Optimal design of non-dispersive solvent extraction processes

Computers and Chemical Engineering 25 (2001) 267– 285 www.elsevier.com/locate/compchemeng

Optimal design of non-dispersive solvent extraction processes Ana I. Alonso, Andre´ Lassahn, Gu¨nter Gruhn * Department of Process and Plant Engineering, Technical Uni6ersity Hamburg-Harburg, Schwarzenbergstr. 95, 21073 Hamburg, Germany Received 22 May 2000; received in revised form 1 November 2000; accepted 1 November 2000

Abstract The purpose of this paper is to carry out the optimal design of a non-dispersive solvent extraction (NDSX) process for the removal and recovery of chromium from waste waters of surface treatment industries. The objective is to design a process that can separate the waste stream into an environmentally acceptable stream with a low chromium concentration and a stream in which chromium is concentrated for further processing. A superstructure of the problem which is rich enough to account for all potential configurations and connectivity of the system is proposed. The problem is formulated as a MINLP optimisation problem to minimise the total cost of the process subject to design specifications and is solved using an Outer Approximation algorithm. The nonconvexivity of the problem due to bilinear terms in the model equations makes necessary the reformulation of the nonconvex equations into linear inequalities. Three different approaches are tried for this reformulation. A bound tightening strategy based on the application of branch and bound methods to the Outer Approximation algorithm is shown to be the most effective approach. © 2001 Elsevier Science Ltd. All rights reserved. Keywords: MINLP; Membrane extraction; Process optimisation; Process synthesis

1. Introduction The non-dispersive solvent extraction (NDSX) process is an alternative method to conventional solvent extraction processes such as mixer – settler arrangements or continuous countercurrent contacting equipment. The conventional solvent extraction presents many disadvantages: need for dispersion, coalescence and density difference between fluids, problems of emulsification, flooding and loading limits in continuous countercurrent devices. NDSX processes overcome such disadvantages. They are characterised by the stabilisation of the aqueous – organic interface at a porous material, avoiding dispersion or mixing of the two phases. Therefore, they elude the subsequent separation phases which avoids problems such as emulsion formation and the requirement of different densities. The two fluid flows are independent avoiding the problems due to flooding and loading. Ho and Sirkar (1992) presented a general review of the NDSX technology and its applications which can * Corresponding author. Tel.: +49-40-428783041; fax: + 49-40428782992. E-mail address: [email protected] (G. Gruhn).

be found in areas such as metal extraction, organic pollutant extraction, aromatics extraction, pharmaceutical extraction, fermentation product extraction and extractive bioreactors. More recently, Gabelman and Hwang (1999) provide a review of this technology using hollow fibre modules including recent commercial uses of gas/liquid applications. In the area of metal recovery from waste waters, the objective of an extraction process is not only to remove the metal to meet disposal requirements but to recover it due to its value. A semicontinuous process is the most suitable way to achieve both objectives (Alonso et al., 1999). A semicontinuous NDSX process (Fig. 1) comprises at least two hollow fibre modules, one for the extraction and the other for the stripping. In addition, a storage tank for the concentration of the stripping stream is required. The dotted lines represent the replacement of the stripping solution by a fresh one at the end of each batch. In hollow fibre modules, the aqueous and the organic solutions flow continuously, one through the lumen of the fibres and the other one by the shell side. Both phases make contact through the pores of the fibre

0098-1354/01/$ - see front matter © 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 9 8 - 1 3 5 4 ( 0 0 ) 0 0 6 5 2 - 9

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wall. The feed solution enters the extraction module and the carrier in the organic phase extracts the solute at the aqueous –organic interface. The new species formed at this interface diffuses across the microporous wall filled with organic solution (hydrophobic fibres) to the outer side of the fibres. The decontaminated aqueous stream exits the extraction module. The organic stream containing the solute – carrier complex is sent to the stripping module. The solute – carrier complex diffuses across the microporous wall filled with organic solution (hydrophobic fibres) to the stripping – organic interface. At the interface, the carrier is regenerated releasing the solute into the aqueous stripping solution. After that, the organic phase is recycled to the extraction module. The solution coming out of the stripping module is stored in the stripping tank. A semicontinuous process means that after a certain period of time the stripping solution is replaced by a fresh one, while the organic solution remains the same and the extraction solution flows in a continuous mode. It is always good to work with a constant batch time and with a recurrent behaviour of the batches. This facilitates the system control and the working mode. To have a recurrent behaviour, the concentration of solute in the organic solution at the end of a batch must be the same as the concentration at the beginning of the batch. In an optimal situation, the concentration of solute in the organic solution should remain constant during the batch allowing a real continuous run of the extraction process. A more detailed description of the process and its semicontinuous behaviour can be found in Alonso et al. (1999). While the application of the NDSX technology and the behaviour of individual NDSX hollow fibre modules have been extensively studied, very little attention has been directed towards the task of designing systems of multiple modules. The works on design (Prasad & Sirkar, 1992; Reed et al., 1995) are based on the basic chemical engineering equation L =HTU × NTU where L is the module length, HTU is the height of the transfer unit, and NTU is the number of transfer units. This equation is used to size hollow fibre modules for a given extraction application. When the length required is high, modular configurations are proposed, either prefixing the number of modules in series (Prasad & Sirkar, 1992) or the number of parallel lines (Reed et al., 1995). In these modular configurations, the connections between modules are identical. Since an industrial application requires not only the number of needed modules for the extraction and the stripping processes for a fixed particular system configuration but also the best arrangement for the modules and the best distribution of the streams for this arrangement, the problem of optimally designing these systems should be considered. Besides, the operation of a NDSX process involves several important decisions, such as the total duration

