3126
Ind. Eng. Chem. Res. 1997, 36, 3126-3139
Thermodynamic Efficiency, Process Flow-Sheet Design, and Membrane Selection in Membrane Gas Separation Jianguo Xu Air Products and Chemicals, Inc., 7201 Hamilton Boulevard, Allentown, Pennsylvania 18195
This paper discusses several ways of improving thermodynamic efficiency in membrane separation other than pressure ratio optimization, such as optimal process flow-sheet design and membrane selection, and explains the relationship between the various factors affected by these measures and thermodynamic efficiency. It also describes the relationship between stage and separator thermodynamic efficiency and local thermodynamic efficiency and permeation rates. It discusses the definition of the extent of separation and stage and local separation ability and explains why, when the type of membrane and the pressures on the two sides of a membrane are fixed, the more thermodynamically efficient membrane-separation process typically also needs less membrane area for separation tasks with specified purities for both permeate and retentate. It then briefly discusses cases with mixing of streams with different compositions and the objective function for optimal design of membrane separators, as well as the effects of backdiffusion, mass-transfer resistances, and cross flow. 1. Introduction Xu and Agrawal (1996a) introduced into membraneseparation analysis the concept of local thermodynamic efficiency of permeation η, making it possible to evaluate the thermodynamic efficiency of the permeation process at any point inside a membrane separator. For ideal gases, the local thermodynamic efficiency of permeation is N
η)
fi ln(xil/xih) ∑ i)1 ln γ
(1)
in which xil, and xih are the mole fractions of component i on the lower and higher pressure sides; γ is the ratio of the pressure on the higher pressure side, ph, to that on the lower pressure side, pl (γ ) ph/pl); N is the total number of components; and fi is the normalized rate of permeation of component i:
fi )
ri N
(2)
rj ∑ j)1 where ri and rj are the permeation rates of components i and j, respectively. Appendix A presents a detailed derivation of eq 1. The above-mentioned paper also introduced the optimal pressure ratio and the maximum efficiency for a given membrane selectivity and compositions on the two sides of the membrane. This enables one to identify more efficient and less efficient points inside the membrane separator and the potential for efficiency improvement. The paper also showed the relationships between the optimal pressure ratio, maximum efficiency, membrane selectivity, and compositions on the two sides of the membrane. Several conclusions were drawn from this analysis: (1) If the permeation rate is proportional to the partial pressure difference between the higher pressure side and the lower pressure side, the local thermodynamic efficiency of permeation is a function of the membrane selectivity, the pressure ratio, and the compositions on S0888-5885(96)00617-3 CCC: $14.00
the two sides of the membrane, but it is independent of the pressures on the two sides of the membrane and of the permeance of the membrane. (2) For binary systems, the optimal pressure ratio increases as the selectivity increases; it decreases to unity as the ratio of the mole fraction of the more permeable component on the higher pressure side to that on the lower pressure side increases and approaches unity. (3) For binary systems, the maximum local thermodynamic efficiency increases as the membrane selectivity increases and is a weak function of the ratio of the mole fraction of the more permeable component on the higher pressure side to that on the lower pressure side. This analysis led to the conclusion that product reflux which makes the compositions on both sides of a membrane identical, e.g., at the ends of a continuous membrane column, makes the thermodynamic efficiency at such locations become zero and should therefore be avoided if practical. The paper also showed that for separations yielding very high-purity permeate or retentate, the thermodynamic efficiency in a membrane separator can approach zero if the membrane selectivity is not extremely large. Therefore, most membranes are not suitable for such separation tasks. It was also determined that if the pressure energy in the product streams could be fully utilized, the pressures inside the membrane unit should be as high as possible to increase the rate of permeation and thus reduce the membrane area and size. This results from the fact that the membrane-separation efficiency is a function of pressure ratio but is independent of the absolute values of the pressures provided the permeation rate of each component is proportional to its partial pressure difference between the two sides of the membrane. Xu and Agrawal also described a permeate compression strategy which could be used to increase the efficiency of countercurrent membrane separators in which the pressure energy in the product streams could be fully utilized. They also briefly discussed the effect of sweep streams on the local thermodynamic efficiency. Local thermodynamic efficiency is a function not only of the pressure ratio but also of the membrane selectivity and of the compositions on the two sides of the membrane. Due to limits on the size and the scope of © 1997 American Chemical Society
Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997 3127
Figure 1. Continuous membrane column.
Figure 2. Scheme LP1-3a.
