling. 5.2.1. Open-Loop Response. The open-loop response of a three-stage adiabatic reactor system with interstage cooling to a 20% increase in


Figure 8. Results for a +20% FR change (adiabatic with interstage cooling).

the recycle flow rate FR is shown in Figure 6. The reactor inlet and exit temperatures of each reactor are shown. This system with heat feedback and large reactor gains is open-loop unstable. These responses are similar to the response of a single-stage adiabatic reactor system except now the upstream reactors affect the downstream reactors. For example, the increase in the exit temperature of the first reactor at 30 min produces an increase in the exit temperature of the second reactor, which then produces an even larger increase in the exit temperature of the third reactor. Thus, these disturbances are amplified as they move down the reactor train. Similar results occur for a decrease in the recycle flow. 5.2.2. Closed-Loop Response. Figure 7 shows the response to increases in the three reactor inlet temper-

ature controller setpoints. As discussed earlier, these changes are made so that the exit temperatures of all three reactors come to the same value at the new steady state (512.4 K). The required setpoint inlet temperature changes are 4.1 K for T1, 6.3 K for T2, and 9.0 K for T3, which all enter as 5-min ramps starting at a time equal to 1 min. These changes in the inlet temperature increase the production rate by 25%. In this open-loop unstable system, ZN tuning gives tighter control than TL tuning. When the reactor inlet temperatures are decreased (4.9 K for T1, 7.1 K for T2, and 9.6 K for T3), the production rate is decreased by 25%, and the new steady-state exit temperature of all reactors is 485.6 K. Figure 8 shows results for a 20% increase in the recycle flow rate FR. The production rate is changed by


only about 10%, which is less than that observed for the single-stage adiabatic reactor. Thus, the recycle flow rate is not an effective manipulating variable for changing the production rate in this multistage adiabatic reactor system with intermediate cooling. The new steady-state exit temperatures of each reactor are not the same. The first reactor exit temperature is more sensitive to changes in the recycle flow rate. For a 20% decrease in FR, the first reactor exit temperature increases to 521 K while the second and third reactor exit temperatures only increase to 503 and 501 K. If it is desired to maintain the same exit temperatures in all reactors, the inlet temperatures would have to be adjusted. These results illustrate an inherent disadvantage of this system. The first reactor is the most sensitive to disturbances. The changes in its exit temperature will be larger than those of the other downstream reactors. 5.3. Multistage Adiabatic Reactor System with Cold-Shot Cooling. 5.3.1. Open-Loop Response. The open-loop response of the seven-bed adiabatic reactor system with cold-shot cooling to a 20% increase in the recycle flow rate FR is shown in Figure 9. The inlet temperature of the first reactor decreases because of the larger flow through the preheat system. However, the reactor inlet temperatures of the other reactors increase because the cold-shot flows are fixed. Then the first reactor inlet temperature starts to increase at 5 min because of the increase in the exit temperature of the seventh reactor. The pressure initially increases gradually because of the lower temperature in the first reactor. At about 12 min, the pressure drops rapidly because of the large increase in the temperature in the first reactor. 5.3.2. Closed-Loop Response. Figure 10 shows the response to increases in the setpoints of all reactor inlet temperatures. The setpoint change is different for each reactor, as previously discussed. We could not find any choice of tuning parameters that would stabilize this seven-bed cold-shot system. Neither the aggressive ZN tuning nor the more conservative TL tuning produced a stable closed-loop system. The changes in reactor exit temperatures keep increasing and cold-shot valves eventually saturate. There are six cold-shot valves in the system. Moving these valves causes changes in the inlet flow rate of the first reactor F1 because the total recycle flow is fixed. With many beds and many coldshot flows, the interaction between the flows and the temperatures leads to amplification of disturbances and limit-cycle behavior. There are several alternative suboptimal designs that could be considered to overcome this problem. Reducing the number of beds is explored below. A second alternative would be to design the system with a recycle flow rate that is much larger than the steady-state optimum. This would reduce the sensitivity of the first reactor (higher inlet temperature and lower reactor gain). A third alternative is to design the system for liquid cold shot. These other alternatives will be the subject of future work. It could also be claimed that some more advanced control structure may stabilize this system. We have not explored this possibility but feel that it is very unlikely.

Figure 9. Open-loop response of a seven-bed adiabatic reactor system with cold-shot cooling (+20% FR).

