Role of Nonlinear Effects in Benzene–Toluene–Xylene Dividing Wall

Article pubs.acs.org/IECR

Role of Nonlinear Effects in Benzene−Toluene−Xylene Dividing Wall Column Control System Design Rishabh Gupta and Nitin Kaistha* Department of Chemical Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India S Supporting Information *

ABSTRACT: Steady-state bifurcation analysis is applied to a ternary benzene−toluene−xylene (BTX) dividing wall column (DWC) to synthesize robust temperature inferential controlled variables (CVs). The coupling between the prefractionator and the main column causes steady-state multiplicity in conventional tray temperature CV loops. Using physical insights into the DWC behavior, differential tray temperature across the side-draw and double differential temperature control across the prefractionator rectification section are shown to significantly mitigate the steady-state multiplicity. Control system variants (with and without prefractionator liquid split manipulation) with alternative temperature-based CVs exhibit nonlinear dynamic phenomena of seeking an infeasible temperature/purity set point and input multiplicity induced “wrong” control action leading to a low purity steady-state transition under closed loop operation. These nonlinear dynamic phenomena correlate with the bifurcation analysis results. The recommended control structure with robust temperature inferential CVs is shown to effectively reject very large feed composition changes, including for on-target product purity operation via temperature inferential CV update. The work brings out the importance of robust CV design for effective control of the highly coupled and nonlinear ternary DWC process.

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INTRODUCTION The ternary dividing wall column (DWC) emulating the Petlyuk sequence1 in a single column shell has been known to be significantly more energy efficient and cheaper than the conventional light/heavy-out-first distillation sequence. For a ternary mixture of components A (light), B (intermediate), and C (heavy), the feed side of the dividing wall (see Figure 1) acts as the prefractionator (TS1 and TS2) and does the easy AC split. The tray sections above and below the side-draw (TS4 and TS5) prevent, respectively, light A and heavy C leakage in the side-draw. The main column rectifying (TS3) and stripping (TS6) sections prevent B leakage up the top and down the bottoms, respectively. With the prefractionator performing the easy AC split, the intermediate boiler B distributes evenly in the prefractionator top and bottom mitigating its remixing and thus inherent thermodynamic irreversibility, compared to a conventional two-column sequence. Consequently, significant energy saving in the range of 20−40% can be achieved.2−5 Despite its high energy efficiency, reports of industrial DWC applications have only recently appeared in the literature, BASF being the industry pioneer.6 More recently, extractive DWC distillation7 and multiple DWCs8 have been reported in the literature. The reluctance on part of the industry to adopt DWC technology is due to the control system design being significantly more complex compared to conventional simple distillation. Specifically, the highly nonlinear interaction between the prefractionator and the main column makes tight quality control on the intermediate boiler (B) side-draw stream around a stringent target (e.g., >99% purity) difficult. In one of the earliest papers on DWC operation for specified purity targets on the three product streams, Wolff and Skogestad9 showed an infeasible operation zone with respect to the liquid split to the prefractionator (the fourth degree of © 2015 American Chemical Society

freedom, dof). Reasons for the infeasibility were not investigated. More recently, in the first comprehensive dynamic simulation study, Ling and Luyben10 show that near minimum boil-up operation can be achieved for large disturbances to a high purity benzene−toluene−xylene (BTX) DWC using direct composition control of the prefractionator top tray xylene mole fraction, and the principal impurities in the three product streams. The problem with direct composition control structures is that in industrial settings, composition measurements are usually available only on product streams. Also, the typical frequency of product composition measurements is quite low, say once or twice a shift. Thus, additional expensive online analyzers need to be installed to make the direct composition control scheme work, as otherwise a composition measurement every 4 or 8 h implies essentially blind column operation. A more practical alternative is to apply temperature inferential control to hold the temperature profile shape in the DWC and then make minor once or twice a shift adjustments to the temperature-based controlled variable (CV) set points to compensate for any measured product purity deviations. This is the industry norm and its success requires that the temperature inferential CVs be robust to large disturbances. The ternary DWC control literature indicates that replacing the four composition measurements by sensitive tray temperatures causes the control system to fail for moderate disturbances implying poor robustness. Replacing temperature measurements with differential/double differential temperature measurements somewhat improves performance.11,12 In particReceived: Revised: Accepted: Published: 9407

May 23, 2015 August 28, 2015 September 8, 2015 September 8, 2015 DOI: 10.1021/acs.iecr.5b01907 Ind. Eng. Chem. Res. 2015, 54, 9407−9420

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Figure 1. Dividing wall column base-case: (a) schematic with design and operating conditions; (b) column temperature profile; (c) column composition profile. Black line, main column; gray line, prefractionator.