the paper mentioned above, it did not discuss how to manipulate factors other than the pressure ratio, such as the compositions on the two sides of the membrane and membrane selectivity, to improve the thermodynamic efficiency, nor did it address another important factor of production cost, the membrane area, other than suggesting that the highest possible pressure be used in order to reduce the membrane area. This paper discusses these and several other aspects of the subject, such as how to design membraneseparation processes with high thermodynamic efficiencies and how to select membranes with optimal values of selectivity and permeance. It also discusses how these measures will affect membrane area. Following these discussions, another parameter is introduced that may be useful in describing the separation ability of the membrane at any given point inside the separator. 2. Cycle Selection and Thermodynamic Efficiency As was mentioned in the Introduction, as the ratio of the mole fraction of the more permeable component on the higher pressure side to that on the lower pressure side is reduced to unity, the optimal pressure ratio is also reduced, and it too approaches unity. In a countercurrent membrane separator, this mole fraction ratio increases from the end at which the retentate exits to that at which the feed enters. This relationship has been used (Xu, 1993; Xu and Agrawal, 1996a) to develop a permeate compression strategy for cases in which the pressure energy contained in the product streams can be fully utilized. Since the rate of permeation is typically proportional to the difference in the partial pressures on the two sides of the membrane, an increase in the pressure of the lower pressure side means an increase in the partial pressure on that side, which reduces the driving force for permeation and results in a decrease in permeation rate. Xu and Agrawal (1996a) presented an example in which an increase of 16% in the membrane area resulted in an energy reduction of 37%. A bigger increase in the membrane area will be needed to reduce a unit of energy consumption as the mole fraction ratio approaches unity. At the extreme case, when that mole
fraction ratio is equal to unity, the optimal pressure ratio is also equal to unity. When the pressure ratio approaches unity, the driving force for permeation approaches zero, or the rate of permeation approaches zero: no separation occurs! For any permeation to take place, the pressure ratio has to be significantly greater than unity. Thus, the efficiency has to become zero at that point. The way out of such a dilemma is to avoid such situations: that is, to use a better process. 2.1. Avoiding Product Reflux. Matson et al. (1983) compared the continuous membrane column, which has product reflux, with several other schemes and showed significant improvement in both membrane area and power consumption by using schemes without product reflux. The schemes used by Matson et al. have significant mixing losses in most circumstances. Laguntsov et al. (1992), Xu (1993, 1994a,b), Xu and Agrawal (1996b), and Agrawal and Xu (1996a) described several schemes that can avoid such mixing losses for membrane-separation schemes with one or two compressors. Agrawal and Xu (1996b) introduced the methodology of drawing such schemes for multicompressor cases. The following example shows how the choice of a scheme without product reflux can greatly reduce power consumption while reducing membrane area at the same time. Example 1. Assume that a mixture of CO and hydrogen with a hydrogen mole fraction of 21.52% at a feed pressure of 1.01 bar is to be separated into a hydrogen stream with 80% H2 at 1.01 bar and a CO stream with 99% CO and a pressure of 27.8 bar. The membrane used for separation has a selectivity of 38, with hydrogen as the more permeable component and hydrogen permeance of 9.5 × 105 barrer/cm. Let us compare the performance of the continuous membrane column (with the throttle valve on the retentate end closed) shown in Figure 1 (Pfefferle, 1964; Hwang and Thorman, 1980) with that of LP1-3a in Xu and Agrawal (1996b), shown in Figure 2. The alphanumeric system introduced in Xu and Agrawal (1996b) is used in naming the different membrane separator processes in this paper. The letters “HP” or “LP” indicate that the feed enters the separator at the higher pressure side (HP)
3128 Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997
Figure 3. x1h, x1h/x1l, ln γ, ln γ*, η, and η* vs fractional membrane area plots. (a, top) Profiles in membrane stage S1 of the schemes in Figures 1, 2 (examples 1, 2, and 3), and 8. (b, bottom) Profiles in membrane stage E1 for continuous membrane column in Figure 1 in example 1. Table 1. Membrane Area and Power Comparison between Processes in Figures 1 and 2 for Example 1 (100 kmol/h Feed Rate)
process
membrane area, m2 S1 E1 E2
Figure 1 Figure 2
295 295
2783 365
N.A. 713
rel area
compressor flow, kmol/h
rel sep power
1 0.446
444 108
1 0.095
or lower pressure side (LP) of the separator, the first number stands for the number of compressors used in the process, the next number indicates the number of membrane stages, and the last letter is the serial number of the cascade scheme. See Xu and Agrawal (1996b). Zero pressure drop and countercurrent plug flow will be assumed in this and in the later analyses. The results of the simulation are listed in Table 1. From this table, we see that the work (or rather power) and membrane area needed for this separation job using scheme LP1-3a are only 9.5% and 45% of those needed for the continuous membrane column. Note that separation power is the power consumed in the separation process. The power used to increase the product pressure is not consumed. Therefore, it is not a part of the separation power. The profiles of the x1h/x1l ratio, the thermodynamic efficiency of permeation, the logarithm of the optimal pressure ratio, and the maximum thermodynamic efficiency for each stage of the two processes are shown in Figure 3a,3b, Figure 3a, Figure 4a, and Figure 4b, respectively. The values of the optimal pressure ratio, γ*, are calculated by setting the derivative of the thermodynamic efficiency with respect to the pressure ratio to zero. For binary separation, it has the form shown in eqs 17 and 18 in Xu and Agrawal (1996a). The maximum thermodynamic efficiency, η*, is the value of the thermodynamic efficiency at such an
Figure 4. x1h, x1h/x1l, ln γ, ln γ*, η, and η* vs fractional membrane area plots for scheme LP1-3a in Figure 2 in example 1. Profiles in membrane stages (a, top) E1 and (b, bottom) E2.
