Plantwide Design and Control of Processes with Inerts. 1. Light Inerts

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Ind. Eng. Chem. Res. 1998, 37, 516-527

Plantwide Design and Control of Processes with Inerts. 1. Light Inerts Paul W. Belanger and William L. Luyben* Chemical Process Modeling and Control Research Center and Department of Chemical Engineering, Lehigh University, Iacocca Hall, 111 Research Drive, Bethlehem, Pennsylvania 18015

This is the first paper of a series that explores the design and control of processes containing inert components. Three strategies for the removal of light inerts from a process are compared both from the standpoint of steady-state economics as well as from the standpoint of dynamic controllability. It is shown that the best structure for the case of light inerts (for the inert volatility and range of inert loadings considered) consists of a reactor followed by a full column with side draw where inert is removed as the overhead product, the column bottoms is the plant product, and the side draw is recycled back to the reactor. It is also demonstrated that for certain plant structures, a decrease in production rate can cause long periods of time when the purge valve saturates open. During this time a substantial amount of raw material and product are lost through the purge stream, wasting a considerable amount of money. It is demonstrated that this problem can be eliminated by changing the setpoint of the purge composition controller whenever production rate changes occur. Introduction In the past, chemical engineers have approached process design in a hierarchical fashion. Design problems were solved initially by developing very simple solutions and then adding successive levels of detail to the process. For this reason, the dynamic properties of the process were not investigated until the final stages of the process design, when the control system was developed. With the advent of high-speed computers came a revolution in optimization techniques. The application of these techniques to the problem of chemical process design leads to the development of highly complex, highly integrated processes that are more efficient than ever before. Unfortunately, these processes are also notorious for having very poor dynamic properties, which make them difficult to control. One approach to this problem is the application of advanced control techniques such as model predictive control (MPC) and feedforward control. Because all feedback control systems share some common limitations (i.e., poor control always results in the neighborhood of the crossover frequency), the application of advanced feedback control techniques can be expected to yield only moderate improvements in dynamic performance over a well-designed classical feedback control system. This was reported by Ricker and Lee (1995). The use of feedforward control typically results in large improvements in dynamic performance; however its application is also limited due to the availability of load measurements and the need for accurate dynamic models. Since the dynamic performance of a plant under automatic control is ultimately determined by the plant design, another approach to the problem is to break with tradition and consider dynamic properties in the early * Author to whom correspondence should be addressed. Telephone: (610)758-4256. Fax: (610)758-5297. E-mail: wll0@ lehigh.edu.

stages of plant design. This approach (the “plantwide design and control” approach) leads to processes that are highly profitable as well as easy to control (Nishida and Ichikawa, 1975; Lenhoff and Morari, 1982; Fisher et al., 1984; Douglas, 1988; Luyben, 1993; Lyman, 1995; Cantrell et al., 1995; Elliott and Luyben, 1995). Unfortunately, due to the current capabilities of simulation software, limitations in computer speed, the need to minimize engineering man hours, and lack of adequate dynamic information for dynamic modeling, detailed dynamic studies during the conceptual design phase are not always practical (many engineers currently infer dynamic controllability from steady-state information). For this reason, much of the research in the area of plantwide design and control has focused on the development of heuristics for certain classes of chemical plants (a review of the literature can be found in a dissertation written by Belanger (1997)). There is very little in the literature on the subject of designing or controlling processes with inerts. Luyben (1981) studied the energy consumption aspects of using distillation columns to purge off light or heavy components. Douglas (1988) mentions some of the steadystate economic aspects of selecting a purging strategy. Verykios and Luyben (1978) studied a reactor/column process with a purge stream to remove a light inert. Joshi and Douglas (1992) discuss adding exit points in the flow sheet to remove trace components. This series of papers focuses on the important problem of designing and controlling processes containing inert components. Most chemical processes have inert chemical components that are introduced into the process through impure feed streams or are generated by irreversible chemical reactions. In processes containing recycle streams, the inert components can accumulate and cause undesirable effects. For this reason, it is necessary to purge these materials at some point in the process. The purpose of this series of papers is to

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Ind. Eng. Chem. Res., Vol. 37, No. 2, 1998 517

Figure 1. Reactor/stripper process with recycle and inert ventingssystem L1.

identify the “best” way to purge inert components from both a steady-state economic design standpoint and a dynamic controllability standpoint. Basic Assumptions The main purpose of this paper is to provide insight into how to design a process that has favorable economic and dynamic properties in the face of light inert loading. Unfortunately, covering all of the possible situations that can arise as a result of the presence of inerts in a plant is not possible. For this reason some basic assumptions are made to limit the scope of the investigation. This section presents the set of assumptions that are made so that the reader may be aware of the range of applicability of the treatment that follows. All of the processes have three components: component “I” is the inert, component “A” is the reactant, and component “B” is the product. Each process uses a single liquid phase CSTR to transform reactant A into product B. The reaction taking place in the reactor is first order and irreversible:

