Systematic Synthesis of Mass Exchange Networks for Multicomponent

Article pubs.acs.org/IECR

Systematic Synthesis of Mass Exchange Networks for Multicomponent Systems Linlin Liu,† Mahmoud M. El-Halwagi,‡,§ Jian Du,†,* José María Ponce-Ortega,∥ and Pingjing Yao† †

State Key Laboratory of Fine Chemicals, Chemical Engineering Department, Dalian University of Technology, Dalian 116024, China Chemical Engineering Department, Texas A&M University, College Station 77843 Texas, United States § Adjunct Faculty at the Chemical and Materials Engineering Department, King Abdulaziz University, Jeddah 21589, Saudi Arabia ∥ Chemical Engineering Department, Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Mich 58060, Mexico ‡

ABSTRACT: The synthesis of mass exchange networks (MENs) is an important systematic tool for screening mass-separating agents (MSAs) and satisfying the mass transfer demands while considering process, environmental, and economic requirements. Most of the MEN research has focused on single-component problems. Much less attention has been given to the multicomponent problem. Therefore, the aim of this paper is to develop a systematic method to deal with the multicomponent MEN synthesis problem. The concept of interception is introduced to account for compatible mass transfer of the multiple components in a way that can be practically realized in mass exchange units. A mixed-integer nonlinear programming (MINLP) model is established to embed potential configurations of interest. Linearization, disjunctive programming, and relaxation are used to enhance the solvability of the optimization program, which is aimed at minimizing the total annualized cost (TAC) of the MEN. Two case studies from the literature, the recovery of copper from an etching plant and the simultaneous removal of H2S and CO2 from coke-oven gas (COG), are solved to illustrate the application of the proposed method.

1. INTRODUCTION The negative environmental impact of discharging pollutants from industrial facilities and the need to conserve natural resources are increasing problems facing the process industries. Various systematic methodologies have been developed to aid industry in developing sustainable designs. Mass exchange (e.g., absorption, adsorption, extraction, stripping, and ion exchange) is commonly used in the process industry to reduce pollution, recover valuable materials, and meet product specifications. Driven by the need to design networks of mass exchangers and to systematically screen the multitude of candidate massseparating agents, El-Halwagi and Manousiouthakis1 introduced the problem of synthesizing the mass exchange network (MEN), where the “MEN is a (cost-ef fective) network of mass exchangers with the purpose of preferentially transferring certain species f rom a set of rich streams to a set of lean streams.” El-Halwagi and Manousiouthakis1 also proposed a two-step approach to solve the synthesis problem. They obtained the minimum consumption of external MSAs by locating the position of a mass exchange “pinch point”, identified minimum operating cost targets, and then synthesized a network that meets this target with minimum number of units. Some subsequent investigations addressing the targeting of total annualized cost (TAC) were developed by Hallale and Fraser.2−7 In addition to the graphical pinch approach, El-Halwagi and Manousiouthakis8,9 also developed mathematical programming techniques aiming at the automatic synthesis of MENs. The regeneration of the recyclable lean streams during the MENs synthesis was considered in their study. Then, El-Halwagi and Srinivas10,11 investigated the problem of reactive MENs. Chemical MSAs and physical MSAs were incorporated in their work to remove the target components from rich streams. Some extended works that consider the © 2013 American Chemical Society

combined mass and heat exchange networks synthesis problem were addressed by Srinivas and El-Halwagi,12 Papalexandri and Pistikopoulos,13 Sebastian et al.,14 Isafiade and Fraser,15,16 Du et al.,17 and Liu et al.18 The batch MENs have been addressed by Foo et al.19 and Majozi.20 Wagialla21 presented an integrated approach for optimizing the stream matching of a MEN for a metal packing process. Recently, Chaiwattanapong et al.22 incorporated controllability analysis for synthesizing MENs, whereas Wagialla et al.23 incorporated property constraints in the synthesis of MENs. A recent systematic survey of MEN synthesis can be found in the books by Foo,24 El-Halwagi,25 Foo et al.,26 Majozi,20 El-Halwagi,27 and Smith.28 Several research efforts have contributed to the MEN synthesis problem with single components. Because of the multicomponent nature of many industrial applications, it is important to develop systematic tools for such applications. In this work, the MEN synthesis problem is investigated for multicomponent systems. One of the challenges to be addressed is the need to account for the thermodynamic and operational bases for transferring multiple solutes from a set of rich streams to a set of MSAs/lean streams. Additionally, it is necessary to account for the interrelationships among the multiple species in the actual mass exchange units. The first work on the multicomponent MEN problem was introduced by El-Halwagi and Manousiouthakis.29 They developed an optimization approach based on simultaneous transfer of multiple components over a number of composition intervals that guarantee thermodynamic and operational feasibility and Received: Revised: Accepted: Published: 14219

March 12, 2013 June 12, 2013 August 26, 2013 August 26, 2013 dx.doi.org/10.1021/ie400807m | Ind. Eng. Chem. Res. 2013, 52, 14219−14230

Industrial & Engineering Chemistry Research

Article

compatibility. Later, Alva-Argaez et al.30 identified the utility targets by incorporating the mass-transfer capacities of the transferred solutes. The superstructure method used for the heat exchange network (HEN) synthesis was extended into the MENs problem by Chen and Hung.31 A mixed integer nonlinear programming (MINLP) formulation was established to evolve the network synthesis by including the maximum number of trays among those needed by the different components in a mass exchanger. The source-interception-sink concept proposed by ElHalwagi et al.32 and Gabriel and El-Halwagi33 is an alternative approach for solving mass integration problems. Interception corresponds to the adjustment of stream (source) characteristics (e.g., concentration, temperature, properties) to meet desired targets before being fed to the process units (sinks). In this method, the allocation of streams and the application of unit operations are investigated by the tracking, assigning, and adjusting concentrations and flows throughout the process. Recently, this approach has been further developed and evolved in the fields of water allocation and property integration (e.g., Lira-Barragán et al.,34 Rubio-Castro et al.,35 Kheireddine et al.,36 Ponce-Ortega et al.,37 Nápoles-Rivera et al.,38 Rubio-Castro et al.,39 Lovelady, and El-Halwagi40). In this paper, multiple components are simultaneously considered to ensure the compatibility of MSAs from the view of components. A new formulation for the multicomponent MEN synthesis problem is proposed by introducing the concept of interceptors to decrease the complexity associated, and a piecewise linearization approach is used to replace the nonconvex terms in the optimization formulation. This proposed approach accounts for the process constraints and the environmental requirements.

Figure 1. Sketch of MEN synthesis problem.

