Globally Optimal Networks for Multipressure Distillation of

Article pubs.acs.org/IECR

Globally Optimal Networks for Multipressure Distillation of Homogeneous Azeotropic Mixtures Paul G. Ghougassian and Vasilios Manousiouthakis* Department of Chemical & Biomolecular Engineering Department, University of California, Los Angeles, California 90095, United States ABSTRACT: In this article, a methodology for the globally optimal synthesis of a network of vapor−liquid equilibrium flash separators that can operate at multiple pressures and separate an azeotropic mixture is presented. The objective function minimized is the total flow entering the network flashes. The proposed synthesis methodology employs the infinite-dimensional state-space (IDEAS) conceptual framework, which is shown to be applicable to the problem under consideration. The resulting infinite linear programming (ILP) IDEAS formulation is shown to have several properties that allow its simplification. The approximate solution of this IDEAS ILP is pursued through the solution of a number of finitedimensional linear programs (FLPs) of ever increasing size, whose optimum values form a sequence that converges to the ILP’s infimum. The proposed optimal design methodology is general in nature and can be used to separate any number of pressure-sensitive azeotropic mixtures, with or without use of an entrainer. The method is demonstrated on a first case study involving the dual-pressure separation of a methyl acetate/methanol binary mixture, which exhibits a minimumboiling azeotrope, without using an entrainer, and a second case study involving the dual-pressure separation of a ternary mixture of water, methanol, and acetone that also exhibits a minimum-boiling azeotrope for the methanol/acetone binary mixture, again without using an entrainer. The IDEAS-generated globally optimal design is shown to be 31.54% better than an optimized, dual-pressure, traditional, two-column design for the binary mixture (case 1) and 15.15% better for the ternary mixture (case 2).

1. INTRODUCTION Knowledge regarding the sensitivity of azeotropes to pressure effects dates back to the 19th century. 1−3 Azeotropic mixtures are typically separated by either homogeneous or heterogeneous azeotropic distillation, in batch 4 or continuous 5 configuration. The separation process requires the addition of an entrainer and/or the use of pressure swing distillation (PSD), which exploits the dependence of the azeotrope on pressure.5 The main advantage of the PSD process is that it does not require an entrainer, that is, a substance that facilitates the considered separation but requires additional separation steps downstream and might itself be a toxic chemical or degrade into byproducts harmful to the environment. The systematic generation process synthesis paradigm has been predicted to expand in the foreseeable future, 6 with the separation of an azeotropic mixture listed as an important problem to address in the 21st century. 7 A large list of pressuresensitive binary azeotropes prevalent in the chemical industry that can be separated through the PSD process has been tabulated.8 The PSD process involves the operation of multiple columns at different pressures to bypass pressuredependent azeotropic pinch points and recover high-purity products. A two-column PSD is an especially interesting process, because its two columns operate at two different pressures and can be readily thermally integrated, by matching the heat removed by the high-pressure condenser with the heat required by the low-pressure boiler or with the heat required by the feed preheater.9,10 Several algorithms have been developed to optimize azeotropic columns based on energy and cost considerations, using © 2012 American Chemical Society

trial-and-error simulations at specific design parameters.5,11−16 However, these techniques do not guarantee global optimality and consider only a limited number of alternatives. Geometric approaches to identify and optimize azeotropic separation processes through homotopy, arc length continuation, residue curve maps, distillation boundaries, shortcut methods, and distillation lines have also been proposed, 1 7 − 2 2 but they have visualization limitations and also do not guarantee global optimality. Unusual and nontraditional variant designs have also emerged.23,24 Recently, researchers have begun using more refined optimization techniques relying on the formulation of complex mixed integer nonlinear programs (MINLPs).25−33 Although the MINLP methodology provides improvements, solutions are locally optimal and only as good as the initially supplied superstructure, because of the nonlinearity and nonconvexity that naturally arises from the superstructurebased MINLP problem formulation. In addition, commercially available globally optimal nonlinear programming software cannot handle the large number of variables resulting from the complexity of distillation column (super)structures. This article demonstrates the use of the infinite-dimensional state-space (IDEAS) framework in simultaneously synthesizing a network that can separate azeotropic mixtures Received: Revised: Accepted: Published: 11183

February 16, 2012 June 16, 2012 July 7, 2012 July 8, 2012 dx.doi.org/10.1021/ie300423q | Ind. Eng. Chem. Res. 2012, 51, 11183−11200

Industrial & Engineering Chemistry Research

Article

and minimizing the total flash inlet flow in the aforementioned network. The IDEAS framework is a generalized methodology that allows for the generation of globally optimal flow sheet designs. The IDEAS framework decomposes a process network into an operator network (OP), where the unit operations (reactors, distillation columns, heat exchangers, etc.) occur, and a distribution network (DN), where the flow operations (mixing, splitting, recycling, and bypass) occur. The optimal process network structure is identified through solution of an infinite linear program (ILP) that is formulated within the IDEAS framework. The ILP’s solution is approximated by finitedimensional linear programs of ever-increasing size. The solution of these linear programs is guaranteed to be globally optimal. IDEAS has been successfully applied to numerous globally optimal process network synthesis problems, such as mass-exchange network synthesis, 34 complex distillation network synthesis,35−37 power cycle synthesis,38 reactor network synthesis,39,40 reactive distillation network synthesis,41 separation network synthesis,42 attainable region construction,43−46 and batch attainable region construction.47 In this work, IDEAS is shown to be applicable to the globally optimal synthesis of networks that employ steadystate vapor−liquid equilibrium flash separators, operating at multiple pressures, to separate azeotropic mixtures. The representation of a theoretical tray of a distillation column as a combination of two separate devices, namely, a mixer that can receive vapor and liquid inlets at different temperatures and an equilibrium (flash) separator that has as its inlet the mixer’s outlet and as outlets a vapor stream and a liquid stream that are in equilibrium with one another, has been suggested. 48 A schematic representation of this concept is shown in Figure 1. Because the possibility

2. APPLICABILITY AND MATHEMATICAL FORMULATION OF IDEAS FOR MULTIPRESSURE FLASH SEPARATOR NETWORK SYNTHESIS In this work, the isothermal, isobaric flash separator shown in Figure 2 is considered. The flash separator’s vapor and

Figure 2. Representation of a flash separator.

liquid exit streams are considered to be in phase equilibrium with one another. Because the main goal of the synthesis task to be undertaken is to establish the feasibility of separation of an azeotropic mixture using multipressure distillation and to identify a reasonably sized distillation network design that can carry out the aforementioned separation, only mass/component balance and phase equilibrium relations are incorporated in the flash separator’s mathematical model, and the considered optimization goal is the minimization of the total flow entering the network’s flashes. This objective function can be thought of to be directly minimizing the network capital cost (flash separator total volume) by considering that each flash separator has the same residence time. It can also be thought of to indirectly minimize energy consumption, because distillation network energy costs are typically related to the network’s flows. As long as no constraints are imposed on the network’s energy consumption, energy balances need not be incorporated in the flash separator and distribution network models for the minimum total network flow to be correctly identified over all feasible networks. Indeed, by omitting all energy balances, the optimization problem has a larger feasible region than if the energy balances were included. Thus, the obtained minimum is less than or equal to the minimum obtained with the energy balances included. These two minima are actually equal, because energy balances can always be written a posteriori for the network minimizing total flow to identify the network’s heating and cooling needs. The lack of any specifications on these needs suggests that the network would be feasible even if energy balances were considered. On the other hand, the synthesis of an optimally heatintegrated multipressure distillation network would require that energy consumption specifications be imposed on the optimization problem, thus necessitating the use of energy balances. This latter problem will be the subject of future research efforts. It is true that distillation is an inherently nonlinear process with nonlinear constraints. The beauty of IDEAS is exactly that it overcomes the inherently nonlinear nature of the distillation process optimization problem, not through some kind of Taylor series approximate linearization, but rather by exploiting the decomposition and linearity properties that naturally exist in the distillation process model, based on knowledge of the intensive properties and design parameters and independently of the extensive properties.

Figure 1. Mixer + flash separator representation of a distillation column tray.48

of mixing is incorporated within the DN of the IDEAS framework and the equilibrium separator can be made part of the OP of IDEAS, a distillation network can be represented within the IDEAS framework as a network of vapor−liquid equilibrium flash separators. Thus, IDEAS can employ this representation to synthesize distillation networks. The rest of the article is structured as follows: First, the applicability of IDEAS to the multipressure flash separator network synthesis problem is established, and the resulting IDEAS mathematical formulation is presented. Properties that facilitate the solution of this mathematical formulation are then presented. This is followed by two case studies in which the IDEAS framework is used to generate an optimal network for the separation of a binary mixture (case study 1) and a ternary mixture (case study 2). The obtained results are then discussed, and conclusions are drawn. 11184

dx.doi.org/10.1021/ie300423q | Ind. Eng. Chem. Res. 2012, 51, 11183−11200


Article

The considered equilibrium separator model employs the Gamma−Phi vapor−liquid equilibrium formulation49 and mass and component balances fkP − xkLF L − ykV F V = 0

∀ k = 1, ..., n

Ψ1: Rn + 2 → R2, ⎡P⎤ ⎢ ⎥ ⎢T ⎥ ⎢ L⎥ Ψ1: u1 = ⎢ x1 ⎥ → Ψ1(u1) ⎢⋮⎥ ⎢ L⎥ ⎢⎣ xn ⎥⎦

(1)

n

∑ xkL = 1

(2)

k=1

n ⎡ ⎤ ⎢ ⎥ ∑ xkL − 1 ⎢ ⎥ k=1 ⎥ ≙⎢ ⎢ n xkLγk({xlL}ln= 1, T ) Pksat(T ) ⎥ − 1⎥ ⎢∑ P ⎣ k=1 ⎦

n

∑ ykV

=1 (3)

k=1

ykV ϕk ({ylV }ln= 1, T , P)P − xkLγk({xlL}ln= 1, T ) Pksat(T ) = 0 ∀ k = 1, ..., n

Ψ2 : Rn + 2 × Rn + 2 → Rn, ⎛⎡ u1 ⎤⎞ ⎡ u1 ⎤ Ψ2 : ⎢ ⎥ → Ψ2⎜⎢ ⎥⎟ ⎣u2 ⎦ ⎝⎣ u 2 ⎦⎠

(4)

A variety of thermodynamic models can be employed in quantifying the functions ϕk({yVl }nl=1,T,P) and γk({xLl }nl=1,T),Psat k , ∀k = 1, ..., n. In the illustrative case studies outlined below, ideal gas behavior is assumed [ϕk({yVl }nl=1,T,P) = 1, ∀k = 1, ..., n]; the Wilson equations (eqs 5 and 6) are used to model the nonideal liquid-phase coefficients γk({xLl }nl=1,T), ∀k = 1, ..., n, and the extended Antoine equation (eq 7) is used to model the vapor pressure Psat k (T), ∀k = 1, ..., n