R ) kVRxAF where R is the reaction rate in (lb mol)/h, k is the reaction rate constant in h-1, VR is the holdup of the reactor in lb mol, and xAF is the mole fraction of A in the reactor. It is assumed that the reactor is run isothermally at a temperature that sets k ) 0.34 h-1. These assumptions should not restrict the applicability

of the study because the main effect of the inert in the reactor is to reduce the concentration of reactant. Therefore the selection of reaction type is not critical. It is assumed that the physical properties of the components are the same except for the relative volatilities. The relative volatilities of A and B are fixed at RA) 2.0 and RB ) 1.0. For column calculations, equimolar overflow, theoretical trays, and a saturated liquid feed are assumed. Partial reboilers and partial condensers are used. It is assumed that columns are run at pressures that allow cooling water to be used to condense the overhead vapor. For applications where refrigerant is required, the reader should consider the resulting increase in energy costs. The production rate of the plant is specified to be 239.5 (lb mol)/h of product. The product is to be 98.95 mol % pure. The composition of the feed is zI0, zA0, and zB0. Light Inert Removal Strategies In this paper the steady-state economic and dynamic properties of systems utilizing three different light inert removal strategies are investigated. The first strategy (system L1) involves designing the separation section to only separate reactant from product (see Figure 1). A small vapor purge is removed from the condensate drum in order to remove inert from the system. In this case the inert is allowed to build up in the plant in order to prevent excessive raw material

518 Ind. Eng. Chem. Res., Vol. 37, No. 2, 1998

Figure 2. Reactor/full column with side drawssystem L2.

losses. Note that the distillation column in system L1 is a simple stripper rather than a full column. It was found during the course of economic study that this was the best economic alternative. The second strategy (system L2) involves designing the separation section to separate both reactant from product and inert from reactant (see Figure 2). This strategy is similar to the first strategy in that a portion of the material to be recycled is removed from the process; however, since the system is able to more effectively separate inert from reactant, a smaller purge stream is required. This reduces raw material losses. The third strategy (system L3) that is investigated involves the design of a separation unit to pretreat the fresh feed (see Figure 3). Here, the fresh feed is fed to a distillation column where inert is separated from reactant. The reactant is then sent to a reactor/ separation system. In the reactor/separation system the reactant reacts to form product, the product is separated from the reactant, and the reactant is recycled back to the reactor. A small portion of the recycle is removed from the reactor/separation system and is fed back to the preseparation column. It is important to realize that the introduction of the gaseous purge from the CSTR/stripper subprocess to the pretreatment column requires either that the stripper be run at a higher pressure than the pretreatment column or that a small compressor be used. Another alternative would be to draw a liquid purge from the CSTR/stripper subprocess rather than a vapor and to pump the liquid into the pretreatment column.

Two main factors affect the choice of purging strategy and the selection of design parameters. The first is the relative volatility of the inert component (RI), which determines how difficult it is to separate inert from reactant. The second factor is the amount of inert that is introduced to the process (zI0). For the sake of brevity, the presentation in this paper is limited to the case where the volatilities of the components are RI ) 10.0, RA ) 2.0, and RB ) 1.0. For the case where the relative volatility of the inert component is very high (the inert is essentially incondensible) the presence of inert is not a plantwide problem (inert will not accumulate in the recycle). For the case where the volatility of inert is low (close to the volatility of the reactant) it was found that the inert behaves more like an intermediate inert. For a detailed analysis of the effects of inert volatility on steady-state economics the reader is referred to a thesis written by Belanger (1997). Analysis of Steady-State Economics Process Design Method. The method used to determine the optimal economic steady-state designs (maximize the discounted cash flow rate of return or DCFROR) for the flow sheets given in Figures 1-3 for different values of RI and zI0 is described in this section. For a detailed discussion of the DCFROR the reader is referred to Douglas (1988). The terms involved in the application of the DCFROR (on-site costs, revenue, utility costs, and raw material costs) are calculated from the following set of correlations and assumptions:


Figure 3. Reactor/stripper system with pretreatment of fresh feedssystem L3.

1. For the calculation of column diameters, an F factor of 1.0 is assumed. The diameter of the column can be calculated from F ) VfaceF0.5 v where Vface is the maximum superficial vapor velocity and Fv is the vapor density. Assuming a molecular weight of 50 lb/lb mol, a pressure of 44.7 psia, and a temperature of 760 °R, the diameter of the column can be calculated as follows: Dc ) 0.1838V0.5 where V is the vapor rate in the column. 2. The column height can be calculated by assuming a 2 ft tray spacing and allowing 20% more for base level volume: Lc ) 2.4NT. 3. A diameter to height ratio of 0.5 is assumed for the calculation of reactor dimensions. 4. Reboiler and condenser heat-transfer rates are determined from the value of V using a heat of vaporization of 250 Btu/lb. The reboiler area Ar is calculated assuming an overall heat-transfer coefficient U ) 100 Btu/(h °F ft2) and a log mean temperature difference of

50 °F. Condenser area Ac is calculated using an overall heat-transfer coefficient U ) 150 Btu/(h °F ft2) and a log mean temperature difference of 20 °F. 5. Energy costs are calculated at a rate of $5/106 Btu. 6. Capital costs of the reactor, column, and heat exchangers are estimated using correlations given by Douglas (1988):

reactor cost ) (3.0)(1917)D1.066 L0.802 r r column tray cost ) 36.26D1.55 NT c L0.802 column shell cost ) 1917D1.066 c c heat exchanger cost ) 1557[A0.65 + A0.65 ] c r 7. Materials of construction are stainless steel. Design pressures are 300 psig. A Marshall and Swift index of 800 is used.