optimal pressure ratio. In these figures and those to follow, the end at which the higher pressure stream leaves the membrane stage is defined as the fractional membrane area of “0” and that at which the permeate side stream leaves the membrane stage is defined as the fractional membrane area of “1”. Apparently, the thermodynamic efficiency of permeation is significantly improved by using scheme LP1-3a. It can be seen that the stripping sections of the two processes (S1) have the same performance. The difference between the two processes is in the enriching sections. It is also interesting to note from Figure 3b that the thermodynamic efficiency curve near the η ) 0 end is rather flat; more than half of the enriching section (stage E1) of the scheme in Figure 1 has efficiencies of less than 5%. The reason for this is that the x1h/x1l ratio is very close to unity in the left half of stage E1 in the continuous membrane column. The logarithm of the optimal pressure ratio, ln γ*, is far from that of the real pressure ratio, ln γ, in this section. That explains why the continuous membrane column is thermodynamically inefficient. In comparison, in the process of Figure 2, the lowest efficiency is greater than 16%. The logarithm of the optimal pressure ratio is never far from that of the real pressure ratio in both stages E1 and E2 so that the efficiency (η) is not far below the maximum values. It can also be seen that ln γ* crosses that of ln γ in both stages E1 and E2 of the process in Figure 2 (Figure 4a and 4b), as well as in stage E1 of the continuous membrane column in Figure 1 (Figure 3b). The real thermodynamic efficiency at such a point is equal to the maximum efficiency. The fact that ln γ* crosses ln γ twice for the process in Figure 2 brings the ln γ* value close to the ln γ value in most of the enriching stages. On the other hand, the rapid decline of the optimal pressure ratio (or rather its logarithm) in stage E1 of the continuous membrane column means that the
Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997 3129
Figure 5. Scheme HP1-3b with a feed compressor. Table 2. Membrane Area and Power Comparison between Processes in Figures 2 and 5 for Example 2 (100 kmol/h Feed Rate) compressor flow, kmol/h
membrane area, m2 process
S1
E1
E2
E3
E4
rel membrane area
C1
C2
rel sep power
Figure2 Figure5 Figure 8
316 1419 316
2412 137 233
94 94 931
N.A. N.A. 73
N.A. N.A. 92
1 0.585 0.583
0 100 103.1
361 54 38.7
1 0.265 0.222
optimal pressure ratio is far below the real pressure ratio in most areas of that stage. 2.2. Adjusting the Flow on the Permeate Side. Xu and Agrawal (1996a) observed that the optimal pressure ratio declines from the end at which the retentate exits to that at which the feed enters the membrane. This occurs because the flow on the lower pressure side has a diluting effect, which causes the ratio of the mole fraction of the more permeable component on the higher pressure side (x1h) to that on the lower pressure side (x1l) to increase. Since the pressure ratio in a real membrane stage is fixed, as the change in composition on the higher pressure side increases, the optimal pressure ratio will eventually decrease to far below the real pressure ratio at the end at which the permeate exits, making the permeation process very inefficient. Any externally introduced sweep stream can only accelerate this decline in efficiency. It is therefore desirable to avoid this high x1h/ x1l situation. One way to do this is to reduce the flow on the permeate side. This can be done by reducing or eliminating the sweep stream or by compressing the permeate stream and feeding it to the higher pressure side. This causes the slope of the optimal pressure ratio dγ*/dx1h to become less steep so that the optimal pressure ratio can remain relatively close to the real pressure ratio throughout the stage, so the overall efficiency of the membrane stage is higher. On the other hand, when the optimal pressure ratio is significantly greater than the real pressure ratio and this real pressure ratio cannot be changed for some reason, increasing the sweep flow can accelerate the decline of the optimal pressure ratio to bring it closer to the real pressure ratio on average. The next example shows how adjusting the sweep flow can greatly reduce power consumption and membrane area. Example 2. Consider the same separation problem with the same type of membrane as in example 1 except that the hydrogen content in the permeate is increased in this example from 80% to 99%. Scheme HP1-3b of Xu and Agrawal (1996b), (shown in Figure 5) is used for comparison with LP1-3a in Figure 2. In this case, another compressor is necessary to compress the feed gas to the pressure of the higher pressure side for scheme HP1-3b.
The results of the simulations with LP1-3a and HP13b are listed in Table 2. It can be seen that the separation power and membrane area using HP1-3b are 73% and 46% smaller than those using LP1-3a. The reason is that the x1h/x1l ratio at the feed end on the higher pressure side of the middle stage is reduced from 0.979 to 0.838 by eliminating the feed flow as the sweep stream in that section. This results in elimination of the membrane section with thermodynamic efficiencies of less than 5%. See the profiles of the x1h/x1l ratio, thermodynamic efficiency, logarithm of the optimal pressure ratio, and maximum thermodynamic efficiency in Figures 3a, 6a,6b, and 7a-c, respectively for these two processes. Notice that for LP1-3a, more than 40% of the membrane area in the middle stage (E1) has efficiencies of less than 5%; the close-to-unity x1h/x1l ratio brings the optimal pressure ratio far below the real pressure ratio. The situation in Figure 7b is much improved in comparison with that in Figure 6a. The cost of all these savings is an additional compressor to compress the feed gas to the pressure on the higher pressure side. Notice that the total energy required for the two compressors in scheme HP1-3b is actually less than that of the single compressor when scheme LP13a is used. For a separation process of sufficiently large capacity, the energy and membrane area savings may well pay for the cost of this additional compressor. Since a second compressor is used, the permeate stream with a composition similar to that of the feed can be combined with the latter and used as the sweep stream in the section in which the optimal pressure ratio is greater than the real pressure ratio. The permeate that has been combined with the feed is then compressed and fed to the higher pressure side. In so doing, the permeate flow is increased in the stage in which the optimal pressure ratio is greater than the real pressure ratio on the right end of the stage, resulting in an accelerated decline of the optimal pressure ratio. This decline makes the optimal pressure ratio closer to the real pressure ratio for that stage in this particular case. On the other hand, the flow on the lower pressure side is reduced in areas where the mole fraction of hydrogen (x1h) is high so that the flow on the lower pressure side (x1l) can be higher than when there is a larger sweep flow. This further reduces the size of the
3130 Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997
Figure 6. x1h, x1h/x1l, ln γ, ln γ*, η, and η* vs fractional membrane area plots for scheme LP1-3a in Figure 2 in example 2. Profiles in membrane stages (a, top) E1 and (b, bottom) E2.