⎡ x Lγ ({x L}n , T ) P1sat(T ) V ⎤ ⎢ f P − x1LF L − 1 1 l l = 1 F ⎥ ⎢ 1 ⎥ P ⎥ ⋮ ≙ ⎢⎢ ⎥ L L n sat ⎢ ⎥ γ x ({ x } , T ) P ( T ) 1 n l l = n n V⎥ ⎢ f P − xnLF L − F ⎣ n ⎦ P

⎤ ⎡ n L n L ⎢ ⎣ ⎦ ⎡ ⎤ ln γk({xl }l = 1, T ) = 1 − ln ∑ xj Λk , j(T )⎥ ⎥⎦ ⎢⎣ j = 1 ⎛ x L Λ (T ) ⎞ i i,k ⎟ − ∑ ⎜⎜ n L ∑ x Λ (T ) ⎟⎠ i=1 ⎝ j=1 j i,j

y = [ y1T | y2 T ]T = [T y1V ··· ynV P x1L ··· xnL | f1 f2 ··· fn F L F V ]T

n

⎛ − Ak , j ⎞ Λk , j(T ) = L exp⎜ ⎟ ⎝ RT ⎠ Vk V jL

∀ k = 1, ..., n

Φ1: Rn + 2 × Rn + 2 → R2n + 2,

(5)

Φ1: (u1 , u 2) → y1 = Φ1(u1 , u 2) ⎡ ⎤ T ⎢ L ⎥ L n sat ⎢ x1 γ1({xl }l = 1, T ) P1 (T ) ⎥ ⎢ ⎥ P ⎢ ⎥ ⋮ ⎢ ⎥ ⎢ L ⎥ L n sat ⎢ xn γn({xl }l = 1, T ) Pn (T ) ⎥ =⎢ ⎥ P ⎢ ⎥ P ⎢ ⎥ ⎢ ⎥ L x1 ⎢ ⎥ ⎢ ⎥ ⋮ ⎢ ⎥ ⎢⎣ ⎥⎦ xnL

∀ k = 1, ..., n; ∀ j = 1, ..., n (6)

ln[Pksat(T )] = Ak +

Bk + Dk ln(T ) + Ek T Fk T + Ck

∀ k = 1, ..., n

(7)

The equations above lead to the conclusion that the aforementioned flash separator model can be employed to construct the following input−output information map Φ: Rn + 2 × Rn + 2 → R2n + 2 × Rn + 2, Φ: u → y = Φ(u) = [[Φ1(u1 , u 2)]T | [Φ2(u1 , u 2)]T ]T

Φ2 T : Rn + 2 × Rn + 2 → Rn + 2,

where

Φ2 T : [ u1T u 2 T ]T → y2 T = Φ2 T([ u1T u 2 T ]T ) = [ f1 ··· fn F L F V ]T

u = [ u1T | u 2 T ]T = [ P T x1L ··· xnL | f1 f2 ··· fn F L F V ]

The engineering importance of the above information maps can best be understood as follows: Consider that u1 = [ P T x1L ··· xnL ]T such that Ψ1(u1) = 0 ∧ u1 ≥ 0 is known. This can be ascertained by first considering [ P T x1L ··· xnL− 2 ]T ≥ [0 0 0 ··· 0]T to be known and

u ∈ D = {u = (u1 , u 2) ∈ Rn + 2 × Rn + 2: Ψ1(u1) = 0 ∧ Ψ2(u1 , u 2) = 0 ∧ u1 ≥ 0 ∧ u 2 ≥ 0} 11185



Article

u1,Ψ3(u1) is a linear operator and thus Ψ2(u1,u2) = Ψ3(u1)u2 is linear in u2. At this point, it should be noted for future reference that the incorporation of energy balances in the problem formulation does not change these characteristics of the flash separator model. Indeed, given a predictive thermodynamic model that accounts not only for phase equilibrium but also for molar enthalpy, calculation of all of the vapor/ liquid product mole fractions and molar enthalpies requires only knowledge of the flash P, T, and {xLl } n−2 l=1 and does not require any knowledge of either the feed component flows or the feed enthalpy flows or the total product molar flows. It has been established that, for a fixed vector u1, the employed flash separator model is defined by an input− output map Φ3(u1) whose domain is defined as the set u1 = [ P T x1L ··· xnL ]T ∈ Rn+2: Ψ 1(u1) = 0 and by an identity operator Φ4(u1) whose domain is the null space of Ψ3(u1), which, for fixed u1, is a linear operator. To incorporate the variation of u1 in the optimization process, so as to avoid suboptimal solutions, an infinite number of flash separator units each with a fixed value of u1 is considered such that the union of the considered u1 values is dense in the set over which u 1 can vary. A one-to-one correspondence is then created between the infinite sequence {u1(i)}∞ i=1 consisting of all possible values of u1 and the infinite sequence {Ψ3[u1(i)]}∞ i=1 of corresponding linear maps used to define the domain of the identity operator for each flash separator unit. These sequences are then used to define the domain and action of a linear operator (termed IDEAS OP) that quantifies the effect of all flash separator units and has its domain and range be subsets of infinite-dimensional spaces. This relation gives rise to the linear OP constraints, which hold true around every OP unit. To account for all possible multipressure distillation flowsheets, the (OP) operator needs to be interfaced with a distribution network (DN) where all stream splitting and mixing and pressure adjustment occurs, as shown in Figure 3. Each of the cross-flow streams in the DN is characterized by

then solving for all q physically meaningful solutions [ xnL− 1 xnL ]T ≥ [0 0]T of the system of equations n

n

∑

xkL

− 1 = 0,

k=1

∑

xkLγk({xlL}ln= 1, T )Pksat(T ) P

k=1

−1=0

q can be either zero or a positive integer. For each of the q solutions, resulting physically meaningful vectors u1 = [ P T x1L ··· xnL ]T{yVk }nk=1 can then be computed through eq 4. In turn, this suggests that, for any u1 such that Ψ1(u1) = 0 ∧ u1 ≥ 0, one can actually evaluate the image Φ1(u). Having defined the maps Ψ1, Ψ2, Φ1, and Φ2, the following properties can be readily verified. IDEAS property 1 is expressed as ∃Φ3: Rn+2 → R2n+2 such that Φ1(u1,u2) = Φ3(u1), ∀(u1,u2) ∈ D. This map is ⎡ ⎤ T ⎢ L ⎥ L n sat ⎢ x1 γ1({xl }l = 1, T ) P1 (T ) ⎥ ⎢ ⎥ P ⎢ ⎥ ⎡T ⎤ ⋮ ⎢ ⎥ ⎢ ⎥ ⎢ L ⎥ ⎢P⎥ L n sat ⎢ xn γn({xl }l = 1, T ) Pn (T ) ⎥ ⎢x L⎥ Φ3: u1 = ⎢ 1 ⎥ → Φ3(u1) ≙ ⎢ ⎥ P ⎢ ⎥ ⎢⋮⎥ P ⎢ ⎥ ⎢ L⎥ ⎢ ⎥ ⎢⎣ xn ⎥⎦ L x1 ⎢ ⎥ ⎢ ⎥ ⋮ ⎢ ⎥ ⎢⎣ ⎥⎦ xnL ∀ u1 ∈ {u1 ∈ Rn + 2: Ψ1(u1) = 0 ∧ u1 ≥ 0}

This implies that y1 ≙ Φ1(u1,u2) = Φ3(u1) can be evaluated based only on knowledge of u1 (intensive properties and design parameters) and independently of u2 (extensive properties). IDEAS property 2 is expressed as ∃Φ4: Rn+2 → R(n+2)×(n+2) such that Φ2(u1,u2) = Φ4(u1)u2 ∀(u1,u2) ∈ D. Indeed, this map is Φ4: u1 → Φ4(u1) ≙ I ∈ R(n+2)×(n+2), and thus, y2 ≙ [ f1 f2 ··· fn F L F V ]Φ4: u1 → Φ4(u1) ≙ I ∈ R(n+2)×(n+2), and thus, y2 ≙ [ f1 f2 ··· fn F L F V ]T = Iu2 = u2. This implies that, for fixed u1, Φ4(u1) is a linear operator (the identity operator) and y2 ≙ Φ2(u1,u2) = Φ4(u1)u2 is linear in u2. IDEAS property 3 is expressed as ∃Ψ3: Rn+2 → Rn×(n+2) such that Ψ2(u1,u2) = Ψ3(u1)u2 ∀(u1,u2) ∈ D. Indeed, this map is Ψ3: u1 → Ψ3(u1) ⎡ ⎢ 1 0 ··· ⎢ ⎢ ⎢ ⋮ 1 ··· ≙⎢ ⎢ ⎢0 ⋮ ⋱ ⎢ ⎢ ⎢⎣ 0 0 ···

x1Lγ1({xlL}ln= 1, T ) P1sat(T ) ⎤ ⎥ ⎥ P ⎥ L L n sat x 2 γ2({xl }l = 1, T ) P2 (T ) ⎥ − ⎥ P ⎥ ⋮ ⎥ ⎥ L L n sat xn γ ({xl }l = 1, T ) Pn (T ) ⎥ − n ⎥⎦ P

0 −x1L − 0 −x 2L 0

⋮

1 −xnL

This implies that Ψ3(u1) can be evaluated based only on knowledge of u1 and independently of u2 and that, for fixed

Figure 3. IDEAS representation of a (multipressure) flash separator network. 11186



Article

a flow rate variable and fixed destination and origin conditions. This information takes the form of the following sequence triplets: DN inlet to DN outlet (FOI,zO,zI), DN inlet to OP inlet (FPI,zP,zI), OP liquid outlet to DN outlet (FOL,zO,xL), OP vapor outlet to DN outlet (FOV,zO,yV), OP liquid outlet to OP inlet (FPL,zP,xL), and OP vapor outlet to OP inlet (FPV,zP,yV). A linear objective is considered in the proposed IDEAS formulation. It can be generally presented as M

