Efficient Optimization-Based Design of Distillation ... - ACS Publications

Ind. Eng. Chem. Res. 2009, 48, 6749–6764

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Efficient Optimization-Based Design of Distillation Processes for Homogeneous Azeotropic Mixtures Korbinian Kraemer, Sven Kossack, and Wolfgang Marquardt* AVT-Process System Engineering, RWTH Aachen UniVersity, 52056 Aachen, Germany

Rigorous optimization is a valuable tool that can support the engineer to tap the full economic potential of a distillation process. Unfortunately, the solution of these large-scale discrete-continuous optimization problems usually suffers from a lack of robustness, long computational times, and a low reliability toward good local optima. In this paper, the rigorous optimization of complex distillation processes for azeotropic multicomponent mixtures is achieved with outstanding robustness, reliability, and efficiency through progress on two levels. First, the integration within a process synthesis framework allows a reduction of the complexity of the optimization superstructure and provides an excellent initialization by shortcut evaluation with the rectification body method. Second, the reformulation as a purely continuous optimization problem enables a solution with reliable and efficient NLP solvers. Moreover, the continuous reformulation considers a particular tight column model formulation such that the introduction of special nonlinear constraints to force integer decisions could be largely avoided. A careful initialization phase and a stepwise solution procedure with gradually tightened bounds facilitate a robust and efficient solution. Different superstructures for the tray optimization of distillation columns are tested. The methods are illustrated by three demanding case studies. The first case study considers the conceptual design as well as the rigorous optimization of a curved boundary process for the complete separation of an azeotropic four-component mixture. The rigorous optimization of a pressure swing process for the separation of a highly nonideal five component mixture is presented in the second case study. Finally, the third case study covers the rigorous optimization of an extractive separation within a complex column system. All case studies could be robustly solved due to the favorable initialization phase. The continuously reformulated problems required significantly less computational time and identified local optima of better quality as compared to the mixed-integer nonlinear programming techniques (MINLP) solution. 1. Introduction The costs of a chemical process are often dominated by the costs for the separation and purification of the products. Distillation, which is still the major separation and purification method as stated by Widagdo and Seider,1 is an expensive unit operation in terms of capital and operating costs.2 It is therefore desirable to design economically optimized distillation processes, which can potentially save significant amounts of financial and energetic resources. Distillation process design in industrial practice is usual conducted by tedious simulation studies that require detailed design specifications in an early design phase. Guided by heuristics like those from Douglas,3 these iterative solution procedures result in a high manual effort, and in addition, no guarantee concerning the quality of the solution can be given. An economically optimal design of a distillation process can be obtained by rigorous process optimization, where the columns are represented by detailed tray-by-tray models. The resulting large-scale nonlinear optimization problems are discretecontinuous by nature and are usually solved with mixed-integer nonlinear programming techniques (MINLP). While the optimization of a single distillation column is nontrivial due to the discrete nature of the tray-by-tray column model and the nonlinearity and nonconvexity of the underlying thermodynamics, the optimization of distillation processes with recycles poses an even more difficult challenge. Large and complex superstructures have to be defined and solved if all possible splits and column configurations are to be considered.2 Given this complexity, it becomes clear that these optimization problems * E-mail: [email protected].

are computationally expensive and that the quality of the final solution strongly depends on the specified initial values.4 As a consequence of these difficulties, the examples of rigorous optimization of distillation processes in the literature are confined to different assumptions, simplifications, or limitations. Viswanathan and Grossmann5 were the first to publish a general MINLP formulation for the tray optimization of single columns. They apply this formulation to an ideal binary mixture and later to azeotropic mixtures.6 Duennebier and Pantelides4 have extended this method to include multicolumn systems and an economic objective function but only consider ideal separations. Bauer and Stichlmair2 have developed a rigorous optimization procedure for azeotropic separations that first generates separation splits based on preferred separations and complements this sequence generation with a MINLP tray optimization. However, they only look at ternary mixtures and report long computational times. In a series of papers, Barttfeld and Aguirre7,8 develop a method for the optimal synthesis of ternary zeotropic distillation processes based on the concept of reversible separation. They solve the MINLP problem efficiently due to a preprocessing phase but cannot handle sharp splits, which are not allowed in reversible separation schemes. The tray optimization formulations in the works mentioned above all suffer from the numerical difficulties inherent to largescale MINLP optimization: lack of robustness, long computational times, and sensitivity to local optima. As a consequence, efforts have been made to apply alternative optimization approaches to the column optimization problem. Yeomans and Grossmann9 formulate general disjunctive programming (GDP) representations for the economic optimization of distillation columns for reversible separations, and Barttfeld et al.10 compare

10.1021/ie900143e CCC: $40.75  2009 American Chemical Society Published on Web 06/16/2009

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Ind. Eng. Chem. Res., Vol. 48, No. 14, 2009

its solution properties to MINLP formulations. They claim that the GDP representation increases the robustness in the solution because nonexistent trays are not included in the subproblems. Still, the GDP has to be reformulated and solved as an MINLP since the development of logical solvers that are capable of handling logical constraints has not yet progressed sufficiently. The numerical results of their case studies for nonsharp separation of ternary mixtures in a single column suggests that the GDP formulation requires less solution time but is more sensitive toward local optima than MINLP formulations. Farkas et al.11 reformulate GDP representations of complex distillation systems as MINLP problems and apply a modified outer approximation algorithm that provides good initial values for the NLP subproblems. They optimize a complex distillation process for an azeotropic four-component example but still report solution times larger than 1 h. In general, the GDP representations modeling the column size by existing or nonexisting (i.e., bypassed) trays cannot benefit from the tight relaxations of the MINLP formulations, where the column size is modeled by a variable reflux/reboil location, variable condenser/ reboiler location, or variable product stream location. Various authors, e.g., Barkmann et al.,12 have solved rigorous column optimization problems by genetic algorithms. While this approach benefits from good robustness, the computational times proved to be significantly longer than for MINLP optimization. Recently, continuous reformulations of MINLP problems that can be solved with robust NLP solvers have gained increased attention due to the remaining drawbacks of discrete optimization. Lang and Biegler13 proposed a column optimization formulation, where the discrete decisions, i.e. number of trays and feed tray location, are modeled by continuous variables. The authors place the continuous decision variables on bellshaped curves with the help of a differentiable distribution function in order to locate optimal regions for the feed and the reflux/reboil streams. The optimization is then carried out in a series of continuous NLP problems with reduced dispersion factors. While this approach is very promising, some of the following simplifications apply for each of their published case studies: linear objective functions, nonsharp splits, fixed feeds, single columns, or ideal mixtures. In addition, they only obtain a narrow distribution of the decision variables instead of an integer solution. Neves et al.14 presented an alternative strategy for the continuous optimization of tray optimization problems, where they replace the differentiable distribution functions by numerically easier to handle nonlinear constraints that force the continuous decision variables to integer values. Like Lang and Biegler, they solve the continuous problem in a sequence while reducing relaxation parameters. The robustness is increased due to the continuous approach, a preprocessing phase based on shortcuts, and the addition of slack variables. However, the published case studies are confined to the tray optimization of either single columns or distillation processes with a fixed number of trays. The rigorous optimization of distillation processes becomes an even more difficult task when degrees of freedom on the flowsheet level are considered, i.e., when columns and streams within the flowsheet are optional. These problems suffer from their large scale, combinatorial complexity, and singularities that are introduced when columns become nonexistent. Grossmann et al.15 have solved superstructures for the synthesis of complex column configurations for ternary zeotropic and azeotropic mixtures. Still, the inclusion of rigorous column models in flowsheet superstructures especially for azeotropic systems of more than three components remains an open question. Simi-

