Analysis of Local Recycle for Membrane Pervaporation Systems

Article pubs.acs.org/IECR

Analysis of Local Recycle for Membrane Pervaporation Systems Anton Santoso, Cheng-Ching Yu, and Jeffrey D. Ward* Department of Chemical Engineering, National Taiwan University, Taipei 10617, Taiwan S Supporting Information *

ABSTRACT: We present an analysis of local recycle in membrane pervaporation processes. Local recycle (reheating and recycling part of the retentate from a membrane unit back to the inlet) can mitigate the problem of temperature drop in pervaporation units. The excess fluid acts as a thermal carrier, increasing the temperature. However, a trade-off occurs because this recycle also decreases the concentration of the more permeable species on the retentate side of the membrane. We present a method based on dimensional analysis that can be used to quickly determine whether local recycle around a single membrane unit is desirable. We show that membrane modules can be classified into one of three types: local recycle is not desirable, local recycle is desirable with an intermediate recycle ratio, and local recycle is desirable with the maximum possible recycle ratio. The method is illustrated using three case studies, two of which are based on hybrid distillation/pervaporation processes. The results indicate that the correct application of internal recycle can significantly improve efficiency and reduce cost. back to the inlet of the same module.9,12 The recycled fluid acts as a thermal carrier reducing the temperature drop through the module. However there is an inherent trade off associated with this design alternative: recycling effluent from the membrane unit back to the membrane inlet tends to reduce the concentration of the more permeable species on the permeate side, which adversely affects performance. Although several researchers have reported process designs that incorporate local recycle in pervaporation processes, we are not aware of any that analyze the trade-off or offer guidelines on when local recycle will improve process performance. That is the purpose of this contribution. In the remainder of this article, we develop a mathematical model of a pervaporation process and express it in terms of dimensionless groups. Using the model, we show that each membrane module within a process can be classified into one of three groups according to the values of certain dimensionless groups. In some cases, the maximum amount of recycle is preferred, in some cases an intermediate amount of recycle is preferred, and in some cases no recycle is preferred. Finally, the method is illustrated using three case studies based on processes described in the literature.

1. INTRODUCTION Membrane pervaporation has attracted increased attention recently as an alternative technology for separations, particularly as an alternative to extractive distillation for breaking azeotropes.1,2 The basic principle is that liquid at elevated pressure flows through a tube composed of a selectively permeable membrane. The outside of the membrane (the permeate side) is maintained at low pressure so species on the permeate side are in the gas phase. Thus, as species permeate through the membrane, they evaporate, hence the term pervaporation. Because the permeate side is in the gas phase, the concentration of species is low and this promotes diffusion across the membrane. The biggest drawbacks of membrane pervaporation are that suitable membranes are not available for all liquid mixtures and the cost of the membrane material is typically high. Therefore it is important that the process be designed so that the most efficient use is made of the membrane units. Typically, for azeotropic distillation applications, conventional distillation will be used to bring the composition of a process stream near to the azeotropic composition, and then pervaporation will be used to cross the azeotrope. A process in which distillation and membrane pervaporation are employed together is called a hybrid process. Hybrid processes have been designed using both conceptual methods3−6 and mathematical programming methods.7−11 An inherent feature of pervaporation units is that the temperature will decrease down the length of the unit because pervaporation (like evaporation) is endothermic. Since the diffusivity of species will decrease with decreasing temperature, the efficiency of the membrane unit will also decrease. Therefore, when multiple membrane modules are arranged in series, a heat exchanger or “inter-heater” is often placed between them to increase the temperature of the fluid to the maximum temperature that can be tolerated by the membrane material. Even with the use of these interheaters, if the membrane module is large enough the temperature drop within each module may be enough to adversely affect performance. A seldom-discussed design alternative is to reheat and recycle part of the effluent from the retentate side of a membrane module © 2012 American Chemical Society

2. PROCESS DESCRIPTION AND MODELING The modeling of membrane pervaporation processes can be divided into two parts: a microscopic model for the local diffusion rate must be combined with a macroscopic model of the membrane pervaporation unit. Models for the diffusion rate of varying complexity have been published in the literature.13−17 In a complex model, the driving force for mass transfer is based on the activity of species including interactions between the diffusing species and the membrane.18,19 The diffusion rate will also depend on temperature; the most common assumption is that the temperature dependence is Arrhenius-type. Received: Revised: Accepted: Published: 9790

December 27, 2011 June 8, 2012 June 18, 2012 June 18, 2012 dx.doi.org/10.1021/ie202967j | Ind. Eng. Chem. Res. 2012, 51, 9790−9802

Industrial & Engineering Chemistry Research

Article

Figure 1. The perfect plug flow model of a pervaporation membrane unit.