The propagation of thermal waves through the reactor beds is an inherent dynamic property of the system that is independent of the type of controller. The situation is similar to that discussed by Shinnar et al.9 in cooled reactor systems. If the number of reactors is reduced, the interaction is less and the system can be made closed-loop stable. This is illustrated by considering a system with three adiabatic cold-shot reactors. The optimal steady-state design of the three-stage system is studied (steady-state conditions are given in a previous paper1 and sketched in Figure 1C). The three-stage system is still open-loop unstable, as shown in Figure 11. The same amplification and limitcycle behavior as those found in the seven-stage system are observed. However, when the pressure, composition, and temperature controllers are put on automatic, the three-stage process can be made closed-loop stable.



Figure 10. (A) Results for increases in inlet temperatures (seven-bed adiabatic reactor system with cold-shot cooling). (B) Results for increases in inlet temperatures (seven-bed adiabatic reactor system with cold-shot cooling).

Figure 12 shows the response to ramped increases in the setpoints of the inlet reactor temperature controllers: 3.1 K for T1, 4.6 K for T2, and 5.8 K for T3. The production rate is increased by 25%, and the new steady-state exit temperature is 510.9 K. When the setpoints of the inlet reactor temperature controllers are reduced (4.4 K for T1, 5.9 K for T2, and 7.1 K for T3), the production rate is decreased by 25% and the new steady-state exit temperature is 487.3 K. Using ZN tuning gives oscillatory responses in the inlet temperatures, but using TL tuning gives effective

control. Remember that ZN tuning was required for the single-stage adiabatic reactor system, and it gave better response for the multistage reactor system with intermediate cooling. Those systems are open-loop unstable and require tight control to stabilize. Large changes in the steam pressure in the intermediate coolers can be tolerated because they do not affect the rest of the process. The cold-shot system is also open-loop unstable, but it cannot tolerate large changes in the cold-shot flows because of the effect on the flow to the first reactor (with


Figure 11. Open-loop response of a three-bed adiabatic reactor system with cold-shot cooling (+20% FR).

a fixed recycle flow). Therefore, ZN tuning is too aggressive and gives oscillatory results for this system. Figure 13 shows the results for +20%, -15%, and -18% changes in the recycle flow rate FR, using TL tuning. The 20% increase in recycle produces a production rate increase of about 10%. When FR is decreased by 15%, the first reactor exit temperature increases significantly to 525 K. The second and third reactor exit temperatures increase to 504 and 502.5 K, respectively. The system can handle a 15% decrease in the recycle flow, but when a larger decrease (18%) is made, as shown in Figure 13B, the system becomes unstable. The first reactor exit temperature increases when FR is decreased. The cold-shot valve controlling the inlet temperature of the second reactor opens, which decreases the inlet flow rate to the first reactor even more. After about 40 min, the cold-shot valve (controlling F2 shown in Figure 13B) is wide open, and the system loses control of the inlet temperature. These results illustrate one of the inherent problems with cold-shot systems. If the total gas recycle is fixed, as it would be in most practical situations because of compressor limitations, changing cold-shot flows produces interaction. The systems studied in this work used gas recycle. If the recycle were liquid, the cold shot could be a liquid stream. This would mean that less cold-shot flow would be required because the latent heat of vaporization would be utilized for cooling. The gas flow to the first reactor, probably coming from an upstream vaporizer, would not be affected by changes in the liquid cold-shot flows to the downstream reactors, which would come from a liquid surge tank somewhere in the system. A system with liquid recycle would naturally occur when the vapor-liquid equilibrium is such that a simple

flash drum cannot be used, and a distillation column (or some other separation unit) is required. This system would probably be more expensive from a steady-state economic standpoint because of the energy consumption in both the distillation column and the vaporizer. However, it may have superior dynamics. This system will be studied in future work. 5.4. Single-Stage Cooled Reactor System. 5.4.1. Open-Loop Response. The open-loop responses of the cooled reactor system for (20% changes in the recycle flow rate FR are shown in Figure 14. Unlike all of the adiabatic reactor systems, the cooled system is openloop stable. The internal heat-transfer provides some self-regulation so that disturbances do not grow. The reactor gain (∆Tout/∆Tin) is only about 1, compared to the adiabatic reactor with gains of 3 or 4. Cooling the reactor makes it less sensitive to changes in the inlet temperature and recycle flow rate. Increasing the recycle flow reduces the inlet, peak, and exit temperatures of the reactor. Pressure builds until the higher partial pressures of the reactants compensate for the lower specific reaction rate due to the lower temperatures. The higher velocities in the reactor tubes also increase the heat-transfer coefficient, which means the heat-transfer rate does not decrease directly with a decrease in reactor temperatures. Remember, steam pressure (and temperature) is held constant in the open-loop run. The net result of the various effects is that, with the fresh feed flow rates fixed, the reactor comes to a new steady-state condition, which has lower reactor temperatures but higher pressure. The net reaction rate and the heat transfer in the reactor remain the same. The decrease in the temperature differential is compensated for by the increase in the heat-transfer coefficient.