larly, the consideration of variable column pressures or different entrainer choices within the rigorous optimization would result in problems far too complex to solve. We believe that these complex problems can only be solved by a decomposition of the design problem into sequential design steps with adapted modeling depth. In this work, we therefore integrate the rigorous optimization of distillation processes into a synthesis framework, which allows a reduction of the process optimization superstructure a priori by an economic screening of process variants with shortcut models prior to the rigorous optimization. Thus, elementary design decisions like the selection of splits, the flowsheet structure, the column pressures, or the choice of entrainer are already made before the rigorous optimization is set up. It will be shown that optimal solutions are identified with the help of shortcut calculations that rely on rigorous thermodynamics, which might not be the case when heuristics are put to use. In addition, the shortcut evaluation provides an excellent initialization for the following rigorous optimization. We solve these rigorous tray-optimization problems with outstanding efficiency and robustness by formulating a particular tight continuous representation. Because of the tight optimization formulation, the local optima of simple columns are located on integer points in the continuous space, and thus, the introduction of special nonlinear constraints to force the integer decisions can be largely avoided. We will show that we can solve continuously reformulated tray-optimization problems significantly faster than the corresponding MINLP problem. The reduction of the computational time is of great benefit when varying specifications for product purities, pressures, feed compositions, or cost parameters necessitates numerous design evaluations. In our previous works,16,17 we have successfully applied these methods to single columns and simple processes. In this paper, we extend the methodology to complex distillation processes with multiple columns for the separation of homogeneous azeotropic multicomponent mixtures. 2. Overview on the Paper In Section 3, we give a review of the process synthesis framework, in which the rigorous optimization of distillation processes is integrated. The shortcut evaluation methodology for the preselection of flowsheets and the initialization of the rigorous optimization problems is outlined in brief in Section 4. We then state the rigorous optimization models, their continuous reformulation, and the initialization and solution procedures in Section 5. Three large-scale case studies for illustrating the methodologies presented in Section 5 are then given in Section 6. 3. Process Synthesis Framework Kossack et al.17 and Marquardt et al.18 have proposed a systematic synthesis framework for the design of distillation processes composed of a sequential combination of both shortcut evaluation and rigorous optimization in order to benefit from the strengths and minimize the shortcomings of each individual design tool (cf. Figure 1). With this combination of synthesis methods of increasing level of detail, separation processes for homogeneous azeotropic multicomponent mixtures can be efficiently evaluated on the basis of rigorous thermodynamics, and the optimal distillation flowsheet, the optimal process operating point, and the optimal column specifications (number of trays, location of feeds, and reboiler/condenser duties) can be determined.


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Figure 1. Process synthesis framework for the design of distillation processes.

The steps of the design framework are performed at different levels of model refinement. The level of model refinement of each step in the framework is adapted to the specific task of the design step in order to meet the model requirements and facilitate an efficient design procedure. In the first step of the proposed framework, possible flowsheet alternatives for the desired separation task are generated, and if needed, suitable entrainers are identified. The generation of flowsheets can be automated for zeotropic multicomponent distillation with simple columns, as presented by Harwardt et al.19,41 However, for azeotropic mixtures, we need to create flowsheet alternatives manually based on heuristics and thermodynamical insight because, to date, no examples of generally applicable automatic flowsheet generation procedures for highly nonideal mixtures are known from the literature. The second step of the synthesis framework involves the rapid evaluation of the flowsheet variants and possible entrainer alternatives with respect to feasibility and minimum energy demand. For the evaluation of distillation flowsheets in the literature, usually shortcut methods like Underwood’s method21 or the boundary value method20 are used, which are confined to different limitations like binary mixtures, ideal thermodynamics, or graphical feasibility checks. In this work, the screening procedure is efficiently performed with the rectification body method (RBM, Section 4), a reliable shortcut method for the distillation of azeotropic multicomponent mixtures based on rigorous thermodynamics. The RBM is algorithmically accessible and, therefore, allows an optimization of the process operating point, while the feasibility is guaranteed by an algebraic feasibility test based on the calculation of pinch distillation boundaries. As a consequence, we are able to study azeotropic multicomponent mixtures and processes with multiple columns and a recycle with excellent robustness and efficiency. The only specifications needed to inspect feasibility and optimize the minimum energy demand in this step are pressure and product purities. A selection of promising flowsheet variants can then be rigorously modeled and optimized in the third step with the help of bounds and initial values from the previous shortcut evaluation. The aim is to determine the process and column specifications that yield the lowest total annualized costs, taking into account capital and operating costs. This involves the simultaneous optimization of recycle and intermediate streams, column energy duties, column diameters, heat exchanger areas, column tray numbers, and feed tray locations, with the latter two being discrete variables. The complex and large-scale discrete-continuous optimization problem can be solved with excellent robustness and efficiency because of the favorable initialization provided by the preceding shortcut calculations. In addition, the MINLP problems are

reformulated as easier-to-solve NLP problems to further speed up the solution procedure. To summarize, the process synthesis framework as a procedure of incremental refinement and successive initialization allows for a rapid synthesis of the cost optimal process while taking into account multiple flowsheet alternatives. Although the economic optimization is confined to local optimization, the favorable initialization and bounding of variables within the stepwise procedure results in very good local optimal solutions. The methodology has recently been extended by Harwardt et al.22,41 such that heat integration between the reboilers and condensers of the columns can be considered from the conceptual design phase on. The column pressures are design variables within the shortcut and rigorous optimization steps in this case. While the focus of this work is on the rigorous optimization of distillation processes, the method of the preceding shortcut evaluation step is reviewed in the next section because it will be employed in the case studies for a preselection of flowsheets and the initialization of the rigorous optimization. 4. Shortcut Evaluation For the fast evaluation of the cost of separation sequences, numerous shortcut evaluation methods have been developed. The most well-known is Underwood’s.21 A review of shortcut methods for distillation is given by Bausa et al.23 These methods are confined to different limitations like binary mixtures, ideal thermodynamics, or graphical feasibility checks. In this work, however, we use the rectification body method (RBM) as developed by Bausa et al.,23 because it is able to determine the minimum energy demand algorithmically for nonideal mixtures with an arbitrary number of components. More recently, Lucia et al.24 presented the shortest stripping line method, which is also applicable to nonideal multicomponent mixtures. While the shortest stripping line method requires tray-to-tray calculations to determine the minimum energy demand, the RBM relies solely on the information gained by pinch points. Consequently, the shortest stripping line method gives more accurate results when the linearized rectification bodies of the RBM are poor approximations of the column profiles. The RBM, however, requires less computational time than the shortest stripping line method and the automation is more straightforward. Because an in-depth description of the RBM is available elsewhere, only the main features of the RBM are outlined here. In the first step of the RBM, the pinch lines for a given feed and given products are calculated, where the exact locations of the pinch points on these pinch lines depend on the separation energy. Rectification bodies are constructed in a next step by

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connecting the pinch points linearly. The resulting rectification bodies can be seen as an approximation of all possible column profiles. When the rectification bodies of the stripping and rectifying sections intersect in a single point, the minimum energy of the separation has been found. Since the intersection of the rectification bodies is checked with an algebraic criterion, no visual inspection is necessary. The RBM has been extended to handle complex columns25,26 and extractive columns27 more recently. For a numerical flowsheet optimization, an algorithmic formulation of a feasibility criterion is necessary in addition to the RBM. This feasibility test has to be algebraically accessible and needs to provide information on the distance of a selected point to the distillation boundary. Both of these criteria are met by the so-called pinch distillation boundary (PDB),28,29 which marks the distillation boundary at minimum reflux. It is, therefore, very well suited to be used in conjunction with the RBM, since here the columns are working at minimum energy demand. In contrast, the simple distillation boundary marks the distillation boundary at infinite reflux. The mathematical formulation of the PDB is based on the (pitchfork) bifurcation phenomena of the pinch lines, which can be detected using a test function.29 The mathematical details can be found in the paper by Bru¨ggemann and Marquardt;29,30 here it suffices to say that the combination of RBM and PDB allows the numerical optimization of distillation processes with a recycle for nonideal mixtures as long as they do not show liquid phase splitting. 5. Rigorous Optimization The evaluation of a distillation process with the rectification body method as portrayed in Section 4 serves as a good approximation to inspect the feasibility and compare different process flowsheets by the minimum energy demand. However, no conclusions can be made regarding the optimal tray numbers, the optimal location of the feeds and side streams, and the capital costs. This information can be gained by a rigorous optimization of the distillation process with an economic objective function. The resulting optimization problem is of discrete-continuous nature due to the discrete decisions concerning the tray numbers and stream locations and the continuous values of energy duties and flowsheet stream rates and compositions. Considering the large scale and complexity of these optimization problems when several columns have to be optimized in a recycle loop and adding in the nonlinearity of the underlying nonideal thermodynamics, it becomes clear that these problems are particularly hard to solve. In this work, substantial progress has been made toward the robust and efficient solution of these problems through measures on two levels: • On the one hand, the resulting MINLP problems are reformulated as purely continuous problems that are solved as a series of a few easier-to-solve NLP problems with successively tightened bounds as presented in Sections 5.2 and 5.4. • On the other hand, the integration of the rigorous optimization into the process synthesis framework allows a reduction of the process superstructure and a favorable initialization, which is outlined in Section 5.3. 5.1. Rigorous Optimization Model. In general, a tray-totray optimization problem can be formulated as a GDP, where the column size is modeled by existing or nonexisting (i.e., bypassed) trays or as a MINLP, where the column size is modeled by variable column ends. Contrary to the GDP column

