Steady-State and Dynamic Effects of Design Alternatives in Heat

Ind. Eng. Chem. Res. 2000, 39, 3335-3346

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Steady-State and Dynamic Effects of Design Alternatives in Heat-Exchanger/Furnace/Reactor Processes Francisco Reyes and William L. Luyben* Department of Chemical Engineering, Lehigh University, Bethlehem, Pennsylvania 18015

Feed-effluent heat exchangers (FEHE) are widely used in industry to preheat the feed to adiabatic tubular reactors. The hot reactor effluent is passed through a FEHE to recover heat. The positive feedback of energy introduces the potential for open-loop instability. Heat-exchanger bypassing is typically used to control the reactor inlet temperature. Previous papers1-4 have explored the control of this type of process. A furnace or heater following the FEHE may or may not be required under normal operation but is always needed for startup. Therefore, a design alternative exists in which both the reactor inlet temperature and the furnace inlet temperature are controlled, using the two manipulated variables: heat-exchanger bypassing and furnace firing rate. This paper explores the impact of these alternative designs on both the steady-state economics and the dynamic controllability. The exothermic, irreversible, gas-phase reaction A + B f C occurs in an adiabatic tubular reactor. A gas recycle returns unconverted reactants from the separation section. Steady-state economics favor the use of only a FEHE with bypassing. Dynamic controllability strongly favors the use of both FEHE bypassing and furnace firing, particularly when the reactor gain (KR ) ∆Tout/∆Tin) is large. 1. Introduction Tubular reactors present challenging design and control problems. One of the unique features distinguishing them from continuous stirred tank reactors (CSTRs) is that the inlet feed temperature to a tubular reactor is a critical parameter. Luyben4 discusses some of the many issues in the design and operation of tubular reactor systems. Because of the need to provide a desired inlet temperature in plug-flow reactor (PFR) systems, the cold feed stream needs to be heated. Also the hot effluent stream from the reactor needs to be cooled before sending it to the separation section. A heat-exchanger network is typically used to preheat the cold feed with the hot reactor effluent. 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.,5 Anderson,6 and Silverstein and Shinnar.7 More recent work is reported by Tyreus,1 Tyreus and Luyben,2 and Luyben.3,4 The potential for chaotic behavior in a similar system has recently been reported by Bilden and Dimian,8 but they examined only open-loop, uncontrolled systems. Very little has appeared in the literature on the steady-state economic design of these types of processes. The startup of these systems requires that some type of heater or furnace 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 usually be achieved in the FEHE, provided its area is large enough and the temperature rise through the reactor is large enough. Under these conditions no heat input is required in the furnace. However, because this * To whom correspondence should be addressed. E-mail: [email protected]. Phone: 610-758-4256. Fax: 610-758-5297.

Figure 1. Inlet temperature control with no furnace.

startup furnace is available, it can potentially be used as an additional manipulated variable. A comparison of these alternative process and control structures is the subject of this paper. 2. Coupled FEHE/Furnace/Reactor Process Figure 1 shows a typical chemical process in which a feed-effluent heat exchanger is 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 exchanger. Split-ranged valves are used to manipulate the bypass flow rate and the flow rate through the FEHE. A simple separation section is assumed that consists of a separator drum. All of the reactants go into the gas phase and are recycled. All of the product C is removed in the liquid stream. Design details are given by Luyben9 and in Appendices A and B.

10.1021/ie9905411 CCC: $19.00 © 2000 American Chemical Society Published on Web 08/12/2000

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Ind. Eng. Chem. Res., Vol. 39, No. 9, 2000

A

Figure 2. Inlet temperature control with furnace.

B

Figure 2 shows the design alternative in which a furnace is installed between the FEHE and the reactor. The furnace inlet temperature is controlled by manipulating the flow rate of the stream FB that bypasses around the heat exchanger and is blended with the gas leaving the FEHE at temperature TC,out, giving a temperature of Tmix. Reactor inlet temperature Tin is controlled by manipulating the heat input to the furnace QF. In the next section the optimum steady-state economic design of this type of system is developed. 3. Optimum Steady-State Design The optimum steady-state design involves an economic tradeoff between (1) the capital costs of the reactor vessel, catalyst, heat exchanger, furnace, and compressor and (2) the operating costs of the compressor work and furnace energy. The exit temperature of the reactor is fixed at its maximum value of 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. At the operating stage, using the maximum reactor exit temperature maximizes capacity or minimizes recycle. The temperature rise through the reactor ∆TR and the ratio of the furnace energy to the total energy transferred into the feed stream (QF/QTOT) are the two design optimization variables. The production rate of C is set at 0.12 kmol/s, which sets the fresh feed flow rate at 0.24 kmol/s of an equimolal mixture of reactant components A and B. To design the heat exchanger, the heuristic is used that the minimum approach temperature (∆TH ) Tout - TC,out) is 25 K, which is reasonable for the temperature level in this process. This pinch temperature differential occurs at the “hot end” of the heat exchanger. An overall heat-transfer coefficient U ) 0.142 kJ‚m-2‚K-1‚s-1 (25 Btu‚h-1‚°F-1‚ft-2) is used in this gas/ gas system. Fixing ∆TR sets the reactor inlet temperature Tin and the recycle flow rate FR because of the energy balance around the reactor. The larger ∆TR, the lower Tin and the smaller the recycle flow rate required to give the 500 K exit temperature. Thus, increasing ∆TR reduces compression costs. However, the lower Tin results in a larger reactor.