document closed loop performance of control system variants with no attempt to understand the reasons behind why a particular variant handles disturbances, while a seemingly similar one fails for the same. Further, to the best of our knowledge, the issue of control system robustness to very large disturbances remains largely untested. A systematic study that investigates the reasons behind the performance of various temperature inferential control system variants and utilizes those insights to design better temperature inferential measures to improve robustness thus remains essential and topical. Such a study must necessarily consider and address the highly nonlinear DWC prefractionator-main column interaction; an issue alluded to in the literature but not addressed systematically. In this work, nonlinear bifurcation analysis of a BTX DWC is used to synthesize a robust temperature inferential control system and to interpret the observed robustness/

ular, temperature-based feedback adjustment of the liquid split to the prefractionator is tricky. This is because all trays across the prefractionator contain the three components and the attribution of tray temperature change to a change in the mole fraction of a specific component is unclear. For example, a decrease in TS1 tray temperature could either be due to less C (heavy) moving up or more A (light) moving up. The challenge in DWC control is devising a robust temperature inferential regulatory control system that ensures proper component material balance control for large disturbances and tracks temperature CV set point changes demanded for ensuring ontarget product quality. There then exists an imperative need for DWC control research on understanding and improving the robustness of temperature inferential control systems. If we look at the extant DWC temperature inferential control literature, the large majority are hit-and-trial studies that simply 9408

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Temperature measurements, on the other hand, are cheap, rugged, fast, and reliable. The use of infrequent product quality measurements, available from, for example, a quality control laboratory, to update the temperature-based CV set points is however not precluded. It is also desirable that the chosen temperature-based CVs ensure small product purity deviations so that large excursions away from the target product purities are avoided and infrequent (say once a shift) temperature-based CV set point adjustments suffice toward acceptably tight quality control. The purities of the three main column product streams are to be tightly controlled. Three temperature-based CVs are then needed to regulate these three quality targets. In the simplest control structure, the condenser duty maintains the column pressure (P) with the distillate (D) and bottoms (B) flow regulating the reflux drum and bottom sump levels, respectively, for liquid and vapor inventory control. The reflux rate (R), side-draw rate (S), and reboiler duty (QR) maintain appropriate temperature-based CVs for BTX component material balance control with the prefractionator liquid split (βL) held constant at its base-case value. This simplest control structure is shown in Figure 2a, and is referred to as CS1 hereafter. A possible enhancement to CS1 is manipulation of βL to hold another temperature-based CV constant. The control structure is shown in Figure 2b and is referred to as CS2 hereafter. It is noted that most control structure variants in the DWC control literature are either of the CS1 type or of the CS2 type. The main differences are in the CVs with some reports controlling compositions, others controlling tray temperatures, and still others controlling differential/double differential temperatures. The CVs paired with R, S, and QR close the BTX component balances respectively on the DWC. The fourth dof (βL) fixes the xylene spillover from the prefractionator top and thus indirectly affects the xylene impurity in the side-draw. At the base-case minimum reboiler duty steady-state in Figure 1, the principal impurity in the liquid side-draw is xylene as it is heavy and thus prefers the liquid phase.10 Benzene is the minor sidedraw impurity. There are two sources of the side-draw xylene impurity: (i) the xylene that spills over from the prefractionator top; and (ii) the xylene that leaks up TS5, (tray section below side-draw). Of these, the latter can be directly compensated for by adjusting S while compensating for the former requires appropriate βL manipulation. Strong interaction exists between the two sources of the principal side-draw impurity, and proper temperature inferential CVs are necessary to maintain their balance. To limit the scope of the study, only tray temperature, differential tray temperature, and double differential tray temperature CVs, which have found industrial acceptance, are considered. Tray Temperature Control. The simplest temperaturebased CVs are direct tray temperature measurements, avoiding any combinations such as differential/double differential tray temperatures. A conventional sensitivity analysis with respect to the four manipulated variables (MVs), R, S, QR, and βL is performed to obtain sensitive tray temperature CVs. The temperatures of prefractionator tray 7 (T7P), main column tray 7 (T7M), tray 32 (T32M) and tray 39 (T39M) exhibit high sensitivity with respect to one or more MVs (temperature sensitivity plot provided in Supporting Information). Pairing these sensitive tray temperatures with their respective MVs, conventional T7M-R, T32M-S, and T39M-QR pairings are recommended for CS1 (constant βL). CS2 additionally uses

