Plantwide Control of Continuous Multiproduct Processes - American

Ind. Eng. Chem. Res. 2003, 42, 2809-2825

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Plantwide Control of Continuous Multiproduct Processes: Three-Product Process Kulchanat Kapilakarn‡ and William L. Luyben* Process Modeling and Control Center, Department of Chemical Engineering, Lehigh University, Bethlehem, Pennsylvania 18015

A previous paper (Kapilakarn, K.; Luyben, W. L. Ind. Eng. Chem. Res. 2003, 42, 1890-1904) explored the design and plantwide control of a continuous process in which two reversible reactions produced two desired products M and D. Because chemical equilibrium established the amounts of the products leaving the reactor, the use of product recycle was required to operate over a range of product ratios M/(M + D). This paper extends the previous work to consider the more complex system in which there are three reactions producing components M, D, and T. Desired product ratios can span the entire ternary distribution space (M/D/T), depending on market conditions. Both steady-state design and dynamic control are explored for this threeproduct process, which features one reactor, four distillation columns, and two recycle streams. Several conventional control structures are studied in which the flow rates of the fresh feed streams are fixed or manipulated by level or composition/temperature controllers and the production rates are not directly set. An alternative “on-demand” control structure for “agile” manufacturing is also developed in which all three product streams are flow-controlled. The control system adjusts the conditions in the plant and the fresh feed streams to achieve the set product flow rates. This control structure requires no reactor composition measurement. The ratio of the fresh feeds is adjusted to give the desired production rates of M and T, and the recycles of D and T are adjusted to give the desired production rate of D. Appropriate liquid levels in the process are used as feedback information to trim the fresh feed ratio so that reaction stoichiometry is precisely satisfied. 1. Introduction The literature covering the design and control of many types of simple and complex chemical processes involving reaction sections, separation sections and recycles was reviewed in our previous paper.1 The process considered in the previous work has two reversible consecutive reaction that produce desired products M and D

A+BSM+C A+MSD+C Figure 1. General flowsheet for the three-product process.

The process consists of a reactor, three distillation columns, and two recycle streams. Both the design of the process and its control system depend on the desired production rates of the two products. The desired product ratio M/(M + D) depends on market conditions. Because chemical equilibrium limits the composition of the reactor effluent, the recycle of products back to the reactor is required for some desired product ratios. Both conventional and on-demand control structures were developed for this two-product process. This paper extends these studies to consider a more complex reaction system and process in which there are three consecutive reactions and three desired products (M, D, and T). The reactions are * To whom correspondence should be addressed. E-mail: [email protected]. ‡ Current address: Department of Chemical Engineering, Prince of Songkla University, Thailand 90112.

A+BSM+C A+MSD+C A+DST+C Two fresh feed streams of component A (at rate F0A) and component B (at rate F0B) are fed into the reactor, along with one or more recycle streams (at rates D1, RD, or RT). (See Figure 1.) The relative volatilities of components are RA ) 16, RB ) 8, RM ) 4, RC ) 2, RD ) 1, and RT ) 0.5. A direct distillation separation sequence is assumed, so unreacted A and B in the reactor effluent F are recycled back to the reactor from the top of column 1 in stream D1. Some of component M is also recycled in this stream, since the steady-state design calls for about 10 mol % M in the D1 stream. This “implicit” M recycle is somewhat less expensive than the alternative of recycling M in an “explicit” recycle stream from the top of column 2.

10.1021/ie030058x CCC: $25.00 © 2003 American Chemical Society Published on Web 05/10/2003

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Ind. Eng. Chem. Res., Vol. 42, No. 12, 2003 Table 1. Design Parameters for Each Case

Figure 2. Triangular diagram for the three-product process.

Byproduct C comes off the top of column 3 in stream D3, product D off the top of column 4 in stream D4, and product T off the bottom of column 4 in stream B4. The steady-state economic designs of the process at some of the design points call for explicit recycle streams of either D or T (from the top or bottom, respectively, of column 4). Parameter values and design assumptions are the same as those used in the two-product process,1 with the third reaction having the same equilibrium constant and reaction rates as reactions one and two. 2. Steady-State Design Ideally, the process should be able to produce any distribution of the three products. Several specific cases are initially explored at different product ratios (M/D/ T), as shown in the triangular diagram in Figure 2. The abscissa of this diagram gives the fraction of the total products produced that is component D. The ordinate is the fraction that is M. The three corners of the triangle correspond to the three pure products. The four points shown in Figure 2 correspond to four different design cases with four different product distributions. The base case is chosen to have a ratio of M/D/T equal to 30/30/40 (in mole percentages), which is near the middle of the ternary diagram. The three other cases have the following molar product ratios: case A, 20/60/ 20; case B, 20/20/60; and case C, 50/20/30. These cases represent situations in which more is required of one product than of the other two products. The line going through each point corresponds to the product distribution when only F0A is varied with F0B and recycle rates RD and RT fixed at their values for the particular case. The arrows show the direction of increasing F0A. As more A is introduced, less M is produced because there is more A available to react with the M and D produced in the first two reactions. Thus, the product distribution moves away from the top of the ternary diagram (100% M) toward the base. In some regions of the product-distribution space, the optimum steady-state design indicates that some of the product D produced in the reactor should be recycled back to the reactor. In other regions, the optimum design calls for a recycle of some of the product T (See Table 1.) These regions are not overlapping, i.e., both D and T are not recycled simultaneously. In some regions, there is a recycle of M back to the reactor from the top of the first distillation column along with the reactants A and B. If the production rate of T is high, the reactions require higher concentrations of components M and D in the reactor. The excess M is recycled by increasing the impurity of M in the top of the first

product split M/D/T (mol %) F0A (lb‚mol/h) F0B (lb‚mol/h) RD (lb‚mol/h) RT (lb‚mol/h) F (lb‚mol/h) zA zB zM zC zD zT D1 (lb‚mol/h) xD1A xD1B xD1M NT1 (optimum) NT1 (design) VS1 (lb‚mol/h) NT2 (optimum) NT2 (design) VS2 (lb‚mol/h) NT3 (optimum) NT3 (design) VS3 (lb‚mol/h) NT4 (optimum) NT4 (design) VS4 (lb‚mol/h) DC1 (ft) DC2 (ft) DC3 (ft) DC4 (ft)

