Ind. Eng. Chem. Res. 2003, 42, 1731-1738
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Optimal Design of Membrane/Distillation Column Hybrid Processes Ioannis K. Kookos† Department of Chemical Engineering, UMIST, P.O. Box 88, Manchester, M60 1QD, United Kingdom
In this work a mathematical programming methodology is proposed for the efficient optimization of hybrid separation systems consisting of distillation columns and membrane separation units. A superstructure representation of the alternative solutions is first presented. This superstructure is obtained by modifying an existing superstructure for distillation columns and combining with a recently proposed superstructure for gas permeation networks. The final morphological representation is then used for the structural and parametric optimization of a hybrid system for the propylene/propane separation (C3 splitter). The results are noticeably different from the results of previous case studies on the same separation and show that the economic potential for using hybrid systems can be significant. 1. Introduction Distillation is among the most commonly used separation processes in the chemical industry despite the fact that it is an energy and capital intensive process. Distillation is not particularly attractive for “difficult” separations involving mixtures of components with close boiling points or mixtures forming azeotropes.1,2 However, distillation has been used extensively even in these cases and significant academic and industrial research effort has been directed toward the development of efficient distillation-based systems for difficult separations. Hybrid processes, where distillation columns are combined with membrane units, have been proposed as a viable alternative that can reduce significantly the installation and/or operating cost of separation. A hybrid process has been defined as “a process package consisting of generally different unit operations, which are interlinked and optimized to achieve a predefined task”.3 Lipnizki et al.,3 have reviewed most of the applications of hybrid processes involving membranes and distillation column processes. The use of membranes as a means of overcoming thermodynamic limitations (such as the existence of azeotropes) is by far the most common application of hybrid systems involving membranes. The first pervaporation-distillation hybrid process was presented in the late 1950s by Binning and James4 where the dehydration of 2-propanol-ethanol mixtures was considered. The production of ethanol by separating ethanol-water mixtures is among the most characteristic examples of hybrid processes on the commercial scale5 where an impressive reduction in operating cost has been reported (a simplified layout is given in Figure 1). Other applications are summarized in Lipnizki et al.,3 and Baker.6 Lelkes et al.,7 and Szitkai et al.,8 have presented studies where rigorous optimization techniques have been applied to the systematic synthesis of ethanol dehydration systems (see Figure 1). The problem of simultaneous structural and parametric optimization of a distillation column and a membranebased unit is simplified in this case by exploiting the fact that the top composition (for a minimum boiling azeotrope) is fixed by thermodynamic limitations (azeo† Phone: +44 (0) 161 200 4346. Fax: +44 (0) 161 200 4399. E-mail:
[email protected].
Figure 1. Hybrid process for ethanol production-simplified process layout.3,5
tropic composition). At the same time the composition of the feed stream to the membrane unit is also fixed for the same reasons. In addition, from the structural optimization point of view, there is little incentive for combining the two units in a more complex morphological manner than the one shown in Figure 1. Moganti et al.,9 Stephan et al.,10 and Pettersen et al.,11 have presented a methodology for designing hybrid separation systems using membranes and distillation columns for difficult separations such as the olefin/ paraffin separations. However, their methodology is only applicable to the case of ideal binary separations. In addition, their aim is to minimize the number of trays in the distillation column, which may not be a meaningful objective for the cases where the operating and annualized capital costs are dominated by the utilities cost. The aim of this paper is to present a formal mathematical methodology for the structural and parametric optimization of hybrid systems consisting of membranes and distillation columns. For demonstration purposes, the case of gas permeation will be considered. The superstructure representation of distillation columns proposed by Viswanathan and Grossmann12,13 will be used as a starting point for developing a more general representation pertinent to the hybrid systems considered in this work. This superstructure is combined with a recently proposed superstructure for synthesizing cost optimal gas permeation networks. The separation of propane and propylene (a mixture with an average
10.1021/ie020616s CCC: $25.00 © 2003 American Chemical Society Published on Web 03/06/2003
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relative volatility that is close to 1) will be investigated in detail and results of the application of the proposed methodology will be presented and discussed. 