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TIME OPTIMAL CONTROL OF A HIGH-DIMENSIONAL,
NONLINEAR BINARY DISTILLATION COLUMN USING
THE LUUS-JAAKOLA OP'MMIZATION PROCEDURE
Manoj Rajagopalan
A thesis submitted in confomiity with the requirements
for the degree of Master of Applied Science.
Graduate Department of Chernical Engineering and Applied Chemistry,
ia the University of Toronto
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The author retains ownership of the L'auteur conserve la propriété du copyright in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantial extracts ftom it Ni la thèse ni des extraits substantiels may be printed or otherwise de cde-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation. Title: Time Optimal Control of a High-Dimensional, Nonlinear Binary Distillation Column using the Luus-Jaakola Optimization Procedure Degree and year of convocation: Master of Applied Science (MASc), 2001 Full Name: Manoj Rajagopalan Graduate ûepartment: Department of Chernical Engineering and Applied Chemistry University: University of Toronto
ABSTRACT
The dynarnic model of a methanol-isopropanol binary distillation column is set up from first principles with the aim of perfonning a time optimal control study. With 21 stages, the state of each king described by 2 variables, the time optimal control problem is 42 dimensional in its state space, with two control variables: the reflux flowrate and the reboiler heat duty. The model poses many computational challenges in its complex fomulation and the nature and number of constraints on the control and the state variables.
It is desired to take the system from an initial steady state to a desired steady state in minimum possible time. The Luus-Jaakola (U)optimization procedure is used to solve this tirne optimal control problem using piecewise constant control and flexible tirne-stage
lengths. The complex mode1 is directly used without any simplifications or transformations.
In light of the speciai computational challenges involved, the LJ optimization procedure is
found to perform very well: the best result obtained takes the system to within 4.319~10~(1-
nom) of the desired state in 52.494 minutes employing a &stage control policy. The large
computation time requirements limit experimentation with this model and possible avenues
for improvements are identifiai. ACKNOWLEDGEMENTS
This work was performed under the guidance of Prof. Rein Luus whose advice and suggestions are highly valued, and encouragement, gratefully acknowledged.
1 also express my gratitude to my colleague Mr. Ramasubramanian Sundaraiingarn for his technical advice. corrections and support.
Profuse th& go out to my fiends Mr.Harpreet Singh Dhariwal and Mr. Parmjit Singh Kanth for their immensely kind and timely help, and advice on cornputational issues.
Financial support from the Naturai Science and Engineering Research Council of Canada and from the University of Toronto in the form of the Differential Fee Waiver. Mary H. Beatty Fellowship and University of Toronto Open Fellowship are gratefuily acknowledged. 1 thank Prof.Mark Kortschot for his invaluable assistance at the time of need. TABLE OF CONTENTS
ABSTRACT ......
ACKNOWLEDGEMENTS ......
TABLE OF CONTENTS ......
LIST OF TABLES ......
LIST OF FIGURES ......
MTRODUCTION ......
Part 1: Distillation Column Model
DISTILLATION COLUMN DYNAMICS ...... 2.1 Ordinary stages ...... 2.2 Extraordinary stages ...... 1.2.1 Condenser ...... 2.2.2 Feed stage ...... 2.2.3 Reboiler ......
3 PROPERTY MODELS ...... 3.1 Vapor-liquid equilibrium...... 3.2 Enthdpy ...... 3.2.1 Saturateci Iiquid phase entfialpy ...... 3 -2.3 Mixing eEects ...... 3.2.3 Composition derivative of saturated liquid phase enthalpy ...... 3.2.4 Samted vapor phase enthaipy ...... 3.3 Liquid molar overflow ...... Part 11: Time Optimal Contml
4 . THEORETICAL DEVELOPMENT ...... 25 4.1 The tirne optimal contrd pmblem ...... 25 4.2 The Luus-Jaakola optimization procedure ...... 27
5 . THE DISTILLATION TIME OPTIMAL CONTROL PROBLEM ...... 30 5.1 Solving for the steady states ...... 31 5.2 Constraint handling ...... 35
6. NUMERICAL RESULTS ...... 6.1 Prelirninary computations ...... 6.1.1 Effect of number of random points ...... 6.1.2 Effectofthepenaltyfunction factor ...... 6.1.3 Effectofnumberoftimestages ...... 6.2 intense computations ...... *...... *...... * 6.2.1 Effeçt of number of random points ...... 6.2.2 Effect of penalty function factor ...... 6.2.3 Effect of number of tirne stages ...... 6.2.4 The best result obtained ......
7. DISCUSSION ...... 63
8. CONCLUSIONS ...... 65
APPENDDC A: Degree of freedom analysis of the culumn at steady state 66
APPENDiX B: FORTRAN program ...... 68
NOTATION ...... 81
REFERENCES ...... 85 LIST OF TABLES
Table 1.1: Summary of assurnptions ......
Table 3.1. Antoine constants for the components ......
Table 3.2. Enthalpy seference data ......
Table 33: Coefficients in the specific heat polynomial in temperature ......
Table 3.5. Heat of mixing at 298.15 K [3] ......
Table 3.5: Heat of mixing [34] ......
Table 3.6. Coefficients in the enthalpy of vaporization polynomial in temperature ...
Table 5.1. Conditions within the column during the initial steady state ......
Table 5.2. Conditions within the column during the final steady state ......
Table 6.1. Effect of the number of random points on result obtained ......
Table 6.2. Effect of penalty Function factor on preliminary results obtained ......
Table 63: Effect of the number of time stages on the resuit obtained ......
Tabk 6.4. Effect of the number of random points on result ......
Table 6.5. Effect of the number of initially chosen tirne stages ......
Table 6.6. Control policies hmruns with 400 random points .differing in P ...... LIST OF FIGURFS
Fig .2.1: Schematic diagram of an ordinary stage
Fig. 6.1. Effect of nurnber of randorn points on convergence for 8= 106 ......
Fig .6.2. Effect of number of random points on convergence for 8= Io8 ......
Fig. 63: Effect of 8on convergence for R = 150 ......
Fig. 6.4. EFfèct of 80n convergence for R = 300 ......
Fig. 6.5. Final time and convergence profile for the case 8= 10' and R = 300 ......
Fig. 6.6: Cornparison of state trajectories (mole fraction) for different cases of P and R ......
Fig .6.7. Cornparison of state tmjectories (liquid holdup) for different cases of P .....
Fig. 6.8. Cornparison of reflux policy for various runs leading to the best resdt ......
Fig .6.9: Cornparison of reboiler heat input policy for various cases leading to the best result ......
Fig. 6.10. State trajectory (mole fraction) obtained using the best control policy ......
Fig .6.1 1: State trajectory (holdup) oblained using the best control policy obtained
Fig. 6.12. Dynamic constraint satisfaction ...... 1 INTRODUCTION
Time optimal control pmblerns are fiequently encountered in the areas of engineering and applied sciences. The dynamics of a system king known, it is frequently required to determine how to manipulate its inputs so that a desired change in its state is effected in minimum possible time: trajectory correction in aviation and change of operating condition for chernical plant equipment are typical examples from diverse fields.
Distillation is one of the most energy consuming processes in chemicai plants. Sometimes a transient behavior is to be effected in distiliation columns to change the steady state operating point so that purity requirements are met ancüor economic benefits are made. Product from the column during transience does not meet specifications; it is sornetimes reusable in cases where the high cost of doing so is not as significant as the overall cost of ninning the whole piant process. in cases where the feed is expensive, or in cases where profits from high vaiued product justify incremental sales. The desire to minirnize this period of transience motivates time optimal control of the distiIlation column that is the focus of this work. Time optimal control study is aiso desirable for fastest possible recovery of the system fiom a disturbance upsetting steady state operation.
The dynamic mode1 of the distillation colmhas to be known prior to the formulation of the tirne optimal control problem. Modeling of distillation columns and processes is an area that has been thoroughly studied and a wealth of literature is available [25, 281. With the availability of powerful, economical computational resource at the desktop, interest in complex distillation rnodels and odie control has grown considerably. Present da): computers are powerfùl enough for onJine oprimal control of linear rnodels [20].
Numerous papers and books deal with the dwelopment of rigomus distillation process rnodels hmfirst principles 1281. The pmss is modeled as a chab of stages to each of which materiai and energy baiances are applied. in addition to these differential equations, several aigebraic equations are required to describe the dependence of physicai properties on column/stage conditions and some, like ihe vapor-liquid equilibrium (VLE) relationships. can not be tackled analytically. The result is an extremely complex but rigomus process model. No references to the direct solution of the rigorous modei have been cited [BI. Cornmon assumptions like equimolal overflow. constant tray efficiency, constant relative voiatility and quasi-equilibrium behavior, are classified as those simplimg (a) vapor flow dynamics, (b) liquid flow dynamics, or (c) the energy balance. The awmption of equirnola: overfiow between the liquid and vapor phases does away with the energy balance, reducing the state dimensionality and considerably simplifj4ng the model. However, this assumption is vaiid only when the specific heats of the components are similar [28], which is mostiy not the case.
This work presents the model of a simple distillation colurnn avoiding the ubiquitous assumption of equimolal overflow. This introduces the question of how to determine either the energy dynarnics or the vapor upflow so that the other may be calculated. Vapor upflow is in fact a function of the liquid molar holdup and the pressure drop. which in tm is govemed by the cesistance of the tray and other transport phenomena [28]. But the energy balance is usually employed to solve for the vapor upflow [6, 81. With the assurnption of rapid energy interactions, the energy balance ODE is expressed as an algebtaic equation ES], which is then used to solve for vapor upflow fiom each stage.
Vapor-liquid interactions are modeled under the assumptions of equilibriurn behavior and Murphree vapor phase tray eficiency. The Wilson equation gives the expression of the vapor mole hction as a product of pure functions of temperature and composition. With the use of Newton's method for solution of nonlinear algebraic equations. bubble-point calculations are canied out much faster than in the more cornmon case where the Wilson equation is not used, and temperature and composition are inseparable in the expression for vapor mole Eraction. The assumptions regarding equilibrium and my efficiency apply less effectively to transient simulation than to steady state simulation and this is the limitation of the model. Nonequilibrium models for distillation are now available [IO, 361. The model developed in this work also discounts the vapor and liquid flow lag between stages. usually of the order of 3-6 seconds [27l, and assumes that changes in the column are reflected everywhere else instantly. Table 1.1 summarizes the assumptions in the model developed in this work. Optimal control methods fall into three broad categories [5]. Variational methoh involve the use of the Pontryagh's maximum principle, which leads to a two-point boundary value problem (TPBVP). This problem can be solved using techniques such as control vector iteration (CVI)and boundary condition iteration (BCI).Control parametrization merhods express the control variables as piecewise polynomials or simple functions in the or the States. The static nodinear optimization problem formulated can be solved using control vector parametrization (CVP)procedures such as iterative dynamic programming (IDP) [19] and the Luus-Jaakola optimization procedure (LJ) [14.16,17]. Confrol and state parametrimiion methoh employ techniques like orthogonal collocation to express the differential algebraic system as a nonlinear programming problem which cm then be solved.
The Luus-Jaakola (U) optimization procedure is a direct search optirnization procedure that uses randomly chosen points about a suitably chosen initial guess within an initial search region, improving the solution obtained by systematically contracthg the size of the search region after every iteration. This procedure is performed in a mdtipass fashion where the region sizes for the control and stage lengths are calcdated from the extent of variation of the variables over the previous pass [l6]. The success of this procedure in solving difficult fed- batch reactor problems [Il] and the cecent success of a celated procedure, iterative dynamic programrning, in online simulation of a linear distillation model [20] motivate us to examine tirne optimal conmi of a nonlinear distillation column model using the U optimization procedwe. Table 1.1 : Summary of assumptions
1. Pressure drop across the ûays is smd compared to the colurnn pressure and therefore the entire column is assumeci to operate at atmospheric pressure (101.325 kPa). 2. At every instant of time, the liquid in a stage and ihe vapor passing through it are in equilibrium. 3. Vapor phase is ideal and the nonideality in the liquid phase can be modeled by the Wilson equation. S. Vapor composition is governed by the Murphree vapor phase eficiency de. 5. Energy interactions occur on a rnuch smaller time scale than flow and composition dynamics. and can therefore be incorporated in the form of algebraic equations. 6. Dpamics of the condenser and reboiler are smail in cornparison to the flow and composition dynamics within the column [27] and cm therefore be neglected. 7. Levei is tightiy controlled in the reboiler and condenser. 8. Vapor and liquid flow Iags cm be neglected. 9. The Generalized Principie of Correspondmg States provides a good approximation to the specific heat and the enthalpy of vaporization of the mixture. 10. The binary interaction parameter in the cdcdation of the mixture critical temperature is unity. 1 1. The condenser is a total condenser and condensate is sarurated liquid at every instant. 1 3. Feed is saturated liquid: an equimolar mixture of methanol and isopropanol. 13. No chernical reactions occur anywhere within the column. 14. Heat losses to the surroundings are negligible. 2 DISTILLATION COLUMN DYNAMICS
The dynamic mode1 of a methanol-isopropanol distillation column is developed begiming with the materiai and energy balances. The state of each stage is described by the liquid phase mole tiaction, molar holdup and enthaipy. VLE calculations incorporate vapor phase tray eficiency. Liquid flow dynarnics are simplitTed to suit data available in literature. Vapor upflow is solved for using the energy balance. Following the nomenclature of Rademaker [25], al1 trays except the feed tray are termed ordiistages. The feed tray, condenser and reboiler are classified as extraordiiary stages.