fragility of alternative temperature inferential measures. To the best of our knowledge, this is the first report on the role of nonlinear effects in DWC control system design. In the following, an energy efficient base-case design of a BTX DWC is briefly described. A bifurcation analysis with respect to the available control dofs is then performed to show the existence of multiplicity in the steady-state input−output (IO) relations. The physical phenomena behind the multiplicity behavior are interpreted and alternative differential/double differential temperature combinations are designed to mitigate the multiplicity for significantly improved control system robustness. Quantitative closed loop dynamic results evaluating the improvement in control system performance due to the choice of the controlled variables (CVs) are then presented. We also discuss the observed nonlinear dynamic phenomena in light of the bifurcation analysis results. A brief summary of the main findings of the work concludes the article. BTX Dividing Wall Column Design. A benzene− toluene−xylene (BTX) ternary DWC is studied in this work. The steady-state and dynamic modeling is performed in Aspen Hysys version 7.2 using the Soave−Redlich−Kwong equationof-state thermodynamic package. Figure 1 shows the base-case design and operating conditions, including temperature and composition profiles, of the DWC for the equimolar fresh feed. This design has been adapted from a recent article from our group.13 For more realistic simulations, we consistently use pressure driven dynamic simulations, in which the internal column pressure profile changes with operating conditions. Theoretically, the DWC has four steady-state dofs corresponding to the liquid split (βL) to the prefractionator, the reflux rate (R), the side-draw rate (S) and the reboiler duty (QR). The vapor split (βV) is not a dof and is fixed by the main column shell cross-sectional area partitioning due to the dividing wall. In the base-case design, βV has been fixed for minimum QR. Of the four steady state dofs, R, S, and QR get fixed by product purity constraints (99 mol % purity) on the distillate, side-draw, and bottoms product streams. The remaining dof, βL, is an unconstrained (free) steady-state operation dof which may be adjusted for, as an example, improved robustness to large disturbances or improved product quality control. At the base-case design, βL value corresponds to minimum QR. Temperature Inferential Control Structure Variants. We are to design a robust temperature inferential control system that can handle large disturbances in the DWC feed composition as well as throughput changes. We consider a ±20% throughput step change and a ±6 mol % step change in a fresh feed component mole fraction (other two components remain equimolar) as the routine nominal disturbance space, for which the control system must work. Larger feed composition changes are possible but are infrequent and it is desirable that the control system also reject such large feed composition changes. The routine feed composition changes are conveniently referred to as component lean/rich. Thus, for example, a benzene lean feed (B-) refers to a 6 mol % step decrease in the fresh feed benzene mole fraction. The temperature inferential regulatory control system should be such that any product purity deviations for the nominal disturbances can be reliably compensated for by predictable adjustments in the temperature-based CV set points. The use of composition CVs for regulatory control is avoided as composition measurements are usually expensive, high maintenance, fragile, unreliable, and have large dead-times. 9409

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To understand the relationship between the above CVs and MVs, Figure 3 plots the steady-state CV−MV relation that each pairing “sees” under open loop (all other temperature loops open) and closed loop (all other temperature loops closed) operation. Both open loop and closed loop IO relations are plotted since in an interacting multivariable system, the control action of one control loop can dramatically alter the input− output (IO) relation that the other loop “sees”. In all plots, the symbol “δ” denotes deviation from the base-case value. The open loop T7M-R, T32M-S and T39M-QR CV−MV relations (constant R, S, QR, βL) are monotonic and do not exhibit severe nonlinearity in the form of steady-state multiplicity. The T7P-βL CV−MV relation, on the other hand, exhibits input multiplicity (multiple inputs for a given output). A crossover with respect to the base-case T7P value does not occur. Under closed loop CS1T operation, the T7M-R (constant βL, T32M, T39M) CV-MV IO relation remains monotonic (no multiplicity), while the T39M-QR (constant βL, T7M, T32M) and T32M-S (constant βL, T7M, T39M) CV−MV relations exhibit input multiplicity. Output multiplicity (multiple outputs for given input) is also observed in the T32M-S IO relation. Although output multiplicity is not an issue as a feedback controller with the appropriate set point and action (reverse/ direct) would stabilize T32M at the desired value, input multiplicity in T32M and T39M can potentially result in control system failure due to set point infeasibility14 or “wrong” control action.15,16 In both IO relations, since the slope at the crossover point marked with the asterisk (∗) has the same sign as the base-case, the “wrong” control action can lead to a closed loop steady state transition for sufficiently large disturbances. Under closed loop CS2T operation, the input multiplicity in the T7P-βL (constant T7M, T32M, T39M) gets exacerbated. The closed loop T7P-βL IO is plotted for the base-case equimolar and benzene lean feeds in Figure 3. The action of the three main column temperature controllers causes T7P to exhibit lower variability with respect to βL compared to open loop operation, as evident in the noticeable shrinkage of the curve’s y-axis span. The base-case temperature of 99 °C is only ∼1.5 °C above the minimum temperature in the IO relation. For the benzene lean feed, the IO relation shifts up by ∼3 °C and the base-case temperature set point of 99 °C becomes infeasible. This suggests that the T7P-βL loop would cause the control system to fail as it seeks an infeasible steady-state for large feed composition disturbances. The large vertical shift in the T7P-βL IO relation with the three temperature loops on the main column closed also suggests that any temperature-based CV in the prefractionator must allow the prefractionator temperature profile to drift appropriately. Attempting to hold the temperature profile in the prefractionator in the face of large feed composition disturbances is then not a good control policy. Differential/ double differential temperatures on the prefractionator are potential CVs that allow the temperature profile to float and are explored later. In light of set point infeasibility of the T7P-βL loop, the recommended control structure using only tray temperatures as CVs for regulatory control is CS1T with βL held constant at its base-case value. For any temperature inferential control system, small steady state deviations in the product purities are expected and CS1T is no exception. To drive the deviations to zero, one can update the three temperature set points under 3 × 3 decentralized product composition cascade control. The set