2. Membrane Separations Membrane-based processes are today finding widespread use in the petrochemical, pharmaceutical, and food industries, in biotechnology, and in a variety of environmental applications.6,14 This is due to the advantages of membrane processes over their conventional counterparts such as energy savings, capital investment reduction, flexibility, modularity, and portability. A membrane is a permeable or semipermeable phase, often in the form of a thin film, made from a variety of materials ranging from inorganic solids to polymeric materials. The aim of the membrane is to regulate the exchange of material between two adjacent fluid phases. To achieve this, a membrane acts as a barrier which separates different species either by sieving or by controlling their relative rates of transport. As a result, the upstream fluid (retentate) is depleted from some of its original components whereas the downstream fluid (permeate) is concentrated in these components. Transport across the membrane is a result of the existence of chemical, pressure, temperature, and electrical potential differences. The ability of a membrane to separate a given mixture is characterized in terms of its permeability and selectivity. The permeability is defined as the flux normalized with respect to the membrane thickness and the driving force. The selectivity is typically defined as the ratio of the individual permeabilities and characterizes the ability of a given membrane to separate two given components. Membranes are categorized according to6,14 (a) whether the permselective layer is dense or porous, (b) the type of membrane material (polymeric, metal, inorganic, etc.), and (c) whether they have a symmetric (homogeneous) or asymmetric structure. Porous membranes are made of polymers, ceramics, and microporous carbon whereas dense inorganic and polymeric membranes are commonly used for molecular-scale separations involving gas and vapor mixtures. Homogeneous membranes are used when the membrane material has the necessary mechanical stability to be self-supporting. In any other case the membrane layer is deposited on a porous support, which can be made from a different material that gives the necessary mechanical stability to the membrane (without increasing the resistance to mass transport). The choice of porous vs a dense film and the type of material used depends strongly on the type of mixture to be separated, the operating conditions, and the driving force used for the separation. The current status in membrane gas separation systems has been discussed in a recent review paper by Baker.15 H2 recovery was the first major application of membrane gas-separation technology, followed by the CO2/CH4 separation and the production of N2 from air.6,15-17 Figure 2 presents a building block for creating superstructure representations for membrane gas separations with countercurrent flow. This structure has recently been proposed18 based on the most commonly used structures in gas permeation networks. A combination of a number of such blocks can generate complex structural representations for achieving high-purity gas separations at acceptable cost. This superstructure is restricted to gas separation networks. Alternative mor-
Figure 2. Gas separation network superstructure.18
phological representations such as the ones proposed by Qi and Henson19 can also be used. 3. Superstructure for Hybrid Systems As discussed by Pettersen et al.,11 when a distillation column is combined with a membrane separation unit to form a hybrid separation system, a number of potential structural alternatives can be generated. The most important alternatives are summarized in Figure 3. In the structure shown in Figure 3a the membrane unit and the distillation unit work in parallel by sharing the separation load whereas the final product purity is achieved by the distillation column. In the structures shown in (b) and (c) of Figure 3 the membrane unit is used in series with the distillation column and achieves the required product purity. To be able to optimize the parameters and the structure of the hybrid system, a representation has to be devised that incorporates all potential alternatives. The main difficulty in achieving this is due to the fact that the existing superstructures for distillation column synthesis, such as the ones proposed by Viswanathan and Grossmann,12,13 cannot be used to generate alternatives such as the one shown in Figure 3a. A more general representation is needed where potential interconnections between each potential tray in the distillation column and the membrane separation network can be postulated. The representation proposed in this work is shown in Figure 4. It should be noted at this point that the membrane unit shown in Figure 4 corresponds to the gas-separation network presented in Figure 2 (details not shown explicitly for clarity of presentation). In this representation, the number of trays and the feed tray location can be optimized based on the same general principles described by Viswanathan and Grossmann,12,13 In addition, a number of streams are introduced for representing all potential interconnections between the distillation column and the membrane separation network. Three additional potential streams per potential tray are defined. The first stream emanates from the potential tray j and ends at the feed of the membrane network (see also Figure 5). This stream, denoted as MFj (where j denotes the jth tray), serves as a potential feed for the membrane network. The second potential stream, denoted by MPj, emanates from the membrane permeate stream and is fed back to the potential tray j. The last potential stream, denoted by MRj, emanates from the membrane retentate and is also fed back to the potential tray j. In other words, in the superstructure shown in
Ind. Eng. Chem. Res., Vol. 42, No. 8, 2003 1733
Figure 3. Alternative configurations of membrane/distillation column hybrid systems.11
methodology to the cases where the streams fed to the membrane unit are liquid streams (pervaporation) involves obvious modifications. In this case a direct feed stream from the main feed to the membrane unit can also be considered without any further complication of the methodology. Finally, it should be observed that since the vapor streams are all saturated streams condensation is expected to occur in the high-pressure side of the membrane unit. Either using a heat exchanger to superheat the vapor streams or allowing a drop in the pressure between the distillation column and the membrane separation unit can avoid condensation. In this work, it is assumed that pressure and temperature are constant and as a result potential condensation is not taken into consideration.