2.1 ORDINARYSTAGES
By nurnbering the stages from top to bottom, hom an overail material balance amund the ?' stage (Fig. 2.1 ), we obtain
where Mi denotes the liquid molar holdup in the stage; LI and Y,respectively denote the liquid molar overfiow and the rnolar vapor uptlow hmthe stage i.
Cornponent balance around this stage yields
where xi and Yi denote the mole fiactions of methanol in the liquid and the vapor phases respectively. Since
using Eqns. (2.1) and (2.2) in Eqn. (2.3) we get Similady, the energy balance around the $ stage becornes
where hi denotes the molar enthalpy of the Iiquid at composition x, and fi, denotes the molar enthalpy of the vapor at composition Yi, both at the temperature T, of the stage. We therefore have three differential equations involving seven variables at each stage: M L, Y,
xi Yi h, Hl
With the assumption of instmtaneous equilibrium in the stages. y, cm be determined fiom x, througfi vapor-liquid equilibrium cdculation and the application of the Murphr~evapor- phase eficiency def241:
where y, and y,! denote the mole fractions of methanol in the vapor upflow streams hm stage i and stage (i+l) respedvely; denotes the equilibrium value of vapor phase methanol mole Fraction correspondmg to x,. Murphree efficiency of q = 0.7 was used in this work.
The values of L,' V,, and H, rnust be determined prior to computing the values of the derivatives in Eqns.(2.1), (2.4) and (2.5) and integrating them to perfonn the dynamic simuiation. initial conditions for Mi, xi and hi are obtained 6om steady state simulation.
With the assumption that energy interactions occui on a much shorter time-sale than composition and flow dyinamics, the tray temperature T, is now the equilibrium temperature conespondhg to x, at the column pressure P. and the liquid hotdup at the tray is always saturated. Le., at its bubble point. Therefore its enthalpy can now be determined fiom an algebraic fùnction in x, and T, of the form,
As a result. the need for an ODE, Eqn. (2.5), to describe energy dynarnics at each stage is eliminated. Similarly,
Since the L, are computed from the Mi using the molar overflow equation denved Chapter 3. it is VIthat remains to be determined before evaluating the RHS of Eqns.(2.1) and (2.4). The actuai formulae that Eqns. (2.7) and (2.8) represent are derived in Chapter 3.
Various methods for detemination of Vfhave ken proposed [6,8]. We follow the suggestion of Howard [8] in using the energy balance to solve for VI. Differentiation of Eqn.(2.7) with respect to time gives
since Tl is a function of x,. The G, are calculated as outlined later in Chapter 3. Using Eqns.(Z.lO), (2.4) and (2.5) to eliminate dhhand wlviag for Y,, we get
Due to the dependence of V, on VeI (Eqn.(2. I 1)) and of y, on y,i (Murphree efficiency), stage-to-stage calculations need to be carried out beginning at the reboiler. Having computed the V,'s, we may calculate the RHS of Eqns. (2.1) and (2.4). Pnor to integrating the differential equations it is necessary to consider the behavior of the extraordinary stages where input-output configurationsdiffer from the ordinary stages. Fig. 2.1 : Schematic diagram of an ordinary stage 2.2 EXTRAORDINARYSTAGES
22.1 Condenser
There is no vapor upflow fiom the total condenser. Reflux is drawn fiom the condensate before it entets the first tray. Due to tight level control:
where R is the reflux flow rate. The methanol component balance yields
We assume the liquid outfiow hmthe condenser to be saturated. Since the values of di variables in Eqns. (2.12) and (2.13) are known, the derivatives can be imrnediately computed.
2.2.2 Feed stage
The balance equations for the feed stage are derived from those of an ordinary stage by incorporating the effect of the feed: dM, = LF-1 -LF+ FF+, - VF+ F. (2.14) dt
where F is the feed flow rate, zhed is the mole fraction of methanol in the feed and hhd is the feed enhalpy. it is assurned that the feed is saturated liquid.
Using Eqns(2.16) and (2.10) to eliminate *& we get The value of GF is computed in tbe exactly the same manner as Gi is in ordinary stages; the procedure is outlined in Chapter 3. The resuIting value of VFmay be used to cornpute the derivatives in Eqn.(Ll4) and (2.15).
2.23 Reboiler
The reboiler is assumed to be an equilibrium stage. Since there is no vapor entering the reboiler and Iiquid level is tightly controlled. the balance equations take the form
n-dM - Ln-,- B - V,, = 0. dt
where B is the bonoms flow rate and QR is the heat input to the reboiler.
Using Eqn.(Z.ZO) and (2.10) to eliminate dhAwe get
Now. the derivatives in Eqns.(2.18) and (2.19) can be caiculated and the integrahon perfomed nurnerically. 3 PROPERTY MODELS
Equations are deveioped to describe the variation of physical properties of the cornponents with column conditions, principaily temperature. These include VLE. saturated liquid snthdpy, saturated vapor enthalpy and liquid molar overflow. in addition to saturated liquid enthalpy, one section is devoted to explaining the insigaiticance of mixing effects and another is devoted to the composition derivative of liquid phase enthalpy, Gi, that is the key intermediate in the conversion of the energy baiance ODE to an algebraic equation.
We assume ideality in the vapor phase mixture and nonideaiity in the liquid phase mixture. Therefore, the modifiai Raoult's law can be used to predict the equilibrium vapor phase methanol mole fiaction y,, fiom the given liquid phase composition x, at its equilibtium temperature Tl. Let the symbols y,,' and y2,* denote the vapor phase mole hactiom of methano1 and isopropanol respectively, at the ?' stage.
where y,, is the activity coefficient of component j (l=methanol, 2=isopropanol) at composition x, and temperature Ti, and PJtm' is the vapor pressure of the pure component j at temperature Tl.The vapor pressure Pjim can be caiculated fiom Antoine's equation [A where P,,ILI*are in Daand Ti is in K. The Antoine constants are given in Table 3.1.
The Wilson equation is most suitable for the caiculation of activity coefficients in systems of Low molecular-weight alcohols [33]:
It cm be observed that the tempemture-independence of the Wilson constants AI? and All makes the activity coefficients pure tiinctions of composition x,.
VLE data sets for the methmol-isopropanol system are available in 171, 1321 and [34]. The Wilson constants taken from [7] are:
Since the Wilson equation permits us to express the vapor phase mole ûaction as the product of x,. pure hctions of x,. and P,;"'. pure functions of Tl,y,, cmbe computed hman initial guess for T, by Newton's method as follows. It is required that
Z(x, Tl)= y i,* (1, 'I;) + ktL(xb Tt) - 1 = o. ComputationalIy. the folIowing adaptation of the above equation is imposed.
1 z(-rh Tt] 1 < a (3.7) where 6 is suitable tolerance. Since x, is known. and and thetefore Z. are dependent only on T,.
Applying Newton's method. the value of Tl in iteration (k+l) is related to its value in iteration (k) Differentiating Eqn.(3.6) with respect to T,,
YIP,41"' B1 + Y z,o - -Tl )P*,"' B2 x ln(1 O). (T, = L P (T, +cl)? P +c$ 1
The value of Tl obtained by performing iterations till the satisfaction of Eq~(3.7)is a close approximation to the tme value of the equilibrium temperature. Applying Lhis value to Eqn.(3 3) and then to Eqns.(3 .I ) and (3.2) we arrive at the equiiibnum values of the vapor- phase compositions and y,'. In stage-to-stage calculations. it is this value ofyil'that is
applied at each stage as JI,' io the Murphree vapor phase efficiency de(Eqn. (2.6)) to determine the methmol mole Fraction y,. A strict tolerance of 6= 10"' was used to minimize error propagation in stage-to-stage calcuiations.
3.2.1 Saturateci liquid phase enthalpy
The molar enthaipy of the liquid mixm at composition xi and temperature T, is given by the sum of the ided liquid enthdpy (molar average) and the heat of mixing: where hi and hz are the pure component tnthalpies. Since the heat of mixing is negligible, as discussed in Sec. 3.2.2,
As a reference point the enthalpy of ideal gas If. at 298.1 5 K and 101.325 kPa, was taken to be zero. The molar enthalpy of the sahuated pure component liquids are then given by
The fmt term denotes the deviation of the enthalpy of the vapor of component j fiom that of the ideal gas at the reference condition. The second tem is the latent heat of condensation at that condition. The third term is the sensible heat effect accompanying the change of temperature of the liquid from the reference to the given value T,.
The values for the vapor phase enthalpy deviation and the heat of condensation for methanol and isopropanol were obtained fiom 1371 and are given in Table 3.2
The sensible heat effect is given by
where C,, is the molar specific heat of pure component j in its saturated state at temperature
The pure component saturated Iiquid specific heat data fiom [381 were fit, by linear least squares regression. to a fifth degree polynomial in temperature T so that the maximum relative error on the data points was less than 1%:
The above coefficients a, are given in Table 3.3. Since the specific heat of the saturated liquid and that of subcooled liquid at the same temperature differ negligibly 1383, the latter may also used with insignificant error. The polynomid expression for C,, gives us an analytical expression for the integral in Eqn. (3.14), letting us precompute its coefficients.
The saturated moiar Iiquid enthalpy cm be obtained by using the composition and its corresponding equilibrium temperature in Eqn. (3.12).
3.2.2 Mixing effects
The mixing effect expressed in Eqn.(3.11) is the surn of rnixing effects in the ideal solution at the reference condition, the correction to this quantity to account for nonidedity. and the excess sensible heat effect in change of temperature hmrefewce to the given value T:
where HE,, j = 1,2. denote the excas enthalpies of the compooents in the mix~and C: denotes the excess specific heat capacity of the mixture.
The enthalpy of mixing of the ideal solution is identicaliy zero. Heat of mixing data are given in Table 3.4 and Table 3.5 for several values of temperature and composition..
Since hi(298.1 5 K) = -3.7967~10' Idho1 and hl = -4.5481~10' kllkmol, at 298.1 5 K the
heat of mwigis at most 0.185% (at x = 0.4) of the molar average enthalpy of the solution. As the temperature rises, the magnitude of heat of mixuig decreases. Though the mohr average enthdpy becornes kss negative since the C,, increzx with temperature [38], the magaitude of the latent heat of condensation at 298.15 K stiII remains large enough to dwarf the kat of mixiig and the excess sensible heaî effect. Therefore, the molar average entbalpy of the liquid phase is a very good approximation to its true value.
By negiecting the excess molar specific heai, the mixture molar specific enthalpy is effectively the molar average of the specific heats of the pure components. This result is also obtained by applying the Generalized Principle of Corresponding States [30] taking the reference fluids to be the pure components themsetves and the mixture acentric factor to be the molar average of those of the reference fluids. Therefore, we expect good approximation to the true mixture enthalpy.
3.2.3 The composition derivative of saîurated liquid phase enthalpy
[n Eqn. (2.10) in Sec. 2.1. we defmed
We need this value of G,to calculate V, fim Eqn. (2.1 1). Since Eqn. (3.12) is an analytical expression for liquid phase enthaipy, differentiating it with respect to x, as in Eqn. (3.17) gives an expression for Gi:
The pure component enthalpies and specific heats are known for any temperature From the
equations in Sec. 3.2.1. We compute as follows. Let Z be the VLE constraint in Egn.
