Znd. Eng. Chem. Res. 1993,32, 1154-1162
1154
Dynamics and Control of Recycle Systems. 4. Ternary Systems with One or Two Recycle Streams Bjorn D. Tyreus E. I . du Pont De Nemours & Co., Newark, Delaware 19714-6090
William L. Luyben’ Department of Chemical Engineering, Iacocca Hall, Lehigh University I l l , Bethlehem, Pennsylvania 18015
This paper is the fourth in a series of papers that explore the challenging problems associated with the dynamics and control of recycle systems. The reactions considered in previous papers were fairly simple. Only first-order reactions were considered, so there was only one fresh feed stream. In this paper, second-order kinetics are considered with two fresh-feed makeup streams. Two cases are considered: (1)instantaneous and complete one-pass conversion of one of the two components in the reactor so there is an excess of only one component that must be recycled and (2) incomplete conversion per pass so there are two recycle streams. It is shown that the generic liquid-recycle rule proposed by Luyben applies in both of these cases: “snowballing” is prevented by fixing the flow rate somewhere in the recycle system. An additional generic rule is proposed: fresh feed makeup of any component cannot be fixed unless the component undergoes complete single-pass conversion. In the complete one-pass conversion case, throughput can be set by fixing the flow rate of the limiting reactant. The makeup of the other reactant should be set by level control in the reflux drum of the distillation column. In the incomplete conversion case, two workable schemes were found: (1)Both recycle flow rates are fixed and both fresh-feed makeups are brought in on level control. Throughput is controlled by changing either the reactor temperature or the recycle flow rates. (2) One fresh-feed makeup controls reactor level and the other controls the composition in the reactor. Throughput is controlled by setting reactor temperature or reactor effluent flow rate.
Introduction In previous papers, simple reactions have been considered binary systems with the single reaction A B in a process consisting of one reactor and one column, and ternary systems with first-order consecutive reactions A B C occurring in a reactor and separation occurring in two distillation columns. Unreacted component A was recycled back to the reactor from one of the columns. There was a single fresh-feed stream. In this paper we consider reactions in which two components,A and B, react to form product C. Therefore there are two fresh-feed streams to be accommodated in the control strategy. In addition, if there is incomplete one-passconversionof both reactants, two recycle streams must be handled. We assume there is a single, isothermal, perfectly mixed reactor followed by a separation section. One distillation column is used if there is only one recycle stream. Two are used if two recycle streams exist. A number of alternative control structures will be studied. All composition controllers are tuned by using the relay-feedback method to determine the ultimate gain and ultimate frequency for the composition loop with the unit isolated from the rest of the plant. Then detuned Ziegler-Nichols settings were used in most loops. In the higher purity loops the controllerswere tuned by assuming that the process consisted of an integrator and deadtime and using the tuning procedure proposed by Tyreus and Luyben (1992). All the dynamic simulations for this paper were carried out on an interactive dynamic simulator developed at Du Pont. The column dynamics are simulated with models similar to those described in Luyben (1990). Equilibrium at each stage was determined from a bubblepoint calculation (with vapor pressure assumed such that relative volatiles were constant). The vapor rates were calculated from an energy balance (with physical properties assumed
--
-
Table I. Physical Propemties of Components A, B, and C A B C component 1 2 3 component number molecularweight (lb/fts) 50 50 100 heat of vaporization at 32 O F 18000 18000 18000 (Btdlb-mol) 0.6 0.6 0.6 liquid heat capacity (Btu/(lb OF)) vapor heat capacity (Btu/(lb OF)) 0.3 0.3 0.3 Antoine constantsa 16.9963 15.5000 16.1931 4 -5Ooo -5000 -6000 273.16 273.16 273.16 Cj a The Antoine equation for pure-component vapor pressures is In q = Aj + [Bj/(Cj+ 271, where q = vapor pressure of component j (psia) and T = temperature (“C).
such that equimolal overflow applied). The physical properties for the three hypothetical components A, B, and C are given in Table I. In the first section we will consider the case where one of the reactants is completely consumed in one pass through the reactor. In the second section we will look at the more realistic case where conversion per pass is incomplete for both components.