of the batch for concentration in the stripping phase, the flowrates of the organic and stripping streams, the volume of the stripping solution as well as the total concentration of the carrier. The process is also subject to several restrictions, including physical constraints, maximum and minimum flowrates and product quality constraints, minimum concentration of the stripping solution at the end of the cycle as required for reuse and maximum concentration of the feed solution for disposal. The design and operating policy of this type of processes is not a trivial task, especially due to the interactions between the two processes, extraction and stripping through the organic phase. Therefore it is necessary to apply systematic mathematical methods of optimisation to carry it out. The case of study selected in this work is the extraction and concentration of Cr(VI) present in waste waters of some surface treatment industries using Aliquat 336 as selective carrier. Aliquat 336 is a quaternary ammonium salt commercialised as a mixture of tri-nalkylammonium chlorides. The selected system has been previously analysed by Ortiz et al. (1996a,b)and Alonso et al. (1999) working in laboratory scale and pilot plant scale. Theoretical analysis through the simulation of the system was done by Alonso and Pantelides (1996) and the optimal selection of operating variables for a NDSX pilot plant is reported by Eliceche et al., 2000. These previous works provide the values of the design parameters and the insight on the influence of the variables on the process performance necessary to face the optimal design of this process. The aim of optimising both the configuration of modules and the operating conditions using nonlinear models gives rise to nonlinear optimisation problems which involve integer and continuous variables. These problems are denoted as Mixed-Integer Nonlinear Programming (MINLP) problems. This paper presents the application of the program system PSANO (Noronha et al., 1997) to solve the MINLP problem for the optimal design of a NDSX process for the removal and recovery of Cr(VI) from wastewaters of surface treatment industries. PSANO uses an extension of the Outer Approximation algorithm (Duran & Grossmann, 1986). The nonconvexity of the problem due to the sum of bilinear terms force the approximation of the nonconvex equations by linear inequalities. This slows down the process of finding the optimum and therefore a new search procedure based on the application of the Outer Approximation on tighter bounds of the variables is proposed.

1.1. Problem description It is desired to synthesise a NDSX process that can separate a waste feed stream into an environmentally

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Fig. 1. Schematic diagram of a semicontinuous NDSX process.

acceptable stream and a stream in which the pollutant is concentrated for further processing and exploitation. The waste feed stream is characterised by a known flowrate, Fe, and a known concentration of a pollutant, Ce,in. An upper bound on the concentration of the pollutant at the outlet of the extraction stream is given by environmental regulations, Ce,out 5upper bound,

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and the concentration of the pollutant in the stripping stream required for reuse is used as lower bound on the stripping concentration, Cs,final ] lower bound. The optimal design task associated with this problem requires identification of the optimal configuration, number and connectivity of the various hollow fibre modules and the optimal operation conditions such as flowrates, volumes and concentrations. To carry out this optimal design task it is necessary to devise an adequate superstructure of the network of modules which is rich enough to account for all potential configurations and connectivity (Fig. 2). In this case, a maximum amount of four modules for the extraction and four modules for the stripping is considered in the superstructure. This limit in the number of modules is due to considerations of clarity in the explanation of the strategy used to solve the optimisation problem. A more general case of M modules can be developed observing the guidelines presented in this work. Each module includes an organic and aqueous inlet and outlet. Commercially available modules with 130 m2 effective surface area (Liqui – Cel Extra-Flow) from the company Celgard LLC are considered in the optimisation problem.

Fig. 2. Superstructure with four extraction modules and four stripping modules.

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Additionally, splitters and mixers are introduced to enable the choice of the optimum structure configuration. This choice is implemented through the use of integer variables, yei, ysi, yoei, yosi. The integer variables are used as switches to enable or disable certain units or connections within the flowsheet. These integer variables are introduced to avoid convergence failures with NLP algorithms when flow rates are allowed to take values equal to zero. The splitters and mixers for the aqueous phases are not only logical, they would be real splitters and mixers depending on the values of the integer variables. For example, the extraction stream ( ——) enters the superstructure through a splitter. In this splitter, the choice between one module (ye1 or ye2=1) or two modules in parallel (ye1 and ye2= 1) has to be done. If one module is chosen the splitter will not be necessary in the final structure while if two modules are chosen, this will be a real splitter dividing the extraction stream into two identical streams. The aqueous outlet solution of both modules is mixed. This will be a real mixer if both modules are present in the chosen structure. The second splitter allows to chose between a module (ye3 or ye4=1), two modules (ye3 and ye4=1) as in the previous case and besides it allows not to chose any more modules (ye5= 1) being the exit of the first two modules the final extraction outlet solution. This possible stream is mixed with the possible exits of module 3 and 4 in the superstructure. The mixer will be a real mixer only if both modules are selected from the superstructure. Therefore to the final structure is allowed to have in total any possible combination from one to four modules with a maximum of two modules in series and in parallel. The organic stream (…) enters the extraction subprocess through a splitter. This first splitter allows the entrance to each module of organic feed coming from the stripping subprocess through the values of yoe10, yoe20, yoe30, yoe40. If any of these organic solutions are not fed into a module, the module can take organic solution from the exit of the other modules. The two first from the two second or viceversa. Both possible organic feeds are mixed in the logical mixers at the entrance of each module. The organic outlet solution from the first two modules is mixed in the superstructure. This mixer will be real if both modules are present in the final structure. After the mixer, a splitter allows the organic solution to fed module 3 or/and 4 (yoe31, yoe41) or to go directly to the exit (yoe50=1). In the same way, the outlet organic solution of modules 3 and 4 is first mixed and then redirected to modules 1 or/and 2 or to the exit. Both possible exists (yoe50 and yoe51) are mixed composing the organic inlet solution to the stripping solution. The superstructure for the stripping subprocess is defined in a similar way. This superstructure allows the flow direction of the organic phase from one module to the next one in co- or counter-current to

the flow direction of the aqueous phase and allows as well the use of organic solution coming directly from the other subprocess in every module.

2. MINLP formulation The presence of discrete decisions represented by the binary variables to account for different modules configurations from the superstructure together with the nonlinear models which describe this type of process leads to a MINLP optimisation problem. The solution of this problem provides the optimal arrangement of modules, mixers and splitters as well as the optimal distribution of streams and it also provides the optimal set of operating conditions. The resulting model can be formulated as the following MINLP model. Z= min f (x,y)

(1)

x,y

subject to h(x,y) = 0 g(x,y) 5 0 y{0,1}m xRn Z constitutes the objective function where vector x represents continuous variables and y corresponds to binary variables. The binary variables, y, are used as switches to enable or disable certain units or connections within the flowsheet. They are allowed to take the 0 or 1 values. The constraints of this problem correspond to the modelling equations of the modules, tank, mixers and splitters, h(x,y), as well as design specifications and logical conditions, g(x,y). The objective of the design problem is to minimise the total cost of the network. For simplicity the total capital cost is considered proportional to the sum of the areas of all modules on the final design so proportional to the total number of modules since the area of a module is fixed to its commercial value, 130 m2. The operating cost of the process are considered as function of the flowrate of the organic phase (flowrates of the other two streams are fixed). A reduction in the total cost due to a reduction in the necessities of raw material in the main process is directly related to the amount of removed solute. Therefore the maximisation of the amount of removed solute is included in the objective function. The maximisation of the amount of removed solute is expressed as the minimisation of the extraction outlet concentration. This concentration is multiplied by a factor of 103 on the expression of the objective function to avoid it to be negligible without conferring it a big significance in comparison with the other two

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terms of Eq. (2). According to these considerations, the objective function is expressed through: 4

