Time-Optimal Control of Dividing-Wall Distillation Columns - American

Ind. Eng. Chem. Res. 2010, 49, 9195–9208

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Time-Optimal Control of Dividing-Wall Distillation Columns Alexandru Woinaroschy* and Raluca Isopescu Department of Chemical Engineering, “Politehnica” UniVersity of Bucharest, 1-5 Polizu Street, Bucharest, Romania

Time-optimal control of startup traditional distillation columns by iterative programming proposed by Woinaroschy for ideal [Ind. Eng. Chem. Res. 2008, 47, 4158] and nonideal mixtures [Ind. Eng. Chem. Res. 2009, 48, 3873] is extended to the case of dividing-wall distillation columns. The minimization of distillation startup time is performed by iterative dynamic programming employing randomly chosen candidates for admissible control. The control variables are the reflux ratio, the reboiler heat duty, and the side-draw flow rate. The dynamic distillation model proposed by the author in the previous papers is applied. Two illustrative case studies for the separation in a dividing-wall column with sieve trays and lateral downcomers are presented as follows: the separation of an ideal benzene-toluene-ethylbenzene ternary mixture and the separation of a nonideal methanol-ethanol-1-propanol mixture. In another case study, a conventional two-column system is presented in comparison to the dividing-wall column. As in the cases of traditional distillation columns, the startup time decrease and the corresponding reboiler energy savings are significant for each of the control variables. Introduction Distillation is the most commonly used separation process in chemical and petrochemical plants. Its main disadvantage is high energy consumption, and this is the reason why thermally coupled distillation systems are studied to provide challenging solutions for industrial implementations. The dividing-wall distillation column (DWC), which represents the Petlyuk column built in a single shell, is a very promising alternative for both energy and cost savings. Several theoretical studies have reported important energy savings in thermally coupled columns, including the DWC, as compared to classical separation schemes.1-5 Some practical applications of this technology are also reported, underlying the thermal efficiency and cost reduction.6-8 The most important parameters that are mentioned to influence the energy consumption in a DWC are the feed composition and pressure.8 The DWC led to about 40% energy savings, and the efficiency of DWC increases when the middle component is in a large amount in the feed.9 A case study for a common hydrocarbon mixture separation in oil refiners demonstrates that the heat transfer units reduction and the use of a single column shell provide a decrease up to 23% of the capital cost as compared to a classical two-column separation sequence.10 Despite the advantages of the DWC, the industry is still reluctant to introduce it on a large scale due to not yet wellestablished design procedures and fear of control issues.5 The design of a full thermally coupled column can not follow the conventional multicomponent design procedures when information about interlinking streams is unknown.11 A design procedure based on a three-column model was developed by Triantafyllou and Smith.12 In their model, the final structure has to be iteratively adjusted, as the number of trays on both sides of the dividing wall must be the same. This problem also arises when a commercial simulator is used to establish the final column topology. A structural design methodology for full thermally coupled distillation columns is developed by Kim,11 based on tray to tray calculations, assuming very large reflux ratios in the main column and ideal trays. From an operational * To whom correspondence should be addressed. Tel: +40-214023902. Fax: +40-21-3185900. E-mail: A_WOINAROSCHY@ chim.upb.ro.

point of view, the thermally coupled columns seem to raise more problems than simple distillation columns. The operational complexity of a DWC derives from the increased number of freedom degrees as compared to a simple column with a side draw. Halvorsen and Skogestad13 found that the liquid and vapor split ratios, considered as two additional freedom degrees, are important for the thermal efficiency of the DWC. An optimal solution can be established only for certain correlations of the liquid and vapor split. Moreover, Wang and Wong14 demonstrated that for an infinite number of stages column, the energy efficiency depends on the liquid and vapor split values, with a trade-off between energy efficiency and controllability of end products composition. Dividing-wall columns still have rare applications, due to some uncertainties when dealing with the dynamics of the process. In general, the heat integration brings supplementary difficulties when column dynamics and start up policies are implied. High values of startup time have been found to characterize a heat-integrated two-pressure column system, implying higher costs.15 The dynamic simulation of a distillation column has proved to be an efficient theoretical tool in providing startup policies.16,17 The comparison with experimental studies concerning startup and transient regimes is also of crucial importance when operating conditions change and a new steady state must be reached.17,18 Commercial simulators, such as HYSYS or ASPEN, provide reliable environments for steady state and dynamic simulation of distillation columns, including conventional PID controllers. A special structure as the DWC can also be implemented using thermodynamic equivalent schemes. For such a scheme, concentration control loops19 and temperature control loops20 were defined to handle feed composition or feed flow variations. When special control or optimization procedures are applied for the analysis of heat-integrated structures, the simulators are used to provide steady state solutions in the frame of an optimization loop21 or stand-alone simulation programs are built, using a convenient mathematical model for the column dynamics, efficient in terms of reliability and computation time for the optimization method or controller type adopted.22,23

10.1021/ie100090p  2010 American Chemical Society Published on Web 08/25/2010

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Ind. Eng. Chem. Res., Vol. 49, No. 19, 2010

In the present work, the DWC topology was designed in the frame of HYSYS simulator, a dynamic model was elaborated based on this structure, and the optimal startup time was defined, using iterative dynamic programming as the optimization procedure. Dynamic Simulation Model The dynamic distillation models (DDMs) presented in literature are algebraic-differential models. The dynamic of DWC was simulated with a rigorous DDM proposed by Woinaroschy.24,25 The advantage and originality of this model consist in the fact that the iterative algebraic equations are avoided, the model being a good compromise between the complexity degree and the correctness. The following simplification assumptions are present in this model: (i) The molar vapor holdup is negligible as compared to the molar liquid holdup. (ii) Interphase heat transfer is considered much more intense than interphase mass transport; consequently, the liquid and vapor leaving each plate are in thermal equilibrium at the boiling temperature, corresponding to the liquid composition. (iii) The liquid and vapor are perfectly mixed on each plate, and Murphree plate efficiency can be used for real trays applications. (iv) Entrainment and weeping rates, plate flooding, downcomer holdup, and delay time between plates are neglected. Thus for a given tray, j, when neglecting the molar vapor holdup, mass-balance equations are defined as: - total mass balance around plate j dNj ) Lj-1 + Vj+1 - Lj - Vj ( FL,j ( FV,j dt

dTj )dt

∑ i)1

(

)

xi,j dγi,j dxi,j γi,jPi,j 1+ pj γi,j dxi,j dt m

∑ i)1

xi,jγi,j dPi,j pj dTj

(2)

m

dTj )dt

i,j

i)1 m

∑ i)1

(4)