In this work, we use a simpler engineering model20 in which the diffusion rate of species i (Ni) depends on a concentration driving force of component i in retentate (CR,i) and permeate (CP,i): Ni = ADi(C R, i − C P, i)

Table 1. System Parameters for Separation Ethanol−Water and IPA−Water System

(1) 2

where A is the mass transfer area in m unit. The diffusion coefficient (Di in m/h) of each species is assumed to be temperature-dependent according to the Arrhenius equation: Di = Do , i e(−Ea,i /(RT ))

(2)

To model the macroscopic pervaporation unit, the perfect plug flow model is employed, as shown in Figure 1. The total molar flow rate on the retentate side is given by dFR = −s(DA (C R,A − C P,A ) + DB(C R,B − C P,B)) dx

dx

= −sDA (C R,A − C P,A )

(4)

The enthalpy change in the system is caused by the amount of energy removed from the system when permeate vaporizes: FR

ethanol/water

IPA/water

300 16.65 57.4 375 0.85 145.058 37237 2.202 × 106 1.158 × 109 9385 9385

300 13.57 161 375 0.738 203.9219 37646 4.673 × 105 1.158 × 109 9385 9385

on the retentate side acts as a thermal carrier. The flow rate on the permeate side also increases, for two reasons. Since the retentate side composition decreases (the concentration of the more permeable species increases) the driving force for mass transfer also increases. Also, since the average temperature in the membrane module increases, the diffusion constants have a higher average value. The center column shows the effect of changing the inlet temperature. As the inlet temperature increases, the product (retentate) purity increases and the permeate flow rate also increases. This indicates that the membrane unit is performing better. For this reason, it is usually preferable to increase the inlet temperature to the maximum value that can safely be tolerated by the membrane. Finally, the right column shows the effect of changing the inlet concentration of less permeable species. As the inlet concentration increases, the outlet concentration naturally increases as well. The permeate flow rate decreases since there is less of the species that diffuses more easily. When the feed composition approaches 1, the rate of diffusion across the membrane becomes very small, and consequently the temperature drop is very small as well. The results from the analysis of a membrane unit without recycle suggest that there may be a benefit from using local recycle, at least in some instances, and hint at the trade-off that is likely to be encountered. If part of the effluent from the membrane unit is reheated and recycled back to the inlet, the extra fluid will act as a thermal carrier, increasing the average temperature in the membrane unit. This should improve performance since diffusion rate depends on temperature. However, at the same time, recycling part of the effluent from the membrane unit will decrease the average concentration of the more permeable species on the retentate side, which should decrease performance.

(3)

where s is the length of the perimeter of the membrane. The second subscript index A refers to retained species and B refers to permeable species. The change in flow rate of the selectively retained species on the retentate side is given by d(FR z R,A )

property A (m2) Cavg (kmol/m3) Fin (kmol/h) Tin (K) zin Cp (kJ/kmol/K) λ (kJ/kmol) Do,A (m/h) Do,B (m/h) Ea,A/R (1/K) Ea,B/R (1/K)

d(hR ) = −s(λA DA (C R,A − C P,A ) + λBDB(C R,B − C P,B)) dx (5)

To investigate the properties of the process and the model, a sensitivity analysis was conducted on a single membrane module without recycle. System parameters were taken from a case study by Luyben20 based on the separation of an ethanol−water mixture as shown in Table 1. The effect of changing three key inlet variables (the inlet flow rate Fin, the inlet temperature Tin, and the concentration of the less permeable species in the inlet zin) on four key outlet variables (outlet retentate concentration of less permeable species zout, outlet permeate concentration of the less permeable species zp, outlet temperature Tout, and permeate flow rate P) was studied. Unless specified otherwise, all concentrations z refer to mole fraction of the less permeable species (the alcohol). The results are shown in Figure 2. The left-most column shows the effect of changing the inlet flow rate. As the inlet flow rate increases, the retentate composition decreases, reflecting the fact that as the residence time in the membrane module decreases, the product purity that the unit is able to achieve also decreases. The outlet temperature also increases with increasing flow rate, because the excess fluid 9791

dx.doi.org/10.1021/ie202967j | Ind. Eng. Chem. Res. 2012, 51, 9790−9802


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Figure 2. Sensitivity analysis for three inlet variables, Fin, Tin, and zin.