Figure 4. Generalized tray model used in the proposed superstructure.
4. Mathematical Model of the Distillation Column Following the approach of Viswanathan and Grossmann,12 we define the set J ) {1, 2, ..., NT} to denote the trays. This set consists of the following subsets: reboiler JR ) {1}, stripping section JSTR ) {2, 3, ..., NF-1}, feed tray JF ) {NF}, rectifying section JREC ) {NF+1, ..., NT-1}, and condenser JC ) {NT}. Let also I ) {1, 2, ..., NC} denote all components present in the feed. The model of the distillation column is as follows:
Figure 5. Generalized tray model used in the proposed superstructure.
Figure 4 the membrane separation network can be fed by using an appropriate sidestream from any potential tray in the distillation column. In addition, the product streams from the membrane network can be directed to the top product of the superstructure (denoted as point D in Figure 4), or they can be directed to the bottom product of the superstructure (denoted as point B in Figure 4) or they are fed back to the distillation as an additional feed stream to any of the potential trays. All streams that might enter or leave a potential tray are summarized in Figure 5. RLj is the reflux that returns at tray j, RVj is the boilup that returns at tray j, and Fj is the main feed at tray j. It is important to note that in the superstructure representation of the hybrid process it is assumed that all streams taken from or returning to the distillation column are vapor streams. This restricts consideration to membranes performing gas separation or vapor permeation. In these cases the structures shown in (b) and (c) of Figure 4 need not to be included in the superstructure since they would involve condensation and then vaporization of the same streams (before feeding to the membrane unit). Extending the proposed
(a) phase equilibrium f Lij ) f Vij ,
i ∈ I, j ∈ J
(1)
(b) mole fraction summations xij ) 1, ∑ i∈I
j∈J
(2a)
yij ) 1, ∑ i∈I
j∈J
(2b)
(c) component material balances Vj-1yij-1 ) (
∑
RLk)xij + Dxij,
i ∈ I, j ∈ JC (3a)
k∈JREC
RLjxiNT + Lj+1xij+1 + Vj-1yij-1 + MPjyPi + MRjyRi ) Ljxij + Vjyij + MFjyij, i ∈ I, j ∈ JREC (3b) Fizi + Lj+1xij+1 + Vj-1yij-1 + MPjyPi + MRjyRi ) Ljxij + Vjyij + MFjyij, i ∈ I, j ∈ JF (3c) RVjyi1 + Lj+1xij+1 + Vj-1yij-1 + MPjyPi + MRjyRi ) Ljxij + Vjyij + MFjyij, i ∈ I, j ∈ JSTR (3d) Lj+1xij+1 ) (
∑
k∈JSTR
RVk)yij + Bxij,
i ∈ I, j ∈ JR (3e)
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Ind. Eng. Chem. Res., Vol. 42, No. 8, 2003
(d) energy balance Vj-1Hj-1 ) Qcond + (
∑
RLk)hj + Dhj,
j ∈ JC
k∈JREC
(4a)
RLjhNT + Li+1hj+1 + Vj-1Hj-1 + MPjHP + MRjHR ) Lihj + VjHj + MFjHj, j ∈ JREC (4b) FihF + Lj+1hj+1 + Vj-1Hj-1 + MPjHP + MRjHR ) Ljhij + VjHj + MFjHj, j ∈ JF (4c) RVjH1 + Lj+1hj+1 + Vj-1Hj-1 + MPjHP + MRjHR + MRjHR ) Ljhj + VjHj + MFjHj, j ∈ JSTR (4d) Lj+1hj+1 + Qreb ) (
∑ RVk)Hj + Bhj, k∈JSTR
j ∈ JR (4e)
where fi is the fugacity coefficient of component i in the phase denoted as a superscript (L, liquid; V, vapor), xi (yi) is the mole fraction of component i in the liquid (vapor) phase, Lj (Vj) is the liquid (vapor) molar flow rate from tray j, D (B) is the top (bottom) product molar flow rate, hj (Hj) is the molar specific enthalpy of the liquid (vapor) on tray j, and Qreb (Qcond) is the heat input at the reboiler (condenser). We define the binary variables δj, j ∈ JSTR ∪ JF ∪ JREC, to denote the existence (δj ) 1) or not (δj ) 0) of the corresponding tray in the final flow sheet (the feed tray is assumed to exist at any feasible alternative and as a result δj ) 1, j ∈ NF). Then, any solution must satisfy the following:
Figure 6. Simplified representation of a gas permeation module in countercurrent flow.
5. Mathematical Model of the Membrane Separator
δj - δj-1 e 0,
j ∈ JREC
(5a)
To describe the behavior of hollow fiber modules with countercurrent flow, we consider the simplified representation shown in Figure 6. We divide the membrane unit into NP (KNP ) {1, 2, ..., NP}) elements and we make the assumptions of uniform properties at each element, ideal gas behavior, steady-state isothermal operation, constant permeabilities independent of composition, negligible axial diffusion and pressure drop, negligible concentration polarization effect, and negligible deformation of the fibers under pressure. On the basis of these assumptions, we obtain the following model of a single-stage hollow fiber membrane module,20,21
δj - δj+1 e 0,
j ∈ JSTR
(5b)
(a) component material balance (tube side)
In addition, if RLU and RVU are estimated upper bounds on the reflux flow and vapor boilup, then
t f tik ) f ik+1 - Jik∆Ak,
i ∈ I, k ∈ KNP - {1,NP} (11a)
RLj - RLU(δj - δj+1) e 0,
j ∈ JREC
(6a)
f tik ) MFyFi - Jik∆Ak,
RVj - RVU(δj - δj-1) e 0,
j ∈ JSTR
(6b)
t - Jik∆Ak, MRyRi ) f ik+1
that is, all reflux and boilup enter exactly on one tray. Feed to the membrane module can be obtained from any tray and the permeate and retentate can be fed to any tray as well:
MFj - MFUδj e 0,
j ∈ JREC ∪ JF ∪ JSTR
(7a)
MPj - MP δj e 0,
j ∈ JREC ∪ JF ∪ JSTR
(7b)
MRj - MRUδj e 0,
j ∈ JREC ∪ JF ∪ JSTR
(7c)
U
The model is complete if we add the physical property relations and pressure drop constraints:
hj ) h(Tj, Pj, x1j, ‚‚‚, xNCj)
(8a)
Hj ) H(Tj, Pj, y1j, ‚‚‚, yNCj)
(8b)
f
L ij
f
V ij
L
(9a)
V
(9b)
) f (Tj, Pj, x1j, ‚‚‚, xNCj) ) f (Tj, Pj, y1j, ‚‚‚, yNCj) Pj ) Pj+1 + ∆Pj
(10)
i ∈ I, k ) NP (11b) i ∈ I, k ) 1
yRi - 1 ) 0 ∑ i∈I
(11c) (11d)
(b) component material balance (shell side, i ∈ I, k ∈ KNP) s + Jik∆Ak, f sik ) f ik-1
i ∈ I, k ∈ KNP - {1,NP} (12a)
s MPyPi ) f ik-1 + Jik∆Ak,
f sik ) Jik∆Ak,
i ∈ I, k ) NP i ∈ I, k ) 1
yPi - 1 ) 0 ∑ i∈I
(12b) (12c) (12d)
where MF is the total molar flow rate fed to the membrane network and yFi is the mole fraction of component i in the feed of the membrane network, fik is the molar flow rate of component i coming out from element k, Jik is the rate of transport of pure component i per unit area of the membrane at element k, and ∆Ak
Ind. Eng. Chem. Res., Vol. 42, No. 8, 2003 1735
is the active area of the membrane at element k calculated by
∆Ak ) Ntπ dlmdzk
(13)
where Nt is the number of membrane tubes, do is the outer diameter, di is the inner diameter, and dlm ) (do - di)/ln(do/di) is the logarithmic mean diameter. The rate of transport of pure component per unit area of the membrane can be calculated as follows,
f (x,δ) ) rCI + CU
Qi Jik ) (ytikPt - ysikPs) lm
(14)
where Qi is the permeability of component i, lm is the thickness of the membrane, Pt (Ps) is the pressure in the tube (shell) side, and ytik (ysik) is the mole fractions of component i in the tube (shell) side. The mole fractions are calculated as follows:
ytik
f tnk ) f tik ∑ n∈I
(15)
ysik
f snk ) f sik ∑ n∈I
(16)
The model is completed by considering the component material balances at the feed of the membrane network:
∑
MF )
MFj
(17a)
j∈JREC∪JF∪JSTR
MFyFi )
∑
MFjyij,
i∈I
(17b)
j∈JREC∪JF∪JSTR
The generalization to the case of multistage membrane networks is straightforward and is omitted for brevity. In addition, due to the assumptions used (negligible pressure drop and negligible condensation), the cost of the membrane network is always going to be an optimistic estimation of the actual cost. This is further discussed in a previous publication by the same author.18 6. Optimization Problem The mathematical model of the hybrid systems consisting of eqs 1-17 together with the bound constraints on the mole fractions and the nonnegativity constraints of the flow rates, temperatures, and so forth, can be written as follows:
h(x) ) 0 g(x) e Aδ
(18)
x ∈ X, δj ∈ {0,1} The aim is to optimize the parameters and the structure of the overall system so as to minimize an index of the economic performance f (x,δ):
min f (x,δ) x,δ
h(x) ) 0 g(x) e Aδ x ∈ X, δj ∈ {0,1}
nonlinear programming (MINLP) problem22,23 where the binary variables appear in the constraints in a linear and separable form. Algorithms for solving this class of problems have been proposed in the open literature and they have been summarized by Floudas23 and Grossmann.24 For preliminary cost estimation and especially when the aim is to compare alternative designs, a simplified cost index (annualized cost) can be used,25
(19)
This mathematical formulation is a mixed, integer
(20)
where CI is the installed cost, CU is the utility cost, and r is the depreciation rate. In the case of distillation columns the cost of equipment involves the cost of trays, cost of tower (bare module), cost of heat exchangers (reboiler and condenser), cost of reflux drum, and the cost of pumps. In the case of membranes the equipment cost involves the cost of membrane (bare) module, the cost of membrane, and the cost of compressor.17 The utilities cost involve the cost of cooling water (or refrigerant), cost of steam, cost of electricity, and the membrane replacement cost.17 Appendix A summarizes the cost correlations used to estimate the annualized cost of the proposed superstructure (taken from Douglas26). 7. Case Study: Propylene/Propane Separation The separation of propylene/propane is considered in this case study, which is based on the data given by Gokhale et al.,27 on an industrial C3 splitter; 0.3 kg‚mol/s of saturated liquid mixture, 70 mol % in propylene and 30 mol % in propane, is to be separated into a top product, 99 mol % in propylene, and a bottom product, 99 mol % in propane. The aim of this case study is to compare the annualized cost of an optimum distillation column with the annualized cost of a hybrid separation system so as to investigate the economic potential of using hybrid systems for difficult separations. The propylene and propane are assumed to form ideal liquid and vapor mixtures. Constant molar overflow is also assumed to be valid for this system. The vaporliquid equilibrium is modeled using the McWilliams28 correlations. The maximum number of trays is 250 trays and the minimum top temperature is 30 °C. The total molar flow rate to the membrane network is restricted to be