The liquid flow rate is obtained on the base of Francis’ correlation for a plate weir, leading to

Lj )

1.84lwFL,j m

∑Mx

i i,j

(

)

1.5

m

Nj

∑Mx

i i,j

i)1

- zW

εL,jApFL,j

(5)

i)1

To avoid iterative calculations, the vapor flow rates and compositions are computed in the j ) n, n - 1, ..., 1 order, while the liquid flow rates are calculated in the j ) 1, 2, ..., n order. The total pressure variation during one time integration step is much smaller than the composition and temperature variations. To simplify the procedure, the pressure is considered constant along the time integration step; it is recomputed at the beginning of each new time step pj ) pj+1 - ∆pj

(6)

∆pj ) -

0.00683 FG,j

(

∑

)

2

m

QG,j

yi,j Mi

i)1

Ao,j

where

σj do 0.07352FL,jzst,j

- 0.02175

(3.1)

(3.2)

zst,j ) 0.65zW + 0.8

( )

(7)

0.667

m

dxi,j dt

dκi,j dTj

]

Lj

This equation is applied for ideal and nonideal mixtures in the liquid phase. For nonideal mixtures, in liquid and/or vapor phase, the temperature variation can be computed with the equation.26

∑κ

[

1 L (h - hj) + Vj+1(Hj+1 - hj) ( FL,j(hF,j - hj) ( Hj j-1 j-1 dhj dTj FV,j(HF,j - hj) - qj - Nj dTj dt

where the pressure drop is calculated on the base of hydraulic correlations, specific for the plate type. For the sieve tray, this is

The model original feature (due to the fact that iterative solving of nonlinear algebraic equations is avoided; for details, see Woinaroschy24,25) consists in the temperature calculation on each plate with the equation m

Vj )

(1)

- mass balance around plate j for component i dxi,j 1 ) [Lj-1(xi,j-1 - xi,j) + Vj+1(yi,j+1 - xi,j) ( dt Nj FL,j(xF,i,j - xi,j) ( FV,j(yF,i,j - xi,j)]

Therefore, knowing the liquid composition and temperature on each tray, the vapor composition is obtained by straightforward computation, without iterative calculations. The model core consists of n equations (eq 1), n(m - 1) equations (eq 2), and n equations (eq 3), therefore, a total number of n(m + 1) differential equations. The rest of the variables involved in these equations are explicitly calculated as follows: The vapor flow rate is obtained from the total energy balance:

∑x

i,j Mi

i)1

lW,jFL,j

(8)

The state variables are Ni, xi,j, and Tj for i ) 1, ..., m and j ) 1, ..., n. The control variables considered in the case studies are the reflux ratio, the reboiler heat duty, and the side-draw flow rate. These control variables are implied in DDM as follows: (i) The reflux ratio is involved in eq 1, which describes the total mass balance around plate 1 in eq 2, expressing the mass balance around plate 1 for each component i, and in eq 4, when calculating the vapor flow rate over plate 1. If the total condenser type is considered, the material balance over tray 1, the condenser, and condensate splitter gives L0 )

R V R+1 1

(9)


Therefore for j ) 1, eqs 1, 2, and 4 become dN1 V1 ) V 2 - L1 dt R+1 dxi,1 1 R V (x - xi,1) + V2(yi,2 - xi,1) ) dt N1 R + 1 1 i,D

[

(1.1)

] (2.1)

V1 )

V2(H2 - h1) - N1

dh1 dT dT dt

R (h - hD) + H1 - h1 R+1 1

(4.1)

The value of the liquid flow rate L1 (eq 5 for j ) 1) is established according to the dependence of the liquid holdup per plate 1 versus the reflux ratio. (ii) Reboiler heat duty is involved in eq 4, giving the vapor flow rate for the bottom (j ) n), where qn ) -qB. (iii) The side-draw flow rate is -FL,j in eqs 1, 2, and 4 for the side-draw tray j. Lower and upper bounds are imposed for each control variable. The equilibrium, thermodynamic data and physical properties correlations are selected according to the mixture nature. Details and model validation for traditional distillation columns are presented by Woinaroschy.25 Startup Simulation The startup of distillation columns is a very challenging control and simulation problem, due to theoretical and practical aspects. In fact, the startup is the most “dynamic” operation stage, involving important economic and safety considerations. Several researchers have studied this subject: Fieg and Wozny,17 Woinaroschy,24-26 Yasuoka et al.,27 Ruiz et al.,28 Barolo et al.,29,30 and Han and Park.31 None of these papers consider the case of thermally coupled distillation columns. Recently, the minimization of startup time in a heat and mass integrated two columns system was analyzed by Verbanov et al.,32 as a generalization of the two heat integrated-two pressure column system.15 An investigation on optimal startup control of DWC was briefly reported by Woinaroschy and Isopescu.16 A general sequence of actions, forming the basis for different startup procedures for sieve plate columns, was formulated by Ruiz et al.28 Step 0: The column is empty, and the liquid feed is introduced (only liquid feed is considered). Step 1: The liquid starts to weep to the plate below through the plate holes rather than through the downcomer. The liquid reaches the bottom of the column (or reboiler), thus increasing the liquid level there. Step 2: Heat is introduced in the reboiler, and vapors start to go up. Step 3: The condenser starts to operate as the vapor phase reaches the top plate, and the reflux drum starts to fill up. Step 4: The reflux is introduced into the column, and operation at total reflux begins. Step 5: The vapor flow through the plate holes seals the plates in terms of liquid weeping through the plate holes. This starts to increase the liquid holdup on the plates. Step 6: All plates have enough liquid holdup so that the liquid can start to fall down the downcomers. The

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downcomers are sealed, and vapors cannot pass upward through them. Step 7: The column operation is changed from the total reflux. Distillate, bottom product, and the possible side streams are taken out. In the frame of the present paper, the effective start-up transient operating regime procedure begins at the end of step 7. This is the objective of the simulation. The liquid composition is considered the same on all plates at the beginning of simulation, being equal to the feed composition. Traditionally, the column is operated at constant values of control parameters (feed rate, feed thermal state, reflux ratio, reboiler heat duty, bottom pressure, etc.). These values are those corresponding to the final desired stationary regime. All state variables values (concentrations, temperatures, pressures on plates, and in output material fluxes) vary in time. The fluctuations amplitude decreases in time and will vanish at steady state regime. The products compositions are, of course, different from the desired values up to the stationary regime, the products being collected, mixed, and recycled to the feed. Apart from the chemical reactors case, the momentum, energy, and mass transfer processes, which are not accomplished by chemical reactions, do not have multiple or nonstable stationary states. Time-Optimal Control (TOC) The TOC problem is to minimize the final time, tf, and to determine the corresponding control variables, ui(τk), i ) 1, 2, ..., nu; k ) 1, 2, ..., S, and length of time stages V(k), k ) 1, 2, ..., S restricted to si(tf) ) ssi i ) 1, 2, ..., ns