Figure 12. Results for increases in inlet temperatures (three-bed adiabatic reactor system with cold-shot cooling).

Decreasing the recycle flow has the opposite effects: temperatures increase, the pressure decreases, and the heat-transfer coefficients decrease. Thus, the cooled reactor system has some inherent self-regulatory properties that make it open-loop stable, at least for the set of kinetic and design parameters used in this example. In the next section, a “hot” reaction system will be discussed that has a higher activation energy and larger specific reaction rate. As we will see, this new system is open-loop unstable. 5.4.2. Closed-Loop Response. Figure 15 shows the response of the cooled reactor system to ramped increases and decreases of 10 K in the setpoint of the inlet temperature controller T1. Raising the inlet temperature

produces a decrease in the reactor exit temperature of about 2.5 K. The production rate only increases by 3%, which indicates that the inlet temperature is a poor manipulated variable for production rate changes in this system. The results shown use TL tuning because the system with ZN tuning was found to be very oscillatory. Because the system is open-loop stable, aggressive controller tuning is not required. Aggressive tuning increases the interaction between the inlet temperature controller and the peak temperature controller. Changing the steam temperature affects the temperature profile throughout the reactor, and the positive feedback



Figure 13. (A) Results for +20 and -15% FR changes (three-bed adiabatic reactor system with cold-shot cooling). (B) Results for -15 and -18% FR changes (three-bed adiabatic reactor system with cold-shot cooling).

of heat from the reactor exit to the FEHE can amplify oscillations. When the reactor inlet temperature is decreased 10 K, the production rate is decreased by only 4% and the

exit temperature increases by 1.6 K. Notice that the furnace heat input goes to zero at about 4 min, and the inlet temperature is maintained by using bypass flow around the FEHE.


Figure 14. Open-loop response of a cooled reactor system.

Figure 16 shows that the closed-loop system can handle 20% increases and decreases in the recycle flow rate FR. However, the production rate only changes about 4%, indicating that changing the recycle flow is not effective for changing the production rate. Figure 17 shows the responses to ramped changes of +10.4 and -12.1 K in the setpoint of the peak temperature controller Tpeak. The production rate is changed by 25%, which indicates that the peak temperature provides an effective production rate handle in the cooled reactor system. 5.5. Summary. Table 8 summarizes the results of these dynamic studies. All of the adiabatic reactor systems are open-loop unstable. The cooled reactor system is open-loop stable. The seven-bed adiabatic reactor system with cold-shot cooling is closed-loop unstable. Reducing the number of beds to three produces a system that can be stabilized by feedback control. The aggressive ZN tuning provides better control of the single-stage adiabatic reactor system and the adiabatic reactor system with interstage cooling. The more conservative TL tuning provides better control of the adiabatic reactor system with cold-shot cooling and the cooled reactor system. The single-stage adiabatic reactor system can achieve production rate changes by changing either the inlet temperature or the recycle flow rate. Multistage adia-

batic reactor systems can achieve production rate changes by changing inlet temperatures. The cooled reactor system can achieve production rate changes by changing the peak temperature. The cooled reactor system has the shortest settling time (15 min). The single-stage adiabatic reactor system has the longest settling time (40 min). The settling times of multistage adiabatic reactor systems with interstage cooling and cold-shot cooling are about 20 and 18 min, respectively. 6. Cooled Reactor with Hot Reaction The impact of reaction kinetics on the steady-state design and dynamic controllability of the cooled reactor system is studied in this section. All of the results presented in the previous section used a moderate activation energy (69 710 kJ kmol-1) and a moderate specific reaction rate. Now the activation energy is doubled (E ) 139 420 kJ kmol-1). In addition, the reaction rate at 500 K is increased by a factor of 4 [the preexponential factor R ) 1.46 × 107 kmol s-1 bar-2 (kg of catalyst)-1]. These changes in reaction kinetics make the system highly nonlinear and very sensitive to changes in temperature. 6.1. Steady-State Design. Table 9 shows the optimal economic steady-state design for the “hot” reactor


