Recycle

758

Ind. Eng. Chem. Res. 1996, 35, 758-771

Analysis of Control Structures for Reaction/Separation/Recycle Processes with Second-Order Reactions Michael L. Luyben and Bjorn D. Tyreus DuPont Central Science and Engineering, Experimental Station, P.O. Box 80101, Wilmington, Delaware 19880-0101

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

This work analyzes the effect of the process design on control structure for a system with a reactor, two distillation columns, and two recycle streams. The reaction A + B f C occurs in a reactor, and since component C is assumed to be the intermediate boiler, the two distillation columns recycle components A and B back to the reactor. A previous paper presented two workable control structures for this process. One fixed the flow rates of the two recycle streams and brought in makeup fresh feeds of components A and B on level control. The other control structure fixed the reactor effluent flow rate, controlled the composition of one reactant in the reactor by manipulating one fresh feed, and brought in the other fresh feed on reactor level control. These two structures have the undesirable feature of not being able to set directly the production rate and, in the second structure, requiring a reactor composition measurement, which can be difficult due to the hostile environment and can require expensive instrument maintenance. Studies of other more complex processes have led to similar results: measurement of composition somewhere in the reaction section is necessary for stable operation. In this paper we present an analysis that explains the fundamental problem with control structures in which one fresh feed is fixed and no reactor composition is measured. We show that this control structure can work if modifications are made in the design from the steady-state economic optimum. This highlights the potential trade-off between steady-state economics and dynamic controllability and illustrates considerations that ought to be included during the conceptual design procedure. A modified control structure is proposed that provides effective control of the economically optimal process design. It permits throughput to be directly set and does not require a composition measurement. The basic idea is to use the flow rates of the recycles from the separation section to infer reactor compositions. Dynamic simulation studies on both simplified and rigorous models are used to evaluate the performance of the proposed control system over a wide range of reactor sizes. Introduction This paper extends the investigations of recycle systems by Tyreus and Luyben (1993) in which the reaction of components A and B to form product C is considered. The control strategy must accommodate two fresh feed streams and two recycle streams since there is incomplete one-pass conversion of both reactants. We assume there is a single, isothermal, perfectly mixed reactor followed by a separation section. The volatilities of the A, B, and C components dictate what the recycle streams will be. If components A and B are both lighter or heavier than component C, a single column can be used, recycling a mixture of components A and B from either the top or the bottom of the column back to the reactor and producing product at the other end. In this paper we consider the case where the volatility of component C is intermediate between components A and B. This produces a process flowsheet with two columns and two recycle streams as sketched in Figure 1. The component volatilities are assumed to be RA ) 4, RB ) 1, and RC ) 2. Component B, the heaviest, is recycled from the bottom of the first column back to the * Author to whom correspondence should be addressed. Email: wllφ@lehigh.edu.

0888-5885/96/2635-0758$12.00/0

Figure 1. Process with two recycle streams.

reactor. Component A, the lightest, is recycled from the top of the second column back to the reactor. Figure 1 shows the nomenclature used in this paper. The alternative flowsheet (recycling A from the top of the first column and recycling B from the bottom of the second column) would give similar results. We use the first flowsheet because it is typical to try to keep column base temperatures as low as possible, and this is accomplished by removing the heaviest component first. The reaction rate is assumed to be first-order in each © 1996 American Chemical Society

Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996 759

Figure 2. Control structure CS1.

reactant:

RC ) VRkzAzB

(1)

where RC ) reaction rate (lb‚mol of component C produced/h), VR ) holdup in reactor (lb‚mol), k ) specific reaction rate (h-1), zA ) concentration of component A in the reactor (mole fraction A), and zB ) concentration of component B in the reactor (mole fraction B). Note that “moles” are not conserved in this system because the reaction is not equimolar. The usual simplifying assumptions are made for the distillation columns: constant relative volatilities, theoretical trays, saturated liquid feeds and refluxes, partial reboilers, total condenser, and equimolal overflow. Control Structures The eleven manipulated variables in this process are the makeup fresh feeds (F0A and F0B); reactor effluent (F); first column reflux, vapor boilup, bottoms, and distillate (R1, V1, B1, and D1); and second column reflux, vapor boilup, bottoms, and distillate (R2, V2, B2, and D2). The eleven potential controlled variables must include five liquid levels: reactor holdup (VR), first column base and reflux drum holdups (MB1 and MD1) and second column base and reflux drum holdups (MB2 and MD2). There are two product-quality controlled variables: the impurities of components A and B in the product stream B2: xB2,A and xB2,B. This leaves four other variables that can be controlled. One of these must be used to set production rate. The remaining three can be specified in a variety of ways, which vary from control structure to control structure: fixing flow rates, controlling recycle purities, controlling reactor compositions, etc. Figures 2 and 3 show the two control structures that were found in the previous paper (Tyreus and Luyben, 1993) to provide effective control. Using their nomenclature, Figure 2 is control structure CS1 and Figure 3 is control structure CS4.

