A Disjunctive Programming Formulation for the Optimal Design of

Article pubs.acs.org/IECR

A Disjunctive Programming Formulation for the Optimal Design of Biorefinery Configurations José María Ponce-Ortega,*,† Viet Pham,‡ Mahmoud M. El-Halwagi,‡,§ and Amro A. El-Baz⊥ †

Department of Chemical Engineering, Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Michoacán, México, 58060 Department of Chemical Engineering, Texas A&M University, 3122 College Station, Texas 77843-3122, United States § Adjunct Faculty at the Chemical and Materials Engineering Department, King Abdulaziz University, Jeddah, Saudi Arabia ⊥ Environmental Engineering Department, Zagazig University, Zagazig, Egypt, 44519 ‡

ABSTRACT: This paper proposes a new general systematic approach for the design of optimal pathways for a biorefinery. The approach includes a set of sequential steps for the superstructure generation and a set of disjunctive programming models for the pathway optimization. The approach allows identification of optimal biorefinery configurations for a given criterion (for example the economic, environmental, or safety, etc). The proposed approach allows the user to solve easily (through a set of subproblems) a difficult problem, and it can be applied to any case study; in this paper, it is applied to a case study for the optimization of the bioalcohols production from lignocellulosic biomass, and the best economic route identified corresponds to the use of biomass using mixed-culture fermentation to yield ammonia carboxylates, then to obtain ester via esterification and finally to obtain the bioalcohols via hydrogenolysis with a cost of $15.14/GJ.

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INTRODUCTION Nowadays, the concept of a biorefinery, a processing facility that receives biomass feedstocks and produces several chemicals (including biofuels) through a system of physical/chemical/ biological processes, has gained a lot of attention because of the current oil depletion and the reduction in the greenhouse gas emission that this can yield. In this regard, many pathways have been developed and analyzed in the laboratories for the biofuel production; however, most of them have not been implemented in large scales because of techno-economic criteria. A common approach to designing configurations for a new biorefinery is to select first a bioconversion technology (e.g., pretreatment, hydrolysis, fermentation, digestion, gasification, pyrolysis), and then consider preprocessing and postprocessing technologies for feedstock preparation and product separation. This approach has the drawback that is not systematic and usually does not consider several pathways that may be economically and environmentally attractive (making difficult to include new technologies that may be desirable). Recently, several reports have been presented for the potential options of products for a biorefinery. The papers by Huber et al.,1 Saxena et al.,2,3 and Goyal et al.4 present reviews for different pathways for the production of different biofuels and biochemicals. The works of Kamm and Kamm5 and Fernando et al.6 review the products that can be produced by the future biorefinery considering lignocellulosic feedstock biorefinery, green biorefinery, whole corn biorefinery, and a biorefinery with integration of thermo-chemical and biochemical platforms. In addition Fernando et al.6 presented an approach to integrate biorefineries into petroleum refineries to produce several chemicals and fuels. The National Renewable Energy Laboratory has reported a list of the top-value chemicals that can be obtained from bioresources (Werpy and Petersen7 considering sugar and syngas as feedstock, and Holladay et al.8 considering lignin as feedstock). © 2012 American Chemical Society

On the other hand, several techniques have been reported for the reaction pathways synthesis using evolutionary and optimization approaches. In this context, Agnihotri and Motard9 presented an approach for the reaction path synthesis in the industrial chemistry; whereas Nishida et al.10 presented a review for the process synthesis at that time, where approaches for the reaction pathway were presented. Govind and Powers11 proposed the retro-synthesis approach for the reaction pathways. May and Rudd12 presented a practical method for synthesizing chemical reactions sequences considering thermodynamically feasibility, and Rotstein et al.13 proposed a graphical approach for the production of desired chemicals based on their properties, then Formari et al.14 extended the approach to consider two degrees of freedom. An approach for the synthesis of environmentally friendly reactions was reported by Crabtree and El-Halwagi,15 whereas Pistikopoulos et al.16 and Buxton et al.17 presented methods for the environmental impact minimization in the reaction networks. In the same context, Li et al.18 presented a hierarchical optimization procedure for a reaction path synthesis considering economic, thermodynamic, and kinetic criteria. In addition Ng et al.19 and Bao et al.20 presented hierarchical approaches for the synthesis and analysis of integrated biorefineries. In addition, several techno-economic analyses have been recently reported for the production of bioethanol (Adem et al.;21 Zhu and Jones;22 Phillips et al.;23 Dutta et al.;24 Kazi et al.25), biodiesel (Pokoo-Aikins et al.;26 Myint et al.27), different transportation biofuels (Pham et al.;28 Holtzapple and Granda;29 Jones et al.30), and bioenergy (Mohan and El-Halwagi;31 Qin et al.32). There are also other approaches that consider the minimization of the energy Received: Revised: Accepted: Published: 3381

July 23, 2011 January 23, 2012 January 25, 2012 January 25, 2012 dx.doi.org/10.1021/ie201599m | Ind. Eng. Chem. Res. 2012, 51, 3381−3400

Industrial & Engineering Chemistry Research

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Figure 1. Schematic representation for the addressed problem.

consumption (Alvarado-Morles et al.;33 Gosling;34 ChouinardDussault et al.35), scheduling issues (Elms and El-Halwagi36), and safety metrics (Pokoo-Aikins et al.37) for different pathways for the biofuels production. Previous works have the disadvantage that they do not provide a systematic analysis to determine the best route for a biorefinery, that is, to determine the optimal feedstock, intermediates, products, and processing routes among a lot of options. In this regard, recently Pham and El-Halwagi38 have presented a new systematic approach based on dynamic programming for biorefinery configurations. However, this approach is based on several heuristic rules to determine optimal routes. On the other hand, Raman and Grossmann39 reported integer programming techniques to optimize logical decisions expressed in terms of disjunctions, then Lee and Grossmann40 presented algorithms to solve nonlinear disjunctive programming problems, whereas Vecchietti et al.41 reported techniques to reformulate algebraically a disjunctive problem, and recently Grossmann and Lee,42 Lee and Grossmann,43 and Sawaya and Grossmann44,45 presented improved relaxations for nonlinear cases. In this context, recently the application of disjunctive programming techniques to different areas have gained a lot of attention because its potential to solve effectively different optimization problems; examples include application in the area of heat exchanger network,46−49 separations,50−53 and mass exchange networks,54−57 among others. Therefore, this paper proposes a general systematic approach based on a new general mathematical programming formulation for the optimal configuration of a biorefinery. The new model is based on a disjunctive programming formulation that allows the consideration of several possibilities that may represent the optimal pathways for a biorefinery. Several rules for the generation of the superstructure and the solution of the remaining problems are presented. The proposed approach allows the identification of different scenarios and to consider several objectives, and this is applied to a case study for the bioalcohols production from lignocellulosic biomass. The paper is organized as follows: first, the definition for the addressed problem; second, the general proposed approach followed by the superstructure and optimization formulations; then the application of the proposed approach shown in a case study; and finally conclusions of the paper.

PROBLEM DEFINITION The problem addressed in this paper can be stated as follows: Given a set of bioresources with specific characteristics, there is a desire to produce a given product (with specified characteristics). The problem consists then in identifying the best pathways including intermediate steps and technologies required to convert the bioresources to the final products. The addressed problem is challenging because there are a lot of pathways and the best one must be selected with respect to economic, environmental, and safety criteria (see Figure 1). In Figure 1, each conversion step represents a reactor (or a set of reactors) in addition to separation and treatment units. The following indexes are used in the model formulation presented in this paper, k is used to represent any compound in the pathway, t represents any conversion technology, i and j are indexes used to denote any equality and inequality constraints, respectively. m and m′ represent adjacent nodes in the pathway, index Adj is used to refer to adjacent nodes, n and n′ represent nonadjacent nodes in the pathway, and NonAdj is used to indicate nonadjacent nodes. K and T are sets for the compounds and the conversion technologies, respectively. It(k) and Jt(k) are sets used to denote the equality and inequality constraints for the conversion technology t to yield components k. Finally, Am,m′ and An,n′ are sets for adjacent and nonadjacent nodes.

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PROPOSED APPROACH The proposed approach for the optimal pathway biorefinery configuration consists of two sequential steps. In the first step, a general superstructure is developed. This superstructure must contain all the potential configurations that may be attractive and that may contain the optimal pathway for a give criterion. Then, based on this superstructure, a general disjunctive programming model is proposed to determine the optimal configuration. Figure 2 shows the main steps of the proposed approach. This way, the next section presents the steps required to formulate the superstructure and the optimization model is then presented. Superstructure Synthesis. To synthesize a general superstructure that involves the entire potential configurations to find the optimal one, this paper proposes to use the approach developed by Pham and El-Halwagi.38 This approach is constituted by the following steps: (1) Forward synthesis of biomass. For a given biomass, the potential intermediate products and the conversion technologies are considered in this step. (2) Backward 3382

dx.doi.org/10.1021/ie201599m | Ind. Eng. Chem. Res. 2012, 51, 3381−3400


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Figure 2. Steps of the proposed approach for the biorefinery configuration.

mental impact). For each technology (t), there is a set of equality (hit(k)(X)) and inequality (gjt(k)(X)) constraints that depend on the operating parameters (X); these constraints include the mass and energy balances, technical criteria, and economic and environmental constraints. The design parameters include the yield, temperatures, pressures, solvent selection, catalysis, etc. The general disjunctive programming formulation to solve simultaneously this problem is stated as follows:

generated product. For a given final product, the required species and the technologies are identified. (3) Matching. This step corresponds to direct connection of two identical species, if one of the species synthesized in the forward step is also generated by the backward step. (4) Interception. This step corresponds to the process of identifying a conversion step to connect two species in the forward and backward steps. Using the steps mentioned above, a general superstructure can be identified as seen in Figure 3. Optimization Formulation. Once the superstructure is synthesized, then an optimization formulation must be constructed to determine the optimal pathways of producing the desired product from the given feedstock and considering all the technologies available. For this task, two strategies are proposed in this paper, in the first one the overall problem is solved simultaneously, which allows the user to solve the problem automatically and to optimize simultaneously the operating parameters for the different technologies involved; whereas in the second one the problem is solved iteratively, which helps to reduce the problem computational complexity. These two strategies are presented as follows. Simultaneous Optimization Approach. Consider for example the superstructure shown in Figure 4, we are going to use the following notations: Yk is a Boolean variable that represents the existence of the component k in the solution. There are several routes to produce each component, for example, routes 21, 8, and 9 are identified to produce component F. The objective function to select the best route among Rk(X) considers the minimization for the total cost (or other objectives such as the minimization for the greenhouse gas emissions and overall environ-