Figure 15. Results for (10 K T1 (cooled).

system when the catalyst cost is $100 kg-1. The important steady-state design parameters for this hot reaction system are a total catalyst weight of 11 880 kg, a recycle flow of 0.27 kmol s-1, a tube diameter of 0.0592 m, and a heat-transfer area of 401 m2. The design optimization variables used are the same as those discussed in a previous paper.1 The TAC of the optimal design is $0.77 million year-1. A comparison of this design with the moderate reaction design (Table 9) shows that the larger specific reaction rate produces a smaller reactor with less recycle, less heat-transfer area, and lower TAC. This optimal design is found to be uncontrollable, as discussed in the following section. A suboptimal design using a smaller tube diameter (0.0254 m) is also shown in Table 9. The TAC of this system is about 10% higher than the optimum. However, the modified design provides a much larger heat-transfer area (725 m2 versus 401 m2), which improves dynamic controllability. Notice that the modified design has many more tubes (2571) than the optimal design (357). It is interesting to note that this suboptimal design uses less catalyst than the optimal design. This occurs because the average temperature in the reactor is higher because the inlet temperature is higher (483.4 K versus 474.7 K). Figure 18 shows the temperature profiles in

the cooled tubular reactor for the optimal (Dtube ) 0.0592 m) and suboptimal (Dtube ) 0.0254 m) designs. 6.2. Dynamic Simulation Results. 6.2.1. OpenLoop Response. As shown in Figure 14, the cooled reactor with normal kinetics is open-loop stable. The cooled reactor with hot kinetics is open-loop unstable, as shown in Figure 19. Results for two disturbances are shown: a 20% decrease and a 20% increase in the recycle flow rate. The system exhibits limit-cycle behavior with periods that are about 20 min for an increase in the recycle flow and about 40 min for a decrease in the recycle flow. Temperatures go as high as 600 K in both cases. Let us consider the increase in the recycle flow. The higher flow initially decreases temperatures in the reactor. Pressure begins to build. Then the temperatures in the reactor start to increase because of the reaction rate increase due to higher pressure. At about 7 min, the exit temperature increases to a value slightly above its steady-state level. The higher exit temperature increases the reactor inlet temperature through the FEHE, and this starts a temperature wave that moves down the reactor. Temperature TT,2, which is located at about 40% of the way down the reactor, spikes first. Then temperature TT,3, which is located at about 50% of the way down the reactor, spikes. The wave moves


Figure 16. Results for (20% FR (cooled).

Table 8. Summary of Dynamic Simulation Results system

NR

open loop

closed loop

tuning

(25% production

settling time (min)

single-stage adiabatic staged adiabatic with intercooling staged adiabatic with cold-shot staged adiabatic with cold-shot cooled

1 3 7 3 1

unstable unstable unstable unstable stable

stable stable unstable stable stable

ZN ZN

Tin, FR Tin Tin Tin Tpeak

40 20

down the reactor and affects the exit temperature at about 17 min. Then the cycle repeats itself. Decreases in the recycle flow produce similar effects but in the reverse direction. The decrease in flow raises reactor temperatures, and a temperature wave starts to move down the reactor. Because the flow rate is lower, the temperature spike moves more slowly than when the recycle flow rate is increased. This explains the longer period of the cycles. These results demonstrate that the optimal design with the hot reaction is open-loop unstable. The openloop response of the suboptimal cooled reactor system (with smaller tubes) is shown in Figure 20. The much larger heat-transfer area makes this system open-loop stable for a 20% increase or a 20% decrease in the recycle flow (Figure 20A). Notice that the decrease produces an oscillatory response that eventually dies out. However, if the decrease in the recycle flow is 23%, the system becomes open-loop unstable, as shown in

TL TL

18 15

Figure 20B. The system cannot handle large decreases in the recycle flow because of its effect on the heattransfer coefficient U. At normal recycle flow rates, U is 0.31 kJ s-1 m-2 K-1. When the flow rate is decreased by 23%, U is about 15% lower. Control problems are expected with this system in this situation. 6.2.2. Closed-Loop Response. Table 10 gives the tuning parameters for the optimal design. There is difficulty performing relay feedback tests on this optimal design because of its high sensitivity as seen in the openloop behavior. The tuning parameters are obtained using the very early period of relay feedback tests. The system runs away if the relay feedback tests are prolonged. The closed-loop system could not be stabilized for a 5 K T1 ramped setpoint change. Table 11 gives the tuning parameters for the suboptimal design. The ZN peak temperature controller gain is 0.36 compared to 2.67 found for the cooled reactor with moderate kinetics studied in previous sections. The


Figure 17. Results for +10.4 and -12.1 K (cooled).