Figure 3. (A) Total annual cost. (B) Effect of QF/QTOT for the ∆TR ) 40 K case.

Fixing the QF/QTOT ratio sets the furnace firing, the heat-exchanger bypass flow rate, the size of the heat exchanger, and the size of the furnace. Increasing the ratio increases the capital cost of the furnace, decreases the capital cost of the heat exchanger, increases the operating cost of the furnace fuel, and increases the bypass flow rate. The design procedure is the following: 1. Specify the desired production rate and the maximum reactor exit temperature. 2. Pick a value of the temperature rise through the reactor ∆TR (to be optimized). 3. Calculate the inlet reactor temperature and the recycle flow rate from a reactor 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, which also corresponds to producing the desired amount of product (0.12 kmol/s). This gives the reactor size and the amount of catalyst Wcat required. 6. Pick a value of the QF/QTOT ratio (to be optimized). 7. Calculate the temperatures in and out of the furnace and heat exchanger, and calculate the heattransfer area. 8. Evaluate the total annual cost (TAC) of the process. See Appendix A for the economic basis and parameter values used. 9. Vary the QF/QTOT ratio from 0 to 0.2, and repeat steps 7 and 8. 10. Vary ∆TR until the minimum TAC is obtained. Figure 3 shows the results of this optimization procedure. Figure 3A shows that the minimum TAC

Ind. Eng. Chem. Res., Vol. 39, No. 9, 2000 3337

simple first-order lag is used.

A GR(s) )

Tout(s) Tin(s)

KR τRs + 1

)

(1)

In this initial analysis, the dynamics of the heat exchanger are assumed to be fast compared to those of the reactor, so simple gain transfer functions are assumed relating the blended temperature (Tin in FS1 or Tmix in FS2) to the two inputs to the heat exchanger: bypass flow rate FB and reactor exit temperature Tout.

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

B

(2)

Later in the rigorous simulations, the dynamics of the heat exchanger are included because of the significant thermal capacitance of the metal tubes in this very large unit. 4.1. Flowsheet FS1. In this flowsheet there is only one controller whose transfer function we call GC1(s).

GC1(s) )

FB(s)

(3)

E(s)

where E(s) ) Tset in - Tin. Both P and PI controllers are considered below. This controller looks at the coupled open-loop process transfer function GCP1(s) shown in Figure 4A. Figure 4. (A) Block diagrams without furnace (FS1). (B) Block diagrams with furnace (FS2).

occurs with ∆TR ) 40 K and with no energy provided by the furnace. Figure 3B shows how the individual values vary with the QF/QTOT ratio for the ∆TR ) 40 K case. The furnace capital cost, energy cost, and bypass flow rate increase with larger ratios. The heat-exchanger capital cost decreases. The steady-state economics favor the flowsheet in which a furnace is not used. As we will demonstrate in the following section, consideration of dynamic controllability favors the flowsheet in which a furnace is used. Thus, this process provides another important example of the everpresent interaction and conflict between steady-state economics and dynamic controllability. In a later section we show the results of rigorous dynamic simulations of this process. However, it may be useful at this point to show the predictions of a linear model of this type of system. 4. Linear Analysis One of the purposes of this linear analysis is to demonstrate the theoretical superiority of the FEHE/ furnace flowsheet in general. This is confirmed in the next section by the results of a rigorous nonlinear study of a specific typical numerical case. We use the same system as that studied by Luyben.3 Block diagrams of the linear open-loop system are shown in Figure 4. The two alternative flowsheets are FS1, in which no furnace is used, and FS2, in which a furnace is used. The reactor transfer function is GR(s), representing the adiabatic tubular reactor, and the reactor by itself is open-loop stable. In Figure 4A a

GCP1(s) )

Tin(s) FB(s)

)

[( (

)

]

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

)

(4)

The basic dynamic problem is clearly revealed by looking at the pole of this open-loop transfer function. The pole is positive, meaning an open-loop unstable process, if the product of the gains K1KR is greater than 1. The heat-exchanger gain K1 depends on the heattransfer 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. The reactor gain KR has typical values ranging from 2 to 6 (Luyben4). Therefore, these systems are often open-loop unstable. Dynamic controllability degrades as the reactor gain increases, as we demonstrate below. In tuning the controller we assume that three small lags τm exist in the loop, so the controller sees the total open-loop transfer function

[( (

)

]

K2 (τ s + 1) K1 KR - 1 R 1 τR (τms + 1)3 s-1 K1KR - 1

)

[

]

(5)

4.2. Flowsheet FS2. In this flowsheet the presence of the furnace provides an additional control degree of freedom, and there are now two controllers. The first GC1(s) controls Tmix by manipulating the bypass flow FB. The second GC2(s) controls Tin by manipulating the furnace firing QF. The second controller sees the furnace transfer function GF2(s), which we assume to be a furnace first-order

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A

B

Figure 5. (A) Nyquist plots without furnace (FS1). (B) Nyquist plots with furnace (FS2).

lag and three small lags. See Figure 4B.