Figure 2. Alternative superstructures for the tray optimization of distillation columns. The top ends of the columns are variable.

representations, MINLP column formulations exhibit very tight relaxations and are therefore better suited for continuous reformulation. Different column superstructures for the MINLP tray optimization can be found in the literature.2 MINLP formulations for three different superstructure variants as illustrated in Figure 2 were considered for continuous reformulation in this work and compared for robustness, reliability, and efficiency in the case studies. The column feeds are variable and can be introduced on every tray in all superstructure variants. The number of column trays is adjusted by shrinking the column from one end while keeping the other end fixed. In Figure 2, the top ends of the columns are variable. The superstructures for variable bottom ends can be established accordingly. A formulation with two variable column ends and a fixed feed is not derived here as it directly results from the combination of both representations. Specifically, superstructure variant (a) in Figure 2 determines the number of column trays by a variable reflux scheme as proposed by Viswanathan and Grossmann.6 The top tray models the condenser, and as the reflux is moved to lower trays, the trays between the reflux location and the top tray dry up, i.e., the liquid overflow disappears for these trays. Note that the introduction of smoothing functions that Lang and Biegler13 formulate to handle the loss of phases on dried-up plates becomes redundant since pressure drops and heat losses are neglected in this work. Superstructure variant (b) has the condenser heat exchange as variable to control the tray number. Heat is exchanged on the top tray in any case when the distillate product leaves the column as boiling liquid. Comparable to variant (a), the trays above the last existing tray, i.e., above the heat exchange in this case, dry up. Again, no smoothing function for dried up trays is required when the column pressure drop and heat losses are neglected. Superstructure variant (c) models the size of the column by a variable distillate product stream and condenser heat location. The trays above the product draw are still calculated but are of no relevance. The rigorous column optimization model for a single column and superstructure variant (a) is listed as follows: 0)

∑b

F,k,nFkzk,i

k

+ Ln-1xn-1,i + Vn+1yn+1,i - Lnxn,i -

Vnyn,i + bSl,nSlxS,i - bSV,nSVyS,i + bR,nRx1,i, n ∈ N\{1, Nmax}, i ∈ I

(1)


∑b

0)

+ Ln-1hl,n-1 + Vn+1hV,n+1 - Lnhl,n -

F,k,nFkhf,k

k

VnhV,n + bSl,nSlhl,S - bSV,nSVhV,S + bR,nRhl,1,

∑b

0)

n ∈ N\{1, Nmax} (2)

+ Vn+1yn+1,i - Dxn,i - Rxn,i,

F,k,nFkzk,i

k

n ) 1, i ∈ I

∑b

0)

∑b

n)1

(4)

n ) Nmax, i ∈ I

(5)

+ Ln-1xn-1,i - Bxn,i - Vnyn,i,

F,k,nFkzk,i

k

∑b

0)

+ Ln-1hl,n-1 - Bhl,n - VnhV,n + QB,

F,k,nFkhf,k

k

n ) Nmax

∑x

n,i

∑y

) 1,

n,i

i

) 1,

F,k,n

) 1,

n

n∈N

(6)

(7)

i

yn,i ) Kn,i(x, y, T, p)xn,i,

∑b

(3)

+ Vn+1hV,n+1 - Dhl,n, - Rhl,n + QD,

F,k,nFkhf,k

k

0)

∑b

Sl,n

) 1,

n

∑b

SV,n

n ∈ N, i ∈ I ) 1,

n

∑b

R,n

) 1,

(8) k∈K

n

(9)

xn,i g xD,purity, xn,i e xB,purity,

n ) 1, i ) light component

(10)

n ) Nmax, i ) light component

(11)

NT ) Nmax -

∑ ∑b

4 · Vn π·2

R · Tn

(12)

R,n

n

Dcol )

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∑ (y

n

n,iMi)

i

p

,

n ) Nmax (13)

Cop ) f(QB, QD)

(14)

Ccap ) f(NT, Dcol, Areb, Acon)

(15)

min TAC ) Cop · ta + fc · Ccap

(16)

Mass and energy balances for superstructure variants (b) and (c) are given in the Appendix. The remaining equations can be adapted accordingly. The column model is based on the MESH equations, which specify tray-to-tray mass and energy balances and assume vapor-liquid equilibrium (VLE) conditions on every column tray. The streams to and from a tray are visualized in Figure 3. Component mass balances and energy balances are given in eqs 1 and 2 for each tray except the topmost and the lowest trays, which represent the condenser and the reboiler, respectively. For these, component mass and energy balances are given in eqs 3-6. Equation 7 contains the closure relations for the liquid and vapor compositions on each tray. The vapor-liquid equilibrium is added in eq 8. We calculate the liquid activity coefficient γ for the K-values by means of either the Wilson or the UNIQUAC model. The pure-component vapor pressure is determined by the extended Antoine equation, whereas vapor-phase fugacities can be neglected for the mixtures of the case studies in Section 6. DIPPR equations are employed

Figure 3. Equilibrium tray with streams for the variable reflux superstructure variant.

for the calculation of the enthalpies. The addition of slack variables to the balances and the equilibrium condition for better convergence properties as in Neves et al.14 can be neglected due to the sound initialization of the optimization problem (see Section 5.3). Closure relations apply for the binary variables modeling the column feed locations, bF,k,n, side draw and side feed locations, bSL,n,bSV,n, and the reflux location, bR,n (eq 9). Purity constraints are added for the distillate and bottom products in eqs 10 and 11 to ensure product quality. A sharp split for the light component is specified here. Sharp splits for the heavy component or intermediate splits can be specified analogously. Equations 12 and 13 calculate, respectively, the number of trays and the column diameter, which is dependent on the vapor flow rates, temperatures, and molar weights in the column. Here, the flow and the temperature on the lowest tray, which is permanent in the tray optimization, are chosen for the calculation of the column diameter. When the column is fed by a vapor feed, the vapor flow rates and, thus, the tray diameters are significantly higher at the top of the column when compared to the bottom of the column. In this case, the feed flow rate is added to the vapor flow rate of the bottom tray in the calculation of the column diameter, since the calculation of the diameter at nonpermanent trays proved to be numerically challenging. Note that the variable reflux superstructure, however, allows for a robust calculation of the column diameter at the top of the column since the vapor stream of the topmost existing tray bypasses the nonexisting trays on its way to the condenser. Thus, the same diameter is calculated for the nonexisting trays as for the topmost existing tray. In this work, we nevertheless calculate the column diameter at the bottom of the column for all superstructure variants to ensure a fair comparison of the alternative superstructure variants. The objective function reflects the total annualized column cost composed of operating cost (cost for cooling and heating) and capital cost (investment for column shell, trays, and heat exchangers). The capital cost, which depends on the tray number, the column diameter, and the areas of the heat exchangers, is calculated from nonlinear cost models given by Douglas3 and updated by the M&S index. The capital charge factor fc ) 0.25 accounts for a depreciation time of about 5 years including interest. In order to optimize the whole distillation process of several columns, the single-column models are connected by flowsheet streams with variable flow rates and compositions. Furthermore, the purity constraints on column end products that turn into flowsheet intermediate and recycle streams are removed. The objective function is then specified as the minimization of the