(3.6) (Sec. 3.1): The vaiue of a%T is available from the last iteration of Newton's meihod in solving for Tl in the VLE cdculation (Sec 3.1). Differentiating Eqn. (3.19) with respect to x,, we get
Differentiating Eqns. (3.4) and (3.5) with respect to xi,
Therefore. to compute G,. we first calculate the composition derivative of the activities fiom Eqns. (3.22) and (3.23), and use these to calculate the composition derivative of the VLE hction Z hmEqn. (3.21). This is then used in Eqn. (3.20) to calculate the composition derivative of the equilibriurn temperature, which is further used in Eqn. (3.18) to determine G,. 33.4 Saturated vapor phase entbalpy
The molar enthalpy of the saturated vapor phase of the mixture can be expresseci as the sum of the molar liquid enthalpy at the same composition (and at the temperature of the vapor phase), and the mixture enthalpy of vaporization at that composition and temperature:
The liquid enthalpy can be detemined using Eqn.(3.1 S), substituting the value of y, for x,. and Tt for the equilibrium temperature.
Many methods exist for the caiculation of the entbaipy of vaporization of a mixture [22] depending on the type and amount of data available. Since data for the methanol-isopropanol system are scarce, we opted for the Sivaraman mode1 [26] for caiculation of AHq. This result is aiso obtained by applying the Generalized Principle of Correspondhg States [l, 30, 3 11 to the estimation of (AH%K) as in the case of rpecific hais diwussed eadier:
where Tc, denotes the critical temperame of component j and R, = 8.3 14 kJ/kmoVK denotes the gas constant.
Since the acentric factor
O-W 3 ' =1-y,
we get, by fiom Eqn. (3.25), However. the mixture critical temperature Tc, needs to be determined before AH,,, can be çalculated: we use the van der Wads one-flztid iheory (VDW-1) [II:
Here. y, and y? refer to the mole fractions of merhano1 Or,) and isopropanol (1 -y,) in the vapor phase respectively. We assume the value of the binary interaction parameter q = 1 since the liquid mixture is close to ideal in its befiavior and like-interactions are expected between the molecules.
The pure cornponent criticaI data 1291 are as follows:
Therefore. we get
vc,= 0.0~059~?- 0.11242~ +0.220t3.
where Vc,, is in m3/kmol,and The enthalpies of vaporization for the pure components were fitted to a fiAh degree polynomial in temperature T to keep the maximum relative error at the data points 1291 below 1%:
Here. MW,is in kJ/kmol and T is in K. The coefficients bv are given in Table 3.6. Table 3.1 : Antoine constants for the components
Table 3.2: Enthalpy reference data
Table 3.3: Coefficients in the specific heat polynomial in temperature Table 3.4
Heat of mixing at 398.15 K [3]
Table 3.5: Heat of mixing [34]
Table 3.6: Coefficients in the enthalpy of vaponzation polynomial in temperature The liquid molar overflow hmstage i, 2 5 i 5 n-1, is dependent on the height of the clear liquid surface over the top edge of the weir on the tray, hW.,. This beight is in tuni dependent on the volume of liquid held up at the tray which cm be computed hmthe molar holdup,
where v,, is the mixture rnolar volume and is the molar average of the pure component saturated rnolar volumes at the tray temperature:
Despite the nonidedity of the liquid phase mixture, examination of data on mixture volumes in [35] led to the condusion that the real mixture molar volume at any composition diflers insignificantly from the ideal mi- molar volume obtained hmEqn. (3.34).
The pure component molar volumes are caiculated using the Francis- 1 equation [29],
where T, is in K and the molar volumes are in m3hol.Having calculated vi, where A, is the liquid holdup area of the tray, 0.13 m2, and hWis the height of the top edge of the weir fiom the tray surface. 0.05 m, and hm,, is the height of the clear liquid surface above the weir in m. if the clear liquid surface is below the weir's top edge, hm.i < O and therefore, Li = 0.
For positive h,,, ,LI can be determineci hmthe Modified Francis Weir Formula [24]:
where LI is in kmoümin. Iw is the weir length, 0.35 m. 4 THEORETICAL DEVELOPMENT
A generic time optid control problern is presented followed by the Luus-Jaakola optimization procedure For solving such a problem. This chapter begins Part II of this work and the notation used differs fiom Part 1.
4.1 THE TIME OP'CONTROL PROBLEM
Let us consider a system, the dynamics of whose state are govemed by an ordinary differential equation of the hm
where the initiai condition x(0) is given. Hete. x is the (nxl) state vector and u is the (mxl) control vector with the constraints
t)5 ut 5 (t) 1 5 j i m. (4.2)
We wish to reaeh the desired state xd in minimum time. The goal of the time optimal contml problem is to determine the control policy u(r) in the time interval O I t < r/ that minimizes the performance index
subject to the final time state constraints where E, is a suitable tolerance.
To minimize the performance index I in Eqn.(4.3), yec force the final tirne equality state constraints. we formulate the augmented performance index to be minimized [19] as
where the 13,are positive penalty function factors and ihe s, are the shifting tems that are updated afler every pass:
where q is the pass number and x,'(t/) is the final state obtained by applying the best control policy (W.) and stage-lengths (I*) obtained so Far (see algorithm below). A modification of this form of the augmented pertbrmance index. to handle dynamic constraints on the control. was used and is derived in Chapter 5. Al1 8,were assigned the same value. 8. The shifiing terms are assigned initial values of zero except in the case of rems where the shifiing tems from the previous run are directly used.
To determine the optimal control policy, we divide the time interval [O. into P stages each of initial length
With piecewise constant control over the stages. the W optimization procedure will determine their optimum lengths. This introduces the additionai constraints
which can be handled at the time the controls and stage-lengths are chosen in the U optimization procedure. 4.2 THE LUUS-JAAKOLAO~'~~~XTION PROCEDURE
1. Chwse parameters and initiai vaiues: initiai guess for finai time, tp. Number of time stages, P: initiai length of tirne-stage R as in Eqn. (4.8).
initial value for (mx 1) control vector uk fgr tirne-stage k, 1 I k 5 P. Number of random points per iteration. R. Number of iterations per pass, M. Numkr of passes, N. Region contraction factor. y.
Region restoration factor. q (used in the first 4 passes only).
2. Accumulate the values for control in the (Pmx1) vector w
'r T T T w=[u U ... Up]. 12
and the values for the stage lengths in the (Px 1) vector I
I = [ 1, l2 ... lp lT.
Let w" and t0denote the respective initiai vaiues.
3. Let r,,' denote the (Pmx1) vector of initial region sizes for the contml and v,'. the (Px 1) vector of initial region sizes for the stage lengths.
4. Initialize pass index : q = 1, i teration index : j = 1. 1 region sizes : I' = r,,, , 4 = v,1 . 5 Choose R random values for controi and stage-lengths about kirinitial values. d =ww + DI f, (4.13)
i'=P-'+~~v'. (4. I 4)
where the (PmxPm) diagonal matrix DIand the (PxP) diagonal matrix & have randomly chosen elements in the range [-1, 11. Different DIand ih are generated each of the R thes. This leads to a P(m+l) dimensional search.
6. For each of the R values of d and Y. compute the augmented performance index Jas in Eqn. (4.6). Save the values of d and 1 that give minimum J as W.' and lV.These are the initial values for the next iteration.
7. Contract region sizes for use in the next iteration (indexj+l)
- Control: r~*'= yr'.
- Stage Iengths: v~+l= yv~.
increment iteration index:j =j + 1.
8. Steps 5 - 7 constituîe an iteration. Perfonn M iterations.
9. Compute the region sizes for the next pass (index q+I):
a For the fus 4 passes. put vi2 = O and use region size testomion for the control.
Set v, to the chosen initial value (step 1) at the beghhgof the Hth pas r For subsequent passes. use the extent of variation of variables methoci [la],
PI *M 'O rii,,=lw - wi ],i=1,2,..., Pm,
10. Update the shifting terms as in Eqn.(4.7):
where r,'(r,)denotes the value of the 1'" nate variable at the final the. obiained by appiying the bat value of control w'.' and stage lengths I'~" at the end of M iterations. Assigi the values of W." and 1'" of pass q to waOand 1'' of pas (q+1) respectively.
increment pass index: q = q + I
1 1. Steps 8- 10 constitute a pas. Perfom N passes. Interpret the results. 5 THE DISTILLATION TIME OPTTMAL CONTROL PROBLEM
The initial and îïnai operating points of the distillation column used in this wodc are based on the work in [5]. This is a simple. conventional distillation column consisting of 19 sieve trays, a condenser-accumulator system and a reboiler. The feed, an equimolar binary mixture of methanol and isopropanol. is introduced to the top of the 10" tray (1 1' stage) at 0.06840 kmoVmin. There are. therefore, 21 stages. Since the composition and liquid holdup describe the state of each stage, the state of the whole system is described by a 42dimensionai vector:
Henceforth. x, will refer to the components of this state vector (1 5 j 5 21 denoting mole fraction of methanol at stage j, and 22 5 j 5 42 denoting the Iiquid holdup at stage j-21) unless stated othenvise. References to the ODES describing the dynarnics of these state variables are as bilows:
j= 1 Eqn. (2.13) j= 22 Eqn. (2.12) 2 5j I 10 Eqn. (2.4) 23 Ij531 Eqr~(2.1) j= il Eqn. (2.1 5) j= 32 Eqn. (2.14) 12 5 j < 20 Eqn. (2.1) 33IjS41 Eqn.(S.l) j= 21 Eqn. (2.19) j=42 Eqn. (2.18)
The control for this -stem is