Z = % (yei +ysi ) · i=1

F A Ce,out + 0 +103 · . (mol/m3) (m2) (l/h)

(2)

The terms of the objective function are not weighted considering real costs but their economical significance is taking into account through their range of values and weight factors. Only commercial modules are considered for the design so the interfacial area of each module is a fixed parameter, A = 130 m2. The objective function is therefore a linear function of the binary variables and of the continuous variables, organic flowrate, F0, and outlet extraction concentration, Ce,out. The logical conditions for the selection of the process structure from the superstructure (Fig. 2) are expressed by the following constraints. At least one module must exist in the final structure: ye1+ye2]1, Either the second block of modules or the exit must be chosen: ye3+ye4+ye5]1, The aqueous exit (ye5=1) and modules 3 or 4 can not be chosen simultaneously: ye3+ye551, ye4+ ye551, If a module is present (yei=1) it must have an inlet of organic solution, if it is not present, none of the possible inlets can exist either: yoe10+yoe11 =ye1, yoe20+yoe21 = ye2, yoe30+yoe31 =ye3, yoe40+ yoe41= ye4, After the organic outlet of the first two modules, either the inlet to modules 3 or/and 4 or the exit must be chosen: yoe31+yoe41+yoe50]1, If the choice ‘exit’ is chosen in the organic splitters (yoe50 =1, yoe51= 1), the organic solution can not be fed to the modules and vice versa: yoe31+ yoe5051, yoe41+yoe5051, yoe11+yoe5151, yoe21+yoe5151, At least one module must be fed with organic solution coming from the stripping subprocess: yoe10+ yoe20+yoe30+yoe40]1, At least one organic exit must exist in the structure: yoe50+yoe51]1 If module 3 and 4 are not chosen, there is not outlet organic solution from these modules and therefore all the streams after this point do not exist either: yoe115ye3+ye4, yoe215ye3+ye4, yoe515 ye3+ye4, If module 3 or 4 are chosen, there is outlet organic solution from these modules and therefore the inlet to modules 1 or/and 2 or the exit must be chosen: ye35yoe11+yoe21+yoe51, ye45yoe11+yoe21+ yoe51. The logical conditions for the stripping are derived in a similar way.

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The objective is to design the process in such a way that the concentration of the outlet extraction stream is always less than 9.61×10 − 3 mol/m3 (Spanish Law: BOE de 30 April 1986) and the concentration at the end of the batch of the stripping solution in the tank is higher than 76 mol/m3 (Ihobe, 1997). These two constraints are the design specifications. Ce,out 5 9.61× 10 − 3 mol/m3

Cs,final ] 76 mol/m3. (4)

In order to avoid infeasibilities in the solution of the NLP subproblem with fixed values of the binary variables, these design specifications are relaxed. They are allowed to be violated, using penalty variables, E1 and E2, which penalise these violations. The penalty variables are introduced as an additional set of two terms in the objective function. They are multiplied by large weight factors. In this case a value equal to 107 has been chosen as weight factor for the penalty variables. With this value it is assured that structures with violations of the outlet extraction concentration higher than 0.15% have higher objective function values than structures with a module more which do not show violations of the constraints. Ce,out − 9.61× 10 − 3 mol/m3 − E1 5 0 Cs,final − 76 mol/m3 + E2 ] 0

(5)

4

F A Ce,out + 0 + 103 · Z= % (yei + ysi ) · 2 (m ) (l/h) (mol/m3) i=1 +107 ·

E1 E2 + 107 · . (mol/m3) (mol/m3)

(6)

The equalities h(x,y) of problem (1) are the modelling equations of the units in the flowsheet. The model has to provide a good representation of the process. The model will be useful only if it can be solved by current optimisation methods. The model for this type of process has been developed in previous studies (Alonso et al., 1999). It results in a large system of partial differential equations to describe the behaviour of the modules, ordinary differential equations to describe the dynamic behaviour of the tank and algebraic equations for the chemical equilibrium expression (Appendix A). A model with partial differential equations is quite complex for MINLP optimisation purposes. The main objective in the resolution of a MINLP optimisation problem is to obtain an optimal structure of the process. Therefore, the model can be simplified as far as the simpler model is useful to determine the integer vector of the optimal structure. The optimum values of continuous variables can be further refined with NLP formulations of the problem for the optimal structure using more complex models. In order to avoid partial differential equations in the description of the modules, the variation of the concen-

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trations in time is assumed to be negligible compared to the axial variation of the concentration for each phase in each module. So the modules can be described by ordinary differential equations depending on the axial position (Ortiz et al., 1996a,b). (C E,S (C E,S e,s,o e,s,o BB . (t (z

(7)

The second simplification is that in the stripping tank, the variation of the solute concentration in time is considered constant. This simplification is based on the following: as it has been explained in Section 1, in an optimal situation, the extraction process would behave as a continuous process and the outlet concentration of the extraction solution would be constant. This means that the extraction rate is constant, (Ce,in −Ce,out)·Fe = constant (mol extracted/time) and then, according to the global mass balance, the stripping rate is constant and equal to the extraction rate. The stripping rate can be expressed as the difference between inlet and outlet concentrations of the tank times the stripping flowrate, Fs·(C Ts,in −C Ts )=constant. This means that the derivative of the concentration with time in the stripping tank is constant and therefore the equation for the stripping tank can be integrated obtaining this algebraic equation: C

T s,tf

−C

T s,t = 0

F =tf· s ·(C Ts,in −C Ts )tf Vs

(8)



Equilibrium (Alonso et al., 1997)

K= =

C 2Cl · CAlCr · CT 0.6 CCr · C 2AlCl

4(C Ee,in − C Ee (i ))2 · C Eoi(i ) · (CT · 10 − 3)0.6 C Ee (i ) · (CT − 2C Eoi(i ))2

i= 1,...,N

(11)

Stripping Module  Aqueous Solution − (C Ss (i )− C Ss (i− 1)) · Fs · N = A · Km · (C Soi(i )− C So(i )) i= 1,...,N 

Organic Solution

(C So(i )− C So(i− 1)) · Fo · N =A · Km · (C Soi(i )− C So(i )) 

−(C Ee (i )−C Ee (i−1)) · Fe · N =A · Km · (C Eoi(i )−C Eo (i )) i= 1,...,N 

(9)

Organic Solution

(C Eo (i )−C Eo (i−1)) · Fo · N =A · Km · (C Eoi(i )−C Eo (i )) i= 1,...,N

(10)

i= 1,...,N

(13)

Equilibrium (Ortiz et al., 1996a,b)