In CS1 the two fresh feed streams are brought into the process on level control. Component B enters in the F0B stream, and the level in the base of the first column is held by this makeup flow. Component A enters in the F0A stream, and the level in the reflux drum of the second column is held by this makeup flow. The flow rates of the two total recycle streams (B1 + F0B and D2 + F0A) are flow controlled. Reactor level is controlled by reactor effluent flow rate F. The reflux flow rates are fixed in the two distillation columns, and base level in the second column sets bottoms flow rate. The amounts of components A and B impurities in the B2 product stream are controlled by manipulating the vapor boilups in the second and first columns. The level in the reflux drum of the first column is held by manipulating D1, the feed stream to the second column. In the CS4 control structure, the F0B fresh feed is used to control the composition of component B in the reactor zB. The F0A fresh feed is used to control reactor level. Reactor effluent F is flow controlled. Base levels and reflux drum levels in both columns are held by manipulating bottoms and distillate flow rates. Product purity of B2 is controlled the same way as in CS1. Both of these control structures lack the convenient feature of being able to set directly the production rate. Structure CS4 has the additional major problem of requiring a reactor composition measurement, which can be expensive and unreliable in many industrial processes. The process presented by Luyben and Luyben (1995) also requires the same features in a workable control strategy. It would be useful to find a workable control structure that does not have these drawbacks. Figure 4 shows a control structure that overcomes these problems. No reactor composition measurement is used, and throughput is directly fixed by flow controlling the fresh feed F0A. The other fresh feed, F0B, is brought in on reactor level control. This control structure is intuitively appealing and is one that is often

760

Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996



proposed in developing control strategies for this type of process. Unfortunately, as demonstrated by Tyreus and Luyben (1993), it does not work. Figure 5 illustrates what happens when a small increase is made in F0A (see also Figure 14 in the previous paper). We use the same values of parameters employed in the previous paper (see Table 1). When the increase is a very small 2% change (Figure 5a), the system can handle it. However, for a 5% increase (Figure 5B), the process shuts down in about 150 h. Component A has built up in the reactor, and component B has been depleted. The reactor level controller shuts down on the makeup F0B as the reactor level increases because component A is building up.

Note the very slow drift in the reactor compositions zA and zB. The initial concentrations of components A and B are 13.00 and 25.38 mol %, respectively. At about 70 h these concentration trajectories cross, and 70 h later a shutdown occurs. We show later in this paper the very important significance and insights that can be gained from examining these reactor composition effects. Figure 5c shows that a still larger increase (10%) results in more rapid drifts in zA and zB. The composition trajectories cross at 30 h, and shutdown occurs at 65 h. In the previous paper, a 20% increase caused the composition trajectories to cross in 20 h, and a shutdown occurs in 40 h. Thus these numerical simulation studies illustrate


a

b

c

Figure 5. CS2 with increases in fresh feed of component A.

that control structure CS2 does not work. In this paper we explore why this is true and suggest how CS2 can be modified to make it work and still maintain its advantages of direct control of production rate and no reactor composition measurement. Our analysis of the problem begins by looking at the steady-state values of various parameters under certain design conditions. This provides valuable insights into

the physically realizable regions of operation. Then we study the closed-loop stability of a linearized model of a simplified version of the process in which we assume steady-state behavior of the separation section. This analysis provides a stability criterion for the CS2 control structure. A modified control structure is proposed and tested via dynamic simulations of both the simplified model and a full rigorous dynamic model.