∑

min R =

Rk(X ) (1)

k∈K

Subject to ⎡ ⎤ Yk ⎢ ⎥ ⎡ ⎤ t(k) ⎢ Y ⎢ ⎥⎥ ⎢ ⎢ ⎥⎥ ⎢ Rk = Rt(k)(X t(k)) ⎢ ⎥⎥ ⎢ ⎢ ⎥⎥ ⎢ ∨ ⎢ t(k) ⎥⎥ ⎢ t(k) ⎢ hi (X t(k)) = 0, ∀ i ∈ It(k)⎥⎥ ⎢ ⎢ t(k) ⎥⎥ ⎢ ⎢⎣ g j (X t(k)) ≤ 0, ∀ j ∈ Jt(k) ⎥⎦⎥ ⎢ ⎥ ⎢⎣ ⎥⎦

∀k∈K

And a set of logical relationships to model the interconnection between different nodes in the superstructure: yk =

∑ yt(k), t(k)

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k∈K (2)



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Figure 3. Generation of a superstructure for a biorefinery configuration.

Previous disjunction implies that if the component k is selected to appear in the optimal pathway, then the objective Rk (for example the associated cost) must be calculated according to the technology t(k) used, and the corresponding equality hit(k) and inequality gjt(k) constraints must be applied (these include mass and energy balances, technical constraints, environmental constraints, etc.). This formulation allows consideration of simultaneously the optimization for the design parameters Xt(k) (that could be the yield, solvent selection, catalyst, temperatures, pressures, etc.) for each technology employed. To reformulate previous disjunction as an algebraic problem, two approaches can be used, the big M and convex hull relaxations (see Vecchietti et al.41). The big M reformulation involves a lower number of constraints and no additional variables are required (see Raman and Grossmann39); on the other hand the convex hull reformulation involves additional variables and constraints (see Sawaya and Grossmann44,45). This means that the big M reformulation is easier to implement. However, one of the major drawbacks for the big M reformulation is the way to determine the big M parameter for each relationship used, and that the convex hull reformulation is tighter than the big M reformulation (Lee and Grossmann40,43), this situation is very important for the solution approach, because a better lower bound for the relaxed solution improves the solution procedure. Recently, new relationships have been reported for improving the convex hull reformulation to avoid singularities (see Sawaya and Grossmann44,45); however, the major drawback for these convex hull reformulations is that some nonconvex terms still appear; however, for the linear cases, the convex hull reformulation is also linear and this way convex (Vecchietti et al.41 and Grossmann and Lee42). Therefore, on

the basis of the previous analysis, in this paper the convex hull reformulation is used for linear cases (to have tight relaxation and linear-avoiding to determine the big-M parameter) and the big M reformulation is used for nonlinear cases (to avoid additional nonconvex terms and compact relationships). The big-M reformulation for the previous disjunction can be stated as follows. First for the objective function, the equality constraints are stated as two inequalities (greater and lower than): Rk ≤ Rt(k)(X t(k)) + Rmax(1 − yt(k)),

∀k∈K

(3)

Rk ≥ Rt(k)(X t(k)) − Rmin(1 − yt(k)),

∀k∈K

(4)

This way, when the technology t is selected, the binary variable yt(k) is equal to 1 and the last term of relationships 3 and 4 (i.e., (1 − yt(k))) becomes zero eliminating the big M parameter (Rmax); therefore, relationship 3 states that Rk ≤ Rt(k)(Xt(k)) and relationship 4 states that Rk ≥ Rt(k)(Xt(k)), and the only way to satisfy these relationships is that Rk = Rt(k)(Xt(k)), which is equivalent to properly applying the equation in the disjunction. On the other hand, when the technology t is not selected, the binary variable yt(k) is equal to zero and the last term of relationships 3 and 4 are equal to the big M parameters (Rmax and Rmin are upper and lower bounds for the corresponding relationships deterring before the optimization); then, the relationships 3 and 4 are relaxed (i.e., the value of Rk is not constrained by relationships 3 and 4 because the big M parameters help to relax these relationships). 3384



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Figure 4. Superstructure for synthesized.

For the equality constraints (i.e., mass and energy balances and technical equations for each technology) for the technology selected, the following relationships are stated: hit(k)(X t(k)) ≤ Rmax(1 − yt(k)),

A similar explanation applies for this case, but only one relationship is required. When the technology t is selected, the binary variable yt(k) becomes one and the right-hand of relationship 7 is zero and the relationship applies properly. On the other hand, when the technology is not selected the binary variable yt(k) is 0; and the big M parameter Rmax relaxes relationship 7. On the other hand, this paper suggests the use of the convexhull reformulation for the cases when the relationships involved in the disjunction are linear:

∀ i ∈ It(k), k ∈ K (5)

hit(k)(X t(k)) ≥ −Rmin(1 − yt(k)),

∀ i ∈ It(k), k ∈ K (6)

⎡ ⎤ Yk ⎢ ⎡ ⎤⎥ ⎢ Y t(k) ⎢ ⎥⎥ ⎢ ⎢ k ⎥⎥ T ⎢ r X C = + R t(k) t(k) t(k)⎥⎥ ⎢ ⎢ ⎢ ⎥⎥ ∨ h ⎢ h ⎢ A t(k)X t(k) = b t(k) ⎥⎥ ⎢ t(k) ⎢ ⎥⎥ ⎢ g g ⎢ A X ⎥⎥ b ≤ ⎢ t ( k ) t ( k ) t(k) ⎣ ⎦⎥ ⎢ ⎥ ⎣ ⎦

In this case, when the technology t is selected, the binary variable yt(k) becomes 1 and the right-hand side of relationships 5 and 6 are zero; this way, both equalities 5 and 6 together become an equality (i.e., hit(k)(Xt(k)) = 0). On the other hand, when the technology t is not selected, the binary variable yt(k) is zero and the right-hand side of relationships 5 and 6 is equal to the upper (Rmax) and lower (Rmin) limits; and this way, relationships 5 and 6 are relaxed. For the inequality constraints inside the disjunction (i.e., technical and practical feasible limits for the operation of the technology), the following relationship is used: g jt(k)(X t(k)) ≤ Rmax(1 − yt(k)),

∀k∈K

In previous relationships, Rk is the objective to be optimized (i.e., cost, greenhouse gas emissions, etc), rt(k) is the constant value to determine the performance of a given technology and this is determined previous to the

∀ j ∈ Jt(k), k ∈ K (7) 3385



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variables rt(k) and xt(k) associated to the technology t are max max min limited between upper (rt(k) and xt(k) ) and lower (rt(k) and min xt(k) ) bounds, for all other cases the continuous disaggregated variables are zero because of relationships 13 and 14. This way, the original continuous variables (Rk and Xt(k)) are equal to the disaggregated variables associated to the technology t selected (because all others are equal to zero for relationships13 and 14). Then the linear relationships inside the disjunction (relationships 10−13) are stated in terms of the disaggregated variables, but remember that just the disaggregated variables associated to the technology selected are different than zero and equal to the original continuous variables applying proper relationships inside the disjunction; whereas for the cases when the technology is not selected, the associated disaggregated variables are set as zero and relationships 10−13 are relaxed. Previous relationships allows optimization of the operating conditions for each technology depending on the other technologies selected; which can be important to reduce the overall cost for the pathway. However, these relationships can be very complicated and involve a lot of binary variables. Alternatively, the sequential optimization approach presented in the next section can be used. Sequential Optimization Approach. Consider for example the superstructure shown in Figure 4, notice that there are several adjacent routes (i.e., different technologies are available to obtain the same product from the same feedstock, see for example node A to node D). In this regard, this paper proposes first to solve disjunctive programming formulations for an optimum technology between the adjacent nodes that involve multiple technologies. Then, to simplify the superstructure, the problems for the nonadjacent nodes are solved through disjunctive programming formulations. Nonadjacent nodes are for example the node G to node O in Figure 4, where there is one technology to go directly from G to O and two sequential steps that go from G to O via J. The solution for these disjunctive formulations yields a simplified superstructure than can then be solved using a general disjunctive programming formulation. Those model formulations are presented in the next sections. Disjunctive Programming Model for Adjacent Nodes. First, policies between all adjacent nodes are optimized to determine the best route from one node m to the adjacent one m′. To do this task, this paper proposes the use of the following general disjunctive programming formulation:

optimization process, Xt(k) is the vector for the optimization variables (for example solvent selection, catalysis, temperatures, pressures, etc), Ct(k) is a vector for the performance of h a given technology, and At(k) and bht(k) are the matrix and vector of coefficients for the equalities constraints (this can be obtained from the mass and energy balances, etc.), and g g finally At(k) and bt(k) are matrix and vector coefficients for the inequalities constraints (these can be obtained from technical, environmental, and safety limits). The convex hull reformulation of previous linear disjunction is stated as follows. First, the continuous variables (Rk and Xt(k)) are disaggregated (Drt(k) and Dxt(k)) for each disjunctive term as follows: Rk =

∑ Drt(k),

∀k∈K

(8)

t(k)

X t(k) =

∑ Dxt(k),

∀k∈K

(9)

t(k)

Then, the relationships for each disjunction are stated in terms of the disaggregated variables as follows: Drt(k) = r t(k)Dxt(k) + Ct(k)yt(k) , ∀t(k) ∈ T (k), k ∈ K

(10)

A ht(k)Dxt(k) = bht(k)yt(k) ,

∀t(k) ∈ T (k), k ∈ K

g A t(k)Dxt(k) ≤ b g t(k)yt(k) ,

∀t(k) ∈ T (k), k ∈ K

(11)

(12)