H=

C S(i ) CCr = Ss CAlCr C oi(i )

i= 1,...,N

(14)

tf · Fs · (C Ts,in − C Ts,tf) Vs

(15)

i= 1,...number of input streams

(16)

Stripping Tank C Ts,tf − C Ts,t = 0 =

T s,tf

where tf is the time of batch and C is the stripping concentration obtained at the end of the batch. With these two simplifications (7) and (8), we have a steady state model of ordinary spatial differential equations where the concentrations vary along the length of the module. This model can be expressed as a model of algebraic equations through the discretisation of the spatial variation of concentrations. The spatial variation of the concentrations within the module is discretised using a first order backward finite difference scheme. The axial dimension is divided into a series of intervals (N) and the derivatives are expressed in terms of truncated Taylor Series. By this finite difference method each ordinary differential equation of the mathematical model is converted into a system of nonlinear algebraic equations. The final algebraic model is the following: Extraction Module  Aqueous Solution

(12)

Mixers Fout = % Fin,i i

Fout · Cout = % Fin,i · Cin,i i

i= 1,...number of input streams

(17)

Splitters Cout,i = Cin

i= 1,...number of output streams

Fin = % Fout,i

(18)

i= 1,...number of output streams

i

(19) Fout,i = ni · Fin

i= 1,...number of output streams –1. (20)

The mathematical equations in this simplified model decrease in complexity but increase in number (24·(N-1) equations more). The superscripts indicate the module or the tank and the subscripts indicate the phase within the module. Therefore C Ee and C Ss represent the concentration of

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chromium in the extraction and stripping phase. C Eo and C So represent the concentration of chromium in the organic phase in the extraction and stripping module and C Ts represent the concentration of chromium in the stripping tank. F are the flowrates, Vs is the volume of the stripping phase in the tank and A, the interfacial area. In the equilibrium expressions, CCl is the chloride concentration, CAlCl is the free carrier concentration, CCr the concentration of chromium, CAlCr the complex carrier-chromium concentration and CT is the total carrier concentration. The term on the right-hand side of the equations for the solutions in the modules corresponds to the convective transport. The term on the left-hand side, represents the transfer of material from the aqueous phases into the organic phase. This transfer of material is given by the product of a mass transfer coefficient, Km, and the concentration difference between the organic-aqueous interface and the bulk of the organic phase. A value for the mass transfer coefficient of Km = 2.2× 10 − 8 m/s was obtained in a previous work from the comparison of the simulated results and the experimental data (Ortiz et al., 1996a,b). As it has been mentioned, the complex species concentration at the interfaces are considered to be the equilibrium complex species concentration. Therefore, these concentrations can be obtained from the chemical equilibrium expressions (11) and (14). Previous studies (Alonso et al., 1997), involving the comparison between simulated and experimental data, showed that the extraction chemical equilibrium can be well described through (11) with a value of 0.2 for the equilibrium constant. For the stripping process, a simpler equation for the equilibrium expression can be obtained. The chloride concentration in the stripping solution is always high in order to favour the stripping process. The experimental studies of the stripping show that the chemical equilibrium can be well described by a distribution coefficient defined as the ratio of the equilibrium concentrations in the aqueous and organic phase. The study of the stripping process gives a value of 3.5 for this parameter when NaCl, 1 mol/l, is used as stripping agent (Ortiz et al., 1996a,b). Numerical simulations with different number of nodes, N, were carried out to determine the value of N which leads to accurate results without requiring excessive computations. The values of the outlet extraction concentration and the final stripping concentration from different values of N were compared with the values obtained with a high N value, N= 200. A difference less than 1% between these results for both concentrations is achieved when N = 12. In order to check the validity of the proposed algebraic model, the simulated results from this model were compared with the simulated results obtained

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using the partial differential model. The accuracy of the partial differential model was verified extensively using actual data from a pilot plant operating with two modules, one for the stripping and one for the extraction, with an interfacial area equal to 19.3 m2 (Alonso et al., 1999). The dynamic behaviour with the algebraic model is calculated as successive steady states at different final times. The simulations with the partial differential model need the value of the initial solute concentration of the organic phase, Co,initial. In order to compare both models, this value is as well introduced as an input data in the algebraic model. The comparison of models was done for two different set of values of the operation variables (Fig. 3). In case (a), Fig. 3a, the graph shows a difference between the extraction and organic results of both models. Under the operation conditions of case (a), the outlet extraction concentration changes with time and therefore the simplification with regard to the variation in time of the solute concentration in the stripping tank (8) is not very accurate. For case (b), Fig. 3b, in contrast, the outlet extraction concentration remains almost constant and there is a very good agreement between both models, partial differential model and algebraic model. Therefore, the algebraic model is valid for simulations in which the extraction and stripping rate are constant. This is the optimal behaviour we are looking for and therefore, the algebraic model is valid for our optimisation purposes. The continuous optimisation variables are selected on the basis of preliminary studies of the performance of the process (Eliceche et al., 2000). The flowrate of the organic phase is shown having major influence on the outlet concentration while the volume of the stripping tank and stripping flowrate do not present a high influence. Therefore, in this work, a value of 2000 l/h equal to the value of the extraction flowrate has been set for the stripping flowrate. The volume of the stripping tank, Vs, is combined for simplicity with the batch time, tf, in only one variable, tf/Vs, which can be estimated through the total mass balance of the process. The total amount of extracted chromium must be equal to the total amount of chromium in the stripping tank. Fe·(Ce,in − Ce,out)=

tf ·(C T − C Ts,t = 0) Vs s,tf

(21)

being Ce,in = 1.234 mol/m3, Fe = 2000 l/h, C Ts,t = 0 =0 mol/m3 and assuming that the optimal solution is going to give values for Ce,out and C Ts,t = tf very close to the design values, 9.61×10 − 3 and 76 mol/m3, respectively, a value of tf/Vs equal to 0.031 h/l can be estimated.

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Fig. 3. Comparison of algebraic, N= 12 and partial differential model. (a) CT= 600 mol/m3; Co,initial =56 mol/m3; Ce,in =1.234 mol/m3; Fe = 22 l/h; Fo = 90 l/h; Vo = 15 l; Fs = 40 l/h; Vs = 22.5 l; tf =120 h. (b) CT= 600 mol/m3; Co,initial =81.85 mol/m3; Ce,in =1.234 mol/m3; Fe = 87.6 l/h; Fo =180 l/h; Vo = 90 litres; Fs = 40 l/h; Vs = 21.6 litres; tf = 16 h.