u = [R QRI~, where the reflux aow rate R is constrained to maintain the condenser-accumulator liquid level, and the reboiler duty QR, which affects the reboiler vapor upflow, is coastrained to maintain the reboiler liquid level: OIVz,(~ The optimal control problem is to determine the R and QRso that the distillation column cm be taken From its initial state, the steady state corresponding to XD = 0.981 and x~ = 0.079 (Table 5.1), to the steady state corresponding to XD = 0.930 and XB = 0.040 (Table 5.2) in minimum time. the feed flow rate F and its composition zfied remairihg constant. Therefore. we seek to minimize the performance index f=tJ subject to the final time equality state constraints where the desired final values x,d are given in Table 5.2. nie absolute value nom of deviation of the final state From its desired value. S = 11 x(r/) - d IIi was used as a measure to judge results. 5.1 SOLVCNG FOR TFiE STFADY STATES ,411 x,'s in Sec. 5.1 nfer to methanol mole fractions. From the degree of freedom dysis of the column (Appendix A), it is seen that for the steady state case. by speci@ing the top and bottom composition the state of the entire column is fixed for constant feed rate and composition. Overall steady state material, component and energy balances around the column give The steady state material-energy balances on an envelope around the reboiler and a stage i below the feed stage, 12 5 i 5 2 1, are L,,, - V,= B, (5.10) L,_ix,-i- Vfi,= Bx,~. (5.1 1) L,.ih,.i - KHI = Bhs - QR. (5.12) The steady state material-energy balances on an envelope around the reboiler and a stage i above the feed stage, 2 5 i 5 1 1. are (S. l Oa) (5.1 la) (5.12a) The steady state material-energy balances on an envelope around the condenser. i = 1. give Eqns. (5.7) through (5.15) need to be solved simdtaneously to determine the state of the column at steady state. We adopt the following procedure of solving a system of nonlinear algebraic equations. outlined in [17]. Equations are grouped into two classes: simple and dificult. Each simple equation cm be solved for a single variable. The solution obtained is used to solve other simple equations in a seriatirn fashion. Difficult equations are implicit in their variables and need to be solved iterativety, using the values of al1 variables obtained by solving the simple equations. Using this procedure, the three equations at each stage, Eqns. (5.10) - (5.12) or Eqns. (5.10a) - (5,12a), are reduced to a single nonlinear equation in one irnplicit variable, a simple equation that can be solved iteratively. The simple equations fiom al1 stages are solved in a seriatim fashion and the determination of the steady state is reduced to one difficult nonlinear equation in a single variable. From Eqns. (5.7) and (5.8) we obtain the simple equatiom, Since x~ (= xi)and XE (= .qi)are specified, hi, Hz, hzi. yzi and Hzican be caiculated from the equations denved in Chapter 3. Assuming an initial value of R. we obtain Qc from Eqn. (5.15). We calculate reboiler duty by converting Eqn. (5.9) into the following simple equation: For a stage i below the feed stage, 12 r i 2 21. using Eqn.(5. IO) to eliminate K From Eqns. (5.1 1) and (5.12), we get Let Dividing Eqn. (5.19) by Eqn. (5.20) to eliminate and rearranging, we get where y, and H, can be calculated if x, is known. Then p, and q, are known. and Eqn. (5.23) is then a nonlinear equation in a sinde variabte x,!, which can be solved using Newton's method with the initial estimate for x,, king xi. LI,! cm then be determined fiom Eqn. (5.19) and Y,, from Eqn. (5.10). Since x2, is known, we calculate the values of Iiquid composition beginning with the reboiler, working ail the way up ti11 the stage 12. below the feed stage. For stages above and iacluding the feed stage, we solve Eqns. (5.19a) through (5.22a), instead of Eqns. (5.19) through (5.22), prior to solving Eqn. (533). For 2 I i I11. Thus we calculate the liquid phase compositions in a seriatim fashion till we calculate xi from stage 2. However, this value of xi depends on the variable R whose value was assumed prior to computing Qc hmEqn. (5.15) (for calculating QR hmEqn. (5.18)). Therefore. the value o€xl computed fiom this procedure will not be the same as xo. We now need to solve a difficuft nonlinear equation in a single, implicit variable R: The following inequality is imposed instead of Eqn. (5.25) for computational purposes: Applying Newton's method with a suitable initial guess. the value of R in iteration (k+l) is obtained from its value in iteration k hm Iterations are perfonned till the satisfaction of Eqn (5.26). The derivative in the denominator of the second term in the RHS of Eqn. (5.27) is obtained by the approximation: Terms in the numerator are computed by solving the entire set of simpIe equations (stage-to- stage -GIcalcdations beginning with Qc and QRdetennination) using the procedure outiined earlier for each perturbed value of R. Ai? was chosen to be RX 1O+. The tolerance was chosen to be E= lod. Once Eqn. (5.26) is satisfied, Ml are back-calculated hm the values of L, by applying Newton's method to the overflow formula as presented in Eqn. (3.38). Here, the toIerance used was 10-'O. Tables 5.1 and 5.2 show the liquid and vapor flow rates in the column in addition to its sbte at the initial and final steady states respectively. We revert to the notation prevalent in Part iI: al1 xi's denote state variables as in Eqn. (5.1) and not just mole fractions. This problem bas two types of constraints. There are 42 find- tirne equality state constraints that can be tackled using the method of shifting tems as discussed earlier. The constraints on the control are dynamic (Eqns. (5.3) and (5.4)) and are handled using the stute constraint variable method of Mekarapiruk and Luus [23]. We introduce two state constraint variables! x43 and x~,such that lit I O. with the initial condition The constraint R 2 O can be handieci at the time the control is chosen. The augmented pertbrmance index that was used for this problem aiso incorporated the state consûaint variables: Table 5.1 : Conditions within the colurnn during the initial steady state Table 5.2: Conditions within the column at the desired steady state 6 NUMERICAL RESULTS Results for this probiem were obtained in two steps. Preiiminary computations were made with srnail values of the number of random points, nurnber of iterations and number of passes in order to develop a feel for these. This was followed by the second step where a series of computationaily intense runs were made to rninimize both the transition the and the deviation of the fuiai state from its desired value. Minimum region sizes of 104 min for stage-lengths, 0.0001 kmoUmin for reflux and 1.O kJ/min for reboiIer duty were maintained. Smail values for the initial region sizes for the controi and stage-lengths had to be chosen in order to avoid numericd difficulties. Computations were canied out in double precision in FORTRAN using the Watcom FORTRAN 9.5 and the GNU-GCC2.95.2 (MingW32) compilers on htel Pentiurn 400, 500 and 866 MHz platforms running Windows 98 and Windows 2000. The built-in randorn number generators. URAND (Watcom) and RAND (gnu) were used in the respective cases. The subroutine DVERK [9] was used to perfonn numerical integration of the system over [O, In with a LocaI error tolerance of IO*. A maximum step size of 1.0 min was set to avoid floating point overfiow. Values of the shiKig tems obtained hmprevious nuis were taken to be the initia1 values for the shifiing tenns in rems instead of reinitiaiizing them to zero. Also. extent of variation of variables method was used to determine region sizes right From the first pass in teruns instead of using the region restoration factor for the first 4 passes. The initiai steady state given in Table 5.1 corresponds to R = 0.18692 kmoYmin and QR= 7924.1 Himin. Numericd diiticulties were encountered when these were chosen to be the initiai values for the controls. These difficulties were circumvented with the choice of smdl initiai region size for the control: 2W of the above values, but it seemed impossible to obtain values of xD and x~ simultaneously below 0.960 and 0.080 respectively. However, when the control was initiaiized to the values of R and QR necessary to maintain the desired frnal steady state in Table 5.2 (0.08781 lanoUmin and 4509.5 Himin respectively) dynarnic simulation could readily be carried out meeting al1 constraints. The initial region size for control was set to 25% of its value. A direct means of forcing finai tirne equaiity constraints was first tried: the augmented performance index was constructed from the ha1 time and the penalized absolute value nom of deviation of the final state from its desired value: Using the following set of parameters and initial values, this performance index was found to yield a final time q= 126.498 min with a I -nom of final state deviation of S = 2.8548~10'' in 60 passes (26.88 hrs), on a PIIV866 computer. R=35 y= 0.8 UL= [0.0878 1 4509.51~ tp = 100 min M= 15 rl = 0.85 (1 SklP) P=5 N= 100 O= lo5 r,' = 0.25 w v,' = 2 min The case of shitiig terms in a quadratic penalty function as in Eqn. (5.32) yielded a result of g= 87.277 min with S = 1.94 1 1 x 10.' in 70 passes. Another run was made with the following parameters and initial values R = 50 y= 0.8 uf = i0.0878 1 4509.51T ~p= 100 min LM=15 TI= 0.85 ( IlklP) P=5 I IV= 100 e= io6 r,' = 0.25 w v, = 2 min The cun incorporating the absolute value nom into the augmented performance index achieved a final tirne of r/ = 99.613 min with S = 3.0559~10~~in 32 passes (8.22 hrs) on a PW866 computer, yielding a 4 stage control policy. This result was taken as the initial value for a rem with initial region sizes of 0.01 kmoVmin for reflux, 600 klfrnin for reboiler heat input and 2 min for stage lengths. The result obtained in 28 passes (7.09 hrs) was r/= 99.765 min with s=2.94~10.