Considering the complication of the mass transfer operation, some assumptions, listed below, are made to simplify the synthesis work. (1) The mass flow rates of all streams remain constant throughout the network. (2) The stream recycling is not allowed in the network. (3) In the composition range involved, the distribution of a concerned component between any pair of rich and MSA streams is linear and independent of other components. (4) The mass is transferred counter-currently and is only allowed from the rich streams to the MSAs. (5) The network operates at a constant pressure and no temperature effects are considered in the problem. (6) The diameter is not considered in the capital cost functions for the mass exchange units. To determine the capital costs for the units, the fixed as well as the variable charges (as a function of the height for the transfer units) are considered. Notice that the diameter for the unit depends mainly on the flow rates manipulated in the units, and also notice that the height for the transfer units depends on the flow rates manipulated. Therefore, an indirect way to consider the effect of the diameter on the capital costs for the transfer units is through the height of the units.

2. PROBLEM STATEMENT The multicomponent MEN synthesis problem can be stated as follows: given are an NR set of process rich streams, an NS set of MSAs (lean streams including process MSAs and external MSAs), and several transferable components NC to be removed from the rich streams to the MSAs. Given also are the flow rate of rich streams, Gi, the inlet and outlet concentrations for rich streams, yini,c and yout i,c , where i ∈ NR and c ∈ NC, the upper flow rate limit of MSAs, Lup j , and the inlet and outlet upper concentrations for MSAs, xini,c and xup j,c , where j ∈ NL and c ∈ NC. In addition, the linear equilibrium relationships for the distribution of the considered components between rich streams and the MSAs, y*i,c = mi,j,cxj,c + bi,j,c, where i ∈ NR, j ∈ N, and c ∈ NC, are also given. To avoid the negative driving forces, which would lead the mass transfer from the MSAs to the rich streams, the feasibility constraints of yi,c > yi,c* are needed at the two ends of each mass exchange unit. The objective of this work is to synthesize a network of mass exchangers with the simultaneous consideration of minimizing the total annualized cost and satisfying the demanded process and environmental regulations among the enormous feasible alternatives. Figure 1 is a general representation of the problem statement. The solution should provide the network elements, including the following: the matches between rich streams and MSAs, the existence of the mass transfer units and their size, the compositions of the concerned components around the devices, the network configuration, and the mass flow rates of all tributary streams.

3. THE SYNTHESIS METHOD Figure 1 is the general representation of an MEN synthesis problem, where the potential matches between rich streams and MSAs, the stream matching options, the mixing and splitting of branches, the mass exchange units, and the column sizes are considered. The corresponding mathematical model is an MINLP. The existence of nonconvex terms hinders the identification of the global solution. Therefore, piecewise linearization is used by using discrete interceptors with specific tasks as shown by Figure 2. 3.1. Network Structure. The proposed network of Figure 2 involves several stages of interceptors (mass exchangers). Mass exchange in each interceptor is based on a certain unit option which is constituted by a fixed stream match option and a fixed operating parameter option. Stream match options indicate the potential pair of rich and lean streams where mass exchange can occur. In this paper, the MSA stream in each match comes directly from the resource, with the composition of xinj,c. In each operating option, the size of the exchange device and the mass flow rate ratio of MSA stream to rich stream are provided. 14220

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used for describing the mass exchange operation around a traycolumn. The tray number, N, is employed to describe the exchange device. N+1

( ) l mg

y in − mx in − b = y out − mx in − b

l mg

−1 −1

in

Moving and distributing y and y with rf, then eq 2 can be obtained, y

out

=

rf −1 m 1 N + rf − m

×y

()

in

1

(1) out

in eq 1 and replacing l/g

rf N + 1 m rf N + 1

() + () m

−

rf m

−1

× (mx in + b)

(2)

Let α=

rf −1 m + N 1 rf − m

() () β= ()

rf N + 1 m rf N + 1

,

and

1 −

rf m

× (mx in + b)

−1

m

(3)

And using α and β to replace the corresponding items in eq 2, then the linear relationship between yin and yout can be represented as eq 4 shows. y out = αy in + β

(4)

Extend the relationship of eq 4 to the various components between the rich streams and the MSAs; then the general formulation can be expressed as follows.

Figure 2. MEN synthesis problem involving interceptors.

NR rich streams and NL MSA streams are involved in Figure 2. Assuming that NPi (∀i ∈ NR) interceptor/unit options are available for the mass exchange of each rich stream i, consider, for example, rich stream R1: there are NP1 unit options, each of which includes a match option and an operating parameter option, from P1 to PNP1. Therefore, NP1 matching options (tributaries) are generated for R1 as potential matches with the interceptor options. To linearize the problem, the stream mixing between stages is not allowed. Therefore, NPi × NPi interceptors, following directly to an R1 substream from stage 1, are placed in stage 2.The allocation of the interceptors and the connection streams between stages is shown in Figure 2. The extension of more stages applies the same regulation. At the end of the MEN, all the same-source streams should mix together. The Ri branch streams out of the last stage should mix into the tank TankRi, and the Sj tributary streams involving interceptors mix into tank TankSj. Notice that avoiding mixing and recycling of MSA may involve several mass exchange units that may increase the total cost and risk; however, this approach simplifies significantly the solution procedure. 3.2. Linear Relationship. Once the match option and the operating parameter option of a unit option are known, the corresponding linear conversion relationship between the inlet and outlet compositions of the rich stream around an interceptor can be obtained as follows: 3.2.1. For Tray-Column Interceptors (Mass Exchangers). The Kremser equation expressed as eq 1 is a function commonly

yiout = αi , j , cyiin, j , c + βi , j , c , ,j ,c

∀ i ∈ NR ,

j ∈ NS ,

c ∈ NC (5)

3.2.2. For the Packed-Column Interceptors (Mass Exchangers). The mass transfer element method is used to calculate the height, H, of a continuous-contact packed column in this study. Equation 6 is the gaseous phase based representation, in which the HTUOG, detailed by eq 7, is the overall height of the mass transfer unit and the NTUOG, detailed by eq 8, is the number of the transfer units. H = HTU OG × NTU OG HTU OG =

NTU OG =

(6)

g K yaS

(7)

⎡⎛ mg ⎞ y in − mx in − b mg ⎤ 1 ⎜ ⎟ ⎥ + mg ln⎢ 1 − l ⎠ y out − mx in − b l ⎦ 1− l ⎣⎝ (8)

Following the treatment for tray-column interceptors, eq 8 can be converted into eq 9 through setting l/g = rf. y out =

(

rf m

)

−1

rf (1 − m )NTU OG e rf m

⎛ ⎜ y in + ⎜1 − ⎜ −1 ⎝

(

rf m

)