low-efficiency section in which the x1h/x1l ratio is high. Figure 8 shows such a scheme (scheme LP2-5b in Agrawal and Xu (1996a)). Table 2 also shows the simulation results using this scheme. As can be seen from the table, the separation power is reduced by 16%, relative to HP1-3b, while the membrane area is slightly smaller. From the table, it can also be seen that the flow of the first compressor (C1) is increased by only 3.1% by using the scheme in Figure 8 and the reduction in the flow of the second compressor (C2) is substantial. That reduction occurs because compressor C1 in the scheme in Figure 8 picks up the permeate with relatively low hydrogen content. To neutralize its diluting effect, a larger amount of hydrogen-rich gas (richer than that fed to the stage from which the permeate product is taken) has to be permeated to the lower pressure side if it is not taken out from the permeate side, such as in the case of using scheme HP1-3b (Figure 5). The increased demand for higher purity hydrogen requires additional membrane area in the section in which the x1h/x1l ratio is high (close to unity) and the thermodynamic efficiency is low. The x1h/x1l ratio, thermodynamic efficiency, logarithm of the optimal pressure ratio, and maximum thermodynamic efficiency vs fractional membrane area plots using the scheme in Figure 8 are shown in Figure 3a and Figure 9a-d. The logarithm of the optimal pressure ratio is greater than that of the real pressure ratio, which means that a decrease in permeate pressure in this stage could further improve the thermodynamic efficiency of this stage. In both part a and part b of Figure 9 (for stages E1 and E2 of scheme LP2-5b), the logarithm of the optimal pressure ratio crosses the real pressure ratio and is never far from the real pressure ratio so that the thermodynamic efficiency value in these stages is not very far from the maximum values. The maximum x1h/ x1l ratio in this process (in stage E3 as shown in Figure
Figure 7. x1h, x1h/x1l, ln γ, ln γ*, η, and η* vs fractional membrane area plots for scheme HP1-3b in Figure 5 in example 2. Profiles in membrane stages (a, top) S1, (b, middle) E1, and (c, bottom) E2.
9c) is also reduced further, to less than 0.8. This reduces the size of the section with high x1h/x1l values and very low values of the optimal pressure ratio (or its logarithm). It should be mentioned that this process is not fully optimized. It is possible that eliminating stage E3 (resulting in scheme LP2-4e, shown in Figure 10) in this process would be a lower cost option. For complete optimization, these processes and that of scheme LP2J-5b in Figure 11 should be considered (Letter “J” in the previous sentence is used to indicate that the first recycle stream jumps across the membrane stage from which the second recycle stream is withdrawn; see Agrawal and Xu (1996a)). For that purpose, the unit costs of energy and membrane area, as well as the cost functions of the compressors, separator shells, skids, etc., will have to be known. For preferred schemes with one or two compressors at other conditions, see Agrawal and Xu (1996a). Apparently, increasing the number of compressors and the number of stages can further reduce the flow on the lower pressure side. In the extreme case, any
Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997 3131
Figure 8. Scheme LP2-5b.
Figure 9. x1h, x1h/x1l, ln γ, ln γ*, η, and η* vs fractional membrane area plots for scheme LP2-5b in Figure 8 in example 2. Profiles in membrane stages (a, top let) E1, (b, top right) E2, (c, bottom left) E3, and (d, bottom right) E4.
Figure 10. Scheme LP2-4e.
amount of permeate is immediately compressed and fed to the higher pressure side at the position with the same composition, and the pressure ratio in each membrane element is set to its optimal value. Such a design will probably result in thermodynamically optimal membrane separator. In reality, the cost of compressors is a major component of production cost. Increasing the number of compressors can quickly lead to a large capital penalty. An economically optimal membrane
process scheme for gas separation cannot afford to have a large number of compressors. 3. Membrane Selection and Thermodynamic Efficiency For a finite membrane selectivity and sweep/feed flow ratio, the x1h/x1l ratio at and near the higher pressure side feed end increases to close to unity as the difference
3132 Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997
Figure 11. Scheme LP2-J-5b.
between x1h at the retentate end and x1h at the feed end of the membrane stage increases. The reason is that as that difference increases, a greater amount of gas has to be permeated to the lower pressure side of the membrane, which results in an overwhelming diluting effect. As x1h/x1l approaches unity, the optimal pressure ratio decreases to close to unity (see Xu and Agrawal (1996a)). This is undesirable for two reasons: (1) Since the x1h/x1l ratio at the end where the retentate exits is relatively low, the optimal pressure ratio is relatively high, while the optimal pressure ratio is very low at the end at which the feed is introduced. Thus, the optimal pressure ratio within a stage differs greatly from one end to the other. Since a membrane stage can only operate at one pressure ratio, the presence of very different optimal pressure ratios within a membrane stage implies that large sections of the membrane stage will be operating with pressure ratios far from their optimal values. The thermodynamic efficiency in those areas will be low. (2) When the permeate compression strategy is used so that the pressure ratio is brought close to its optimal value at and near the end with higher x1h/x1l, the pressure ratio across the membrane is lower. In so doing, the driving force for permeation is reduced at and near that end so that a larger membrane area is needed to achieve a given amount of separation. This disadvantage is in addition to the cost of introducing more stages of compression and more membrane modules. Reducing the difference in mole fraction of the more permeable component on the high-pressure side, x1h, between the feed and the retentate ends of a membrane stage can reduce the x1h/x1l ratio at the feed end of that stage. One way of doing this is to reduce the mole fraction of the more permeable component in the retentate of this stage. Another way is to increase the mole fraction in the feed to that stage. Choosing the proper membrane for the each section of a separator can often decrease the change in x1h without causing a negative impact on the total membrane area of the separator. From the paper described in the Introduction (Xu and Agrawal, 1996a), we know that a higher membrane selectivity generally results in a greater maximum efficiency. This also means that the thermodynamic efficiency in a membrane separator with a greater selectivity is generally greater for a given pressure ratio and gas composition on the two sides of the membrane. The problem is that a membrane with a greater value of selectivity typically has a smaller permeance or is more expensive to make. Therefore, both power cost and capital cost (cost of membrane and associated equipment parts) have to be considered in choosing the optimal membrane. In the following section, we will learn that in different locations of the membrane separator, the optimal membrane is also different. The
