MULTI-LOOP CONTROL STRUCTURE FOR DIVIDING-WALL DISTILLATION COLUMNS Salvador Tututi-Avila, Center for Biotechnology and Interdisciplinary Studies, Rensselaer Polytechnic Institute, Troy, NY, Arturo Jiménez-Gutiérrez, Departamento de Ingeniería Química, Instituto Tecnológico de Celaya, Celaya, Gto., México and Juergen Hahn, Biomedical Engineering and Chemical & Biological Engineering, Rensselaer Polytechnic Institute, Troy, NY. Abstract This work investigates the dynamics and control of a dividing wall column (DWC) used for the separation of a ternary mixture. A rigorous, first-principles-based, dynamic model was developed in gproms. Step responses were simulated and a transfer function model was developed. RGA analysis was performed and a multi-loop control scheme based on PI controllers, tuned using IMC, was developed and implemented on the rigorous dynamic model. The results show that the standard DB- LSV control configuration was able to reject commonly-occurring disturbances and also achieved good set point tracking performance. A comparison with other conventional control configurations based on PI control loops is presented. Keywords: Dividing-wall column, Distillation, Petlyuk, IMC. Introduction Separation by distillation, the most common separation process in the chemical and petrochemical industries, consumes a significant amount of energy. New distillation structures, which hold the perspective to be more energy-efficient, have been recently considered. The fully thermally coupled structure, or Petlyuk column 1, has received special attention. The Petlyuk arrangement consists of a prefractionator coupled to the main column, using two recycle streams. Dividing-wall columns (DWCs) are thermodynamically equivalent to the Petlyuk system 2. DWCs split the middle section of a single vessel into two sections by inserting a vertical wall, thus implementing the Petlyuk configuration in a single shell 3. Dividing-wall columns represent a typical example of process intensification since they can bring significant reductions in both capital investment and energy costs of up 30% 4-6. In the last couple of decades, there have been over 100 DWCs found in industrial use worldwide 2. The benefits of DWC technology can only be achieved with proper control structures that provide a stable and robust operation of the separation process. Compared to a conventional distillation system, the control of a DWC can be more difficult due to increased interactions among the controlled and manipulated variables. Nevertheless, it has been reported that thermally coupled sequences have good controllability properties 7,8, providing that an appropriate control structure is selected 5. One key aspect for evaluating the performance of control systems applied to DWCs is the rigor of the model that the controllers are applied to. The most commonly used dynamic models incorporate several simplifying assumptions, which may not be the most realistic scenario for evaluating control structures. This work addresses this last point in that a rigorous, first-principles-based dynamic model of a DWC used for separation of a ternary mixture is developed. This model is implemented in gproms and is subsequently used to develop a control scheme. The PI/PID controllers are tuned using IMC tuning. The developed control structure is tested on the rigorous model for several commonly found disturbance
rejection and set point tracking tasks and then it is compared with various control structures based on a multi-loop framework. Finally a brief discussion and conclusions are presented. Modeling of DWCs Model Equations In contrast to simplified models 9,10, a rigorous dynamic first-principles based model is developed in this work and implemented in gproms. The dynamic model for the DWC was developed assuming a generic equilibrium stage. The column contains a total of N T theoretical trays. The liquid holdup on each tray, including the downcomer is M j. The liquid on each tray is assumed to be perfectly mixed with composition x i,j. The feed to stage j is a single or two phase feed of molar flow rate F j, with overall mole fraction composition z i,j of component i and corresponding overall molar enthalpy h Fj. Also entering stage j is interstage liquid from stage j 1 above, if any, of molar flow rate L j 1, with mole fraction x i,j and enthalpy h j 1. Similarly, from stage j + 1 below, interstage vapor of molar flow rate V j+1, with mole fraction y i,j+1 