8. Raw material cost is calculated on a basis of $20/ (lb mol). 9. Revenue is calculated on a basis of $23/(lb mol) of product. 10. The plant is assumed to operate for 8400 h/yr. 11. A 10 yr plant life is assumed. With the set of assumptions above used to calculate on-site costs (sum of the equipment costs), energy costs, raw material costs, and revenue, the DCFROR is calculated by solving eq 1 for i (i represents the discounted cash flow rate of return).

{(1.81on-site )[(1 + i) - 1] + 4 0.535ion-site}(1 + i) ) [0.52(0.975revenue 4

N

1.03(raw matl. + util.) - 0.186on-site - 4.26 × 105) + 0.8688on-site][(1 + i)N - 1] + (0.454on-site)i (1)

The first step in optimizing the plant designs is the selection of the design degrees of freedom, which will be a set consisting of both continuous variables and integer variables. A discussion of the selection of design degrees of freedom is given in Luyben (1996). Once the set of design degrees of freedom is identified, the optimal plant designs can be found using the following procedure: 1. Guess values for the integer degrees of freedom (number of stages in each column, feed tray locations, and draw tray locations). 2. Guess values for the continuous degrees of freedom (e.g., reactor holdup, various flow rates). 3. Solve for the remaining variables (i.e., find compositions, temperatures, and pressures and determine draw rates, boilup rates, reflux rates, etc. that satisfy product purity specifications). 4. Check for optimality with respect to the continuous degrees of freedom (has the nonlinear programming problem been solved?). If not then reguess the values of the continuous degrees of freedom and go back to no. 3. Otherwise continue. 5. Check for optimality with respect to the integer degrees of freedom (has the integer programming problem been solved?). If not then reguess the values of the integer degrees of freedom and go back to no. 2. Otherwise finished. The two inner loops correspond to the solution of the constrained nonlinear programming problem. It is possible to solve the constraining equations and maximize the objective function simultaneously; however, it was found during the course of our research that the solution space is typically plagued with relative minima. For this reason we solve the set of constraining equations in the innermost loop. There are a variety of methods (both simultaneous and sequential) for solving the system of constraining equations. We have used several in our research and have found that the method of Broyden (1965) and methods of continuation offer a good compromise between speed and stability. Several methods are also available for the solution of the nonlinear programming problem (a good overview of methods can be found in a book by Press et al. (1992)). We have found that the downhill simplex method of Nelder and Mead (1965) works well. The outermost loop corresponds to the solution of the integer programming problem. Many techniques exist

Table 1. Economic Impact of Changes in Feed Inert Level on System L1 zI0 NT xAF xIF F xAD xID D yAP yIP P V VR Dc (ft) Dr (ft)

0.01 16 0.300 0.171 814.1 0.422 0.240 570.9 0.236 0.670 3.63 574.6 2113

0.05 18 0.224 0.348 865.3 0.310 0.473 610.8 0.112 0.849 15.0 625.8 2952

Reactor and Stripper Diameters 4.41 4.60 11.0 12.3

0.1 20 0.070 0.438 645.0 0.111 0.679 377.0 0.031 0.940 28.5 405.5 9976.6 3.70 18.5

reboiler condenser

Heat Exchanger Areas (ft2) 1436 1565 2394 2608

1013.7 1689.6

reactor cost col. cost

Capital Costs ($1000) 886.2 1091.2 5998 6510

2327.3 513.1

Utility and Raw Matl. Costs ($1000/yr) energy cost 301.6 328.5 212.9 raw matl. cost 40 845.6 42 752.6 45 025.5 DCFROR

0.405 65

0.249 29

0.108 20

for the solution of this type of problem. We use a biased creeping random search. In this method a point is selected around the search center (biased toward the direction last moved). If the new point improves the objective function, then it becomes the new search center; otherwise it is rejected and a new trial point is selected. This method works well; however, we have found that the application of heuristics greatly improves the speed of convergence (e.g., if the number of trays in a distillation column are much greater than twice the minimum number estimated via the Fenske equation, then the number of trays are reduced). Design Results. The optimal designs of systems L1, L2, and L3 for feed inert levels ranging from zI0 ) 0.01 to zI0 ) 0.10 are summarized in Tables 1-3. Figure 4 illustrates the steady-state profitabilities of these designs. It can be seen from Figure 4 that system L1 is the least profitable for all levels of inert loading considered. System L2 is slightly more profitable than L3 for small feed inert levels and slightly less profitable for higher inert loading; however, the profitabilities of these two systems are so close that it would be difficult to justify the selection of one design over the other without the use of some other criterion (e.g., dynamic performance). What is responsible for the differences in profitability is the relative ability of each system to separate inert from reactant. Since raw material cost is the dominant factor among the factors determining profitability (in general this is the case, as reported by Douglas (1988)), it is very important to ensure that raw material losses will be small (note the small purge flow rates for each design in Tables 1-3). On the other hand, if the purge flow rate is too small, then the buildup of inert in the plant will be large and a larger reactor will be required in order to compensate for the diluting effects of the inert. This increases the capital cost of the plant. In addition, if a larger recycle flow rate is required to compensate for the reduced one-pass conversion in the reactor, then it may be necessary to increase the size of