~. Using the parameters for the first nui above but now using a quadratic penalty fiinction with shifting terms, we found r/ to Vary between 62.578 min and 11 1.656 min with no definite convergence pattern. Using R = 50. M = 15 and N = 50 and augmented performance index as in Eqn. (6.1), now wilh P = 6 the stages. we obtained 4 = 92.586 min with S = 4.6620~IO-' at the 5oL"pass (10.37 hrs) on a Pm866 computer. For the same parameters except for P = 5 time stages and ip = 60 min (previously 100 min), we obtained a result of r/= 80.627 min and S = 1.009 1 x 10- ' in 48 passes (9.82 hrs) on the PiW866 computer. But with shifting terms, using the same parameters as the last run mentioned above (rp = 60 min R = 50. ,M = 15. N = 50) except for P = 4 time stages, we were able to achieve r/ = 57.587 min with S = 5.9667~IO-' in 18 passes (8.77 hrs) on a PW500 cornputer. A nurnber of runs with various values of R. shown in Table 6.1, revealed that the use of shifiing terms was very efficient in forcing al1 42 equaiity constraints in less time. While the augmented performance index of the form in Eqn. (6.1) is an undeniably good means of reducing S. it ohen does so at the expense of IF Based on the recent success of the use of a quadratic penalty function with shifiing terrns in the augmented performance index in solving tirne optimal control problems using the Luus-Jaakola optimization procedure [14], it was decided to use tp = 60 min, J of the form given in Eqn. (5.32), and controls corresponding to the desired steady state as the initial guess in al1 subsequcnt runs (exception to latter are reruns). 6.1.1 Effect of the number of rnndom points Table 6.1 shows results obtained fiom runs differing in the number of random points for one set of procedure parameters. There is no particular pattern in the results. Cases where R < 100 hint at a possible tradeoff between the final theand the nom while cases wbere R = 100 and R = 150 suggest the possibility of attaining good values of both using a iarger number of random points. Values of R upwards of 150 were used in the computationally intense search phase. 6.1.2 Effect of the penalty function factor Table 6.2 shows the effect of 8 on the results obtained for two cases of R. The run with R = 50 and 8 = 10' contributes to the tradeoff in Table 6.1. But with 8 = 108 and R = 150. the result obtained is good in S but poor in r/. Fmm Table 6.1 it is seen that with 8 = 106 it is possible to do better than this. For the two nins with 8 = 10'' shown in Table 6.2. the best values ofS were obtained in the fmt 4 passes of constant stage-lengths. Mer the 4" pas. no definite convergence pattern was observed. The value of S fluctuated with each pass but never got bener than the values in Table 6.2. This shows that 8= 10'' is too small a value to ensure convergence. 6.13 Efiect of number of time stages A Iarger value of P suggests better result for qat the expense of S (Table 6.3). The latter may be the result of increased search space dimensionality. Based on the effect of the number of random points, P = 8 stages was chosen as the initial value for the m comprishg the computationally intensive search phase with a large number of random points per iteration to handIe the search space dimensionaiity. Table 6.1 : Effect of the nurnber of random points on result obtained y= 0.80 uk = [0.0878 1 4509.51 ' tp = 60 min q = 0.85 (ISkSP) P=4 8= 106 r,n'= 0.25 w v,~'= 2 ," Comp Time .. .at best CPU clk -.. (tu) pass no. (MHz) Table 6.2: Effect of penalty function factor on preliminary results obtained R r/ S .rg(l/) ) Comp Time (min) .. . (tu) 50 54.565 6.1655~105 0.92958 0.04046 1 1.62 150 70.785 2.5085~IO" 0.93012 0.03990 9.96 Table 6.3: Effect of the number of thestages on the result obtained R = 50 y= 0.80 uk = [0.0878 1 4509.5]~ i, = 60 min iU= 15 tt = 0.85 ( IrklP) N=50 O= lo6 r,,,' =0.25 w vifil = 2 min - P r/ S Comp Time .. .at best CPU clk (min) .. . (hr) pass no. (MHz) 3 62.320 2.6503~lu3 17.74 50 500 4 57.587 5.9967~lw3 5.18 18 500 5 55.838 t -1623~10-2 19.62 45 500 6.2.1 Effect of number of random points Table 6.4 shows results obtained for various vdues of R for two values of 6. Besides a large number of random points per iteration, there were bite as many iterations per pass (M = 30) as in the preliminaq phase. increasing the number of random points improves equality state constraint satisfaction as evident in the decreasing vdues of îhe final state deviation nom S. This is observed in Table 6.4 and in Figs. 6.1 and 6.2. However. the value of t/ increases. This suggests its tradeoti with the quality of attainable final state as with the preliminary runs. It can also be seen that too large a nurnber of random points does not necessarily yield better results as in the case of R = 500 for 6 = 10' and R = 400 for 8 = 108. Large values like R = 500 aiso consume a lot of time. However. it is possible to obtain better results by reninning the problem with smaller values of R. such as 300, using the results hma previous run as presented later. 62.2 Effect of penalty function factor The role of the penalty iùnction factor 0 as a weight between final time and proximity to the desired final state is significant in this problem. It can be seen in Fig. 6.3 and Fig. 6.4 that for a higher value of 8 the search is conducted in a region characterized by lower values of S. From Table 6.4, it is observed that this oçcurs at the expense of r/, al1 other procedure parameters remainine the same, Therefore, the fact that better values of and norm can be simuitaneously achieved as show in Sec. 6.2.4 suggests thai 8= 108 is hi@; it favors lower values for the norm at the expense of f'inal the. From Fig. 6.5 it can be seen that when 8 = 10: the procedure rapidly decreases r/wiiile keeping S at a value much larger than that in the case of 6 = 10'- This kind of behavior was also observai in the preliminary m.This indicates that leis too small a value for 8. Therefore, acceptable vdues of 8 lie between 10' and 10'. 1 o6 being a good value. Tabie 6.4: Effect of the nurnber of random points on result S Comp Time .. .at best CPU clk rr(min) .. . (ru) pas no. tm) 3 -744~1o'~ 39.52 37 500 R r/ S Comp Time .. .at best CPU clk (min) .. . (lu) pstss no. (MHz) 1 50 62.54 1 7.145~10~ 35.46 21 400 300 76.663 6.601x10J 47.2 1 17 400 400 73.048 4,451 x10" 58.40 15 400 Fig 6.1 : Effkct of number of randorn points on convergence for O= 1o6 Every second data point has ken rnarked O IO 20 30 40 Pass Fig. 6.2: Effwt of number of random points on convergence for O= 10' Every second data point has been marked K \A Al) 1: Pass Fig. 6.3 : Efféct of 8 on convergence for R = 150 Every second data point has been marked Pass Fig. 6.4: Effixt of B on convergence for R = 300 Every second data point has been marked Pass Fig. 6.5: Final theand convergence profile for the case 8= 104 and R = 300 Every second data point has been marked Final theprofile 1E-O01 p w 8 t- G s< L C CD 1E-002 g 'I 3 O 9 -b 0.s s L ZS 1E-003 8 a CD< zm. O' 3 1 E-004 O 4 8 12 16 20 Pass 6.23 Effect of the number of time stages A larger number of thestages offers more flexibility in the time-stage lengths provided the nurnber of random points is &cient to handle the search space dimensionality. As seen in Table 6.5, use of a larger numkr of time stages yields a better value for the final time at comparable norms. Using fie parameters in this table, a run with R = 300 random points and P = 7 stages was made for cornparison with a previous run using R = 300 and P = 8. While the nom to which the final state constraints were satisfied was definitely better, the final time result was poor (cases 2 and 4 in Table 6.5). The number of random points was increased to R = 400 to see if a better value for 9 codd be obtained. The result obtained (case 3) was a poorer value of r/ but at a bener value of S. This result compares the sarne way with the run using R = 400 points and P = 8 stages (case 5). as do the nins with R = 300 points discussed above: while the nom is marginally pater in the case with larger P. the final Ume result is significantly better. It is fond in cases where the initiai number of time stages exceeds 7, some of the stages collapse to zero length to yield a 7-stage control policy. Table 6.6 shows 3 cases of P for R = 400. An initiai 7-stage control policy is not changed in the number of time stages. But in cases with P = 8 and P = 9, a 7-stage control policy is reached through collapse of stages. Fig. 6.6 and 6.7 compare the state trajectories obtained from control policies resulting hm the run using R = 60 with P = 4 (Table 6.1) and R = 150 with P = 8 (Table 6.4). The former has a final time of t! = 50.417 min while the latter has a tj = 50.137 min: the norms are comparable. It is interesting to note that in any case, the approach to the desired state is smooth to readily be able to maintain steady state once the controls correspondhg to it are applied at the final time. With a greater number of time stages, some of them conûibute to hastening the upward curve of the trajectories aerthe initial downward growth while others conûibute to a smoother approach to the desired thal state. With fewer time stages, the trajectories oscillate about the desired States and the rate of change of the state variables at the tinai time are farther fiom zero. Table 6.5: Effect of the number of initially chosen time stages 0= 106 ri,,' = 0.25 w v,,' = 2 min Case P Comp Tirne .. .at best CPU clk No. ... (hr) ( pass no. 1 (Mk) 1 1 4 2 7 3 7 4 8 5 8 6 9 Table 6.6: Control policies tiom runs with 400 random points. differing in P k [A (min) Reflux R Reboiler duty QR Table 6.6 (contd.) 