3. LOCAL RECYCLE Having studied the behavior of a single membrane module without recycle, we now consider the possibility of using local recycle to improve process performance. The basic idea is shown in Figure 3 which also shows the definition of recycle ratio RR which is used in this work.

lim P =

RR →∞

z in + RR·z R 1 + RR

(7)

zM =

2

Infinite recycle is not possible in practice, and therefore we assume that there is some maximum possible practical value of the recycle ratio. In this work, the maximum value of RR is set to be 10 for case studies where cost analysis is not performed and 5 when cost analysis is performed. In principle, if there are several membrane units in series it is possible to design a process with recycle around two or more membrane units, as shown on the right side of Figure 4. However, it is always the case that local recycle around a single membrane unit performs better than recycle around two or more units. The left side of Figure 4 shows how the composition of the less permeable species at the outlet of the last module changes as the recycle ratio changes for different configurations. The greatest benefit occurs when the recycle loop includes only a single membrane module. If the recycle loop includes three or four membrane modules, then the use of recycle actually reduces performance. The reason is that the penalty for remixing is greater as the number of membrane units in the recycle loop increases, but the benefit from increasing the average temperature is the same. This is a general result: the greatest benefit from using local recycle always occurs when the recycle loop is around a single membrane module. Next we investigate whether local recycle is equally effective around all membrane units. Figure 5 shows the effect of using recycle around each of the four membrane units. As shown on the left side of Figure 5, using recycle around the first membrane unit significantly improves the performance of the overall process, but using recycle around the last membrane unit significantly reduces the performance of the overall process. This significant difference motivates us to investigate the behavior of these units more closely. Detailed information about the change of process variables as the recycle ratio increases is shown in Figure 6 for recycle around membrane units 1 (left column) and 4 (right column).

The flow rate and concentration of membrane feed can be found by material balance at mixing point: (6)

(cavgADB + Fin)2 − 4cavgADBFin(1 − z F)

(8)

Figure 3. Schematic of local recycle around a single module.

FM = (1 + RR)Fin

cavgADB + Fin −

where zR can be found by numerical integration of eqs 3, 4, and 5 with the initial values FM, zM, and Tin. However FM and zM also depend on zR, therefore iteration is required to find the value of zR that satisfies eq 3, 4, 5, 6, and 7. In the general case of finite RR it is not possible to solve the model analytically because the equations are too complex. However, some analytical results are possible in the limit as the recycle ratio approaches infinity. In this case, the composition change of the retentate down the length of the membrane module is negligible, and the permeate flow rate P is given by 9792



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Figure 4. Effect of local recycle around different numbers of membrane units.

Figure 5. Effect of local recycle around different membrane units.

first unit than in the last unit, which in turn means that the temperature drop (in the absence of recycle) is much greater in unit 1 than in unit 4. This means that in unit 1, even though the inlet concentration is increased (which tends to reduce performance) this effect is more than compensated for by the increase in temperature in the membrane unit. By contrast, in unit 4, because the pervaporation rate is much less (because the feed is already quite pure), the temperature drop in the absence of recycle is small (less than 3 degrees) and therefore the benefit from increasing the average temperature in the unit does not compensate for the penalty associated with the remixing. A similar phenomenon can also be observed if one considers only a single membrane module (as in Figure 3) but different values of the diffusion activation energy, as shown in Figure 7. Keeping other system parameters constant, if the activation energy is small, then recycle always impairs performance.

This corresponds to cases 1 and 4, respectively, in Figure 5. For both membrane units, increasing the recycle ratio increases the flow rate at the mixing point (FM) and the composition at the mixing point (zM). The outlet temperature also increases with increasing recycle ratio and approaches the inlet temperature (375 K) in both cases. However, the trend in performance of the membrane unit with recycle ratio is markedly different for the two units. In unit 1, the retentate flow rate to the next unit (R) decreases, while the permeate flow rate (P) and outlet retentate concentration (zM) increase, indicating that the performance of the membrane unit improves with increasing recycle ratio, while those trends are reversed for unit 4, indicating that performance decreases with increasing recycle ratio. The reason for this dramatic difference is that the feed composition to unit 1 is much less than the feed composition to unit 4. As a consequence, much more pervaporation occurs in the 9793



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Figure 6. Sensitivity of process variables to recycle around unit 1 and unit 4.