A

B

base case

C

20/60/20

20/20/60

30/30/40

50/20/30

511.01 256.67 0 89.59 2193.52 0.196 66 0.264 78 0.171 35 0.233 32 0.070 20 0.063 69 1336.25 0.323 18 0.434 91 0.241 89 16 28 2705.91 30 35 715.55 31 32 1305.09 29 29 608.91 8.99 5.70 8.71 6.75

613.07 258.21 107.26 0 2288.54 0.394 09 0.112 16 0.089 08 0.268 54 0.070 26 0.065 87 1310.00 0.689 07 0.195 92 0.115 01 16 28 2269.57 30 35 851.50 30 32 1505.38 29 29 558.95 8.24 1.21 9.36 6.47

536.27 257.18 19.74 0 1751.94 0.381 59 0.116 87 0.084 31 0.306 66 0.056 09 0.057 48 938.12 0.713 17 0.218 01 0.068 82 17 28 1619.73 30 35 813.67 32 32 1253.90 29 29 346.40 6.96 6.07 8.54 5.09

459.22 256.27 57.54 0 1488.03 0.234 02 0.124 30 0.117 60 0.309 33 0.073 95 0.050 80 715.00 0.487 47 0.446 39 0.066 05 28 28 1430.00 35 35 823.82 29 32 1136.37 29 29 364.12 6.54 6.11 8.13 5.22

Figure 3. Reactor size (lb‚mol) for a product split of 30/30/40.

column (raising the set point of the column 1 tray temperature controller). The excess D is recycled by increasing the flow rate of stream RD. If the production rate of T is low, excess T is recycled in stream RT. The implications of these various recycles on the control structure necessary to achieve different product splits are profound, as we will see later in this paper. Figure 3 shows the effects of reactor size on energy and capital costs for the base case. The reactor size of 8000 lb‚mol gives the lowest total annual cost. Table 1 shows the optimum steady-state design for each of the desired product splits specified in cases A-C and the base case. The worst-case design parameters are used for each column, i.e., we select for each of the columns in the final process design the column with the largest diameter and number of trays found in any of the design cases. In addition, the diameter of each column is made 10% larger than the worst-case diameter. The four columns are design with diameters of 10.8, 7.5, 11.2, and 7.8 ft, respectively. The corresponding largest reboiler and condenser sizes are selected.

Ind. Eng. Chem. Res., Vol. 42, No. 12, 2003 2811 Table 2. Sensitivities for CS1 RD, RTa (lb‚mol/h) change

2% ∆F0A

2% ∆F0B

40 RD, 0 RT

0 RD, 20 RT

∆F (%) ∆D2 (% M) ∆D4 (% D) ∆B4 (% T)

3.48 -8.00 2.00 4.15

-0.73 9.04 0.28 -2.45

0.97 7.23 -13.91 5.44

2.18 -10.52 23.18 -8.91

a

Figure 4. Relationship between the ratios F0A/F0B) and M/(M + T). Notation: A is for a product M flow rate of 75 lb/mol, and B is for a product M flow rate of 100 lb‚mol/h.

3. Constraints and Degrees of Freedom In the two-product process, fixing the fresh flow rates of A and B produces specific amounts of the two products M and D, i.e., there is a unique relationship between the pure fresh feeds and the net production rates of the two products. In the three-product process, fixing the two fresh feeds does not uniquely set the amounts of the three products produced. Many distributions of M, D, and T can be achieved for the same F0A and F0B flow rates. The stoichiometry of three reactions requires that

F0A ) M + 2D + 3T F0B ) M + D + T where M, D, and T represent the flow rates of the three products. To solve for D, T, and M, one variable must be specified. For example, if D is specified, then solving for M and T gives

M ) (3F0B - F0A - D)/2 T ) (F0A - F0B - D)/2 According to reaction stoichiometry, the relationship between the ratio of the two fresh feeds (F0A/F0B) and the ratio of the production rate of M to the sum of the production rates of M and T [M/(M + T)] is given by

F0A M ) [-2(1 - D/F0B)] + (3 - D/F0B) F0B M+T

(

)

The D/F0B ratio can vary from 0 to 1. Figure 4 shows this relationship for different production rates of D. These curves are generated by fixing the production rates of all products and then calculating the operating conditions required to achieve these flow rates while minimizing the energy consumption. Two variables are adjusted to achieve these objectives: the impurity of M in the distillate of the first column, xD1M (the implicit recycle of M back to the reactor) and the recycle rate of either D or T (RD or RT). Each line in Figure 4 represents a production rate of product D. The results of these calculations show that the lines are almost collinear. This means that the F0A/ F0B ratio affects primarily the production rates of M and T, and that the production rate of component D is not strongly affected by the fresh feed ratio. If we compare

RD at steady state is 19.74 lb‚mol/h, and RT is 0.