GF2(s) )

Tout(s) QF(s)

)

KF2 (τFs + 1)(τms + 1)3

Figure 6. (A) Nyquist plots without furnace (PI control). (B) Nyquist plots with furnace (PI control). Table 1. Controller Tuning Constants

(6)

The first controller GC1(s) sees a coupled “open-loop” transfer function in which the second controller is nested, i.e., controller GC2(s) is on automatic. This controller was tuned by finding the ultimate gain and period and using the Tyreus-Luyben settings. 4.3. Nyquist Plots. To compare the two flowsheets and the two control structures, a numerical case is used. Parameter values are τR ) K1 ) K2 ) 1 and τm ) 0.1. Several values of reactor gain KR are explored. Figure 5A gives Nyquist plots of the open-loop coupled process for the process without a furnace (FS1). The closed-loop system is conditionally stable because one counterclockwise encirclement of the (-1, 0) point is required (because of the one open-loop pole in the righthalf plane). Therefore, if the controller gain is above a maximum gain Kmax or below a minimum gain Kmin, the closed-loop system is unstable. The controller gain must be such that the (-1, 0) point lies between the ω ) 0 point and the ω ) ωu point (the crossover frequency) on the negative real axis. Figure 5A shows that the distance between these two points shrinks as the reactor gain increases. For values of KR ) 2 and 3, there is a stable region. However, for KR ) 5, it is impossible to obtain a counterclockwise encirclement, so a P controller cannot stabilize the FS1 system. Figure 5B gives Nyquist plots for the process with the furnace (FS2). First, note that the system is now openloop stable for values of the reactor gain KR ) 2 and 3,

flowsheet

KR

Kc

τI

FS1

2 5 2 5

2.05 1.50 2.55 2.56

0.876 1.04 0.795 0.801

FS2

which was not the case for the FS1 flowsheet. Second, observe that even for reactor gains up to above KR ) 8 it is possible to use a P controller to stabilize the system. Remember that we are talking about the GC1(s) controller, with the GC2(s) controller on automatic. The settings of this PI controller are Kc2KF2 ) 2.13 and τI ) 1.94. Figure 6 demonstrates the improvement obtained when PI controllers are used instead of P controllers for GC1(s). The controller tuning constants used are given in Table 1 for the two flowsheets and for two values of KR. This improvement is true for both flowsheets, as a comparison of Figures 5 and 6 reveals. For example, in the process without the furnace, a P controller could not stabilize the system for KR ) 5. However, the PI controller can stabilize the system. Figure 7 gives time-domain results that verify the frequency-domain predictions. Figure 7A shows that without a furnace the control performance degrades as KR increases. The responses are for a unit step change in setpoint. Results from two tuning methods are shown: the Tyreus-Luyben settings and the optimum settings (Luyben4). Figure 7B shows that much larger reactor gains can be handled by the process with the furnace. The Tyreus-Luyben settings are used (Kc1K1 ) 2.5 and τI ) 0.80), which were almost constant for all values of KR .


A

A

B B

Figure 7. (A) Reponse without furnace. (B) Reponse with furnace.

Thus, the linear analysis predicts that the dynamic controllability of the FS2 process is much better than the FS1 process. Even though steady-state economics favor not using a furnace, the actual profitability of the process may be higher if a furnace is used. This is particularly true for systems that exhibit large reactor gains. 5. Rigorous Simulation To verify the results of the linear analysis, we develop a rigorous dynamic model of the process and study two different flowsheets. Figure 8A shows the FS1 flowsheet without a furnace. The FEHE area is 2261 m2, and the bypass flow rate is 0.184 kmol/s. Figure 8B shows the FS2 flowsheet with a furnace. The FEHE area is smaller (1712 m2), and the bypass flow rate is larger (0.365 kmol/s). 5.1. Dynamic Model. The dynamic model is an extension of that developed by Luyben,9 who considered reversible reactions, with and without inerts in the system. The reaction considered in this paper is the gasphase, irreversible, exothermic reaction A + B f C occurring in a packed tubular reactor, without inerts in the system. Details of the nonlinear differential and algebraic equations for the reactor, recycle, and separator are given by Luyben9 and are summarized in Appendix B. Dynamic models for the heat exchanger and furnace are developed and combined with the model of the reactor to simulate the entire system. The reactor and heat exchanger are both distributed systems, which are rigorously modeled by partial differential equations. Lumped-model approximations are used to capture the important dynamics with a mini-

Figure 8. (A) Base case (no furnace). (B) Furnace supplies 10% of heat load.

mum of complexity. There are no sharp temperature or composition gradients in the reactor because of the low per pass conversion and high recycle flow rate. Luyben9 found that a 10-lump model was adequate for the reversible reaction case he considered. However, in this work, which considers irreversible reactions, a larger number of lumps are required in the reactor. For example, the rigorous steady-state reactor exit temperature is 500 K, with a 460 K inlet reactor temperature. When a 10-lump model is used, with the same amount of catalyst, the steady-state exit temperature is 505 K. This occurs because of the “numerical diffusion” or effective backmixing that is inherent with a lumped model. A 50-lump reactor model is used in all of the simulations. It gives a steady-state reactor exit temperature of 501 K. The differences in the dynamic responses of the 10-lump and 50-lump reactors are given later in this paper. There are no phase changes or small pinch temperatures in the heat exchanger. Therefore, a 10-lump model is used for this unit. The steady-state exit temperatures predicted by the lumped model are close to those calculated by the countercurrent model used in the steady-state design. Table 3 compares the steadystate conditions for the process obtained from the dynamic model with those obtained in the steady-state design. The thermal capacitance of the gas in the reactor is assumed negligible compared to that of the solid catalyst. Therefore, a single dynamic energy balance is used for each lump, and the gas temperatue is assumed to