∞

∞

M

[zkO(i)]l F O(i) ≤

j=1

+

+

∀ i = 1, ..., N ; ∀ k = 1, ..., n

(12)

fkP (i) − xkL(i) F L(i) − ykV (i) F V(i) = 0

∞

∀ i = 1, ..., ∞ ; ∀ k = 1, ..., n

(13)

i=1 j=1 ∞

M

∑ ∑ γi ,jF PV(i , j)

fkP (i) − F P(i) zkP(i) = 0 ⇒ fkP (i) −

M

∞

∞

∞

−

i=1 j=1

i=1 j=1

F I ≥ 0,

∞

∞

∑ ∑ F PL(i , j)

i=1 j=1 ∞ ∞

i=1 j=1

∑ ∑ F PV(i , j) i=1 j=1

subject to N

F I(j) −

∞

∑ F OI(i , j) + ∑ F PI(i , j) = 0 i=1

∀ j = 1, ..., M

i=1

(8) M

F O(i) −

∞

∞

∑ F OI(i , j) − ∑ F OL(i , j) − ∑ F OV(i , j) = 0 j=1

j=1

j=1

∀ i = 1, ..., N N

F L ( j) −

(9) ∞

∑ F OL(i , j) − ∑ F PL(i , j) = 0 i=1

∀ j = 1, ..., ∞

i=1

(10) N

F V ( j) −

∞

∑ F OV(i , j) + ∑ F PV(i , j) = 0 i=1

∀ j = 1, ..., ∞

j=1

F O ≥ 0,

F P ≥ 0,

(14)

F L ≥ 0,

F OI ≥ 0,

F PI ≥ 0,

F OL ≥ 0,

F PL ≥ 0,

F PV ≥ 0,

fkP ≥ 0

F V ≥ 0,

F OV ≥ 0,

Constraints 8 and 9 represent the DN inlet (splitting) and outlet (mixing) total flow balances, respectively, graphically appearing on the left and top of the DN block in Figure 3. Constraints 10 and 11 correspond to the OP total liquid and vapor outlet flow balances, respectively, which undergo splitting operations in the DN and graphically appear on the bottom of the DN. Constraint 12 correspond to the bounds on the quality variables of the DN outlet and is represented by component balances, which undergo mixing operations in the DN and graphically appear on the top of the DN. Constraint 13 represents the separator model, and finally, constraint 14 corresponds to the OP inlet component flow balances, which undergo mixing operations in the DN and graphically appear on the right of the DN. In addition, all flow variables are nonnegative. Close examination of this formulation reveals that it gives rise to an infinite linear program (ILP). The aforementioned linearity of the infinite program ν stems from the linearity of the IDEAS OP established earlier, and the linearity of the constraints that quantify the IDEAS DN: (1) The DN total flow mixing and splitting balances appearing in constraints 8−11 are inherently linear. (2) The DN component flow mixing balances appearing in constraints 12 and 14 are linear, because the vapor (liquid) composition of vapor (liquid) streams entering or leaving the DN at any junction are known and fixed. Two propositions are next presented to simplify the structure of the above IDEAS formulation by reducing the number of variables considered without compromising optimality. Proposition 1. The OP inlet and outlet flow variables f Pk (i), L F (i), and FV(i), ∀i = 1, ..., ∞, can be eliminated from the IDEAS formulation (IF1) through variable substitution without compromising optimality.

∞

which represents the total flow entering the network’s flashes (total network unit inlet flow). The resulting mathematical formulation of IDEAS (IF1) is M

∑ xkL(j) F PL(i , j) − ∑ ykV (j) F PV(i , j) = 0

∀ i = 1, ..., ∞ ; ∀ k = 1, ..., n

i=1 j=1

ν ≙ inf ∑ ∑ F PI(i , j) +

∞

j=1

∑ ∑ F PI(i , j) + ∑ ∑ F PL(i , j) + ∑ ∑ F PV(i , j)

∞

∑ zkI(j) F PI(i , j) j=1

∞

This objective function can be used to realize a wide array of objectives, through appropriate selection of the cost coefficients αi,j, βi,j, and γ i,j, associated with each of the problem’s flow variables. In the case study below, the minimization of total flash volume is considered. Under the assumption that each flash has the same residence time as any other flash (i.e., assuming that the same time is needed to reach equilibrium), the common residence time shared by all flashes can be factored out. This yields αi,j = βi,j = γ i,j = 1, which then yields the following objective function

+

∑ ykV (j) F OV(i , j) ≤ [zkO(i)]u F O(i) j=1

i=1 j=1

∞

j=1

∞

∑ ∑ αi ,jF PI(i , j) + ∑ ∑ βi ,jF PL(i , j) i=1 j=1 ∞

∞

∑ zkI(j) F OI(i , j) + ∑ xkL(j) F OL(i , j)

i=1

(11) 11187



Article

Proof. Substitution of f Pk (i), ∀i = 1, ..., ∞, from eq 13 into eq 14 and subsequent substitution of FL(i) and FV(i), ∀i = 1, ..., ∞, from eqs 10 and 11 into the resulting equation yields M

M j=1

∑

PI

F ( i , j) +

j=1

∑

xkL(j)

PL

F ( i , j)

∞

∑ ykV (j) F PV(i , j) − xkL(i) F L(i) − ykV (i) FV(i)

⎤ ⎡ ⎥ ⎢N + ∑ F PL(j , i)⎥ − ykV (i)⎢∑ F OV(j , i) + ⎥ ⎢ j=1 ⎥ ⎢ j=1 j≠i ⎦ ⎣

j=1 (10),(11)

M

∞

∞

= 0 ==== ⇒ ∑ zkI(j) F PI(i , j)+ ∑ xkL(j) F PL(i , j) j=1

j=1

⎡N + ∑ ykV (j) F PV(i , j) − xkL(i)⎢∑ F OL(j , i) ⎢⎣ j = 1 j=1 ∞

∞

+

∑ j=1

+

∞

⎤

j=1

⎦

∀ i = 1, ..., ∞ ; ∀ k = 1, ..., n

Constraint 15 involves only the flow variables occurring inside the DN, namely, FPI(i,j), FPL(i,j), FPV(i,j), FOL(i,j), and FOV (i,j), and replaces eqs 10, 11, 13, and 14. Furthermore, the non-negativity constraints f Pk (i) ≥ 0, FL(i) ≥ 0, and FV(i) ≥ 0, ∀i = 1, ..., ∞, are ensured by the non-negativity of FPL(i,j), FPV(i,j), FOL(i,j), FOV(i,j), xLk (i), and yVk (i), ∀i = 1, ..., ∞; ∀j = 1, ..., ∞; ∀k = 1, ..., n. In addition, the variables f Pk (i), FL(i), and FV(i), ∀i = 1, ..., ∞, do not appear in the objective function. Thus, they can be omitted. QED Proposition 2. The self-recycling liquid flows and vapor flows FPL(i,i) and FPV(i,i), ∀i = 1, ..., ∞, can be eliminated from the IDEAS formulation (IF1) without compromising optimality. Proof. The variables FPL(i,i) and FPV(i,i), ∀i = 1, ..., ∞, appear only in equality constraint 15 and in their respective non-negativity inequalities. Constraint 15 can be rewritten as

∑

PI

F ( i , j) +

∞

+

j≠i

∞

∞

∀ i = 1, ..., ∞ ; ∀ k = 1, ..., n

F O ≥ 0,

F OI ≥ 0,

F PL ≥ 0,

F PI ≥ 0,

F PV ≥ 0

F OL ≥ 0, (18)

An infinite-dimensional linear program cannot be solved explicitly. However, its solution can be approximated by a series of finite linear programs of increasing size, whose sequence of optimum values converges to the infinitedimensional problem’s infimum. In particular, consider the aforementioned IDEAS formulation with optimum value ν, which aims at the synthesis of a multipressure flash distillation network with M inlets and N outlets and allows for the possible use of an infinite number of multipressure flash separator units. By considering an ever-increasing number G of multipressure flash separator units, the optimum objective function values of the resulting finite linear programs form a nonincreasing sequence {ν G} ∞ 1 that converges to ν. The DN of these finite-dimensional formulations contains (M + 2G) × (N + G) cross-flow streams. Proposition 2 reduces the number of those streams to (M + 2G) × (N + G) − 2G streams. These cross-flow streams are distributed as follows: M × N DN inlet to DN outlet streams, M × G DN inlet to OP inlet streams, G × N OP liquid outlet to DN outlet streams, G × N OP vapor outlet to DN outlet streams, G2 − G OP liquid outlet to OP inlet streams, and G2 − G OP vapor outlet to OP inlet streams.

j=1

⎤ ⎥ PV PV ∑ F (j , i) + F (i , i)⎥⎥ = 0 j=1 ⎥ j≠i ⎦

∞

F OV ≥ 0,

∑ ykV (j) F PV(i , j) + xkL(i)F PL(i , i) + ykL (i) F PV(i , i)

⎡ ⎢N V − yk (i)⎢∑ F OV(j , i) + ⎢ ⎢ j=1 ⎣

i=1 j=1 j≠i

∑ ∑ F PV(i , j)

F I ≥ 0,

j≠i

⎤ ⎥ PL PL ∑ F (j , i) + F (i , i)⎥⎥ j=1 ⎥ j≠i ⎦

∞

∑ ∑ F PL(i , j)

subject to constraints 8,9,12, and 17

∑ xkL(j) F PL(i , j)

⎡ ⎢N L ⎢ − xk (i) ∑ F OL(j , i) + ⎢ ⎢ j=1 ⎣

(17)

i=1 j=1 j≠i

∞

+

∞

i=1 j=1

j=1

j=1

M

∞

ν ≙ inf ∑ ∑ F PI(i , j) +

∞

zkI(j)

⎤ ⎥ ∑ F PV(j , i)⎥⎥ j=1 ⎥ j≠i ⎦ ∞

Equation 17 no longer contains the variables FPL(i,i) and FPV(i,i) ∀i = 1, ..., ∞. This is also the case for all other equality constraints. Thus, these variables appear only in the objective function and in their respective non-negativity constraints. In addition, the weight coefficient of each of these variables in the objective function is positive (actually equal to 1). Because ν is a minimization problem, the value of each of these variables at the optimum is zero. QED Based on these two properties, the mathematical formulation of IDEAS (IF2) becomes

(15)

M

∀ i = 1, ..., ∞ ; ∀ k = 1, ..., n

=0

⎤ ⎡N F PL(j , i)⎥ − ykV (i)⎢∑ F OV(j , i) ⎥⎦ ⎢⎣ j = 1

∑ F PV(j , i)⎥⎥ = 0

xkL(j) F PL(i , j)

j = 1/ j ≠ i

∞

j=1

+

∑

⎡ ⎢N + ∑ ykV (j) F PV(i , j) − xkL(i)⎢∑ F OL(j , i) ⎢ j=1 ⎢ j=1 j≠i ⎣

∞

zkI(j)

∞

⇔∑ zkI(j) F PI(i , j) +

(16) 11188



Article

captured in Figure 4, which illustrates the T−x−y diagram for the mixture under study at the two pressures considered.

Next, the proposed IDEAS framework is illustrated on a case study involving the dual-pressure distillative separation of an azeotropic mixture.