1 k 1 4 (min) / R (kmoumin) / Qn(ld/min) 1 Fig. 6.6: Cornparison of state trajectories (mole fraction) for different cases of P and R 20 30 t (min) Legend solid line: R = 60 and P = 4 dashed line: R = 150 and P = 8 56 Fig. 6.7: Cornparison of state trajectories (liquid holdup) for different cases of P and R O 10 20 30 40 50 Time (min) Legend solid Iine: R = 60 and P = 4 dashed Iine: R = 150 and P = 8 6.2.4 The best result obbined Previously, a final time of r/= 5 1.966 min was obtained with S = 2,835~ IO-^ using R = 300. P = 8 and 8 = 10'. The resuiting 7-stage control policy and shifting temis were taken as initial conditions for a remwith R = 300 and initial region sizes of 0.5 min for stage lengths, 0.005 kmoVmin for reflux and 500 kJ1mi.n for reboiler kat duty. This gave a final tirne r/ = 5 1.843 min at S = 4.337~10~.The maximum absolute deviation in composition was 6.695~10-'. in .YI, (bottoms composition). The maximum relative deviation in composition, also in QI, was 0.167%. The maximum absolute deviation in holdup was 7.301x IO& kmol, at stage 6. The maximum relative deviation. also at this stage, was 0.006% . One of the time stages in the resulting control policy was found to be of length 0.167 min and was discarded. The 6-stage control policy was taken to be the initial condition for yet another rem with R = 150 and the same region sizes as its predecessor. A final time of I./ = 52.494 min was obtained at S = 4.3 19x10~which is by far, the closest that the system has come to the desired final -te. The maximum absolute deviation in composition was 7.342~10". in XII. The maximum relative deviation in composition aiso in xz 1, was 0.183%. The maximum absolute deviation in hoidup was 6.599~10~kmol. at stage 16. The maximum relative deviation. at stage 20. was 0.007% . Figs. 6.8 and 6.9 compare the control policies of the above cases. The control policy of the run with R = 60 in the preliminary phase is also included for cornparison. Figs. 6.10 and 6.1 1 show the state trajectones obtained by applying the best conml policy obtained so fa.. As can be seen hm Fig, 6.12, there is no constraint violation by the control though there a~ instances where the reflux and reboiler vapor upflow are tight against their constraints. 58 Fig. 6.8: Cornparison of reflw poticy for various nrns leading to the best result O. 14 --Original run First rerun Second rerun -----R=60case 0.12 - 111111111111 20 30 40 t (min) 59 Fig. 6.9: Cornparison of reboiler heat input policy for various cases leading to the best result 3000 lilllllIIlllllllllllllll O 10 20 30 40 50 t (min) Fig. 6.10: State trajectory (moLe fraction) obtained using the best control policy 20 30 t (min) Fig. 6.1 1: State trajectory (holdup) obtained using the best conml poiicy obtained 1111 1111~1111(1111~1111~ - C - M,,(feed stage) - - 1111 1111 111111111111111 1 O 10 20 30 40 50 t (min) Fig 6.12: Dynarnic constraint satisfaction Every third point has been marked t (min) DISCUSSION The direct approach to solving this problem, without any shpli%ng assumptions (besides those in the formulation) and transformations, presents one more step towards tackiing real world problems accurately especially in cornparison to the solution of the linearized model. The large computation tirne necessitates cethinking of strategies for solving real world optimal control problems. the ultimate aim king online optimal control. Speed up in computation. especially in light of this problem, reguires modifications in The model: rnodel reduction techniques could be used to simplify the problem structure. The method: other rnethods such as iterative dynamic programmin6 (IDP) and successive quadratic programming (SQP) could prove to be rnuch faster, permitting us to use different initial conditions and parameters. For many -dard optimization / optimal control problems, IDP has been show to be at Ieast twice as fast as U optirnization [14. 153. The computation: the U optimization procedure has the wonderhl property of linear speedup in a data parailel mode of execution [4], Techniques like workload distribution, domain decomposition and their hybcids can be applied to partition the task of choosing a number of random points (by ushg different seed values) and evaluating the augmented performance index: synchnization is required only at the end of the iteration. With a loosely packed ciuster of 4 workstations, each quivalent to a PIIY866. on a common 100-base-T Ethernet each nui can be made in a single day as opposed to 4 days on a single computer. The cornputers: computer chip/memory access speeds and computer arithrnetic. One notable development in the ment past is the use of the LJ optirnization procedure for parameter estimation for emrs-in-variables data [12]. Dependence of properties like specific heat and enthalpy of vaporization of pure cornponents on temperature in this work has been modeled by linear least squares tegression which is based on the assumption that there is no mrin the measurement of the independent variable. This is mostly not the case in engineering problems. This development is weii-suited for overcoming this problem since the proposed property mode1 equarionç can be used to duce the nwnber of variables for optimization. in order to arrive at a good resdt, it is not merely suficient to carry out runs with varying procedure parameters. As with the cancer chemotherapy problern by Luus et ai. [Zl], it is the sequence of remusing the control policy, stage-Iengths and shifling tem from previous runs as initiai values. that has ultimately led us to better results with each run till the answer is finally obtained. Analysis of the resuIts obtained der each run by the user is essential to get close to the optimum. 8 CONCLUSIONS A nonlinear mode1 of a simple distillation column, within the limits of data availability, was set up and tirne optimal control ushg the Luus-Jaakola optimization procedure was established. The mode1 circumvents cornmon assumptions such ris equimolal overflow and temperature independence of physical properties. Avoiding the assumption of equirnolal overîlow makes the problem complicated. The unknowns introduced into the state dynamics are handled with the assumption of quasi-equilibrium behavior to reduce the dimension of the state vector. To obtain the initial and final condition during steady state calculation the set of nonlinear algebraic equations was solved eficiently by grouping the equations appropriately. in the time optimal control problem, a speciai challenge is the necessity to meet the 42 fuial tirne equaiity state constraints. in addition to this. there are two dynamic constraints on the control. one of which is implicit in the control variable (reboiler heat input). The Luus-laakola optimization procedure was successful in tackling this problem: the best result obtained achieves strict constraint enforcement within a good fmal time. The typical approach to solving time optimal control problems using this method is to make a number of runs with varying initial conditions and procedure parameters: repeatability of the result under these variations is key to determining if the result obtained is the (global) optimum. The main irnpedirnent to this analysis is the computation tirne requirement. Despite the present computation times, granted the assorted special challenges that this problem poses, the Luus-Jaakola optimization procedure performs very well, especially in meeting dl constraints. Dynamic constraints on the control were successfully handled using the stale constraint variable method. The use of a quadratic penalty function with shifting terms is particularly good in forcing equality state constraints within reasonable fùial tirne results. APPENDK A: DEGWE OF FREEDOM ANALYSlS OP THE COLUMN AT STEADY STATE At steady *te, the material energy balances for an ordinary stage become The M, are implicit in these equations since the L, are computed fiom Mi. At the feed stage, LF-[ - LF + VF+I- VF + F = 0. LF[xF-~- LFXF + VF+1~j=l- Vw+FfIéCd= 0. LF-Ih.1- LfhF+ VF+~HF+I- VFHF -+ Fh/,,d = 0. (A.2) At the condenser. Vz-R-D=O, vyr-Rrl-Dxl =o. V?&- Rhi - Dhl -&=O. and at the rebiler. Therefore. the number of variables in this system is: 190t'dÎÎstage~ IM,,L,, 6,x,,yi. h,, H,: 19x7 = 133 Condenser R. D.xi, hi, Qc: 5 0 Reboiler Total number of variables: 145 The number of independent equations in this system are: 19 ordinary stages Material-energy balances: 19x3 VLE relation: 19x 1 Vapor enthalpy equation: 19 x 1 Liquid enthalpy equation: 19 x 1 Tray hydraulics: 19x 1 Condenser Material-enew balances: 3 Liquid enthalpy: 1 Reboiler Material-energy balances: 3 VLE relation: 1 Vapor enthalpy: 1 Liquid enthalpy: 1 6 Total number of independent equations: 143 Therefore. the number of degrees of kedom for the column = 145 - 143 = 2. APPENDIX B: FORTRAN PROGRAM FORTRAN code for time optimal control of the ciistiilation colm using the LJ optimization procedure. Written in WATCOM FORTRAN 9.5 PROGRAM DISTIL. F IMPLICIT DOUBLE PRECISION (A-HlO-Z 1 EXTERNAL RDISTIL,DVERK PARAMETER (NSTG=21,NSTG2=42, N=45, NI=4 4 ) PARAMETER (NR=3OO,NIT=30r NPASS=50, NPASS1=51,NU=2, THETA=l .D6; 2ARWTER (GAMMA=O.gDO, CTA=O .95DO,TF=6O. DO,NT=7 ) PARAMETER (TREGINIT=2.DO,EPS=1.OD-6,TOL=1.D-6) PPLWETER (URlLIM=l. D-4, UR2LIM=l. DO ) DIMENSION X(N) ,XIN (N), XS (N),XDTF(N) , DX(N) DIMENSION U(NU, 40),US(NU,NT) ,UIN(NU,NT) ,UR(NU,NT) ,UP(NUrNT) DIMENSION UA(NU,NT),TA(NT) CIMENSION T (NT),TIN (NT),TS (NT),TP(NT) ,TR(NT) DIKENSION C(24),W(N,9) ,S(NSTG2)3A(NSTG2) DOUBLE PRECISION Y (NSTG), L (NSTG), MINNORM CHARACTER FILENAME* l4 COMMON / PAF#,M/R,QR, HF1, QC COMMON /DIFFEQ/U,KK, hF COMMON /COLUMN/L,V (NSTG), Y,HL (NSTG), HV (NSTGl,TEMPR (NSTG) DATA S/NSTGZfO.ODO/ ND1 FF=N FILENAME='NR300\NR30ONT7' CALI, GETTIM(IH,IM,IS,IlOO) TIMEI = IiOO - iOO*IS + 6000CIM + 36@000+IH ISLED=37 ETIME=O.DO do 456 i=1,9 c (i)=O .dO c(6)=L.dO MINNOW=l.D30 R=O.l86SôDC CALL STEADY(C.S8lDO,O.O79DO,XIN) hF=HFl 8=5.0875DO CALL STEADY [O.93OD0,O. 04OD0,XDTF) RO=R QR=QR/ 1. D3 QRO=QR DO 101 K=l,N KS (KI =XIN (K) X (K)=XIN (K) CONTINUE DO 310 I=I,NT JIN(I,I)=RO us(r,r) =RO UP(I,I) =RO GIN (2,II=QRO US(2,I) =QRO UP(2,I) =QRO TIN (1)=TF/DBLE (NT) -'/) WRITE (6,330) NR, NIT,NPASS WRITE(3,330) NF!, NIT, NPAÇS FORMAT(' NR = ',I5,' NIT =',I4,' NPASS =', 141 WRITE (6,331) GAMMA,ETA, THETA HRITE ( 3,3 31 ) GPMMA,ETA, THETA FOWJ-T(' GAMMA= ',F4.2,' ETA = ',F4.2,4X1'THETA= ',2E9.2) WRITE[6,332) UIN(1,l),UIN(2,1) ,USTARTl1USTMT2 WRITE!3,332) üIN(lr1),UIN(2,l),USTART1,i1START2 FORMAT(/' U(0) = ',F10.5,F15.5/,' UREG(0) = ',F10.5,DL5.5) WRITE(6,333) TF,TREGINIT,NT WRITE(3,333) TFfTREGINITfNT FORMF.T(/'TF(O) =',F10.6,5X,'TREG(O) =',F10.6,5X,'NT =',14/j WRITE(6,335) ' XD( O) = ',X(l),' XB( 0) = ',X(NsTG) WRITE(3,335) ' XD( O) = ',X(l),' XB( O) = ',X(NSTGI WRXTE(3, '1 WRITE(6,335) ' XD(TF) = ',XDTF(l),' XBiTf) = ',XDTF(NSTGI