762


Table 1. Base-Case Parameter Values VR, lb‚mol k, h-1 RC, lb‚mol of component C produced/h F0A, lb‚mol/h F0B, lb‚mol/h F, lb‚mol/h zA, mole fraction component A zB, mole fraction component B B1, lb‚mol/h xB1,A xB1,B, mole fraction component B xB1,C, mole fraction component C D2, lb‚mol/h xD2,A, mole fraction component A xD2,B xD2,C, mole fraction component C B2, lb‚mol/h xB2,A, mole fraction component A xB2,B, mole fraction component B xB2,C, mole fraction component C

a 2971 1 98 99 99 160 0.1300 0.2538 40 0 0.99 0.01 20 0.99 0 0.01 100 0.01 0.98 0.01

b Steady-State Analysis The following assumptions are made for values of recycle compositions (xB1 and xD2) and losses of reactants in the product stream B2. (1) There is no component A leaving the bottom of the first column (xB1,A ) 0). Therefore, xB1,B ) 1 - xB1,C. (2) There is no component B leaving the top of the second column (xD2,B ) 0). Therefore, xD2,A ) 1 - xD2,C. (3) The composition of component C in B1 is xB1,C ) 0.01. (4) The composition of component C in D2 is xD2,C ) 0.01. (5) The losses of reactants A and B in the product stream B2 are Aloss ) 1 lb‚mol/h and Bloss ) 1 lb‚mol/h based on a production rate of component C of 98 lb‚mol/h (xB2,A ) xB2,B ) 0.01 mole fraction). A. Steady-State Analysis for Fixed Production Rate. We begin by exploring the effects of operating the system with a constant production rate and with a fixed reactor volume but with varying values of reactor composition. This leads to varying values of recycle flow rates. The procedure used is as follows: (1) Fix the value of reactor holdup (VR ) 2971 lb‚ mol), production rate of component C (RC ) 98 lb‚mol/ h), specific reaction rate (k ) 1 h-1), Aloss ) 1 lb‚mol/h, and Bloss ) 1 lb‚mol/h. (2) Specify a value of reactor composition zB (to be varied later). (3) Calculate zA from eq 2.

zA ) RC/(VRkzB)

(2)

Figure 6. Steady-state parameter values for base case VR with fixed production rate: (a) zA dependence on zB; (b) flow rates of reactor effluent F and recycles B1 and D2.

recycle D2. Component A balance:

F0A + D2xD2,A ) FzA + RC (6)

Component B balance:

F0B + B1xB1,B ) FzB + RC (7)

Total molar balance: F0A + F0B + B1 + D2 ) F + RC (8) Combining eqs 6-8 gives RC - F0B RC - F0A + - RC xB1,B xD2,A 1 - zA/xD2,A - zB/xB1,B

F0A + F0B + F)

(4) Calculate values for the fresh feed makeups and product streams.

F0A ) RC + Aloss

(3)

F0B ) RC + Bloss

(4)

B2 ) F0A + F0B - RC

(5)

(5) From the two steady-state component balances and steady-state total molar balance (remembering that the reaction is nonequimolar) around the reactor, we can solve for the three unknowns: the reactor effluent flow rate F, the heavy recycle B1, and the light

(9)

D2 )

FzA - F0A + RC xD2,A

(10)

B1 )

FzB - F0B + RC xB1,B

(11)

All variables on the right side of eq 9 are known, so F can be calculated. Then eqs 10 and 11 give the two recycle flow rates. Note that the design is physically realizable if the flow rates are all positive and the compositions zA and zC (zC ) 1 - zA - zB) are between 0 and 1 for the specified value of zB. Not all regions of the zB-zA plane give operable plants.


a

a

b b

Figure 7. Steady-state parameter values for base case VR with variable production rate: (a) zA dependence on zB; (b) flow rate of reactor effluent F.