3. CASE STUDY 1: DUAL-PRESSURE DISTILLATION OF A BINARY MIXTURE OF METHYL ACETATE (1) AND METHANOL (2) In this case study, the dual-pressure (1 and 3 bar) distillative separation of an equimolar mixture of methyl acetate (species 1) and methanol (species 2) is considered. Indeed, mixtures of methyl acetate (1) and methanol (2) exhibit azeotropic behavior at 65.5% mole fraction of methyl acetate at 1 bar and 56.1% mole fraction of methyl acetate at 3 bar. This behavior is captured by the Gamma− Phi vapor−liquid equilibrium model outlined in section 2. The vapor phase is considered to be an ideal gas, whereas the liquid-phase activity coefficients are quantified by the Wilson equations 5 and 6 and the vapor pressure of the various species is quantified by the Antoine equation7. The molar flow rate of the stream to be separated is 2 mol/ s. The specifications of the two desired product streams are 20% methyl acetate mole fraction for the first product and 80% methyl acetate mole fraction for the second product. A traditional two-column PSD design method and the aforementioned IDEAS design method are compared. To this end, both design methods were emplyed to minimize the total network unit inlet flow. The traditional design was pursued within the UniSim software platform, whereas the IDEAS design was carried out with in-house-developed IDEAS software. Both UniSim and IDEAS employed the aforementioned Wilson and Antoine equation thermodynamic models. Associated coefficient values are already built-in UniSim and can also be found in the literature (Gmehling et al.50 for the Wilson equation coefficients and Poling et al. 51 for the Antoine equation coefficients). The values of these coefficients are summarized in Tables 1 and 2. The thermodynamic behavior of these models is

Figure 4. T−x1−y1 diagrams of methyl acetate (1)/methanol (2) at (top) 1 and (bottom) 3 bar.

Given that UniSim’s coefficients are already built into the software, for purposes of a fair comparison between the traditional and IDEAS methods, UniSim’s values were also employed in IDEAS. Next, the optimization of a traditional two-column PSD design is carried out (section 3.1), followed by the IDEAS optimization (section 3.2). 3.1. Optimized Traditional Two-Column PSD Design. A traditional two-column PSD design and its UniSim representation are shown in parts a and b, respectively, of Figure 5. The objective function minimized here (FTotal) is the total network unit inlet flow, which is FTotal = FT1 + FT2, where FT1 and FT2 are the sums of vapor and liquid molar flow rates entering each plate in columns T1 and T2, respectively. To determine the number of degrees of freedom associated with the flow structure of the separation system in Figure 5, the mass and component mass balance equations (using methyl acetate as the reference species) outside columns T1 and T2 and the inlet and outlet specifications are considered

Table 1. Wilson Equation Coefficients from UniSim and Gmehling et al.50 Gmehling et al. UniSim

coefficient A11 (cal/mol) A12 (cal/mol) A21 (cal/mol)

0 −31.19 813.18

0 9.117 776.6

coefficient

Gmehling et al. UniSim

A22 (cal/mol) V1 (cm3/mol) V2 (cm3/mol)

0 79.84 40.73

0 80.26 40.76

Table 2. Antoine Equation Coefficients from UniSima and Poling et al.51 b k=1 coefficient

Poling et al.

UniSim

Poling et al.

UniSim

Ai Bi Ci Di Ei Fi

14.240 −2662.78 219.69 0 0 0

96.52 −7050 0 −12.38 1.137 × 10−5 2

16.578 −3638.27 239.50 0 0 0

59.84 −6283 0 −6.379 4.617 × 10−6 2

a

⎧ F = B1 + B2 ⎫ ⎪ ⎪ ⎪ D1 = D2 + B2 ⎪ ⎪ ⎪ ⎪ F1 = D1 + B1 ⎪ ⎨ ⎬, {F , z , xB1 , xB2} ⎪ Fz = B1xB1 + B2 xB2 ⎪ ⎪ ⎪ ⎪ D1xD1 = D2xD2 + B2 xB2 ⎪ ⎪ Fz = D x + B x ⎪ ⎩ 11 ⎭ 1 D1 1 B1

k=2

≙ {2, 0.5, 0.2, 0.8}

T (K), P (kPa). bT (°C), P (kPa). 11189



Article

Figure 5. (a) Traditional two-column azeotropic separation system. (b) UniSim representation of the azeotropic separation system.

Figure 6. Two views of iso-xD1 lines of FTotal as a function of xD2 in (xD1,xD2) space.

This is a set of 10 equations involving 12 variables. Therefore, there are two degrees of freedom (which can be chosen as xD1 and xD2) associated with the traditional design’s flow structure. These are augmented by four degrees of freedom associated with column internals, namely, the number of plates and the feed plate location for each of the two columns, thus giving rise to six degrees of freedom in total for the traditional design. The solution to the minimization problem is carried out using an exhaustive search in the triangular feasible

region of the (xD1,xD2) two-dimensional space defined by the physical inequality constraints xAZ1 ≥ xD1 ≥ xD2 ≥ xAZ2 . For every point (xD1 ,xD2 ) in this region, the aforementioned mass and component mass balances are solved, and the input−output flow/composition information for each of the two columns is determined. Then, the optimization problem is reduced to two separate single column optimization problems, each of which involves identifying the optimum number of plates and optimum feed plate location and is carried out by systematically 11190



Article

the separation of a binary azeotropic mixture. The grid discretization used for the temperature range can be selected to be uniform or nonuniform. A uniform grid generates flashes at equal temperature intervals, whereas a nonuniform grid generates more flashes for some temperature intervals and fewer flashes for others. Both uniform (section 3.2.1) and nonuniform (section 3.2.2) grids are used in the following development to acquire globally optimal solutions. Exact product specifications on the network outlet are achieved by setting the upper and lower bounds of the outlet composition vector to be the same for each of the outlet streams 1 and 2 such that [zO(1)]l = zO̅ (1) = [zO(1)]u and [zO(2)]l = zO̅ (2) = [zO(2)]u, where zO̅ (1) and zO̅ (2) are vectors of fixed mole fractions for the network outlet streams 1 and 2 corresponding to zO̅ (1) = [0.8 0.2] and zO̅ (2) = [0.2 0.8]. The network’s two outlet flows are such that FO(1) = FO(2) = 1, and the network’s inlet mole fractions and flow are zI(1) = [0.5 0.5]T and FI(1) = 2 . The solution process for the IDEAS formulation involves the sequential solution of finite linear programs of increasing size G, until the values of the finite optima do not change significantly (less than 0.25%). 3.2.1. Uniform Temperature Grid Discretization Strategy. Figure 8 shows the convergence of the IDEAS optimum

varying these integer-valued variables in UniSim. Every time the number of plates or the location of the feed plate changes, UniSim automatically adjusts the reflux and reboil ratio to ensure feasible column operation. Following this procedure, the globally minimum FTotal value as a function of xD1 and xD2 is shown in Figure 6 and is found to be 202.93 mol/s at (xD1,xD2) optim = (0.629,0.586). 3.2. IDEAS-Generated Globally Optimal Azeotropic Separation Design. According to the phase rule52 for binary mixtures in equilibrium at a fixed pressure, knowledge of temperature can yield a finite number of corresponding mole fractions, thus in the context of IDEAS u1 = [ P T ]T. Using the procedure outlined below, it can be determined that, for P = 1 bar, if T ≤ 330 K, there exist two xL1 solutions (q = 2), whereas if T > 330 K, there exists only one solution xL1 (q = 1). Similarly, for P = 3 bar, if T ≤ 365.7 K, there exist two xL1 solutions (q = 2), whereas if T > 365.7 K, there exists only one solution xL1 (q = 1). The following numerical/graphical procedure captures all feasible flash separators corresponding to the two operating pressures considered and a discretization of the feasible temperature range. (1) For each pressure, select a temperature T from the considered temperature grid and then evaluate Psat k , ∀k = 1, 2, from eq 7. (2) Express first γk, ∀k = 1, 2, and subsequently yVk , ∀k = 1, 2, as a function of xL1 from eqs 5 and 6 and from eq 4, respectively (3) Generate the graph of the function ∑2k=1yVk versus xL1 plot for the chosen P and T and identify all values of xL1 ∈ [0, 1]at which ∑2k=1yVk = 1 holds (as illustrated in Figure 7). Each of these values corresponds to a

Figure 8. IDEAS convergence: νG as a function of G. Optimized traditional design (squares), three IDEAS approximations (ovals): 0.15 K, 0.075 K, and 0.0375 K.

objective function ν G as G increases. As previously mentioned, the number of units made available to IDEAS depends on the level of discretization of the temperature space. Starting with an initial discretization level of 0.15 K, the discretization levels chosen are uniform, cover the entire temperature space, and correspond to 0.15 K/i, ∀T (G = 193 units for i = 1, G = 386 units for i = 2, and G = 772 for i = 4). At the smallest temperature discretization (0.0375 K) the generated flow sheet has a converged total network unit inlet flow value of 138.9072 mol/s. Because a temperature discretization of 0.0375 K contains as a subset in its universe of flashes all flashes generated at a temperature discretization of 0.075 K and because the total network unit inlet flow between these last two points differs by only 0.2%, convergence can be declared. This represents a 31.54% reduction over the optimized twocolumn design. The actual number of units employed at the IDEAS optimum is only a small fraction of the number of units made

Figure 7. ∑2k=1yVk versus xL1 plots: (▲) (P, T) = (1 bar, 328 K), where q = 2; (●) (P, T) = (1 bar, 333 K), where q = 1.

feasible flash separator with known values P, T, {xLk }2k=1, and {yVk }2k=1. (4) Return to step 1 and repeat steps 1−3 until all temperatures and pressures have been considered. All flashes created for a given pressure are part of an ensemble of flash separator units called a pressure universe. The above procedure is therefore repeated for several different pressure levels, to create as many pressure universes as desired. At least two pressure universes are required for 11191



Article

Figure 9. IDEAS optimum for G = 193 available units (uniform discretization). Liquid and vapor states indicate the phase of only the flash outlets but not the inlets.