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Table 2. Parameter ValuessBase Case -23 237 69 710 3.309 × 10-8

heat of reaction λ (kJ/kmol of C) activation energy E (kJ/kmol) k at 500 K [kmol‚s-1‚bar-2‚(kg of catalyst)-1] heat capacities (kJ‚kmol-1‚K-1) cp,A cp,B cp,C molecular weights (kg/kmol) MWA MWB MWC

30 40 70 15 20 35

Fo (kmol/s) FB (kmol/s) QF (kJ/s) Tmix (K) Tin (K) Tout (K) area (m2)

FS2

design

dynamic

design

dynamic

0.24 0.184 0 460 460 500 2261

0.248 0.232 0 460 460 501.3 2261

0.24 0.365 1,023 445.3 460 500 1712

0.248 0.391 978 446 460 501.3 1712

Table 4. Heat-Exchanger Parameters (FS2

1712 4291 10.9 10.9 44 500

a Tubes: 1 in. i.d., 0.133 in. wall thickness; 5 m length; c ) p 0.05 kJ‚kg-1‚K-1 (0.12 Btu‚lb-1‚°F-1).

be equal to the catalyst temperature. The heat exchanger is quite large because of the low heat-transfer coefficient found in these gas-phase systems. Therefore, the mass of metal in the tubes is quite significant in terms of thermal capacitance. Table 4 gives design details of the heat exchanger. It is modeled by assuming the cold gas has a temperature TC,n in the nth lump, into which heat is transferred at a rate QC,n from the hotter tube metal at temperature TM,n.

QC,n ) UCAH,n(TM,n - TC,n)

(7)

The hot gas flows countercurrently and has a temperature TH,n in the nth lump. Heat is transferred from the hot gas into the tube metal at a rate QH,n

QH,n ) UHAH,n(TH,n - TM,n)

Favcp,CVC,n

dTC,n ) FCcp,CTC,n-1 - FCcp,CTC,n + QC,n dt (10) dyH,j,n ) FH,n+1yH,j,n+1 - FH,nyH,j,n dt

(8)

The heat-transfer area in each lump is the total area divided by the number of lumps. The overall heattransfer coefficient used in the steady-state design calculations was 0.142 kJ‚s-1‚m-2‚K-1. The two heattransfer coefficients UC and UH are set at 3.5 times the overall coefficient so that the exit stream temperatures of the dynamic model are approximately the same as those of the steady-state model. Each lump of the cold and hot sides of the exchanger is described by an energy balance and three composition balances because both temperatures and compositions can change with time and axial position. Note that the cold side is always a 50/50 mixture of A and B, but the hot side can contain varying amounts of component C as conditions in the reactor change dynamically with time.

(11)

where j ) A, B, and C and yH,j,n ) mole fraction of component i in the hot-side lump n. The stage parameter n is numbered starting from the cold inlet. An average molar gas density Fav is used in the component balances. There is flow in and out of each lump. Pressure drops are neglected. The hot and cold side volumes are assumed to be equal. Each lump of the metal is described by an energy balance.

cp,mWn

Case)a

area (m2) no. of tubes volume of the tubes (m3) volume of the shell (m3) weight of the tubes (kg)

dTH,n ) FHcp,HTH,n+1 - FHcp,HTH,n - QH,n dt (9)

FavVH,n

Table 3. Design and Dynamic Model Steady States FS1

Favcp,HVH,n

dTM,n ) QH,n - QC,n dt

(12)

where the weight of tube metal in each lump is the total tube weight divided by the number of lumps. The furnace is model with a simple energy balance, with the molar holdup MF adjusted to give a 2 min time constant.

cp,CMF

dTin ) QF + FCcp,C(Tmix - Tin) dt

(13)

The flow rate into the furnace is the blended stream FC with a temperature Tmix. 5.2. Controller Structure. The control structure used has the following loops: 1. The gas loop pressure P is controlled by manipulating the fresh feed flow rate Fo. 2. The liquid level in the separator is controlled by manipulating the liquid flow rate. 3. The recycle gas flow rate FR is held constant by fixing compressor speed. 4. The temperature Tmix of the blended stream of gas from the heat exchanger and gas bypassing the heat exchanger is controlled by manipulating the bypass flow rate FB. 5. In the FS2 flowsheet with the furnace, the reactor inlet temperature Tin is controlled by manipulating the furnace heat input QF. All controllers are PI except for the drum level controller, which is P only. Two 0.5 min lags are assumed in the pressure loop. Three 0.1 min lags are assumed in the temperature loops. Relay feedback tests are conducted to get the ultimate gain and period, and the Tyreus-Luyben settings are used. 5.3. Results. Figure 9 gives the open-loop responses of the heat exchanger and the reactor to step changes in the inlet temperature. These results are for each of the units operating in total isolation from the rest of the plant. In Figure 9A the temperature of the hot stream entering the heat exchanger is increased 20 K at time equal to zero. The temperature of the cold exit stream responds with a time constant of about 3 min. The temperature of the hot exit stream responds with a time constant of about 10 min. Most of the temperature dynamics are due to the tube metal thermal capacitance because the holdup of gas is small. Note