Figure 4 with the equation above, each D production rate gives a ratio D/F0B that is almost constant. Thus, the basic stoichiometry indicates that the relationship between the fresh feed flow rate ratio and the rate ratio M/(M + T) should be close to a linear function. When the desired M/(M + T) ratio is high, the process must produce either more product M or less product T. This means that the system requires less A. Therefore, the F0A/F0B ratio decreases. A simple straight-line correlation between the F0A/F0B ratio and the M/(M + T) ratio is used later in one of the control structures. Table 2 shows the sensitivities of the production rates of M, D, and T to changes in the fresh feed flow rates (F0A and F0B) and recycle stream flow rates (RD and RT). These results again demonstrate that changing F0A or F0B has more effect on the production rates of M and T than on that of D. In addition, the results show that changing the recycle flow rates has more effect on the production rate of D. Increasing F0A increases the production rates of D and T but decreases the production rate of component M because M is consumed by component A to produce components D and T. However, D is also consumed by component A to produce T. The net effect is larger changes in M and T (in opposite directions) than in D. Increasing F0B has the reverse effects. Changes in recycle flow rates RD and RT have more effect on the production rate of D than on the production rates of M and T. Increasing RD puts more of component D back into the reactor, drives the equilibrium reactions in the direction of decreasing the production of D, and also results is smaller increases in the production of T and M. Increasing RT puts more of component T back into the reactor, drives the equilibrium reactions in the direction of increasing the production of D, and also results in smaller decreases in the production of T and M. These sensitivity tests provide process insight into which variables need to be adjusted as the desired production rates of the three products change. We use this insight to develop control structures in the next section. 4. Control Structures In the two-product process, we explored both conventional and on-demand control systems. The conventional control structures developed for the three-product process in this paper are similar to those used in the twoproduct process, but some interesting differences in performance are found. The on-demand control structure developed in this paper for the three-product process is more complex than that used in the twoproduct process and requires override controls to achieve the required switching of control loops as product production rates change and the process moves around in the product-distribution space. For example, sometimes the reflux drum level in a column is controlled

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Ind. Eng. Chem. Res., Vol. 42, No. 12, 2003

Figure 5. CS1 conventional control structure.

by an appropriate recycle flow, and sometimes this level is controlled by manipulating the feed to the column. A. Conventional. The flow rates at the front end of the process (fresh feeds and/or reactor effluent) are fixed, as are the flow rates of the recycles. The process then generates certain amounts of products, which come off the columns on level control. B. On-Demand. The three products are flow-controlled. The control structure must adjust conditions in the process (recycle flow rates) and bring in the correct amounts of the two fresh feeds to achieve the set production rates. Several control loops are used in all of the control structures discussed below, both conventional and ondemand: (1) Reactor temperature is controlled by the reactor cooling water flow rate. (2) Tray temperature controllers in each column adjust reboiler heat inputs. (3) Column reflux flows are ratioed to column feeds. (4) Column pressures are controlled by condenser heat removals. (5) The distillate streams D1 and D3 control the reflux drum levels of the first and the third columns, respectively. 4.1. Conventional Control Structures. In the several conventional control structures discussed below, the product stream flow rates are not set. The product flow rates are manipulated to control liquid levels in the corresponding distillation columns: the column 2 reflux drum level sets stream D2 (product M), the column 4 reflux drum level sets stream D4 (product D), and the column 4 base level sets stream B4 (product T). The fresh feed streams are set directly or are manipulated by some other controller (level, composition, or column temperature) near the front of the process. The RD and RT recycle streams are flow-controlled. Note that, in the base-case design, the RT recycle rate is 0. Column base levels manipulate the bottom flow rates, and reflux drum levels control the distillate flow rates.

Three alternative conventional control structures are shown in Figures 5-7. These structures are similar to those used in the two-product process. 4.1.1. CS1. The flow rates of feeds F0A and F0B, and recycles RD and RT are flow-controlled. The RT recycle rate is 0 for the base case. The reactor effluent is controlled by the reactor level. The structure is shown in Figure 5. This simple, straightforward intuitive control structure was shown in the previous paper to not work well for the two-product process. The opposite finding is observed for the three-product process. This unexpected difference is due to the strict relationship between fresh feeds and product production required in the two-product process, which is no longer true in the three-product process. Thus, the three-product process is less restricted and can accommodate changes in fresh feeds without running into equipment constraints. 4.1.2. CS2. Reactor effluent and product recycles are flow-controlled. The reactor level is controlled by manipulating F0B. Two alternatives are explored for manipulating the fresh F0A feed. In the first (CS2A, Figure 6A), a reactor composition measurement is used. In the second (CS2B, Figure 6B), a temperature in the top of column 1 is used to infer the amount of A in the system. This structure eliminates the need for an expensive composition measurement. 4.1.3. CS3. Figure 7 shows this structure in which reactor effluent is flow-controlled and the flow of F0B is ratioed to the reactor effluent flow rate. The reactor level is controlled by manipulating F0A. No composition measurement is used. The performances of these conventional control structures are discussed in section 5. All of them provide stable basic regulatory control of this complex process. However, none permits the direct setting of any of the three product flow rates.

Ind. Eng. Chem. Res., Vol. 42, No. 12, 2003 2813


4.2. On-Demand Control Structure. One of the main concepts in “agile manufacturing” is the ability of the process to rapidly deliver high-quality products at whatever flow rate the customer requires. The ideal way to achieve this is to permit the downstream customer

to set the flow rates of the product streams. This “ondemand” control is inherently more difficult than conventional control2 and more challenging for the process and control engineer, but its value in supply-chain management and enterprise integration can be signifi-

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Figure 8. CS4 on-demand control structure.

cant. These on-demand control structures have been used for many years in plant utility systems (steam, power, and refrigeration) and in the production of industrial gases (oxygen, nitrogen, hydrogen, etc.). Figure 8 shows an on-demand control structure that was developed for this complex three-product process. All three of the product streams are flow-controlled. This structure features variable loop configurations as determined by an override control system. The override

control structure in the last column is used because some product ratios require RD recycle while others require RT recycle. The override controls on the other columns are used to handle product ratios that require the production of large amounts of M, which leads to no recycling of component M from column 1. 4.2.1. Normal Conditions. Under normal conditions near the base case, the base level in each column is held by manipulating the feed flow rate to that column. The