(10)

where the desired final stationary state values, ssi, are computed from the differential equations of the continuous dynamic system dsi ) V(k)Sfi(s1, s2, ..., sns, u1, u2, ..., unu) i ) 1, 2, ..., ns; dτ k ) 1, 2, ..., S (11) Practically, eq 11 corresponds to eqs 1-3, after introducing the normalized time variable, τ, so that τ ) t/tf, takes discrete values τk ) k/S, k ) 0, 1, ..., S. Each stage is of equal length 1/S in the transformed time domain. To minimize the final time, tf, subject to the constraint given in eq 10, the penalty performance index was formulated as ns

I ) tf + ω

∑ i)1

|

1-

si(tf) ssi

ns

|

(12)

where ω > 0 is a penalty coefficient. The application of TOC is the same as in the previous papers,25,26 using the algorithm proposed by Bojkov and Luus,33 based on the IDP procedure given by Bojkov and Luus,34,35 which employs randomly chosen candidates for the admissible control.36 The system of differential equations was numerically integrated by a fourth order Runge-Kutta-Gill method, which automatically chooses the time step to satisfy local error tolerance. It can be appreciated that the corresponding computer time increase due to the small values of the integration step is justified, thus avoiding the iterative solution of algebraic equations, which are even more computer time consumers.25,26 The computer programs were coded in FORTRAN. Case Study 1: A Ternary Ideal Mixture Configuration of DWC. The ternary ideal mixture considered is a common one in oil refineries: benzene, toluene, and

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Figure 1. DWC topology: Simulation flowsheet in HYSYSsCase Study 1.

Figure 2. DWC topology: Designed structure of the DWCsCase Study 1.

ethylbenzene. The feed compositions, in mole fractions, were 0.6/0.3/0.1. An estimation of the number of trays in each section of the DWC, feed tray, and side-draw location as well as the position of interlinking vapor and liquid streams were obtained using the short-cut design option available in HYSYS, which applies Fenske-Underwood-Gilliland-Kirkbride equations. This structure is largely used in the design step, as generally mentioned in the literature.12 A four-column equivalent structure was derived from this initial design (Figure 1). This structure is more appropriate for dynamic simulations due to its capabilities of reflecting the internal streams and geometrical characteristics of the tray sections, to be used in the dynamic simulation and in the TOC procedure. The final structure, corresponding to the DWC, consists of a top tray section placed above the dividing wall, the trays in the prefractionator, the trays in the side-draw region, and the trays below the dividing wall. The number of trays in each section, the feed tray, side-draw location, the position of thermal coupling streams, and the reflux ratio as identified by short-cut design were slightly modified by repeated rigorous simulations aiming to fulfill the purity requirements (more than 0.95 mol fraction) at a low reboiler duty. This final design was considered a good DWC design, but this is not an optimal one in terms of minimum energy consumption. The Antoine model was used to calculate the thermodynamical properties of the system. Figure 1 presents the final topology, as resulting after rigorous simulation in the frame of HYSYS. As shown in Figure 2, the number of trays on either part of the DWC is not the same, unlike the generally used approach, with an equal number of trays in both sides of the wall.12,20 From the constructive point of view, an equal trays number would be preferable. In our analyses, the dividing wall is meant to ensure the same cross-section in the prefractionator, as in the side-draw region. The liquid split, defined as the ratio

Table 1. General Data for Case Study 1 parameters and operating conditions feed flow rate (kmol/h) feed composition (mole fractions) side-draw flow ratea (kmol/h) thermal state of the feed stream thermal state of the side-draw stream condenser type pressure in the top of column and in condenser (mm Hg) reflux ratioa reboiler type reboiler heat dutya (kW) bottom liquid volume (m3) tray area for top and bottom sections (m2) tray area for prefractionator and side-draw (m2) hole area per plate for top and bottom sections (m2) hole area per plate for prefractionator and side draw (m2) weir height (m) hole diameter (m) tray number of top section tray number of prefractionator tray number of side draw tray number of bottom section feed tray position side-draw tray position liquid split fraction in prefractionator a

value 126 benzene, 0.6 toluene, 0.3 ethylbenzene, 0.1 36 liquid at boiling temperature liquid at boiling temperature total 760 2 total 2095 0.911 2.8143 1.4071 0.2815 0.1407 0.025 0.005 11 14 25 13 12 30 0.6

Prescribed values for the desired stationary regime.

between the reflux flow rate in the prefractionator and the total reflux flow rate in the column, was adjusted during the dynamic simulation step, to provide the products purities. The vapor split is free to adjust by imposing the same pressure drop on both


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Table 2. Initial State and Desired State for the Case Study 1 initial state

desired state

tray j

x1,j

x2,j

x3,j

Tj

pj

x1,j

x2,j

x3,j

Tj

pj

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 B

0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6

0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

94.87 94.94 95.01 95.09 95.16 95.24 95.31 95.38 95.46 95.53 95.60 95.81 95.89 95.97 96.06 96.15 96.24 96.32 96.41 96.49 96.57 96.66 96.74 96.83 96.91 95.49 95.55 95.61 95.67 95.73 95.79 95.86 95.92 95.98 96.04 96.10 96.16 96.22 96.28 96.34 96.40 96.46 96.52 96.58 96.64 96.70 96.76 96.82 96.88 96.94 97.00 97.07 97.14 97.21 97.27 97.34 97.41 97.48 97.55 97.62 97.69 97.76 97.82 97.89

867.3 869.1 871.0 872.8 874.7 876.5 878.4 880.2 882.1 883.9 885.7 890.8 893.0 895.2 897.4 899.6 901.8 904.0 906.2 908.4 910.6 912.8 915.0 917.1 919.3 882.8 884.4 885.9 887.5 889.1 890.6 892.2 893.7 895.3 896.8 898.4 899.9 901.5 903.0 904.6 906.1 907.7 909.2 910.7 912.3 913.8 915.4 916.9 918.4 920.0 921.5 923.3 925.1 927.0 928.8 930.6 932.4 934.2 936.0 937.8 939.6 941.4 943.2 945.0