Conversely, if the activation energy is large, then recycle always benefits performance. If the activation energy has an intermediate value, then there is an optimal finite and nonzero value of the recycle ratio. This is not surprising, since the activation energy is directly related to the penalty associated with the temperature drop in the membrane module. If the activation energy is small, then the diffusion rate is not strongly affected by the temperature drop, and the penalty associated with the temperature drop is reduced. On the basis of this observation, we classify membrane units into one of three categories based on the usefulness of local recycle. If local recycle is always undesirable, we classify the membrane unit as type I. If local recycle is desirable and the optimal recycle ratio is finite (or less than the maximum achievable recycle ratio) then we classify the membrane unit as type IIA. If local recycle is desirable and the optimal recycle ratio is infinite (or greater than the maximum achievable recycle ratio) then we classify the membrane unit as type IIB. Figure 7. Effect of recycle for systems with different activation energy.

4. SIMPLIFIED DIMENSIONLESS MODEL

as density, heat capacity, and latent heat of vaporization are constant (independent of temperature). The first assumption can be justified because membranes used for pervaporation are usually sufficiently selective so that the rate of mass transfer of the more permeable component is much larger than that of the less permeable component. The expression for the temperature dependence of the diffusion constant is rewritten as

To further investigate the effect of process variables on the classification of membrane modules, we construct a simplified dimensionless model of a membrane pervaporation process and explore the role of dimensionless groups that appear in the expressions. To simplify the expressions, the following assumptions are employed: (1) Only the permeate component can cross the membrane, so zA = 0 and PA = 0. (2) Physical properties such 9794



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⎛ Ea,B ⎛ 1 1 ⎞⎞ DB = Din,B exp⎜⎜ − ⎜ − ⎟⎟⎟ Tin ⎠⎠ ⎝ R ⎝T

DZ′ =

Fin

(9)

so that the diffusion constant at the inlet temperature will remain the same if the activation energy Ea,B changes. All of the dimensional variables in the pervaporation model equation system are replaced by dimensionless variables, which are shown in Table 2, and the other variables are grouped into dimensionless groups.

E B′ =

λ′ =

Table 2. Dimensionless Variables Used in Dimensional Analysis variable name

reference variable

FR

Fin

Fx =

FR Fin

zR

zin

Z=

zR − z in 1 − z in

T

Tin

θ=

x

L

ζ=

dimensionless variable

and 1 − Z =

1 − zR 1 − z in

T Tin x L

⎛ dFx ⎛1 ⎞⎞ = −DZ′ exp⎜ −E B′ ⎜ − 1⎟⎟(1 − z in)(1 − Z) ⎝θ ⎠⎠ ⎝ dζ

(13)

Ea,B RTin

(14)

λ c pTin

(15)

The physical meanings of the dimensionless groups are as follows: D′z is the dimensionless initial rate of mass transfer without the concentration driving force. It consists of the membrane area, the average molar density of the system, and the initial diffusivity divided by the inlet flow rate. E′B is the dimensionless activation energy. λ′ is the dimensionless enthalpy of vaporization. The dimensionless variable, zin, is the inlet mole fraction of the less permeable species. For the ethanol−water system with feed condition as in Table 1, the values of the dimensionless groups are: D′z = 1.33, E′B = 25.03, and λ′ = 0.68. On the basis of the observation from Figure 7 that the effect of recycle on process performance changes with changing activation energy, we define two dimensionless transition activation energies E′R and E′T. E′R is the dimensioless activation energy above which some amount of recycle is preferable to no recycle and ET′ is the dimensionless activation energy above which the maximum possible amount of recycle is preferred. Using these concepts, we investigate how the classification of membrane modules changes as the values of dimensionless groups change. Figure 8 shows how the two transition activation energies change as the inlet concentration changes for different values of D′z for λ′ = 0.68. The left panel shows the change in the first transition activation energy (ER′ ). The dashed line shows the dimensionless activation energy for the ethanol/water system (E′B = 25). If the inlet concentration is less than about 0.95, then the activation energy of the ethanol/water system is greater than the first transition activation energy, and at least some amount of

The following simplified dimensionless equations can be derived by substituting dimensionless variables:

Fx

ACavgDin,B

(10)

⎛ z in ⎞ ⎛ ⎛1 ⎞⎞ dZ =⎜ + Z ⎟DZ′ exp⎜ − E B′ ⎜ − 1⎟⎟(1 − z in)(1 − Z) ⎝ ⎠⎠ ⎝ θ dζ ⎝ (1 − z in) ⎠ (11)