Figure 9. CS4 on-demand control structure with override controllers.

reactor level is controlled by manipulating F0B. Thus, under normal operation, level control is in the opposite direction of flow. When product flows are increased, the control system increases the feed flow rates to the columns and to the reactor. The fresh feed flow rate F0A is ratioed to F0B. As discussed below, this ratio is adjusted by the output signal from the function f shown at the left side of Figure 8. The function f looks at the flow rates of products M and T (streams D2 and B4) and predicts the required F0A/F0B ratio. The predicted ratio is then trimmed by two level signals (reflux drum level in column 2 and base level in column 4), which provide feedback information about whether too much or too little A or B is being fed. This is discussed in more detail in the next section. The reflux drum level in column 2 is controlled by changing the set point of the tray temperature controller in column 1 (giving an implicit recycle of M). The reflux drum level of column 4 is controlled by the RD recycle rate, and the RT recycle rate is 0 under these normal conditions. 4.2.2. Override Conditions. Figure 9 shows the control structure with the overrides included for the entire process. Dotted lines represent control signals under the normal conditions, and dashed lines represent control signals under override conditions. High selectors are used to smoothly transition between normal and override conditions. The low limiter on the temperature set point of the first column prevents a situation in which significant amounts of component B are lost out the bottom of the first column. When a higher production rate of M is demanded, the signal from the reflux drum level of column 2 decreases the temperature set point of the first column to drop more M out the bottom and send it into

column 2. If the low-temperature limit is reached and if the level continues to drop, a low reflux drum level override controller takes over control of the column 2 feed rate through the high selector (HS) on the valve in the stream B1. The base-level control structures in columns 2 and 3 are then automatically realigned so that the high base-level override controllers take over control of each of the bottom valves through the appropriate high selectors. Thus, the control structure under this override condition has levels controlled in the direction of flow, as in a conventional control structure. Figure 10 shows the override control structure for column 4 by itself to present more detail of the setup by showing explicitly each of the override controllers. Dotted lines represent control signals under normal conditions, and dashed lines represent override conditions. When more product D is required and the set point of the flow controller on D4 is increased, the reflux drum level in column 4 drops. The level controller starts decreasing of recycle RD back to the reactor. When the flow rate of RD goes to zero, the reflux drum level continues to drop. A low-level override controller detects this low level and takes over control of the feed to column 4 by using a high selector on the valve in stream B3. This is accomplished by having the output signal of the override controller (OR4) increase as the reflux drum level decreases. This signal is sent to a high selector (HS in Figure 10) that selects the highest of three signals: one from the normal base-level controller on column 4 (NB4), one from the high base-level override controller on column 3 (OB3), and one from the low reflux drum level override controller on column 4 (OR4). Increasing the feed to column 4 will start to increase the base level, and the output signal (OB4)

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Figure 10. Override control structure for column 4.

Figure 11. Normal and override level controller signals for column 4.

from the high base-level override controller will increase to start recycling RT back to the reactor. Figure 11 shows how the normal-level controller signals and the override controller signals vary as the column 4 reflux drum (MD4) and base (MB4) levels change. The trajectories of the signals as the process transitions from normal control to override control are shown by numbers in this figure. 4.2.3. Adjustment of Fresh Feed Ratio. Figure 12 provides details about how the F0A/F0B ratio is set by the amounts of products M and T. The relationship between the F0A/F0B and M/(M + T) flow rates, as given in Figure 4, shows that the fresh feed ratio has a large effect on the production rates of M and T but less of an effect on the production rate of D. The flow rate of D2 (product M) and the flow rate of B4 (product T) are measured, and the ratio M/(M + T) is calculated and fed into function f1, whose output is

the calculated fresh feed ratio. The correlation derived from the results shown in Figure 4 is

F0A/F0B ) -1.438[M/(M + T)] + 2.72 If this correlation were perfect, we would only have to use this “open-loop” setting of the fresh feed ratio. Even if the correlation were perfect, flow measurement inaccuracies or changes in the fresh feed compositions would inevitably result in some misbalance between the amounts of A and B fed into the system. Because no A is lost in any product stream and only a small amount of B is lost in D2, the amounts of A and B fed must be precisely balanced, down to the last molecule. This balance can only be achieved by some type of feedback of information about the component inventories in the system. Thus, the ratio calculated in f1 must be adjusted. This is accomplished by block f2, which receives signals from


Figure 13. (10% changes in F0A or F0B for CS1.

Figure 12. Adjustment of fresh feed ratio.

the level transmitters of the column 2 reflux drum and column 4 base. The changes in these two levels from their steady-state values tell us if too much or too little A is being fed into the system. The empirical f2 relationship used is

f2 ) 1 + k1

(

MD2

MsD2

) (

- 1 - k2

MB4

MsB4

)

-1

If too much A is fed into the system (for the fixed production rates of M, D, and T), the process produces more T than desired and less M. This is detected by an increase in the base level in column 4 (MB4) and a decrease in the reflux drum level in column 2 (MD2). Thus, these two signals provide some feedback information to trim up the fresh feed ratio calculated from the correlation. The values of k1 and k2 are empirically derived, as discussed in the next section, by on-line tuning.

Figure 14. 10% changes in F for CS2A and CS2B.