Finally, limits are imposed to the disaggregated variables as follows: rtmin yt(k) ≤ rt(k) ≤ rtmax yt(k) , (k) (k)

∀ t(k) ∈ T (k), k ∈ K (13)

xtmin yt(k) ≤ xt(k) ≤ xtmax yt(k) , (k) (k)

∀ t(k) ∈ T (k), k ∈ K (14)

To explain the previous convex hull relaxation for the linear relationships, we have the following. First, if the technology t is selected, then just the binary variable associated to this technology is equal to 1 (and all others are zero); then from relationships 13 and 14, the continuous

Adj ⎡ ⎤ min R m , m ′ ⎢ ⎥ ⎡ ⎤⎥ Adj ⎢ Y ⎢ ⎥⎥ t Adjm . m ′ ⎢ ⎢ ⎥⎥ ⎢ Adjm . m ′ ⎢ ⎥⎥ Adj ⎢ R m . m ′ = R Adj (X Adj ) ⎢ ⎥⎥ t m.m′ t m.m′ ⎢ ⎢ ⎥⎥ , ∨ ⎢ Adj Adj (X Adjm . m ′) = 0, ∀ i ∈ I Adjm . m ′ ⎥⎥ ⎢ t m , m ′ ⎢ h Adj t ⎢ i ,t m.m′ t ⎥⎥ ⎢ ⎢ ⎥⎥ ⎢ ⎢ g Adj ⎥⎥ ≤ ∀ ∈ ( X ) 0, j J ⎢ Adj Adj Adjm . m ′ t m . m ′ t m . m ′⎥⎦ ⎢ , j t ⎣ ⎢ ⎥ ⎢⎣ ⎥⎦

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∀ m , m′ ∈ A m , m ′



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where RAdjm,m′ is the objective function to be minimized. The function can be total cost, energy required, and environmental impact, among others. Its value depends on the technology used and can be optimized through the design parameters. For each processing technology, a set of equality Adj Adj (hi,tAdjm.m′ ) and inequality (gj,tAdjm.m′ ) constraints must be satisfied (i.e., mass and energy balances, feasibility, and technical constraints, between others). To explain the previous

disjunction, we have the following. For two adjacent nodes m and m′, only one technology t Adjm.m′ must be selected. If a certain technology t Adjm.m′ is selected, then the associated Boolean variable is true, and the associated relationships are applied. On the other hand, if a technology is not selected, the Boolean variable is false and the associated constraints are relaxed. For the cases where it is possible to formulate the performance equations as linear relationships, the following disjunction can be stated:

Adj ⎡ ⎤ min R m , m ′ ⎢ ⎥ ⎡ ⎤⎥ Adj ⎢ Y Adj ⎢ ⎥ t m.m′ ⎢ ⎥ ⎢ ⎥⎥ ⎢ Adj ⎢ R Adjm , m ′ = r Adj ⎥ T ⎢ Adjm . m ′X Adjm . m ′ + C Adjm . m ′⎥⎥ ⎢ t t t ⎢ ⎥ ⎢ ⎥⎥ , ⎢ Adj∨ h ⎢ ⎥ Adj ⎢ t m,m′ ⎥ = bhAdj A Adj X Adj ⎢ ⎥⎥ t m.m′ t m.m′ t m.m′ ⎢ ⎢ ⎥⎥ ⎢ g ⎢ ⎥⎥ g Adj ⎢ ≤ b Adj A Adj X Adj ⎢⎣ ⎥⎦⎥ t m.m′ t m.m′ t m.m′ ⎢ ⎢⎣ ⎥⎦

Adj

∀ m , m′ ∈ A m , m ′

node:

where rtAdjm.m′ is a constant performance parameter for technology t Adjm.m′ connecting two adjacent nodes m and m′ (i.e., unitary costs optimized previously, or the unitary GHG emissions, between others); XtAdjm.m′ is a vector for the performance of the parameters of the technology tAdjm.m′ (for example yield, temperatures, pressure, solvent selecAdjg Adjh Adj are coefficient matrices; and tion, etc); AtAdjm.m′ and At m.m′ g h btAdjm.m′ and btAdjm.m′ are vector parameters (for the mass and

∑ t Adjm . m ′

Adj

y Adj = 1, t m.m′

∀ m , m′ ∈ A m , m ′ (15)

Previous disjunction can be reformulated as an algebraic mathematical programming problem using the big-M or the convex-hull reformulations (see Raman and Grossmann;39 Vecchietti et al. 41). The same approach shown in relationships 2−7 for the big M and relationships 8−14 for the convex hull reformulation is implemented for this case. Disjunctive Programming Model for Nonadjacent Nodes. After the routes for the adjacent nodes are optimized, the routes for nonadjacent nodes from n to n′ are optimized through the following disjunction:

energy balances as well as for the technical and economic limits). Then, the Boolean variables are reformulated as binary variables, when the Boolean variable is true the associated binary variable is 1. To select only one technology, the following constraint is required for each adjacent

NonAdjn , n ′ ⎧ ⎫ min R ⎪ ⎪ ⎡ ⎤ NonAdjn , n ′ ⎪ ⎪ Y NonAdjacent ⎢ ⎥⎪ ⎪ n.n′ t ⎢ ⎥ ⎪ ⎪ ⎢ ⎥⎪ NonAdjn , n ′ ⎪ NonAdjn , n ′ ⎢ R = R NonAdjacent (X NonAdjn , n ′) ⎥⎪ ⎪ n.n′ t ⎪ t ⎢ ⎥⎪ ⎨ ∨ ⎢ ⎥⎬ , NonAdjn , n ′ ⎪ t NonAdjn , n ′ ⎢ ⎪ NonAdjn , n ′ = R NonAdjacent (X NonAdjn , n ′) ⎥⎪ R ⎪ ′ n n . t t ⎢ ⎥ ⎪ ⎢ NonAdj ⎥⎪ ⎪ ⎪ n,n′ ⎢g (X NonAdjn , n ′) ≤ 0, ∀ j ∈ J NonAdjn , n ′⎥⎪ ⎪ t ⎢⎣ j , t NonAdjn , n ′ t ⎥⎦ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭

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∀ n , n′ ∈ A n , n ′



Article NonAdj (gj,tNonAdjn.nn.n′ ′)

NonAdj (hi,tNonAdjn.nn.n′ ′)

where RNonAdjn,n′ is the objective function to be minimized (economic, environmental, etc.). The explanation for this disjunction for the nonadjacent nodes is similar to the one corresponding to the adjacent nodes considering the equality

and inequality constraints for two nonadjacent nodes n and n′. For the cases where it is possible to formulate the performance equations as linear relationships, the following disjunction can be stated:

NonAdjn , n ′ ⎡ ⎤ min R ⎢ ⎥ ⎡ ⎤⎥ NonAdjn , n ′ ⎢ Y ⎢ ⎥⎥ NonAdjn , n ′ ⎢ n,n′,t ⎢ ⎥⎥ ⎢ ⎢ NonAdjn , n ′ NonAdjn , n ′ ⎥⎥ ⎢ T ⎢ RNonAdjn , n ′ = r ⎥ + X C ⎢ NonAdjn , n ′ NonAdjn , n ′ NonAdjn , n ′⎥⎥ ⎢ t t t ⎢ ⎥ ∨ ⎥⎥ , ⎢ t NonAdjn , n ′ ⎢ NonAdjh ⎢ ⎥ ⎢ ⎥ A NonAdj X NonAdjn , n ′ = bhNonAdj ⎢ ⎥⎥ n,n′ t n,n′ t t ⎢ ⎢ ⎥ ⎢ ⎥ g NonAdj g ⎢ ⎥⎥ ⎢ ≤ A X b NonAdj NonAdjn , n ′ t NonAdjn , n ′ n,n′ ⎢⎣ ⎥⎦⎥ t t ⎢ ⎢⎣ ⎥⎦

NonAdj ′ is a constant performance parameter for where rtNonAdjn,nn,n ′ technology t NonAdjn,n′ connecting two nonadjacent nodes n and n′; XtNonAdjn,n′ is a vector for the performance of the h

parameters Xtroute; and for each processing technology a set of equality (htroute) and inequality (gtroute) constraints must be satisfied. The general explanation for the disjunction is similar to the previous one. For the cases when it is possible to formulate the performance equations as linear relationships, the following disjunction can be stated:

NonAdj g

NonAdjn.n′ and At NonAdjn.n′ parameters for technology t NonAdjn,n′; AtNonAdj g are coefficient matrices; and bthNonAdjn.n′ and btNonAdjn.n′ are vector parameters.

The Boolean variables are reformulated as binary variables. When the Boolean variable is true the associated binary NonAdjn.n variable (ytNonAdjn.n′ ′) is 1. To select only one technology the following relationship is required for each pair of nonadjacent nodes:

∑ t NonAdjn . n ′

NonAdj

y NonAdjn . n ′ = 1, n.n′ t

∀ n , n′ ∈ A n , n ′

min R

⎤ ⎡ Y t route ⎥ ⎢ ⎢ R = r routeXT + C route ⎥ t t ⎥ ⎢ ∨ ⎢ ⎥ h route t ⎢ A t routeX = bt route ⎥ ⎥ ⎢ ⎢ A t routeX ≤ b groute ⎥ ⎦ ⎣ t

∀ n , n′ ∈ A n , n ′ (16)

And previous disjunctions can be reformulated as algebraic relationships in a similar way than relationships 3−7for the big M; and using relationships 8−14 for the convex hull for the linear case. Disjunctive Programming Model for Simplified Superstructure. Finally, a disjunctive programming model is formulated for the remaining problem as follows:

In previous disjunction, rtroute is a constant performance parameter for technology t route, XTtroute is a vector for the performance of the parameters for the technology t route, A i,troute and Aj,troute are coefficient matrices, whereas btroute h and btgroute are vector parameters. Only one technology can be used, and this is stated as follows:

min R

⎡ ⎤ Y t route ⎢ ⎥ ⎢ ⎥ R = R t route(X t route) ⎥ ∨ ⎢ route ⎢ ⎥ route route route = ∀ ∈ h X i I ( ) 0, t t t ⎢ t ⎥ ⎢ g route(X route) ≤ 0, ∀ j ∈ J route ⎥ ⎣ t ⎦ t t