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cumulated annualized operating and capital costs of all columns in the process. 5.2. Continuous Reformulation of the Rigorous Optimization Model. As stated in Section 5.1, the MINLP tray optimization formulations exhibit very tight relaxations. Because of this property, these MINLP problems are perfectly suited for continuous reformulation, where the tight relaxations can be ideally exploited. In order to gain a purely continuous formulation, the binary variables modeling the reflux location, i.e., the number of column trays, bR,n, and the feed and side stream locations, bF,k,n,bSL,n,bSV,n, are replaced by the respective continuous decision variables cR,n, cF,k,n, cSL,n, and cSV,n. When MINLP problems are reformulated, integer values for the continuous decision variables are usually reached by the introduction of special nonlinear constraints to force integer decisions (cf. Stein et al.31 or Neves et al.14) or by the addition of penalty terms in the objective function. Yet in this work, the need for these illconditioned nonlinear constraints or penalty functions is largely avoided by carefully exploiting the properties of column optimization problems. Hence, integer values for the continuous decision variables modeling the tray numbers are obtained by formulating the column model in a particularly tight way, where the economic objective function is locally minimized when these continuous decision variables take on integer values. The tight formulation is achieved by replacing the purity constraint in eq 10 by Nmax

∑c

R,n

· yn,i g xD,purity,

i ) light component

(17)

n)1

where the vapor purity is formulated as a weighted sum over the trays that denote the variable column end. Note that the vapor composition on the topmost existing tray forms the liquid distillate after leaving the condenser in superstructure variants (a) and (b). The vapor composition yn,i in eq 17 is substituted by the liquid composition xn,i when superstructure variant (c) is employed. In a finite column, the vapor or liquid compositions cannot be identical for adjacent trays. As a consequence, according to eq 17, the purity on the topmost tray (when the column is modeled with a variable top end) needs to be larger than the required purity when the decision variable for the column end is spread among two or more trays. Since higher purities involve higher costs for sharp splits, columns with an integer number of trays are favored in the optimization and the introduction of special constraints like eq 18 below forcing an integer number of column trays becomes redundant. Column formulations with variable bottom ends can be treated analogously. As far as the feed distribution is concerned, Viswanathan and Grossmann6 already observed that a single column feed is optimally distributed when it is introduced on a single column tray. In our own work (Kossack et al.16), we have interpreted this property as a maximization of effective trays in each column section: the largest sections are obtained when the impure feed is introduced on a single tray and thus placed farthest away from both column ends. We have observed, however, that multiple feed columns do not typically have discrete feed trays in the optimal solution of the relaxed problem. The costs for a two-feed column, for example, are usually minimal when one feed is introduced on a single tray while the other feed is distributed among several trays. Here, an optimal integer solution is reached by the addition of a special nonlinear constraint in the form of the Fischer-Burmeister function (Jiang and Ralph32),

cF,k,n + (1 - cF,k,n) e √cF,k,n2 + (1 - cF,k,n)2 + µ, k ∈ K, n ∈ N

(18)

which forces the continuous variables cF,k,n representing the feed distribution to integer values when µ ) 0. Given that ∑n cF,k,n ) 1 and cF,k,n g 0, an optimal discrete feed tray is determined. The relaxation parameter µ is added to the right-hand side of eq 18 in order to improve the numerical properties of the Fischer-Burmeister function. In a series of three solution steps, µ is reduced from µ ) 0.5 to µ ) 0.2 and µ ) 0, where an integer solution is reached (cf. Section 5.4). Still, it is desirable to add as few as possible of these illconditioned and nonconvex constraints to allow for a robust and reliable solution of the reformulated optimization problem. Through the tight column formulation in this work, the use of special constraints like eq 18 could be reduced to a minimum as they only have to be introduced for additional feeds in multiple feed columns. All other integer decisions (number of trays and feeds of single feed columns) converge to integer values in the local optima even without being constrained to integrality. 5.3. Initialization of Rigorous Optimization. The complex large-scale nonlinear tray optimization problem requires a sound initialization to allow for a robust, reliable, and efficient solution. Different initialization concepts are known from the literature that typically suffer from the drawback that a priori knowledge about the distillation process is required to specify initial values and bounds. Various authors (e.g., Barttfeld et al.,10 Neves et al.14) have reported, however, that a sound initialization of the column optimization improves both the robustness and the probability to identify good local optima. In order to identify appropriate initial values, different proposals for preprocessing phases have been published. Fletcher and Morton33 generate initial values by studying the limiting column condition at infinite reflux. Energy-efficient columns, however, operate close to the minimum reflux condition instead of the infinite reflux condition. Barttfeld et al.7,8,10 use the theory of reversible distillation of Koehler et al.34 to identify the energy-efficient “preferred separation” for initial values. As a drawback, this concept usually leads to nonsharp splits for azeotropic mixtures. In this work (see also Kossack et al.16), the integration of the rigorous optimization in the synthesis framework offers excellent prospects for initialization because initial values and bounds can be directly retrieved from the preceding shortcut evaluation with the RBM. The initialization procedure not only provides a feasible starting point for the rigorous optimization but indeed an excellent one due to the proximity of the minimum reflux condition to the real column operating condition. The initialization procedure for the rigorous optimization of distillation processes is carried out in several steps as illustrated in Figure 4. It is our experience that a stepwise initialization of the optimization problem with gradually refined models helps both robustness and efficiency, although more optimization runs have to be carried out. For the initialization, the process is therefore disaggregated into single columns at first that are initialized separately with the column feed and product streams being fixed at the optimal values from the preceding shortcut evaluation. Initial linear column composition and temperature profiles can be derived for every column from the linear piecewise combination of the pinch points calculated by the RBM in the shortcut step, providing very good approximations of the actual column profiles. These linear profiles then serve as initialization for a rigorous column simulation for which the tray number is fixed at a user-specified maximum value and


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Figure 4. Steps of the initialization procedure. For illustration purposes, the rectification bodies and flowsheets at the respective initialization step of an extractive distillation process are included.

energy balances are neglected in order to facilitate easy convergence of this initializing column model. Note that the feed tray location is set free to prevent an infeasible specification by the user. In the next step, energy balances are introduced and the relaxed feed tray location variable is optimized by a minimization of the reboiler duty to provide excellent initial values for the rigorous tray optimization. Finally the columns are connected by the flowsheet streams to form the process. The compositions and flow rates of these streams are optimized by a minimization of the process energy duty to adjust the optimal process operating point from the shortcut evaluation for the representation with rigorous models. The initialization procedure is now completed, and the tray optimization with the economic objective function can be robustly performed (see next section). Because of the initialization, this last step will converge quickly to local optimal solutions of good quality. 5.4. Solution Procedure of Rigorous Optimization. Now that the rigorous optimization is initialized by the procedure described in the previous section, we are able to solve the continuously reformulated process optimization problems for the alternative superstructure variants (cf. Figure 2) robustly and efficiently. However, the tight column models as specified in Section 5.1 tend to yield local optimal solutions of bad quality when solved right away after the initialization due to the abundance of nonlinearities within the model. Therefore, a solution procedure in two steps was implemented. Loose formulations of the distillation columns are solved in the first step. Thus, the decision variables (feed locations and reflux, condenser, or product draw locations, respectively) are optimally distributed among several column trays and the problem converges reliably to favorable solution regions. The formulations are then tightened in the second step to retrieve the tight continuous formulations described in Section 5.1, where the cost

is minimized when the decision variables take on integer values. In the following, these two steps are outlined for the alternative column superstructures. 5.5. Variable Reflux/Reboil or Variable Condenser/ Reboiler. The problem formulations are relaxed in the first step by replacing the tight purity constraint in eq 17 by the looser purity constraint in eq 10. Thus, a relaxed underestimation of the continuous-discrete problem, i.e., an optimal distribution of the reflux or the condenser among several trays, is obtained at first. Integer solutions are then obtained in the second step where the tight purity constraint in eq 17 is applied. 5.6. Variable Product Draw. The relaxation is accomplished by disaggregating the product draw Dn from the associated continuous decision variable cD,n as follows: Dn ·

1 e cD,n · M

N

∑D

j

e Dn · M,

n∈N

(19)

j)1

M is the relaxation parameter that is reduced in a succession of five NLP problems from 100 to 50, 20, 5, and 1 such that the tight model formulation is restored in the last step and integer solutions are obtained. As already mentioned in Section 5.2, the Fischer-Burmeister relaxation parameter µ in eq 18 was successively reduced from 0.5 to 0.2 and 0 in the stepwise solution of the optimization problems to diminish the effect of the nonlinearity inherent to the Fischer-Burmeister function. The successive steps of the solution procedure are comprehensively shown in Figure 5. 6. Case Studies The optimization-based design procedure is illustrated by three large-scale case studies. All examples involve multicomponent azeotropic distillation resulting in processes with multiple columns and a recycle.