A

B

Figure 9. (A) Open-loop response of the FEHE. (B) Open-loop responses of the reactor.

that the heat-exchanger gain for this numerical case is ∆TC,out/∆TH,in ) 18/20 ) 0.9. Figure 9B gives the responses of the reactor exit temperature using the 10-lump and 50-lump models. Step changes in the reactor inlet temperature of several magnitudes are made at time zero. The 10-lump model shows more sensitivity than the 50-lump model. Note that the reactor gain (∆Tout/∆Tin) increases as the magnitude of the input step size increases. This is an important effect, which we discuss in a later section when a “hotter” reaction system is considered. There is a very slight amount of “wrong-way” or inverse response observed in this numerical example. This type of response can occur for certain parameter values (Tyreus1) and complicates the control problem. These types of reactors will be the subject of a future paper. During the initial simulation studies, considerable difficulty was experienced in getting the entire process simulation to stabilize. We started with the FS1 flowsheet without the furnace and tried to converge the model to a steady state from some approximate initial conditions predicted by the steady-state design. However, unless our guesses for the initial conditions were very close to the correct ones, the system would quench. The reactor inlet temperature decreased, which decreased the reactor exit temperature. Despite the bypass valve closing while trying to return the inlet temperature to the setpoint, the drop in the reactor exit temperature decreased the reactor inlet temperature even more. So, the system shut down: pressure built up and fresh feed was cut off. However, when we switched to the FS2 flowsheet, the presence of the furnace prevented the quench, and the

A

B

Figure 10. (A) FS2 response; change in the recycle flow. (B) FS2 response; change in Tset in .

simulation was rather easily converged to a steady state. These startup experiences indicate that the FS2 flowsheet is much more robust than the FS1 flowsheet. 5.3.1. FS2 with a Furnace. Figure 10 illustrates the effective dynamic performance for the FS2 flowsheet. In Figure 10A changes in the recycle flow rate FR of (20% are easily handled. Note the interesting phenomenon that an increase in recycle flow rate decreases the production rate (the fresh feed flow rate Fo decreases). This is somewhat counterintuitive until one remembers that what we are doing is holding the reactor inlet temperature constant. This means that an increase in the flow rate results in a decrease in the reactor exit temperature because the reactor is operating adiabatically and the increase in flow provides more thermal sink. This lowers the temperature rise in the reactor. See the lower left graph in Figure 10A. The lower temperatures in the reactor yield a smaller reaction rate. We are considering an irreversible reaction in this paper. For reversible reactions, Luyben9 demonstrated the opposite effect of the recycle flow rate on the production rate; i.e., increasing the recycle flow rate increases the production rate. With reversible reactions the impact of temperature is less important than the impact of concentrations because of the reaction equilibrium constraints. Figure 10B demonstrates that changes in the setpoint of the reactor inlet temperature controller are smoothly handled. When the reactor inlet temperature is increased, for a constant recycle flow rate, there are

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A

Table 5. Optimum Design Parameters with Hot Reaction base case E (kJ/kmol) λ (kJ/kmol) Wcat (kg) QF (kJ/s) QH (kJ/s) FR (kmol/s) FB (kmol/s) Tin (K) Tout (K) Tmix (K) area (m2) TAC (106 $/yr)

hot reaction

design SS

design SS

dynamic SS

69 710 -23 237 45 100 1023 8200 1.75 0.365 460 500 445.3 1712 3.01

132 450 -44 150 62 980 2770 24950 4.81 0.640 470 500 454.3 5121 5.53

132 450 -44 150 62 980 2667 24950 4.81 0.749 470 501.8 454.3 5121 -

FS2 Flowsheet; Ratio (QF/QTot) ) 0.1

B

by using furnace heat input when the bypass valve approaches the fully closed position. In all of these schemes, bypass control would be used during normal operation. This would achieve the desired steady-state economic objective of using no energy. Such schemes may be possible if the heating is done using highpressure steam or a high-temperature medium (such as Dowtherm). However, if the heat is being provided by a furnace, which is the most common case because of the high temperature levels typical of many industrial reactors, these control structures are not viable. A furnace cannot be conveniently operated with no firing rate. So, some minimum base load (about 10% of capacity) is required. 6. Hot Reaction Case

Figure 11. (A) FS1 response; change in the recycle flow. (B) FS1 set . response; change in Tmix

increases in the production rate, bypass flow rate, reactor exit temperature, and furnace heat input. 5.3.2. FS1 without a Furnace. It is more difficult to operate the system without the furnace. Quenches occur because of bypass valve saturation. Figure 11A illustrates such a case. A 20% decrease in the recycle flow rate is handled because the bypass valve can open more because of the higher production rate and higher reactor exit temperature. However, a 20% increase in the recycle flow rate causes a quench. The reactor exit temperature drops, decreasing the reactor inlet temperature. The bypass valve closes but is unable to maintain the reactor inlet temperature. The reaction rate decreases, pressure increases, and fresh feed is gradually shut off. It takes about 1 h for the shutdown to occur. Figure 11B shows that small changes in the temperature controller setpoint can be handled, but a large decrease causes the bypass valve to saturate shut, leading eventually to a quench. Thus, the absence of the furnace to provide heat input when the bypass valve is completely shut produces a process that is not as robust and not as easy to operate. Another problem with just using heat-exchanger bypassing is the impact of changes in heat-transfer coefficients over time due to fouling. The stability of the process will degrade as the FEHE fouls. One might consider using a flowsheet like FS2 but setting the setpoints of the two temperature controllers at the same value (or using one temperature controller with a “split-range valve” setup), or a low-temperature override controller could be used to prevent quenching