∑ t route

yt route = 1

(17)

and previous disjunctions can be reformulated as was presented by relationships 3−7 and 8−14 for the nonlinear and linear cases, respectively. It is noteworthy that the major advantage of the sequential approach is that this can be easily implemented, avoiding large and complex problems that must be solved simultaneously. However, the sequential approach does not allow the consideration of the interactions between successive nodes, because this sequential approach only

where R is the objective function to be minimized. It can be the total cost, the energy required, and the environmental impact, among others, and this depends on the technology used than what can be optimized through the design 3388



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obtain esters via two routes, one is first acid springing and then esterification and the other one is via ketonization. Notice in Figure 4 that the upper half includes biological conversions while the lower half involves chemical conversions. Once the superstructure is identified, the two optimization approaches proposed in this paper are used: the simultaneous optimization approach and the sequential optimization approach. For both cases, the data for the different identified routes are shown in Tables 1, 2, and 3. This information

considers the effect for the interactions between adjacent nodes before to obtain the simplified superstructure. For linear and nonlinear cases, this sequential approach discharges the possibility to simultaneously optimize the performance parameters for successive nodes. For the sequential approach, the interaction between successive nodes only is considered in the last simplified superstructure obtained (where several interactions have been discharged). This situation could provide significant differences between these two solutions, and these differences could be more significant when nonlinear relationships are involved. On the other hand, for the case when the performance parameters are previously optimized, the sequential approach yields the same solution as the simultaneous solution.

Table 1. Low Yield Conversion Steps (Pham and El-Halwagi)38

■

CASE STUDY Two cases are presented in this paper to show the applicability of the proposed approach. In the first case, a linear case is presented, whereas in the second case some nonlinear equations for the cost functions to account for the economies of scale are considered. Case a: Linear Case. In this paper, we have considered the case of determining the optimum pathway for the production of fuel-grade alcohols from lignocellulosic biomass. The objective is the minimization of the route cost based on the dollars per KJ produced by the bioalcohols. The solution starts with the synthesis of a superstructure. For this task, first the forward branching tree is constructed. Notice in Figure 4 that there are identified five products for biomass (ammonia carboxylates, calcium carboxylates, sugar, bioalcohols, syngas, and methane) through different technologies (mixedculture fermentation, mixed-culture fermentation, acid hydrolysis, cellulose hydrolysis, ABE fermentation, pyrolysis, gasification, landfill, and digestion). Second, the backward branching tree is developed. The bioalcohols can be obtained by ester, acid carboxylic, ketone, sugar, biomass, syngas, ethylene, bromo-ethane, aldehyde, and chloro-methane using several technologies as it can be seen in Figure 4 (hydrogenolysis, hydrogenation, Grignard synthesis, hydrogenation, aqueous phase reforming, ethanol fermentation, ABE fermentation, methanol catalytic synthesis, syngas fermentation, mixed alcohol catalytic synthesis, indirect hydrolysis, hydration, hydrolysis, hydrogenation, hydrolysis). Next step consists in indentifying the matching of two species of the forward and backward branching trees; this is the case of route 5 in Figure 4. Then, the interception step is considered; for example, this step identifies that the ammonia carboxylates can be used to obtain esters via esterification; also calcium carboxylates can be used to

conversion steps

feed

product

(6) pyrolysis

biomass

syngas

(14) aqueous phase reforming (15) aqueous phase reforming (19) syngas fermentation

sugar

(24) chlorination

yield

yield base

ketones

max 29.2% 23.7%

biomass weight fed carbon weight

sugar

alcohols

8.7%

fed carbon weight

syngas

alcohols

53.1%

methane

chloromethane

max 12%

carbon monoxide weight methane weight

corresponds to experimental data reported in the literature. In addition, each route identified is characterized technologically and economically, and this way the disjunctive programming models for linear relationships can be applied. The mathematical programming problems resulting from the disjunctions reformulation were coded in the software GAMS (Brooke et al.58) and were solved using the solver CPLEX. Simultaneous Approach. Based on the information provided in the superstructure shown in Figure 4 and the simultaneous optimization model presented in this paper, the following disjunctive programming formulation is obtained, min R = RO + R J + RL + RM + RN + RK + RG + RH + RI + RB + RC + RD + RE + RF + R A (18)

Subject to the following disjunctions. First for the product O, there are the several alternative routes that can be represented by the following disjunction:

⎤ ⎡ YO ⎥ ⎢ ⎤ ⎡ ⎡ ⎤ D H ⎥ ⎢ Y Y ⎡ ⎤ J ⎤ ⎢ ⎥ G ⎥ ⎡ YA ⎤ ⎢ ⎥ ⎢ Y30 ⎥ ⎡⎢ Y 26 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 5 H D H D ⎢ ⎢ ⎥ ⎥ ∨ ∨ ⎢ ⎢ Y27 ⎥ ⎢ Y28 ⎥⎥ ∨ ∨ ⎢ ⎢ Y15 ⎥ ⎢ Y16 ⎥⎥ ∨ ... ⎥ ⎢ O O ⎢ ⎥ O ⎢ ⎥ ⎢ ⎥ ∨ ∨ ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ ⎣ R = r30 ⎦ ⎣ R = r26 ⎦ ⎢ ⎢ RO = r ⎥ ⎢ RO = r ⎥⎥ ⎣ R = r5 ⎦ ⎢ ⎢ RO = r ⎥ ⎢ RO = r ⎥⎥ ⎣ ⎦ 27 ⎦ 28 ⎦ 15 ⎦ ⎣ 16 ⎦ ⎦ ⎥ ⎢ ⎣⎣ ⎣⎣ ⎥ ⎢ ⎡ ⎤ ⎡ ⎤ E K ⎥ ⎢ Y Y ⎥ ⎢ ⎥ ⎡ Y L ⎤ ⎡ Y M ⎤ ⎡ Y N ⎤⎥ ⎢ ⎢⎡ ⎤ ⎡ ⎡ ⎤ ⎡ ⎤ ⎤ ⎤ ⎡ E 35 ⎥ ⎢ 36 ⎥ ⎢ 37 ⎥ K K E E ∨ ∨ ⎢ ∨⎢ ⎢ Y18 ⎥ ⎢ Y19 ⎥ ⎢ Y20 ⎥⎥ ∨ ⎢ ⎢ Y31 ⎥ ⎢ Y32 ⎥⎥ ∨ ⎢⎢ O ⎥ ⎢ O ⎥ ⎢ O ⎥⎥ ⎥ ⎢ ∨ ∨ ∨ ⎥ ⎢ ⎢⎢ O ⎥⎦ ⎢⎣ RO = r ⎥⎦ ⎢⎣ RO = r ⎥⎦⎥ ⎢ ⎢⎣ RO = r ⎥⎦ ⎢⎣ RO = r ⎥⎦⎥ ⎣ R = r35 ⎦ ⎣ R = r36 ⎦ ⎣ R = r37 ⎦⎥ ⎢ r R = ⎣ 31 32 ⎦ 18 19 20 ⎦ ⎣ ⎥⎦ ⎣⎢ ⎣

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⎡ YN ⎤ ⎢ ⎥ ⎢ YF ⎥ 24 ⎢ ⎥ ⎢ N ⎥ ⎣ R = r24 ⎦

Then, for the other component we have the following disjunctions: ⎡ ⎤ YJ ⎢ ⎥ ⎢⎡ B ⎤ ⎡ Y G ⎤⎥ Y 25 ⎥⎥ 10 ⎥ ⎢ ⎢⎢ ⎢⎢ J ⎥⎥ ⎥∨⎢ J ⎢⎣⎣ R = r10 ⎦ ⎣ R = r25⎦⎥⎦

Notice in this case that the performance (R) is constant for each technology and this represents the associated cost (see Tables 1, 2, and 3). These are linear relationships and therefore the convex−hull reformulation can be used. First the logical relationships can be stated as follows:

⎡ YB ⎤ ⎢ ⎥ ⎢ YA ⎥ 1 ⎢ ⎥ ⎢ B ⎥ ⎣ R = r1⎦

J + yG + y H + y H + y A + y D + y D + y E yO = y30 26 27 28 5 15 16 18

⎡ ⎤ YG ⎢ ⎥ ⎢⎡ C ⎤ ⎡ Y D ⎤⎥ ⎢⎢ Y ⎥⎥ ⎥∨⎢ ⎢⎢ G ⎥ ⎢ RG = r ⎥⎥ R r = 13 ⎦⎦ 11⎦ ⎣ ⎣⎣

E + y E + yK + yK + yL + y M + y N + y19 20 31 32 35 36 37

⎡ C ⎤ ⎥ ⎢ Y ⎢ YA ⎥ 2 ⎥ ⎢ ⎥ ⎢ C ⎣ R = r2 ⎦ ⎡ ⎤ YH ⎢ ⎥ ⎢⎡ C ⎤ ⎡ Y D ⎤⎥ Y ⎢⎢ ⎥⎥ ⎥∨⎢ ⎢⎢ RH = r ⎥ ⎢ RH = r ⎥⎥ 14 ⎦⎦ 12 ⎦ ⎣ ⎣⎣

⎡ ⎤ YD ⎢ ⎥ ⎢⎡ A ⎤ ⎡ Y A ⎤⎥ Y 3 4 ⎢⎢ ⎥∨⎢ ⎥⎥ ⎢⎢ D ⎥ ⎢ D ⎥⎥ = = R r R r ⎢⎣⎣ 3⎦ ⎣ 4 ⎦⎥ ⎦ ⎤ ⎡ YE ⎥ ⎢ ⎢⎡ ⎤⎥ D ⎤ ⎡ Y A ⎤ ⎡ YA ⎤ ⎡ F Y Y 7 6 ⎢⎢ ⎥∨⎢ ⎥∨⎢ ⎥∨⎢ ⎥⎥ ⎢⎢ E ⎥ ⎢ E ⎥ ⎢ E ⎥ ⎢ E ⎥⎥ ⎢⎣⎣ R = r17 ⎦ ⎣ R = r6 ⎦ ⎣ R = r7 ⎦ ⎣ R = r22 ⎦⎥⎦