6756


Figure 6. Distillation boundary of the acetone, chloroform, benzene, toluene mixture at 1.013 bar.

Figure 5. Steps of the solution procedure for the variable reflux/reboil and variable condenser/reboiler superstructures on the left. The steps for the variable product draw superstructure are shown on the right.

The first case study covers the whole design procedure of the process synthesis framework (cf. Section 3) with generation of alternative flowsheets, shortcut-evaluation with the RBM, and rigorous optimization for the complete separation of an azeotropic four-component mixture in a curved-boundary process. It is shown that significant cost reductions in both capital and investment costs can be achieved by selecting the best flowsheet and rigorously optimizing the column specifications. The second example considers a three-column pressure-swing process for the separation of a highly nonideal five-component mixture with nine azeotropes. The separation task and flowsheet was taken from Wasylkiewicz.35 Because of the complexity of the mixture and the flowsheet, this example is well-suited to study the potential of the synthesis methods and demonstrate that such a large-scale complex problem can be robustly and efficiently handled when initialized by the shortcut evaluation and reformulated as a continuous optimization problem. The costs for the flowsheet from Wasylkiewicz are compared to the costs for a simpler two-column flowsheet. In the last case study, the rigorous optimization of a complex column system is presented. Specifically, an extractive distillation process involving a side column for the separation of a ternary azeotropic mixture is considered. The costs for this integrated process are compared to the costs for a comparable process consisting of simple columns. In all case studies, the column pressures are fixed at constant values in each run and pressure drops along the column trays are neglected. The structural and operational degrees of freedom in the rigorous optimization are the number of column trays, feed tray locations, side stream locations, recycle flow and compositions, and column energy duties. Solutions are presented for all three alternative column optimization superstructures introduced in Section 5.1 to allow for a comparison of the respective optimization properties. Furthermore, the continuously reformulated optimization problems are confronted with the respective MINLP solutions of the variable reflux and variable condenser superstructures. The MINLP solutions for the variable product draw superstructure either yielded optima of bad quality (tight formulation, see Appendix) or did not converge reliably

Figure 7. Flowsheet proposed by Thong et al.39 with a recycle of azeotropic composition. Table 1. Recycles and Minimal Process Reboiler Duties for the Flowsheet Variants of Figure 8; Compositions Are Given As Molar Fractions of Acetone, Chloroform, Benzene, and Toluene, Respectively flowsheet variant 1 2 3 4

recycle flow rate and composition 7.38 mol/s 5.44 mol/s 4.81 mol/s 5.59 mol/s

[0, 0.08, 0.65, 0.27] [0, 0, 0, 1] [0, 0, 0.5, 0.5] [0, 0, 0, 1]

Qbmin 1.399 1.103 1.201 1.209

MW MW MW MW

(relaxed formulation, eq 19). The results for these formulations are, therefore, not included in the results tables. It should be also noted that all MINLP problems were solved with the branch-and-bound solver SBB.36 The problems did not converge reliably when an outer-approximation solver (DICOPT) was applied. While the shortcut calculations were executed in C, all rigorous optimization problems were formulated and solved on the optimization platform GAMS 22.7.37 The SQP-based solver SNOPT 7.2-438 was employed for the solution of the continuously reformulated problems on a PC with a 3 GHz Dual-Core CPU. External functions are used in GAMS to calculate the thermodynamic properties (liquid activity coefficients and enthalpies) and the required derivatives. While this approach raises the required solution time because of the communication overhead between the solver and the external function, the robustness of the optimization problem is increased. 6.1. Complete Separation of an Azeotropic Four-Component Mixture. An equimolar mixture of acetone, chloroform, benzene, and toluene is to be separated into its pure components in a multicolumn process at atmospheric pressure (1.013 bar).


6757

Figure 8. Flowsheet variants for the shortcut evaluation: (1) four-column flowsheet of Thong et al.,39 (2) three-column flowsheet with toluene recycle, (3) direct sequence with benzene/toluene recycle, and (4) direct sequence with toluene recycle. Table 2. Costs, Column Configurations, and Operating Point for All Columns of Flowsheet 2 after Rigorous Optimization; The Results Are for the Continuous Reformulation of the Variable Reflux Column Superstructure acetone column toluene column TAC capital cost operating cost reboiler duty condenser duty number of trays feed tray recycle feed tray diameter recycle

Figure 9. Rectification bodies, linear column profile approximations, and initial column profiles of the initialization for the rigorous optimization of flowsheet 2.

The flow rate of the saturated liquid feed is set to 10 mol/s, and all products are specified at 99% purity. The Wilson model is chosen for the calculation of the liquid activity coefficients of the homogeneous mixture. As illustrated in Figure 6, the mixture exhibits a maximum boiling binary azeotrope on the acetone/chloroform edge and an associated distillation boundary between the azeotrope and the benzene/toluene edge. The boundary can be crossed by placing the bottom product of the acetone-producing column on the boundary and exploiting the distinct curvature of the boundary toward the chloroform vertex. Thong and Jobson39 generate feasible flowsheets for this separation task based on heuristic rules and a search among

165869 euro/a 110089 euro/a 55781 euro/a 453 kW 462 kW 51 14 8 68.6 cm 8.27 mol/s

chloroform/benzene column

134516 euro/a 76819 euro/a 57696 euro/a 511 kW 339 kW 25 13

117227 euro/a 83340 euro/a 33887 euro/a 266 kW 417 kW 40 18

72.5 cm [0, 0, 0.01, 0.99]

64.3 cm

possible recycle options. They propose the flowsheet shown in Figure 7, where a recycle stream of 2.5 mol/s (R/F ) 0.25) and azeotropic composition is returned back to the first column. A minimum process reboiler duty of 1.576 MW is quickly determined when we evaluate this flowsheet with the RBM (cf. Section 4) at the operating point suggested by Thong and Jobson. As described in Section 4, the RBM allows an automatic optimization of the process operating point, which leads to a reduction of the minimum process reboiler duty by 12.6% to 1.399 MW in this case. Interestingly, the recycle flow rate increases to 7.38 mol/s at the optimum (cf. Table 1). The energy savings, however, result from a shift of the bottom product of the acetone column toward the benzene/toluene edge by an addition of benzene and toluene to the recycle. Thus, the shape of the distillation boundary can be fully exploited: the boundary coincides with the chloroform/benzene/toluene plane for higher concentrations of benzene or toluene, allowing a complete

6758


Table 3. CPU Times and Objective Values (TAC) for the Rigorous Optimization of Flowsheet 2 by Continuous Reformulations of the Alternative Superstructure Variants; The Optimization Properties for the MINLP Solution with a Branch-and-Bound Solver Are Given for Comparison continuous reformulation-NLP

MINLP (SBB)

superstructure

var. reflux location

var. condenser location

var. product location



CPU time initialization CPU time optimization TAC

146 s 417612 euro/a

184 s 417498 euro/a

72 s 347 s 417775 euro/a

1202 s 417768 euro/a

2977 s 437187 euro/a

Table 4. Singular Points of the (A)cetone, (C)hloroform, (M)ethanol, (E)thanol, and (B)enzene System at Normal Pressure (1.013 bar), Calculated by the Wilson Model azeotrope/pure comp.

composition

T [°C]