5. Results and Discussion Dynamic simulations of the entire process with the various control structures were performed. The controller tuning methods used were the same as those discussed in the two-product process.1 5.1. Conventional Control Structures. 5.1.1. Control Structure CS1. Figure 13 shows the responses of the system for several 10% step changes in F0A and F0B when CS1 is used. The process has no trouble with these disturbances. Increasing F0A results in the production of more D and T but less M. Increasing F0B results in the production of more M, less T, and slightly more D. It should be noted that the two-product process with this type of control structure cannot handle modest decreases in F0A or increases in F0B. This occurs because of the unique relationship between the flow rates of the fresh feeds and the production rates of the two products. For given fresh feed flow rates, the process must be able to adjust itself for the precise amounts of products dictated by the stoichiometry. In trying to achieve this,

Figure 15. 10% impurity of B in F0A for CS2A and CS2B.

constraints in the reactor or columns can be encountered for some fresh feed flow rates. In contrast with the two-product process, the threeproduct process can vary the relative amounts of the three products for given fresh feed flow rates. This increased flexibility results in the CS1 control structure being able to handle feed disturbances without hitting process constraints. 5.1.2. Control Structure CS2. Results for the two different versions of CS2 are presented in Figures 14 and 15. In the first (CS2A), the composition of A in the reactor is measured and controlled by the fresh feed flow

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Figure 16. Change in F for CS3; F0A or F0B ratioed to F.

rate F0A. In the second (CS2B), the temperature at the top of column 1 is measured and controlled by F0A. The temperature is used to infer the amount of component A at the top of the column, and this provides information about the inventory of A in the system. Inferring composition from temperature works in this column because the material near the top of column 1 is essentially a binary mixture of mostly A and B. The lower the temperature, the higher the concentration of the lighter component A. Obviously, the use of a temperature measurement is preferred to the use of a composition measurement because of lower cost and higher reliability. Note that two temperature controllers are used in column 1 with the CS2B control scheme. The results given in Figure 14 are for a +10% step increase in reactor effluent flow rate F. The increase in F lowers the reactor level, and the level controller increases F0B. The composition of A in the reactor drops, and the reactor composition controller increases F0A when CS2A is used. When CS2B is used, the increase in F immediately affects the first column, causing an initial decrease in the top temperature, and the top temperature controller reduces the flow of F0A. As more A is consumed in the reactor, less is fed into the column, so the temperature at the top starts to increase (less light A and more heavy B). Then, F0A is increased by the top temperature controller. The changes in F0A for control structure CS2B are considerably larger than those encountered when CS2A is used. However, both control structures provide stable regulatory control of the process. The performance of CS2A is better, but it requires a composition measurement. Note that the final steady-state reactor compositions, the fresh feed flow rates, and the production rates of the various products are different for the two structures. Structure CS2A ends up producing more M and D but about the same amount of T. Structure CS2B ends up producing less M but more D and T. Another disturbance is considered in Figure 15. The composition of fresh feed F0A is changed from 100% A to 90% A and 10% B. Structure CS2A sees the change in feed composition immediately and adjusts F0A quickly to compensate. Structure CS2B does not see this disturbance for some time because the reactor composition changes slowly (reactor residence time is 6 h). It takes about 10 h for the top temperature loop to respond. Note that the feed composition disturbance does not change the final steady-state production rates when

CS2A is used because the net molar flow rates of A and B entering the process remain the same. 5.1.3. Control Structure CS3. The responses of the system to step changes in reactor effluent flow rate F are shown in Figure 16 for control structure CS3 (see Figure 7). In the two-product process, we found that ratioing F0A to F worked. In the three-product process, we found that ratioing F0A to F does not work, as illustrated in Figure 16. The increase in F0A results in a gradual accumulation of A in the system, which eventually shuts down the process. However, ratioing F0B to F does work. This difference in the effectiveness of these control structures for the two similar processes is easily explained by considering the difference in the steady-state amounts of A in the reactor. The two-product process has a reactor composition of 20 mol % A and a D1 recycle composition of 36 mol %. (See Table 2 in the two-product process paper.1) The three-product case has much higher amounts of A (zA ) 0.38 and xD1A ) 0.71; see Table 1). This makes it more difficult for the reactor to compensate for changes in zA by changing zB such that the reaction rate (which depends on the product of the compositions zAzB) can consume all of the reactants fed. If F0B is ratioed to F, the control system works well for the three-product process. The amount of B in the reactor is less than the amount of A (zA ) 0.38 and zB ) 0.12). Therefore, zB can change enough to balance the reaction rates with the components fed. 5.1.4. Comparison of Conventional Control Structures. Figure 17 compares the responses of CS1, CS2B, and CS3 when the composition of the fresh feed F0A is changed from pure A to 90 mol % A and 10 mol % B. All three structures can handle this disturbance, but the response times for CS2B are lower than those for the other structures. When CS1 is used, the change in the F0A fresh feed composition is the same as the response to a decrease in flow rate. The lower amount of A increases the production rate of M but decreases the production rates of both D and T. The process would respond in the reverse direction if the composition of F0B were changed from pure B to contain some A impurity. The responses of CS2A and CS2B were discussed in the previous section. When CS3 is used, changes in the F0A composition have no effect on the flow rate of F0B because F0B is ratioed to the reactor effluent flow rate F. When less A enters in F0A, there is less A leaving the reactor and


Figure 17. Comparison of 10% impurity in F0A for CS1, CS2B, and CS3.