0.9190 0.8562 0.7708 0.6729 0.5803 0.5045 0.4493 0.4122 0.3881 0.3724 0.3609 0.5071 0.4705 0.4173 0.3481 0.2697 0.1935 0.1296 0.0821 0.0499 0.0294 0.0165 0.0090 0.0046 0.0022 0.1794 0.0803 0.0319 0.0118 0.0042 0.0015 0.0009 0.0008 0.0007 0.0007 0.0007 0.0008 0.0008 0.0008 0.0008 0.0008 0.0008 0.0008 0.0008 0.0008 0.0008 0.0008 0.0008 0.0008 0.0008 0.0008 0.0004 0.0002 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0810 0.1438 0.2291 0.3270 0.4195 0.4949 0.5494 0.5850 0.6058 0.6147 0.6124 0.4159 0.4508 0.5017 0.5680 0.6433 0.7165 0.7781 0.8238 0.8546 0.8736 0.8857 0.8931 0.8975 0.9000 0.7907 0.8868 0.9334 0.9528 0.9600 0.9621 0.9620 0.9613 0.9604 0.9592 0.9578 0.9563 0.9545 0.9524 0.9501 0.9475 0.9446 0.9414 0.9379 0.9340 0.9297 0.9250 0.9198 0.9141 0.9079 0.9011 0.8975 0.8894 0.8740 0.8458 0.7968 0.7177 0.6046 0.4669 0.3281 0.2115 0.1274 0.0730 0.0402

0.0000 0.0000 0.0000 0.0001 0.0002 0.0006 0.0013 0.0028 0.0061 0.0129 0.0267 0.0770 0.0787 0.0810 0.0839 0.0871 0.0900 0.0924 0.0941 0.0955 0.0970 0.0978 0.0980 0.0979 0.0978 0.0298 0.0329 0.0347 0.0354 0.0358 0.0364 0.0371 0.0379 0.0389 0.0400 0.0414 0.0430 0.0448 0.0468 0.0491 0.0517 0.0546 0.0578 0.0613 0.0652 0.0695 0.0742 0.0794 0.0851 0.0913 0.0981 0.1022 0.1105 0.1260 0.1542 0.2032 0.2823 0.3954 0.5331 0.6719 0.7885 0.8726 0.9270 0.9598

86.05 87.52 89.67 92.35 95.09 97.51 99.39 100.72 101.64 102.34 102.99 98.59 99.94 101.95 104.68 107.97 111.40 114.46 116.85 118.51 119.57 120.20 120.51 120.64 120.62 110.44 115.34 117.94 119.02 119.40 119.55 119.59 119.60 119.59 119.58 119.58 119.57 119.58 119.57 119.59 119.60 119.64 119.70 119.75 119.83 119.91 120.01 120.11 120.24 120.37 120.52 120.50 120.62 120.88 121.45 122.47 124.19 126.82 130.27 134.06 137.54 140.24 141.95 143.00

757.3 761.1 765.0 768.8 772.7 776.5 780.3 784.1 787.9 791.7 795.5 798.3 804.2 810.1 816.0 822.0 828.0 834.0 840.0 846.1 852.3 858.4 864.6 870.8 877.0 800.7 804.5 808.4 812.2 816.1 819.3 822.5 825.8 829.0 832.2 835.4 838.6 841.8 845.0 848.2 851.4 854.6 857.8 860.9 864.1 867.3 870.5 873.7 876.8 880.0 883.2 887.9 892.7 897.4 902.1 906.8 911.6 916.3 921.0 925.7 930.4 935.2 940.1 945.0

sides of the dividing wall, which implied a different number of trays. In practice, this issue of nonequal number of trays can be overcome by using packing elements of different heights.37 Apart from the main design and operating parameters obtained from the HYSYS simulation, several geometrical and mechanical characteristics are also considered. Table 1 presents the detailed data used in the dynamic analysis. The vapor and liquid flow rate values calculated with DDM at the stationary regime are practically the same as the HYSYS values. This comparison was necessary to validate the solution obtained with the proposed mathematical model. As the HYSYS simulation was performed using an equivalent scheme without including heat transfer through the dividing wall, the same hypothesis was adopted in DDM, where no heat transfer between

the prefractionator and the side draw was considered. The heat rate losses per plate, qj, were also neglected. Startup Simulation of DWC. The dynamic model was applied for DWC startup simulation. The stationary state was considered, as in all case studies presented in this paper, when the average absolute value of the derivatives is less than 10-6. During the startup, all control variables were maintained constant at the values indicated in Table 1. The initial state and the desired state are indicated in Table 2. The final steady state values for product concentrations are very close to those that resulted from HYSYS simulation (Table 3). The evolutions during the startup transient regime of benzene concentration in the top product, toluene in the side product, and ethylbenzene in the bottom product are presented in Figure 3. As shown, the

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Figure 3. Case Study 1: Evolutions during the traditional startup transient regime of benzene concentration in top product (- - -), toluene in side product (s), and ethylbenzene in bottom product (- · -). Table 3. Mole Fractions of Product Components for Case Study 1 benzene

toluene

ethylbenzene

HYSYS DDM at stationary state

top product 0.9613 0.9632

0.0387 0.0368

∼10-6 ∼10-6


side product 0.0106 0.0042

0.9612 0.9600

0.0282 0.0358

bottom product ∼10-8 0.0463 ∼10-8 0.0402

0.9537 0.9598


steady state is reached in more than 630 min, and reaching the stationary regime is decided by the bottom response. The explanation consists of the bottom position and its volume, which is much greater than the trays volume. TOC of DWC. A set of DDM simulations indicated that the best control parameters for the optimal startup of the DWCs are the reflux ratio, reboiler heat duty, and the side-draw flow rate. The bounds on these control variables are as follows: 1.8 e R e 2.5

(13)

1.676 × 106 e qB e 2.514 × 106

(14)

25.77 e Fs e 37.14

(15)

These bounds are in agreement with technological and control reasons; at the same time, the control domains limited by these restrictions are large enough. For a total number of 64 trays (including the bottom), the number of state variables is 256.

Figure 4. Case Study 1: The best reflux control for 5 stages ( · · · ) and 10 stages (s).