−1


⎞ ⎟ in ⎟(mx + b) − 1 ⎟⎠

(9) 14221



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given. The operating parameters and the number of units involved are determined prior to the optimization process according to the experience, typical results reported, or the theoretical guidance from literatures. Also, simulation by software can be used to determine the performance of the mass transfer units and to select the discrete operating condition for the units. Furthermore, if a big number of options are considered, better results can be obtained; however, the optimization procedure becomes more complicated. This way, the designer has to compensate between the use of a big number of discrete operating conditions that can provide better results, with respect to the use of a small number of discrete operating conditions that can be solved easily. Then, the disjunctive representation of the proposed superstructure is presented stage by stage. 3.3.1. For the First Stage. Flow Rate Balance for Each Rich Stream. As shown in Figure 2, a list of potential unit options should be stated in the model. The existence of a unit is indicated by the (0−1) binary variable zli,p, where i ∈ NR, p ∈ NPi. Each rich stream is split into several potential matching options through the potential mass exchange units. So the overall mass flow rate balance of the rich stream is needed and can be expressed as follows.

Let γ=

(

rf m

)

−1

(1 − rfm )NTU OG

rf e m

⎛ ⎜ θ = ⎜1 − ⎜ ⎝

(

,

and

−1 rf m

)

−1


⎞ ⎟ in ⎟(mx + b) ⎟ − 1⎠

(10)

Then, the linear relationship can be expressed as follows: y out = γy in + θ

(11)

Extending eq 11 to all concerned components, then the general representation for the linear relationship between yin and yout around a packed mass exchange column is represented by eq 12. yiout = γi , j , pyiin, j , c + θi , j , c , ,j,c

∀ i ∈ NR ,

j ∈ NS ,

c ∈ NC

(12)

3.3. Mathematical Model. In the superstructure of Figure 2, two network stages with the potential splitters for rich streams between them are preset. As more stages can generate more network possibilities, the number of stages for the network should be considered specific to each case. Two mass exchanger modes, the tray-column exchanger and the packed-column exchanger, are considered in the model. Notice that the choice merely depends on the properties of the MSAs. On the basis of this, the MSA set NL can be separated into two parts: the NLtray subset and the NLpacked subset, where the MSA streams in the former set use tray-column exchangers and packed-column exchangers are required by the MSAs in latter set. One match option and one operating parameter option for ith rich stream are packaged in a unit option p, where p ∈ NPi. If p refers, in particular, to any unit option between the ith rich stream and the jth MSA stream, it can be expressed as p ∈ NPi,j. Corresponding to each operating option, a certain mass flow rate ratio of MSA over rich, rf, and a certain number of column plates to the tray-column mass exchange devise, NT, or a certain height of a transfer unit to the packed-column mass exchanger, NTUOG, are given. This way, the corresponding linear conversion coefficient α or γ, and the parameter β or θ can be obtained through eq 3 or 10. A unit option package is shown with its corresponding elements by Figure 3. It is noteworthy that, though there is no subscript j involved in the operating option column, the unity of the i−j match, specifically the solubility equilibrium relationship between them, has already been implicated by the operating option. Additionally, the inlet concentration of MSA j, Xinp,c, is also

Gi =

∑

g1i , p ,

∀ i ∈ NR (13)

p ∈ NPi

g1i , p ≤ g max z1i , p ,

∀ i ∈ NR ,

∀ p ∈ NPi

(14)

Design Equations for Mass Exchangers. The mass exchange process around a potential exchange unit can be described with a set of design equations. These design equations are applied when the corresponding unit exists (zli,p = 1); otherwise (zli,p = 0) the process variables are equal to zero. For the matches involving the MSA streams in set NLtray, the disjunctive formulations can be stated as below: ⎤ ⎡ z1i , p = 1 ⎥ ⎢ ⎥ ⎢ in in ⎥ ⎢ y1i , p , c = Yi , c , ∀ c ∈ NC ⎥ ⎢ in in ⎥ ⎢ x1i , p , c = Xi , c , ∀ c ∈ NC ⎥ ⎢ ⎥ ⎢ out in y y c NC = α + β ∀ ∈ 1 1 , i ,p,c i ,p,c i ,p,c i ,p,c ⎥ ⎢ ⎥ ⎢ in out y1i , p , c − y1i , p , c ⎥ ⎢ out in x x c NC = + ∀ ∈ 1 1 , ⎥ ⎢ i ,p,c i ,p,c rf i ,p ⎥ ⎢ ⎥ ⎢ l1i , p = rfi , p g1i , p ⎥ ⎢ ⎥ ⎢ N1i , p = NTi , p ⎥ ⎢ ⎥ ⎢ in out ⎥ ⎢ y1i , p , c ≥ y1i , p , c , ∀ c ∈ NC ⎥ ⎢ out in ⎥ ⎢ x1i , p , c ≥ x1i , p , c , ∀ c ∈ NC ⎥ ⎢ ⎢ y1in − m x1out − b > 0, ∀ c ∈ NC ⎥ i p c i p c i p c i p c , , , , , , , , ⎥ ⎢ ⎥ ⎢ out in ⎣ y1i , p , c − mi , p , c x1i , p , c − bi , p , c > 0, ∀ c ∈ NC ⎦ ⎡ ⎢ ⎢ ⎢ y1iin, p , c ⎢ ⎢ x1in ⎢ i ,p,c ∨ ⎢ y1out ⎢ i ,p,c ⎢ out ⎢ x1i , p , c ⎢ ⎢ ⎢ ⎣

Figure 3. Unit option package p for the ith rich stream. 14222

z1i , p = 0 = 0, ∀ c ∈ = 0, ∀ c ∈ = 0, ∀ c ∈ = 0, ∀ c ∈ l1i , p = 0 N1i , p = 0

⎤ ⎥ ⎥ NC ⎥ ⎥ NC ⎥ ⎥ NC ⎥⎥, ∀ i ∈ NR , ∀ p ∈ NPi ⎥ NC ⎥ ⎥ ⎥ ⎥ ⎦

(15)