The continuous optimisation variable is therefore the total organic flowrate, Fo (minimum value 100 l/h, maximum value 4000 l/h) and furthermore, we have two penalty functions, E1 and E2. The binary optimisation variables are ye1, ye2, ye3, ye4, ye5, yoe10, yoe20, yoe30, yoe40, yoe50, yoe51, ys1, ys2, ys3, ys4, ys5, yos10, yos20, yos30, yos40, yos50, and yos51.

3. MILP reformulation and solution Grossmann and Kravanja (1995) presents an overview of different MINLP techniques. They point out that the design of separation systems is normally solved with the Generalised Benders Decomposition algorithm or with the Outer Approximation algorithm. In this work, a solution strategy based on an extension of the Outer Approximation algorithm through the program system PSANO has been used (Noronha et al., 1997). PSANO is based on common computer aided tools which are normally used in the total process of design and engineering (Gruhn & Noronha, 1998). In the Outer Approximation approach, the MINLP problem is divided into Nonlinear Programming (NLP) and Mixed Integer Linear Programming (MILP) subproblems. These subproblems are alternately solved until the final solution is attained. In PSANO, the NLP subproblem is solved by using a SRQP method within the equation-oriented flowsheeting package SPEEDUP (Aspen Technology, Inc.). With this, the active structure is evaluated and optimised. The software CPLEX

(ILOG, Inc.) has been used for solving the MILP subproblem. By the solution of the MILP subproblem a new structure is proposed, which is defined by a specific combination of the topological variables yi. Since the MILP subproblem can only consist of a set of linear equations, an intermediate step is required. In this step, the nonlinear equations of the NLP subproblem are linearised. The first NLP subproblem is solved by relaxing the binary variables in order to enable an adequate approximation of the whole superstructure. This large nonlinear problem often leads to numerical difficulties to converge. In order to overcome these numerical difficulties, good starting points and adequate bounds for the binary variables avoiding integer solutions have been used. Subsequent NLP-subproblems do not lead to such numerical difficulties because they optimise fixed structures proposed by the MILP subproblems. The required information for the formulation of the MILP problem is extracted from the NLP-problem solved by SPEEDUP (equations, derivatives, Lagrangian multipliers). All nonlinear equations are linearised by first-order Taylor series approximation at the current operation point and are relaxed according to the values of the correspond Lagrangian multipliers. The resulting solution of the MILP subproblem is then evaluated. This solution yields a lower bound and the next NLP subproblem is solved with fixed binaries yi. This solution is an upper bound. The program terminates when the lower and upper bounds are within a specific tolerance. The software SPEEDUP and CPLEX were used on a work station, IBM 3AT RS6000 and on a personal

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computer, Pentium II, 233 MHz, respectively. The NDSX optimisation problem is formulated in 717 algebraic equations with 22 binary decision variables (eight binary variables are fixed by the values of the others (3)) and 718 continuous variables which mainly result from the axial discretisation of the modules. The problem is nonconvex owing to the bilinear constraints (9), (10), (12), (13), (17) and (20). The nonlinear equation (11) is convex (CT is fixed and equal to 600 mol/m3) and Eq. (15) is linear (tf/Vs is fixed and equal to 0.031 h/l and Fs is fixed and equal to 2000 l/h). In the formulation of each MILP problem, the nonconvex constraints have not been included to avoid that the linearisations of these nonconvex equations cut into the feasible region of candidate integer points. A reformulation of these constraints is therefore required to handle the nonconvexity of the constraints and make the problem linear. The reformulation will expand the MILP problem in terms of number of equations but the problem becomes more focused. The bilinear equations which describe the splitters and mixers (17) and (20) can be replaced by a series of linear inequalities. In the splitters, the split fraction, ni, is always defined by the values of the integer variables because the outlet streams are assumed to have the same flowrate. For the first aqueous splitter, Fout1 will be zero when ye1 is zero, it will be equal to Fin when ye1 is 1 and ye2 is zero and will be half Fin when both ye1 and ye2 are 1. These three possibilities are included in the bilinear equation (22). Fout 2 can be calculated from the linear equation of the total mass balance (19).

275

2 · Fout1 ] Fin − M1 · (1− ye1) −M1 · (1− ye2) 2 · Fout2 5 Fin + M1 · (1− ye2)+M1 · (1− ye1) 2 · Fout2 ] Fin − M1 · (1− ye2)−M1 · (1− ye1) Fout1 5 M1 · ye1 Fout2 5 M1 · ye2.

(23)

When ye1=1 and ye2= 0, the first inequality forces Fout1 to be equal to the inlet value, Fin, while the other inequalities except the last one are inactive. The last one forces Fout2 to be equal to zero. Something similar happens when ye1 is 0 and ye2 is 1. When both integer variables are one, only from the third to the sixth inequalities are active forcing both outlet flowrates to be half the inlet flowrate. The bilinear equation of the mixers, can be reformulated in a similar way. The value of the outlet concentration depends on the existence of the input streams which will be determined by the values of the binary variables. Therefore, the following nonlinear equation (24) can be replaced by big-M formulations (25) which will be active depending on the value of the integer variables. (M2 = 5).

Fout ·Cout = Fin1·Cin1 + Fin2 · Cin2

(24)

Cout 5 Cin1 + M2 · (1− ye1)+ M2 · ye2 Cout ] Cin1 − M2 · (1− ye1)− M2 · ye2 Cout 5 Cin2 + M2 · (1− ye2)+ M2 · ye1 Cout ] Cin2 − M2 · (1− ye2)− M2 · ye1 2 · Cout 5 Cin1 + Cin2 + M2 · (1− ye1)+ M2 · (1− ye2) 2 · Cout ] Cin1 + Cin2 − M2 · (1− ye1) − M2 · (1− ye2).

Fout1 ·(ye1+ye2)=ye1·Fin

(22)

The values of the outlet streams are defined by the values of the binary variables and therefore the bilinear equation (22) can be replaced by big-M formulations (23). M1 is a big parameter which makes the inequalities redundant for certain values of the binary variables. In Eq. (23), M1 =3 × 103. Fout1 ]Fin − M1 · (1− ye1) −M1 · ye2 Fout2 ]Fin − M1 · (1− ye2) −M1 · ye1 2 · Fout1 5Fin + M1 · (1 −ye1) +M1 · (1 −ye2)

(25)

All mixers and splitters in the superstructure can be reformulated using Eqs. (23) and (25). In the model of the extraction module, there are two bilinear equations (9) and (10). The nonlinearity of the equation which refers to the aqueous phase (9) arises from the product of the aqueous flowrate and the aqueous concentrations. −(C Ee (i )− C Ee (i− 1)) · Fe · 12 = 10.296 · (C Eoi(i )− C Eo (i ))

i= 1,....12.