Figure 18. Temperature profile in the cooled reactor with a hot reaction.

low controller gain of the peak temperature controller is required because of the high sensitivity of the reactor with the hot reaction. The lower gain of the pressure controller is due to the smaller gas volume of the hot reactor system.

Figure 21 shows the responses to +10 and -10 K T1 ramped setpoint changes using TL tuning. The system with ZN tuning is unstable. The production rate is increased by only 3%. Figure 22A shows the results for a 20% increase or decrease in the recycle flow rate. The production rate is changed by no more than 4%. Figure 22B shows the response to an even larger decrease in the recycle flow (25%). The system is now at the limit of closed-loop stability. Remember that this system is open-loop unstable for a 23% decrease in recycle. The effect of flow rate on U is so large that adding a controller cannot stabilize the system. Figure 23 shows the responses to changes in the setpoint of the peak temperature controller (+6.0 and -6.8 K). The production rate is changed by 25%. The settling time of the system is more than 40 min when the peak temperature is decreased. The long settling time is the result of the low controller gain used for the peak temperature loop. 7. Conclusion The design and control of alternative tubular reactor systems have been studied. The single-stage cooled reactor system is found to be the best for both steady-


Figure 19. Open-loop response to (20% FR (optimal design with a hot reaction). Table 9. Steady-State Design Results for the Cooled Reactor System case TAC Dtube Ltube Ftube yA/yB Wcat Ntube FR Fby ∆P Ps TC ) Tin Tout XA U V DR AR

$106 year-1 m m kmol s-1 103 kg kmol s-1 kmol s-1 bar bar K K kJ s-1 m-2 K-1 m s-1 m m2

moderate optimum

hot optimum

hot suboptimum

1.57 0.1113 8.51 0.0033 1.017 31.38 190 0.38 0 0.75 46.75 477.33 495.25 0.38 0.246 0.276 2.17 564

0.77 0.0592 6.03 0.0014 1.024 11.88 357 0.27 0 1.23 46.27 474.73 494.83 0.46 0.36 0.42 1.58 401

0.84 0.0254 3.53 0.0002 1.025 9.20 2571 0.33 0 0.50 47.00 483.42 491.09 0.42 0.31 0.36 1.82 725

Table 10. Controller Tuning (Optimal Design with a Hot Reaction) ZN tuning control loop P, F0A Tmix, FB T1, QF Tpeak, Tst

TL tuning

KC

τI (min)

KC

τI (min)

0.25 -5.30 6.96 0.42

2.14 1.00 2.81 2.63

0.17 -3.64 4.79 0.29

5.64 2.64 7.42 6.93

state economics and dynamic controllability. The cooled reactor is the best from a dynamic point of view because

case AHX AC Wcomp QR QHX QC QF

m2 m2 kW 103 kW 103 kW 103 kW 103 kW

capital cost

$106 reactor catalyst compressor furnace FEHE condenser $106 year-1 compressor furnace fuel steam

operating cost

moderate optimum

hot optimum

hot suboptimum

942 217 89.46 2.40 3.35 0.61 0.15

772 178 73.26 2.43 2.74 0.50 0.09

838 197 70.61 2.64 2.98 0.55 0.34

0.90 3.13 0.17 0.15 0.63 0.72

0.72 1.19 0.15 0.10 0.55 0.64

1.06 0.92 0.14 0.28 0.58 0.68

0.05 0.02 -0.41

0.04 0.01 -0.40

0.04 0.05 -0.47

of its self-regulatory nature, particularly when plenty of heat-transfer area is used. Dynamic simulation results show that the adiabatic reactor systems are open-loop unstable because of the reactor high sensitivity and the feedback of heat. The steady-state optimal design of the cooled tubular reactor system with moderate activation energy is found to be open-loop stable. This is because cooling the reactor makes it less sensitive to changes in the inlet temperature and flow rate, which prevents it from running away.