The linear analysis predicted that the FS2 flowsheet should be able to handle systems with reactor gains up to about 6. To test this prediction, the “worst-case” kinetic parameters found by Luyben4 to give reactor gains of this magnitude are used in a new design. Both the activation energy and the heat of reaction are increased by a factor of 1.9. The specific reaction rate at 500 K is kept constant by adjusting the preexponential factor as the activation energy is changed. The optimum steady-state economic design was determined with these new kinetic parameters, and parameters are given in Table 5. The FS2 flowsheet is used with a ratio QF/QTOT ) 0.1. The impact of the kinetic parameters on the optimum design is striking. The hotter reaction requires a much larger recycle flow rate and a higher reactor inlet temperature for the same reactor exit temperature Tout ) 500 K. These lead to a larger reactor, heat exchanger, compressor, and furnace. The process is more expensive (TAC increases from 3.01 × 106 to 5.53 × 106$/yr). The dynamic simulations of the reactor by itself are shown in Figure 12. A very small increase in the reactor inlet temperature produces a reactor runaway because of the high activation energy and high heat of reaction. Compare these results with those of the base-case kinetic parameters shown in Figure 9B. Figures13 and 14 give results using the FS2 flowsheet with the furnace for this hot reaction case. Figure 13 shows that a 10% decrease in the recycle flow rate can be handled, but a 20% decrease produces a reactor runaway. This occurs despite the fact that the reactor inlet temperature increases only slightly (about 0.5 K) during the transient. Figure 14 gives results for changes in the setpoint of the reactor inlet temperature control-

Ind. Eng. Chem. Res., Vol. 39, No. 9, 2000 3343 Table 6. Hot Reaction: Nonlinear Reactor Gain

Figure 12. Open-loop response of the hot reactor.

Figure 13. FS2 response with a hot reaction; change in the recycle flow.

Figure 14. FS2 response with a hot reaction; change in Tset in .

ler. Rather surprisingly, the inlet temperature can be increased by 2 K without a runaway. This is unexpected because the isolated reactor (Figure 12) showed a runaway with a +2 K change in Tin. The difference may be due to the effect of pressure. In the isolated reactor simulation, pressure is held constant at 50 bar. In the simulation of the whole process, pressure drops as the reactor temperature increases because of the increased consumption of reactants. Because the reaction rate depends on the square of the total pressure (P2), the decrease in pressure lowers the reaction rates. However, a 3 K increase cannot be handled. Note that the reactor gain reported by Luyben4 for this system is 6, which the FS2 flowsheet should be able

∆Tin

KR

1 2 3 3.5

6.4 7.9 11.9 21.1

to handle. However, the large activation energy makes the process very nonlinear, and the effective reactor gain increases very significantly with the magnitude of the input change, as shown in Table 6. The new kinetic parameters drastically increase the sensitivity of the reactor to the inlet temperature. Decreases in Tin result in modest decreases in the reactor exit temperature Tout. However, even small increases in the inlet temperature (>1 K) result in reaction runaways. This extreme parametric sensitivity to the inlet temperature occurs because of the large activation energy and heat of reaction and because of the high reactant concentrations (low per pass conversion). Remember that the feed to the reactor is a 50/50 molar mixture of pure reactants. There are large amounts of reactants available to fuel the reaction runaway. These results indicate that a process change would probably be required to handle the dynamic problems. There are several alternatives. A cooled nonadiabatic reactor should reduce the sensitivity because more heat will be removed as temperatures increase. The design of a cooled tubular reactor is quite complex and involves important tradeoffs between the tube diameter (to achieve sufficient heat-transfer area), tube length (to keep pressure drop reasonable, which is particularly important in gas-phase systems where compression costs for recycle are large), reactor size, and recycle flow rate. Probably a more practical solution would be to design for a lower concentration of one of the reactants. This mode of operation would prevent reaction runaways because the reaction rate would drop off quickly as the concentration of the limiting reactant declined. The economic penalties would include requiring a larger reactor and more recycle than in the equimolar pure reactant feed mode of operation. Alternatively, the concentrations of both reactants could be reduced by recycling an inert (probably product C). This would also increase the reactor size and recycle flow rate. All of these considerations again demonstrate the strong effect of process design on process control. In almost all processes there are important tradeoffs and conflicts that must be considered to produce an easyto-operate, stable process with little product quality variability. Such a process will be more profitable in the operational stage than a process that may look better from just a steady-state point of view. Techniques are available (Elliott et al.10) for quantitatively evaluating this tradeoff between steady-state economics and dynamic controllability. 7. Conclusion This paper has studied processes with gas-phase, exothermic, irreversible reactions carried out in adiabatic tubular reactors with gas recycle from the separation section back to the reactor. Two alternative structures for feed preheating are explored: feed-effluent heat exchanger only and heat exchanger plus furnace. Steady-state economics favor the use of only heat