⎛ −DZ′ λ′ ⎛1 ⎞⎞ dθ = exp⎜ −E B′ ⎜ − 1⎟⎟(1 − z in)(1 − Z)) ⎝ ⎠⎠ ⎝ θ dζ Fx (12)

The equations include three dimensionless groups and one additional dimensionless variable. The dimensionless groups are

Figure 8. The values of ER′ and ET′ for different Dz′ and Zin at λ′ = 0.68 for the ethanol/water process. 9795



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Figure 9. An illustration of the relationship between the transition activation energies and the transition compositions.

recycle is preferred. The value of D′z does not have a strong effect on the value of the first transition activation energy. The right panel shows how the second transition energy changes as the inlet concentration changes. In this case the effect of changing D z′ is more pronounced, and increasing Dz′ significantly increases the transition activation energy. If D′z = 4 (the highest value), then the maximum possible recycle flow rate is desired only if the inlet concentration is less than about 0.82. By contrast, if D′z = 0.25 (the smallest value) then the maximum possible amount of recycle is preferred if the inlet concentration is less than about 0.93. These results indicate that the inlet concentration is also a key design variable affecting the classification of membrane modules. Consider the situation where multiple membrane units are connected in series, as shown on the right side of Figure 5. In this case, most properties are similar for all four modules. The inlet temperature is likely to be the same, the inlet flow rate is likely to be similar, and the physical properties of species would also be the same. The variable that is most likely to change significantly from module to module is the inlet concentration. In particular, the concentration of the retained species will increase from module to module. Thus we might expect from Figure 5 that recycle would be attractive for modules where the inlet concentration is lower (i.e., those early in the train) and unattractive for modules where the inlet concentration higher (i.e., those later in the train). This is consistent with the results shown on the left side of Figure 5 and discussed earlier. Therefore, we also define two transition compostions zRin and T zin. zRin is the inlet composition below which some amount of recycle is preferable to no recycle, and zTin is the composition below which it is optimal to use the maximum possible amount of recycle. The relationship between ER′ and ET′ and zRin and zTin can be seen in Figure 9. The left panel shows the optimal recycle ratio versus dimensionless activation energy when λ′, Dz′ and zin are constant. When the activation energy is less than ER′ , then the optimal recycle ratio is zero. After ER′ , the optimal recycle ratio increases until it reaches the maximum possible value when E′ = E′T. When E′ ≥ E′T, the optimal recycle ratio is the maximum possible recycle ratio. The center panel shows how ER′ and ET′ change with changing zin. zRin and zTin are the values of zin for which ER′ and ET′ , respectively, cross the horizontal line corresponding to the dimensionless activation energy of the species in the process. Finally, the right panel shows how the optimal recycle ratio changes with changing zin. If zin < zTin, then the optimal value of

the recycle ratio is the maximum possible value of the recycle ratio. As zin increases beyond zTin, the optimal value of the recycle ratio decreases until it equals zero when zin = zRin. For values of the composition higher than zRin, the optimal value of the recycle ratio is zero. Table 3 shows a summary of the classification of membrane modules. Table 3. Classification of Membrane Modules type I use recycle? optimum recycle ratio

no RRopt = 0

values of dimensionless process variables (both criteria are equivalent by definition)

0 < EB′ < ER′ or Zin > ZRin

type IIA

type IIB

yes 0 < RRopt < RRmax ER′ < Eb′ < ET′ or ZTin < Zin < ZRin

yes RR = RRmax Eb′ > ET′ or Zin < ZTin

5. A SYSTEMATIC METHOD FOR INCORPORATING LOCAL RECYCLE INTO MEMBRANE PERVAPORATION PROCESSES The results developed so far naturally suggest a systematic method for incorporating local recycle into the of design processes with membrane pervaporation. As stated previously, pervaporation processes are usually designed with multiple units in series with heat exchangers in between. Consider a process with four membrane units in series as shown in Figure 11. The objective is to determine which modules should be augmented with internal recycle. The values of zRin and zTin for the system must be determined first. To do this, the value of RRopt (the value of RR that maximizes zR) must be determined as a function of zin. For a given value of zin, the value of RRopt can be determined by plotting zR as a function of RR. (Recall that for given values of zin and RR, zR can be determined by solving eqs 3−7 iteratively.) When the value of RRopt as a function of zin is known, then zRin can be identified as the smallest value of zin for which RRopt is zero, and zTin can be identified as the smallest value of zin for which RRopt is less than RRmax. As seen before, internal recycle is most likely to be beneficial if the inlet concentration is low. Therefore, the best candidate for internal recycle is the first unit and thus the first unit should be classified first. If the first module is Type I, then local recycle is not desirable for this module. Since the inlet concentration of all further modules will be greater than the first module, all further modules must also be type I and internal recycle is not attractive for this process. 9796