Figure 20. (A) Open-loop response to (20% FR (suboptimal design with a hot reaction). (B) Open-loop response to -23% FR (suboptimal design with a hot reaction).


Figure 21. Results for (10 K T1 (suboptimal design with a hot reaction).

Table 11. Controller Tuning (Suboptimal Design with a Hot Reaction) ZN tuning control loop P, F0A Tmix, FB T1, QF Tpeak, Tst

TL tuning

KC

τI (min)

KC

τI (min)

0.29 -5.39 11.56 0.36

2.05 1.00 2.56 2.75

0.20 -3.71 7.95 0.25

5.42 2.64 6.76 7.25

The optimal steady-state designs of alternative systems with moderate activation energy can be stabilized by using conventional PI controllers, except for the seven-bed adiabatic reactor system with cold-shot cooling. The first reactor in all of the multistage adiabatic reactor systems is very sensitive to the flow rate and inlet temperature. In the cold-shot system, changing one of the cold-shot flows produces a change in the flow to the first bed because the total recycle gas flow rate is fixed. The seven-bed system cannot be stabilized because the design has too many cold-shot valves and moving these valves generates a disturbance in the flow rate to the first bed. A suboptimal design with three reactor beds gives good dynamic controllability. The cold-shot design with gas recycle is found to be sensitive to reduction in the recycle flow rate. The cold-

shot design with three beds cannot handle a 18% decrease in the recycle flow due to the very high exit temperature of the first reactor. The impact of the activation energy and specific reaction rate on the cooled reactor system has been studied. The optimal steady-state design is open-loop unstable and could not be stabilized by a conventional PI control structure. A suboptimal design using smaller tube diameters yields open- and closed-loop stability. However, the limitations of the analysis presented in this work must be kept in mind. We have made the plugflow assumption: no radial gradients in composition or temperature and negligible axial conduction and diffusion. For cooled reactors, in which heat must be transferred from the process inside the tubes to the coolant outside the tubes, radial temperature gradients can sometimes be significant. In addition, flow maldistribution through multiple parallel tubes in a cooled reactor can degrade both steady-state and dynamic performances. These problems do not occur in adiabatic reactors. Loading and unloading catalysts is also much more difficult in the small tubes of a cooled reactor than in the large beds of the adiabatic systems. These considerations may make the adiabatic reactor systems more practical for some chemical systems.


Figure 22. (A) Results for (20% FR (suboptimal design with a hot reaction). (B) Results for -25% FR (suboptimal design with a hot reaction).


Figure 23. Results for +6.0 and -6.8 K Tpeak (suboptimal design with a hot reaction).

Nomenclature A ) reactant component AC ) heat-transfer area in the condenser (m2) AHX ) heat-transfer area in the heat exchanger (m2) AR ) heat-transfer area in the reactor (m2) As ) heat-transfer area of the shell side of FEHE (m2) At ) heat-transfer area of the tube side of FEHE (m2) B ) reactant component C ) product component C ) gas concentration (kmol m-3) ccat ) heat capacity of the catalyst (kJ kg-1 K-1) CF ) steam-cost factor c ) heat capacity of the gas (kJ kg-1 K-1) cm ) heat capacity of the heat exchanger tube (kJ kg-1 K-1) CPJ ) heat capacity of component J (kJ kmol-1 K-1) DP ) catalyst-pellet diameter (m) DR,n ) diameter of the nth reactor or bed (m) Dtube,n ) tube diameter of the nth reactor (m) E ) activation energy (kJ kmol-1) f ) friction factor Fby ) bypass flow rate around FEHE (kmol s-1) FC,n ) flow rate of product C entering the nth reactor (kmol s-1) FEHE ) feed-effluent heat exchanger FJ ) flow rate of component J (kmol s-1) Fm ) flow rate of cold-shot stream entering the mth reactor (kmol s-1) FR ) recycle flow rate (kmol s-1) Fst ) flow rate of generated steam (kmol s-1)