3344


exchangers. Dynamic controllability favor the use of both heat exchangers and furnaces. Separate preheating and cooling systems on the reactor feed and effluent streams would provide very effective dynamic control but would be very uneconomical because of high energy consumption. The use of some coupled heating and cooling represents a compromise between the two extremes of a completely independent heating/cooling system and complete dependence on reaction heat for feed preheating. The contribution made in this paper is providing some indication of the degree of interdependence that can be tolerated before control problems occur. The impacts of kinetic and design parameters are also illustrated. There are many open research issues in the area of tubular reactor systems. Nonadiabatic reactors and more complex reaction systems will be the subjects of future papers. Appendix A Process Studied. The process is adapted from that presented by Jones and Wilson.11 An exothermic, gasphase irreversible reaction A + B f C occurs in an adiabatic tubular reactor with a reaction rate R [kmol‚s-1‚(kg of catalyst)-1].

R ) kPAPB

(A1)

where Pj ) partial pressure of component j and the specific reaction rate k is temperature dependent [kmol‚s-1‚(kg of catalyst)-1‚bar-2].

k ) Re-E/RT

(A2)

components A or B, and the gas from the separator contains no component C. The fresh feed contains only pure reactants A and B, and following Jones and Wilson,11 we assume that 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. Because 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.12 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. Economics and Sizing. The optimization of this process involves finding the best values of the reactor temperature difference (∆TR) and QF/QTOT ratio. The objective function to be minimized is the TAC (106$/yr), which is the sum of the annual capital cost (reactor, catalyst, heat exchanger, furnace, and compressor capital investment divided by a payback period of 3 yrs) and operating costs (work to drive the compressor and fuel consumed in the furnace).

TAC ) CompressorWork + FurnaceFuel + (Reactor + Furnace + Compressor + HeatExchanger + Catalyst)/PaybackPeriod (A6)

where E is the activation energy (kJ/kmol). Parameter values used in the base case are given in Table 2. The heat of reaction is λ ) -23 237 kJ/kmol, and the activation energy is E ) 69 710 kJ/kmol. 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 the reactor catalyst weight W, with the limits of integration between W ) 0 and W ) Wcat.

The 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).

dFC )R dW

CompressorWork (106$/yr) ) 0.227FR

(A3)

(A4)

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)

[( )

γRT1 P2 γ - 1 P1

(γ-1)/γ

]

-1

(A7)

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

(A8)

where FR is in kmol/s. The capital costs of the compressor and reactor vessel are

with the boundary value FC(0) ) FCo.

λR dT )dW cPAFA + cPBFB + cPCFC

Wcomp )

(A5)

A simplified separation section is assumed in which the liquid from the separator drum contains no reactant

CompressorCost (106$) ) 0.345FR0.82

(A9)

ReactorVesselCost (106$) ) 0.035DR1.066LR0.802 (A10) 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. The capital cost of the heat exchanger is

HeatExchangerCost (106$) ) 0.00730AH0.65 (A11) with area in square meters. The annual cost of fuel


consumed in the furnace is based on $5/106 Btu.

FurnaceEnergyCost (106$/yr) ) 0.150QF

(A12)

with furnace heat duty QF in 103 kJ/s. The capital cost of the furnace is

log10[CostFurnace (106$)] ) -2.49 + 0.762 log10(QF) (A13) Appendix B Dynamic Model. The thermal capacitance of the gas in the reactor is assumed negligible compared to that of the solid catalyst. Therefore, a single dynamic energy balance is used for each lump, and the gas temperatue is assumed to be equal to the catalyst temperature. Energy balance for the nth lump:

cp,catWcat,n

dTn ) Fn-1cp,n-1Tn-1 - Fncp,nTn - λRC,n dt (B1)

where Wcat,n ) Wcat/NR and NR ) number of lumps. The heat capacity of the catalyst is 0.837 kJ/kg‚K. The reaction rate is given by

RC,n ) Wcat,nqkF,nPn,APn,B

(B2)

where Pn,j ) Pynj and ynj is the mole fraction of j in lump n. The dynamic component balances on the gas phase in each lump assume a constant average molar density of Fav ) 1.124 kmol/m3 (at 535 K and 50 bar). Component balances for the nth lump (j ) A, B)

VgasFav

dynj ) Fn-1yn-1,j - Fnynj - RC,n dt

(B3)

Component balances for the nth lump (j ) C)

VgasFav

dynj ) Fn-1yn-1,j - Fnynj + RC,n dt

(B4)

where Fav is the average molar density. Note that the reaction is nonequimolar, so the total flow rate decreases down the length of the reactor. The total pressure in the gas loop is calculated from an approximate dynamic total mass balance. Gas enters in the fresh feed, and gas is converted into liquid component C in the heat exchanger following the reactor. An average density is used assuming an average molecular weight of 17.5 kg/kmol and an average temperature of 500 K. Total mass balance on gas loop:

17.5(VFEHE + VReactor + Vdrum) dP ) dt (0.08314)(500) Fo(17.5) - (35)FNRyNR,C (B5) Total mass balance on the drum:

dMD ) FNRyNR,C - L dt

(B6)

where MD is the molar liquid holdup in the separator and L is the liquid flow rate (kmol/s). Nomenclature A ) reactant component B ) reactant component