C/M A/M A A/C/M/B A/C/M M/B C/E C A/C/E A/C M E/B E B

[0, 0.663, 0.337, 0, 0] [0.783, 0, 0.217, 0, 0] [1, 0, 0, 0, 0] [0.267, 0.194, 0.456, 0, 0.083] [0.334, 0.235, 0.431, 0, 0] [0, 0, 0.611, 0, 0.389] [0, 0.856, 0, 0.144, 0] [0, 1, 0, 0, 0] [0.356, 0.469, 0, 0.175, 0] [0.346, 0.654, 0, 0, 0] [0, 0, 1, 0, 0] [0, 0, 0, 0.450, 0.550] [0, 0, 0, 1, 0] [0, 0, 0, 0, 1]

53.99 55.38 56.24 57.33 57.37 58.06 59.41 61.14 62.90 64.22 64.64 68.04 78.34 80.14

separation of acetone in the first column and, consequently, an acetone-free mixture in the downstream columns. The special curvature of the distillation boundary therefore enables a complete separation of the four-component mixture in three columns. A recycle is still required for the given feed mixture, since the mass balance line of the acetone column that stretches out to the boundary needs to be shifted toward benzene/ toluene. Significant cost savings can be assumed for any threecolumn process when compared to processes that comprise four columns. In light of these findings, the evaluation of further flowsheet variants is confined to flowsheets with three simple columns. These variants are shown in Figure 8 together with the original flowsheet. The respective minimum process reboiler duties after evaluation with the RBM and optimization of the operating points are given in Table 1. The product streams of all flowsheet variants are withdrawn as saturated liquid. However, the distillate streams that are fed into another column are not condensed but transferred as saturated vapor as a measure of heat integration. This actually penalizes the flowsheet variants 3 and 4 for the absence of intermediate distillate streams. Variant 4 considers both a benzene and a toluene recycle with variable flow rates. The benzene recycle, however, vanishes in the optimization. A pinch distillation boundary constraint had to be considered only for the first column as the following columns are acetonefree and, therefore, not restricted by distillation boundaries. As a drawback, the optimization runs turned out to be computationally costly. The optimization times amounted to ∼1000 s for each flowsheet variant. By far the largest portion of the computational time is spent on the calculation of the PDB in every iteration step. It is expected that this computational time can be significantly reduced by a piecewise linearization of the PDB as suggested by Ryll et al.40 The PDB linearizations can be calculated and stored in advance since the PDBs only depend on pressure. Flowsheet variant 2 exhibits the lowest minimum process reboiler demand (∼43% less than in the original work of Thong) within the shortcut evaluation and is, therefore, chosen for

further evaluation with rigorous optimization to determine the cost-optimal column configurations. At first, the initialization procedure for the rigorous optimization out of the shortcut evaluation as presented in Section 5.3 is applied step by step. The linear column profile approximations along the relevant edges of the rectification bodies are shown in Figure 9 together with the profiles of the initializing process optimization (penultimate step of the initialization procedure), where the tray numbers are fixed to 60, 40, and 60 for the acetone, toluene, and chloroform/benzene columns, respectively. It can be observed that the process reboiler demand for a finite number of trays in this step (1.107 MW) corresponds very well with the minimum reboiler demand in the shortcut evaluation (1.103 MW). The difference is particularly small due to the fact that the 99% product purity specification for the rigorous tray optimization allows minor impurities in contrast to the pure product streams of the shortcut calculations. Because of this thorough initialization phase, excellent initial values and bounds are provided for the following rigorous optimization. Here, all three columns are modeled with a variable top end and a fixed bottom end. The maximum tray numbers were set to the same values as in the initialization phase, yielding optimization problems of about 4000 variables, including 380 decision variables. Fischer-Burmeister functions, eq 18, were introduced only for the recycle feed of the first column to obtain an integer feed location. All other continuous decision variables took on integer values in the optimal solution without being forced by a Fischer-Burmeister function or similar constraint as discussed in Section 5.2. The solution of the optimization problem was carried out by the solution procedure proposed in Section 5.4 and shown in Figure 5. The optimization results for all three columns of flowsheet 2 are displayed in Table 2. These figures were obtained with the continuous reformulation of the variable reflux superstructure (cf. Section 5.1). Interestingly, the feed tray of the high-boiling toluene recycle is located well above the tray of the fresh feed of equimolar composition in the optimal solution. Apparently, the toluene recycle has an extractive effect on the distillation of acetone in the first column. For a comparison of the optimization properties of all three superstructure variants and a comparison of continuous as well as MINLP solutions, Table 3 lists the respective solution times and objective values. The initialization phase, which is identical in all cases, took 72 CPU s. The computational times of the continuously reformulated rigorous optimization problems were significantly lower than the computational times of the corresponding MINLP problems, which also benefited from the favorable initialization. The variable reflux superstructure converged quicker than the alternative superstructures for both the continuous and the MINLP problems. When solved as continuous problems, all superstructure variants identified very good local minima, i.e., TAC, at or close to the global minimum. In the case of the MINLP solutions, which are used to compare our results, only the variable reflux scheme produced a good solution of low cost.


6759

Figure 11. Two-column flowsheet for the pressure-swing process. Table 6. Costs, Column Configurations, and Operating Point for All Columns of the Three-Column Pressure-Swing Process after Rigorous Optimization; The Results Are for the Continuous Reformulation of the Variable Reflux Column Superstructure Figure 10. Three-column flowsheet for the pressure-swing process as proposed by Wasylkiewicz. 35

Note that the whole process, i.e., all three columns, was optimized simultaneously. Thus, no specifications needed to be made for the intermediate streams because their purities were optimized. Alternatively, the first two columns can be optimized separately from the chloroform/benzene column, which is not part of the recycle loop. In that case, the optimization problem can be solved faster, but on the other hand, solutions of lower quality are identified since the purity of the intermediate stream needs to be fixed. 6.2. Pressure Swing Distillation Process for an Azeotropic Five-Component Mixture. The second case study considers the homogeneous five-component mixture of acetone, chloroform, methanol, ethanol, and benzene, for which the Wilson model calculates six binary, two ternary, and one quaternary azeotrope (cf. Table 4). Wasylkiewicz35 has generated a distillation process with the help of the synthesis software ASPEN Distil for this mixture and the following separation task: • feed of 25 mol % acetone, 40 mol % chloroform, 25 mol % methanol, 5 mol % ethanol, and 5 mol % benzene; • complete separation of pure benzene and pure ethanol; • ethanol- and benzene-free residual as a recycle to the reactor. A complete separation of ethanol and benzene in two simple columns at normal pressure is not possible because of the azeotropic behavior of the mixture with multiple separation regions. Hence, Wasylkiewicz proposed a pressure-swing process as shown in Figure 10: after the removal of the benzene in a first column, all ethanol is separated in the second column with the help of excess methanol that is recycled from the third column. This column operates at low pressure (10 mbar) to shift the distillation boundary that prohibits a methanol recycle at normal pressure. Alternative possibilities for distillation of this mixture include extractive distillation and heteroazeotropic distillation. However, we have confined this case study to pressure-swing distillation since the application of the process synthesis framework to extractive distillation is demonstrated in the third case study and in a work by Kossack et al.16 We consider a feed of 10

benzene column ethanol column low-pressure column TAC capital cost operating cost reboiler duty condenser duty number of trays feed tray recycle feed tray diameter recycle