Figure 2. Basic temperature inferential control structures for DWC.

the T7P-βL pairing. For brevity, these control structures with only single tray temperature control are referred to as CS1T and CS2T. 9410

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Figure 3. Open and closed loop CV−MV steady state relationships: solid line, fixed βL, R, S, or QR; dashed line, fixed T7M, T32M, T39M or βL; dotted line: fixed T7M, ΔTM, βL.

points T7M SP and T39M SP get updated for distillate and bottoms toluene impurity (xtolD and xtolB) control, respectively, to 1 mol % each. The set point T32M SP gets updated to hold the sidedraw toluene purity (xtolS) at 99 mol % (total side-draw impurity: 1 mol %). Figure 4 plots the variation in xtolS with T32M holding xtolD and xtolB at 1 mol % each for the base-case equimolar fresh feed and a benzene lean feed at the base-case βL. The plot clearly shows that for the benzene lean feed, a side-draw toluene purity of 99 mol % is infeasible. Similarly, we found that the xtolB-T39M IO relation (constant xtolB, xtolS, βL) exhibits input multiplicity

and an xtolB of 1 mol % becomes infeasible for particular nominal feed composition changes. In contrast, the xtolD-T7M IO relation (constant xtolB, xtolS, βL) does not exhibit multiplicity for the tested T7M range. These IO relations imply that CS1T cannot simultaneously achieve the target purity of all product streams for large feed composition disturbances. If the top and bottom (side) purities are held constant, the side-draw (bottom) purity must be allowed to float. Attempting to eliminate the side-draw (bottom) purity offset can cause control system failure due to steady-state infeasibility. Alternative Temperature-Based CVs. Overall, the results suggest incentive for seeking alternative temperature-based CVs to replace T7P, T32M, and T39M. As explained subsequently, the closed loop input multiplicity behavior of T32M and T39M is due to the same underlying physical phenomena and therefore correlated. Thus, appropriately replacing T32M significantly mitigates T39M multiplicity. Accordingly, CV replacements are sought only for T7P and T32M. T32M may not effectively close the toluene component balance due to input multiplicity. The multiplicity in the xtolST32M IO relation also makes side-draw purity based T32M SP compensation unreliable due to the possibility of side-draw composition set point infeasibility. With T7P as CV, the problem is that controlling it (using βL) may further significantly degrade control performance as the possibility of control system failure due to temperature set point infeasibility lurks even when ontarget product purity operation is not attempted. Alternative CV for T32M. To seek a CV alternative to T32M, we first consider differential temperature as a candidate. The primary motivation is improving the rangeability15 of the CV-