Figure 6 gives results for the case where VR ) 2971 lb‚mol and RC ) 98 lb‚mol/h. Figure 6a shows the hyperbolic relationship between the selected value of zB and the required value of zA. The process can be operated at any point on this curve, giving the same production rate for the given reactor volume, but the reactor effluent and recycle flow rates will be different, as shown in Figure 6b. For small values of zB, there is more component A in the reactor, so the D2 recycle is large. For large values of zB, there is more component B in the reactor, so the B1 recycle is large. The reactor effluent flow rate F becomes large for either large or small values of zB. There is a value of zB that gives a minimum reactor effluent flow rate, and this occurs in the region where zA and zB are similar in value. As eq 1 shows, the reaction rate depends on the product of the two concentrations, so designs with similar zA and zB concentrations would be expected to be economically optimal. As we show later, these designs do not provide the best dynamic controllability when control structure CS2 is used, so there is a trade-off between design and control. Note that the heuristic optimum design studied in the previous paper featured a reactor effluent flow rate near this minimum (zA ) 0.1300, zB ) 0.2538, F ) 160). Figure 6 also demonstrates that there are two different values of zB that give the same production rate in the same reactor for the same value of reactor effluent

Figure 8. Steady-state parameter values for twice base case VR with variable production rate: (a) zA dependence on zB; (b) flow rate of reactor effluent F.

flow rate F. These two alternative operating points have different light and heavy recycle flow rates, D2 and B1. This type of multiplicity was pointed out in the previous paper. B. Steady-State Analysis for Variable Production Rate. Now we vary the production rate (RC) with reactor volume held constant and look at the physically realizable ranges of parameters. Figure 7a shows how zA must increase as production rate is increased for a given value of zB. Figure 7b gives the required changes in reactor effluent flow rate F. Remember that control structure CS2 fixes the flow rate of the reactor effluent. If F is fixed at the design point of 160 lb‚mol/h, Figure 7b clearly shows that there is no way that a production rate of 120 lb‚mol/h can be attained. To get to this production rate, F must be at least 210 lb‚mol/h. These steady-state analyses provide a simple explanation of why control structure CS2 cannot handle throughput changes and confirm the dynamic simulation results. Of course, this suggests that it may be possible to modify the process operating conditions (recycle flow rates) or the process design parameters (reactor holdup) and move away from the economic steady-state optimum point to be able to use control structure CS2 with its advantages of not requiring a composition measurement and being able to set production directly. Figure 8 gives results for a design with a larger reactor (twice the base case). Now higher production

764


three algebraic equations.

VR

dzA ) F0A + D2xD2,A - FzA - VRkzAzB dt

(12)

VR

dzB ) F0B + B1xB1,B - FzB - VRkzAzB dt

(13)

F0A + F0B + D2 + B1 ) F + VRkzAzB

(14)

D2 )

FzA - Aloss xD2,A

(15)

B1 )

FzB - Bloss xB1,B

(16)

Substituting eqs 14-16 into eqs 12 and 13, linearizing around the steady-state values zjA and zjB, and Laplace transforming give the characteristic equation of the closed-loop system.

s2 + s[kzjB + F/VRxB1,B] +

[

]

zjA kF zjB )0 VR xB1,B xD2,A

(17)

The linear analysis predicts that the process will become unstable whenever

zjB xB1,B

Figure 9. CS2 with twice base case VR for increases in fresh feed of component A: (a) F0A + 10%; (b) F0A + 20%.

rates can be achieved for the same reactor effluent flow rate. Dynamic simulation of the system with a larger reactor demonstrates the improved rangeability of the process. Figure 9 shows that a 10% increase in F0A can be handled with CS2. But the system cannot handle a 20% increase. Linear Stability Analysis of Control Structure CS2 The dynamic simulation study of the system with control structure CS2 indicates that a reactor shutdown occurs when the disturbance in F0A drives the reactor compositions into a region where zA becomes greater than zB. To understand the fundamental reason for this observed phenomenon, we linearize the simplified process model. The separation section is assumed to be at steady state and the only dynamics are in the reactor compositions. Perfect reactor level control is assumed. The disturbance is the fresh feed flow rate F0A. Reactor effluent flow rate F is fixed. State variables are zA and zB. Algebraic dependent variables at any point in time are the flow rates B1, D2, and F0B. To simplify the analysis, we assume that the losses of components A and B (Aloss and Bloss) in the product B2 stream are constant. Note that we are looking at the closed-loop system with control structure CS2 in place. Therefore, we are exploring closed-loop stability of the system described by two nonlinear ordinary differential equations and

zjA

Recycle

Recommend Documents