The TOC algorithm was applied for 5 and 10 time stages. The grid point numbers were ns ) 5, nu ) 9, and nV ) 9. The region contraction factor was set to 0.8, and the total number of IDP iterations was 10. The penalty function values and the tolerance in having reached the desired state are given in Table 4. The tolerance was calculated as the average relative differences between computed and desired values of components concentrations on all DWC trays. The tolerance values are the consequence of the stopping criterion (the average absolute value of the derivatives less than 10-6). The best control policies are presented in Figures 4-6. The 5 or 10 steps cannot always be distinguished, because two or more successive policy values are very close or identical. The corresponding evolutions of benzene concentration in the top product, toluene concentration in the side product, and ethylbenzene concentration in the bottom product for the 10 time stages policies are presented in Figures 7-9. These figures show that the control policy is very much dependent on the number of time stages. If another number of time stages is used, it would be possible to obtain different results. Therefore, the “mathematical” optimal control has not been reached. For practical reasons of DWC control, the “best” (not necessary “optimal”) solutions obtained are also useful. It can be observed that the responses in the bottom concentrations are determinant for the values of startup time, as expected. The performances of best control policies are given in Table 5. The decreases of the startup time and the reboiler energy savings are related to the “traditional” startup at time constant values of control variables (indicated in Table 1). The performances of the 5 stages side-draw flow rate control and 10 stages side-draw flow rate control are very close. The first policy is easier and therefore preferred. Among the policies with one

Table 4. Values of Penalty Function and Tolerance in Having Reached the Desired State for Case Study 1

control policy

number of stages

penalty function at the end of first IDP iteration

penalty function at the end of the 10th IDP iteration

tolerance at the end of the 10th IDP iteration

reflux ratio reflux ratio reboiler heat duty reboiler heat duty side-draw flow rate side-draw flow rate reflux ratio and reboiler heat duty

5 10 5 10 5 10 10

28396.73 22910.74 22442.24 25706.15 25667.71 22623.48 17288.19

9928.07 7899.85 13889.87 11798.57 15347.52 15126.25 7366.22

9.88 × 10-4 9.39 × 10-4 3.51 × 10-3 3.28 × 10-3 1.25 × 10-3 1.21 × 10-3 1.19 × 10-3


Figure 5. Case Study 1: The best reboiler heat duty control for 5 stages ( · · · ) and 10 stages (s).

Figure 6. Case Study 1: The best side-draw flow rate control for 5 stages ( · · · ) and 10 stages (s).

Figure 7. Case Study 1: Evolutions during transient regime for the best reflux control (10 stages) of benzene concentration in top product (- - -), toluene in side product (s), and ethylbenzene in bottom product (- · -).

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Figure 8. Case Study 1: Evolutions during transient regime for the best reboiler heat duty control (10 stages) of benzene concentration in top product (- - -), toluene in side product (s), and ethylbenzene in bottom product (- · -).

Figure 9. Case Study 1: Evolutions during transient regime for the best side-draw flow rate control (10 stages) of benzene concentration in top product (- - -), toluene in side product (s), and ethylbenzene in bottom product (- · -).

control variable, the 10 stages reflux control is the best. For this policy, the decrease in startup time (and the corresponding reboiler energy saving) is 81.65% from the time of the traditional startup procedure. An improved result is given by the simultaneous reflux and reboiler heat duty control. The TOC algorithm was applied for 10 time stages, and the grid point numbers were ns ) 5, nu1 ) 9, nu2 ) 9, and nV ) 9. As in the case of policies with one control variable, the region contraction factor was set to 0.8, and the total number of IDP iterations was 10. The decrease in startup time for this policy is 83.70% from the time of the traditional startup procedure, and the corresponding reboiler energy saving is 83.23%. This control policy is presented in Figure 10. The corresponding evolutions of benzene concentration in the top product, toluene concentration in the side product, and ethylbenzene concentration in the bottom product are presented in Figure 11. A slight improvement of these results is expected when all three controls will be used simultaneously. In this case, the estimated computer execution time is overwhelming (approxi-

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Table 5. Performances of the Best Control Policies for Case Study 1 control policy traditional procedure reflux ratio reflux ratio reboiler heat duty reboiler heat duty side-draw flow rate side-draw flow rate reflux ratio and reboiler heat duty

reduction of reboiler energy reboiler energy computer execution number of stages startup time (min) startup time (%) consumption (kW h) savings (%) time (h) 5 10 5 10 5 10 10

632 149 116 173 142 235 232 103

matively 140 h for 10 IDP iterations). The attempt to reduce the computer execution time by decreasing the grid point numbers ns, nu1, nu2, nu3, and nV did not give good results. In fact, the corresponding decrease of startup time and reboiler energy consumption is expected to be minor, because the sidedraw flow rate is the control variable with the smallest effect (Table 5). Although there are systems for which more than one state grid point is necessary, Luus36 showed that for most systems that he investigated, a single state grid point is a good choice. Therefore, the state grid in the present work was reduced to

76.42 81.65 72.63 77.53 62.82 63.29 83.70

22067 5202 4050 6005 5078 8078 8100 3700

76.42 81.65 72.79 76.98 62.82 63.29 83.23

8.35 15.79 7.48 13.24 8.78 14.01 46.90

one point, which gave the possibility to increase the number of points in the control grid and time grid to 25. The number of time stages was 10, the region contraction factor was set to 0.8, and the total number of IDP iterations was increased to 20. The best solution obtained for the reflux control corresponds to a startup time of 116 min, a value identical to that previously obtained (Table 5). The computer execution time was 29.47 h for 20 IDP iterations, about twice the time for a double number of IDP iterations. The time evolution of control variable is different from the best reflux control policy presented in Figure 4 for 10 time stages. The best solution obtained for the reboiler heat duty control corresponds to a startup time of 162.5 min, which is worse than the solution previously obtained (Table 5). The computer execution time was 37.99 h for 20 IDP iterations, about 2.87 times larger for a double number of IDP iterations. These results show that despite the use of a higher number of points in the control grid and in the time grid and a double number of IDP iterations, one state grid point is not a good choice for the TOC of DWC. A similar result was obtained for the piecewise linear control of distillation columns,26 when the use of one state grid point was also considered. Case Study 2: Comparison with a Conventional Two-Column System

Figure 10. Case Study 1: The best simultaneous reflux (thick line) and reboiler heat duty (thin line) control.