Complete One-Pass Conversion of One Component Figure 1 shows the process. Pure component B is fed into the reactor on flow control. The concentration of B in the reactor, z(2), is zero because we assume complete one-pass conversion of B. A large recycle stream of component A is fed to the reactor. The reactor effluent is a mixture of unreacted component A and product C. This binary mixture is separated in a distillation column. We assume that the relative volatility of A is greater than that of C, so the distillate product is recycled back to the reactor. This simple system is encountered in a number of commercial reaction systems. It occurswhen reaction rates
Q888-5885193/2632-1154$Q4.oO/Q 1993 American Chemical Society
Ind. Eng. Chem. Res., Vol. 32, No. 6, 1993 1155 RECYCLE
R
Under the assumption of complete one-pass conversion of component B, in theory both the recycle flow rate D and the holdup of the reactor V R Cbe~set at any arbitrary values. Once these are selected, the system can be designed. The feed flow rate F and composition Z j to the column can be calculated once D has been specified.
T FRESH
FRESH
'-8=
FOB
Figure 1. Process with one recycle stream. RECYCLE R
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Figure2. Scheme A both reactant makeup streams flow controlled. F R E S H FEED
RECYCLE R
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Figure 3. Scheme B limiting reactant makeup stream flow controller; other reactant brought in on level control.
are so fast that B reacts quickly with A, but a large excess of A is needed. There are several reasons why a large recycle of one component is needed. One important one is to prevent the occurrence of undesirable side reactions. The alkylation process in petroleum refining is a common and important example. Another reason for recycle may be the need to limit the temperature rise through an adiabatic reactor by providing a thermal sink for the heat of reaction. A. Steady-State Design. The purity of the product stream B leaving the bottom of the distillation column xg(1) is set at 0.01 mole fraction component A, with the concentration of component C being 0.99. There is no component B in the feed to the column because complete one-pass conversion has been assumed. The column is 1 10.95 mole fraction designed for a distillate purity ~ ~ ( of component A. In the base case the fresh feed flow rate is 100 lb-mol/h (FOB),and it is pure component B. Since component B is completely converted, the amount of component C in the product stream must also be 100 lbmol/h at steady state. Therefore the total flow rate of the product stream and the flow rate of the makeup fresh feed of component A, FOA,can both be calculated.
The separation in the distillation column is binary between A and C, so the design of the column is straightforward. The reflux ratio was set at 1.2 times the minimum, and tray-to-tray calculations gave the total number trays NT and the optimum feed tray NF. Table I1 gives steady-state design results for a range of values of recycle flow rate D . As D increases, the concentration of A in the reactor z(1)increases. This causes the reflux flow rate to increase initially, but then decrease. At very high recycle flow rates, the reflux rate goes to zero, indicating that the distillation column becomes just a stripping column. As D increases, energy consumption,capital investment, and total cost all increase. Thus the recycle flow rate should be kept as low as possible, subject to the constraints on the minimum recycle flow, e.g., preventing undesirable side reactions from occurring or limiting adiabatic temperature rise through the reactor. B. Dynamics and Control. Two alternative control schemes were evaluated. In the first, both of the fresh feed streams, FOAand FOB,are flow controlled (or one is ratioed to the other). This is a control strategy that is quite commonly seen in plants, but it has major weaknesses as will be demonstrated below. One of the objectives of this paper is to clearly point out these problems and to illustrate them quantitatively by means of a numerical example. As sketched in Figure 2, reactor level is held by column feed. Column base level is held by bottoms. Reflux drum level is held by distillate recycle back to the reactor. Reflux flow rate is flow controlled. Distillate composition is not controlled since the recycle is an internal stream in the unit. Bottoms product purity is controlled