and enthalpy H j+1. Leaving stage j is vapor with y i,j, H j. This stream can be split into a vapor sidestream of molar flow rate W j and an interstage stream of molar flow rate V j to be sent to stage j 1. Also leaving stage j is a liquid stream given by x i,j, h j in equilibrium with the vapor (V j + W j ). This liquid is split into a sidestream of molar flow rate U j and an interstage stream of molar flow rate L j to be sent to stage j + 1. Heat can be transferred at a rate Q j from or to stage j to simulate stage intercoolers, interheaters, intercondensers, interreboilers, condensers, or reboilers. The mass and energy balances for any stage (2 < j < N T 1 ) including the feed streams are given by: d M j x i,j = L dt j 1 x i,j 1 + V j+1 y i,j+1 L j + U j x i,j V j + W j y i,j (1) + F j z i,j for i = 1,, N C d M j h j = L dt j 1 h j 1 + V j+1 H j+1 L j + U j h j V j + W j H j + F j h fj + Q j (2) Mole fraction summations (2 < j < N T ): NC NC x i,j = 1 y i,j = 1 i=1 i=1 Each tray and the base of the column use an equilibrium relationship (2 < j < N T ): L y i,j = K i,j x i,j = φ i,j V x i,j for i = 1,, N C (4) φ i,j where K is the vapor-liquid equilibrium constant and φ is the liquid or vapor fugacity coefficient. The liquid flow rates throughout the column will depend on the fluid mechanics of each stage and will not be constant throughout the column. The modified Francis weir formula (Equation 5) provides a relationship between the tray liquid holdup 11, M j, and the liquid molar flow rate leaving the j-th stage L j, (2 < j < N T 1 ): L j = 1.84 L weir v j L M L 3/2 Ljv j h A weir tray (3) (5)
where L weir is the weir length, v j L is the liquid molar volume, A tray is the tray active area and h weir is the liquid height on weir. An equation linking the pressure driving force to the vapor flow is given by 12 (2 < j < N T 1 ): P j+1 P j = α V j+1v j+1 V A holes 2 V ρ j+1 + βgρ j L M Ljv j L A tray (6) where A holes is the total area of all active holes; α and β are parameters; and ρ refers to molar density. Tray geometry and sizing were configured as in Georgiadis et al. 11. Equations (1-6) describe the model of the main column; the prefractionator equations are similar to the main column, but do not contain a condenser or a reboiler. The equations for the prefractionator and the main column must be solved simultaneously because of the recycle streams. Equation (7) provides the liquid and vapor relationships between the main column and the prefractionator. β L = L P β L V = V P (7) R V S where L P and V P is the liquid or vapor flow rate fed to the prefractionator, respectively. L R is the total liquid leaving the bottom tray in the rectifying section and V S is the total vapor leaving the top tray in the stripping section. Enthalpies, fugacity coefficients and molar volumes of liquid and vapor streams are calculated as functions of temperature, pressure, and composition. Vapor hold-up is neglected for this model; however, it can be important for columns operating at very high pressure 13. Additionally, a dynamic energy balance is used in this model. A common simplification found in the literature is to use an algebraic form 9,14. The rigorous dynamic model was implemented in gproms and solved for steady state as well as for dynamic conditions. Case Study Simulation studies require that a column design and mixture are specified for dynamic simulations. A mixture of n-pentane (nc 5 ), n-hexane (nc 6 ) and n-heptane (nc 7 ) has been used in several studies from the literature 9,15 and was considered for this work. The feed flow rate is 12.6 mol/s of a saturated liquid, with a molar composition 40%, 20%, and 40% for the components nc 5, nc 6, and nc 7, respectively. Specified product purities of 98 mol % for all components were assumed. The pressure design was limited by the assumption that the condenser could use cooling water. Thermodynamic properties were estimated using the SRK equation of state. The design methodology of the DWC used in this work is reported in Hernández and Jiménez 14. The details of the resulting design and operating conditions in both sections of the column are presented in Figure 1, where liquid composition profiles of the column are also shown. Composition profiles obtained with Aspen Plus, AspenTech, are basically the same to those obtained with gproms; however, composition profiles obtained with the simplified model clearly differ from the rigorous model in several points.