354983 euro/a 214216 euro/a 140766 euro/a 1217 kW 909 kW 64 39 110 cm 0.95 mol/s

33075 euro/a 205583 euro/a 24222 euro/a 84842 euro/a 8852 euro/a 120740 euro/a 65 kW 50 kW 49 kW 477 kW 34 11 1 10 8 23 cm 158 cm [0.001, 0.003, 0.427, 0.568, 0]

mol/s and purities of 99.99 mol % for benzene, 99.9 mol % for ethanol, and 99.9 mol % for the residual. The distillate of the benzene column is drawn as saturated vapor and fed to the pressure-swing sequence. Here, the vaporous distillate of the ethanol column is introduced in the low pressure column as superheated vapor, and the recycle is transferred as subcooled liquid due to the pressure difference. In the shortcut evaluation of the process, the feasibility of the flowsheet is verified and the recycle flow rate and composition are optimized by a minimization of the reboiler energy of the ethanol column and the condenser energy of the low pressure column. The remaining energy duties of the pressure swing sequence can be neglected as they are conveniently provided by inexpensive cooling water. While a distillation boundary restricts the separation at the light end of the ethanol column, the pressure reduction to 10 mbar allows for an unrestricted separation in the low pressure column. Hence, PDB constraints had to be considered only for the light end of the ethanol column. The operating point as suggested by Wasylkiewicz, i.e., a recycle of 2.769 mol/s of methanol, was chosen as the starting point for the optimization. The recycle flow rates and compositions as well as the minimum reboiler duties of all columns at the operating point proposed by Wasylkiewicz and at the optimized operating point are given in Table 5. A significantly lower recycle flow rate and, consequently, lower minimum energy duties are achieved by adding ethanol to the recycle. By a comparison of the energy duties at the optimal operating point to the respective values at the operating point proposed by Wasylkiewicz, the potential of the flowsheet optimization within the shortcut evaluation can be highlighted: the reboiler duty of the ethanol column could

Table 5. Recycle and Minimal Reboiler/Condenser Duties at the Operating Point Proposed by Wasylkiewicz and at the Optimized Operating Point; Compositions Are Given As Molar Fractions of Acetone, Chloroform, Methanol, Ethanol, and Benzene, Respectively

operating point Wasylkiewicz optimized operating point

recycle flow rate and composition

Qbmin benzene column

Qbmin ethanol column

Qdmin low-pressure col.

2.769 mol/s [0, 0, 1, 0, 0] 0.8369 mol/s [0, 0, 0.36, 0.64, 0]

887.2 kW 887.2 kW

433.5 kW 53.9 kW

544.3 kW 456.1 kW

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Table 7. CPU Times and Objective Values (TAC) for the Rigorous Optimization of the Three-Column Pressure-Swing Process by Continuous Reformulations of the Alternative Superstructure Variants; The Optimization Properties for the MINLP Solution with a Branch-and-Bound Solver Are Given for Comparison continuous reformulation-NLP

MINLP (SBB)

superstructure







271 s 227 s 593640 euro/a

262 s 596457 euro/a

997 s 596522 euro/a

1721 s 594080 euro/a

12256 s 596422 euro/a

Table 8. Costs, Column Configurations, and Operating Point for the Complex-Column System and the Simple-Column Process after Rigorous Optimization; The Results Are for the Continuous Reformulation of the Variable Reflux Column Superstructure complex-column system main column TAC capital cost operating cost reboiler duty condenser duty number of trays feed tray entrainer feed tray liquid side stream tray vapor side stream tray diameter entrainer recycle

320546 euro/a 170131 euro/a 110502 euro/a 813 kW 496 kW 49 30 15 40 40 113 cm 28.93 mol/s

simple-column process

side column 34716 euro/a 5197 euro/a 317 kW 6

74 cm [0, 0, 1]

be reduced by 87.6%, and the condenser duty of the low pressure column could be reduced by 16.2%. Interestingly, Wasylkiewicz did not consider a two-column process, where the separation of ethanol from the residual acetone, chloroform, and methanol is accomplished in a single low-pressure column operated at 10 mbar as shown in Figure 11. When evaluated with the RBM, this simplified flowsheet proved to be feasible since the feasibility check revealed no active PDB constraint. In addition, with a minimum condenser duty of the low-pressure column of 430.9 kW, the two-column process seems to be slightly more energyefficient than the original three-column process. Note that the distillate of the benzene column is condensed here as saturated liquid in order to save expensive cooling utility in the low-pressure column. Both flowsheet alternatives are rigorously optimized in the following. It can be expected that the three-column process offers lower capital costs due to the reduced amount of low-pressure trays with large diameters. In the rigorous optimization step, all columns of the threecolumn process are optimized simultaneously to avoid a specification of intermediate streams. The benzene column is modeled with a variable top end and a maximum of 100 trays. The ethanol column is modeled with a variable bottom

extractive column

recycle column

259489 euro/a 91058 euro/a 45308 euro/a 374 kW 467 kW 53 33 15

60015 euro/a 63108 euro/a 445 kW 352 kW 18 7

55 cm 23.74 mol/s

70 cm [0, 0, 1]

end and a maximum of 50 trays, and the low-pressure column is modeled with a variable top end and a maximum of 20 trays. Both columns are allocated a maximum of 40 trays each. Optimization problems with about 4400 variables, including 390 decision variables, were obtained. While low pressure steam (3 bar) and cooling water serve as hot and cold utilities for the benzene and ethanol columns, a cold utility of -50 °C is used for the low-pressure column. Water at 25 °C is supplied as hot utility there. In this case study, no Fischer-Burmeister constraints were required. All decision variables, even both feeds of the ethanol column, took on integer values in the optimal solution when the tight formulations of Section 5.2 are applied. The optimization results for the three-column process are given in Table 6. The continuous reformulation of the variable reflux/reboil scheme was used to generate these figures. It can be observed that the vaporous feed of the ethanol column is optimally placed at the top, yielding a small column diameter and low capital costs for this column. The low-pressure column contains only 11 trays in the optimal solution as the large diameter raises the capital costs. Table 7 gives a comparison of the computational times and the identified cost minima for the alternative column super-

Figure 12. Flowsheets for the extractive distillation process. Complex column system on the left and simple column process on the right.


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Table 9. CPU Times and Objective Values (TAC) for the Rigorous Optimization of the Complex-Column System by Continuous Reformulations of Different Superstructure Variants; The Optimization Properties for the MINLP Solution with a Branch-and-Bound Solver Are Given for Comparison continuous reformulation-NLP

MINLP (SBB)

superstructure







76 s 47 s 320546 euro/a

57 s 321305 euro/a

89 s 321052 euro/a

99 s 330794 euro/a

316 s 338787 euro/a

structures for both continuous reformulation and MINLP solution algorithms. The continuous reformulation of the variable reflux/reboil scheme offers the best combination of both quick computational times and low costs in the local minimum. The MINLP solutions are characterized by considerably longer computational times and higher costs. The rigorous optimization of the two-column process (Figure 11) yields a cost minimum of 607186 euro, which is slightly larger than the cost for the three column process because of a higher amount of low-pressure trays (21 trays) with large diameters (169 cm). Note that operators would presumably still prefer the two-column process for its operability advantage due to the absence of recycles. 6.3. Extractive Distillation of Acetone from Methanol in a Complex Column Process. Complex column systems have the potential to lower energy costs as well as capital costs when compared to sequences of simple columns. At the same time, both the assessment of the savings potential and the design of a complex column process is particularly difficult because of little practical experience as well as an increased number of degrees of freedom. Von Watzdorf et al.25 have extended the RBM such that complex columns can be included in the shortcut evaluation. In the case of complex column systems, however, the rigorous optimization with detailed models is all the more useful because shortcut calculations give no information about the location of sidestreams and the capital cost difference to simple column processes. On the other hand, the integration of complex columns also leads to additional challenges for the mixedinteger optimization. In this case study, the rigorous optimization of an extractive distillation process involving a side column as an example for a complex column system is presented. The structural integration of the side column in the main column is not considered in this work. Instead, the side column is treated as a separate column for the calculation of capital costs. The azeotropic mixture of acetone and methanol is to be separated into its pure components via extractive distillation at normal pressure. A feed of 10 mol/s and a molar acetone fraction of 0.77 is considered. The VLE is calculated by the UNIQUAC model. In an earlier work by our group,17 suitable entrainer candidates for this separation were discriminated by computer-aided molecular design and evaluated by their selectivity at infinite dilution and by shortcut calculations with the RBM. Chlorobenzene, although a harmful substance, was determined to be the entrainer with the highest selectivity toward methanol. An evaluation with the RBM (energy duties) as well as with a rigorous optimization (costs) of a flowsheet with simple columns (cf. Figure 12) confirmed the optimal entrainer choice. In this work, it is studied whether this separation can be performed at a lower cost in a complex column setup with an extractive main column and a side rectifier as shown in Figure 12. A minimum reboiler duty of 793 kW was calculated with the RBM for the flowsheet of simple columns. The shortcut calculations for the complex column process are carried out

by an application of the RBM to an equivalent sequence of simple columns as described by von Watzdorf et al.25 An insignificantly larger reboiler duty of 803 kW is determined. Hence, we cannot draw a conclusion which process is more economical based on the results of the shortcut evaluation. The insight gained by a rigorous optimization, where the capital costs are included in the comparison of the economics, is essential here. Again, the results of the shortcut evaluation are used as initialization of the rigorous optimization step. We have observed that the rigorous optimization of complex column systems tends to locate local optima of bad quality when we apply the same routines as in the previous case studies. This may be attributed to the complexity induced through the multitude of streams that leave or enter the extractive column. The fact that these stream locations are all interdependent apparently hinders the column from shrinking to the optimal size in the course of the optimization. For this reason, we introduced the differentiable distribution function (DDF)