Article

While for j ∈ NLpacked

max in y1iout (1 − z1i , p ), , p , c ≤ αi , p , cy1i , p , c + βi , p , c + y

∀ p ∈ NPi ,

⎤ ⎡ z1i , p = 1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ y1iin, p , c = Yiin, c , ∀ c ∈ NC ⎥ ⎢ ⎥ ⎢ x1iin, p , c = X pin, c , ∀ c ∈ NC ⎥ ⎢ ⎥ ⎢ in out y y c NC γ θ 1 1 , = + ∀ ∈ i p c i p c i p c , , , , , , i p c , , ⎥ ⎢ ⎥ ⎢ y1iin, j , p , c − y1iout ⎥ ⎢ out ,p,c + x1iin, p , c , ∀ c ∈ NC ⎥ ⎢ x1i , p , c = rf i ,p ⎥ ⎢ ⎥ ⎢ l1i , p = rfi , p g1i , p ⎥ ⎢ ⎥ ⎢ g1i , p ⎥ ⎢ OG HTU 1 = i ,p ⎥ ⎢ K a S y i j , ⎥ ⎢ ⎥ ⎢ OG OG H HTU NTU 1 1 = × ⎥ ⎢ i ,p i ,p i ,p ⎥ ⎢ y1iin, p , c ≥ y1iout c NC , ∀ ∈ ⎥ ⎢ ,p,c ⎥ ⎢ in ⎥ ⎢ x1iout , p , c ≥ x1i , p , c , ∀ c ∈ NC ⎥ ⎢ ⎢ y1in − m x1out − b > 0, ∀ c ∈ NC ⎥ i ,p,c i ,p,c i ,p,c i ,p,c ⎥ ⎢ ⎥ ⎢ out in ⎣ y1i , p , c − mi , p , c x1i , p , c − bi , p , c > 0, ∀ c ∈ NC ⎦

y1iout ,p,c

∀ c ∈ NC

z1i , p ,

∀ i ∈ NR ,

∀ p ∈ NPi ,

∀ p ∈ NPi ,

max y1iout z1i , p , ,p,c ≤ y

∀ i ∈ NR ,

x1iout ,p,c ≤

y1iin, p , c − y1iout ,p,c rfi , p

∀ i ∈ NR , x1iout ,p,c ≥

−

y1iout ,p,c

rfi , p

∀ i ∈ NR ,

∀ p ∈ NPi ,

∀ i ∈ NR ,

∀ p ∈ NPi ,

∀ p ∈ NPi ,

≥ X pin, c − x max(1 − z1i , p ),

∀ i ∈ NR ,

∀ p ∈ NPi ,

∀ i ∈ NR ,

∀ p ∈ NPi

l1i , p ≥ rfi , p g1i , p − l max(1 − z1i , p ),

∀ i ∈ NR ,

∀ p ∈ NPi

∀ i ∈ NR ,

∀ p ∈ NPi

(6) Determination of equipment conditions of mass exchangers • For the tray-column exchangers The column tray number of a tray-column exchanger N1i , p ≤ NTi , p + N max(1 − z1i , p ),

∀ p ∈ NPi ,

max

(1 − z1i , p ), ∀ i ∈ NR ,

∀ i ∈ NR ,

∀ p ∈ NPi

∀ i ∈ NR ,

∀ p ∈ NPi

∀ p ∈ NPi (23)

• For the packed-column exchangers The overall height of a transfer unit in a packed-column exchanger HTU1OG i ,p ≤

∀ i ∈ NR ,

∀ c ∈ NC

l1i , p ≤ rfi , p g1i , p + l max(1 − z1i , p ),

l1i , p ≤ l maxz1i , p ,

∀ c ∈ NC

∀ p ∈ NPi ,

z1i , p ,

∀ p ∈ NPi ,

(22)

∀ i ∈ NR ,

∀ i ∈ NR ,

≤x

∀ i ∈ NR ,

(5) Mass flow rate of lean stream

(16)

(1 − z1i , p ),

max

∀ c ∈ NC

(21)

∀ c ∈ NC x1iin, p , c

∀ c ∈ NC

+ x1iin, p , c − x max(1 − z1i , p ),

∀ p ∈ NPi ,

max x1iout z1i , p , ,p,c ≤ x

∀ c ∈ NC x1iin, p , c

∀ c ∈ NC

+ x1iin, p , c + x max(1 − z1i , p ),

∀ p ∈ NPi ,

y1iin, p , c

N1i , p ≤ N maxz1i , p ,

+x

∀ p ∈ NPi ,

(4) Outlet concentration of lean stream The outlet concentration of the lean stream is deduced from the following equation.

(2) Assignment of the inlet concentration of lean streams ≤

∀ i ∈ NR ,

(20)

N1i , p ≥ NTi , p − N

max

∀ i ∈ NR ,

∀ c ∈ NC

(17)

X pin, c

∀ c ∈ NC

∀ c ∈ NC

in max y1iout (1 − z1i , p ), , p , c ≥ γi , p , cy1i , p , c + θi , p , c − y

∀ c ∈ NC

x1iin, p , c

∀ p ∈ NPi ,

in max y1iout (1 − z1i , p ), , p , c ≤ γi , p , cy1i , p , c + θi , p , c + y

∀ c ∈ NC

∀ i ∈ NR ,

∀ i ∈ NR ,

• For the packed-column exchangers

yield the following set of equations. (1) Assignment of the inlet concentration of rich streams

y1iin, p , c ≤ y max z1i , p ,

≤y

max

+ βi , p , c − y max (1 − z1i , p ),

(19)

On the basis of the big-M reformulation, these disjunctions

y1iin, p , c ≥ Yiin, c − y max (1 − z1i , p ),

∀ c ∈ NC

αi , p , cy1ini , p , c

∀ p ∈ NPi , y1iout ,p,c

⎤ ⎡ z1i , p = 0 ⎥ ⎢ ⎥ ⎢ in ⎢ y1i , p , c = 0, ∀ c ∈ NC ⎥ ⎥ ⎢ ⎢ x1in = 0, ∀ c ∈ NC ⎥ i ,p,c ⎥ ⎢ ⎢ y1out = 0, ∀ c ∈ NC ⎥ i p c , , ⎥, ∀ i ∈ NR , ∀ p ∈ NPi ∨⎢ ⎥ ⎢ out ⎢ x1i , p , c = 0, ∀ c ∈ NC ⎥ ⎥ ⎢ l1i , p = 0 ⎥ ⎢ ⎥ ⎢ OG HTU1i , p = 0 ⎥ ⎢ ⎥ ⎢ H1i , p = 0 ⎥⎦ ⎢⎣

y1iin, p , c ≤ Yiin, c + y max (1 − z1i , p ),

≥

∀ i ∈ NR ,

∀ c ∈ NC HTU1OG i ,p ≥

(18)

g1i , p K yai , pS g1i , p K yai , pS

+ HTU max(1 − z1i , p ),

∀ i ∈ NR ,

∀ p ∈ NPi

− HTU max(1 − z1i , p ),

∀ i ∈ NR ,

∀ p ∈ NPi

max HTU1OG z1i , p , i , p ≤ HTU

(3) Outlet concentration of rich stream • For the tray-column exchangers

∀ i ∈ NR ,

∀ p ∈ NPi

(24) 14223



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The height of a packed-column exchanger max OG H1i , p ≤ HTU1OG (1 − z1i , p ), i , p NTUi , p + H