Table 3. Flash Outlet Flow Rates and Mole Fractions for IDEAS Design unit no.

xL (methyl acetate)

flow

destination

yV (methyl acetate)

flow

destination

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 inlet

0.5628 0.5117 0.4444 0.3751 0.2957 0.2830 0.2035 0.1956 0.6249 0.6653 0.6938 0.7360 0.7681 0.7818 0.8060 0.8167 0.500

13.992 7.690 4.860 3.320 0.209 0.795, 0.876 0.548 0.452 18.449 5.263 4.536 2.729 0.206 0.750, 0.446 0.113 0.441 2.00

2 3 4 6 7 7, 8 outlet 1 outlet 1 10 11 12 14 15 16, outlet 2 outlet 2 outlet 2 3

0.6047 0.5792 0.5454 0.5082 0.4590 0.4501 0.3844 0.3766 0.5894 0.6143 0.6333 0.6631 0.6883 0.6997 0.7211 0.7312 −

14.827 14.977 8.682 3.857 0.248 2.071 0.456 0.423 13.856 17.466 4.267 3.540 0.195 1.536 0.0926 0.309 −

9 1 2 3 4 4 5 6 1 9 10 11 12 12 13 13 −

IDEAS infimum ν provides the globally optimal solution to the posed optimal design problem and, hence, represents a lower bound to the cost of any alternative design employing the same technologies (flash separators) and operating pressures. This is confirmed in Figure 8, where the optimum value of the traditional design (202.93 mol/s) is seen to be above the IDEAS value of ν G = 138.9072 mol/s for G = 772 available units. When a stream portion is indicated as leaving a flash and entering another flash, its thermal condition is known at the point of departure but not at the point of arrival.

available to the IDEAS formulation. Indeed, the IDEASgenerated flow sheet corresponding to 193 available units (0.15 K) is shown in Figure 9 and consists of a total of 16 units. This particular IDEAS design has an optimum objective function corresponding to 152.4843 mol/s, which represents a 24.85% decrease over the optimized twocolumn design. Every flow and corresponding quality vector for that design is detailed in Table 3. Because all approximating problems are linear programs, the global optimality of their solution and associated objective function values {ν G}∞ 1 is guaranteed. Thus, the 11192



Article

This suggests that although, a very large number of units and interconnections is made available for consideration, only a select few participate in the optimal network. In addition, the optimal networks are physically meaningful and can be realistically constructed. Other interesting similarities between the IDEAS optimum networks representation and the traditional azeotropic separation structure are as follows: (1) High-purity methyl acetate leaves the high-pressure (3 bar) sequence, whereas high-purity methanol leaves the low-pressure (1 bar) sequence, both in liquid form. (2) There exists only one stream connecting the low-pressure sequence to the high-pressure sequence and only one recycle stream connecting the high-pressure sequence back to the low-pressure sequence. (3) The feed enters the low-pressure sequence at a single unit, whose exit compositions straddle the feed composition. (4) Vapor streams always flow against the temperature gradient of flashes (hot to cold) belonging to a pressure universe, whereas liquid streams always flow in the direction of the temperature gradient (cold to hot) of flashes belonging to a pressure universe. In contrast, interesting differences between the IDEAS representation and the traditional azeotropic separation structure are as follows:

The model considers that an appropriate cooling/heating apparatus and possibly an appropriate pressure-altering apparatus are associated with each DN flow from one flash to another. What is important to realize is that the networks identified by the present formulation can always be realized a posteriori through the use of the aforementioned apparatuses, and as previously mentioned, the identified minimum total flow is correct over all networks regardless of whether the formulation includes energy balances. Close examination of the generated optimum IDEAS networks reveals that they have a number of properties: (1) They employ only a small percentage ( 327 ≤ 359.6 > 359.6

T discretization (K) 0.01875 (i = 16) 0.3 (i = 1) 0.01875 (i = 16) 0.3 (i = 1)

(1) Communication between low- and high-pressure sequences occurs at the last stage of each sequence in IDEAS

Figure 10. IDEAS optimum for G = 150 available units (nonuniform discretization). Liquid and vapor states indicate the phase of only the flash outlets but not the inlets. 11193



Article

Table 5. Flash Outlet Flow Rates and Mole Fractions for IDEAS Design 2 unit no.

xL (methyl acetate)

flow

destination

yV (methyl acetate)

flow

destination

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 inlet

0.5654 0.5365 0.5303 0.4890 0.4256 0.3809 0.3443 0.3134 0.2866 0.2630 0.2420 0.2231 0.2059 0.1902 0.6272 0.6539 0.6852 0.7323 0.7652 0.7919 0.8146 0.500

8.831 3.396 3.444 5.739 4.518 0.107 2.211, 0.244 0.149 0.231 1.165 0.102 0.0741 0.127 0.695 9.857 6.609 5.751 2.628 0.952, 0.853 0.644 0.355 2.00

3 4 4 5 7 8 10, 11 12 13 14 outlet 1 outlet 1 outlet 1 outlet 1 16 17 18 19 20, 21 outlet 2 outlet 2 5

0.6060 0.5915 0.5884 0.5679 0.5357 0.5114 0.4902 0.4708 0.4526 0.4353 0.4187 0.4025 0.3866 0.3711 0.5907 0.6071 0.6272 0.6603 0.6858 0.7084 0.7292 −

14.794 4.440 5.380 7.833 6.732 0.1308 3.247 0.136 0.238 1.044 0.141 0.0754 0.103 0.469 13.823 8.866 5.615 4.755 1.322 0.307 0.498 −

15 1 1 2 4 5 5 5 6 7 7 8 8 9 1 15 16 17 18 18 19 −

Table 6. Wilson Equation Coefficients coefficient A11 A12 A13 A21 A22 A23

(cal/mol) (cal/mol) (cal/mol) (cal/mol) (cal/mol) (cal/mol)

value

coefficient

value

0 469.55 1448.01 107.33 0 583.11

A31 (cal/mol) A32 (cal/mol) A33 (cal/mol) V1 (cm3/mol) V2 (cm3/mol) V3 (cm3/mol)

291.27 −161.88 0 17.88 40.76 74.47

Table 7. Antoine Equation Coefficientsa

Figure 11. Isopurity lines of the total flow entering network flashes as a function of pressure.

designs, while it occurs at intermediate stages in the traditional design. (2) The high-pressure sequence has no rectifying section. Analysis of the above flow sheets indicates that the flash inlet flow quantity is largest for network flashes within close proximity of the azeotropic pinch point (lower temperatures). Therefore, a nonuniform temperature grid is introduced in the next section to allow the generation of more flashes around to the azeotropic pinch point. This approach considerably reduces the problem size while still resulting in globally optimal flow sheets. 3.2.2. Nonuniform Temperature Grid Discretization Strategy. Starting with an initial discretization level of 0.3 K, the discretization level chosen is nonuniform, with its specifics displayed in Table 4 corresponding to G = 150 units. The resulting objective function is equal to 138.6528 mol/s, which is effectively identical to the converged globally optimal value of 138.9072 mol/s determined in the previous section using a uniform temperature grid

a

coefficient

k=1

k=2

k=3

Ai Bi Ci Di Ei Fi

65.93 −7228 0 −7.177 4.031 × 10−6 2

59.84 −6283 0 −6.379 4.617 × 10−6 2

71.30 −5952 0 −8.531 7.824 × 10−6 2

T (K), P (kPa).

discretization strategy and G = 772 units. The IDEASgenerated flow sheet corresponding to this number of flashes is shown in Figure 10, with the flow and the corresponding quality vector detailed in Table 5. Therefore, employing a nonuniform temperature grid discretization can potentially serve as an intelligent allocation of limited computational resources while still yielding globally optimal designs. In the next section, the effects of the operating pressure of the second universe of flashes (while the operating pressure of the first universe of flashes is kept equal to 11194



Article

given product purity, a design operating at lower pressure requires more total unit inlet flow than a design operating at a higher pressure. This intuitive trend manifests itself in IDEAS designs as well. Figure 11 shows the total unit inlet flow as a function of pressure at three different product purities for a uniform discretization of 0.15 K. This discretization level was selected because it leads to readily realizable flow sheets and a smaller computational burden than that needed for smaller discretization levels. The obtained IDEAS optimization results indicate that, after an initial sharp decrease in total unit inlet flow at low pressures, the impact of further increasing pressure is minimal. This trend repeats itself at all product purities, albeit at lower product purities the curves begin to level off at lower pressures (earlier) than for higher product purities. Therefore, IDEAS can help determine not only optimal design structures and the optimal total unit inlet flow but also appropriate levels of operating pressure for a desired level of product purity.

Figure 12. T−x2−y2 diagram of methanol (2)/acetone (3) at (top) 0.2 and (bottom) 1 bar.

4. CASE STUDY 2: DUAL-PRESSURE DISTILLATION OF A TERNARY MIXTURE OF WATER (1), METHANOL (2), AND ACETONE (3) In this case study, the dual-pressure distillative separation of a mixture of water (species 1), methanol (species 2), and acetone (species 3) is considered. The mixture’s molar flow rate and the mole fractions of species 1, 2, and 3 are 3 mol/s and 0.3, 0.35, and 0.35, respectively. Binary mixtures of methanol (2) and acetone (3) exhibit a minimumboiling azeotrope at 79.07% mole fraction of acetone (3) at 1 bar and 328.5 K and 97.5% mole fraction of acetone (3) at 0.2 bar and 288.7 K. Similarly to the previous case study, the mixture’s thermodynamic behavior is captured by a Gamma−Phi vapor−liquid equilibrium model. The vapor phase is considered to be an ideal gas, whereas the liquid-phase activity coefficients are quantified by the Wilson equations (eqs 5 and 6), and the vapor pressures of the various species are quantified by the Antoine equation7. Similarly to case study 1, a traditional two-column PSD design method and the aforementioned IDEAS design

Table 8. Design Specifications (Component, Mole Fraction, Flow Rate) product 1 product 2 product 3

specification set 1

specification set 2

water, 0.80, 1.0 mol/s methanol, 0.80, 1.0 mol/s acetone, 0.80, 1.0 mol/s

water, 0.80, free methanol, 0.80, free acetone, 0.80, 1.2 mol/s

atmospheric) and product purity on the IDEAS optimum is quantified. 3.3. Effects of Pressure and Purity on the Total Flash Inlet Flow. The IDEAS framework makes it possible to systematically and rigorously evaluate the effects that product purity and pressure have on the optimum objective function value of the total inlet flash flow. Consider the traditional azeotropic separation structures described in section 3.1. For a given pressure, a design specifying lower product purity requires less total unit inlet flow than a design specifying higher product purity. Similarly, for a