The aim of this case study is to compare the optimal startup results of DWC to those for a conventional two-column system (CCS). The same mixture, feed composition, and flow rate as in Case Study 1 were considered. In the first column, benzene is separated as a top product, while toluene and ethylbenzene are separated in the second column. The general data for this case study are given in Table 6. Products compositions and flow rates are presented in Table 7. The structures of the two columns Table 6. General Data for Case Study 2 parameters and operating conditions feed flow rate (kmol/h) feed composition (mole fractions)

thermal state of the feed

Figure 11. Case Study 1: Evolutions during the transient regime for the best simultaneous reflux and reboiler heat duty control of benzene concentration in top product (- - -), toluene in side product (s), and ethylbenzene in bottom product (- · -).

condenser type reboiler type reboiler heat dutya (kW) pressure in the top of column and in condenser (mm Hg) reflux ratioa column diameter (m) bottom liquid volume (m3) tray area (m2) hole area per plate (m2) weir height (m) hole diameter (m) number of trays feed tray position a

column 1

column 2

126 benzene, 0.6 toluene, 0.3 ethylbenzene, 0.1 liquid at boiling temperature total total 2100 760

49.82 benzene, 0.0231 toluene, 0.7240 ethybenzene, 0.2529 liquid at boiling temperature total total 1250 760

2.6 2 0.911 2.8143 0.2815 0.025 0.005 20 15

1.66 0.9 0.364 1.257 0.1126 0.025 0.005 35 24



Figure 12. Case Study 2: Evolutions during the traditional startup transient regime of concentrations of benzene (- - -), toluene (s), and ethylbenzene (- · -). Table 7. Products Compositions and Flow Rates for Case Study 2 top product

bottom product

column 1 composition (mole fractions): benzene 0.9777 toluene 0.0223 ethylbenzene ∼0 flow rate (kmol/h) 76.18

0.0231 0.7240 0.2529 49.82

column 2 composition (mole fractions): benzene 0.0311 toluene 0.9617 ethylbenzene 0.0072 flow rate (kmol/h) 37.06

∼0 0.0333 0.9667 12.76

were established iteratively with the DDM to obtain at final stationary state products compositions similar to those considered in the DWC (Case Study 1). The total number of trays in the CCS was close to the number of trays in the DWC. These considerations allow a proper comparison of the two cases. The corresponding values of the reflux ratios of the two columns resulted from these conditions and not from an optimum design. As for Case Study 1, the Antoine model was used to calculate the thermodynamical properties of the system. The evolutions during the traditional startup transient regime of products concentrations are presented in Figure 12. The reflux ratio was selected as the control variable for TOC in the CCS. The bounds on this control variable are: - For the first column:

(16.1)

1.5 e R e 4 - For the second column:

(16.2)

1.3 e R e 2.2

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Figure 13. Case Study 2: The best reflux control for column 1 (s) and column 2 ( · · · ).

The startup and TOC of the second column are applied when the first column reaches the stationary regime. This startup procedure is also applied in industrial operations. For the total number of 20 trays in the first column (including the bottom), the number of state variables is 80, and for 35 trays in the second column, there are 140 state variables. This difference of the number of state variables is accountable for the higher value of computer execution time for the second column (Table 8). The TOC algorithm was applied for each column considering 10 time stages. The grid point numbers were ns ) 5, nu ) 9, and nV ) 9. The region contraction factor was set to 0.8, and the total number of IDP iterations was 20. The best reflux control policies of the two columns are presented in Figure 13, and the corresponding evolutions of products composition are given in Figure 14. The performances of the best control policies are given in Table 8. The comparison of these results to the best reflux control of DWC for 10 time stages (Table 5) indicates that the decreases by TOC of startup time (and corresponding energy savings) have close values: 86.20% for the first column, 75.91% for the second column, and 81.65% for the DWC. The total startup time for the best reflux control of the CCS is 93 min, which is 20% shorter than the corresponding startup of the DWC (116 min). This result does not affect the use of DWC, which has the advantages of energy savings in stationary regime and lower capital cost. The reboiler duty for steady state in DWC (2095 kW in Table 1) is about 63% from the total reboilers duties in CCS (3350 kW, the sum of the reboiler duty in column 1 and reboiler duty in column 2 in Table 6) as expected due to the thermal coupling that reduces energy consumption. During the startup, the energy consumption is less in CCS because the startup time is also shorter (632 min for DWC in traditional operating mode and 116 min for DWC in reflux ratio control policy in Table 5, as compared to 510 min for CCS in traditional operating mode and 93 min reflux ratio policy in Table 8). The

Table 8. Performances of the Best Control Policies for Case Study 2

control policy column column column column

1 1 2 2

traditional procedure reflux ratio traditional procedure reflux ratio

number of stages 10 10

startup time (min) 290 40 220 53

reduction of startup time (%) 86.20 75.91

reboiler energy consumption (kW h) 10150 1400 4583 1104

reboiler energy savings (%)

computer execution time (h)

86.20

8.54

75.91

20.48

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Ind. Eng. Chem. Res., Vol. 49, No. 19, 2010 Table 9. General Data for Case Study 3 parameters and operating conditions feed flow rate (kmol/h) feed composition (mole fractions) side-draw flow ratea (kmol/h) thermal state of the feed stream thermal state of the side-draw stream

Figure 14. Case Study 2: Evolutions during the transient regime for the best reflux control of concentrations of benzene (- - -), toluene (s), and ethylbenzene (- · -).

higher energy consumption in DWC during the startup is overcome by the advantage of DWC at steady state operation when important energy savings are possible. The comparison between DWC startup (Case 1) and CCS startup (Case 2) aimed to prove that the startup policies bring comparable energy savings in both cases (81.65% for DWC, Table 5, and 83% for CCS, Table 8). Case Study 3: A Ternary Nonideal Mixture A ternary nonideal mixture of alcohols, respectively, methanol, ethanol, and 1-propanol, is separated in a DWC. The feed flow rate and feed composition are given in Table 9. The design of DWC was done using HYSYS in a similar manner to Case Study 1. The general data resulted from the design procedure are presented in Table 9. Because of higher values of the number of trays and reflux ratio, the separation degrees of the components (Table 10) are better than in Case Study 1. The equilibrium constants for the mixture methanolethanol-1-propanol for eq 3.2 were obtained by regression of equilibrium data, calculated in the frame of HYSYS, using the UNIQUAC model. This procedure is similar to that used by Woinaroschy,26 for the equilibrium data of propene-propane mixture using the Soave, Redlich, and Kwong equation of state. The regressed expressions of the equilibrium constants and corresponding regression statistics parameters are as follows: κ1,j ) 5173.230 - 46.87533Tj + 8.8297247 × 10-2Tj2 + 3.3168898 × 10-4Tj3 - 1.4083716 × 10-6Tj4 + 1.3958698 × 10-9Tj5 (17) where the coefficient of determination ) 0.998 and standard error ) 0.0174. κ2,j ) 4933.838 - 0.3914401Tj2 + 2.2053993 × 10-3Tj3 4.6604342 × 10-6Tj4 + 3.5028422 × 10-9Tj5 (18) where the coefficient of determination ) 0.997 and standard error ) 0.0177. κ3,j ) 6504.007 - 0.5150276Tj2 + 2.8969788 × 10-3Tj3 6.1087665 × 10-6Tj4 + 4.5788536 × 10-9Tj5 (19)