Figure 4. Variation of xtolS with T32M at constant xtolD, xtolB, and βL. 9411

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reduces reflux dramatically, which in turn causes T32M to increase and the low liquid rate pulls xylene up TS5. A corollary to the input multiplicity in the T32M-S IO relation (constant T7M, T39M, βL) around points 2 and 3 is the input multiplicity in the T39M-QR IO relation (constant T7M, T32M, βL; see Figure 3). As QR is reduced below its base-case value, more toluene leaks out the bottoms causing T39M to reduce (positive slope), as expected. Upon large QR reduction, however, much of the feed toluene drops down the prefractionator and is pulled into TS5 causing T32M to decrease. The T32M-S loop, in order to bring T32M back to set point, increases S dramatically pulling up more xylene. With significant xylene being pulled up the side-draw, the TS6 (main column stripping section) toluene mole fraction decreases. T39M thus increases with decreasing QR (positive slope) at significantly reduced QR values. The first turning point in the closed loop T39M-QR IO relation is then the direct counterpart of point 2 in the closed loop T32-S IO relation. Similarly, when the QR reduction is very large, benzene starts dropping down into TS5 with a very large reduction in R by the T7M controller. The significantly reduced liquid rate in the main column causes toluene to drop down the bottoms causing T39M to again decrease with QR. The second turning point in the closed loop T39M-QR IO relation is thus the direct counterpart of point 3 in the closed loop T32M-S IO relation. The same physical phenomena of changes in the specific component being pulled into the side-draw are the root-cause of the input multiplicity in the closed loop T32M-S and T39M-QR IO relations. It is then reasonable to expect that replacing T32S by a CV that adjusts S in a manner that mitigates input multiplicity would also mitigate the multiplicity in the T39M-QR IO relation. To replace T32M with a better CV, consider again the T32M-S IO relation at constant βL, T7M, and T39M (Figure 5). Around the base-case point 0, T32M changes due to xylene leakage up TS5 and does not have information on the prefractionator top xylene spillover. Clearly, if the spillover changes, for example, because of a large feed composition disturbance, the TS5 xylene leakage must also be counter adjusted via appropriate changes in T32M SP in order to mitigate deviations in the side-draw principal impurity content. Since essentially all of the spilled over xylene moves down, an increase in the spillover can be inferred from an increase in the tray temperatures above the side-draw (TS4). The clearest indication of a change in the xylene spillover would be a TS4 temperature near the sidedraw, since those trays have negligible benzene and the tray composition is binary (toluene−xylene, no benzene). If in response to an increase in a TS4 tray temperature implying increased xylene spillover, T32M SP is increased, it would pull more xylene up TS5, which in turn would cause the bottoms toluene impurity to decrease (since heavier xylene is being pulled up). The bottoms xtolB-T39M cascade would cause QR to decrease to maintain xtolB at 1%, which would cause the vapor into prefractionator to decrease. The prefractionator top xylene spillover would eventually decrease back toward its base-case value, as desired, via the nested action of the bottoms impurity controller. On the basis of the above arguments, we consider ΔTM = TS5 − TTS4, the difference between a TS5 and a TS4 tray T temperature, as a candidate CV for replacing T32M. We looked at different TTS5 and TTS4 combinations and selected ΔTM = T27M − T22M, with T22M located just above the side-draw. Figure 5 also shows the T27M − T22M versus S IO relation (constant βL, xtolD, xtolB) along with IO relations for T32M − T22M, T24M −

MV IO relation using the alternative CV. Consider the points marked 1, 2, and 3 in the T32M-S IO relation (constant βL, T7M, T39M) in Figure 5. At point 1, the slope is ∞ while at points 2

Figure 5. Variation of alternative temperature-based CVs for pairing with S.

and 3 the slope is 0. The three points are indicative of large shifts in DWC behavior due to specific physical phenomena, as interpreted below. The slope change around point 1 corresponds to the opposing trend between the xylene spillover from the prefractionator top and the xylene leakage up TS5 (tray section below side-draw). More specifically, as T32M SP is increased above point 1, the TS5 xylene leakage increases noticeably, while the prefractionator top xylene spillover remains negligibly small. This causes S to increase for DWC overall toluene material balance closure (positive slope). On the other hand, as T32M SP is decreased below point 1, the prefractionator top xylene spillover increases noticeably while the TS5 xylene leakage remains negligibly small. This again causes S to increase for a negative slope (i.e., turning point 1). We found that a T32M SP decrease too far below point 1 is not feasible at constant βL, xtolD, and xtolB. This is because the prefractionator xylene spillover cannot be made too large as the prefractionator performs the very easy benzene−xylene split. Since distillation is one of the most efficient separation techniques, too sloppy a split is not feasible.17 The slope sign change around point 2 corresponds to a change in the component being pulled into TS5. To the left of point 2, as S is increased, more xylene gets pulled up, which, being the heaviest component, causes T32M to increase (positive slope). To the right of point 2, increasing S causes more and more toluene to be pulled in from the prefractionator. This is because since S is large, significant xylene leaks out in S resulting in significant reduction in QR and hence the vapor to prefractionator. Most of the toluene then drops down the prefractionator and moves up TS5 causing T32M to decrease (negative slope) as toluene is lighter than xylene. The slope sign change around point 3 corresponds to benzene being drawn out the side-draw. Since S is very large beyond point 3 and both top and bottom temperatures are held constant (i.e., near constant purity), component material balance closure forces benzene to leak out in S. Since benzene must drop down TS4 for a very large S, the T7M controller 9412