Figure 1. Divided wall column flowsheet and comparison of composition liquid profiles among gproms, Aspen Plus and the simplified model. RGA Analysis and PI Controller Tuning Relative gain array (RGA) was used 16 to determine the pairing of the controlled and manipulated variables. In order to identify the parameters of the process dynamics, step changes of 0.1% of the steady-state value in the input variables were implemented and the open loop dynamic responses were recorded. The dynamic responses were fitted to transfer functions and arranged into a transfer function matrix and then the RGA was calculated. Table 1 gives the RGA analysis for the DB control stabilization; these values suggest that LSV is the most favorable pairing. Table 1. Relative Gain array (RGA) for DB stabilization control. L 1 (Reflux) S (Side stream flow) Q R (Reboiler duty) β L (Split ratio) x A 1.1938-0.0015-0.1880-0.0043 x B 0.2409 0.8407-0.0688-0.0127 x C -0.3074 0.1629 1.3019-0.1574 y P11-0.1273-0.0021-0.0449 1.1743 To compare the common DB-LSV structure with other different control structures, only practical configurations are considered. Thus, the multi-loop control structures considered here are DB-LSV, DV- LSB, LB-DSV and LV-DSB 17, all of them with an additional loop additional control loop which is given by manipulating the liquid split ratio (β L ) in order to control of the heavy component composition (y P11(nC7) ) in the top of the prefractionator 18. PI controller settings were determined by using IMC tuning rules 19. Proportional controllers are used for the level controllers as no integrating action is required for level control 18.
Results and discussion Due to space limitations, we present here the dynamic response only for the best two control structures: DB/LSV and DV/LSB. Figure 2 shows the responses of the DWC system for the DB-LSV configuration. Figure 2a illustrates the dynamic responses of the system under a regulatory test by assuming a disturbance of +10% in the feed flow rate, at time = 0.5 hr. Figure 2b displays the dynamic results obtained for a feed composition disturbance, which was set by changing the composition of the light component by +10%, with a proportional adjustment of the other two components. The three product compositions with this control configuration return to their set-point within five hours or less. Figure 2. Dynamic response of DB-LSV control structure, under a disturbance of a) +10% in the feed flow rate and b) +10% x A in the feed composition. The dynamic responses of the DV-LSB structure shown in Figure 3 produce similar settling times to the DB-LSV structure for the disturbances discussed above. Figure 3. Dynamic response of DV-LSB control structure, under a disturbance of a) +10% in the feed flow rate and b) +10% x A in the feed composition. For comparison purposes of the effectiveness of the control structures, Figure 4 gives a summary of the overall performance by evaluating the integral of the absolute error (IAE) for the regulatory tasks. The superior performance of the DB-LSV control structure is supported by the lower IAE values.
Figure 4. Comparison of the performance of the control structures in terms of Integral Absolute Error (IAE) for a disturbance of a) +10% in the feed flow rate and b) +10% in the light component composition. The dynamic simulations show that the control actions can reject disturbances in the feed flow rate and in the feed composition, although the LB-DSV and LV-DSB control structures showed longer settling times than the DB-LSV and the DV-LSB structures. Conclusions This work presented a rigorous dynamic model of a DWC and developed a control scheme for the process. The model was implemented in the dynamic simulator gproms and then compared with a simplified model reported in the literature. A noticeable difference was found when the profiles of the rigorous model were compared with the simplified model. The results of the dynamic simulations were used to derive transfer functions between the controlled and manipulated variables. RGA was used for selecting the variables for control loop pairing and the controllers were tuned using IMC. For the common DB stabilization, the chosen control loop structure was the same as the widely known DB-LSV structure, with the exception that an additional loop has been used to implicitly minimize energy requirements. The closed-loop system using this control structure exhibited good control performance. A comparison with other commonly-used control structures was made. The DB-LSV and the DV-LSB control structures showed similar dynamic performance, with a slight advantage of the DB-LSV structure. That being said, both of these two control structures were able to satisfactorily reject disturbances in the feed flow rate and feed composition for the mixture considered in this work. Acknowledgement S. Tututi appreciates the financial support for scholar visiting from Conacyt, Mexico, through program "Estancias Posdoctorales y Sabáticas al Extranjero para la Consolidación de Grupos de Investigación 2011-2012 (Register No. 184961)". References 1. Petlyuk FB, Platonov VM, Slavinskii DM. Thermodynamically optimal method for separating multicomponent mixtures. Int. Chem. Eng. 1965;5(3):555-561. 2. Dejanović I, Matijašević L, Olujić Ž. Dividing wall column A breakthrough towards sustainable distilling. Chem. Eng. Proc.: Proc. Int. 2010;49(6):559-580.
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