⎡ ⎤ YF ⎢ ⎥ ⎢⎡ A ⎤ ⎡ ⎤⎥ A ⎤ ⎡ E ⎢⎢ Y8 ⎥ ⎢ Y9 ⎥ ⎢ Y21 ⎥⎥ ⎢⎢ F ⎥∨⎢ F ⎥∨⎢ F ⎥⎥ ⎢⎣⎣ R = r8 ⎦ ⎣ R = r9 ⎦ ⎣ R = r21⎦⎥⎦

⎡ YK ⎤ ⎥ ⎢ ⎥ ⎢ YI 29 ⎥ ⎢ ⎥ ⎢ K ⎣ R = r29 ⎦

⎡ YI ⎤ ⎥ ⎢ ⎢ YF ⎥ 23 ⎥ ⎢ ⎥ ⎢ I R r23⎦ = ⎣ ⎡ YL ⎤ ⎥ ⎢ ⎢ YK ⎥ 33 ⎥ ⎢ ⎥ ⎢ L = R r33⎦ ⎣ ⎡ YM ⎤ ⎥ ⎢ ⎥ ⎢ YK 34 ⎥ ⎢ ⎥ ⎢ M = R r ⎣ 34 ⎦ 3390

(19)

J y J = y30

(20)

B + yG y J = y10 25

(21)

C D yG = y11 + y13

(22)

G + yG yG = y25 26

(23)

C + yD y H = y12 14

(24)

H + yH y H = y27 28

(25)

y D = y3A + y4A

(26)

D + yD + yD + yD + yD y D = y13 14 15 16 17

(27)

D F y E = y17 + y6A + y7A + y22

(28)

E + yE + yE + yE y E = y18 19 20 21

(29)

E y F = y8A + y9A + y21

(30)

F + yF + yF y F = y22 23 24

(31)

y B = y1A

(32)

B y B = y10

(33)

yC = y2A

(34)

C + yC yC = y11 12

(35)

F y I = y23

(36)

I y I = y29

(37)

I y K = y29

(38)

K + yK + yK + yK y K = y31 32 33 34

(39)

K y L = y33

(40)

L y L = y35

(41) dx.doi.org/10.1021/ie201599m | Ind. Eng. Chem. Res. 2012, 51, 3381−3400


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Table 2. Data for the Pathway Nodes (Pham and El-Halwagi)38 nodes

feed

product

routes

A→F

biomass

methane

A→E

biomass

syngas

H→O

ketones

alcohols

K→O

ethylene

ethanol

(7) gasification and (21) methanation (8) landfill (9) digestion (7) gasification (9) digestion and (22) autothermal reforming (27) grignard synthesis (28) hydrogenation (31) indirect hydrolysis

G→O

acid carboxylic

alcohols

E→O

syngas syngas

methanol alcohols

key comments

(32) hydration (33) hydrobromination and (35) hydrolysis (34) hydroformylation and (36) hydrogenation (25) esterification and (30) hydrogenolysis (26) hydrogenation

production cost: $8.53/GJ of methane production cost: $1.90−$3.79/GJ of methane production cost: $0.20−$0.55/GJ of methane energy efficiency: 82.8% energy efficiency: 63% yield is 82−88% yield is 100% well developed and commercialized in 1960s but phased out due to less economic than hydration simple, direct, and most costly effective pathway involve many more steps than hydration pathway involve many more steps than hydration pathway include mild esterification (203 kPa and 50 °C) and hydrogenation (160 °C and 405 kPa) involve furnace, intense hydrogenation (230−270 °C and 4.1−7.1 MPa), and expensive molecular sieve. production cost ($2010): $19.98/GJ methanol production cost ($2010): $19.98/GJ methanol

(18) methanol synthesis (20) mixed alcohol synthesis

Table 3. Cost Data for the Pathways (Pham and El-Halwagi)38 base year

pathways A→B→J→O A→C→H→O A→D→G→J→O A→D→O A→O A→E→O A→F→I→K→O

year 2011

capacity MMGPY

product cost $/gal

energy density MJ/gal

date

45

1.21

92

PPI

alcohols cost $/gal

$/GJ

αt(k)

GHG kg/JG

2007

213.7

1.39

15.14

0.90

−20

description

product

mixed alcohols production via acid fermentation and esterification mixed alcohols production via acid fermentation and ketonization ethanol production via acetongen fermentation and ester synthesis ethanol production via hydrolysis and yeast fermentation mixed butanol and ethanol production via ABE fermentation methanol production via biomass gasification ethanol production via syntheses of methane, acetylene, and ethylene

mixed alcohols mixed alcohols ethanol

35

1.44

101

2009

229.4

1.54

15.28

0.92

−22

n/a

n/a

n/a

n/a

n/a

n/a

n/a

0.99

−19

ethanol

50

1.03

79

2003

161.8

1.56

19.69

0.87

18

mixed alcohols methanol

n/a

1.50

110

2007

214.8

1.71

15.47

0.85

−12

24

0.61

59

2001

151.8

1.18

19.98

0.88

5

ethanol

13

2.74

79

2008

245.5

2.73

34.43

0.99

−10

K y M = y34

(42)

G G RG = DR11 + DR13

(49)

M y M = y36

(43)

H + DR H RH = DR12 14

(50)

F y N = y24

(44)

RD = DR3D + DR 4D

(51)

N y N = y37

(45)

RC = DR2C

(52)

RB = DR1B

(53)

E + DR E + DR E + DR E RE = DR17 6 7 22

(54)

F RF = DR8F + DR 9F + DR21

(55)

I RI = DR23

(56)

(47)

K RK = DR29

(57)

(48)

L RL = DR33

(58)

y A = y1A + y2A + y3A + y4A + y6A + y7A + y8A + y9A (46)

Then, the continuous variables must be disaggregated as follows: O + DRO + DRO + DRO + DR O + DR O RO = DR30 26 27 28 5 15 O O O O O O + DR16 + DR18 + DR19 + DR20 + DR31 + DR32 O O O + DR35 + DR36 + DR37 J

J

R J = DR10 + DR25

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M RM = DR34

(59)

N RN = DR24

(60)

Then, the relationships are stated in terms of the disaggregated variables as follows:

E D DR17 = r17y17

(86)

DR 6E = r6y6A

(87)

DR7E = r7y7A

(88)

E F DR22 = r22y22

(89)

O J DR30 = r30y30

(61)

O G DR26 = r26y26

(62)

O H DR27 = r27y27

(63)

DR8F = r8y8A

(90)

O H DR28 = r28y28

(64)

DR 9A = r9y9A

(91)

DR5O = r5y5A

(65)

F E DR21 = r21y21

(92)

O D DR15 = r15y15

(66)

I DRK = r29y29

(93)

F DRI = r23y23

(94)

O D DR16 = r16y16

(67)

O E DR18 = r18y18

(68)

O E DR19 = r19y19

(69)

K DRL = r33y33

(95)

O E DR20 = r20y20

(70)

K DRM = r34y34

(96)

O K DR31 = r31y31

(71)

F DRN = r24y24

(97)

O K DR32 = r32y32

(72)

O L DR35 = r35y35

(73)

O M DR36 = r36y36

(74)

O N DR37 = r37y37

(75)

J

B DR10 = r10y10

J

Notice that in this case, there is no requirement of upper bounds because of the previous relationships. Then, the model is a mixed-integer linear programming problem and consists of 30 binary variables, 55 continuous variables, and 80 constraints. The model is solved in a CPU time of 30 s and (including the constraint for the existence of the existence of O (i.e., yO = 1)) the solution implies that the best route corresponds to the pathway ABJO with a total cost of $15.14/GJ. To avoid the complexities associated to the construction of the general disjunctive programming formulation and at the same time the number of binary variables and logical relationships, the following section presents an efficient way to solve this problem sequentially. Sequential Approach. For the adjacent nodes AD (biomass−sugar), there are two available technologies (i.e., 3 (acid hydrolysis) and 4 (cellulose hydrolysis)) and the following mathematical programming problem must be solved:

(76)

G DR25 = r25y25

(77)

DR1B = r1y1A

(78)

G C DR11 = r11y11

(79)

G D DR13 = r13y13

(80)

DR2C = r2y2A

(81)

H C DR12 = r12y12

(82)

H D DR14 = r14y14

(83)

DR3D = r3y3A

(84)

DR 4D = r4y4A

(85)

min R AD s. t . y3AD + y4AD = 1 R AD = r3ADy3AD + r4ADy4AD

(98)

AD

where R is the economic objective ($/KJ) for the node AD; y3AD and y4AD are binary variables associated to the routes 3 and 4; and r3AD and y4AD are constant parameters for the economic measure for the routes 3 and 4. zThe data for the economic measure for the different routes are given in Table 1. 3392



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For the adjacent nodes AE (biomass−syngas) two technologies are available (pyrolysis and gasification), and the following mathematical programming problem is formulated:

The solution for the mathematical programming problems for all adjacent nodes yields the superstructure shown in Figure 5; this representation is simpler than the original one because several routes have been eliminated. Then, based on the simplified superstructure shown in Figure 5 and the general disjunctive programming model, the problems for the nonadjacent nodes are solved considering the information provided in Tables 1, 2, and 3. First, the optimization for the nonadjacent nodes AE (biomass-syngas) via gasification or digestion-autothermal reforming is formulated using the following disjunctive programming:

min R AE s. t . y6AE + y7AE = 1 R AE = r6AEy6AE + r7AEy7AE

(99)

Two technologies are available (landfill and digestion) for the adjacent node AF (biomass-methane). Then, the mathematical programming problem is formulated as:

min R A → E s. t .

min R AF s. t .

y AE + y AFE = 1

y8AF + y9AF = 1 R

AF

= r8AFy8AF + r9AFy9AF

R A → E = r AEy AE + r AFEy AFE

Similarly for the adjacent node HO (ketone-bioalcohols through Grignard synthesis or hydrogenation):

For the nonadjacent node AF (biomass-methane) through digestion or gasification-methanation is optimized using the following disjunctive programming model:

min RHO s. t .

min R A → F s. t .