condenser type pressure in the top of column and in condenser (mm Hg) reflux ratioa reboiler type reboiler heat dutya (kW) bottom liquid volume (m3) tray area for top and bottom sections (m2) tray area for prefractionator and side-draw (m2) hole area per plate for top and bottom sections (m2) hole area per plate for prefractionator and side-draw (m2) weir height (m) hole diameter (m) tray number of top section tray number of prefractionator tray number of side draw tray number of bottom section feed tray position side-draw tray position liquid split fraction in prefractionator a

value 126 methanol, 0.25 ethanol, 0.45 1-propanol, 0.3 56.7 liquid at boiling temperature liquid at boiling temperature total 760 9 total 2864 0.410 1.2636 0.6318 0.1266 0.0633 0.025 0.005 16 36 36 9 25 79 0.59


Table 10. Mole Fractions of Products Components for Case Study 3 methanol

ethanol

1-propanol

top product 0.9967

0.0033

∼10-8

side product 0.0043

0.9759

0.0198

bottom product ∼10-6

0.0036

0.9964

where the coefficient of determination ) 0.997 and standard error ) 0.0092. The evolutions during the traditional startup transient regime of products concentrations are presented in Figure 15. As this figure shows, the steady state is reached in 760 min. For TOC the control parameters were reflux ratio and reboiler heat duty. The bounds imposed on these control variables are as follows: 8.5 e R e 9.5

(20)

2.30 × 106 e qB e 3.44 × 106

(21)

As in Case Study 1, these bounds are in agreement to the technological and control reasons, but at the same time, the control domains, limited by these restrictions, are large enough. For the total number of 98 trays (including the bottom), the number of state variables is 392. The TOC algorithm was applied for 10 time stages. The grid point numbers were ns ) 5, nu ) 9, and nV ) 9. The region contraction factor was set to 0.8, and the total number of IDP iterations was 20.


Figure 15. Case Study 3: Evolutions during the traditional startup transient regime of methanol concentration in top product (- - -), ethanol in side product (s), and 1-propanol in bottom product (- · -).

Figure 16. Case Study 3: The best reflux control.

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Figure 18. Case Study 3: Evolutions during transient regime for the best reflux control of methanol concentration in top product (- - -), ethanol in side product (s), and 1-propanol in bottom product (- · -).

Figure 19. Case Study 3: Evolutions during transient regime for the best reboiler duty control of methanol concentration in top product (- - -), ethanol in side product (s), and 1-propanol in bottom product (- · -).

top product, ethanol in side product, and 1-propanol in bottom product are given in Figures 18 and 19. The ethanol concentration in the side product reaches a minimum value after 15 min in the case of the best reflux control and at 30 min in the case of the best reboiler duty control. The performances of the best control policies are similar (Table 11) but less significant than for Case Study 1. Despite the higher number of state variables and a double number of IDP iterations, the computer execution time is shorter than for Case Study 1. The explanation consists of the simpler expressions of equilibrium constants and their derivatives. Discussion

Figure 17. Case Study 3: The best reboiler heat duty control.

The best control policies are presented in Figures 16 and 17, and the corresponding evolutions of methanol concentration in

The computer execution times are given for a computer with a I7-965 processor (3.2 GHz), 3 GB DDR2 (677 MHz) memory, and Microsoft Windows XP Professional SP3 operating system. This computer is approximately 3.4 times faster than the IBM Intel Pentium IV computer (2.66 GHz processor, 1 GB memory, and Microsoft Windows XP Professional SP2 operating system) that we have used in previous TOC work.26 With traditional

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Table 11. Performances of the Best Control Policies for Case Study 3 control policy

number of stages

startup time (min)

reduction of startup time (%)

reboiler energy consumption (kW h)

reboiler energy savings (%)

computer execution time (h)

traditional procedure reflux ratio reboiler heat duty

10 10

760 297 309

60.92 59.34

36277 14177 14144

60.92 61.01

5.97 8.25

nonparallel computer programming, the CPU usage of this 4-core processor is only 13%. By parallel, computer programming as shown by Keil and Mandel,38 the computation time can be reduced, since many calculations in IDP can be done in parallel fashion. The main goal of this paper is to present the possibility to use IDP for TOC of DWC. Detailed investigations concerning the best choice of IDP parameters (number of time stages, the grids points numbers, region contraction factor, penalty coefficients, total number of IDP iterations, etc.) were not considered. The total number of state variables and differential equations of the DDM in the TOC of DWC is too high for these investigations. Long computing times would have been required despite our access to a high-performance computer. There are excellent works36,39 in the literature that performed this research with simple differential models characterized by a low number of state variables. In many cases, a good selection of IDP parameters improves the performance index mathematically but with very low or no practical consequences. For example, in an application presented by Dadebo and McAuley40 by increasing the number of time stages from 5 to 17 and the number of state grid points from 10 to 20, the CPU time increased 7.29 times, and the performance index improvement was 1.77%. Dadebo and McAuley,41 in a parallel reactions problem with two state variables and one control parameter, managed to improve the performance index only with 0.22%, by increasing the number of time stages from 10 to 80. In a more recent work, for a fed-batch reactor control with four state variables and one control parameter, Luus42 obtained an improvement of the performance index from 20842.278 to 20842.282, by increasing the number of time stages from 44 to 324, and the best result for an increased number of state grid points from 1 to 19 led to an improvement of 0.33% of the performance index. For each case study, the widely accepted thermodynamical model type was selected. The benzene-toluene-ethylbenzene mixture, at normal pressure, can be considered an ideal one, and the use of Antoine model is reasonable. UNIQUAC (as in Case Study 3) or NRTL models are usually used for vapor-liquid equilibrium concerning nonideal mixture of alcohols, such as methanol-ethanol-1-propanol. It was not the aim of the present work to use several thermodynamic models for vapor-liquid equilibrium in the frame of each case study to compare the influence of the thermodynamic model on the results. The DDM and the application of IDP for TOC of DWC accept any vapor-liquid equilibrium model, and their proper selection is a thermodynamic problem. At the industrial level, it is important to consider the time delay. According to the simplification assumption (iv) of DDM, the time delay between plates was neglected in our study. To include the time delay, a DDM with more detailed tray hydraulics would be required. It is impractical to include complex hydraulic calculations in the TOC procedure due to excessive CPU time. The selected control variables (reflux, reboiler heat duty, and side-draw flow rate) can be easily manipulated in practical applications. Unlike the cases of reflux or side-draw flow rate control, when the switch off the control variables values can