Ind. Eng. Chem. Res., Vol. 37, No. 2, 1998 521 Table 2. Economic Impact of Changes in Feed Inert Level on System L2

Table 3. Economic Impact of Changes in Feed Inert Level on System L3

zI0

zI0

NT FT DT xAF xIF F xAD xID D yAP yIP P V R VR Dc (ft) Dr (ft) reboiler condenser reactor cost col. cost

0.01 16 4 3 0.384 0.006 842.5 0.535 0.004 600.0 0.201 0.788 3.08 603.0 600.0 1646.5

0.05 14 4 3 0.338 0.098 915.9 0.462 0.116 663.7 0.007 0.993 12.7 676.4 663.7 1954.5

Reactor and Stripper Diameters 4.51 4.78 10.1 10.7 Heat Exchanger Areas (ft2) 1508 1691 2513 2818 Capital Costs ($1000) 758.8 844.3 618.0 643.3

0.10 15 6 5 0.071 0.060 525.5 0.133 0.018 259.3 0.003 0.997 26.7 286.0 259.3 9885.0 3.11 18.4 715 1192 2314.0 384.0


0.422 76

0.285 84

0.128 87

the distillation equipment and higher utility costs may be incurred. Thus there is a tradeoff between capital and energy costs and raw material cost that determines the optimal mole fraction of inert in the purge for each process. The capital, energy, and raw material costs at this optimum are determined by the ability of the system to separate inert from reactant. System L2 and L3 have similar raw material costs for all zI0. This stems from the fact that both systems are able to effectively separate inert from reactant. System L2 and L3 also have similar capital and energy costs. This explains the similarity in profitability of the two systems seen in Figure 4. System L1, on the other hand, does not have the same potential to separate inert from reactant as systems L2 and L3. There is only one equilibrium stage separating the vapor purge and the liquid recycle. A large buildup of inert is required in this system to prevent an excessive loss of raw material, necessitating the use of a large reactor for all zI0. Despite this large buildup of inert at steady state, the raw material losses for system L1 are considerably higher than those of the other two systems. This causes the profitability of system L1 to be substantially lower than the profitabilities of systems L2 and L3. It is interesting to note how the optimal plant designs change as a function of feed inert levels. As the amount of inert in the feed increases, the reactor size tends to increase and the recycle flow rate tends to decrease. It becomes more difficult (requires more capital and energy) to obtain a recycle that contains a reasonable amount of reactant. Thus it makes more sense to build a larger reactor (to obtain a higher one-pass conversion) and cut back on the recycle flow rate. Another factor that contributes to this trend is a nonlinearity of the DCFROR vs operating (raw material plus utility) cost curve. When the plant operating costs are high, the DCFROR is much more sensitive to a change in operat-

NT1 FT1 NT2 xAF xIF F xAF1 xIF1 F1 xAD xID D yAP yIP P yAP2 yIP2 P2 V1 R1 V2 VR Dc1 (ft) Dc2 (ft) Dr

0.01 5 5 16 0.360 0.058 805.0 0.899 0.008 244.8 0.509 0.080 560.2 0.027 0.973 2.5 0.457 0.358 5.3 50.9 53.7 565.5 1756.4

0.05 5 5 18 0.271 0.059 686.8 0.923 0.016 250.1 0.411 0.084 436.7 0.022 0.978 12.9 0.379 0.388 10.6 102.5 100.2 447.3 2438.4

Reactor and Stripper Diameters 1.31 1.86 4.37 3.89 10.4 11.6

reboiler 1 condenser 1 reboiler 2 condenser 2

Heat Exchanger Areas (ft2) 127 256 234 471 1414 1118 2356 1864

reactor cost col. 1 cost col. 2 cost

Capital Costs ($1000) 790.0 968.8 109.4 170.1 593.9 529.6

0.10 7 5 20 0.070 0.020 482.7 0.914 0.018 267.6 0.121 0.023 215.1 0.010 0.990 26.9 0.182 0.173 28.1 100.1 101.3 243.2 9994.6 1.84 2.87 18.5 250 534 608 1013 2329.9 184.5 375.5


0.416 07

0.282 13

0.131 74

Figure 4. Steady-state profitability of systems L1, L2, and L3 vs feed inert level (RI ) 10).