HO + y HO = 1 y27 28 HO HO HO HO RHO = r27 y27 + r28 y28

(101)

y AF + y AEF = 1

For the adjacent node DO (sugar-bioalcohols via aqueous phase reforming or ethanol fermentation):

R A → F = r AFy AF + r AEFy AEF

min RDO s. t .

min RK → O s. t .

(102)

Three technologies are available (methanol catalytic synthesis, syngas fermentation or mixed alcohol catalytic synthesis) for the adjacent nodes EO (syngas-bioalcohols), and the disjunctive programming model is stated as: min R s. t .

(106)

Similarly, the nonadjacent nodes KO (ethylene-bioalcohols) via hydration or hydrobromination-hydrolysis or hydrofromylation-hydrolysis is optimized as:

DO + y DO = 1 y15 16 DO DO DO DO RDO = r15 y15 + r16 y16

(105)

(100)

y KO + y KLO + y KMO = 1 RK → O = r KOy KO + r KLOy KLO + r KMOy KMO

(107)

EO

Finally, for the nonadjacent nodes GO (acid carboxylicbioalcohols) via esterification−hydrogenolysis or hydrogenation is optimized as

EO + y EO + y EO = 1 y18 19 20 EOy EO + r EOy EO + r EOy EO REO = r18 19 19 20 20 18

min RG → O s. t .

(103)

And finally for the adjacent node KO (ethylene-bioalcohols via indirect hydrolysis or hydration) the disjunctive programming problem can be stated as follows:

yGJO + yGO = 1 RG → O = r GJOyGJO + r GOyGO

KO

min R s. t .

After these mathematical programming problems for the nonadjacent nodes are solved, the simplified superstructure shown in Figure 6 is obtained. Finally, a mathematical programming model is formulated for the simplified superstructure shown in Figure 6. The model is based on the general disjunctive programming problem

KO + y KO = 1 y31 32 KO KO KO KO RKO = r31 y31 + r32 y32

(108)

(104) 3393



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Figure 5. Superstructure after solving the adjacent nodes.

tation to yield calcium carboxylates then using ketonization to obtain ketone and finally to obtain bioalcohols via hydrogenation) yields a solution which is more expensive than the optimal solution by only 0.9% ($15.28/GJ). Also, the direct route AO (i.e., biomass to bioalcohols via ABE fermentation) yields a solution 2.13% ($15.47/KJ) more expensive than the optimal pathway. These additional pathways identified using the proposed approach are very important because these can be attractive respect to additional criteria (for example environmental or safety metrics). Notice that both approaches, the sequential and simultaneous, yield the same optimal pathway, this is because the same cost data for the cost function is used and because the interaction between successive nodes is not considered in this case study. However, the simultaneous approach is able to find better solutions when the interaction between two successive nodes is considered. Case b: Nonlinear Case. This case considers the same example of the previous case, but this includes nonlinear relationships for the associated costs for the processing steps to account for the economies of scale. In addition, this example also includes the associated greenhouse gas (GHGt(k)) emissions for each processing step based on the life cycle analysis (see Chouinard-Dussault et al.35). The information for this case is presented in Table 2, where αt(k) represents the exponent for the cost function (R = rt(k)(Yield)αt(k)) that makes the problem nonlinear, Yield represents the gallons produced, and to determine the energy produced, the energy density in GJ per gallon (EDrt(k)) reported in Table 2 for each route is used. The problem consists in determining the pathway with the minimum

proposed in this paper (see eqs 8−14, 17) and stated as follows: min R

s.t. y ABJO + y ACGJO + y ACHO + y ADGJO + y ADHO + y ADO + y ADEO + y AO + y AEO + y AFIKO + y AFNO = 1

and R = r ABJOy ABJO + r ACGJOy ACGJO + r ACHOy ACHO + r ADGJOy ADGJO + r ADHOy ADHO + r ADOy ADO + r ADEOy ADEO + r AOy AO + r AEOy AEO + r AFIKOy AFIKO + r AFNOy AFNO

(109)

Notice that this problem is considerably simpler than the original one. This way, on the basis of the techno-economical information given in Table 3, the optimal solution is the pathway ABJO that corresponds to biomass using mixedculture fermentation to yield ammonia carboxylates, then to obtain ester via esterification, and finally to obtain the bioalcohols via hydrogenolysis. This optimal pathway represents a cost of $15.14/GJ (which is shown in Figure 7). Notice that this solution is the same as the one obtained with the simultaneous approach. In addition, the proposed approach allows the user to find that the pathway ACHO (biomass via mixed-culture fermen3394



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Figure 6. Superstructure after solving the nonadjacent nodes.

costs to produce an amount of total energy of 1 GJ. The simultaneous approach is represented as follows:

OGHG ≤ GHGt(k) + M GHG(1 − y t(k))

(114)

(110)

OGHG ≥ GHGt(k) − M GHG(1 − y t(k))

(115)

min R

Subject to the following constraints. First the total energy produced is of 1 GJ:

Energy = 1

Notice that, for this case only, the greater than constraint is required for the cost and only the lower than constraints is required for the energy because of the objective function. This MINLP problem was solved using the solver BARON and the optimal solution is the route A−O (i.e., mixed butanol and ethanol production via ABE fermentation) with a total cost of $11.164/GJ; which is significantly smaller than the one obtained with the linear case because of the consideration of the economies of scale (see Figure 7b). This solution is obtained for the case that considers the economies of scale because this pathway involves a minor number of processing steps. It is noteworthy that the proposed approach is useful to identify the solution with the minimum GHG emissions, which corresponds to the pathway A−C−H-O (i.e., mixed alcohols production via acid fermentation and ketonization) with overall GHG emissions of −22 kg GHG/GJ (based on the life cycle analysis35) and a cost of $20.543/GJ. This approach is also useful to determine the solution with the minimum number of processing step (in this case the direct pathway A−O), which is very important to intensify processes. Finally, for this nonlinear case, both the simultaneous and sequential approaches yield the same optimal solution; this is because both used the same relationships and cost functions. However, the simultaneous approach may yield better solutions than the sequential one for the cases when the interactions of

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and the following disjunction states that only one route t(k) is selected, and the associated nonlinear cost is determining, whereas the corresponding energy density is used to determine the total energy produced based on the amount of bioalcohols produced (Yield). In the same way, when the technology is selected, the associated GHG emissions are determined. ⎡ ⎤ Y t(k) ⎢ ⎥ ⎢ R = r (Yield)αt(k) ⎥ t(k) ⎥ ∨⎢ ⎢ t(k) Energy = YieldEDrt(k)⎥ ⎢ ⎥ ⎢ OGHG = GHG ⎥ t ( k ) ⎣ ⎦

And previous disjunction is reformulated as follows: R ≥ rt(k)(Yield)αt(k) − MR (1 − yt(k))

(112)

Enery ≤ YieldEDrt(k) + MEnergy(1 − yt(k))

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Figure 7. Optimal pathway for the case study: (a) linear case minimum cost, (b) nonlinear case minimum cost, (c) minimum GHG.

Table 4. Results Summary for the Cases Analyzed concept pathway cost, $/GJ GHG, kg GHG/GJ

minimum cost linear case mixed alcohols production via acid fermentation and esterification 15.14 −20

minimum cost nonlinear case mixed butanol and ethanol production via ABE fermentation 11.164 −12

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two successive nodes are considered. Table 4 shows the main results obtained for the cases analyzed, where different pathways where obtained for the minimum cost linear, minimum cost nonlinear, and the minimum greenhouse gas emissions.

minimum GHG emissions mixed alcohols production via acid fermentation and ketonization 20.545 −22

CONCLUSIONS

This paper presents a new general systematic approach for the optimal design of the pathway for a biorefinery. The proposed approach decomposes the problem in several subproblems (first 3396



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Parameters

to find the general superstructure and then to determine the optimal pathway) that allow the user to find the optimal configuration without any problem. The paper also presents a general disjunctive programming formulation for the optimization of the developed superstructure that can be applied to any case study. The application of the proposed approach to a case study for the configuration of a biorefinery for production of bioalcohols from lignocellulosic biomass shows that the proposed approach is applicable. This way, the approach allows the user to solve systematically several subproblems without any complication and avoid solving simultaneously very difficult problems. The proposed approach allows the user zto find the optimal configuration and also to identify additional configurations that can be attractive considering additional criteria.

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A troute = matrix of coefficients for the equality constraints for the pathway A troute = matrix of coefficients for the inequality constraints for the hpathway Adj AtAdjm.m′ = matrix of coefficients for the equality constraints for the adjacent node m to m′ NonAdjh A tNonAdjn.n′ = matrix of coefficients for the equality constraints for theAdjnonadjacent node n to n′ g = matrix of coefficients for the inequality A tAdjm.m′ constraints for the adjacent node m to m′ NonAdjg A tNonAdjn.n′ = matrix of coefficients for the inequality constraints for the nonadjacent node n to n′ At(k) h = matrix of coefficients for the equality constraints for the technology t to produce component k g At(k) = matrix of coefficients for the inequality constraints for the technology t to produce component k bhtroute = vector of parameters for the equality constraints for the pathway bhtAdjm.m′ = vector of parameters for the equality constraints for the technology tAdjm.m′ for the adjacent node m, m′ bhtNonAdjn.n′ = vector of parameters for the equality constraints for the technology tNonAdjn.n′ for the nonadjacent node n, n′ h = vector of parameters for the equality constraints for bt(k) the technology t to produce component k g btroute = vector of parameters for the inequality constraints for the technology troute for the pathway g btAdjm.m′ = vector of parameters for the inequality constraints for the technology tAdjm.m′ for the adjacent node m, m′ g btNonAdjn.n′ = vector of parameters for the inequality constraints for the technology tNonAdjn.n′ for the nonadjacent node n, n′ g bt(k) = vector of parameters for the inequality constraints for the technology t to produce component k Ctroute = vector of parameters for the technology troute for the pathway Adj Ad ′ CtAdjm.m′ = vector of parameters for the technology t jm.m for the adjacent node m, m′ NonAdj NonAdjn.n′ CtNonAdjn.n′ = vector of parameters for the technology t for the nonadjacent node n, n′ Ct(k) = vector of parameters for the technology t to produce component k Drt(k) = disaggregated objective for the technology t to produce component k Dxt(k) = disaggregated design variables for the technology t to produce component k EDrt(k) = Energy density for biofuel produced for technology t to produce component k route = upper bound for the equality constraints for the pathway htmax Adjmax hi,tAdjm.m′ = upper bound for the equality constraints for the adjacent node m to m′ max hi,tNonAdj NonAdjn.n′ = upper bound for the equality constraints for the nonadjacent node n to n′ route = upper bound for the Inequality constraints for the gtmax pathway max gj,tAdjAdjm.m′ = upper bound for the Inequality constraints for the adjacent node m to m′ max = upper bound for the Inequality constraints for the gj,tNonAdj NonAdj n.n′ nonadjacent node n to n′ GHGt(k) = greenhouse gas emissions for processing technology t to produce component k

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: +52 443 3273584. Fax. +52 443 3273581. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Support from NSF (Project Number OISE 0710936 and its associated project by the US-Egypt Board of Science and Technology) as well as CONACYT is gratefully acknowledged.