NRITE!3,335) ' YI(TE) = ',XDTF(I),' XB!Tf) = ',XDTF(NSTGI FORMAT (~10,~12.6,5X,A10, F12.6) WRITE(6,lO36) WRITC(3,10361 CLOSE ( 3) DO 999 IPASS -i,NPASSl CALL GETTIM(IH,IM, IS, 1100) TIME1 = 1100 + 100'TS t 6000fIM + 360000'IH OPEN (UNIT=7,FILE=FILENAME//' . PI' ,ACCESS='APPEND', RECL=1551 OPEN (UNIT=3,FILE=FILENAME// ' .OUT',AcCESS='APPEND' ) WRIT5(3,129) IPASS WRITF(6,129) IPASS nRMAT(//3X1'PASS = ',ï3) DO 26 I=l,NT U(1,I)=UIN(l, 1) O(2, i)=UIN(2, I) T(T)=TIN(I) IF(IPASS.LT.5) THEN UR ( 1,I ) =USTARTl UR ( 2, I ) =USTART2 TRII)=O.DO ELSE IF(IPASS.EQ.5) THEN TR ( 1 ) =TREGINIT ENDIF CONTINUE USTPJTl=USTARTl'ETA USTART2=USTART2'ETA DO 25 K=l,N X(K) =XIN(K) TIM=O .ODO WRITE(Sl3IaO) X(1J,X(NSTG) 00 50 KK=l,NT INP2 TIMEF = TIM + TIN(KK) CXL OVERKfNDIFF, RDISTIL, TIM,X, TIMEF,TGL, IND, C,NDIFF,W) TIM=TIMEF WRIff (3,338) TIX(KK),TR(KK),U~I~~)~~~(~~~CI~~~(~~~~~~~~~~~~~ -X(l) ,X(NSTG) WRITE(6,330) TIN(KK),TR(KECl,U(I,KK)rUR(1tKK)ru{2rKK)tUR(2tKK)r -K ( 1), X (NSTG) 50 CONTINUE 3380 FOW?T(6OX, 2F8.5) 338 FOWAT(4F9.5,2D12.5,2FB.51 CXL GETNORMS [X,XDTF, N, NSTG, DX1, IXINF, DXINF, IXKEL,DXML, DM1 -, IMINF,DMINF, IMREL,DMREL) WRITE(3,L59) X(N) , IPASSA,DXL, IXfNF,DXINFl IXREL,DXREL WRITE (3,160) x {NSTG~+~),X (NSTG2+2) , DM1, IMINF, DMINF, IMREL,DMREL WRITE{6,l59) XiV),IPASSA,DXi,IXINF,DXINF,IXREL,DXREL WRITF 16,160) X [NSTG2+1),X (NSTG2+2),OM1, IMINF,DMINF, IMREL,DMREL 159 FORMAT(' Tf=',FlO.6,' IPASSA=',I3,4XID12.6,2(' {', 12, ', ',D12.6, -') '1) 160 "MT{'<',2013.6,'> ',DL2.6,2{' (',12,~,',DI2.6,'~'1) IF(IPASS.GT. 1) NRITE (7,1164) 13x1, IXINF, DXINF, IXREL,DXREL, DMI, IMINF -,DMINF, IMREL,DMREL 1164 FORMAT(2iG'14.6,2[' ('112t'1',Di2.6,'~'~)) IF(IPASS.GT.NPASS) CYCLE TESf?=l.OD40 JO 99 LOOP = l,f.IIT 30 60 IR=i,NR DO 161 E=l,NT A CWN = 2.DO*(URAND(ISEEDl-0.500) IF(IR.EQ.11 RAN = O.gDO ü(1,;) = UIN(I,I) + RANtURI1,I1 ;a[u(1,1) .LT.O.DO) U(I,I)=O.DO 2 W = 2.3O*(UWC(ISEED)-O.5DO) ït(1R.EQ.i) RAN - O.ODO 'J(2,I) = UIN(2,I) t W*UR(2rIl 3 MN - ~.EO*(URAND(ISEZD)-~.~DOI IF(1R.EQ.L; &Y = 0.000 T(I] = TIN(l) + WtTR(I) rrcrc~i.Lr.o.no) T(II=EPS 6 CONTINUE 00 125 K=l,N 125 X(K) =XIN(K) TIM = 0-ODO DO 621 KK=I, NT IND = 2 TIMEF-TIMiT (KK) CALL DVERK (XDIFF,BDISTIL, TIM, X,TfmF, TOL,IND, Cl NDIFFIw) TIM=TIMEF 621 CONTINUE DNORM=O. DO DO 9E7 I=T, NSTGZ 987 DNORM=DNORM+(X(I)-XDTF(I)-Ç[I))~~ ?ERF=X (N)+THETA* (DNORM+X(NSTG2+I) +X (NSTG2+2)) IF(PERF.GT.TEsTP) GO TO 30 00 164 I=l,NT US(l,II=U(I,L) US(2, I)=U(2, Il TS(Il=T(I) CONTINUE ANORMS=O. DO DO 165 K=L,NSTG2 XS (K)=X I KI -XOTF (KI ANORMS-WORMS+ DABS (XS[K CONT 1 NUE XS(N-~)=x(N-2! -XDTF(N-2 XS(N-~]=X[N-~)-XDTF!N-1 XS (N)=X (NI TESTP = 3ERF PERFS = FERF CONTINUE CONTINUE IF (TIMEF. LE. TIMEI) ETIME= €TIMX+ 1.4403 WRITE!?, 124) TIME, ETIME WRITE [ 6,124) TIME, ETIME FOEWAT ( ' ?P.SS TIME = ' ,F10.3, ' MIN, ELAPSED TIME = ' ,F10.3, ' MIN' ) WAITE (7,2821 IPASS,XSINl ,iWORMS FORMAT(lX,15, fi5.8,d15.5\) WRITE (3,128) XS (N) WRITE(6,lZS) XS [NI FOMT( /lx, 'FINAL VALUE OP PERFOP.MANCE INDEX, I= ' ,Fl7.6) WRITE(3,128i) (XS(K) , K=1, NI) WRITE(6,1282) (XS(K),K=l,NI! FORMAT (lx, 'STATES [Tf1 = ' ,5D12.4 1 NRITE(6, ' (lx,13HVIOLATIONS = ,2015.61 '1 XS(NSTG2+1) ,XS(MSTG2-21 WRITE (3,' (iX,13kiVIOLATIONS = ,2Dl5.6) 'i XS (NSTG2+1),XS {NSTG2+2] WRITE(3,1282) [S(K),K=1,NSTG2] WRITE(6,1282) (SiK) ,K=i,5) FORMAT(IX, 'S (Tf) = ',5D13.6] WRITE (3,1283) (-2.DO"THCTA+S (K) ,K=i,NSTGZ) h'RIYE(6,1283) [-Z.DO+THETA+S(K) ,K=1,5] FORMAT ( '-2THhS(TF] =', 5DL3.6) CLOSE ( 7 ) CLOSE(3) CONTINUE OPEN (UNIT=7,LILE=FILENAME/ / ' , PI ' ,ACCESS=' A?PEND1,RECL=l55 ) OPEN (gNIT=3,FILE=FI LENAME/ / .OUT1,ACCESS='APPEND' ) WRITE(7,'1 WRITE(7,*1 'OPTIMUM: ',IPASSA,' PASS' CLOSE (7) WRITE(3, iO3S) 'ENC TIME ', TH, IM, IS,1100 WRITE(3, '! WRITE(3,*} 'OPTIMUM: ',I?ASSA,' PASS' CLOSE (31 OPEN (UNIT=9,FILE=FILENAME/ / ' .CTL' ,ACCESS= ' APPEND' ) WRfTC [8,+! 'OPTIMUM: ' , IPASSA, ' PASS' TIM = O.CD0 DO 163 I=L,NT WRITE(8,iEZ) TA(1) ,TIM,UA[l,I),UA(=) TIM = TIF + TA( I) 'NRITE(8,l82l} TIbl,UA(i, Il ,!JA(2,II CONTINUE wrize(8, -1 'fmITE (8,1036) TIM = O.ODO DO 181 I=i,NT WRITE(a,Z82) T(I),TIM,UIN(1,T),UIEl(2,I) TIM = TIM - TII) -XRITE(8,Lâ21] TIM, UIN(i, 1), UIN(2,I) CONTINUE FORMAT (lX,4FlZ.6) FORMAT (l3X,3FIZ.6} CLOSE ( 8 1 OPEN (UNIT=4,FILE=FILENAME// ' -ST1,ACCESS='APPEND1, -3ECY=500) WRiTE (4,*) 'OPTIMUM: ' ,IPASSA, ' ?ASs' :RITE (4,'! DO 29 I=i,NT O(I, I]=UA(l,Il U(2,I)=UA(2,1) TtT\ =TAIT1 * (-1 6-1 29 CONTINUE ZXECUTE TRAJ OPEN (üNIT=3,FILE=FILENPME// ' .END1,ACCESS='APPEND' ) WRITE(3,+) 'OPTIMUM: ',IPFSSA,' FASS' WRITE (3,') DO 1289 K=l,Nl 1289 WRITE(3,12891) K,X(K)-XDTF(K),DX(K) 12891 FORMAT (lx,I3,2DLS. 6) WRITE(3,1036) WRITE(4,1036) DO 27 T=l,NT U(1, I)=UIN(l,1) U (2,I)=UIN (2,Ij T!I)=TIN(I) 27 CONTINUE ZXECüTE TRAJ CLOSE (4) 30 1290 K=i,Nl 1290 WRITE(3,L2891) K,X(K)-XDTF(K),DX(K) CLOSE ( 3 ) RZMOTF BLOCK TRAJ 30 28 K=l,N 2a X(K) =XIM(K) TIM = 0. ODO WRITE(4,338) TIM,X(l),X(NSTG),X(NSTG2cl),X(NSTG2&2) DO 55 KI(=l,NT 3T=T (KK)/S.ODO DO 56 II=1,5 TIMEF=TIM+DT IND=2 CUL DVERK(NDIFF,RCISTIL, TIM,X,TIMEF,TOL, IND,Cf NDïF'F,W) TIM = TfMEF WRITE(4,338) TIM,X(i),X(NSTI),X(NSTG2tl)IX(EfSTG2-2)lU(l~KK~, -U(Z,KK) 56 CONTINUE z c CONTINUE WRITE (4,') WRITE(4,338) TIM,X(L),X(NSTG) ,X(NSTG2+1) ,XiNSTG2+2) ,U(l, NT) , -CJ(2,NT) KK= 1 G(l,KK)=a0 iT(2,~KK)=QRO CALL RDISTIL(N,O.DO,X, DXI END aLOCK OPEN (üNIT=3,FILE=FILENAME/ / '. SiiF' ,ACCESS='APPENDr) WRITE ( 3, * ) 'AT BEST ?ASS' WRITE(3, ') DO 1663 K=l,NSTG2 1663 -RITE(3, ') K,SA(K) , -2.DO'THETA*SA(K) WRITE(3,1036) WRITS (3,') 'AT LAS?' ?ASS1 WRITE(3, ') 30 1664 K=l,Nl 1664 WRITE (3,') KIS (K), -2.DOtTHETAtS (KI CLOSE ( 3 ) STOP END SVHROUTINE GETNORMS (X, XDTF, N, NSTGfDX1, IXINF, DXINFI IXREL, DXRELlDMl, -ïMINF, DMINF, IMAEL, DMREL) IMPLICIT DOUBLE PRECISION (A-H,O-2) DIHENSION X(N) ,XDTF [N) DX1=0. DO DMl=O. DO IiCINF=-1 DXINF=-1.040 IXREL=- 2 DXREL=- 1 -040 IMINF=-I DMINF=-1. D4 0 IMREL=- 1 DMREL-- 1. D4 0 DO 20 I=L,NSTG axl=dxl+aans (xfil-xdtf(i)) TEMPA=UABS ( X ( 1 ) -XDTF ( I 1 1 IF[?EMPA.GT.DXINFI THEN DXINF=TEMPA IXINF=I ZNDIF TEMPR=TEMPA/X i 1 ! iF(TEMPR.GT. DXREL) PHEN DXREL=TEMER fXREL=f END1 L J=IrNSTG dml=dmltaabs (x(J) -xdtf {JI ) TIMPA=DABS (X(J; -XDTF'( 2) f F (TEMPA-GT.DMINF) THEN DMIMF=TEMPA IMf NF=I ZNDIF TEMFR=TEMPA/X (J) IF(TEMPR.GT.DMFGL) THEN DKREL=T EMPR IMREL=I ZNDIF 20 CONTINUE RETURN END SLOCK DATA TEM? FARAMETER [NSTG=21) DOUBLE PRECISION L (NSTG}, V (NSTG], S (NSTG), HL (NSTG], IfV (NSTG), -TEHPR (NSTG) COMMON /COLUMN/L,VI Y, AL, HV, TEMPR 3ATA TEMPB/NSTG+352.15DO/ END SUDROUTINE STEAYY (XD,XB, FD) IMPLICIT DOUBLE PRECISION (A-H,0-2) PmWTER (NSTG=~~,NSTG~=~~,P=~~~.DO,NF=~I,N=~~,EFF=~~.~DO] ?ARPMETER (EFFl=0.3DOtvleTOL=1.13-20) DICENSION X (NSTG), T (NSTGI,Y (NSTG?, HL (NSTG], HVf NSTG] ,V (NSTGI, ED (NI DOUBLE PRECISION L (NSTG), L2O COMMON /TEST/L20tVST, V2 COMMON /PARAM/R,QR, HF. QC COK4ON /COLUMi/L,V, Y, HL, HV, TEMPR (NSTGI F=0 +O684DO ZF=O.5DO TF=352. DO CUL VLE(P,TF,ZF,YF,vleTOLi HF=HLIQ (ZF,TF) D=F' (ZF-X8) / [XD-XB) B-F-D L (NSTG)=B X(l)=XD T(li=352.1500 CPL: VLE(P,T(l] ,X(il ,Y(l) .vleTOL) HL(LI=HLIQ(X(lI ,T(i)1 HD=t;.L ( 1 ) HViL)=HVP.P(Y(l} ,T(ll) HV1=HV ( 1 ) X (NSTG)=XB T (NSTG)-352.lsDO CAL; VLl(?, T (>ISTG), X (NSTG), Y (NSTGj, vleTOL1 HL(NSTG) -HLIQ(X(?JSTGl, T (EISTG]) HB=HL [ NSTG) HV (NSTG)=HVAP [ Y {NSTG), T t NSTGH ùO 67 LOOP=L,200 LXECUTE GETXD Z=Xi 1)-Xn fF(D?BS(Z].LT.l.D-lf) GO TO 30 PRINT *,LOOP,':',Lil),~(i) RR=R R=1.00001'RR EXECUTE CETXD XlA=X (1) R=0.99999'RR EXECUTE GETXD Xia=X ( 1) 3Z= (XlA-XiBj / (3, D-j*RR] R=RR-Z/DZ 67 COClTINUE ?RINT +,'STEADY CONVERGENCE WARNING: ' ,CABS [X(1) -XD) 30 DO 40 I=l,NSTG tDiI]=X(I) CD(I+NSTG)=GETM{X(I),T(I),L(I) ,Il 40 CONTINUE FD (NSTG2cl)=O. 30 ED (NSTG2&2)=O. DO FE !N)=O. DO QC= CI, ( i) *D) + [HVl-HD) QR=B*HB+D+HD+QC-F+HF 3=L(Il L20=L (20) '12i=V (2i) VZ=V ( 2 1 RETLTRN REMOTE BLOCK GETXD L (2)=R QC=(L (L)+Dl ' (HVI-HD) QR=BfHB+D'HD+QC-F'HF DO 10 I=NSTG,2,-1 IF ( 1. NE. NSTG) THEN T(I)=T(I+l) CALL VLE(P,TiI) ,X(I) ,Y!I) ,vIeTGL) Y (1)=EFFfY (1) + EFFlfY (I+I) HV(I)=HVAP(Y!I) ,T(I) 1 ENDIF Pl=B+(HE-HV(1)) - QR P2=Ef (XE-Y( 1) ) IF(I.LE.NF) THEN Pl=Pl-F+ (HF-HV(1) ) P2=P2-F' !ZF-Y (1)! ENDIF P3=P2'HV(I) -Pl*Y (1) X(I-l)=X(I) HL(1-l)=HL(I) CALL DNEWTON(Pl1P2,P3,X(I-1),BL(I-11 1 ;(1-1)=?2/ (X(I-l)-Y(Ij1 V(1)=L(Z-1)-B IF(I.LE.MF) V(I)=V[I)*F iO CONTINUE END BLOCK =ND SiJBROUTIME DNEWTON (Pl,PZ, F'3,K, BI IMPLICIT DOUBLE PRECiSION (A-!i,O-Z) PAWETER (P=760.DO,vLeTOL-1.D-10) T=3SZ. DO DO 10 LOOP=l, 150 F=Fl'X - PZfH *P3 IF(DRBS(F).LT.i.D-9) GO TO 30 xA=i.o001+x CUL VLE (PlTl XA, Y, vieTOL) Bl=HLIQ (XA, Tl XA=O.9999*X CFLL VLL ( PlTl XA, Y, vleTOL 1 H2=HLIQ (XA, T 1 DF-Pl-?2+ (Hi-H2)/ (2. D-4+X! X=X-F/DF C.UL VLF(?,T,X, Y,vleTOL! Fi=HLIQ (XIT) iO CONTINUE ?RINT *, 'DNEWTON CONVERGENCE NRKNING : ' ,F 30 RETURN CND DOUBLE PRECISION FUNCTION GETM (XIT, DL,N) IMPLICIT DOUBLE PRECISION (A-HlO-Z) DOUBLE PRECISION Ml, H2 PARPNETER (fr~l=i.07176DO,frBI=8.43939D-4,frC1=8.19166DO) FARPflETER (frA2=1.04812DO,frB2=5.93086D-4, frC2=25.0269DO) PARAMETER (frEl=5.42587D2,frE2=5.76666D2,DLw=O.35DO) PARENETER ~~1=32.042DO,M2=60.096DO,At=O.13d0,Hw=O.GSDO) IF(N.EQ.l) THEN GETM=l .l46ODO XTURN ELSE IE'(N.EQ.21) THEN GETM=4.5517DO ETURN ENDI F = KI / ;frAl-fr21CT-frC1/(fzEI-T!! V2 = M2 / (frA2-frB2'T-frC2/ (frE2-TH X2-1. DO-X Vmix- (X*VI t X2*V2 1 cl. d-3 Ho-DLW 00 10 LCOP=t,lSO C=(DLw-O.~DO'HO) *HO'D~QRT(HO)-DL'Vmix/llO.293DO IF(DABS(F).LT.l.D-12) GO TO 30 DF=O. SDO'DSQRT (Ho)* (3.DOtDLw-Ho) 30-Ho-f / DF 10 CONTINUE PRINT ','GETM: COMRGENCE WAilNING',F 30 CETM= ( SorHw)+At/Vmix RETCRN END SUBROUTINE RDISTIL(N,TIM, XI 0x1 IMPLICIT DOUELE PRECISION (A-Hl0-21 INCLUCE ' RPPAWS .FOR ' OEMENSION X(N),DX(N) ,T (NSTGI ,U (2,401 DIMENSION HV (NSTG),V(NSTG) ,Y (NSTG) ,AWG) DOURLE PRSCISION L (NSTG), HL (NSTGI COMMON /DIFFEQ/U,KK, hF COMMON /VLCDATA/V~~AO,vle~O, vle~i, vle~I, vleC, GAMl,GAM2, ??SAT, P~SAT -,ùZDT data t/ns~g'350.d0/ 3=rJ[I,KK) Q=U{2, KK) '?.ci3 20 10 I=NSTG,!,-L CALL VLE(P,T(Ii ,X(I),U(ij,vleTOL) HL(I)=HLIQ(X(Il ,T(I)I IF[I.NE.NSTG.AVD. 1-NEE1j ?