ing costs. For the case of high levels of inert in the feed, raw material costs are much higher than for the case of low levels of inert in the feed. Thus when zI0 is high, the DCFROR is much more sensitive to utility costs than when zI0 is low. This alters the relative importance of utility costs and capital costs to such an extent that when large amounts of inert are present in the feed stream a very large reactor and small vapor boilup rate


in the distillation equipment are favored over a design with a moderately sized reactor and moderate vapor boilup rate. Dynamic Analysis Figure 4 indicates that system L2 is the most favorable system for zI0 < 0.07 and that system L3 is the most favorable for zI0 > 0.07; however, it is not necessarily the case that these systems will be as profitable in actual operation as the steady-state models predict. Substantial economic penalties can be incurred if the variability of any product stream is high. Even higher penalties may result if the plant shuts down. In this section the dynamic properties of each plant are studied and quantified on an economic scale so that the plant design that is the most profitable can be identified. In order to quantify the load rejection properties of each system on an economic scale, the capacity-based economic approach of Elliott (1996) and Elliott and Luyben (1995) is used. This approach uses the frequencies of the peak closed-loop regulator log moduli to construct a “worst case” load disturbance (a signal designed to excite the peak frequencies in the plant). Acceptable bounds are chosen for product quality, and the plant is simulated with the worst case load disturbance. The revenue of the plant is reduced in proportion to the fraction of product that is outside the specification band (“on-aim” control). The peak closed-loop regulator log moduli are determined through the use of the load auto tune variation (LATV) method of Belanger and Luyben (1996). Control Structures. The control structures used in this study are illustrated in Figures 1-3. All control structures are similar to avoid biasing the results of the study. In each system it is assumed that the production rate is determined by the fresh feed flow rate F0. The reactor effluent flow rates are manipulated to regulate reactor holdups. It was found that a loosely tuned proportional only reactor inventory controller works best (Belanger, 1997). These controllers are tuned such that a 20 (lb mol)/h change in the fresh feed flow rate leads to a 50 (lb mol)/h change in reactor inventory. The justification for this choice can be found in the aforementioned reference. The pressure of each column is controlled by manipulating the cooling water flow rate. Bottoms product flow rates are used to control the liquid level in each column base. The column accumulator inventories are controlled through the manipulation of the flow rate of the largest liquid stream leaving the accumulator (i.e., the recycle flow rate is used for system L1, the reflux flow rate is used for system L2, the reflux flow rate is used for the pretreatment column of system L3, and the recycle flow rate is used for the product column in system L3). The recycle flow rate is used in system L2 to regulate the inventory in the side draw surge tank. Perfect control is assumed for each loop regulating column inventory (pressure and liquid level). Product compositions are controlled through the manipulation of the reboiler steam flow rates in each product column. The purge flow rates are used to control the mole fraction of inert in each purge stream. All of the composition loops use tightly tuned PI controllers. A 3 min deadtime is assumed for composition analysis. Relay-feedback tests (or ATV tests) (Astrom, 1984; Luyben, 1987, 1990) are used to determine TL settings (Tyreus and Luyben, 1992) for each of these quality control loops.

Figure 5. Closed-loop regulator log moduli of systems L1, L2, and L3. Table 4. Recycle Gains system L1 system L2 system L3

0.7058 0.7157 0.7025 zI0 ) 0.01

0.7232 0.7385 0.6513 zI0 ) 0.05

0.6287 0.5442 0.5038 zI0 ) 0.10

Table 5. Reactor Inventory Controller Tunings system L1 system L2 system L3

1.360 1.407 1.345 zI0 ) 0.01

1.445 1.530 1.147 zI0 ) 0.05

1.077 0.8776 0.8061 zI0 ) 0.10

Quantification of Dynamic Properties. With the control structures selected and tuned for each process design, LATV tests were performed to determine the peak closed-loop regulator log moduli from each load disturbance (F0 and zI0) to each output variable of interest (xAB and yIP). The magnitudes of the closedloop log moduli are shown in Figure 5. This figure illustrates some interesting dynamic characteristics of processes utilizing the three different inert removal strategies. Tables 4 and 5 give the recycle gains and reactor inventory control gains for each system for each level of inert loading. The effects of fresh feed flow rate fluctuations on product quality are higher for system L1 than for systems L2 and L3, especially for higher levels of inert in the feed. One factor that contributes to this is the difference in recycle gains. High values of the recycle gain mean a large impact of flow rate recycling on product quality control (Belanger, 1997). For low levels of inert in the fresh feed, the recycle gains are fairly similar for all three systems. For intermediate levels of inert in the feed, the recycle gain of system L3 is substantially lower than the recycle gains of systems L2 and L1. For high levels of inert in the feed, the recycle gains of both systems L2 and L3 are significantly lower than the recycle gain of system L1. The recycle gain is directly related to the vapor to liquid ratio in the stripping section used to purify product. If the fresh feed to the column changes but the column feed composition remains the same, then in order to obtain the same product purity the vapor boilup rate must change to maintain the slope of the stripping line. Since the purge flow rate does not change much under these conditions, the change in the recycle flow rate is roughly equivalent to the change in the vapor boilup rate. This