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Industrial & Engineering Chemistry Research T22M and T32M for an illustrative comparison. For T27M − T22M CV, the IO relation exhibits good base-case sensitivity (slope) and avoids a crossover with respect to the base-case set point value by a large margin. The crossover is avoided even for ±6 mol % changes in each of the fresh feed component mole fractions (data not shown). Also, the IO relation turning point corresponding to point 1 is >1 °C away from the base-case CV set point. This allows operators leeway for making small adjustments in the CV set point to correct for, for example, small sensor biases. Note that reaching CV set points below point 1 requires an impractical change in the controller gain sign, since the IO relation slope sign changes at point 1. In contrast, for T32M − T22M CV, even as the turning point corresponding to point 1 is >2 °C away from the base-case set point, the IO relation hovers very close to a crossover with respect to the base-case set point. In fact, since the middle section temperature section profile floats with changes in the feed composition, a crossover does occur for small changes in the feed composition (data not shown). The IO relation for T24M − T22M is relatively straight with no maximum (input multiplicity). However, its base-case sensitivity (slope) is noticeably lower. Also, the turning point equivalent to point 1 is only 0.45 °C away from the base-case, which severely limits its adjustability by the operator (or product purity controller). ΔTM = T27M − T22M thus represents a good combination of good base-case sensitivity (open loop gain) for tight product purity control and robustness to measurement biases/set point changes, as well as its CV−MV relation avoiding input multiplicity with respect to the base-case set point for robustness to large feed composition changes. As an aside, we also note that at fixed values for the recommended ΔTM, T7M, and βL, no multiplicity is observed in the T39M-QR CV− MV relation. The relation is shown as the gray dashed line in Figure 3, confirming the complementary nature of the multiplicity behavior of T39M-QR and T32M/ΔTM-S IO relations. Subsequent dynamic simulations showed that the recommended ΔTM CV effectively handles significantly higher feed composition changes as well as ±1 °C set point changes. In view of its good performance (discussed later), ΔTM is considered an appropriate replacement of T32M. No further CV refinement is deemed necessary. The basic problem of on-target side-draw purity infeasibility for feed composition changes remains with the recommended ΔTM CV. This is illustrated in Figure 6 that plots the variation of xtolS with ΔTM for the base-case and benzene lean feeds at three values of βL with constant xtolD and xtolB (1 mol % each). The infeasibility of 99 mol % side-draw purity for the benzene lean feed at the base-case βL is clearly evident. The plot also shows that the maximum achievable side-draw purity changes with βL. Thus, even for no change in the fresh feed composition, an inadvertent decrease in βL due to, for example, sensor biases, can result in side-draw purity infeasibility. To avoid it, βL must be changed in an appropriate feedback arrangement. Earlier, the T7P−βL CV−MV pairing was found unsuitable due to temperature set point infeasibility caused by input multiplicity coupled with large vertical drifts for feed composition changes under closed loop operation. We therefore seek alternatives to T7P CV. Alternative CV for T7P. A more robust temperature-based CV for replacing T7P is sought by evaluating the effect of βL on the prefractionator temperature profile for altered feed compositions with the three product purities at set point. Figure 7 plots the TS1 temperature profile for an equimolar,

Figure 6. Effect of βL on xtolS-ΔTM IO relation at constant xtolD and xtolB: black line, base-case feed; gray line, benzene lean feed.

Figure 7. Effect of βL on prefractionator rectification section temperature profile: black line, equimolar feed; dark gray line, benzene-rich feed; light gray line, xylene-lean feed.

benzene lean and xylene lean fresh feed for three different βL values (fixed xtolD, xtolB, and xtolS). The downward drift in the entire TS1 temperature profile for the heavier feeds is evident. The plot also shows that as βL is changed, the temperature profile curvature changes in a predictable manner. An increase in βL decreases the xylene spillover due to higher liquid flow and the profile curves downward. Conversely, a decrease in βL results in more xylene spillover and the profile curves upward. This suggests that βL may be used to hold the TS1 temperature profile curvature. Since double differential temperature across a tray section is a direct measure of curvature, we consider Δ2TP = (T13P − T8P) − (T7P − T1P) as a candidate CV for replacing T7P. Figure 8 plots the Δ2TP-βL CV-MV IO relation at fixed R, S, and QR as well as at fixed T7M, T32M, and T39M for an equimolar feed and a benzene lean feed. While input multiplicity is observed in the IO relation, a crossover with respect to the base-case set point value does not occur for a βL range of 0.2 to 0.4. Also, unlike the T7P-βL IO relation at fixed T7M, T32M, and T39M, the corresponding Δ2TP-βL IO relation does not exhibit large vertical drifts for feed composition changes, which should significantly mitigate the possibility of CV set point infeasibility. 9413