(Il EFFtY(I) * KFPl+'f(IL1l X2=t. DO-X (1) vieA2 = vleA ' vieAO vLeB2 = vieB - vieBO vieCl = -~ieBl*vleB2- vleAl'vleA2 DG~V.I~DX=G~~*(-V~~A~-V~~C+X~~V~~C~~ DGXY2DX=GFM2+(-vl.eBZ-~~leC-X (11 -vieCl 1 ùZDX= i (GAML+x(;) CDGAM~DX]*PISAT+ {-GAMStX2*3GAM2DX)'L2SAT) /F 3TDX = -DZDX/DZDT DELH=T(I)*(~~A+T(I]~[~B+T(I)'(~~C~T(I)*(~~O+T~~~*~~~E~T~I~~~~E)~) -; 1 -3ELH25 A(ï! = GETCF(X(i;,T(I))*DTDX + DELH + hiO-h2C IF(I.CQ-1) CYCLE HV(T)=HVAP(Y(I),T(Il 1 TF ( 1. EQ. NSTG) CYCLE ï(I]=OVFLOW(X(I-NSTGI ,X(I) ,T(I)) iO CONTINUE Lil)=R DX(N) = 1.DO V(NSTG;= (Q+L(NSTG-~~+(E~{NSTG-~)-~L(NSTG)-A(NSTG)*!X(NSTG-~)-X( -NSTG) ] ) ) / (HV(NSTG)-HL(NSTG) -A(NSTGI '"(Y (NSTG)-X(NSTG) 1) 3=L (NSTG-1)-V(NSTG) IF(B,LT.O.DO] THEN DX (NSTG2+2)=-1.E5*8 a=o .DO ZLSt IF(V(NSTG).LT.O.DO) THEN DX(NSTG2+2)=-l.DlO*V(NSTG) B=L(NSTG-1) LLSC OX (NSTG2+2) =O. DO ENDIF L (NSTG) = B DX (NSTG2) = L(NSTG-1) - V(NSTG) - 3 DX (NSTG) = (L(NSTG-1) * (X(NSTG-1) -X (NSTG)) -VINSTG1 + (Y(NSTG) -X (NSTG) -) ) /X (NSTG21 DO 20 I=NSTG- 1,2,- 1 1-1=ï-1 Il-ILl V(I)=(L(I l)'{RL(I 1)-HL(1)-A(I)'(X(I 1)-X[I)))*V(Il)'(HV(n)- -HL(I)-A~I]*(Y ~IL~-X~TJ))))/(HV(I)-HL(II-A(I)*(Y(II-X(I) 11 iF(I.EQ.NF) v(~)=V(I)-F+(hF-HL(I)-A(1)WF-X(Il) )/(HV(I)-HL(1)- -A(I)-(Y(I1-X(1)1 3X(I+iSTG) = L(T-l)-L(I) + V(Il)-V(I) 3x(ï)=(L(:-l)*(X(I -1)-X(1) l-v(Il)*(Y(Il)-X(:) )-TJiI)+(Y(ï)-x(:! 1 )/ -13 ( IANSTG i IF(I.SQ.NF\ THEN DX ( I tNSTG)=DX ( 1 +NSTG)+ F 3X(I)=OX(I)+F+(ZF-X(1))/X(I&NSTG) ZNDIF CONTINUE D=V (2)-R IF(D.LT.O.DO) THEN DX (NSTG2411=-1.35'0 D=G .DO ELSE DX(NSTG2+1;=O.DO ENDIF DX ( l+NSTG)=V (21 -R-0 DX(L)=(V!2)'Y(2)-(R+D)W)) /X(MTG) RETURN END SUBROUTINE VLt (P,Tl XI Y, vieTOL) IMPSICIT DOUBLE PRECISION (A-ElO-Z) DOUBLE PRECISION -MBLZ,LAMB2 1, LOGOF10 PARPmTER (Al= 7.87863D0, 91= l473.liODOI Cl= -43.15130) ?METER (M=6.66040D0, B2= 813.055D0, C2= -140.22DO) PAR.4iMETER (LAMB12= 0.5662iD0, LPMB21= 1.24972DOl PARAiïETER !vle~=O.43379~0, vleB=O .24972DO,LOGOFIO=2.30258509jDO) COMMON /VLYDATA/V~~AO,vle~O, vleAl, vldl,vleC, GIIMI, GAM2, PlSAT,P2SAT -,DZDL X2=!. DO-X 30 10 LOOP=l,lSO Told = T PlSAT = iO.DO** !A1 - BL/ (T+C1)1 PZSAT = 10 -00" (AL - 32/ (T+C2)) vleA0 = ?.DO/:*LAM3L? + vleA*X) vleüO = T.DO/ (1.DO + vLeB'X) vieAl = LAMB12 ' vleAO vieBl = LAM821 ' vleBO vleC = vleAl - vleB1 GAMl = vleAO *wDEXP(X2+~leC) GAM2 = vleBO + DEXP (-X*vleC) Y = GAIYifX-PlSAT/P Y2=GAM2*X2*P2SAT/P Z = Y +Y2 -1.DO IF( DAES(Zj .LT.vleTOL) GOTO 30 DZDT = (YfB1/(T+CL) **2 + YZfE2/ (T+C2)+*2) "LOGOF10 T = Told - Z/DZDT LO CONTINUE 30 RETURN END DOUSLE PRECISION FUNCTION GETCP (XITl 1 IMPLICIT DOUBLE PRECISION [A-HfO-Z 1 PARAMETER (~pZA=-l.19265D6,~p2E=2.37623D6,cp2C=-l.68944D6) PARAMETER (cp2D=5.77813D5,cp2E=-9.36765D4, cp2F=5.81248D3) PARAMETER (dcpA=l.O87865D6,d~pB=-2.035107D6~dcpC=l.432202D6) PARAMETER (d~pD=-4.847282D5,d~pE=7.75919D4,d~pF=-4.711240D3) T=T1/1 .DZ CP2=cp2A+T*(cp2E+T+!cp2C+Tt~cp2O+Tf(cp2E+T*cp2F)1) 1 DCP=dcpA+T' (dcpB+T' id~~Ct?'+(dcpD*TZ (dc?EtT*dcpF) ) ) ) GETC?=CPZ-XtDCP RETURN END DOUBLE PRECISION FUNCTZON HLIQ (X, Tl IMPLICIT DOUBLE PRECISION (A-H,O-Z) PARAMETER (hSA=-1.19265D6,h2B=L. l88ll5d4,h2C=-5.631476'1) PARPMETER (h2D=i.444533d-Lfh2E=-1.87353d-4,.iitF=9.68747d-6) ?ARAiiETER (H2_25=-2.3B5l3D7,hiO=-3. Ï96ÏO2D7, h20=-4.548089D7) ?PHMETER (dhA=l.087865136,unB=-i.O~7554d4,dhC=4.774Oldl) PARAMETER (dhD=-1.21i821d-i,dhE=1.551a38d-4,dhF=-7.85207~i-9) ARAMETER (DELH25=3.7961745D7) B2=T'(H2A+T+(HZa-T*!H2C+T*[H2D+TC(H2E-TC2F-2 25 DH=T' (DHA-T' (DHB+T' (DHC+T4(DHD+Tf(DHE+TbDHE) ) ) ) 1 -DEEH~S !iLIQ=H2 - XCDE + h20 + (h10-h20IbX RETURN 2ND DOUBLE PRECIS ION FUNCTZON HVAP (Y,T) IMPLICIT DOUBLE PRECISION (A-B,O-Z) ?ARPMETER (h~lA=1.42318d8~hvla=-!..68332d6,hvlC=L.11355d4j PARAMETER (hvlD=-3.65697dlrnvlE=5.89891d-2,~vlF=-3.7935ld-5l PPMETER (h~2~=5.58821dB,hv2B=-7.62275d6~h~2C=4.50785d4) FARAMETER ~hv2D=-l.3l326d2,nv2E=l.8768Od-lfhv2F=-l.O6228d-4) PARAMETER (VCll=L.l78D2, VC22=2.2OL3Dî, VCl2=327.3432DO) PARAMETER(VTC11=60338.992DOfVTC22=ll1892.079DO,VTC12=167097.361DO) ?ARAMETER [RTCl=4262.@8903,RTC2=4226.OO6D3, gasR=B. 3l4D3) Y2=1. DO-Y VCMLX=(VC;lfY+VC12*Y2)*YfVC22+Y2*Y2 TCMIX=((VTCll+YtVTC12*Y2}*Y+VTC22*Y2*Y2)/VCMIX ~V1=hvlA+T*!hvla+Tf(hvlC+TtihvlD~Tf(nv~~tTChv~F)1)) HV~=~~ZA+T+(~~~B+T+~~V~C+T+[~V~D+T*(~V~E+T~~V~F)))) HVAP=~~~R*TCMIX*(Y*HV~/RTCICY~~HV~/RTC~;+ !iLIQ!YfT) 2ETURN END SOUBLE PRECISION ETFICTION OVFLOW (H, X, Tl IMPLICIT DOUBLE PRECISION (A-H, 0-2) DOUBLE PRECISION M,Ml, M2, Lw PARAMETER (A1=i.07L7600,31=i3.43939D-4,Cl=8.19166DO,E1=5.42587D2) PARAMETER (AZ=l.O4812DO,B2=5.93006D-4,C2=25.0269D~,22=~.76866D2) FARAMETER (At=O.13dOfHw=0.05D0, Lw=O. 35DO) FARAMETER (Ml=32.04200,M2=60.096DO 1 V1 = Ml / (Al-Bl*T-Cl/(El-T)) VZ = W2 / !A2-52'T-CZ/!E2-T1) VOLmix= (XV1 + (1.DO-X) *V2) * 1. d-3 Ho=M*VOLmix/At - Hw IF(Ho.GT.O.DO1 THEN OVFLOW = (~~O.~~~DO*(LW-~.~EO'RO)~~~.O*DSQRT(HO))/VOhix ZLSE OVELOW=O.DO ENDIF RETURN END NOTATION Antoine coefficients of componentj Coefficients in correlation of Cm as a polynorniai in temperature of degree k Area of tray, 0.13 m' Bottoms flow rate, kmoVmin Coefficients in correlation of AH, as a polynomial in temperahue of degee k Molar specific heat capacity of the liquid mixture. kJhoVK Molar specific heat capacity of pure component j, kJ/kmoüK Molar excess specific heat capacity of componentj in solution, kJ/kmol/K Distillate flow rate, krnoVrnin Feed flow rate. 0.0684 krnoVmin Composition derivative of saturated liquid-phase enthalpy at stage i Ideal gas saturated vapor enthalpy at reference condition. O Idlkmol Real gas saturated vapor enthaipy at reference condition for cornpnent j. Uhol Enthalpy of feed. -3.5750~10' W/kmol Molar excess enthalpy of componentj in solution, kJ/kmol Satmted liquid-phase enthalpy at stage i, kJ/kmol Saturated vapor-phase enthalpy at stage i. klhol Saturated liquid enthalpy of pure component j, Id/kmol Saturated vapor enthalpy of pure component j, Who1 Height of liquid holdup over weir at stage i, m Height of top surface of weir fiom tray surface, 0.05 m Molar enthalpy of mixing in true solution. klhol Molar enthaipy of mixing in ideai solution Idho1 MoIar enthaipy of vaporization of true saturated Iiquid solution klikmol Molar enthalpy of vaporization of pure component j. kJ/kmol Liquid molar overtlow hmstage i, kmol/min Weir length, 0.35 m Liquid molar holdup at stage i, km01 Number of stages in the column. 21 Column pressure, 101.325 kPa Vapor pressure of componentj at the given temperature Tl,kPa Heat extracted hmcondenser, Wmin Heat input to reboiler, kllmin Reflux flow rate, krnoVmin Gas constant, 8.3 14 kJ/kmoUK Critical temperature of pure component j. K Criticai temperature of mixture. K Equilibrium temperature at stage i. K Criticai volume of pure componentj, m3/kmol Critical volume of mixture. m3/kmoi Vapor upflow from stage i, kmovmin Mixture liquid rnolar volume, m3/krnol Liquid molar volume of pure component j at the given temperature. m31kmol Liquid-phase mole fraction of methanol at stage i Actual vapor phase mole Fraction of methanol at stage i Equilibrium vapor phase mole fraction of methanol at stage i VLE calculation: Equilibrium vapor phase mole fraçtion of methano1 at stage i VLE calculation: Equilibrium vapor phase mole fiaction of rnethanol at stage i VLE relation: surn of vapor phase component mole fiactions less 1 Mole hction of methanol in feed, 0.5 Activity coefficient of cornponentj at temperature T, VLE Newton's iteration tolerance. 1O-'' Murphree vapor phase tray efficiency, 0.7 Binary interaction parameter between componentsj and k in determination of VA,= and Tc ,,, 1.O Ai*, A2, Wilson coefficients oi Acentric factor of pure component j O Acentric factor of mixture indices Reboiler Condenser Index of f'd stage with reference to the top stage, 1 1 Strige index, i 5 i In Component index. j = 1.2 Component index. k = !,2 Methanol Isopropanol (PmxPm)Diagonai rnatrix of randomly chosen elements between -1 and +1 (PxP) Diagonal matrix of mndomly chosen eiements between -1 and + 1 l& final time equality consiraint Pertbrmance index Augmented performance index Vector of accumuiated stage lengths of all time stages Best values of stage lengths obtallied at the end of the jh iteration Length of time stage Control vector dimension. 2 Number of iterations per pass Number of state variables. 42 Number of passes Coefficients in the nonlinear equation in column stage i, whose solution is x,-t Number of Mie stages Region size for the control during iterationj Region size for the control at the beginning of pass q Number of random points per iteration Shifting term for the I& ha1tirne equality constraint Absolute value nom of deviation of final state fiom its desired value Time Final tirne mdirnensional control vector Region size for the stage lengths at the beginning of iteration j Region size for the stage lengths at the beginning of pass q Vector of accumulated control variables from ail time stages Best values of control obtained at the end of the jh itmtion ndimensional state vector ih state variable Desired value of the I& state variable at the final time Value of? state at the final time obtained by applying the best known control policy ti11 that instant in the Luus-Jaakola optimization procedure. Dynamic lower bound for control j Dynamic upper bound for control j Region contraction factor Tolerance to which the th final time equality con&nt must be satisfied Region restontion factor Penalty function factor for the ?' final time equality state constraint = 0 Cornmon penalty fûnction factor indices State variable Control variable. or iteration in the U procedure Pass in the W procedure Time stage, iterations in Newton's method Assael, M.J., Trusler. J.P.M. and Tsolakis, T.F., Thermophysical Properties of Fluids: An Infroduction IO their Prediction, imperid Coilege Press, London, 1996, pp. 84-86. Bojkov, B., Hansel, R. and Luus, R., "Application of direct search optimization to optimal control probiems", Hung. J. Ind Chem.. 21, 1993, 177-1 85. Christensen, J.J., Hanks, R.W., and Izatt, R.M., Handbook of Heats of Miring, John Wiley and Sons inc., 1982, p. 1023. Culler. D.E. and Singh. J.P., Parailel Computer Architecture: A HardwareiSofiare Approach. 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