Figure 13. Optimized traditional two-column setup for specification set 2. 11195



Article

method are compared. To this end, both design methods were used minimize the total network unit inlet flow. The traditional design was pursued within the UniSim software platform, whereas the IDEAS design was carried out with in-house-developed IDEAS software. Both UniSim and IDEAS employed the aforementioned Wilson and Antoine equation thermodynamic models. The coefficient values employed in the two methods are summarized in Tables 6 and 7. The ability of these thermodynamic models to capture the azeotropic behavior of the methanol (2)/ acetone (3) mixture is shown in Figure 12, which illustrates the mixture’s T−x−y equilibrium diagram, at the two pressures considered. Two sets of design specifications were considered, as shown in Table 8. For each set of specifications, a traditional two-column design33 was compared to the IDEAS-generated optimal design. The traditional design was carried out in UniSim, and the aforementioned thermodynamic data were used in both the UniSim and IDEAS designs. 4.1. Optimized Traditional Two-Column Design. A UniSim representation of the considered two-column design is shown in Figure 13. This traditional design has been shown to be suitable to separate the water−methanol− acetone mixture.33 The objective function to be minimized is again the same as for the first case study (total network unit inlet flow), FTotal = FT1 + FT2, where FT1 and FT2 are the sums of vapor and liquid molar flow rates entering each plate in columns T 1 (P = 0.2 bar) and T2 (P = 1 bar). respectively. The first distillation column in the considered design has a known feed. Thus, it has four degrees of freedom,48 two of them integer variables (number of plates and feed plate location) and the other two continuous. In this case study, for both specification sets, the two continuous degrees of freedom are chosen to be the specified top product flow rate and specified acetone mole fraction. Once the first column’s degrees of freedom are specified, the second column’s feed is also specified. Thus again, the second column has two integer and two continuous degrees of freedom. For the first set of specifications, the second column must meet four specifications, namely, the flow rates of both of its products are known, and the methanol mole fraction in its distillate and the water mole fraction in its bottom are known. Given the continuous nature of the specifications and the integer nature of two of the four degrees of freedom, it is likely that the considered traditional design might not be able to meet the first set of specifications, because it has four integer degrees of freedom and must meet two continuous specifications. On the other hand, for the second set of specifications, the first column again has only two integer degrees of freedom. The second column, however, must meet two specifications, namely, the methanol mole fraction in its distillate and the water mole fraction in its bottom are known. By selecting these variables as the second column’s continuous degrees of freedom, the second column has two integer degrees of freedom. Therefore, the traditional design has no unmet continuous specifications, and four integer degrees of freedom that can be used to optimize the design’s total network flow.

An exhaustive search over all possible values of the four integer degrees of freedom is carried out through repeated UniSim simulations. For specification set 1, which must meet two continuous specifications, there exists no twocolumn design capable of delivering the desired specifications. For specification set 2 (Figure 13), which has no unmet continuous specifications, the optimized two-column design yields distillate and bottom streams flows equal to 0.7634 and 1.037 mol/s, respectively. The optimized total network flow is equal to 44.487 mol/s. 4.2. IDEAS-Generated Globally Optimal Azeotropic Separation Design. According to the phase rule52 for ternary mixtures in equilibrium at a fixed pressure, knowledge of temperature, and species 1 mole fraction can yield a finite number of corresponding mole fractions for the other two species; thus in the context of IDEAS, u1 can be chosen as u1 = [ P T x1L ]T. Superscript q, when needed, is used to indicate multiple xL2 solutions corresponding to a fixed T,xL1 . A numerical/graphical procedure similar to the one outlined in the first case study is employed to capture all feasible flash separators corresponding to the two operating pressures considered, by discretizing the feasible range of both temperature T and the first species’ liquidphase mole fraction xL1 . Figure 14 illustrates how feasible

Figure 14. ∑3k=1yVk versus xL2 plots for an xL1 discretization of 0.125 at T = 336 K, P = 1 bar. Triangles indicate feasible flash separators.

separators are generated, by identifying the xL2 values for which ∑3k=1yVk = 1 at any given values of xL1 , T, and P. In this case, a nonuniform grid, with an increased refinement strategy at low temperatures, to better capture the change in composition around the azeotropic pinch point, is used. The grid sizes for T and xL1 are displayed in Table 9 for different temperature ranges and pressures. Table 9. Nonuniform Grid Discretizations of T and xL1 for the Two Pressure Levels Considered P (bar) 1 1 0.2 0.2 11196

T range (°C) T T T T

≤ 60 > 60 ≤ 18 > 18

T discretization (K)

xL1 discretization

1 2 0.5 1

0.03125 (1/32) 0.0625 (1/16) 0.03125 (1/32) 0.0625 (1/16)



Article

(1) For the first set of specifications, which is unattainable by the traditional two-column method, IDEAS is able to identify a feasible design. The optimum solution obtained features a total network flow of 43.785 mol/s. The identities of the flashes participating in the optimum network, as well as the optimal interconnecting flows, are shown in Appendix A and Appendix B, respectively. (2) For the second set of specifications, which is attainable by the traditional two-column method with a minimum total network flow of 44.487 mol/s, the IDEAS optimum design features a total network flow of 37.738 mol/s, which is 15.15% lower than the traditional design’s optimum value.

Table A2. Optimal Network Flashes: P = 0.2 bar, T (°C) T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T

5. CONCLUSIONS A methodology is demonstrated for the global minimization of total network flash inlet flow for pressure swing distillation (PSD) systems using the IDEAS framework. IDEAS yields globally optimal designs that minimize the total network flow required to break an azeotrope achieving the desired level of separation. This allows for a rigorous comparison of alternative designs. The IDEAS framework is able to successfully generate optimal distillation networks using flash units as building blocks, without any preconception of a network structure. Only vapor−liquid equilibrium data information is provided a priori to the IDEAS design procedure. Two case studies consisting of a binary and a ternary mixture are considered, and the obtained IDEAS designs indicate that the traditional two-column design can be significantly improved (by 31.54% for the binary case and by 15.15% for the ternary case). IDEAS-generated designs surpass the two-column PSD traditional designs, because the flexibility of the IDEAS framework allows the consideration of all possible flow and unit combinations. Not only can IDEAS lead to improvements when compared to optimized traditional designs but it can also provide vital information in other areas. Indeed, for the binary mixture, the obtained IDEAS designs indicate that, following an initial sharp decrease, the dependence of the optimum objective function value on pressure is minimal, and higher purity level requirements lead to increased optimum objective function values with similar pressure dependence. Also, for ternary mixtures, IDEAS can deliver compositions and outlet flow rates unattainable using the rigid two-column design structure.

■

APPENDIX A The identities of the flashes participating in the optimum network are listed in Tables A1 and A2. Table A1. Optimal Network Flashes: P = 1 bar, T (°C) T T T T T T

= = = = = =

69.0, 77.0, 77.0, 79.0, 81.0, 85.0,

#79, x1 = 0.3125, x2 = 0.6760, y1 = 0.1275, y2 = 0.8319 #98, x1 = 0.6875, x2 = 0.3050, y1 = 0.3167, y2 = 0.6273 #100, x1 = 0.8750, x2 = 0.0870, y1 = 0.3771, y2 = 0.2261 #101, x1 = 0.8750, x2 = 0.0970, y1 = 0.4083, y2 = 0.2722 #102, x1 = 0.8750, x2 = 0.1060, y1 = 0.4417, y2 = 0.3209 #104, x1 = 0.8750, x2 = 0.1210, y1 = 0.5160, y2 = 0.4249

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

16.0, 16.5, 17.0, 17.5, 18.0, 18.0, 18.0, 19.5, 19.5, 19.5, 20.5, 20.5, 20.5, 20.5, 20.5, 21.5, 21.5, 22.5, 22.5, 22.5, 22.5, 23.5, 23.5, 23.5, 23.5, 24.5, 24.5, 24.5, 25.5, 25.5, 25.5, 26.5, 26.5, 26.5, 27.5, 27.5, 27.5, 28.5, 29.5, 29.5, 29.5, 29.5, 30.5, 30.5, 30.5, 31.5, 31.5, 31.5, 33.5, 35.5, 36.5, 38.5, 39.5,

#105, #108, #113, #120, #125, #129, #130, #135, #138, #139, #144, #147, #148, #149, #150, #159, #160, #167, #168, #171, #172, #181, #185, #186, #190, #195, #199, #200, #206, #207, #213, #219, #220, #225, #230, #231, #236, #243, #247, #248, #249, #253, #256, #257, #258, #261, #262, #263, #270, #273, #276, #279, #280,

x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

0.0312, 0.0625, 0.0938, 0.1250, 0.0312, 0.1562, 0.1875, 0.0625, 0.2500, 0.3125, 0.0625, 0.2500, 0.3125, 0.3750, 0.4375, 0.3125, 0.3750, 0.0625, 0.1250, 0.3125, 0.3750, 0.1250, 0.3750, 0.4375, 0.6875, 0.1875, 0.4375, 0.5000, 0.0625, 0.1250, 0.5000, 0.1250, 0.1875, 0.5000, 0.1250, 0.1875, 0.5000, 0.3750, 0.1250, 0.1875, 0.3125, 0.6250, 0.2500, 0.3125, 0.4375, 0.2500, 0.3125, 0.3750, 0.4375, 0.5000, 0.6250, 0.7500, 0.6875,

x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

0.1190, 0.1630, 0.1980, 0.2270, 0.4550, 0.2520, 0.2020, 0.5530, 0.3020, 0.2270, 0.6260, 0.3880, 0.3160, 0.2490, 0.1890, 0.3830, 0.3170, 0.7350, 0.6570, 0.4370, 0.3700, 0.7000, 0.4130, 0.3490, 0.1280, 0.6610, 0.3840, 0.3210, 0.8460, 0.7680, 0.3490, 0.7960, 0.7210, 0.3730, 0.8210, 0.7450, 0.3940, 0.5480, 0.8630, 0.7870, 0.6380, 0.2970, 0.7290, 0.6550, 0.5120, 0.7450, 0.6700, 0.5970, 0.5500, 0.4980, 0.3640, 0.2370, 0.3120,

y1 y1 y1 y1 y1 y1 y1 y1 y1 y1 y1 y1 y1 y1 y1 y1 y1 y1 y1 y1 y1 y1 y1 y1 y1 y1 y1 y1 y1 y1 y1 y1 y1 y1 y1 y1 y1 y1 y1 y1 y1 y1 y1 y1 y1 y1 y1 y1 y1 y1 y1 y1 y1

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

0.0138, 0.0236, 0.0315, 0.0382, 0.0095, 0.0441, 0.0518, 0.0173, 0.0597, 0.0704, 0.0171, 0.0589, 0.0696, 0.0788, 0.0865, 0.0702, 0.0797, 0.0175, 0.0334, 0.0717, 0.0816, 0.0342, 0.0842, 0.0933, 0.1199, 0.0506, 0.0970, 0.1055, 0.0191, 0.0366, 0.1102, 0.0380, 0.0545, 0.1154, 0.0395, 0.0568, 0.1210, 0.1041, 0.0430, 0.0619, 0.0949, 0.1537, 0.0829, 0.0994, 0.1280, 0.0869, 0.1043, 0.1201, 0.1486, 0.1808, 0.2213, 0.2766, 0.2751,

y2 y2 y2 y2 y2 y2 y2 y2 y2 y2 y2 y2 y2 y2 y2 y2 y2 y2 y2 y2 y2 y2 y2 y2 y2 y2 y2 y2 y2 y2 y2 y2 y2 y2 y2 y2 y2 y2 y2 y2 y2 y2 y2 y2 y2 y2 y2 y2 y2 y2 y2 y2 y2