Lj =

∀ i ∈ NR ,

LjX jout ,c =

∑ ∑

(l1i , p x1iout ,p,c ) +

i ∈ NR p ∈ NPi , j

H1i , p ≤ H maxz1i , p ,

∀ i ∈ NR ,

∀ p ∈ NPi

∑

(25)

max y1iin, p , c ≥ y1iout (1 − z1i , p ), ,p,c − y

∀ i ∈ NR ,

(26) −x

max

(1 − z1i , p ),

∀ p ∈ NPi ,

∀ c ∈ NC

∀ i ∈ NR , (28)

in max y1iout (1 − z1i , p ), , p , c > mi , p , c x1i , p , c + bi , p , c − y

(32)

∀ i ∈ NR ,

∀ c ∈ NC

(33)

up X jout ,c ≤ Xj ,c ,

∀ j ∈ NL ,

∀ c ∈ NC

(34)

∀ j ∈ NL

(35)

min TAC = TY

∀ i ∈ NR ,

∑

(LjCjMSA) +

j ∈ NL

∀ p ∈ NPi ,

∀ c ∈ NC

(29)

+

3.3.2. For the Second Stage. Each rich stream flowing through stage 2 is a tributary that passes through the operation unit in stage 1, so each unit in stage 1 can be seen as a stream resource of stage 2 under this condition. The mass integration model of stage 2 can be analogized from stage 1 by increasing the model dimension to indicate the sequence relationship between stages. To avoid repeated description, the formulation for stage 2 is not presented. In the overall model, as subscript p (∀p ∈ NPi) has already been used to indicate the unit options for the ith stream in stage 1, the subscript t (∀t ∈ NPi) is used to indicate the unit options in stage 2 if related parameters (which have the number of “2” in the expressions) are required in the subsequent mathematical expressions. 3.3.3. Overall Equations. (1) Overall Outlet Composition of Rich Streams. The rich substreams with the same source assemble into a main stream after their mass exchange in the second stage. The mass balances of these streams are expressed as follows: Y iout , c Gi =

∑ ∑

(g 2i , p , t y 2iout , p , t , c),

∀ i ∈ NR ,

∀ c ∈ NC

3.3.4. Objective Function. Therefore, as the operating/ interceptor options are given, the core problem of the synthesis work becomes how to find the minimum TAC solution by determining the allocation of the rich streams as well as selecting the operating options. In this study, an MINLP mathematical model is formulated for the purpose. The objective is composed of the annualized operating cost for the use of MSAs and the annualized capital cost for the set of mass exchangers. The tray-column mass exchange unit and the packed-column mass exchange unit are both considered in the objective function. The determination of unit type adopted depends on the involved stream.

(8) Equilibrium constraints for the mass exchange process The mass exchange is constrained by the solubility equilibrium relationship. Transferable components are only allowed from the rich streams to the MSAs. The following relationships are used to ensure positive driving forces for the mass transfer.

∀ c ∈ NC

∀ j ∈ NL ,

up Y iout , c ≤ Yi , c ,

Lj ≤ Ljup ,

(27)

max y1iin, p , c > mi , p , c x1iout (1 − z1i , p ), , p , c + bi , p , c − y

l 2i , p , t x 2iout ,p,t ,c ,

(4) Overall Concentration and Mass Flow Rate Constraints.

∀ p ∈ NPi ,

∀ c ∈ NC

∑ ∑ i ∈ NR p ∈ NPi

t ∈ NPi , j

(7) Feasibility of the transferable component The following equations are set to make sure of the monotonic decrease of species concentrations in the mass exchange unit.

∀ p ∈ NPi ,

∀ j ∈ NL (31)

∀ p ∈ NPi

≥

l 2i , p , t ,

i ∈ NR p ∈ NPi t ∈ NPi , j

(3) Overall Mass Balance of MSAs.

OG max H1i , p ≥ HTU1OG (1 − z1i , p ), i , p NTUi , p − H

x1iin, p , c

∑ ∑ ∑

l1i , p +

∀ i ∈ NR ,

∀ p ∈ NPi

x1iout ,p,c

∑ ∑ i ∈ NR p ∈ NPi , j

∑ ∑

(Cifix , p z1i , p )

i ∈ NR p ∈ NPi

∑ ∑ ∑

(Cifix , t z 2i , p , t )

i ∈ NR p ∈ NPi t ∈ NPi

+

∑

∑ tray

i ∈ NR j ∈ NL

+

∑

∑ tray

i ∈ NR j ∈ NL

+

∑

∑ packed

i ∈ NR j ∈ NL

+

∑

∑ packed

i ∈ NR j ∈ NL

∑

(N1i , p Citray ,j )

p ∈ NPi , j

∑ ∑

(N 2i , p , t Citray ,j )

p ∈ NPi t ∈ NPi , j

∑

(H1i , p Cipacked ) ,j

p ∈ NPi , j

∑ ∑

(H 2i , p , t Cipacked ) ,j

p ∈ NPi t ∈ NPi , j

(36)

3.4. Relaxation of the Nonlinearities in the Model. The proposed model eliminates the most complicated nonlinear and nonconvex relationships involved in the mass exchanger networks related to the Kremser equations or equilibrium relationships. These are the most complex relationships involved in the model because these involve several logarithmic functions of nonconvex terms that involve multiplication and powers of the involved variables, which make it very difficult to find a feasible solution and even impossible to find the optimal one. The elimination of these highly nonconvex terms is one of the major contributions of the paper, and the proposed approach involves the use of a discretization procedure, where the number of variables and equations increases significantly, but all are linear, and this problem can be easily solved. However, the problem is still nonconvex because there still remain some bilinear terms (left-hand of eq 32,

∀ c ∈ NC

p ∈ NPi t ∈ NPi

(30)

(2) Overall Mass Flow Rate Balance MSAs. MSAs are used directly from their sources in each network stage. The mass flow rate of the MSAs from the same origin should be added together to obtain the gross usage. In the equation, NPj indicates the unit option set that involves MSA stream j. 14224



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for example); however, this is an easier problem and there are several approaches reported to solve the bilinear problems (see, for example, Pahm et al.;41 Rubio-Castro et al.42). The general algebraic modeling system43 (GAMS) can be used to find the optimal solution of such problem.