The value of the flowrate is fixed depending on the values of the binary variables: Fe = 1000 l/h (two

A.I. Alonso et al. / Computers and Chemical Engineering 25 (2001) 267–285

276

modules in parallel) (ye1= 1 and ye2=1), Fe =2000 l/h (one module) (ye1=1 and ye2=0 or ye1 =0 and ye2= 1). Therefore a similar approach to the one done for the mixers and splitters can be applied. Linear inequalities with a fixed value for the flowrate are introduced. These inequalities will be active depending on the value of the integer variables (M3 =105). − (C Ee (i )− C Ee (i− 1)) · 1000 · 12 510.296 · (C Eoi(i )− C Eo (i )) + M3 · (1 −ye1) +M3 · (1− ye2) − (C Ee (i )− C Ee (i− 1)) · 1000 · 12 ]10.296 · (C Eoi(i )− C Eo (i )) −M3 · (1 −ye1)

prove with the number of iterations and the MILP solutions yield a nondecreasing sequence of lower bounds. Equations for the organic solution in the modules (10) and (13) are first estimated by the relaxation proposed by Quesada and Grossmann (1995). The procedure consists in applying the bounds of the variables involved in the bilinear terms to the estimators. The bounds of the variables in the bilinear terms in Eqs. (10) and (13) are: 05Co(i )-Co(i-1)5300 mol/ m3, 05 Fo(module)5 4000 l/h. With these bounds the linear estimators for Eq. (10) according to the approach of Quesada and Grossmann are: 10.296/12 · (C Eoi(i )− C Eo (i ))] 0

−M3 · (1− ye2)

10.296/12 · (C Eoi(i )− C Eo (i ))

− (C Ee (i )− C Ee (i− 1)) · 2000 · 12

]300 · Fo + 4000 · (C Eo (i )− C Eo (i− 1))− 300 · 4000

510.296 · (C Eoi(i )− C Eo (i )) +M3 · (1 −ye1)

10.296/12 · (C Eoi(i )− C Eo (i ))5 300 · Fo

+M3 · ye2

10.296/12 · (C Eoi(i )− C Eo (i ))

− (C Ee (i )− C Ee (i− 1)) · 2000 · 12

54000 · (C Eo (i )− C Eo (i− 1)).

]10.296 · (C (i )− C (i )) −M3 · (1 −ye1) E oi

E o

− M3 · ye2 − (C Ee (i )− C Ee (i− 1)) · 2000 · 12 510.296 · (C Eoi(i )− C Eo (i )) +M3 · ye1 +M3 · (1− ye2) − (C Ee (i )− C Ee (i− 1)) · 2000 · 12 ]10.296 · (C Eoi(i )− C Eo (i )) −M3 · ye1 −M3 · (1− ye2)

(26)

For the second nonlinear equation (10), this approach is not possible because the organic flowrate in each module not only depends on the value of the integer variables but on the value of the total organic flowrate. This is an optimisation variable and therefore unknown. A similar reformulation can be done for the nonlinear equations of the stripping modules (12) and (13). The addition of these type of inequalities (23), (25) and (26) will improve the calculations of the MILP subproblem because equations with bilinear terms from splitters, mixers and aqueous solutions in the modules are replaced by systems of linear inequalities which provide the same information to the model. After this reformulation, only the nonlinear equations (10), (11) and (13) remain. The extraction equilibrium equation (11) is convex and it is approximated by its linearization at the NLP solution point in each iteration. The linearisations are accumulated and therefore the approximation will im-

(27)

Another four estimators can be derived similarly for Eq. (13). The lower bound of the organic flowrate in the modules is zero because if a module is not selected from the superstructure, its flowrate is zero. If a module is selected, the minimum flowrate in the module is the lower bound of the total organic flowrate divided by the maximum number of modules (100/ 4= 25 l/h). This lower bound can not be used in Quesada and Grossmann estimation because the third inequality in Eq. (27) would be 10.296/12·(C Eoi(i )C Eo (i ))5 300·Fo + 25·(C Eo (i )-C Eo (i-1))− 300·25. This inequality would force the difference of organic concentration between the interface aqueous –organic and the organic bulk to be less than zero when the module is not selected. This is not feasible and therefore the solution of the MILP subproblem would be forced to have always a structure with the maximum number of modules. This fact cuts out the optimum. Due to the characteristics of the process, it is known that in the extraction modules, the organic concentration increases with the axial position, the organic concentration at the interface decreases with the axial position and the difference of organic concentration between two points decreases with the axial position. In the stripping modules, the organic concentration decreases with the axial position, the organic concentration at the interface increases with the axial position and the difference of organic concentration between two points decreases with the axial position. This behaviour of the system offers more inequalities to restrict the feasible solutions.

A.I. Alonso et al. / Computers and Chemical Engineering 25 (2001) 267–285

C Eo (i )]C Eo (i− 1)

C Eo (i )− C Eo (i− 1)5

C Soi(i )5 C Soi(i− 1) C Eo (i )5C Eo (i− 1)5 C Eo (i −1 ) − C Eo (i −2)

C Eo (i )− C Eo (i− 1)

C So(i ) 5C So(i−1)

5

C Soi(i )] C Soi(i− 1) C So(i −1)−C So(i )5 C So(i −2) −C So(i −1)

4. Bounds tightening strategies In order to speed up the process of optimisation, better approximations of Eqs. (10) and (13) had to be developed, exploiting the special mathematical structure of the problem. Comparing Eqs. (9) and (10), the following equation can be written C Eo (i )−C Eo (i− 1)=

Fe E (C (i −1) −C Ee (i )). Fo e

(29)

Using the possible values of Fe (1000 or 2000 l/h) and the bounds of Fo (Fomin, Fomax), the following type of inequalites can be written (example: first extraction module, M4 =4× 102):

2000 · 4 E (C e (i− 1)− C Ee (i )) Fomin

1000 · 4 E (C e (i− 1)− C Ee (i ))+ M4 · (1− ye2) Fomin +M4 · (1− ye1)

(28)

This reformulation of the MILP problem (23), (25), (26), (27) and (28) increases the number of algebraic equations to 2239 and the number of continuous variables to 2240 due to the necessity of 1558 slack variables for the inequalities. The solution of the MINLP problem with the reformulation of the equations with bilinear terms (23), (25), (26), (27) and (28) gives unsatisfactory results (Table 1). The first MILP solution is far away from the optimum, resulting in a structure that can not fulfil the design specifications (high values of the objective function because the penalty variables, E1 and E2 are not zero). This poor first point will force the procedure to carry out too many iterations before the optimal solution is found. The approximation of Eqs. (10) and (13) does not constrain enough the feasible area of MILP solutions generating a great number of infeasible structures before the accumulation of integer cuts (removal of the last solution structure in the next iteration) through the iterations restricts the feasible area to feasible structures. The reason can be that this type of relaxation is proposed in the frame of a branch and bound algorithm (Quesada & Grossmann, 1995). This algorithm tightens the bounds of the variables involved in the bilinear terms while the procedure goes forward. In the case of the Outer Approximation algorithm, the bounds for the organic concentration and the organic flowrate in the MILP subproblem remain constant.