less A in the D1 recycle. The flow rate of D1 decreases, and the reactor level controller brings in more F0A. Thus, the process ends up making more M and D but less T. This occurs because more B is being fed and the ratio of the net molar flow rates of components A and B in the two fresh feeds changes in the direction of less A. This results in the production of more M and less T. 5.2. On-Demand Control Structure. The ondemand control structure for the three-product process is shown in Figure 8, and its operation was discussed in Section 4. The tuning constants k1 and k2 in the function f2 were determined empirically. Small values of these parameters were used so that the effects of changes in levels during transient periods did not make large undesirable adjustments in the fresh feed ratio. 5.2.1. Empirical Tuning of k1 and k2. Figure 18 shows the effects of k1 and k2 used in f2 (see Figure 11) when the process at the base-case product ratios (30/ 30/40) is disturbed by changing the composition of the fresh feed F0A from 100 mol % A to 95 mol % A and 5 mol % B. When no feedback trim is used (k1 ) k2 ) 0), the imbalance between the A and B fed into the process results in the process slowly being depleted in A and filling up with B. The recycle D1 steadily increases until a constraint shuts the unit down. It is desirable to use small values of these parameters so that they do not produce large changes in the fresh feed ratio immediately after the production rates are changed. Following this logic, the values selected are k1 ) 0.2 and k2 ) 0.1. Figure 19 shows the response of the system for a transition from the base case to case C for two sets of values of k1/k2. Both successfully make the transition, but the larger pair of parameter values produces larger changes in the fresh feeds during the initial transient period, which results in a large change in the purity of the M product leaving in stream D2. Note that, for both sets of parameters, the process ends up at the final steady state with exactly the same fresh feeds (because the three products are the same), but other variables in the process are different, including the reactor compositions and recycle flow rates. 5.2.2. Override Control Structure. Figure 20 shows what happens when we transition from the base case to a case in which a large amount of M is desired (60/ 20/20). When the flow rate of D2 increases, the reflux drum level of column 2 decreases, and the level control-

ler decreases the set point of the column 1 temperature controller. When the low-temperature limit is encountered, the drum level continues to decrease. The lowlevel override controller produces the signal OR2 (shown in the top left graph in Figure 20B), which eventually becomes larger that the signal from the column 2 baselevel controller (NB2), which normally controls the feed to column 2. The high selector then switches the control structure to control the reflux drum level with feed flow B1 (shown in the top right graph in Figure 20B). The higher feed to column 2 raises its base level, which increases the OB2 signal from the high base-level override controller. When this signal is higher than the normal signal NB3 from the column 3 base-level controller, stream B2 is manipulated to control the column 2 base level. The feed to column 4 is normally set by the column 4 base level (signal NB4), but it can be set by signals OB3 or OR4 from the two other override controllers. As the bottom left graph in Figure 20B shows, one of these override signals is only briefly larger than the normal signal. 5.2.3. Transitions from the Base Case to Cases A, B, and C. Figure 21 shows the responses of several important variables as the set points of each of the product flow controllers are changed such that the process moves from the base case to cases A, B, and C. The total production of all products is kept constant in these cases, i.e., the sum of the flow rates of D2, D4, and B4 is constant, which means the final flow rate F0B of fresh B into the system does not change. Switching to Case A. The disturbances are decreasing M (D2 changed from 75 to 50 lb‚mol/h), increasing D (D4 changed from 75 to 150 lb‚mol/h), and decreasing T (B4 changed from 100 to 50 lb‚mol/h). Because a large amount of component D is removed, the level of the reflux drum in column 4 drops. The signal from the level controller shuts off RD, and the low-level override controller brings in more column 4 feed. The base level of this column increases because of the increase in column feed and the decrease in bottom product withdrawal. Therefore, recycle RT is required. In the second column, the reflux drum level increases because of the decrease in D2. The reflux drum level controller in column 2 increases the set point of the tray temperature controller in column 1 to recycle more component M back to the reactor from the first column. The fresh feed flow rate F0B starts increasing because of the decrease in the flow rate of RD and the increase in F resulting from the increases in the flow rates of B3, B2, and B1. The process takes about 60 h to reach the new steady state. Switching to Case B. The disturbances are a decrease in M of 25 lb‚mol/h, a decrease in D of 25 lb‚ mol/h, and an increase in T of 50 lb‚mol/h. The override controls on column 4 do not come into play. Column feed is set by the base-level controller. The higher level in the reflux drum increases the flow rate of RD. The flow rate changes in M and T increase the F0A/F0B ratio. Because more F0A is fed to the reactor, the composition of M in the reactor effluent, zM, is lower than in case A. The control tray temperature in the first column is lower than in case A because there is less M to recycle back to the reactor in D1. Switching to Case C. The disturbances are an increase in M of 50 lb‚mol/h and decreases in D and T of 25 lb‚mol/h each. The signal from the base-level

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Figure 18. Effect of k1/k2 (5% impurity in F0A).

Figure 19. Comparison of k1/k2 for transition to case C.

Figure 20. Override control for transition to the 60/20/20 case.

controller of column 4 decreases the column feed flow rate to maintain the base level. This tends to decrease the reflux drum level of this column. However, the reduction in the flow rate of D4 tends to increase reflux drum level. The net result of these two competitive effects on reflux drum level is less of an increase in the flow rate of RD than occurs in case B in which the two effects act in the same direction. Case C has the highest M/(M + T) ratio of the three cases, which means the fresh feed flow rate ratio F0A/F0B is the lowest in case C. Thus, less F0A is fed to the reactor than in the other cases.

A problem was encountered during this transition. The impurity of B in stream D2 (the M product) increased to an unacceptable level (changed from 0.5 to 2 mol %). This indicates that the simple temperature control in column 1 used to infer compositions cannot handle this situation. It would be necessary to use a direct composition measurement on the bottoms from column 1 or develop an improved inferential control scheme to solve this steady-state product-quality problem. 5.2.4. Attainable Region in Product-Distribution Space. These simulation results demonstrate that the


Figure 21. Transition to cases A-C for CS4. Table 3. Transitions from the Base Case M/D/T (%)

F0A (lb‚mol/h)

F0B (lb‚mol/h)

F0A/F0B

M (lb‚mol/h

D (lb‚mol/h)

T (lb‚mol/h)

RMa (lb‚mol/h)

RD (lb‚mol/h)

RT (lb‚mol/h)

5/5/90 10/10/80 40/10/50 20/20/60b 10/40/50 10/80/10 20/60/20b 50/40/10 30/40/30 25/40/35 20/40/40 30/35/35 40/30/30 50/30/20 50/20/30b 60/20/20 80/10/10