by manipulating heat input. Note that this scheme violates the rule proposed by Luyben for liquid recycles since the streams in the recycle loop (F and D)are both on level control. We call this control structure scheme A. In the second control scheme, Figure 3, the total recycle flow rate to the reactor (distillate plus makeup A) is flow controlled. The makeup of reactant A is used to hold the level in the reflux drum. This level indicates the inventory of component A in the system. We call this control structure scheme B. Results of dynamic simulation studies of the process using these two control structures are given in Figures 4-7. Figure 4 shows what happens using scheme A when ! ,which the flow rate of makeup A is incorrect by only 1% is smaller than any real flow measurement device can achieve in most plants. The recycle flow rate D and the composition of component A in the reactor ramp up with time, and it takes more and more vapor boilup in the column to keep component A from dropping out the bottom. These trends would continue until the column floodsor some other constraint in heat input, heat removal, or pumping capacity is encountered. Figure 5 shows that if a pulse is made in the makeup flow rate of component A, the system goes through a
1156 Ind. Eng. Chem. Res., Vol. 32, No. 6, 1993 Table 11. Steady-State Designs for Complete Conversion Case D (lb-mol/h) 25 50 100 200 . , 0.1965 0.3212 0.4776 0.6346 z(1) (m.f.)" F (lb-mol/h) 151 126 201 301 4.71 2.81 1.22 1.80 RRmin 2.72 2.37 4.08 3.27 Dc (ft) V (lb-mol/h) 219 166 494 316 24/14 24/14 23/13 21/13 NTINF R (lb-mol/h) 141 169 294 216 energy (106 Btu/h) 2.08 2.73 6.17 3.95 capital costa (I@$) column 124 144 169 199 trays 4.1 3.3 5.2 6.7 heat exch 177 148 225 301 total capital 276 325 388 507 energy cost (103$/yr) 91 120 173 270 total cost (103$/p) 228 183 306 439 m.f. = mole fraction.
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400 0.7605 501 0.83 5.20 799 19/13 399 9.99
0.8446 901 0.48 6.53 1262 18/14 462 15.8
lo00 0.8636 1101 0.37 6.97 1439 18/14 439 18.0
1200 0.8770 1301 0.27 7.33 1589 17/15 389 19.9
1400 0.8867 1501 0.19 7.61 1716 17/15 316 21.5
1600 0.8942 1701 0.12 7.85 1825 18/16 225 22.8
238 3.9 412 658 438 657
290 12.0 554 856 691 976
311 13.2 603 928 788 1097
314 13.5 644 971 870 1194
327 14.3 677 1018 940 1279
353 15.9 704 1074 999 1357
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Figure 4. Scheme A 1% over-feed of component A.
Figure 5. Scheme A pulse in component A feed.
transient and lines out with a new recycle flow rate and a new reactor composition. These results quantitatively illustrate the basic flaw in scheme A both reactants cannot be flow controlled because flow measurementinaccuracymakesit impossible
to achieve perfect stoichiometric amounts of the two reactants in an open-loop system. Thus scheme A is not a workable scheme. Somehow the amount of component A in the system must be determined and the makeup of component A must be
Ind. Eng. Chem. Res., Vol. 32, No. 6, 1993 1157 115
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Figure 6. Scheme B reduction in recycle flow.
adjusted to maintain the inventory of component A at a reasonable level. Scheme B is a control structure that provides good control of the system. Figure 6 shows what happens using scheme B when the total recycle flow rate is reduced from 500 to 400 lb-mol/h. The system goes through a transient and ends up a t the same fresh feed flow rate for reactant A. The reflux drum level controller adjusts the flow rate of FOAto maintain the correct inventory of component A in the system. Note that the concentration of component A in the reactor, q l ) ,decreases when the recycle flow rate is decreased (Figure 6b, middle). This has no effect on the reaction rate because we have assumed the instantaneous reaction of component B. Figure 7 shows the response of scheme B for a step increase in component B makeup flow rate, FOB. The control system automatically increases the makeup of component A to satisfy the stoichiometry of the reaction.
Process with Two Recycle Streams We now look at the more common situation in which both reactants are present in the reactor since one-pass
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1158 Ind. Eng. Chem. Res., Vol. 32, No. 6, 1993 RECYCLE
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6. Calculate the feed to the first column:
FRESH FEED R
Figure 8. Process with two recycles.