277

C Eo (i )− C Eo (i− 1)]

1000 E (C (i− 1)− C Ee (i )) Fomax e

C Eo (i )− C Eo (i− 1) ]

2000 E (C (i− 1)− C Ee (i ))− M4 · (1− ye1) Fomax e + M4 · ye2

(30)

being Fomin = 100 l/h and Fomax = 4000 l/h. The value 4 that appears in the first two inequalities corresponds to the maximum number of extraction modules. So being 100 l/h the lower value of the total flowrate, 100/4 l/h is the lower value of the organic flowrate in a module. Similar inequalities can be derived for the stripping modules. These inequalities can be further constrained if the value of the binary variables for the organic streams are considered. For example, if only two extraction modules are present in the structure and both modules receive organic from the stripping subprocess, the lower organic flowrate in each module is 100/2 l/h. When the organic streams is circulated from one module to the other, the lower organic flowrate is 100 l/h. These possibilities are determined by the values of the binary variables yoi. These considerations tighten the constraints represented in Eq. (30) providing a better estimation of Eqs. (10) and (13). The inequalities presented in Appendix B are introduced and in this way, the nonlinear solution space of the entire superstructure is surrounded by a more rigorous outer linear approximation. In order not to increase the number of equations to be solved too much, the inequalities (Appendix B) are applied only to the first and last interval in which the modules are divided. This new reformulation of the MILP problem comprises 3615 algebraic equations and 3616 continuous variables. The solution of the optimisation problem including the inequalities (30) and in Appendix B gives good results (Table 2). The optimum is found in the second iteration but 11 iterations are necessary in order to check that no other configuration can give a better value of the objective function. Fig. 4 shows the optimal structure and in Table 3, the optimum values of the concentrations and

A.I. Alonso et al. / Computers and Chemical Engineering 25 (2001) 267–285

278

flowrates of each module in the optimal structure are presented. Two modules in serie are selected for the extraction and four for the stripping. The organic and aqueous phases flow in countercurrent flow between the modules in both subprocesses. The outlet extraction concentration is 8.316× 10 − 3 mol/m3 and the stripping concentration is 76.080 mol/m3. The inlet organic concentration into the extraction subprocess is 70.514 mol/m3 and the organic flowrate is the minimum flowrate allowed, 100 l/h. Although the optimal solution is found in an early stage during the search, it takes a long time to verify the solution. The most efficient manner to reduce the number of iterations to assure the optimum is to introduce even tighter bounds of the variable total organic flowrate, Fo. Following the procedure of branch and bound methods for global optimization (Ryoo & Sahidinis, 1995; Quesada & Grossmann, 1995; Smith & Pantelides, 1999), the application of the same principles is tried in the solution of the MILP subproblem. Branch and bound methods employ lower and upper bounds of the optimal objective function value over subregions of the search space. Certain subregions are further divided while others are excluded from considerations based on optimality and feasibility criteria. The bounds become tighter as the search is confined to smaller subregions (Ryoo & Sahidinis, 1995). The procedure applied in this work (Fig. 5) consists on resolving the MILP subproblem, after the second iteration, over two subregions: one to the right and one to the left of the solution value of the total or-

Table 1 Solution of the MINLP problem using Quesada–Grossmann approach Iter.

ZNLP×10−3

ZMILP

y-Combinationa

× 10−3 1

0.667342

0.6200

2

22501.217677

0.6200

3

22820.524603

0.6200

4

22831.395464

0.6200

5

22515.094925

0.6261

6

01100 01100 10100 10100 01010 11001 01100 10100 10010 01100

00 00 01 10 00 10 00 10 10 00

01 01 00 00 01 10 01 00 00 01

10 10 10 10 00 00 10 01 00 10

00 00 00 00 10 00 00 00 10 00

10 10 10 11 10 10 10 01 11 10

–

a y-Combination order: ye1, ye2, ye3, ye4, ye5, yoe10, yoe11, yoe20, yoe21, yoe30, yoe31, yoe40, yoe41, yoe50, yoe51, ys1, ys2, ys3, ys4, ys5, yos10 yos11, yos20, yos21, yos30, yos31, yos40, yos41, yos50, yos51.

ganic flowrate from the previous NLP subproblem. Then a new MILP subproblem is solved for each subdivision following by the corresponding NLP subproblem. In the subregions, the approximation becomes tighter and the MILP solution comes closer to the NLP solution. The process is repeated for each subdivision until the subregions present lower bounds that either exceed or are sufficiently close to the best found solution of the problem (best current UB). The search procedure should choose as next region to be divided the region with the lowest upper bound (the lowest solution from the NLP subproblems). In this way, a better upper bound can be introduced in the solution of the MILP subproblems of other regions allowing larger regions to be excluded from further consideration. With this procedure the optimum is found in 5.5 h within five iterations (Table 4). The first nonrelaxed NLP problem gives as solution a value for the total organic flowrate that is the lower bound of this variable. In this case, the first two subregions consist in; one point, Fo = 100 l/h, and a large region with the rest of possible values considering 1% gap between both regions, 1015Fo 5 4000 l/h. In the second iteration, the large region of values of the total organic flowrate can be excluded because according to the solution of the corresponding MILP subproblem, there is no configuration with a lower objective function than the current upper bound (0.888316). This fact confines the search to the minimum region (only one point, Fo = 100 l/h) and only two more iterations are needed to check the optimum found in the second iteration. The optimal solution is the same as the one represented in Fig. 4.

5. Conclusions The optimal design of a NDSX process for the removal and recovery of chromium from waste waters of surface treatment industries has been presented in this paper. The design problem developed in this work allows not only the calculation of the number of modules necessary to carry out a desired extraction but its optimal arrangement and connections. Besides, both extraction and stripping subprocesses are optimised as a whole, calculating the optimal conditions, concentration and flowrate, of the organic phase which connects both subprocesses. The optimisation problem is formulated as a MINLP problem and includes convex equations and nonconvexities introduced by the bilinear terms in the model equations. An Outer Approximation approach with a reformulation of the nonconvex equations has

A.I. Alonso et al. / Computers and Chemical Engineering 25 (2001) 267–285 Table 2 Solution of the MINLP problem adding Eq. (30) and equations in Appendix B Iter.