723.56 686.36 534.55 611.6 614.02 441.58 503.16 392.47 507.25 535.72 564.08 523.78 482.49 426.8 454.38 391.08 306.21

255.71 255.19 258.46 256.25 255.31 216.16 250.44 259.94 257.67 257.75 258.07 259.91 260.59 260.47 260.13 255.9 257.4

2.83 2.69 2.07 2.39 2.40 2.04 2.01 1.51 1.97 2.08 2.19 2.02 1.85 1.64 1.75 1.53 1.19

12.53 25.06 99.55 50.05 25.05 25.06 50.05 122.71 74.76 62.41 50.00 74.76 99.16 122.36 122.60 137.59 152.38

12.69 25.38 25.38 50.76 101.52 203.05 152.28 101.52 101.52 101.52 101.52 88.83 76.14 76.14 50.76 50.76 25.38

227.74 202.44 126.52 151.83 126.52 25.30 50.61 25.30 75.91 88.57 101.22 88.57 75.92 50.61 75.91 50.61 25.30

268.50 238.37 87.78 159.67 150.02 203.43 174.58 69.82 93.61 104.40 116.53 91.86 66.05 42.08 36.90 8.07 0.38

246.9 205.43 126.54 113.20 10.87 0 0 0 0 0 0 12.90 26.57 23.08 56.43 50.35 147.86

0 0 0 0 0 410.39 246.68 91.82 36.54 21.43 4.50 9.06 19.79 48.04 15.42 46.21 193.44

a

RM is the amount of component M in D1. b 20/20/60, 20/60/20, and 50/20/30 are cases B, A, and C, respectively.

proposed on-demand control structure CS4 can effectively handle the large disturbances associated with transitioning from the base case to cases A-C. Additional runs were made to explore the limits of the control system. Table 3 gives steady-state operating conditions for a number of product split ratios. The implicit recycle of M in stream D1 is shown as RM, which has a flow rate of D1xD1M. Some recycle of M is required in all cases. The recycles of D and T are required in some cases, but usually both are not required at the same time. Simultaneous recycle of D and T is needed only in those cases where a large amount of M is desired. As one would expect, operation near the T corner of the product-distribution space requires no RT, and operation near the D corner requires no RD. As the desired operating point moves up toward the M corner, less and less recycle of M is required. This attainable region (see Figure 22) depends on the process design, the control structure, and the tuning of the controllers (particularly the override controllers). These results demonstrate that the given process with the proposed CS4 control structure is capable of covering a very wide range of product production rates. 6. Different Separation Configurations In the case study used in this work, the relative volatilities among the components leaving the reactor are RA > RB > RM > RC > RD > RT. The direct separation

Figure 22. Attainable region for CS4.

configuration removes A and B first, then M, then C, then D, and finally T. Suppose that the volatilities of the components are different. Can the on-demand control structure proposed in this work be applied? The discussion below attempts to address this question. If the chemistry is similar to the chemistry considered in this work (consecutive reversible reactions), we believe that the control structure can be applied for other systems with different relative volatilities. The on-

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Figure 23. CS4 with separation sequence M/T/D.

demand control structure for this chemistry requires that the production rates of M and T be established by adjusting the fresh feed flow rate ratio F0A/F0B and the production rate of product D be established by adjusting the recycle flow rate RD or RT (and implicitly the recycle flow rate of M in stream D1). The reactor sees fresh feed and recycle flows, and it does not care where they come from. Therefore the configuration of the separation section should not affect the reactions in the reactor, at least under steady-state conditions. The hypothesis is that the proposed on-demand control structure is generic and can be applied in other chemical systems with a variety of relative volatilities, provided that the basic chemistry is the same. Verification of this hypothesis will be the subject of future work. However, to illustrate what we expect the control structures to be for systems with different separations sections, six cases are considered. Case 1: RM > RD > RT. This is the situation used in the process considered in this paper, and it has the ondemand control structure shown in Figure 9. Case 2: RM > RT > RD. The control structure, shown in Figure 23, is quite similar to case 1. Now, recycle RD comes from the bottom, and recycle RT comes from the top of the last column. The dotted lines represent the control signals under normal conditions. The dashed lines represent the control signals under override conditions. The override controls are still used on column 4, but now, the normal control structure controls the reflux drum level with the column feed and the column base with recycle RD. The override controls are just the reverse of case 1. Case 3: RD > RM > RT. The separation section now has component D coming off the top of the second column. The recycle of D can occur implicitly by taking it overhead in the first column. Therefore, the normal control structure controls reflux drum level in column 2 by adjusting the set point of the temperature controller in column 1, as shown in Figure 24. This tempera-

ture set point must be limited to some minimum value so that reactant component B is not lost from the bottom of the column under conditions when the reflux drum level controller is calling for more D to be sent to the second column. When this minimum temperature set point is reached, the column 2 reflux drum level will continue to drop. A low-level override controller will then take over control of the feed to column 2, which is normally set by the column 2 base-level controller. Then, the column 3 high base-level override controller takes over control of the column 3 bottoms valve. This is repeated in column 4, except that the high base-level override controller now adjusts the recycle RT valve. However, if more component M is required, the reflux drum level of column 4 drops so the signal of the lowlevel override controller in the drum will be compared with the other signals (the override signal from the base of column 3 and the normal signal from the base of column 4). Thus, there are override controllers on all of the last three columns. Case 4: RD > RT > RM. This system is the same as case 3 except M comes out the bottom of the last column. The column 4 reflux drum level is controlled by the column feed under normal conditions. The override controls are the same as in case 3 on all columns, except they are reversed from top to bottom on column 4. (See Figure 25.) Case 5: RT > RD > RM. Now, T comes from the top of column 2, so under normal conditions, the reflux drum level in this column is controlled by the column feed, as shown in Figure 26. The base levels of columns 2 and 3 are controlled by each of their bottom streams. The base level of column 4 sets the recycle RM, and the reflux drum level sets the recycle RD. When more product D is required and the set point of the flow controller on D4 is increased, the level controller starts decreasing the flow rate of recycle RD back to the reactor. When the flow rate of RD goes to 0, the reflux drum level continues to drop. A low-level override controller detects


Figure 24. CS4 with separation sequence D/M/T.