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Industrial & Engineering Chemistry Research

in lieu of constant βL, the output multiplicity also disappears for a monotonic IO relation. In addition to CS1T, CS2T, and CS2ΔTΔ2T, we also have two other control structure variants that merit evaluation. In CS1ΔT, ΔTM replaces T32M for improved IO relation rangeability with respect to S. In CS2Δ2T, Δ2TP replaces T7P to avoid T7P set point infeasibility for feed composition disturbances. For clarity, Table 1 tabulates the CVs corresponding to the four control dofs for each of these structures with a brief summary of the multiplicity behavior of the temperature-based CVs. Dynamic Evaluation of CS1 and CS2 with Alternative CVs. The closed loop performance of the five CS1/CS2 variants in Table 1 is now evaluated using rigorous dynamic simulations. In addition to basic regulatory temperature inferential control (no product purity control), we also consider decentralized composition controllers on top of the three temperature-based CV loops on the main column for holding the distillate, side-draw, and bottoms product purities. The subscript “CC” is applied on a control structure acronym to conveniently identify product purity control. Thus, for example, refers to CS2ΔTΔ2T with T7M, ΔTM, T39M and Δ2TP CS2ΔTΔ2T CC as the temperature-based CVs and xtolD, xtolS, and xtolB composition controllers manipulating T7M SP, ΔTM SP, and T39M SP respectively. Dynamic Simulation and Controller Tuning. A pressure driven rigorous dynamic simulation is used for control system evaluation. The reboiler and condenser are sized for a base-case residence time of ∼10 min at the 50% level. The DWC shell diameter is chosen such that the maximum vapor superficial velocity in any part of the column does not exceed 1 m/s for a dividing wall cross-sectional area partitioning as per the prefractionator vapor split. The tray resistance to vapor flow is initialized to the base-case column pressure-vapor flow profile. The feed, distillate, side-draw, and bottoms pump-valve combinations are sized for ∼800 kPa pressure drop across the valve at 50% opening at the base-case steady-state for good flow rangeability. The reflux and liquid/vapor to prefractionator flows are directly set in Hysys. The controllers are tuned as follows. All flow controllers are proportional integral (PI) controllers and use a small reset time (∼30 s) and the proportional gain adjusted for a fast nonoscillatory servo response. The column pressure controller

Figure 8. Variation of Δ2TP with βL: black line, constant R, S, and QR; gray line, constant T7M, T32M, and T39M.

From a regulatory perspective, the Δ2TP-βL loop holds the prefractionator top xylene spillover. This is likely to significantly alter the xtolS-T32M/ΔTM CV−MV IO relation at constant βL, that exhibits composition set point infeasibility due to input multiplicity (see Figure 5). Figure 9 compares the xtolS-T32M/ ΔTM and T32M/ΔTM-S IO relations for constant βL operation and constant Δ2TP operation with xtolD and xtolB fixed. Since the TS1 xylene spillover is maintained by holding Δ2TP, the input multiplicity in the xtolS-T32M/ΔTM IO relations disappears and xtolS monotonically decreases with T32M/ΔTM. Holding Δ2TP also causes the output multiplicity in the T32M/ΔTM-S IO relation to disappear. The closed loop input multiplicity behavior of T32M-S remains as this multiplicity is not correlated with TS1 xylene spillover. Purely based on the bifurcation analysis results, the recommended regulatory control structure for robust DWC operation is CS2 with T7M, ΔTM, T39M, and Δ2TP as the temperature-based CVs. This recommended regulatory control structure is referred to as CS2ΔTΔ2T. ΔTM is preferred over T32M because the ΔTM-S IO relation (constant βL, xtolD, and x tol B ) does not exhibit input multiplicity, unlike the corresponding T32M-S IO relation. Further, at constant Δ2TP,

Figure 9. Effect of prefractionator CV on xtols-ΔTM/T32M and ΔTM/T32M-S IO relations at constant xtolD and xtolB: dashed line, constant βL; solid line, constant Δ2TP. 9414

DOI: 10.1021/acs.iecr.5b01907 Ind. Eng. Chem. Res. 2015, 54, 9407−9420

Article

Industrial & Engineering Chemistry Research Table 1. Control Structure Variants Evaluated CS