means that the higher the vapor to liquid ratio in the stripping section at steady state, the higher the recycle gain. Table 5 shows that for cases where the recycle gain is high, the reactor inventory control gain is high. This is because tighter inventory control is required to prevent large fluctuations in reactor holdup for cases where the recycle gain is high. Higher values of Kc lead to a wider range of frequencies over which the full effects of flow rate recycling are seen. This again amplifies the effects of load disturbances on product quality. Another factor contributing to the differences in the effects of fresh feed flow rate fluctuations on product quality is the large buildup of inert in system L1 at steady state. Inert levels in the reactor are more sensitive to fluctuations in the fresh feed flow rate when the inert levels in the reactor at steady state are high. The larger fluctuations in inert levels in the reactor effluent stream cause large fluctuations in the relative amounts of components A and B fed to the separation system. This has a detrimental effect on product quality control. Figure 5 also shows that the effects of fresh feed flow rate fluctuations on purge composition are significantly lower for system L2 than for systems L1 and L3. As was stated above, the fluctuations in inert levels in the reactor effluent stream are smaller for system L2 than for system L1. This means that the inert loading to the distillation column is smaller for system L2 than it is for system L1. Thus, the sensitivity of the purge composition to fresh feed flow rate fluctuations is smaller. The poor performance of system L3 in this regard is due to the location of the disturbance. In system L3 the disturbance is introduced directly to the distillation column being used to purify the purge stream whereas in the other two systems the disturbance is fed to a large reactor that serves as a filter. This also explains why changes in the inert levels in the fresh feed have a more severe impact on purge purity for system L3 than for systems L1 and L2. The effects of fresh feed composition changes on product quality are similar for all three systems for low levels of inert loading; however, for high levels of inert loading, the effects of fresh feed composition changes on product composition are smaller for system L2. One obvious conclusion that can be drawn from the information in Figure 5 is that system L2 is dynamically superior to systems L1 and L3 for all levels of inert loading. The capacity-based economic approach is applied to quantify the benefits of the superior dynamic controllability of system L2. For the application of the capacity-based economic approach the following assumptions are made: 1. Peak fluctuations in F0 are 20 (lb mol)/h for each system. 2. Peak fluctuations in zI0 are 1.0 mol % for zss I0 ) 0.01, 5.0 mol % for zI0 ) 0.05, and 10.0 mol % for zss I0 ) 0.10. 3. The specification band on product quality is (0.1 mol %. 4. The specification band on purge quality is (1.0 mol % (Purge quality does not need to be controlled as tightly as product quality because the purge is fed to a flare). 5. The revenue of the plant is penalized in proportion to the fraction of the product and purge that is offspec.

Figure 6. Response of system L1 to its worst case load disturbanceszss I0 ) 0.01. Table 6. Percentage of Offspec Product and Offspec Purge under Worst Case Loading Conditions system L1 % product offspec % purge offspec system L2 % product offspec % purge offspec system L3 % product offspec % purge offspec

6.9 0.0 0.0 0.0 5.1 0.0 zI0 ) 0.01

21.2 0.0 0.0 0.0 4.0 35.8 zI0 ) 0.05

0.0 0.0 0.0 0.0 0.0 17.3 zI0 ) 0.10

The fraction of revenue lost is equal to 10% of the fraction of offspec product plus the fraction of offspec purge. Using these assumptions and the frequencies obtained from LATV tests, worst case load disturbances were constructed for each system for each level of inert loading. Each process was simulated using its worst case disturbance to determine the percentage of offspec product and offspec purge that is expected for each system under worst case loading conditions. This information is given in Table 6. For illustrative purposes the response of system L1 for low levels of inert loading to its worst case load disturbance is given in Figure 6. The specification bands on xAB and yIP are shown as dotted lines. The data given in Table 6 were used to find new, more realistic values for the return on investment of each process (this takes into account both the controllability of product and purge as well as the need to invest in a larger reactor). The new values of return on investment are shown in Figure 7. A comparison of Figures 4 and 7 illustrates the dramatic effect that dynamic performance can have on overall profitability. When dynamic controllability is not taken into account (see Figure 4), it is unclear which system, system L2 or L3, will bring the highest return on investment. When dynamic controllability is taken


Figure 7. More conservative profitability estimates for systems L1, L2, and L3.

into account (see Figure 7), it becomes clear that system L2 is the most desirable process from an overall economic standpoint for all levels of inert loading. Production Rate Changes. One of the peculiar problems with processes containing inert components is that changes in production rate can bring about long periods of time when the purge valve saturates. This effect is due to a required change in inert levels in the plant (in this case characterized by the product VRxIF) in the transition from one steady-state operating condition to another. This may not present a problem in the case where the inert levels in the plant are required to rise. In this case the purge valve closes fully until the inerts have built up. On the other hand, if the inert levels in the plant are required to drop, then the purge valve may saturate fully open for a period of several hours. During this time a substantial amount of raw material will be lost through the purge stream if the purge purity is not high, as is the case in system L1. This type of behavior is clearly undesirable and should be remedied. One possible solution to the problem is a change of control strategies during a production rate change. For example, the setpoint of the purge composition controller can be changed. Another solution to the problem is to design the plant such that a large change in the total inert holdup is not necessary for a given change in production rate. For the plant designs illustrated in Figures 1-3 the total amount of inert in the plant is approximately equal to VRxIF. This is because the reactor contains most of the material in the plant. For any system that moves from one operating condition to another the total change in inert levels is given approximately by