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If the first module is type IIA, the optimal value of RR can be determined using an optimization algorithm. If the module type is IIB, the recycle ratio is set as maximum. On the basis of preferred RR, the outlet concentration from the first module, which is also the inlet concentration of the second module, is recalculated and the classification procedure is applied to the second module. This procedure is repeated for each module until the last module is reached or a module of type I (no recycle is desired) is encountered. A flowchart in Figure 10 represents the procedure.

from Luyben.20 The last case study involves the separation of isopropyl alcohol and water. Feed conditions are taken from Arifin and Chien.21 It was assumed that the diffusion activation energy of isopropyl alcohol was the same as that of ethanol, and the diffusion coefficient of isopropyl alcohol was estimated to be 0.21 times that of ethanol at 375K.22 6.1. Case Study I: Ethanol−Water Pervaporation Process. The first example considers the design of a train of four membrane units. The objective is to increase the purity of a feed stream with a flow rate of 57.4 kmol/h containing 85% methanol as much as possible by incorporating local recycle where it will be helpful. Figure 11 shows a process flow diagram

Figure 11. Conceptual diagram of a train of membrane modules in series.

Table 4. Inlet and Outlet Concentrations of Each Module before and after Applying Internal Recycle (Case Study 1, Figure 11). Note that zout,1 = zin,2, etc. without recycle recycle at 1st module recycle at 1st, 2nd modules

zin,1

zin,2

zin,3

zin,4

zR

0.85 0.85 0.85

0.9146 0.9330 0.9330

0.9616 0.9727 0.9736

0.9871 0.9917 0.9920

0.9966 0.9979 0.9980

for the process. The transition concentrations for this system are zTin = 0.88 and zRin = 0.952. Table 4 shows the concentration in each membrane inlet before and after recycle is applied to the first and second units. Before recycle is considered, the inlet to the first membrane unit has a concentration less than zTin. Therefore the maximum amount of recycle around this unit is desired. (For this case study RRmax = 10 is assumed.) The inlet concentration of each membrane unit is then recalculated for the case when the first membrane unit has the maximum allowed amount of recycle. The inlet concentrations increase as shown in the second row of Table 4. Next, the second unit is considered. With maximal recycle around the first membrane unit, the inlet concentration to the second unit is 0.9330, which is in between zTin and zRin. Therefore, some intermediate value of the recycle ratio greater than zero but less than the maximum possible amount is desirable. Therefore eqs 3−7 are solved iteratively to determine the optimal value of the recycle ratio, which is determined to be RR = 0.91. Therefore, the second module is equipped with recycle at a recycle ratio of 0.91. The inlet concentrations of all of the modules are then recalculated and the results are shown in the third row of Table 4. The inlet concentration of the third module (zin,3 = 0.9736) is greater than zRin. Therefore, recycle is not desired in the third module or any subsequent modules. Therefore, the procedure ends at this point. The final design incorporating recycle is shown in Figure 12. 6.2. Case Study 2: Ethanol−Water Hybrid Process. The previous case study illustrated how using local recycle can improve process performance when the number and size of the membrane modules is fixed. In this case study, the application of the method at the design stage is illustrated. The product purity is

Figure 10. Flowchart for the application of internal recycle to a train of membrane modules in series.

The basic process design without internal recycle must be provided first before this method can be applied. Since the procedure can be fully automated, it would be possible to include the optimization of local recycle around membrane modules within a larger algorithm that determined optimal values of all plantwide design variables, with the caveat that the complexity of the algorithm would naturally increase.