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

0.0976 0.1250 0.1452 0.1617 0.3074 0.1763 0.1432 0.3747 0.2106 0.1616 0.4342 0.2744 0.2271 0.1831 0.1431 0.2833 0.2393 0.5490 0.4914 0.3359 0.2901 0.5474 0.3385 0.2936 0.1311 0.5448 0.3388 0.2930 0.7269 0.6615 0.3349 0.7209 0.6569 0.3768 0.7825 0.7145 0.4194 0.5790 0.9117 0.8373 0.6963 0.3860 0.8258 0.7534 0.6171 0.8891 0.8120 0.7384 0.7753 0.8044 0.6784 0.5600 0.7180

■

APPENDIX B The optimal interconnecting flows are listed in Table B1. 11197



Article

Table B1. Optimal Interconnecting Flows FP(171,1) = 0.140 FP(185,1) = 0.800 FP(186,1) = 0.010 FP(199,1) = 1.091 FP(213,1) = 0.958 FOL(1,100) = 0.814 FOL(1,101) = 0.086 FOV(1,190) = 0.092 FOL(2,167) = 0.158 FOV(2,206) = 0.043 FOV(2,220) = 0.080 FOV(2,230) = 0.031 FOV(2,248) = 0.685 FOL(3,105) = 0.011 FOL(3,108) = 0.182 FOV(3,113) = 0.191 FOV(3,120) = 0.612 FPL(79,248) = 0.021 FPL(79,256) = 0.173 FPL(79,262) = 0.024 FPL(98,258) = 0.127 FPL(98,276) = 0.287 FPL(100,104) = 1.063

■

FPL(159,148) 0.142 FPL(160,149) 0.342 FPL(168,144) 0.050 FPL(172,148) 0.432 FPL(181,144) 0.703 FPL(185,147) 1.059 FPL(186,148) 0.018 FPL(190,100) 0.028 FPL(199,147) 0.599 FPL(200,148) 0.074 FPL(213,160) 0.289 FPL(219,167) 0.051 FPL(220,168) 0.104 FPL(220,181) 0.392 FPL(225,159) 0.189 FPL(225,172) 0.361 FPL(230,206) 0.027 FPL(231,181) 0.085 FPL(236,185) 1.245 FPL(236,199) 0.918 FPL(245,171) 0.173 FPL(248,207) 0.144 FPL(248,219) 0.042

= = = = = = = = = = = = = = = = = = = = = = =

FPL(273,269) = 0.090 FPL(273,270) = 0.273 FPL(276,243) = 0.617 FPL(276,258) = 0.371 FPL(276,263) = 0.154 FPL(279,213) = 0.273 FPL(279,225) = 0.341 FPL(279,236) = 1.601 FPL(279,245) = 0.087 FPL(279,276) = 0.205 FPL(280,270) = 0.048 FPL(280,273) = 0.430 FPV(100,138) = 0.345 FPV(101,100) = 0.041 FPV(102,101) = 0.026 FPV(105,108) = 0.007 FPV(105,130) = 0.003 FPV(108,120) = 0.055 FPV(108,139) = 0.134 FPV(113,129) = 0.152 FPV(113,139) = 0.186 FPV(120,138) = 0.250 FPV(120,149) = 0.899

FPV(149,160) 0.344 FPV(150,149) 0.156 FPV(159,135) 0.103 FPV(159,199) 0.058 FPV(160,125) 0.236 FPV(160,186) 0.011 FPV(160,200) 0.039 FPV(167,207) 0.198 FPV(167,220) 0.153 FPV(168,243) 0.115 FPV(171,236) 0.180 FPV(172,225) 0.199 FPV(181,258) 0.245 FPV(185,236) 0.382 FPV(190,150) 0.095 FPV(195,243) 0.228 FPV(195,258) 0.365 FPV(200,101) 0.015 FPV(206,219) 0.054 FPV(206,247) 0.025 FPV(207,231) 0.188 FPV(207,249) 0.152 FPV(219,257) 0.035

=

FPL(101,104) = 0.097 FPL(102,104) = 0.114 FPL(104,98) = 0.280 FPL(104,213) = 0.372 FPL(104,253) = 0.028 FPL(104,267) = 0.040 FPL(104,279) = 1.941 FPL(104,280) = 0.259 FPL(135,125) = 0.197 FPL(138,129) = 0.103 FPL(138,139) = 0.291 FPL(139,102) = 0.064 FPL(139,113) = 0.146 FPL(139,190) = 0.032 FPL(144,135) = 0.355 FPL(147,138) = 1.216 FPL(148,149) = 0.533 FPL(149,104) = 0.262 FPL(149,120) = 0.482 FPL(149,150) = 0.098 FPL(150,102) = 0.038 FPL(159,138) = 0.086

= = = = = = = = = = = = = = = = = = = = = =

FPV(125,135) 0.423 FPV(129,138) 0.246 FPV(130,139) 0.005 FPV(135,144) 0.513 FPV(135,168) 0.068 FPV(135,245) 0.088 FPV(135,267) 0.011 FPV(138,100) 0.525 FPV(138,147) 1.599 FPV(139,138) 0.376 FPV(144,167) 0.138 FPV(144,181) 0.477 FPV(144,195) 0.297 FPV(147,171) 0.156 FPV(147,185) 0.994 FPV(147,199) 0.713 FPV(147,213) 0.584 FPV(148,147) 0.406 FPV(148,159) 0.199 FPV(148,172) 0.274 FPV(149,148) 0.746

= = = = = = = = = = = = = = = = = = = =

FPV(230,257) = 0.031 FPV(231,263) = 0.290 FPV(243,279) = 0.962 FPV(247,261) = 0.062 FPV(248,79) = 0.063 FPV(248,256) = 0.188 FPV(248,257) = 0.069 FPV(248,261) = 0.217 FPV(248,262) = 0.254 FPV(249,276) = 0.115 FPV(253,102) = 0.028 FPV(256,270) = 0.229 FPV(257,276) = 0.077 FPV(258,279) = 0.990 FPV(261,269) = 0.083 FPV(261,273) = 0.437 FPV(263,276) = 0.458 FPV(270,98) = 0.134 FPV(273,280) = 0.225 FPV(279,104) = 1.388

=

xLk (i) = kth-species equilibrium liquid composition leaving the ith unit yVk (i) = kth-species equilibrium vapor composition leaving the ith unit Psat k (T) = kth-species temperature-dependent saturated vapor pressure γk({xLl }nl=1,T) = kth-species nonideal liquid activity coefficient Λk,j(T) = Wilson equation temperature-dependent parameter ϕk({yVl }nl=1, T,P) = kth-species nonideal fugacity coefficient

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: +1 310 206 0300. Fax: +1 310 206 4107 420. Notes

The authors declare no competing financial interest.

■ ■

FPL(248,230) = 0.029 FPL(249,195) = 0.181 FPL(253,186) = 0.018 FPL(256,248) = 0.132 FPL(257,220) = 0.174 FPL(258,195) = 0.115 FPL(261,247) = 0.042 FPL(262,220) = 0.089 FPL(262,231) = 0.185 FPL(262,248) = 0.150 FPL(263,248) = 0.019 FPL(267,200) = 0.050 FPL(267,213) = 0.014 FPL(269,261) = 0.151 FPL(269,262) = 0.022 FPL(270,249) = 0.144 FPL(270,257) = 0.116 FPL(270,262) = 0.123 FPL(270,263) = 0.033 FPL(273,79) = 0.145 FPL(273,261) = 0.134

ACKNOWLEDGMENTS Financial support for this work through NSF Grant NSF-CBET 0829211 is gratefully acknowledged.

Traditional PSD Design Variables

NOTATION

F = inlet flow rate to the PSD system F1 = inlet flow rate to the first distillation column, F1 = F + D2 D1 = flow rate of distillate stream leaving the first distillation column for the second column D2 = flow rate of distillate stream leaving the second distillation column, recycled back to the first column B1 = bottom flow rate leaving the first distillation column B2 = bottom flow rate leaving the second distillation column

Thermodynamic Variables

Ai,j = Wilson equation interaction parameters between the ith and jth species Ak, Bk, Ck, Dk, Ek, Fk = Antoine equation kth-species parameters R = universal gas constant P = flash-unit pressure T = flash-unit temperature Vk = kth-species molar volume 11198



Article

FT2 = total flow of vapor and liquid inside the second (highpressure) distillation column FT1 = total flow of vapor and liquid inside the first (lowpressure) distillation column FTotal = sum of the flow rates entering each plate inside each column in the PSD system z = Composition of methyl acetate in the inlet flow rate to the PSD system z1 = Composition of methyl acetate in the inlet flow rate to the first distillation column xD1 = Composition of methyl acetate in the distillate stream leaving the first distillation column xD2 = Composition of methyl acetate in the distillate stream leaving the second distillation column xB1 = Composition of methyl acetate in the bottom stream leaving the first distillation column xB2 = Composition of methyl acetate in the bottom stream leaving the second distillation column xAZ1 = Azeotropic composition of methyl acetate in the lowpressure column (column T1) xAZ2 = Azeotropic composition of methyl acetate in the highpressure column (column T2)