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NOMENCLATURE Indices Am,m′ = set for all adjacent nodes An,n′ = set for all nonadjacent nodes Aroute = set for the complete routes i = index for the equality constraints Itroute = set for all equality constraints for the technology troute for the pathway ItAdjm.m′ = set for all equality constraints for the technology t Adjm.m′ for the adjacent node m, m′ ItNonAdjn.n′ = set for all equality constraints for the technology t NonAdjn.n′ for the nonadjacent node n, n′ It(k) = set for all equality constraints for the technology t to produce component k j = index for the inequality constraints Jtroute = set for all inequality constraints for the technology t route for the technology Jt Adjm.m′ = set for all inequality constraints for the technology t Adjm.m′ for the adjacent node m, m′ JtAdjn.n′ = set for all inequality constraints for the technology t Adjn.n′ for the nonadjacent node n, n′ Jt(k) = set for all inequality constraints for the technology t to produce component k k = index for component K = sets for components l = index for the logical constraints L = set for the logical constraints m, m′ = adjacent nodes n, n′ = nonadjacent nodes t route = technology for the pathway t Adjm.m′ = technology for the adjacent node m, m′ t NonAdjn.n′ = technology for the nonadjacent node n, n′ t(k) = technology to produce component k 3397



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rtroute = vector of parameters for the design variables for the technology troute for the pathway rtAdj Adj = vector of parameters for the design variables for the m.m′ technology tAdjm.m′ for the adjacent node m, m′ NonAdj rtNonAdjn.n′ = vector of parameters for the design variables for the technology tNonAdhn.n′ for the nonadjacent node n, n′ rt(k) = vector of parameters for the design variables for the technology t to produce component k Rmax = upper bound for objective for the pathway Adjmax = upper bound for objective for the adjacent node m Rm,m′ to m′ NonAdjmax = upper bound for objective for the nonadjacent Rn,n′ node n to n′ αt(k) = exponent for the nonlinear cost function for the technology t to produce component k

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Adjm.m′ YtAdj for the Adj = boolean variable for the technology t m.m′ adjacent node m, m′ ytAdj Adj = binary variable for the technology tAdjm.m′ for the m.m′ adjacent node m, m′ YtNonAdj NonAdj = boolean variable for the technology tNonAdjn.n′ for the n.n′ adjacent node n, n′ ytNonAdj NonAdj = binary variable for the technology tAdjn.n′ for the n.n′ nonadjacent node n, n′ Yield = Amount production for the biofuels

REFERENCES

(1) Huber, G. W.; Iborra, S.; Corma, A. Synthesis of Transportation Fuels from Biomass: Chemistry, Catalysts, and Engineering. Chem. Rev. 2006, 106 (9), 4044. (2) Saxena, R. C.; Adhikari, D. K.; Goyal, H. B. Biomass-Based Energy Fuel through Biochemical Routes: A review. Renew. Sustain. Energy Rev. 2009, 13 (1), 167. (3) Saxena, R. C.; Seal, D.; Kumar, S.; Goyal, H. B. ThermoChemical Routes for Hydrogen Rich Gas from Biomass: A Review. Renew. Sustain. Energy Rev. 2008, 12 (3), 1909. (4) Goyal, H. B.; Seal, D.; Saxena, R. C. Bio-Fuels from Thermochemical Conversion of Renewable Resources: A Review. Renew. Sustain. Energy Rev. 2008, 12 (2), 504. (5) Kamm, B.; Kamm, M. BiorefineriesMulti Product Processes. In White Biotechnology; Ulber, R, Sell, D, Eds.; Springer: New York, 2007; Vol. 105, p 175. (6) Fernando, S.; Adhikari, S.; Chandrapal, C.; Murali, N. Biorefineries: Current Status, Challenges, and Future Direction. Energy Fuels 2006, 20 (4), 1727. (7) Werpy, T.; Petersen, G. Results of Screening for Potential Candidates from Sugars and Synthesis Gas. Top Value Added Chemicals from Biomass; Pacific Northwest National Laboratory (PNNL) and National Renewable Energy Laboratory (NREL): Richland, WA, 2004; Vol. 1. (8) Holladay, J. E.; Bozell, J. J.; White, J. F.; Johnson, D. Results of Screening for Potential Candidates from Biorefinery Lignin. Top Value-Added Chemicals from Biomass; Pacific Northwest National Laboratory; Richland, WA, 2007; Vol. 2, PNNL-16983. (9) Agnihotri, R. B.; Motard, R. L. Reaction Path Synthesis in Industrial Chemistry. In Computer Applications to Chemical Engineering; Squires, R. G., Reklaitis, G. V., Eds.; American Chemical Society: Washington, DC, 1980; Vol. 124, 193. (10) Nishida, N.; Stephanopoulos, G.; Westerberg, A.W. A Review of Process Synthesis. AIChE J. 1981, 27 (3), 321. (11) Govind, R.; Powers, G. J. A Chemical Engineering View of Reaction Path Synthesis. In Computer-Assisted Organic Synthesis; Wipke, W. T., Howe, W. J., Eds.; American Chemical Society: Washington, DC, 1977; Vol. 61, p 81. (12) May, D.; Rudd, D. F. Development of Solvay Clusters of Chemical Reactions. Chem. Eng. Sci. 1976, 31 (1), 59. (13) Rotstein, E.; Resasco, D.; Stephanopoulos, G. Studies on the Synthesis of Chemical Reaction PathsI: Reaction Characteristics in the ([Delta]G, T) Space and a Primitive Synthesis Procedure. Chem. Eng. Sci. 1982, 37 (9), 1337. (14) Fornari, T.; Rotstein, E.; Stephanopoulos, G. Studies on the Synthesis of Chemical Reaction PathsII. Reaction Schemes with Two Degrees of Freedom. Chem. Eng. Sci. 1989, 44 (7), 1569. (15) Crabtree, E. W.; El-Halwagi, M. M. Synthesis of Environmentally Acceptable Reactions. AIChE Symp. Ser. 1994, 90 (303), 117. (16) Pistikopoulos, E. N.; Stefanis, S. K.; Livingston, A. G. A Methodology for Minimum Environmental Impact Analysis. AIChE Symp. Ser. 1994, 90 (303), 139. (17) Buxton, A.; Livingston, A. G.; Pistikopoulos, E. N. Reaction Path Synthesis for Environmental Impact Minimization. Comput. Chem. Eng. 1997, 21 (Suppl.1), S959−S964. (18) Li, M.; Hu, S.; Li, Y.; Shen, J. A Hierarchical Optimization Method for Reaction Path Synthesis. Ind. Eng. Chem. Res. 2000, 39 (11), 4315.

Variables or Functions

Drtroute = disaggregated variable for rtroute DXtroute = disaggregated variable for Xtroute Energy = total energy produced for the biofuels ht(k) = equality constraints to obtain product k from technology t htroute = equality constraints for the pathway hi,tAdjAdjm.m′ = equality constraints for the adjacent node m to m′ NonAdj hi,tNonAdjn.n′= equality constraints for the nonadjacent node n to n′ = inequality constraints for the adjacent node m to m′ gj,tAdjAdjm.m′ NonAdj gj,tNonAdjn.n′= inequality constraints for the nonadjacent node n to n′ gjt(k) = inequality constraints to obtain component k from technology t OGHG = total amount of greenhouse gas emissions R = objective for the pathway Rk = objective to produce component k Rt(k) = function to obtain the objective to produce component k from technology t Rtroute = objective for the pathway for technology troute Adj Rm,m′ = objective for the adjacent node m to m′ NonAdj Rn,n′ = objective for the nonadjacent node n to n′ = functionality for the technology tAdjm.m′ for the RtAdj Adj m.m′ adjacent node m, m′ NonAdj RtNonAdj = functionality for the technology tNonAdjm.m′ for the n.n′ adjacent node m, ḿ Xtroute = design variables for the technology troute for the pathway XtAdjm.m′ = design variables for the technology tAdjm.m′ for the adjacent node m, m′ XtAdjm.m′ = vector for the design variables for the technology tAdjm.m′ for the adjacent node m, m′ XtNonAdjn.n′ = vector for the design variables for the technology tNonAdjn.n′ for the adjacent node n, n′ XtNonAdjn.n′ = design variables for the technology tNonAdjn.n′ for the adjacent node n, n′ Xt(k) = design variables for the technology t to produce component k Ytk = boolean variable for the existence of the component k yk = binary variable for the existence of the component k Yt(k) = boolean variable for the existence of the component k obtained from technology t yt(k) = binary variable for the existence of the component k obtained from technology t Ytroute = boolean variable for the technology troute ytroute = binary variable for the technology troute 3398