[( ∑ [ (

n-

exp -

Cn )

exp -

n

∑c

n

·n

n

σ n-

∑c

n

n

σ

)] )] 2

·n

2

,

n∈N

(20)

as proposed by Lang and Biegler13 in the first optimization step, where relaxed formulations are applied (cf. Section 5.4). The DDF, which is imposed only on the entrainer feed with a standard deviation of σ ) 4, is discarded in the second optimization step, such that the tight formulations in this step are able to reach an integer solution. The main column and the side rectifier were modeled with variable top ends and a maximum of 60 and 20 trays, respectively. Fischer-Burmeister functions had to be formulated only for the fresh feed and the extractive recycle feed in order to gain integer solutions. The optimization results for the continuous reformulation of the variable reflux/reboil scheme, are given in Table 8 for both the complex-column and the simple-column process. The optimization of the simple-column process was already published by Kossack et al.17 The results there differ quantitatively from this work, since different economic parameters and functions were used. Apparently, the sidestreams to and from the side rectifier were optimal when located on the same single stage. The complexcolumn system exhibits significantly higher costs than the simple-column process, although the reboiler duties are almost identical, which was already predicted in the shortcut evaluation. The cost difference can be mainly attributed to the large diameter of the main column in the complex system. In addition, the heat for the main column of the complex system needs to be supplied at a higher, costlier temperature level than for the extractive column of the simple-column process.

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The optimization properties for the continuous reformulations of the alternative column superstructures for the complexcolumn system are listed in Table 9 together with the properties for the respective MINLP solution. The optimization times for this case study are shorter than for the previous case studies due to the lower number of components in the mixture and the lower number of column trays. Accordingly, we also observe a smaller difference in computational time between the continuously reformulated problems and the MINLP problem. The benefit of the continuous reformulation is more distinctive in this case study when the qualities of the optimal solutions are compared. The continuously reformulated problems identify cost minima about 10000 euro/a lower than the MINLP problem. This can be attributed to the introduction of the DDF in the continuous problems. 7. Summary and Conclusion We have shown in this work that the rigorous optimization of complex large-scale distillation processes for multicomponent azeotropic mixtures can be accomplished robustly and efficiently when integrated into a process synthesis framework and reformulated as a continuous problem. The shortcut evaluation step preceding the rigorous optimization in the process synthesis framework served two purposes: on the one hand, alternative flowsheet variants for the separation task were screened with the RBM and ranked based on feasibility and minimum energy demand as shown in the first case study. Thus, the flowsheet superstructure could be efficiently reduced. On the other hand, excellent initial values and bounds for the rigorous optimization were generated due to the minimum reflux condition of the RBM and the optimization of the process operating point. The benefit of a rigorous optimization apart from obtaining useful information about optimal column configurations was demonstrated in the second and third case studies. In these studies with complexcolumn connections, the selection of the most economic process alternative could not be based on the comparison of the minimum energy duties alone but required the consideration of combined operating and capital costs. The complex and large-scale tray optimization problems could be solved with unprecedented efficiency, robustness, and reliability with the help of a suitable initialization procedure and a continuous reformulation of the MINLP problems. More precisely, the problems were continuously reformulated in a particularly tight way such that the introduction of special nonlinear constraints to force integer decisions could be largely avoided. The solution was then carried out in a procedure of a few successive NLPs with gradually tightened bounds in order to obtain local optima of good quality. Three different tray optimization superstructures were tested and the solution times and local optima (total annualized costs) were compared with the respective optimization properties for the MINLP solution. Because of the careful initialization procedure, all continuously reformulated problems could be solved with good robustness. The superstructure that models the column tray number by a variable reflux or reboil location offered the best results as far as solution times and quality of the local optima are concerned. By applying the continuous reformulation in the multicomponent examples of the first and second case studies, the computational times could be cut by at least 85% when compared to the respective MINLP solutions, which also benefit from the favorable initialization procedure. In addi-

tion, better local optima were identified by the continuously reformulated problems than by the respective MINLP problems. The difference in solution time between the MINLP and the continuously reformulated problems is less distinct in the third case study due to a lower number of components in the mixture and less column stages. In this case study involving a complexcolumn process with multiple column streams, the introduction of a differentiable distribution function as in Lang and Biegler13 in the relaxed step of the solution procedure for the continuously reformulated problems helped the identification of better local optimal solutions. In our future work, we will extend the optimization of distillation processes to heteroazeotropic distillation, where a shortcut evaluation is essential to assess the separation potential and find suitable entrainers. In addition, a thorough initialization phase of the rigorous tray optimization is here all the more important due to the occurrence of VLLE trays. The methods will also be transferred to the synthesis of hybrid separation processes, which pose a major challenges for both conceptual and detailed design as the coupling of distillation with other unit operations such as membrane or crystallization cascades multiplies the degrees of freedom. Appendix Mass and energy balances for the superstructure variant with a variable condenser. Balances are given for each tray except the topmost and lowest trays. The balances for the topmost and the lowest trays can be established accordingly. 0)

∑b

F,k,nFkzk,i

k


Vnyn,i + bSl,nSlxS,i - bSV,nSVyS,i,

0)

∑b

F,k,nFkhf,k

n ∈ N \{1, Nmax}, i ∈ I (A1)


k

VnhV,n + bSl,nSlhl,S - bSV,nSVhV,S + bQD,nQD,

n ∈ N \{1, Nmax} (A2)

Mass and energy balances for the superstructure variant with a variable product draw. Balances are given for each tray except the topmost and lowest trays: 0)

∑b

F,k,nFkzk,i

k

0)


Vnyn,i + bSl,nSlxS,i - bSV,nSVyS,i - Dnxn,i, n ∈ N \{1, Nmax}, i ∈ I

∑b

F,k,nFkhf,k

(A3)


k

VnhV,n + bSl,nSlhl,S - bSV,nSVhV,S - Dnhl,n + QD,n, n ∈ N \{1, Nmax}

Nomenclature B ) bottoms stream [mol/s] b ) binary variable C ) cost [euro] c ) continuous variable D ) distillate stream [mol/s] Dcol ) diameter [m] F ) feed stream [mol/s] fc ) capital charge factor for objective function h ) enthalpy [kJ/kmol]

(A4)

Ind. Eng. Chem. Res., Vol. 48, No. 14, 2009 K ) equilibrium constant L ) liquid stream [mol/s] M ) Big-M constant/molar mass [kg/kmol] NT ) total number of trays Q ) energy [W] p ) pressure [Pa] R ) recycle stream [mol/s]/gas constant [kJ/kmol · K] S ) side stream [mol/s] T ) temperature [K] ta ) annual operation time [8000 h] TAC ) total annualized cost [euro] V ) vapor stream [mol/s] x ) liquid composition y ) vapor composition z ) total composition Greek Letters γ ) activity coefficient µ ) relaxation parameter σ ) standard deviatio Subscripts B ) bottom cap ) capital con ) condenser D ) distillate F ) feed i ) component i k ) feed stream l ) liquid n ) tray number max ) maximum min ) minimum op ) operating QD ) condenser R ) reflux reb ) reboiler S ) side stream Sl ) liquid side stream SV ) vapor side stream V ) vapor

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ReceiVed for reView January 27, 2009 ReVised manuscript receiVed May 5, 2009 Accepted May 7, 2009 IE900143E

Efficient Optimization-Based Design of Distillation ... - ACS Publications

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