4. CASE STUDIES Two examples are adopted to convey the applicability of the methodology proposed in the paper. A one-component involved MEN synthesis problem is investigated with considering two types of mass exchangers in the first example. Then, a multicomponent problem that refers to two transferable species is illustrated in example 2. The capital cost data of 4552 $ per tray for tray-column exchanger, and 4245 $ per height for packed-column exchanger used by Papalexandri et al.44 are applied here for the both examples. The synthesis of MENs and the minimization of total annual cost are implemented based on the given process and operation conditions. Several interceptor options are individually provided to the two cases. The operation parameters and the number of units for the involved interceptors are determined according to the experience of previous results reported or the theoretical guidance from the literature. Example 1: One-Component Case Study. The first example, originally introduced by El-Halwagi and Manousiouthakis,9 deals with the recovery of copper from an etching plant for manufacturing printed circuit boards in the microelectronic industry. The schematic process is represented by Figure 4, in

for R1−S1:

y1 = 0.734x1 + 0.001

for R 2−S1:

y2 = 0.734x1 + 0.001

for R1−S2 :

y1 = 0.111x 2 + 0.008

for R 2−S2:

y2 = 0.148x 2 + 0.013

The stream data of rich streams and MSAs are listed in Tables 1 and 2. The supply and target concentrations, the mass Table 1. Rich Streams of Copper Recovery Problems*

*

stream no.

description

Gi/kg·s−1

Ysj

Ytj

R1 R2

amm solution rinswater

0.25 0.1

0.13 0.06

0.1 0.02

stream data reported by El-Halwagi and Manousiouthakis9

Table 2. Lean Streams of Copper Recovery Problemsa stream no.

description

Ljup/kg·s−1

Xsj

Xup j

costb/$·kg−1

S1 S2

LIX63 P1

∞ ∞

0.03 0.001

0.07 0.02

0.002 0.024

a b

Stream data reported by El-Halwagi and Manousiouthakis.9 Reduced unit cost of MSAs used by Papalexandri et al.43

flow rates, and the unit annual costs are provided. Following the recent study by Chen and Hung,31 the reduced operating cost of MSAs and operating hours (8150 h·year−1) used by Papalexandri et al.44 are adopted in this paper. In addition, the corrected Kya values, 0.685 kg copper m−3·s−1 for R1 and 0.221 kg copper m−3·s−1 for R2, deduced from the original data of the 2 m diameter column (El-Halwagi and Manousiouthakis9) to the 1 m diameter column by Hallale and Fraser7 are also applied. The proposed MINLP model is used to synthesize the MEN for this copper recovery system. Two network stages are set to construct the potential network configuration. The synthesis procedures can be broken into the following steps. Step 1. Determine the Match Options According to the Given Streams. Four streams, R1, R2, S1, and S2, are taken into account in Example 1; thus, four match possibilities, R1−S1, R1−S2, R2−S1, and R2-S2, are generated as the potential stream match options during the synthesis. Step 2. Form the Unit Options by Providing Operating Options. In Step 1, two match options are formed for each rich stream. In this step, several mass exchange parameter options are provided to each match option in terms of the groups composed by the mass flow rate ratio options of MSA over rich streams and the column tray number options for MSA S1 or the column unit height options for MSA S2. Besides the concentration conversion factors, α and β (or γ and θ) for copper in each pair of match, obtained based on eqs 3 and 12, and other parameters listed in Figure 3 are also given. Each pair of match and operating option constitutes a unit option. In this example, seven unit options are available for each rich stream, respectively. The options in consideration are listed in Table 3. Among them, the options R1−P1 and R2−P1 are used to show the condition of no mass exchange operation involved. Step 3. Build the Network Structure Based on the Unit Options Formed in Step 2. On the basis of the configuration represented in Figure 1 and the unit options formed in Step 2, the potential network superstructure of this example can be described as follows: each rich stream is split into seven 1st-matching

Figure 4. Recovery of copper from the liquid effluents of an etching plant (El-Halwagi and Manousiouthakis9).

which the etchant of ammoniacal solution etches the circuit boards into etched boards by dissolving copper during the operation of the etching line. As the etching efficiency of the ammoniacal solution is the highest during the copper concentration range of 10% (w/w) to 13% (w/w), the regeneration and the recycle of the etching solution is necessary for the sake of a high productivity and cost savings. Water is used in the rinse bath operation to clean the surface of the etched boards. The copper in the effluent rinsewater should be removed under the consideration of environmental and economic reasons. Therefore, the two streams that require moving copper away are obtained from the etching process. To remove the copper in the rich streams of the ammoniacal solution (R1) and rinsewater (R2), two MSA streams are recommended: LIX63 (aliphatic α-hydroxyoxime, S1 in Figure 4) and P1 (aromatic β-hydroxyoxime, S2 in Figure 4). Tray-column mass exchangers are applied to S1 while packed-column exchangers are applied to S2. Among the involved range of concentration, the distribution of copper between rich streams and MSAs is dominated by the following linear equilibrium relations: 14225



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Table 3. Unit Options for Example 1 operating option rich stream

unit option

R1

P1 P2 P3 P4 P5 P6 P7 P1 P2 P3 P4 P5 P6 P7

R2

match option S1 S1 S1 S2 S2 S2 S1 S1 S1 S2 S2 S2

rf

N or NTUOG

α or γ

β or γ

m

Xin

1 0.3 1.5 1.2 0.5 0.7 1.2 1 1.5 2.1 2.5 0.3 0.6 1.3

0 1 2 3 1 2 2 0 2 3 2 0.6 2 2

1 0.709864 0.138506 0.103333 0.397931 0.161121 0.150035 1 0.138506 0.028196 0.062473 0.587824 0.176631 0.153564

0 0.006679 0.019832 0.020641 0.004883 0.006804 0.006894 0 0.019832 0.022371 0.021582 0.005419 0.010826 0.011129

0 0.734 0.734 0.734 0.111 0.111 0.111 0 0.734 0.734 0.734 0.148 0.148 0.148

0 0.03 0.03 0.03 0.001 0.001 0.001 0 0.03 0.03 0.03 0.001 0.001 0.001

Kya

0.685 0.685 0.685

0.221 0.221 0.221

Figure 5. Cost-optimal MEN for Example 1.