C ) product component cp,C ) heat capacity on the cold side of the heat exchanger (kJ‚kmol-1‚K-1) cp,H ) heat capacity on the hot side of the heat exchanger (kJ‚kmol-1‚K-1) cpj ) heat capacity of component j (kJ‚kmol-1‚K-1) cp,m ) heat capacity of tube metal (kJ‚kg-1‚K-1) DR ) reactor diameter (m) E ) activation energy (kJ‚kmol-1) E(s) ) error in the feedback controller F ) flow rate in the reactor (kmol/s) FB ) bypass flow rate (kmol/s) FC ) flow rate through the cold side of the heat exchanger (kmol/s) FC ) molar flow rate of component C at any point in the reactor (kmol/s) Fj ) flow rate of component j (kmol/s) FH ) flow rate through the hot side of the heat exchanger (kmol/s) Fo ) fresh feed flow rate (kmol/s) FR ) recycle flow rate (kmol/s) GC1 ) feedback controller transfer function GC2 ) feedback controller transfer function GCP1 ) transfer function of a coupled exchanger/reactor process without furnace GCP2 ) transfer function of a coupled exchanger/reactor process with furnace GR ) reactor transfer function k ) specific reaction rate [kmol‚s-1‚bar-2‚(kg of catalyst)-1] K1 ) heat-exchanger gain K2 ) heat-exchanger gain Kc ) controller gain KF1 ) furnace gain KF2 ) furnace gain Kmax ) maximum controller gain for a given reset time Kmin ) minimum controller gain for a given reset time KR ) reactor gain Ku ) ultimate gain L ) liquid flow rate leaving the separator drum (kmol/s) LR ) length reactor (m) MF ) furnace molar holdup (kmol) MWj ) molecular weight of component j (kg/kmol) P ) total pressure (bar) Pj ) partial pressure of component j (bar) Pu ) ultimate period P1/P2 ) suction and discharge pressures of the compressor (bar) QC,n ) heat transfer in the nth lump in the heat exchanger between metal and cold gas (kJ/s) QF ) heat transfer in the furnace (kJ/s) QH ) total heat transfer in the heat exchanger (kJ/s) QH,n ) heat transfer in the nth lump in the heat exchanger between hot gas and metal (kJ/s) QTOT ) total preheating in the furnace and heat exchanger (kJ/s) R ) perfect gas law constant (bar‚m3‚kmol-1‚K-1) RC ) rate of production of C (kmol of C/s) T ) temperature in the reactor (K) TAC ) total annual cost (106$/yr) T1 ) compressor suction temperature (K) TC,out ) temperature of the cooler stream leaving FEHE (K) TH,out ) temperature of the hotter stream leaving FEHE (K) Tin ) reactor inlet temperature (K) Tset in ) reactor inlet temperature setpoint (K) Tmix ) blended bypass and heat-exchanger temperature (K) Tout ) reactor exit temperature (K) U ) overall heat-transfer coefficient in FEHE (kJ‚s-1‚m-2‚K-1)

3346


VH,n ) volume on the hot side of the nth lump in the heat exchanger (m3) VDrum ) volume in the separator drum (m3) VFEHE ) volume in the heat exchanger (m3) VReactor ) gas volume in the reactor (m3) Wcomp ) compressor work (kJ‚s-1‚kmol-1) Wcat ) weight of the catalyst (kg) yC,j,n ) composition of the cold gas in the nth lump of the heat exchanger (mole fraction of component j) yH,j,n ) composition of the hot gas in the nth lump of the heat exchanger (mole fraction of component j) R ) preexponential factor ∆TH ) minimum temperature differential in FEHE (K) ∆TR ) temperature differential across the reactor (K) λ ) heat of reaction (kJ/kmol of C produced) γ ) ratio of heat capacities Fav ) average gas density in the heat exchanger (kmol/m3) τI ) reset time constant (min) τm ) temperature measurement lag (min) τR ) reactor time constant ωu ) ultimate frequency

Literature Cited (1) Tyreus, B. D. Control and design applied to a reactor-heat exchange system, Paper presented at the 1992 AIChE Meeting. (2) 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.

(3) Luyben, W. L. External and internal open-loop unstable processes. Ind. Eng. Chem. Res. 1998, 37, 2713. (4) Luyben, W. L. Effect of kinetic, design and operating parameters on reactor gain. Ind. Eng. Chem. Res. 1999, in press. (5) 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. (6) Anderson, J. S. A practical problem in dynamic heat transfer. Chem. Eng. 1966, 97. (7) 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. (8) Bilden, C. S.; Dimian, A. C. Stability and multiplicity approach to the design of heat-integrated PFR. AIChE J. 1998, 44, 12, 2703. (9) Luyben, W. L. Design and control of gas-phase reactor/ recycle processes with reversible exothermic reactions. Ind. Eng. Chem. Res. 2000, 39, 1529. (10) Elliott, T. R.; Luyben. M. L.; Luyben, W. L. Application of the capacity-based economic approach to an industrial-scale process. Ind. Eng. Chem. Res. 1997, 36, 1727. (11) Jones, W. E.; Wilson, J. A. An introduction to process flexibility; Part 2: Recycle loop with reactor. Chem. Eng. Educ. 1998, 32, 224. (12) 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 (6), 1154.

Received for review July 21, 1999 Revised manuscript received February 8, 2000 Accepted June 16, 2000 IE9905411

Steady-State and Dynamic Effects of Design Alternatives in Heat

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