6. CASE STUDIES To illustrate implementations of local recycle, three following case studies are conducted. Physical properties for the case studies are given in Table 1. The first two case studies involve an ethanol−water system where process parameter values are taken 9797



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value of the permeate concentration is typically small (because most membranes are quite selective) and variations in xP do not have a significant effect on the material balances. The optimization is conducted by adjusting the key concentration design variables sequentially and using the equations in Table 1 in the Supporting Information to determine the flow rates in the process. Cost models are taken from Douglas23 and other references.8,24,25 It is assumed that the membrane area is the same in each module and that the maximum area in one module is 371.6 m2 (4000 ft2).26 The results for Design A and Design B are shown in Figures 14 and 15, respectively. The cost of the best Design A is significantly lower than the best Design B ($1,042,000/year vs $2,151,000/ year) because for the ethanol/water system the relative volatility of the species is very small beyond the azeotropic composition, making the second distillation column in Design B impractical. Having determined that Design A is more economical and having determined approximate values of the optimal flow rates and compositions throughout the process, local recycle can now be considered to reduce cost. The optimal flowsheet without local recycle is shown in Figure 14. The process requires six membrane modules, each of which have an area of 296 m2. Table 5 shows the inlet composition to each module, as well as the membrane area in each module and the number of modules, before and after the addition of local recycle. The same procedure used in case study 1 can then be applied to determine where local recycle should be applied, except that after each module is considered, the total number of modules required and the membrane area in each module must be recalculated. The results are shown in Table 5. After incorporation of local recycle in the first membrane unit, the required number of modules decreases from six to five,

Figure 12. The optimal design of the first case study with internal recycle.

fixed at 0.999, and the objective is to determine the process design (including the number and size of membrane modules) that achieves the specification at the lowest cost. In this case study and in the following case study, we consider the design of an entire hybrid process including both distillation and pervaporation. The two most commonly encountered hybrid process designs are shown in Figure 13. In Design A (shown in the left part of the figure) a distillation column enriches the feed to near to the azeotropic composition, and a train of membranes further enriches the product stream, crossing the azeotrope in the process. In Design B (shown in the right part of the figure) a train of membranes is used to cross the azeotrope, and distillation is employed on both sides of the azeotrope. In the first step, the best design is chosen among the two configurations. In both configurations, key variables for optimizing the design are the intermediate concentrations between units. When these are specified, flow rates of different process streams can be determined from a series of material balance equations listed in Table 1 in the Supporting Information. These variables also determine the separation duty for each unit. All of these can be set independently in the process design except the permeate concentration xP since this will depend on in the selectivity of the membrane. However, the

Figure 13. Conceptual sketch of Design A and Design B.

Figure 14. Optimal Design A for the ethanol/water system without local recycles. 9798



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Figure 15. Optimal Design B for the ethanol/water system without local recycles.

Table 5. Inlet Concentration of Each Module before and after Applying Internal Recycle (Case Study 2, Figure 14). zR = 0.999 in All Cases without recycle recycle at 1st module recycle at 1st, 2nd modules

zin,1

zin,2

zin,2

zin,4

zin,5

zin,6

A

n

Dz′

0.8000 0.8000 0.8000

0.8686 0.9072 0.9049

0.9271 0.9579 0.9603

0.9679 0.9859 0.9864

0.9889 0.9962 0.9962

0.9967

296 332 320

6 5 5

1.2672 1.4214 1.3700

Figure 16. Optimal Design A for the ethanol/water system with local recycles.

Figure 17. Optimal Design A for the IPA/water system without local recycles. 9799



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Figure 18. Optimal Design B for the IPA/water system without local recycles.

Table 6. The Inlet Concentration of Each Module before and after Applying Internal Recycles (Case Study 3) without R adding recycle at 1st module adding recycle at 2nd module adding recycle at 3rd module adding recycle at 4th module

zin,1

zin,2

zin,3

zin,4

A

n

Dz′

0.7375 0.7375 0.7375 0.7375 0.7375

0.7900 0.8037 0.7997 0.7977 0.7971

0.8395 0.8481 0.8524 0.8492 0.8476

0.8835 0.8874 0.8892 0.8904 0.8884

348.7 304.4 282.2 270.8 264.2

4 4 4 4 4

0.46 0.40 0.37 0.36 0.35

Figure 19. Optimal Design B for the IPA/water system with local recycles.