(lb-mol),k = specificreaction rate (h-9, z(1) = concentration of component A in the reactor (mole fraction A), and z(2) = concentration of component B in the reactor (mole fraction B). Note that “moles” are not conserved in this system. A. Steady-State Design. The process is described by three component balances for each unit: reactor, column 1,and column 2. There are a number of ways to solve the nine nonlinear algebraic equations, but the following procedure was found to be straightforward and involved no iteration. 1. The flow rate of the product stream B 2 leaving the bottom of column 2 was fixed at 100 lb-mol/h. 2. The composition of this stream was specified to be xB2(1) = 0.01, XB2(2) = 0.01, and xB2(3) = 0.98. 3. The flow rate of the light recycle stream D 2 from the top of column 2 was specified (to be varied later in order to determine the optimum flow rate). The composition of this recycle stream was specified to be xD2(1) = 0.99, xD2(2) = 0, and xD2(3) = 0.01. 4. The flow rate of the heavy recycle stream B 1 from the bottom of column 1was specified (to be varied later in order to determine the optimum flow rate). The composition of this recycle stream was specified to be x ~ 1 ( 1 ) = 0, Z B ~ ( Z ) 0.99, and X B I ( ~ )= 0.01. 5. Calculate the feed to the second column:
7. The rate of production of product C is equal to must be equal to the rate of generation of component C in the reactor (assuming the two fresh feed streams contain no component C). Therefore the reactor volume can be calculated: B 2 ~ ~ 2 ( 3and ) , this
Note that there is a unique reactor size for each selected pair of light and heavy recycle flow rates. 8. Calculate the fresh-feed makeup flows of both component A (FoA)and component B (FOB). FOA
= Fz(l)+ VRkz(1)z(2) - DflD2(1)
(13)
FOB = Fz(2) + VRkz(1)z(2) - B1xB1(2) (14) 9. Calculate the minimum number of trays for each column from the Fenske equation and the minimum reflux ratio from the Underwood equation. Set the actual number of trays equal to 1.5times the minimum, and set the actual reflux ratio equal to 1.2 times the minimum. Results are given in Table 111. Figure 9 shows how several design parameters vary as the heavy and light recycle flow rates are varied. 1. Low flow rates of either light or heavy recycle result in very large reactor size and high capital cost but low energy cost. 2. As heavy cycle B 1 is increased, reactor holdup VR decreases,the concentration of component B in the reactor increases, energy cost increases slightly, and capital cost decreases slightly. There is a minimum in the total cost curve at some heavy-recycleflow rate because of the tradeoff between increasing energy cost and decreasing capital cost (due to the decrease in reactor size).
Figure 9. Effect of light and heavy recycle flow rates: (a, left top) on reactor holdup; (b, left bottom) on composition of component A in the reactor; (c, middle top) on annual energy cost; (d, middle bottom) on total capital cost; (e, right top) on composition of component B in the reactor; (f, right bottom) on total annual cost.
Table 111. Two-Recycle-Process Designs energy
total
203 218 232 245 214 229 243 224 240 253 266 234 242 250 256 263 263 278 291 303
574 551 555 568 554 537 544 548 535 542 555 549 540 539 541 546 570 562 569 581
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0.0906 0.1486 0.1300 0.1156 0.1040 0.1776 0.1661 0.1561 0.1471 0.1392 0.2538 0.2256 0.2030 0.1846
0.1600 0.2707 0.3553 0.4221 0.1541 0.2619 0.3451 0.1486 0.2538 0.3356 0.4010 0.1435 0.1981 0.2461 0.2886 0.3265 0.1300 0.2256 0.3020 0.3646
3. As light recycle D2 increases,reactor holdup decreases, the concentration of component A in the reactor increases, energy cost increases rapidly, and capital cost decreases. There is an optimum pair of light and heavy recycle flow rates. As can be seen in Figure 9f, the minimum total cost ($535500/y) occurs with a light recycle flow rate D2 = 20 lb-mol/h and a heavy recycle flow rate B1 = 40 lbmol/h. Table I11 gives more design details over a range of values of two recycle flows. All of the above calculations used a specific reaction rate k = 1 h-l. If other values of k are used, the optimum light and heavy recycle flow rates and the minimum total cost change. This occurs primarily because the size of the reactor depends directly on the value of k. The smaller the value of k, the larger the optimum recycle flow rates and the higher the total cost. For example, when k = 10 h-1, the optimum recycle flow rates are Dz = 5 and B1 = 10 lb-mol/h, and the minimum total cost is $409 OOO/yr. When k = 1 h-l, the optimum recycle flow rates are D2 = 20 and B1 = 40 lb-mol/h, and the minimum total cost is $535 OOO/yr. When k = 0.1 h-l, the optimum recycle flow rates are D2 = 40 and B1 = 70 lbmol/h, and the minimum total cost is $832 OOO/yr. B. Dynamics and Control. A number of alternative control structures were studied. No control structures that violated the recycle route proposed by Luyben (1992) were found to work. Unless one flow somewhere in the recycle loops was fixed, the recycle flow rate would grow to very high rates when a disturbance occurred or when additional throughput was desired. Two schemes were found to be fairly effective. Four control structures that are typical of the many studied are described below. In all the schemes, the following control structures were used: (a) the reflux flow rates on both columns are fixed; dual composition control was not used in order to keep the column control systems as simple as possible; (b) vapor boilup in the first column controlled the impurity of B in final product stream (the bottoms from the second column) through a composition/composition cascade strategy; (c) impurity of component A in the final product was controlled by vapor boilup in the second column; (d) base level in the second column was controlled by bottoms flow rate; (e) reactor temperature was controlled by coolant flow rate. 1. Scheme 1 (Figure 10): Both recycle flow rates are fied. Reactor level is controlled by reactor effluent. Freshfeed makeup of component A controls the level in the
I
,b-FEED B
ODUCT C
Figure 10. Scheme 1: fixed recycle flow rake.
reflux drum in the second column. Fresh-feed makeup of component B controls the level in the base of the first column. This scheme works well. Throughput can be changed by changing either the recycle flow rates (Figure 11) or the reactor temperature (Figure 12). Increasing recycle flow rates increases the concentrations of both components A and B in the reactor, and this increases the production of component C. Increasing reactor temperature increases the specific reaction rate, k,to increase the production of component C. The concentrations of reactants z(1) and z(2)both decrease slightly (see Figure 12d,bottom). 2. Scheme 2 (Figure 13): Fresh-feed makeup of component A is fixed. Fresh-feed makeup of component B is used to control reactor level. Reactor effluent is fixed. This scheme is similar to scheme B (inthe previous section) in that the limiting component is flow controlled into the reactor. The differenceis that here thelimitingcomponent is not completely converted per pass through the reactor. This scheme does not work. Figure 14 shows what happens when we attempt to increase throughput by increasing FOA. The system begins to fill up with component A, and the concentration of B in the reactor, q2), becomes so low that the reaction rate drops. 3. Scheme 3 (Figure 15): Fresh-feed makeup of component A is fixed. Fresh-feed makeup of component B is used to control the composition of component B in the reactor. Reactor level is controlled by reactor effluent. This scheme does not work because of the usnowball” effect. As shown in Figure 16,the flow rates of reactor effluent, light recycle, and heavy recycle change drastically when the feed rate is changed. Note that this control structure has only liquid level controllers in both recycle feedback loops, so snowballing can easily occur. 4. Scheme 4 (Figure 17): Fresh-feed makeup of component A is used to control reactor level. Fresh-feed makeup of component B is ratioed to the flow rate of freshfeed makeup of component A, and the ratio is changed by a reactor composition controller. Reactor effluent is flow controlled. This scheme works well. Throughput is adjusted by changing the flow rate of reactor effluent as suggested by Downs (1988). Figure 18 shows that as F is ramped up, both fresh feed makeup flows (FoAand FOB)are increased by the level and composition controllers. Notice that the concentration of component A (~(1)) must increase to give the required increase in the production rate of component C since the concentration of component B ( ~ ( 2 ) )is constant. Throughput could also be changed by changing reactor temperature as in scheme 1. It might seem that there is a third way of changing throughput in scheme 4,namely by changing the setpoint of the composition controller (component B composition in the reactor). However, at
*
1160 Ind. Eng. Chem. Res., Vol. 32, No. 6,1993
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Figure 12. Scheme 1: increase in reactor temperature.