(9) Knapp, J.; Doherty, M. Thermal Integration of Homogeneous Azeotropic Distillation Columns. AIChE J. 1990, 36, 969−984. (10) Abu Eishah, S. I.; Luyben, W. L. Design and Control of a TwoColumn Azeotropic Distillation System. Ind. Eng. Chem. Process Des. Dev. 1985, 24, 132−140. (11) Knight, J.; Doherty, M. Optimal Design and Synthesis of Homogeneous Azeotropic Distillation Sequences. Ind. Eng. Chem. Res. 1989, 28, 564−572. (12) Pham, H. N.; Ryan, P.; Doherty, M. F. Design and Minimum Reflux for Heterogeneous Azeotropic Distillation Columns. AIChE J. 1989, 35 (10), 1585−1591. (13) Ryan, P.; Doherty, M. F. Design/Optimization of Ternary Heterogeneous Azeotropic Distillation Sequences. AIChE J. 1989, 35 (10), 1592−1601. (14) Bossen, B. J.; Jorgensen, S. B.; Gani, R. Simulation, Design and Analysis of Azeotropic Distillation Operations. Ind. Eng. Chem. Res. 1993, 32, 620−633. (15) Mortaheb, H. R.; Kosuge, H. Simulation and Optimization of Heterogeneous Azeotropic Distillation Process with a Rate-Based Model. Chem. Eng. Process. 2004, 43 (3), 317−326. (16) Martin, L. L.; Huang, X. Optimized Heat and Power Exchanger Network in Ethanol−Water Pressure Swing Distillation. In Proceedings of the AIChE Annual Meeting; American Institute of Chemical Engineers (AIChE): New York, 2005; Paper 599d. (17) Fidkowski, Z. T.; Malone, M. F.; Doherty, M. F. Computing Azeotropes in Multicomponent Mixtures. Comput. Chem. Eng. 1993, 17 (12), 1141−1155. (18) Pham, H. N.; Doherty, M. F. Design and Synthesis of Heterogeneous Azeotropic Distillations: II. Residue Curve Maps. Chem. Eng. Sci. 1990, 45 (7), 1837−1843. (19) Pham, H. N.; Doherty, M. F. Design and Synthesis of Heterogeneous Azeotropic Distillations: III. Column Sequences. Chem. Eng. Sci. 1990, 45 (7), 1845−1854. (20) Liu, G.; Jobson, M.; Smith, R.; Wahnschaftt, O. Shortcut Design Method for Columns Separating Azeotropic Mixtures. Ind. Eng. Chem. Res. 2004, 43 (14), 3908−3923. (21) Tapp, M.; Holland, S. T.; Hildebrandt, D.; Glasser, D. Column Profile Maps. 1. Derivation and Interpretation. Ind. Eng. Chem. Res. 2004, 43, 364−374. (22) Gutierrez-Antonio, C.; Jimenez-Gutierrez, A. Method for the Design of Azeotropic Distillation Columns. Ind. Eng. Chem. Res. 2007, 46 (20), 6635−6644. (23) Phimister, J. R.; Seider, W. D. Semicontinuous, Pressure Swing Distillation. Ind. Eng. Chem. Res. 2000, 39, 122−130. (24) Modla, G.; Lang, P. Separation of a Ternary Homoazeotropic Mixture by Pressure Swing Batch Distillation. Hung. J. Ind. Chem. 2008, 36, 89−94. (25) Brusis, D.; Frey, Th.; Stichlmair, J.; Wagner, I.; Duessel, R.; Kuppinger, F.-F. MINLP-Optimization of Several Process Structures for the Separation of Azeotropic Ternary Mixtures. Comput.-Aided Chem. Eng. 2000, 8, 109−114. (26) Bauer, M. H.; Stichlmair, J. Design and Economic Optimization of Azeotropic Distillation Processes Using Mixed-Integer Non-Linear Programming. Comput. Chem. Eng. 1998, 20 (9), 1271−1286. (27) Frey, Th.; Bauer, M. H.; Stichlmair, J. MINLP-Optimization of Complex Column Configurations for Azeotropic Mixtures. Comput. Chem. Eng. 1997, 21 (1), S217−S222. (28) Li, C.; Zhang, X. Design of Separation Process of Azeotropic Mixtures Based on the Green Chemical Principles. J. Clear Prod. 2007, 15 (7), 690−698. (29) Caballero, J. A.; Grossmann, I. E. An Aggregated MINLP Optimization Model for Synthesizing Azeotropic Distillation Systems. Comput. Chem. Eng. 1999, 23 (1), S85−S88. (30) Kossak, S.; Kraemer, K.; Marquardt, W. Efficient OptimizationBased Design of Distillation Columns for Homogeneous Azeotropic Mixtures. Ind. Eng. Chem. Res. 2006, 45 (25), 8492−8502. (31) Kraemer, K.; Kossak, S.; Marquardt, W. Efficient OptimizationBased Design of Distillation Processes for Homogeneous Azeotropic Mixtures. Ind. Eng. Chem. Res. 2009, 48 (14), 6749−6764.

IDEAS Variables

FI(i) = ith DN inlet stream FL(i) = ith OP liquid outlet FO(i) = ith DN outlet stream FOI(i,j) = jth DN inlet stream to the ith DN outlet FOL(i,j) = ith DN outlet stream from the jth OP liquid outlet FOV(i,j) = ith DN outlet stream from the jth OP vapor outlet FPI(i,j) = ith OP inlet stream from the jth DN network inlet FPL(i,j) = ith OP inlet stream from the jth OP liquid outlet FPV(i,j) = ith OP inlet stream from the jth OP vapor outlet FV(i) = ith OP vapor outlet G = total number of flashes generated in all pressure universes for different discretizations M = number of IDEAS network inlets N = number of IDEAS network outlets xLk (i) = kth-species, ith OP liquid outlet composition yVk (i) = kth-species, ith OP vapor outlet composition ZIk(i) = kth-species, ith DN inlet stream composition ZOk (i) = kth-species, ith DN outlet stream composition zO(i), [zO(i)]l, [zO(i)]u = composition vector of the ith DN outlet stream and its lower and upper bounds, respectively

■

REFERENCES

(1) Roscoe, H. E. On the Composition of the Aqueous Acids of Constant Boiling Point. J. Chem. Soc. 1860, 13, 146−164. (2) Roscoe, H. E. On the Composition of the Aqueous Acids of Constant Boiling PointSecond Communication. J. Chem. Soc. 1862, 15, 270−276. (3) Roscoe, H. E.; Ditmar, W. On the Absorption of Hydrochloric Acid and Ammonia in Water. J. Chem. Soc. 1859, 12, 128−159. (4) Diwekar, U. Batch Distillation Simulation, Optimal Design, and Control, 2nd ed.; CRC Press: Boca Raton, FL, 2011; p 236. (5) Knapp, J.; Doherty, M. A New Pressure-Swing Distillation Process for Separating Homogeneous Azeotropic Mixtures. Ind. Eng. Chem. Res. 1992, 31, 346−357. (6) Barnicki, S.; Siirola, J. Process Synthesis Prospective. Comput. Chem. Eng. 2004, 28 (4), 441−446. (7) Noble, R.; Agrawal, R. Separations Research Needs for 21st Century. Ind. Eng. Chem. Res. 2005, 44 (9), 2887−2892. (8) Horsley, L. H., Ed. Azeotropic Data III; Advances in Chemistry Series; American Chemical Society: Washington, DC, 1973; Vol. 116. 11199



Article

(32) Linninger, A. A. Industry-wide Energy Saving by Complex Separation Networks. Comput. Chem. Eng. 2009, 33 (12), 2018−2027. (33) McCallum, B. R.; Lucia, A. Energy Targeting and Minimum Energy Distillation Column Sequences. Comput. Chem. Eng. 2010, 34 (6), 931−942. (34) Wilson, S.; Manousiouthakis, V. I. I DEAS Approach to Process Network Synthesis: Application to Multi-component MEN. AIChE J. 2000, 46 (12), 2408−2416. (35) Drake, J. E.; Manousiouthakis, V. I. IDEAS Approach to Process Network Synthesis: Minimum Plate Area for Complex Distillation Networks with Fixed Utility Cost. Ind. Eng. Chem. Res. 2002, 41, 4984. (36) Drake, J. E.; Manousiouthakis, V. I. IDEAS Approach to Process Network Synthesis: Minimum Utility Cost for Complex Distillation. Chem. Eng. Sci. 2002, 57, 3095. (37) Holiastos, K.; Manousiouthakis, V. I. Infinite-Dimensional StateSpace (IDEAS) Approach to Globally Optimal Design of Distillation Networks Featuring Heat and Power Integration. Society 2004, 43 (24), 7826−7842. (38) Martin, L. L.; Manousiouthakis, V. I. Globally Optimal Power Cycle Synthesis via the Infinite Dimensional State Space (IDEAS) Approach Featuring Minimum Area with Fixed Utility. Chem. Eng. Sci. 2003, 58 (18), 4291−4305. (39) Zhou, W.; Manousiouthakis, V. I. Non-Ideal Reactor Network Synthesis through IDEAS: Attainable Region Construction. Chem. Eng. Sci. 2006, 61 (21), 6936−6945. (40) Zhou, W.; Manousiouthakis, V. I. Variable Density Fluid Reactor Network SynthesisConstruction of the Attainable Region through the IDEAS Framework. Chem. Eng. J. 2007, 129 (1−3), 91− 203. (41) Burri, J. F.; Manousiouthakis, V. I. Global Optimization of Reactive Distillation Networks using IDEAS. Comput. Chem. Eng. 2004, 28, 2509−2521. (42) Justanieah, A.; Manousiouthakis, V. I. IDEAS Approach to the Synthesis of Globally Optimal Separation Networks: Application to Chromium Recovery from Wastewater. Adv. Environ. Res. 2003, 7 (2), 547−562. (43) Burri, J. F.; Wilson, S. D.; Manousiouthakis, V. I. InfiniteDimensional State-Space Approach to Reactor Network Synthesis: Application to Attainable Region Construction. Comput. Chem. Eng. 2002, 26 (6), 849−862. (44) Manousiouthakis, V. I.; Justanieah, A.; Taylor, L. A. The Shrink Wrap Algorithm for the Construction of the Attainable Region: An Application of the IDEAS Framework. Comput. Chem. Eng. 2004, 28 (9), 1563−1575. (45) Zhou, W.; Manousiouthakis, V. I. On Dimensionality of Attainable Region Construction for Isothermal Reactor Networks. Comput. Chem. Eng. 2008, 32 (3), 439−450. (46) Posada, A.; Manousiouthakis, V. I. Multi-Feed Attainable Region Construction Using the Shrink-Wrap Algorithm. Chem. Eng. Sci. 2008, 63 (23), 5571−5592. (47) Davis, B. J.; Taylor, L. A.; Manousiouthakis, V. I. Identification of the Attainable Region for Batch Reactor Networks. Ind. Eng. Chem. Res. 2008, 47 (10), 3388−3400. (48) Gilliland, E. R.; Reed, C. E. Degrees of Freedom in Multicomponent Absorption and Rectification Columns. Ind. Eng. Chem. 1942, 34 (5), 551−557. (49) Smith, J. M.; Van Ness, H. C.; Abbott, M. M. Introduction to Chemical Engineering Thermodynamics, 7th ed.; The McGraw-Hill Chemical Engineering Series; McGraw-Hill: New York, 2005; p 545. (50) Gmehling, J.; Onken, U.; Rarey-Nies, J. R. Vapor−Liquid Equilibrium Data Collection; Chemistry Data Series; DECHEMA: Frankfurt am Main, Germany, 1981−1988; Vol. 1, Parts 1a, 1b, 2c, and 2e. (51) Poling, B. E.; Prausnitz, J. M.; O’Connell, J. P. The Properties of Gases and Liquids, 5th ed.; McGraw-Hill: New York, 2001; Appendix A. (52) Smith, J. M.; Van Ness, H. C.; Abbott, M. M. Introduction to Chemical Engineering Thermodynamics, 7th ed.; The McGraw-Hill Chemical Engineering Series; McGraw-Hill: New York, 2005; pp 339−340. 11200


Globally Optimal Networks for Multipressure Distillation of

Recommend Documents