question then is how to select the membranes for the different stages of a separator. 3.1. Local Thermodynamic Efficiency and Stage Thermodynamic Efficiency. The relationship between the local thermodynamic efficiency of permeation and the stage efficiency, ηstage, is as follows:
∫0
W
ηstage )
dw
∫0Aη dA dA
η dw )
W
) nRT0 ln γ N
∫0AηRT0 ln γ∑ri dA
1
i)1
) nRT0 ln γ
N
∫0 ∑ri) dA i)1
n
A
η(
(3)
or
ηstage )
∫0nη dn
1 n
(3a)
in which n is the total number of moles of gas permeated through the membrane stage, W the work (or power) needed for the separation, R the universal gas constant, and A the membrane area of the stage. T0 is the ambient temperature. From eq 3, we know that the thermodynamic efficiency in areas with higher rates of permeation has more impact on the stage efficiency. This same equation can be extended to the whole membrane separator, which may be composed of several stages, if there is no mixing of streams with different compositions. Therefore, we should typically spend less effort in improving the thermodynamic efficiency in locations with lower permeation rates, such as near the retentate end of the membrane separator where the driving force is small, than in areas with higher permeation rates. The reason is that the driving force for the more permeable component is lower at that end, while the permeance for the less permeable component is smaller, although the driving force for that component is greater. The net effect is that the total rate of permeation is always smaller at the retentate end if the pressures on the two sides are constant. If membranes with different values of selectivity are available, the membrane with higher permeance (but lower selectivity) should be used at that end. Therefore, the higher permeance (and lower selectivity) membrane should be considered for use in the stripping section rather than in the enriching section. On the other hand, we should pay more attention to efficiency improvement at the locations with higher rates of permeation, i.e., in the areas with a higher mole fraction of the more permeable component. It is desirable to use membranes with greater values of
Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997 3133 Table 3. Membrane Area and Power Comparison between Processes in Figure 2 with Different Membrane Selection for Different Stages for Example 3 (100 kmol/h Feed Rate) membrane area, m2 R ) 100 membranes in stage
S1
E1
E2
rel membrane area
compressor flow, kmol/h
rel separation power
none E1 E2
316 316 316
2412 9800 1575
94 94 596
1 3.62 0.882
361 330 199
1 0.889 0.426
selectivity in these areas for efficiency improvement. The question then becomes whether the improvement in thermodynamic efficiency will accompany production cost reduction, since membranes with greater selectivities are typically those with lower permeance. The next section shows that the proper use of higher selectivity membranes for the right sections can reduce power consumption and for some applications reduce the total membrane area as well. 3.2. Optimal Membrane Selection for Efficiency Improvement. Let us now consider the case using scheme LP1-3a for example 2. The lower overall efficiency (in comparison to scheme HP1-3b or LP2-5b in Figure 8) is caused by the lower thermodynamic efficiency in the feed end of the middle stage of the membrane separator, E1. Apparently, using a higher selectivity membrane in that section could increase the thermodynamic efficiency of that section. The benefit, however, is limited to that particular section. Moreover, the effect of the relatively large sweep stream will make the flow on the lower pressure side large, therefore maintaining the diluting effect at a relatively high level. The effect of a higher selectivity membrane on the diluting effect of the permeate flow will not be great. In such circumstances, the composition of the feed (on the higher pressure side) to that stage will not change substantially. This means that the optimal pressure ratio at the feed end of the middle stage is still significantly lower than the real pressure ratio. Since a higher selectivity membrane has a lower permeance value, this approach may not necessarily be better than using the lower selectivity, higher permeance membrane throughout the separator. On the other hand, using a higher selectivity membrane in the section at the permeate product end, stage E2, not only puts the higher selectivity membrane at the end with the highest permeation driving force (and thus highest permeation rate) but also reduces the mole fraction of the more permeable component in the feed to this stage (which is the permeate of the middle stage after compression) and that in the retentate to be sent to the middle stage as feed, when the desired permeate purity is fixed. As a consequence, the composition change in the middle stage is significantly reduced, thus largely eliminating the section with low thermodynamic efficiency, even though the type of membrane in the middle stage is not changed. Example 3. Consider the same separation task as in example 2 using scheme LP1-3a. Two types of membranes with different values of selectivity are available. One is the same as that in example 2; i.e., it has a selectivity of 38 and permeance of 9.5 × 105 barrer/cm. The other has a selectivity of 100 and permeance of 2.167 × 105 barrer/cm. This selectivitypermeance relationship satisfies the relation p ) kR-1.5275, similar to that proposed for H2/N2 separation by Robeson (1991). The results of using different membranes in different stages in the enriching section are shown in Table 3. In comparison with the case in which only the lower selectivity membrane is used, using the higher selectivity membrane in the stage at the permeate
Figure 12. x1h, x1h/x1l, ln γ, ln γ*, η, and η* vs fractional membrane area plots for scheme LP1-3a in Figure 2 in example 3 with the higher selectivity membrane in stage E1. Profiles in membrane stages (a, top) E1 and (b, bottom) E2.