Figure 11. Scheme 1: increase in both recycle flow rates.
For example, under normal conditions, z(1) = 0.13 and = 0.2538,giving a product z(1)z(2)= 0.033. Now if z(2) is increased to 0.317,the composition of component A decreases to z(1) = 0.098,giving a product z ( ~ ) z ( z=) 0.031. Likewise if z(2) is decreased to 0.164,the composition of component A increases to z(1) = 0.192,giving a product ~ ( 1 ) ~ (= 2 0.031. ) It should be recognized that an additional composition analyzer is required in scheme 4 compared to scheme 1. ?(2!
fixed reactor effluent flow rate, this is not an option. For both increases and decreases in the composition of component B in the reactor, the production rate decreased. The reason for this is that, with everything else constant, the composition of component A in the reactor, z(1), assumes a new value so as to make the product of z(1) and z(2) decrease.
Ind. Eng. Chem. Res., Vol. 32, No.6, 1993 1161 RCCYCLC R
RECYCLE R
FRESH
FRESH
Figure IS. Scheme 3: component A feed fixed;component B feed controls reactor composition; reactor effluent controls reactor level.
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20
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1162 Ind. Eng. Chem. Res., Vol. 32, No. 6, 1993 RECYCLE R
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This may be a serious drawback of scheme 4 in many processes because of the high capital and operating costa of some analyzers and because of their poor reliability.
Conclusion
Figure 17. Scheme 4 component A feed controls reactor level; component B feed controls reactor composition;reactor effluent flow rate Axed.
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This paper has studied a recycle process in which a second-order reaction takes place in a CSTR and reactor effluent is separated in two downstream distillation columns. One or two recycle streams from the separation section are returned to the reactor. These studies support the rule that the flow rate of one stream somewhere in a liquid recycle loop must be fixed in order to prevent "snowballing". They also support the rule that the flow rate of a makeup stream of a reactant can be fixed if and only if that component is completely consumed in one pass through the reactor. In the incomplete conversion case, reactant makeup streams should be brought into the process to control an appropriate level or composition.
Nomenclature
Time (hours)
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Ac = area of condenser (ft2) Aj = coefficient in Antoine equation AR = area of reboiler (ft2) Bj = coefficient in Antoine equation Bk = bottoms flow rate from kth column (lb-mol/h) C, = coefficient in Antoine equation Dk = distillate flow rate from kth column (lb-mol/h) D C k = diameter of column of kth column (ft) DR = diameter of reactor (ft) F = column feed flow rate = reactor effluent (lb-mol/h) Foj = fresh feed rate of component j to reactor (lb-mol/h) k = specific reaction rate (h-l) &k = feed tray number in kth column = pure component vapor pressure (psia) N T k = total number of trays in kth column Rk = reflux flow rate in kth column (lb-mol/h) 73 = reaction rate (h-1) T = temperature ("C) vk = vapor boilup in kth column (lb-mol/h) VR = reactor holdup (lb-mol) XBkj = bottoms composition in kth column (mole fraction of component j ) X D k j = distillate composition in kth column (mole fraction of component j ) zj = reactor composition (mole fraction of component j ) zoj = fresh feed composition (molb fraction of componentj ) aj = relative volatility of component j
Literature Cited Downs, J. Paper presented at a Workshop on Plant-Wide Control, Lehigh University, 1988. Luyben, W. L. Process Modeling, Simulation and Control for Chemical Engineers, 2nd ed.; McGraw-Hill: New York, 1990. Luyben, W. L. Dynamics and Control of Recycle Systems. 3. Alternative Process Designa in a Ternary System. Znd. Eng. Chem. Res. 1993, preceding paper in this issue. Tyreus, B. D.; Luyben, W. L. Tuning PI Controllers for Integrator/ Deadtime Processes. Znd. Eng. Chem. Res. 1992,31,2625-2628.
o.mm
0.0070
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40
Time (hours)
Figure 18. Scheme 4: throughput increased by increasing flow rate of reactor effluent.
Originally submitted June 22, 1992 Resubmitted for review March 1, 1993 Accepted March 16, 1993