Ftube,n ) inlet flow rate per reactor tube of the nth reactor (kmol s-1) F0J ) fresh-feed flow rate of component J (kmol s-1) F1 ) feed flow rate of the first reactor (kmol s-1) k ) specific reaction rate [kmol s-1 bar-2 (kg of catalyst)-1] LC ) liquid flow rate of product C leaving the separator drum (kmol s-1) LR,n ) length of the nth reactor/bed (m) Ltube,n ) length of reactor tubes in the nth reactor (m) Mav ) average molecular weight (kg kmol-1) MJ ) molecular weight of component J (kg kmol-1) NR ) number of reactors Ntube,n ) number of reactor tubes in the nth reactor P ) total pressure (bar) PA ) partial pressure of reactant A (bar) PB ) partial pressure of reactant B (bar) Pn ) pressure at the inlet of the nth reactor (bar) PR ) recycle or compressor discharge pressure (bar) PS ) compressor suction pressure (bar) Pst ) pressure of steam (bar) QC ) heat transfer in the condenser (kW) QF ) heat transfer in the furnace (kW) QHX ) heat transfer in the heat exchanger (kW) QR ) heat transfer in the reactor (kW) R ) ideal gas constant (bar m3 kmol-1 K-1) rC ) reaction rate of product C [kmol s-1 (kg of catalyst)-1] r′C ) reaction rate of product C (kmol s-1 m-3) RC,n ) rate of generation of C in the nth reactor (kmol s-1) Re ) Reynolds number T ) temperature (K) Tav ) reactor average temperature (K)

Ind. Eng. Chem. Res., Vol. 43, No. 4, 2004 1029 TCS ) temperature of cold-shot stream (K) Tin ) temperature at the reactor inlet (K) Tmax ) maximum allowable temperature (K) Tm ) temperature of the heat-exchanger tube (K) Tn ) temperature at the inlet of the nth reactor (K) Tout ) temperature at the reactor exit (K) TR ) temperature of the recycle stream (K) Ts ) temperature of gas on the shell side of FEHE (K) TS ) compressor suction temperature (K) Tst ) temperature of steam (K) Tt ) temperature of gas on the tube side of FEHE (K) TT,n ) temperature of gas in the cooled reactor measured by the nth temperature sensor (K) T0 ) temperature of the fresh-feed stream (K) thm ) thickness of heat-exchanger tube (m) U ) overall heat-transfer coefficient in the reactor (kJ s-1 m-2 K-1) UFEHE ) overall heat-transfer coefficient in the feedeffluent heat exchanger (kJ s-1 m-2 K-1) UHX ) overall heat-transfer coefficient in the heat exchanger and condenser (kJ s-1 m-2 K-1) Ut ) overall heat-transfer coefficients of the tube side of FEHE (kJ s-1 m-2 K-1) Us ) overall heat-transfer coefficients of the shell side of FEHE (kJ s-1 m-2 K-1) v ) superficial velocity (m s-1) vs ) superficial velocity of gas on the shell side of FEHE (m s-1) vt ) superficial velocity of gas on the tube side of FEHE (m s-1) Vgas ) system gas volume (m3) VHX ) gas volume in the heat exchanger (m3) Vs ) gas volume of the shell side of FEHE (m3) Vt ) gas volume of the tube side of FEHE (m3) w ) weight of the catalyst (kg) Wcat,n ) total weight of the catalyst in the nth reactor (kg) Wcomp ) compressor work (kW) Wtube,n ) weight of the catalyst per tube in the nth reactor (kg) XA ) fractional per pass conversion based on reactant A yA ) composition of reactant A at the first reactor inlet stream (mole fraction) yB ) composition of reactant B at the first reactor inlet stream (mole fraction) yAR ) composition of reactant A in the recycle stream (mole fraction) yBR ) composition of reactant B in the recycle stream (mole fraction) Greek Letters R ) preexponential factor γ ) ratio of heat capacities ∆PHX ) design pressure drop in the heat exchanger (bar) ∆Pn ) pressure drop in the nth reactor (bar) ∆THX ) design differential temperature in the heat exchanger (K) ) voidage of the bed λ ) heat of reaction [kJ (kmol of C produced)-1] µ ) average gas mixture viscosity (kg m-1 s-1) F ) gas density (kg m-3) Fcat ) catalyst bed density (kg m-3) Fm ) density of the tube material (kg m-3) Fs ) density of gas on the shell side of FEHE (kg m-3) Ft ) density of gas on the tube side of FEHE (kg m-3)

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Received for review May 19, 2003 Revised manuscript received November 11, 2003 Accepted December 3, 2003 IE030434D

Dynamic Comparison of Alternative Tubular Reactor Systems

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