CV−MV pairings

CS1T

βL-LP, T7M-R, T32M-S, T39M-QR

CS2T

T7P-LP, T7M-R, T32M-S, T39M-QR

CS1ΔT

βL-LP, T7M-R, ΔTM-S, T39M-QR

CS2Δ2T

Δ2TP-LP, T7M-R, T32M-S, T39M-QR Δ2TP-LP, T7M-R, T32M-S, T39M-QR

CS2Δ2TΔT

comments on suitability of temperature-based CVs T7P as CV −input multiplicity at fixed T7M, T32M, T39M −infeasible temperature set point for large upward/downward drift for feed composition changes T32M as CV −output multiplicity at fixed βL, T7M, T39M. Output multiplicity disappears for fixed Δ2TP. −CV−MV input multiplicity at fixed βL/Δ2TP, T7M, T39M with possible closed loop steady state transition. −possible infeasible xtolS at constant βL, xtolD, xtolB. Infeasibility removed when βL replaced by Δ2TP. ΔTM as CV −output multiplicity at fixed βL, T7M, T39M. Output multiplicity disappears for fixed Δ2TP. −avoids crossover with respect to base−case set point. −possible infeasible xtolS at constant βL, xtolD, xtolB. Infeasibility removed when Δ2TP replaces βL. T39M as CV −input multiplicity at fixed βL, T7M, T32M. Possible closed loop steady state transition. Replacing T32M with ΔTM removes input multiplicity. −possible infeasible xtolD at constant βL, xtolD, xtolS. Infeasibility removed when Δ2TP replaces βL.

Table 2. Product Purity Deviations of Control System Variants to Nominal Disturbances steady state purity deviations disturbance

product stream

CS1T

B+

distillate side bottom distillate side bottom distillate side bottom distillate side bottom distillate side bottom distillate side bottom distillate side bottom distillate side bottom

0.0046 0.0017 −0.0011 −0.0032 0.0026 0.0013 −0.0024 0.0005 −0.0008 0.0046 0.0000 0.0011 −0.0005 0.0001 0.0022 0.0007 0.0016 −0.0020 −0.0014 −0.0022 0.0049 0.0011 0.0025 −0.0030

B−

T+

T−

X+

X−

F+

F−

T

CS2

0.0075 0.0170 −0.0015 fails

fails

0.0062 0.0089 0.0008 −0.0010 −0.0025 0.0024 0.0016 0.0051 −0.0022 −0.0014 −0.0027 0.0049 0.0010 0.0021 −0.0030

CS1ΔT

CS2Δ2T

CS2Δ2TΔT

0.0043 0.0004 −0.0011 −0.0031 0.0022 0.0013 −0.0025 0.0002 −0.0008 0.0047 0.0005 0.0010 −0.0004 0.0009 0.0022 0.0004 −0.0003 −0.0020 −0.0008 0.0002 0.0045 0.0008 −0.0001 −0.0029

0.0055 0.0031 −0.0012 −0.0034 −0.0018 0.0015 −0.0026 −0.0006 −0.0007 0.0052 0.0014 0.0010 −0.0007 −0.0011 0.0023 0.0009 0.0021 −0.0021 −0.0015 −0.0032 0.0050 0.0012 0.0028 −0.0030

0.0051 0.0016 −0.0012 −0.0033 −0.0008 0.0014 −0.0026 −0.0008 −0.0007 0.0053 0.0019 0.0010 −0.0005 0.0000 0.0022 0.0006 0.0000 −0.0020 −0.0009 0.0000 0.0045 0.0008 0.0000 −0.0029

With the controllers tuned as above, the five variants (CS1T, CS1ΔT, CS2T, CS2Δ2T and CS2ΔTΔ2T) with/without product purity controllers are dynamically tested for throughput and feed composition changes. For brevity, we have omitted the dynamic plots and summarize the control system performance in terms of the tightness of product purity control for constant temperature-based CV set point operation as well as robustness to large disturbances. Product Purity Control. Table 2 reports the distillate, sidedraw, and bottoms product purity deviations for the nominal ±20% throughput change and ±6 mol % feed composition changes at constant set point of the temperature-based CVs (i.e., purity controllers on manual). All the disturbances are effectively rejected by all the control systems, except CS2T, with the steady-state product purity deviations being within ±0.5

is PI and uses a large gain for tight pressure control. The reflux drum and bottom sump level controllers are P only and use a gain of 2. In the temperature-based CV loops, a lag of 2 min is applied to account for sensor dynamics. All temperature-based CV controllers are PI and are tuned individually. The initial tuning parameters are obtained using the Hysys autotuner. Subsequently, the controller gains are detuned to ensure a slightly underdamped response to the principal disturbances under closed loop operation. All the product composition controllers use a measurement sampling time and dead-time of 5 min each. These loops use a liberally large reset time with the gain adjusted by hit-and-trial for a slightly under-damped servo response. The salient controller parameters used in the simulations are provided in Supporting Information Table S1. 9415

DOI: 10.1021/acs.iecr.5b01907 Ind. Eng. Chem. Res. 2015, 54, 9407−9420

Article

Industrial & Engineering Chemistry Research Table 3. Maximum Feed Composition Change Handled by Alternative Control System Variants disturbance

CS1T

CS2T

CS1ΔT

CS2Δ2T

CS2Δ2TΔT

T CS1CC

T CS2CC

ΔT CS1CC

Δ2T CS2CC

Δ2TΔT CS2CC

B+ B− T+ T− X+ X−

12% 18% 18% 12% 18% 12%

6%