product end (but lower selectivity membrane in other stages) improves the efficiency by 57%, while the total membrane area is reduced by 12%, despite the fact that the permeance of this second membrane is much smaller. The reasons for the superior performance in this case were already described in the previous section. When the higher selectivity membrane is used in the middle stage (E1), on the other hand, the reduction in separation power is only 11%, while the total membrane area is increased by 262%. This example shows how a proper selection of membrane can substantially improve the performance of a membrane separator. Thermodynamic efficiency analysis can guide such a selection. The x1h/x1l ratio, thermodynamic efficiency, logarithm of the optimal pressure ratio, and maximum thermodynamic efficiency vs fractional membrane area for the cases summarized in Table 3 are shown in Figures 3a, 6a, and 6b, 3a, 12a and 12b, and 3a, 13a, and 13b, respectively. As can be seen from Figure 13a, when the higher selectivity membrane is used in the stage near the permeate product, E2 (but not in other stages), less than 15% of the membrane in the middle stage E1 has an efficiency of less than 5%, with the minimum at 3%, despite the fact that the peak value is less than 30%. In comparison, when the higher selectivity membrane is used in the middle stage E1 (but not in other stages), more than 40% of the middle stage has an efficiency of less than 5%, with the minimum close to 0.6%, although
3134 Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997
Figure 13. x1h, x1h/x1l, ln γ, ln γ*, η, and η* vs fractional membrane area plots for scheme LP1-3a in Figure 2 in example 3 with the higher selectivity membrane in stage E2. Profiles in membrane stages (a, top) E1 and (b, bottom) E2.
2, the same strategy can also be successfully applied to the two compressor schemes: HP1-3b with a feed compressor in Figure 5 and LP2-5b in Figure 8. Notice that when a membrane with a selectivity of 100 is used in the stage on the left side of LP2-5b, the stage next to it is no longer necessary for the problem in the example. Therefore, LP2-J-5b (in Figure 11) should be used instead. In such a scheme, the higher selectivity membrane can be used in the two stages on the lefthand side, E3 and E4. A similar situation may arise for HP1-3b: when a higher selectivity membrane is used at the end at which the permeate product exits, HP13c (in Figure 14) may be the preferred process. It can be seen from Figures 12b and 13b that the optimal pressure ratio in the stage at the permeate product end (left-hand end) is significantly lower than the real pressure ratio in the processes in examples 2 and 3. If the pressure on the permeate side can be efficiently utilized, the permeate side pressure can be increased to improve the efficiency in that stage. However, varying the pressure ratio is beyond the scope of this paper and will not be discussed in detail here. Membranes with different values of permeance (and therefore different values of selectivity) are not difficult to obtain. Even if only one membrane material is available, changes in operating temperature often result in changes in the properties of the membrane. For example, for polymeric membranes, the values of permeance typically increase with temperature, while those of selectivity typically decrease. 4. Discussion
the peak value is close to 40%. The reason is that the maximum x1h is reduced to less than 0.5 from approximately 0.7 so that the section with an x1h/x1l ratio of greater than 0.9 is eliminated. The section with the lowest optimal pressure ratio (which is furthest from the real pressure ratio), and therefore the lowest thermodynamic efficiency, is thus eliminated. See Figures 12a and 13a. In stage E2, the efficiency is in the17-20% range when the higher selectivity membrane is used in that stage (Figure 13b), while it is in the 7.7-8.4% range when the higher selectivity membrane is used in stage E1 (Figure 12b). This is because the higher selectivity increases not only the maximum efficiency but also the optimal pressure ratio, following the observation that the optimal pressure ratio is greater when the selectivity is higher. Increasing the optimal pressure ratio brings it closer to the real pressure ratio. While using appropriate membranes in the right locations improves the efficiency greatly without paying penalties in membrane area for the scheme in Figure
Figure 14. Scheme HP1-3c with a feed compressor.
4.1. Extent of Separation, Separation Ability, and Thermodynamic Efficiency. We have learned from experience that for a fixed separation job with purity of both permeate and retentate products specified, with the pressures on the two sides of the membrane kept constant, and with the same type of membrane being used throughout the separator, a lower power membrane scheme typically also needs less membrane area. While we do not have rigorous proof for this relationship and although it seems at first glance to be counterintuitive, an explanation is presented below. Let us begin with a discussion on how to describe the extent of separation. We do not wish to use the change of mole fraction on one side of the membrane since that does not uniquely reflect the change. For example, the mole fraction difference between the feed and the retentate in a membrane-separation process does not reflect the change on the lower pressure side, unless the membrane selectivity, the pressure ratio of the mem-
Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997 3135
brane separator, and the composition and flow rate of the sweep stream are also given. Therefore, it cannot be considered a generally useful measurement, especially when a permeate product is desired at a certain purity. Besides, the difficulty in removing the last drop of impurity (e.g., increasing the purity from 98.999 99% to 99.999 99%) is also different from that in enriching that component from a lower purity, e.g., from 49% to 50%. This is true for any separation process. A true measure of the separation extent should reflect both the changes on each side of the membrane and the difficulty of the separation. The minimum work of separation achieved in the separation process seems to satisfy these two requirements from a thermodynamic point of view. The mathematical expression for minimum work is 2
Wmin ) RT0[
N
N
(Mj∑xi ln xi)2 - F∑(xi ln xi)1] ∑ j)1 i)1 i)1
(4)
in which Mj is the number of moles of gas in product stream j (j ) 1, 2) and F is that in the feed stream: M1 + M2 ) F. Subscripts 1 and 2 represent the states before and after separation. Since eq 4 is a function of the ambient temperature and is an extensive quantity, it is desirable to convert this value into an intensive quantity that is unrelated to the ambient temperature. Therefore, we would like to define the extent of separation ξ by dividing the minimum work of separation by FRT0:
ξ)
Wmin
[∑ 1
) FRT0
2
Fj)1
N
(Mj
∑ i)1
]
N
xi ln xi)2 -
(xi ln xi)1 ∑ i)1
(5)
which is a dimensionless number. Such a definition is useful in all separation processes. Let us now define the separation ability of a membrane, Sstage, the reciprocal of the membrane area, A, needed for a unit of separation.