which 37849 $·year−1 is the cost for the employment of mass separation agents/lean steams and 19744 $·year−1 is for the capital cost of mass exchange units. It should be noted that the configuration for the optimal solution obtained with the proposed approach is the same as the one previously obtained by Chen and Hung31 using a global optimization approach; here, the main advantage of the discretized proposed approach is that this is simpler and easer to implement, as well as that the CPU time required is significantly smaller. Furthermore, between these two solutions, the main differences are in the operating conditions for the mass transfer units; this is because in the proposed approach these are restricted to the discrete conditions proposed before the optimization process. Example 2: Multicomponent Case Study. The MEN synthesis problem about the sweetening of coke-oven gas (COG) was first introduced by El-Halwagi and Manousiouthakis.1 This process involves the removal of two transferable species, H2S and CO2, from the rich streams, R1 (COG), R2 (Claus unit tail gases) to the MSA streams, S1, S2. The data of rich streams and MSAs are listed in Tables 4 and 5, respectively. In these tables, the

options, and each 1st-tributary links with an unit option in network stage 1; then each first-tributary breaks into another seven 2st-matching options, and each 2st-tributary links a unit option in network stage 2. The matching options sourcing from the same rich/MSA resource mix together at the terminal of the network. Step 4. Synthesize and Optimize the MEN with the Proposed MINLP Model. During the synthesis and optimization process, the employment of unit options, the allocation of rich streams, and the determination of the cost-optimal MEN among the given options are simultaneously considered. The synthesis and optimization work are carried out with the proposed MINLP model. The optimal MEN of Examples 1 and 2 are shown in Figures 5 and 6, and the mass flow rate requirements of S1 and S2 are 0.285 kg·s−1 and 0.03 kg·s−1, respectively. Four unit options, R1−P2 and R2−P3 in the first stage, R1−P1 and R2−P5 in the second stage, are selected in the network. Notice that, as option R1−P1 corresponds to no-mass exchange operation, there is no mass exchange unit assigned to R1 in stage 2. This MEN features an annual total cost of 57593 $·year−1, out of 14226



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Figure 6. Cost-optimal MEN for Example 2.

Table 4. Data for Rich Streamsa Gi

ySH2S

ytH2S

ysCO2

ytCO2

kg·s−1

kg H2S·kg−1

kg H2S·kg−1

kg CO2·kg−1

kg CO2·kg−1

0.9 0.1

0.07 0.051

0.0003 0.0001

0.06 0.115

0.005 0.005

R1 R2 a

The concentrations of H2S and CO2 are calculated simultaneously to each unit option. The best solution package is obtained based on the given options. The binary variables for R1−P4, R2−P3 in stage 1, R1−P5 and R2−P6 in stage 2 are identified as 1, while the rest are 0. Then, the network structure is built as Figure 4 shows, where the stream mass flow rate, the concentration distribution of H2S and CO2, the trays of mass exchangers, and the allocation of operation units are all displayed. It is noteworthy that the concentrations of H2S and CO2 obtained with our method are all real values; no compromises problem is considered during the synthesis work. This has demonstrated the suitability of the proposed method to the MEN synthesis problem of multicomponents systems. The total annual cost corresponding to Figure 4 is obtained at 433730 $·year−1, where 342690 $·year−1 is the annualized operation cost and 91040 $·year−1 is the capital cost. Compared with Chen and Hung,31 this cost result is relatively high because only several unit options are taken into account. But at the point of concentrations for H2S and CO2, the solutions in this paper are much more real values than those in the literature. The concentrations in this paper are obtained on the basis of the real refection of mass transfer in exchangers, eqs 5 and 12, so the removal degree difference, which comes mainly from the solubility distinction and the removal requirement distinction between H2S and CO2, can emerge obviously. For example, notice in Figure 4 that the concentrations of CO2 in rich streams reduce drastically. However, the same situation does not appear in the solutions of Chen and Hung,31 because they only paid

El-Halwagi and Manousiouthakis.1

stream mass flow rates, the supply concentrations, the target concentrations, and the unit costs are given. Due to a lack of capital cost data in El-Halwagi and Manousiouthakis,1 this paper considers the case with the reduced operating cost data and the capital cost data given by Papalexandri et al.44 Assuming the distribution constraints of transferable components between rich streams and MSAs is only related to their concentrations in MSAs, then during their ranges of concentration involved, the solubility data of H2S and CO2 in S1 and S2 can be expressed by the following equations, respectively. yH S = 1.45x HS12S

yH S = 0.26x HS22S

2

yCO = 2

2

S1 0.35xCO 2

S2 yCO = 0.58xCO 2 2

The synthesis problem of Example 2 is solved following the same synthesis procedures stated in Example 1. The potential match options are set as R1−S1, R1−S2, R2−S1, and R2−S2. Then, the unit options with the linear conversion factors for both H2S and CO2 are formed by setting several operating options for different stream match options, as listed in Tables 6 and 7. Table 5. Data for MSA Streamsa

S1 S2 a

Lj,max

xSH2S

xtH2S

xsCO2

xtCO2

annual costb

kg·s−1

kg H2S·kg−1

kg H2S·kg−1

kg CO2·kg−1

kg CO2·kg−1

$·kg−1

2.3 ∞

0.0006 0.0002

0.031 0.0035

0 0

0.171 0.103

0.004 0.006

El-Halwagi and Manousiouthakis.1 bDeduced unit cost from Papalexandri et al.44 14227



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Table 6. Unit Options for Example 2 (H2S Section) operating option rich stream

unit option

R1

P1 P2 P3 P4 P5 P6 P7 P1 P2 P3 P4 P5 P6 P7

R2


rf

N

αH2S or γH2S

βH2S or γH2S

mH2S

XinH2S

1 0.5 1.8 2.1 0.4 1.2 1.5 1 1.5 3.4 4.0 0.5 1 1.5

0 7 8 11 3 6 5 0 2 3 5 3 3 5

1 0.655303 0.040225 0.005327 0.117005 8.10 × 10−05 0.000129 1 0.322099 0.046008 0.004000 0.072816 0.013066 0.000129

0 0.000300 0.000835 0.000865 4.59 × 10−05 5.2 × 10−05 5.2 × 10−05 0 0.000590 0.000830 0.000867 4.82 × 10−05 5.13 × 10−05 5.2 × 10−05

0 1.45 1.45 1.45 0.26 0.26 0.26 0 1.45 1.45 1.45 0.26 0.26 0.26

0 0.0006 0.0006 0.0006 0.0002 0.0002 0.0002 0 0.0006 0.0006 0.0006 0.0002 0.0002 0.0002

Table 7. Unit Options for Example 2 (CO2 Section) operating option rich stream

unit option

R1

P1 P2 P3 P4 P5 P6 P7 P1 P2 P3 P4 P5 P6 P7

R2


rf

N

αCO2 or γCO2

βCO2 or γCO2

mCO2

XinCO2

1 0.5 1.8 2.1 0.4 1.2 1.5 1 1.5 3.4 4.0 0.5 1 1.5

0 7 8 11 3 6 5 0 2 3 5 3 3 5

1 0.026218 1.646 × 10−06 2.297 × 10−09 0.401075 0.006628 0.005319 1 0.042278 0.000979 4.680 × 10−06 0.308082 0.092404 0.005319