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(19) Ng, D. K. S.; Pham, V.; El-Halwagi, M. M.; Jiménez-Gutiérrez, A.; Spriggs, D. H. A Hierarchical Approach to the Synthesis and Analysis of Integrated Biorefineries. Paper presented at the 7th International Conference on Foundations of Computer-Aided Process Design (FOCAPD 2009), Design for Energy and the Environment 2009; Breckenridge, Colorado. (20) Bao, B.; Ng, D. K. S.; Tay, D. H. S.; Jiménez-Gutiérrez, A.; ElHalwagi, M. M. A Shortcut Method for the Preliminary Synthesis of Process-Technology Pathways: An Optimization Approach and Application for the Conceptual Design of Integrated Biorefineries. Comput. Chem. Eng. 2011, 35 (8), 1374. (21) Aden, A.; Ruth, M.; Ibsen, K.; Jechura, J.; Neeves, K.; Sheehan, J.; Wallace, B.; Montague, L.; Slayton, A.; Lukas, J. Lignocellulosic Biomass to Ethanol Process Design and Economics Utilizing Co-Current Dilute Acid Prehydrolysis and Enzymatic Hydrolysis for Corn Stover; National Renewable Energy Laboratory: Golden, CO, 2002; NREL/ TP-510-32438. (22) Zhu, Y.; Jones, S. Techno-Economic Analysis for the Thermochemical Conversion of Lignocellulosic Biomass to Ethanol via Acetic Acid Synthesis; Pacific Northwest National Laboratory: Richland, WA, 2009; PNNL-18483. (23) Phillips, S.; Aden, A.; Jechura, J.; Dayton, D.; Eggeman, T. Thermochemical Ethanol via Indirect Gasification and Mixed Alcohol Synthesis of Lignocellulosic Biomass; National Renewable Energy Laboratory: Golden, CO, 2007; NREL/TP-510-41168. (24) Dutta, A.; Dowe, N.; Ibsen, K. N.; Schell, D. J.; Aden, A. An Economic Comparison of Different Fermentation Configurations to Convert Corn Stover to Ethanol Using Z. Mobilis and Saccharomyces. Biotechnol. Prog. 2010, 26 (1), 64. (25) Kazi, F.; Fortman, J.; Anex, R.; Kothandaraman, G.; Hsu, D.; Aden, A.; Dutta, A. Techno-economic Analysis of Biochemical Scenarios for Production of Cellulosic Ethanol; National Renewable Energy Laboratory: Golden, CO, 2010; NREL/TP-6A2−46588. (26) Pokoo-Aikins, G.; Nadim, A.; Mahalec, V.; El-Halwagi, M. M. Design and Analysis of Biodiesel Production from Algae Grown through Carbon Sequestration. Clean Technol. Environ. Policy 2010, 12 (3), 239. (27) Myint, L. L.; El-Halwagi, M. M. Process Analysis and Optimization of Biodiesel Production from Soybean oil. Clean Technol. Environ. Policy 2009, 11 (3), 263. (28) Pham, V.; Holtzapple, M.; El-Halwagi, M. M. Techno-Economic Analysis of Biomass to Fuels Conversion Via the MixAlco Process. J. Ind. Microbiol. Biotechnol. 2010, 37 (11), 1157. (29) Holtzapple, M.; Granda, C. Carboxylate platform: The MixAlco Process Part 1: Comparison of Three Biomass Conversion Platforms. Appl. Biochem. Biotechnol. 2009, 156 (1), 95. (30) Jones, S.; Zhu, Y.; Valkenburg, C. Municipal Solid Waste (MSW) to Liquid Fuels Synthesis. Vol. 2: A Techno-economic Evaluation of the Production of Mixed Alcohols; Pacific Northwest National Laboratory: Richland, WA, 2009; PNNL-18482. (31) Mohan, T.; El-Halwagi, M. M. An Algebraic Targeting Approach for Effective Utilization of Biomass in Cogeneration Systems through Process Integration. Clean Technol. Environ. Policy 2007, 9 (1), 13. (32) Qin, X.; Mohan, T.; El-Halwagi, M. M.; Cornforth, F.; McCarl, B. A. Switchgrass as an Alternate Feedstock for Power Generation: Integrated Environmental, Energy, and Economic Life Cycle Analysis. Clean Technol. Environ. Policy 2006, 8 (4), 233. (33) Alvarado-Morales, M.; Gernaey, K. V.; Woodley, J. M.; Gani, R. Synthesis, Design and Analysis of Downstream Separation in Biorefinery Processes through a Group-Contribution Approach. Comput.Aided Chem. Eng. 2010, Vol 28, 1147. (34) Gosling, I. Process Simulation and Modeling for Industrial Bioprocessing: Tools and Techniques. Ind. Biotechnol. 2005, I (2), 106. (35) Chouinard-Dussault, P.; Bradt, L.; Ponce-Ortega, J. M.; ElHalwagi, M. M. Incorporation of Process Integration into Life Cycle Analysis for the Production of Biofuels. Clean Technol. Environ. Policy, 13 (5), 673.

(36) Elms, R. D.; El-Halwagi, M. M. Optimal Scheduling and Operation of Biodiesel Plants with Multiple Feedstocks. Int. J. Process Syst. Eng. 2009, 1 (1), 1. (37) Pokoo-Aikins, G.; Heath, A.; Mentzer, R. A.; Mannan, M. S.; Rogers, W. J.; El-Halwagi, M. M. A Multi-criteria Approach to Screening Alternatives for Converting Sewage Sludge to Biodiesel. Loss Prevent. Process Ind. 2010, 23 (3), 412. (38) Pham, V.; El-Halwagi, M. M. Process Synthesis and Optimization of Biorefinery Configurations. AIChE J. 2011, in press. DOI: 10.1002/aic.12640. (39) Raman, R.; Grossmann, I. E. Modelling and Computational Techniques for Logic Based Integer Programming. Comput. Chem. Eng. 1994, 18 (7), 563. (40) Lee, S.; Grossmann, I. E. Logic-Based Modeling and Solution of Nonlinear Discrete/Continuous Optimization Problems. Ann. Oper. Res. 2005, 139 (1), 267. (41) Vecchietti, A.; Lee, S.; Grossmann, I. E. Modeling of Discrete/ Continuous Optimization Problems: Characterization and Formulations of Disjunctive and their Relaxations. Comput. Chem. Eng. 2003, 27 (3), 433. (42) Grossmann, I. E.; Lee, S. Generalized Convex Disjunctive Programming Nonlinear Convex Hull Relaxation. Comput. Optim. Appl. 2003, 23 (1), 83. (43) Lee, S.; Grossmann, I. E. New Algorithms for Nonlinear Generalized Disjunctive Programming. Comput. Chem. Eng. 2000, 24 (9−10), 2125. (44) Sawaya, N. W.; Grossmann, I. E. Computational Implementation of Non-linear Convex Hull Reformulations. Comput. Chem. Eng. 2007, 31 (7), 856. (45) Sawaya, N. W.; Grossmann, I.E. A Cutting Plane Method for Solving Linear Generalized Disjunctive Programming Problems. Comput. Chem. Eng. 2005, 29 (9), 1891. (46) Ponce-Ortega, J. M.; Jiménez-Gutiérrez, A.; Grossmann, I. E. Optimal Synthesis of Heat Exchanger Networks Involved Isothermal Process Streams. Comput. Chem. Eng. 2008, 32 (8), 1918. (47) Ponce-Ortega, J. M.; Serna-González, M.; Jiménez-Gutiérrez, A. A Disjunctive Programming Model for Simultaneous Synthesis and Detailed Design of Cooling Networks. Ind. Eng. Chem. Res. 2009, 48 (6), 2991. (48) Ravagnani, M. S. S. S.; Caballero, J.A. A MINLP Model for the Rigorous Design of Shell and Tube Exchangers Using TEMA Standards. Chem. Eng. Res. Des. 2007, 85 (A10), 1423. (49) Mizutani, F. T.; Pessoa, F. L. P.; Queiroz, E. M.; Hauan, S.; Grossmann, I. E. Mathematical Programming Model for HeatExchanger Networks Synthesis Including Detailed Heat-Exchanger Designs. 1. Shell-and-Tube Heat-Exchanger Design. Ind. Eng. Chem. Res. 2003, 42 (17), 4009. (50) Barttfield, M.; Aguirre, P. A.; Grossmann, I. E. A Decomposition Method for Synthesizing Complex Column Configurations Using Tray-by-Tray GDP Models. Comput. Chem. Eng. 2004, 28 (11), 2165. (51) Caballero, J. A.; Grossmann, I. E. Design of Distillation Sequences: From Conventional to Fully Thermally Coupled Distillation Systems. Comput. Chem. Eng. 2004, 28 (11), 2307. (52) Yeomans, H.; Grossmann, I. E. Disjunctive Programming Models for the Optimal Design of Distillation Columns and Separation Sequences. Ind. Eng. Chem. Res. 2000, 39 (6), 1637. (53) Reyes-Labarta, J. A.; Grossmann, I. E. Disjunctive Optimization Design Models for Complex Liquid−Liquid Multistage Extractors. AIChE J. 2001, 47 (10), 2243. (54) Tian, D. Q.; Zhi, L. Y.; Chen, X.; Shao, Z. J.; Qian, J. X. Object Oriented Disjunctive Programming with a Nested Heuristic and Gradient-Based Solver for Chemical Process Synthesis. Ind. Eng. Chem. Res. 2010, 49 (4), 1779. (55) Karuppiah, R.; Grossmann, I. E. Global Optimization for the Synthesis of Integrated Water Systems in Chemical Processes. Comput. Chem. Eng. 2006, 48 (6), 650. (56) Sotelo-Pichardo, C.; Ponce-Ortega, J. M.; El-Halwagi, M. M.; Frausto-Henandez, S. Optimal Retrofit of Water Conservation Networks. J. Clean. Prod. 2011, 19 (14), 1560. 3399



Article

(57) Ponce-Ortega, J. M.; Hortua, A. C.; El-Halwagi, M. M.; JiménezGutiérrez, A. A Property-Based Optimization of Direct Recycle Networks and Wastewater Treatment Processes. AIChE J. 2009, 55 (9), 2329. (58) Brooke, A.; Kendrick, D.; Meeruas, A.; Raman, R. GAMSLanguage Guide; GAMS Development Corporation: Washington, DC, 2006.

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A Disjunctive Programming Formulation for the Optimal Design of

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