Sstage ≡
Fξ A
(6)
Sstage, of course, is the averaged membrane separation ability of a separator. It represents the average separation ability of the membrane separator at different positions. Local separation ability can be defined as follows:
S≡
1
1 dµ ) RT0 dA
N
dn
) [fi ln(xil/xih)]} ∑ dA i)1
{RT0
RT0
N
ri ln(xil/xih) ∑ i)1
(7)
in which µ is the minimum work of separation. As can be seen, it is the local sum of the product of each permeation rate multiplied by the logarithm of the ratio of the mole fraction on the higher pressure side to that on the lower pressure side. This is a better measure of the separation ability of a membrane than the permeation rate of a certain component or the sum of the permeation rates in reflecting the ability of the membrane to perform a particular separation task. For example, a leak in a pipe will give a certain flux of a certain component (equivalent to the permeation rate of that component) and a total rate of the leak (equivalent to the sum of the permeation rates). However, the
leak in the pipe does not result in separation. Therefore, that leak in the pipe has no separation ability. Eq 7 will give a separation ability of zero if the composition on the two sides of the pipe is the same or a negative number if the composition inside the pipe is different from that outside. This negative value of separation ability is relevant since that leak results in mixing of streams with different compositions, and more separation would be necessary to merely recover to the original state. Since dµ ) η dw, the separation ability is related to the thermodynamic efficiency by the following relation:
S)
1 dµ
1 )
RT0 dA
η
dw
RT0 dA
N
ri ∑ i)1
) η ln γ
(8)
That is, the separation ability is the product of the thermodynamic efficiency, the logarithm of the pressure ratio, and the sum of the permeation rates. For a given pressure ratio, γ, it is proportional to the product of the thermodynamic efficiency and the sum of permeation rates. It can be further deduced that the separation ability is proportional to the permeance and the pressure on the higher pressure side (or the lower pressure side), for a given pressure ratio, if the permeation rate is proportional to the difference between the partial pressures on the two sides of the membrane. If a given scheme improves the thermodynamic efficiency to a degree such that its impact on the separation ability is greater than the negative effect of the sum of permeation rates, then the membrane area decreases along with the decrease in power consumption. The above relationship explains the phenomenon mentioned at the beginning of this section: the membrane-separation process with lower power consumption typically also needs less membrane area for a fixed separation job at fixed pressures on the two sides of the membrane and fixed membrane type. For example, when there is a product reflux, such as at the permeate end of the continuous membrane column (and the other end also if the valve is open), the thermodynamic efficiency is zero at that end. The separation ability of the membrane at that particular point is also zero since the product of the logarithm of pressure ratio and the sum of permeation rates cannot be infinitely large. The same conclusion can be obtained from eq 7: since xil/xih is equal to unity and its logarithm is equal to zero, all the terms after the summation sign are equal to zero at the reflux end. The zero value of separation ability where xil/xih ) 1 explains why the continuous membrane column not only consumes more power but needs more membrane area as well, in comparison with other membrane schemes which do not mix streams with different compositions, as shown before. This effect can make the performance of the continuous membrane column even worse than some membrane schemes which do have mixing loss, as demonstrated by Matson et al. (1983). The average separation ability of the membrane stage can be related to the local separation ability by the following equation:
Sstage )
∫0AS dA A
Substituting eq 8 into eq 9 gives
(9)
3136 Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997 N
∫0Aη ln γ∑ri dA i)1
Sstage )
A
(10)
In comparison with eq 3, we have
Sstage )
n ln γ ηstage A
(11)
The same is true for the relationship between the separation ability of a membrane separator and the local separation ability if there is no mixing of streams with different composition. While the local separation ability can be useful in explaining certain phenomena, more can be done to relate this parameter to the optimal design of membrane separators. 4.2. Mixing Loss. It was mentioned in the preceding section that the averaged separator efficiency or separation ability can be related to the integration of the corresponding local values when there is no mixing of streams with different compositions. When that mixing does occur, the exergy loss due to mixing has to be added to the integration term to arrive at the averaged value for the separator:
ηsep ) ηstageWstage/(Wstage + Wmixing)
(12)
in which Wstage is the work consumed in the stage or the sum of the work consumed in the stages, Wmixing is the loss of exergy due to mixing, ηsep is the overall efficiency of the separator, and ηstage is the efficiency of the membrane stage or the average of those of all stages if there are K (K > 1) stages. When there are several stages, ηstage can be calculated by K
ηstage )
∑ i)1
K
Wi) ∑ i)1
ηiWi/(
(13)
In the equation, ηi is the stage efficiency of stage i and Wi is the work consumed in stage i. If the process is not properly chosen or the unit is very small so that a simple system is preferred, the mixing loss term Wmixing can be large as was shown in an example in our previous study (Xu and Agrawal, 1996b). This purely parasitic loss should be minimized or eliminated if possible. Xu and Agrawal (1996b) and Agrawal and Xu (1996a) describe ways of avoiding such mixing losses. For binary separations, the minimum work of separation is fixed when the purities of both the permeate and retentate are fixed or when one of them is fixed and the recovery of one component is also fixed. When only one product purity is specified, such as in the separation of air to produce nitrogen or oxygen-enriched air, an interesting situation arises. In these cases, we only care about the purity of that particular stream, not about the changes on the other side. One may find that a scheme with a higher averaged thermodynamic efficiency is not necessarily the more power-efficient scheme. One of the reasons is that the stream that does not have a purity requirement (waste stream) will be “thrown away.” It is either vented to the atmosphere or consumed in a different unit operation such as combustion. In this throwaway step, there is a mixing loss. It may also involve loss of mechanical energy, such as the pressure contained in a waste retentate stream. A rigorous thermodynamic study has to include that