∆MI ) {xIF}ss∆VR + {VR}ss∆xIF + ∆VR∆xIF

(2)

There are two terms in eq 2 that are of major importance. The third term, the cross term, is usually not large when compared with the other two. The first term represents the change in inert levels in the plant that result from changes in reactor holdup. The second term represents the change in inert levels that result from changes in the reactor composition. Both of these terms can be significant. All of the reactor inventory control gains for the example processes have been selected to give similar reactor holdup changes for

Figure 8. Responses of systems L1 and L2 to a 20% step change in fresh feed flow rate.

similar production rate changes (a 20 (lb mol)/h change in fresh feed flow rate results in approximately a 50 (lb mol) change in reactor holdup). Thus according to the first term in eq 2 the inert levels in systems L2 and L3 will tend to be less sensitive to changes in production rate than the inert level in system L1 for a given production rate change because the mole fractions of inert in the reactor are lowest for systems L2 and L3. Unfortunately, due to the complexity of the systems, there is no way to know in advance, without simulation, what the change in the mole fraction of inert in the reactor of each system will be for a given production rate change. In this ternary system with a first-order reaction in terms of reactant (zAF), we do not know how the concentration of the other two compounds (zIF and zBF) will change at different production rates. Our studies have shown (Belanger, 1997) that the mole fraction of inert in the reactor in system L1 tends to drop for a reduction in production rate (zIF decreases, zAF increases, and zBF increases as production rate decreases). The mole fractions of inert in the reactor for systems L2 and L3 are not sensitive to changes in production rate. It can be concluded that system L1 will be more prone to sustained purge valve saturation than systems L2 and L3. To illustrate this point, consider making a -20% step change in fresh feed flow rate to systems L1 and L2 for the case where RI ) 10 and zI0 ) 0.01. The responses of the two systems are illustrated in Figure 8. This figure shows that once the fresh feed flow rate change has been made the inert mole fraction in the reactor of system L1 begins to increase. Since the mole fraction of inert at steady state in the reactor is higher than the mole fraction of inert in the fresh feed, the steady-state


gain relating fresh feed flow rate changes to changes in inert composition is negative. Thus the reduction in fresh feed flow rate results in a short term increase in the inert mole fraction in the reactor. The inert mole fraction in the reactor for system L2 decreases, but only slightly (the gain relating feed flow rate to the reactor inert mole fraction is small and positive). The control systems must eventually drive the systems to steady state. This results in the drop in the reactor inert mole fraction in system L1 shown in Figure 8 (a solution of the steady-state model gives a reduction of 1.13 mol % from 17.10 to 15.97 mol %). For both systems the reactor holdup decreases at the same rate. This is because the recycle gains and inventory control gains are similar for both systems. The effects of changes in reactor inert mole fraction and reactor holdup manifest as changes in the total amount of inert in the reactor. Note that for system L2 the total change in inert levels is very small (the mole fraction of inert at steady state is very small and the change in the inert mole fraction in the reactor is small). For system L1, however, the inert mole fraction in the reactor at the original steady state is substantial as is the change in the inert mole fraction. As is shown in Figure 8, the change in inert levels in the reactor is fairly large for system L1 (approximately 50 lb mol of inert) while the total change in inert levels for system L2 is almost negligible. The effect that this has on the behavior of the purge control systems is apparent. During the period of time when the inert levels in the reactor of system L1 are decreasing (the first 20 h of simulation), the purge valve of system L1 saturates fully open. The purge composition control system is not able to simultaneously release the inerts from the plant and maintain purge composition. In contrast the purge valve of system L2 does not saturate. It can be concluded from this example that designing a plant such that inert concentrations in the reactor are low at steady state does more than simply improve its sensitivity to small periodic load disturbances. It improves the ability of the system to reject the effects of large sustained changes in production rate as well. Systems with small buildups of inert at steady state do not exhibit the same problems with purge valve saturation as systems with large buildups at steady state. Ideally one would like to avoid the problem of losing valuable material through the purge stream during production rate decreases by choosing a plant design with a small buildup of inerts at steady state. In practice this may not always be an alternative. The following question arises: for a fixed plant structure, what changes should be made in preparation for a production rate decrease such that an excess amount of raw material is not lost through the purge stream? A simple solution would be to increase the setpoint of the purge composition controller. This presents two benefits: (1) for a lower production rate the optimal steady-state mole fraction of inert in the purge is probably higher, and (2) increasing the setpoint of the purge composition controller reduces the necessary drop in plant inert levels (reduces the amount of time when the purge valve is fully open). To illustrate this point, consider making a -20% step change in fresh feed flow rate to system L1. This is illustrated in Figure 9. The solid curves represent the response of the system when the purge composition controller setpoint is held constant at 67 mol % “I”. The

Figure 9. Effect of increasing the purge composition controller setpoint on the response to a -20% step change in feed flow rate.