alcohol and water. These species have an azeotrope at zIPA = 0.6883 (at 1.1 bar). Physical properties of this system are summarized in Table1. The process feed conditions are assumed to be the same as in Arifin and Chien:21 Fin = 100 kmol/h, zin = 0.5. As for the ethanol/water system, the optimal values of key concentration variables for Design A and Design B are determined first, and then these two alternative designs are compared based on their total annual cost. The results are shown in Figures 17 and 18. The cost of Design B, which is $1,004,050/year, is slightly lower than Design A, which is $1,084,320/year. Because the azeotropic composition is lower than that of the ethanol/water system, a large number of membrane modules (nine units) are required to achieve the desired product purity in Design A. Therefore design

which causes an increase in the membrane area per unit to 332 m2. Performance is further improved by the incorporation of local recycle around the second membrane unit. However, local recycle around the third and fourth units does not improve performance. The complete hybrid design with local recycle is shown in Figure 16. The improved design has five membrane modules, the first two of which have local recycle. The first module is type IIB and second module is type IIA. The cost of the improved design is $992,000/year, for a total cost reduction of 4.8% and a membrane cost reduction of about 9.7%. Further details about the cost breakdown are given in Table 2 in the Supporting Information. 6.3. Case Study 3: IPA−Water Hybrid Process. The third case study based on the separation of a mixture of isopropyl 9800



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D′z = dimensionless initial rate of mass transfer without the concentration driving force, Ea = activation energy of diffusion, kJ/kmol EB′ = dimensionless activation energy, E′R = dimensionless first transition activation energy, E′T = dimensionless second transition activation energy, F = flow rate, kmol/h FR = retentate flow rate, kmol/h h = specific enthalpy, kJ/kmol Ni = diffusion rate of species i, kmol/h R = ideal gas constant, kJ/K/kmol RR = recycle ratio, T = temperature, K s = specific area of membrane per unit length, m2/m x = a length in x direction, m zR,i = mole fraction of component i in retentate, zRin = first transition composition, zTin = second transition composition, Z = dimensionless variable for composition, -

B is selected and used as the base case for the implementation of local recycle. The values of the dimensionless variables for this system are Dz′ = 0.46, EB′ = 25.03, and λ′ = 0.49.The transition compositions have values of zTin = 0.91 and zRin = 0.93. Local recycle is considered using the same procedure as in the previous case study. The inlet concentration of each module for design B before and after the addition of recycle is shown in Table 6. In this case, all of the modules are found to be type IIB, meaning that the maximum amount of recycle should be employed around each of the membrane units. However it is still important that the modules be considered sequentially because it is possible in some cases that incorporating local recycle around one membrane unit may increase the inlet concentration to another unit beyond the first transition concentration. The final design with local recycle is shown in Figure 19. The cost of this design is $953,320/year. Incorporation of local recycle reduces the membrane cost by 10.9% and the total cost by 4.8%. Further details about the cost breakdown are given in Table 2 in the Supporting Information.

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7. CONCLUSION If applied correctly, local recycle can improve the performance of membrane pervaporation units and reduce costs. In this article, we show how an engineer can determine in a straightforward and systematic way whether local recycle is desirable for a particular membrane module, and apply local recycle to a series of membrane modules. The method is based on dimensional analysis. On the basis of the values of certain dimensionless groups, the engineer can identify two transition compositions, zRin and zTin. If the inlet composition to a particular membrane module is less than zRin, then the maximum local recycle ratio is preferred. If the inlet composition is between zRin and zTin, then some intermediate value of the recycle ratio is preferred. Finally, if the inlet composition is above zTin, then no recycle is preferred. The method is illustrated with three case studies for two different systems. In all cases, local recycle is preferred for at least some of the modules. Properly incorporating local recycle reduces total process cost by an average of 4.8% and membrane unit costs by an average of 10.3%.

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λ = enthalpy of vaporization, kJ/kmol λ′ = dimensionless enthalpy of vaporization, θ = dimensionless temperature, ζ = dimensionless length, Subscript

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REFERENCES

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ASSOCIATED CONTENT

S Supporting Information *

Additional equations and cost summaries as described in the text. This material is available free of charge via the Internet at http:// pubs.acs.org.

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A = less permeable component B = more permeable component M = at mixing point i = component i in = the inlet of membranes out = the outlet of membrane R = retentate P = permeate

AUTHOR INFORMATION

Corresponding Author

*Fax: +886-2-2369-1314. E-mail: jeff[email protected]. Notes

The authors declare no competing financial interest.

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NOTATION A = mass transfer area, m2 cavg = the average molar density of mixture, kmol/m3 CR,i = concentration of component i in retentate, kmol/m3 CP,i = concentration of component i in permeate, kmol/m3 cp = molar heat capacity, kJ/kmol/K Di = diffusion coefficient, m/h Do,i = pre-exponential factor of diffusivity, m/h 9801



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Analysis of Local Recycle for Membrane Pervaporation Systems

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