be done instantaneously, the modification of the reboiler heat duty has a time delay, and the startup time and state variables will be affected. These effects are not included in the above case studies. In a DWC, the reflux must be high enough to build-up the reflux on both sides of the dividing wall. The value of the reflux ratio has a big impact on the separation on both sides of the dividing wall; hence, it affects the concentration profiles along the trays. Not only the distillate and bottom product will be influenced by the variation of reflux ratio but also the sidedraw stream, which is desired to contain the intermediate component at high concentration. This consideration can lead to the conclusion that a correct reflux policy will bring the DWC in stable operating condition in a reasonable time. In Case Study 1, the values chosen for product purities are not very high. Higher values for product purities would impose a larger number of trays, leading to longer computing time requirements. Similar values for products composition are used in other case studies referring to the separation of hydrocarbon mixtures in a DWC. For instance, Premkumar and Rangaiah43 mention values from industrial sources used as specifications of the main component concentration in the top, side and bottom products of 0.995, 0.91, and 0.92 for the separation of benzene-toluene-styrene mixture, and 0.995, 0.96, and 0.96 for benzene-toluene-ethylbenzene mixture; they are very close with the products purities in Case Study 1. With regard to the separation of methanolethanol-1-propanol (Case Study 3), product concentrations in the main component of 0.9968, 0.9766, and 0.9962 are reasonable. In fact, this is a separation by distillation, not by highpurity advanced separation techniques. The selected case studies refer to mixtures not easily separable at high purities. For benzene-toluene-ethylbenzene mixture, ethylbenzene increases the difficulties to obtain high purities. For methanol-ethanol-1propanol mixture, methanol increases the difficulty to obtain higher degrees of separation. Of course, by increasing the plates number and/or reflux ratio, the products purities can be improved. As the aim of the present paper was to formulate and test a method for startup time minimization in a DWC, the computation effort was not increased by imposing very high product purities. The liquid split, a practical tool for controlling the side-draw stream purity (Mutalib and Smith44), was set at the values obtained in steady state, as the feed composition of DWCs did not vary in the present study. The influence of the liquid split variation along the startup period did not affect significantly the startup time. With regard to the vapor split, it was left to adjust according to the temperature distribution along the trays and pressure drop. Luus45 investigated obtaining a bang-bang optimal control policy for a binary distillation column. The application contains a control model consisting of only 11 linear differential equations. The IDP programming algorithm was different from the “classical” IDP procedure (used in the cited works33-36 and also in many other papers): He used a nested iterative supplementary loop inside the IDP passes to reduce the control variables domain. The use of a bang-bang control for distil-


lation columns with its deep technological implications (e.g., hydrodynamic regime) is a challenging subject. As in the case of startup optimization of classical distillation columns, it is possible to avoid undesirable secondary bang-bang effects of the piecewise constant control by replacing it with piecewise linear control.25 The practical implementation of the optimal policies obtained here can lead to some hydrodynamic problems (flooding, weeping, liquid entrainment, etc). Therefore, it is useful to test these control policies by simulations, using a DDM with more detailed plate hydraulics, taking into account the possibility of including the time delay. This way, some suboptimal control policies, avoiding wrong hydrodynamic regimes, can be identified by suitable corrections. Of course, a more useful but more difficult way to validate these control policies, based exclusively on simulation, is to test the approach on a pilot plant. Conclusions In this work, the ability of IDP to solve high dimensional TOC problems, involving complex models, was demonstrated once more. TOC of startup traditional distillation columns by iterative programming proposed by Woinaroschy for ideal25 and nonideal mixtures26 was extended to the case of DWCs. The proposed DDM proved to represent correctly the separation of an ideal ternary hydrocarbon mixture and of a nonideal ternary alcohol mixture in a DWC. The values of internal flows and temperature distributions along the trays at steady state were in good agreement with the simulations obtained in the frame of commercial simulators. Usage of reflux ratio, reboiler heat duty, or side-draw flow rate as control variables enabled a decrease of the startup time and a corresponding reboiler energy saving up to about 80%, as compared to classical startup procedures. For the case of DWCs, these reductions are of a similar order of magnitude to the traditional distillation columns. Note In the equations, the units of time are seconds, and the units of power are Watts. In the tables and figures, for practical reasons, the units of time are minutes or hours, and the units of power are kiloWatts. Nomenclature A0 ) hole area per plate (m2) Ap ) total area per plate (m2) d0 ) hole diameter (m) F ) feed stream or side-draw streamflow rate (kmol s-1) h ) liquid enthalpy (J kmol-1) H ) vapor enthalpy (J kmol-1) I ) performance index L ) liquid flow rate (kmol s-1) lW ) weir length (m) m ) number of components N ) liquid holdup per plate (kmol) n ) number of plates ns ) dimension of state vector nu ) dimension of control vector nV ) dimension of time vector P ) vapor pressure (mm Hg) p ) total pressure on plate (mm Hg) q ) heat rate losses per plate (W) R ) reflux ratio S ) number of time stages s ) state vector

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T ) temperature (°C) t ) time (s) u ) control vector V ) vapor flow rate (kmol s-1) V ) time stage (s) x ) liquid mole fraction on plate y ) vapor mole fraction on plate zst ) static height of the liquid (m) zW ) weir height (m) Greek Letters γ ) activity coefficient ∆p ) pressure drop (mm Hg) ε ) volumetric fraction κ ) equilibrium constant F ) density (kg m-3) σ ) superficial tension (kg s-2) τ ) normalized time variable ω ) penalty coefficient Subscripts B ) bottom, reboiler D ) distillate F ) feed f ) final i ) component j ) plate k ) time stage L ) liquid S ) side-draw s ) stationary V ) vapor

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ReceiVed for reView January 14, 2010 ReVised manuscript receiVed July 25, 2010 Accepted July 26, 2010 IE100090P

Time-Optimal Control of Dividing-Wall Distillation Columns - American

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