ZNLP ×10

ZMILP×10−3

y-Combination

01010 11110 10010 11110 01010 11110 01010 11110 10010 11110 10010 11110 10010 11110 10010 11110 10010 11110 10010 11110 01100 11110

−3

1

0.667342

0.887232879

2

0.888316

0.88732238689

3

0.888323

0.88736618047

4

0.889472

0.8873869204

5

0.889480

0.88748297864

6

0.888424

0.88749023084

7

0.888429

0.88749946209

8

0.889597

0.88750640062

9

0.889603

0.88753834133

10

0.888419

0.88756783141

11

0.889591

0.88933732466

00 01 01 10 00 01 00 10 01 10 01 10 10 10 10 10 01 10 10 10 00 10

01 01 00 10 10 01 10 10 00 01 00 10 00 01 00 10 00 10 00 10 10 10

00 10 00 01 00 10 00 01 00 10 00 01 00 10 00 10 00 10 00 10 01 01

10 10 10 01 10 10 10 01 10 10 10 10 10 10 10 01 10 10 10 10 00 01

10 10 10 01 11 10 11 01 10 10 10 01 11 10 11 01 10 11 11 11 01 01

Total CPU time: 50628.8 s =14 h

Fig. 4. Optimal structure for the NDSX process.

been applied to the solution of the optimisation problem. The computational work required by the Outer Approximation algorithm to solve nonconvex optimisation problems with linear reformulation of the nonconvex equations has been shown to depend largely on the quality of the linear reformulation. The

279

bounds of the variables involved in the nonconvex terms strongly determine the quality of the reformulation. Therefore, bounds tightening strategies have been applied to the solution of the NDSX optimisation problem to accelerate the Outer Approximation algorithm. The linear reformulation of the nonconvex equations has been carried out making use of the characteristics of the problem and its mathematical structure. First, a general reformulation like the one used by Quesada and Grossmann (1995) has been used. In this particular case, this approximation was not effective and therefore an additional reformulation with tighter bounds for the variables in the bilinear terms had to be implemented. This tightening of bounds is carried out first by introducing stricter linear constraints depending on the values of certain binary variables which account for the structure of the process. The results show that this strategy improves the speed of the search. Further improvements have been obtained with an approach based on the division of the interval of values of the variable involved in the bilinear terms and the solution of the iteration MILP –NLP of the Outer Approximation algorithm in each of these divisions. The results show that by executing the division of the interval of values of the variable involved in the bilinear terms, large parts of the search region over which the objective function takes only values above a known upper bound are eliminated. It is shown that with this approach the total number of iterations and the computational time for the solution decrease to a great extent. The application of the division of the search interval strategy on the optimisation of other type of processes where higher number of variables are involved in the nonconvex equations is in principle possible but its success and advantages should be checked. A higher number of variables would involved a more complex search for each variable and the number of MILP –NLP problems for each internal division may reach high values which could make the application questionable. The success of the division of the search interval has been shown in this paper for the optimal design of non-dispersive solvent extraction processes but a more general study should be done to check the applicability of this strategy to more general optimisation problems.

Acknowledgements The authors are grateful to the Alexander von Humboldt Foundation for the financial support for this work.

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280

Table 3 Optimal values of concentrations and flowrates for the optimal structure in Fig. 4 Inlet

Aqueous S. E1 E2 S1=S2 S3=S4 Organic S. E1 E2 S1=S2 S3= S4

Outlet

Flowrate (l/h)

Concentration (mol/m3)

Flowrate (l/h)

Concentration (mol/m3)

2000 2000 1000 1000

1.234 0.266 76.080 76.632

2000 2000 1000 1000

0.266 8.316·10−3 76.632 77.306

100 100 50 50

75.666 70.514 81.542 95.028

100 100 50 50

95.028 75.666 70.514 81.542

Fig. 5. Schematic representation of the procedure of applying branch and bound methods to the Outer Approximation algorithm. Table 4 Solution of the MINLP problem applying branch and bound methods to the Outer Approximation algorithm Iter.

ZNLP×10−3

Fo Interval

ZMILP×10−3

1 2

0.667342 0.888316

100–4000 100 101–4000 100 100

0.8872322879 01010 00 01 00 10 10 11110 01 01 10 10 10 0.8877135656 10010 01 00 00 10 10 11110 10 10 01 01 01 no solution with a lower value than 0.888316 0.8880658207 01010 00 10 00 10 11 11110 01 10 10 10 10 no solution with a lower value than 0.888316

3 0.888323 4 0.888424 Total CPU time: 19722.5 s = 5.5 h

y-Combination

A.I. Alonso et al. / Computers and Chemical Engineering 25 (2001) 267–285

Appendix A. Partial differential model of NDSX modules — Mathematical model of general application for NDSX processes (Alonso et al., 1999)



−

A V Sms (C Ss (C Ss = + Km(C Soi − C So) FsL (t (z FsL

t=0, C Ss = Cs,initial; z =0, C Ss = C Ts

Extraction Module Aqueous Solution



V Eme (C Ee (C Ee A − = + Km(C Eoi −C Eo ) FeL (t (z FeL

Aqueous Solution



Organic Solution (C So A V Smo (C So =− + K (C S − C So) FoL (t (z FoL m oi

t= 0, C Ee =C Ee,in; z =0, C Ee =C Ee,in

t= 0, C So = Co,initial; z =0, C So = C Eo (z= L) 

Organic Solution V Emo (C Eo (C Eo A =− + Km(C Eoi −C Eo ) FoL (t (z FoL t= 0, C Eo =Co,initial; z =0, C Eo =C So(z =L)





Stripping –Equilibrium H=

C Ss C Soi

Extraction –Equilibrium K=

4C Eoi(Ci − C Ee )2 (CT ×10 − 3)0.6 C Ee (CT −2C Eoi)2

Stripping Module

Stirred Tanks dC Ts Fs T = (C s,in − C Ts ) dt Vs t= 0, C Ts = Cs,

initial

.

281

282

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Appendix B. Inequalities to tighten the upper and lower bounds of the organic flowrate in the modules according to the values of the binary variables of the organic stream. — Inequalities for the first extraction module for the first and last interval of the module M4 = 4×102 i = 1 and 12

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283

284

A.I. Alonso et al. / Computers and Chemical Engineering 25 (2001) 267–285

A.I. Alonso et al. / Computers and Chemical Engineering 25 (2001) 267–285

Similar inequalities can be derived for the other extraction modules and stripping modules.

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