Figure 25. CS4 with separation sequence D/T/M.

this low level and takes over control of the feed valve to the column. This is accomplished by having the output signal of the override controller increase as the reflux drum level decreases. This signal is sent to a high selector, which selects the highest of three signals: one from the base-level controller on column 3 (normal), one from the low reflux drum level override controller on column 4, and one from the low column 4 base-level override controller. Thus, under override conditions, the column 4 feed is controlled by the signal from the reflux drum level override controller.

If more product M is required from the bottom of column 4 and the RM recycle flow rate goes to zero, the column feed is controlled by the signal from the low column 4 base-level override controller. Increasing the feed to column 4 will start to decrease the base level of column 3. A low-level override controller will take over control of the feed to column 3, which is normally set by the column 2 base-level controller. This is also repeated in column 2. When more feed is brought into this column, the reflux drum level starts increasing. The high-level override controller then takes over control of

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Figure 26. CS4 with separation sequence T/D/M.

Figure 27. CS4 with separation sequence T/M/D.

the temperature controller set point in column 1 in order to recycle T back to the reactor. Case 6: RT > RM > RD. This system is the same as case 5 except M comes out the top of the last column. The column 4 base level is controlled by the flow rate of recycle RD under normal conditions. The override controls are the same as in case 5 on all columns, except they are reversed from top to bottom on column 4. (See Figure 27.) Table 4 summarizes the manipulated and controlled variables for all of these cases.

7. Conclusion Several workable conventional control structures are shown to provide effective regulatory control of this complex process. An on-demand plantwide control system is demonstrated to provide effective control over a wide area of product-distribution space. To get the specified flow rates of all products, precise amounts of the fresh feeds are required. Inaccuracies in flow measurements, errors in the predicted ratio, or changes in feed compositions can cause a misbalance between

Ind. Eng. Chem. Res., Vol. 42, No. 12, 2003 2825 Table 4. Control Parameters in Each Case of Product Relative Volatilities variables under normal conditions

variables under override conditions

case

controlled

manipulated

controlled

manipulated

M/D/T

MD1 MD2 MD3 MB1 MB2 MB3 MD4 MB4

D1 TD1 (xD1M) D3 F B1 B2 RD B3


D1 TD1 (xD1M) D3 F B1 B2 B3 RT

M/T/D


D1 TD1 (xD1M) D3 F B1 B2 B3 RD


D1 TD1 (xD1M) D3 F B1 B2 RT B3

D/M/T


D1 TD1 (xD1D) D3 F B1 B2 RM B3


D1 B1 D3 F B2 B3 RM RT

D/T/M


D1 TD1 (xD1D) D3 F B1 B2 B3 RM


D1 B1 D3 F B2 B3 RT RM

T/M/D


D1 B1 D3 F B2 B3 RM RD


D1 TD1 (xD1T) D3 F B1 B2 RM B3


D1 B1 D3 F B2 B3 RD RM


D1 TD1 (xD1T) D3 F B1 B2 B3 RM

T/D/M

components A and B fed into the system. Therefore, the process requires some feedback signals to adjust the fresh feed flow rate ratio F0A/F0B. The signals are from

the levels of the reflux drum of the second column and the base level of the last column. Nomenclature Bi ) bottom flow rate from the ith column (lb‚mol/h) CC ) composition controller CS ) control structure DCi ) diameter of the ith column (ft) Di ) distillate flow rate from the ith column (lb‚mol/h) F ) reactor effluent flow rate (lb‚mol/h) FC ) flow controller FT ) flow transmitter F0A ) fresh feed flow rate of A component (lb‚mol/h) F0B ) fresh feed flow rate of B component (lb‚mol/h) LC ) level controller NBi ) normal signal from the base level of the ith column NRi ) normal signal from the reflux drum level of the ith column OBi ) override signal from the base level of the ith column ORi ) override signal from the reflux drum level of the ith column ratio ) mulitplier Ri ) reflux flow rate of the ith column (lb‚mol/h) RD ) product D recycle flow rate (lb‚mol/h) RM ) product M recycle flow rate (lb‚mol/h) TC ) temperature controller TD1 ) set point of temperature controller in column 1 (°F) TR ) reactor temperature (°F) VR ) reactor holdup (lb‚mol) VSi ) vapor boilup in ith column (lb‚mol/h) xBij ) bottoms composition of the jth component from the ith column (mole fraction) xDij ) distillate composition of the jth component from the ith column (mole fraction) zi ) reactor composition of the ith component (mole fraction) Greek Letters Ri ) relative volatility of the ith component. ∆F ) change in reactor effluent flow rate (lb‚mol/h) ∆F0A ) change in fresh feed rate of component A (lb‚mol/ h) ∆F0B ) change in fresh feed rate of component B (lb‚mol/ h)

Literature Cited (1) Kapilakarn, K.; Luyben, W. L. Plantwide Control of Continuous Multiproduct Processes: Two-Product Process. Ind. Eng. Chem. Res. 2003, 42, 1890-1904. (2) Luyben, W. L. Inherent Dynamic Problems with OnDemand Control Structures. Ind. Eng. Chem. Res. 1999, 38, 2315.

Received for review January 21, 2003 Accepted March 21, 2003 IE030058X

Plantwide Control of Continuous Multiproduct Processes - American

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