dashed curves represent the response when the setpoint is increased to 70 mol % “I”. The notable differences in dynamic behavior are the purge valve positions and the total inert holdups. Note that in the case when the setpoint of the purge composition controller is increased the purge valve saturates closed rather than open. This is due to the abrupt change in setpoint. The error signal to the controller causes the purge valve to snap shut immediately. This causes the inert levels in the reactor to increase over a short period of time. Eventually the decrease in reactor holdup causes a reduction in reactor inert levels (the inert moves from the reactor to the stripper). This causes the inert mole fraction in the purge to increase, which causes the purge valve to reopen. Once the required change in inert level is made the system reaches steady state. It is important to note that the total change in inert level in the reactor is much smaller for the case where the purge composition controller setpoint is increased. This is the main reason that the purge valve does not saturate open in this case. Figure 10 illustrates why the increase in purge composition control setpoint is important during times when the production rate is decreased. During the time that the purge valve remains open (for the case where the purge composition control setpoint is held constant) an excess of valuable material is lost through the purge stream. This results in an hourly profit curve that is substantially lower than for the case where the setpoint is increased slightly. The change in setpoint does not have a strong impact on steady-state profitability. This is because the profit of this plant is not a strong function of the inert mole fraction in the purge. It does, however, have an impact on the average hourly profit. When the setpoint is held constant, the average hourly profit (during this 35 h period) is $552/h whereas the average


Figure 10. Effect of increasing the purge composition controller setpoint on the plant profit during a production rate decrease.

hourly profit for the case where the purge setpoint is increased is $589/h. The total monetary savings during this 35 h time period is $1295 (a substantial savings). A 3 mol % increase in purge composition controller setpoint may not be optimal for the given change in production rate. In fact, it may be that a step in setpoint is not the optimal way to make the change (e.g., a ramp change in setpoint may be more appropriate). For a given system and production rate change optimization studies are necessary in order to maximize the average plant profit. The important thing to realize is that for systems with a large buildup of inert at steady state a purge composition control setpoint increase should accompany a reduction in production rate. This reduces the amount of valuable material that will be lost during a production rate decrease. Conclusions It can be concluded from the studies in this paper that the second strategy for light inert removal (designing the separation section of the plant to be able to purify the purge stream as well as the product stream) is the most favorable from an overall economic standpoint. This could not have been determined by solely examining the steady-state economic properties of each process. From a steady-state standpoint, the second and third strategies were of comparable profitability. From a dynamic controllability standpoint, however, the second strategy proved superior to the other two. The dynamic properties of a process can have a considerable impact on profitability. Thus, considering the dynamic properties of the plant in the early stages of design can greatly increase the profitability of the plant. In general, with regard to light inerts, it is economically desirable to design the plant in such a way that inert can be removed from the plant while avoiding excessive losses of raw material through the purge stream and large buildups of inert within the recycle loop. This can be accomplished by designing the separation section of the plant such that it is capable of purifying the purge stream. This is provided, of course, that this alternative is not too expensive (i.e., requires the use of refrigerant or operation at excessive pressure). The purification of the purge stream not only results in an improvement in steady-state profitability but also gives an improvement in plantwide control-

lability as well. Allowing inerts to build up in the reactor increases fluctuations in the composition of the reactor effluent stream for a given feed flow rate fluctuation. This places a larger load on the separation system. It was also shown that it is very important from a dynamic standpoint to introduce the fresh feed to a location in the plant that has a high capacitance (in this case the liquid CSTR). Introducing the fresh feed directly to a distillation column producing a product stream results in a high sensitivity of product quality to fluctuations in fresh feed conditions. A final conclusion that was drawn was that designing the plant in such a way that a large buildup of inerts at steady state is not required reduces the possibility of purge valve saturation when a production rate change is made. This is important because it reduces the amount of raw material and product that are lost through the purge stream when the production rate of a plant is decreased. In cases where the plant structure is fixed and there is a large buildup of inert at steady state, a purge composition control setpoint increase should accompany a reduction in production rate. Nomenclature Ac ) condenser area (ft2) Ar ) reboiler area (ft2) Bj ) bottoms flow rate of column j ((lb mol)/h) CLLM ) closed-loop log modulus (dB) D ) recycle flow rate ((lb mol)/h) DCFROR ) discounted cash flow rate of return Dc ) column diameter (ft) Dr ) reactor diameter (ft) F ) reactor effluent flow rate ((lb mol)/h) F ) flooding factor k ) reaction rate constant (h-1) Lc ) column height (ft) Lr ) reactor height (ft) P ) purge flow rate ((lb mol)/h) R ) reaction rate ((lb mol)/h) Rj ) reflux flow rate in column j ((lb mol)/h) U ) heat-transfer coefficient (Btu/(lb mol °F)) VR ) reactor holdup (lb mol) Vface ) maximum superficial vapor velocity (ft/s) Vj ) vapor boilup in column j ((lb mol)/h) xij ) liquid mole fraction of component i in stream j yij ) vapor mole fraction of component i in stream j zij ) total mole fraction of component i in stream j Greek Symbols Rij ) volatility of component i with respect to component j Fv ) vapor density (lb/ft3)

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Received for review April 16, 1997 Revised manuscript received September 26, 1997 Accepted September 26, 1997X IE970288+

X Abstract published in Advance ACS Abstracts, December 1, 1997.

Plantwide Design and Control of Processes with Inerts. 1. Light Inerts

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