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0.35 0.35 0.35 0.58 0.58 0.58 0 0.35 0.35 0.35 0.58 0.58 0.58

0 0 0 0 0 0 0 0 0 0 0 0 0 0

undeniable that these two points are worthy to be referred either from practical or economical standpoints. So, these correlational studies need to be considered in the future research.

attention to the maximum-tray-number corresponded component in a mass exchanger, and the real concentrations of the rest components and their affect to later exchangers were not considered. Therefore, it cannot ensure the application to an incompatible system. NEOS (network-enabled optimization system) server (Version 5.0) is a free Internet-based service for solving optimization problems. The two case studies were solved using the GAMS port implemented in the NEOS server; then the CPU times of the CBC solver for cases 1 and 2 were 9.3 and 27.2 s, respectively.

■

AUTHOR INFORMATION

Corresponding Author

*Address: Room D-305, Chemical Dalian University of Technology, Ganjingzi District, Dalian, Liaoning 116024. Tel: +86 411 84986301. E-mail: [email protected].

5. CONCLUSIONS An optimization framework has been proposed for the synthesis of multicomponent MENs. An interceptor-based superstructure was developed to embed potential configurations of interest while ensuring compatibility of the multicomponent mass transfer in each mass exchanger. A disjunctive programming approach coupled with a linearization method and relaxation operation was developed to enhance the solvability of the developed program. Two examples from the literature have been solved to illustrate the applicability of the proposed method. To reduce the synthesis complexity, the mutual affections among components on their solubility equilibriums and the regeneration of MSAs were not included in this paper. It is

Engineering Department, No. 2 Linggong Road, Province, P. R. of China, Fax: +86-411-84986201.

Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors gratefully acknowledge the financial support from the Natural Science Foundation of China (No. 20976022) and the China Scholarship Council. M.E.-H. is thankful to funding from King Abdulaziz University.

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NOMENCLATURE b phase equilibrium constant Cfix fixed cost of each mass exchange unit, $ year−1 i,p MSA C unit cost of MSA stream, $·kg−1 14228


Industrial & Engineering Chemistry Research Cpacked Ctray G g max

g g1 g2

H Hmax HTUmax HTUOG HTU1OG i,p H1 H2 Kya L Lup l lmax l1 l2 m N Nmax NTUOG N1 N2 rf S TY TAC Xin Xout Xup xin xmax xout x1in x1out x2out

Article

per height annual cost of packed column, $·year−1 per plate annual cost of tray column, $·year−1 total mass flow rate of rich stream, kg·s−1 mass flow rate of rich stream to a mass exchange unit, kg·s−1 upper limit for rich streamflow rate, kg·s−1 mass flow rate of rich stream to an unit option in the first stage, kg·s−1 mass flow rate of rich stream to an unit option in the second stage, kg·s−1 overall height of a packed-column exchanger, m upper limit for overall height of a packed-column unit option, m upper limit for gas-phase-based height of a mass transfer unit option gas-phase-based height of a mass transfer unit, m gas-phase-based height of a packed-column unit option in the first stage, m overall height of a packed-column unit option in the first stage, m overall height of a packed-column unit option in the second stage, m total volumetric mass transfer coefficient based on gas phase, kg·m−3·s−1 total mass flow rate of MSA stream, kg·s−1 maximum mass flow rate of MSA stream, kg·s−1 mass flow rate of MSA stream to a mass exchange unit, kg·s−1 upper limit for MSA streamflow rate, kg·s−1 mass flow rate of MSA stream to an unit option in the first stage, kg·s−1 mass flow rate of MSA stream to an unit option in the second stage, kg·s−1 solubility equilibrium coefficient number of the plates used in a tray-column exchanger upper limit for tray number number of the transfer units in a packed-column exchanger plate number for a tray-column unit option in the first stage plate number for a tray-column unit option in the second stage mass flow rate ratio of MSA over rich cross-sectional area of a packed-column exchanger, m2 operating time per year, s·year−1 total annual cost of the MEN, $·year−1 overall inlet concentration of MSA stream, kg·kg−1 overall outlet concentration of MSA stream, kg·kg−1 upper limit to the concentration of MSA stream, kg·kg−1 inlet concentration of MSA stream to a unit option, kg·kg−1 upper limit for MSA stream concentration, kg·kg−1 outlet concentration of MSA stream to a unit option, kg·kg−1 inlet concentration of MSA stream to an unit option in the first stage, kg·kg−1 outlet concentration of MSA stream to an unit option in the first stage, kg·kg−1 outlet concentration of MSA stream to an unit option in the second stage, kg·kg−1

overall inlet concentration of rich stream, kg·kg−1 overall outlet concentration of rich stream, kg·kg−1 upper limit to the concentration of rich stream, kg·kg−1 inlet concentration of rich stream to a mass exchange unit, kg·kg−1 upper limit for rich stream concentration, kg·kg−1 outlet concentration of rich stream to a mass exchange unit, kg·kg−1 inlet concentration of rich stream to an unit option in the first stage, kg·kg−1 outlet concentration of rich stream to an unit option in the first stage, kg·kg−1 outlet concentration of rich stream to an unit option in the second stage, kg·kg−1 binary variable that indicates the existence of unit option in the first stage binary variable that indicates the existence of unit option in the second stage

Yin Yout Yup yin ymax yout y1in y1out y2out z1 z2

Greek Letters

α β θ γ

conversion conversion conversion conversion

coefficient of tray-column constant of tray-column constant of packed-column coefficient of packed-column

Subscript

c i j p t

component, c = 1, 2, ... rich stream, i = 1, 2, ... MSA stream, j = 1, 2, ... unit option, p = 1, 2, ... unit option, t = 1, 2, ...

Sets

NC NL NPi NR

{c/c is a component, c = 1, 2, ...} {j/j is a MSA stream, j = 1, 2, ...} {p or t/p or t is an unit option, p or t = 1, 2, ...} {i/i is a rich stream, i = 1, 2, ...}

Superscript

in MSA out OG packed tray up *

inlet MSA stream outlet gaseous phase packed-column exchanger tray-column exchanger upper limit equilibrium concentration

Acronyms

HEN GAMS MEN MINLP MSA TAC

■

heat exchange network general algebraic modeling system mass exchange network mixed-integer nonlinear programming mass-separating agent total annualized cost

REFERENCES

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Systematic Synthesis of Mass Exchange Networks for Multicomponent

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