mixing loss and any other losses to obtain any meaningful conclusion. The second factor is that the minimum work of separation for obtaining a certain product with a certain purity is different when a different amount of raw feed gas is used. If the pressure drop inside the membrane separator is negligible and if the reject stream comes out at the same pressure as the feed, a case with lower recovery would have a lower minimum work of separation. At the extreme, when the feed stream flow is infinitely large, the minimum work of separation for extracting a certain component from the feed gas is the lowest. Unless such factors are correctly accounted for, the thermodynamic efficiency analysis becomes meaningless. 4.3. Objective Function for Optimization. The purpose of introducing the concept of local thermodynamic efficiency of permeation and local separation ability, together with the analyses based on these concepts, is to understand how these parameters affect membrane-separator design and operation. The knowledge obtained from such a study can then be used to improve the design of membrane separators or to explain certain phenomena. These concepts can be used to screen potential candidates for a particular separation job in order to eliminate inferior process processes and to give direction on what the optimal process should look like. The ultimate policy for the optimal design of the separator, however, should still be based on minimization of the production cost, which is the sum of capital cost and operating cost. Membrane area has a direct impact on the capital cost since it determines not only the cost of the membrane but also that of the separator shells, skids, etc. Power consumption determines the power cost and also impacts the capital cost because it is related to the size of energy-consuming and conversion equipment such as compressors and pipelines. Other factors, such as the number of compressors, the number of membrane modules, and the operating pressures, will also have significant impacts on the production cost. These factors together with other important factors such as pressure drop, back-diffusion mass-transfer resistances, and flow patterns should be considered in process design. Back-diffusion, masstransfer resistances, and flow patterns will be briefly discussed below. Ways of improving the efficiency and reducing membrane area have been discussed separately. An optimal design should consider all these factors simultaneously. Consequently, the optimal design may contain several of the elements discussed in this paper and in the previous papers (Xu and Agrawal 1996a,b): It should have minimal mixing loss. It should operate at the highest possible pressure if the pressures in the product streams can be used efficiently. If membranes with different selectivity values are available, they should be properly chosen according to the guideline given in this paper. Product reflux should be avoided if practical. The permeate compression strategy should be considered when staged compressors are used. The flow of the sweep stream should be manipulated to improve the performance of the separator if such a maneuver is beneficial. 4.4. Back-Diffusion, Sweep, Mass-Transfer Resistance, and Flow Pattern. In our analysis, we also have made simplifications in terms of flow pattern (i.e., countercurrent flow without back-mixing). This is an
Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997 3137
idealized situation. In reality, there will be diffusion in the gas and on the membrane surface, which may significantly affect the concentration profiles on either side of the membrane. The effect of diffusion is more pronounced when the velocity of the gas is low or when the concentration gradient, dx/dl, is large (l is the length in the flow direction in the membrane separator). For a fixed flow area, too small a flow on the permeate side may cause the flow pattern on the lower pressure side to deviate from such a zero back-diffusion assumption and hurt the membrane-separator performance. Sweep on the lower pressure side can reduce or even eliminate the back-diffusion problem and may be preferred in small, simple membrane separators in which the permeate flow without a sweep stream would be small, although such a sweep would not be desirable for a truly countercurrent membrane-separation in which the pressure ratio can be optimized. There are also other solutions to such problems, such as splitting the membrane into several stages and minimizing the flow areas on the permeate sides with small permeate flows in order to reduce the back-diffusion problem. However, these solutions are beyond the scope of this paper and will not be discussed further. Another assumption made in these calculations has been that permeation is the rate-determining step and that the gas phase is well mixed at a particular cross section. When the permeation rate is fast enough, other factors such as diffusion in the nonselective layers in asymmetric membranes and in the gas film surrounding the membrane surface may become important. This requires modification of the permeation rate equation to reflect the additional resistance for mass transfer introduced by these diffusion steps. However, such a modification should not affect the analysis using countercurrent flow patterns. The only effect that we should see would be that the apparent permeance and selectivity of the membrane change as the composition and pressures on the two sides of the membrane change. It is also worth noting that a real membrane separator may have feed and/or permeate come in and/or out through a limited number of nozzles. This introduces cross flow. Cross flow may also be intentionally introduced to increase the mass-transfer coefficient in the gas films surrounding the membrane surface. How cross flow affects membrane performance depends on the exact configuration of the membrane separator, a big subject in its own right. Just as in heat exchangers, the effect of this cross flow on the flow pattern is something to be minimized in a typical membrane separator, in order to achieve a performance close to that with countercurrent flow. 5. Conclusions This paper discussed several ways of improving the thermodynamic efficiency of membrane separators other than varying the pressure ratio across the membrane and eliminating mixing losses. One way is to choose the proper process flow sheet. Two aspects were discussed on this topic. One was elimination of product reflux, such as that in the continuous membrane column. An example was given to show the magnitude of improvement that could be made if such reflux were eliminated. The second was to adjust the rate of decline in the optimal pressure ratio by adjusting the flow of the permeate stream so that the optimal pressure ratio of the membrane stage could be maintained to close to
Figure 15. Permeation in a countercurrent membrane element.
the real pressure ratio throughout the whole membrane stage. Since this can be done without compromising the driving force for the desired permeation, it does not increase the needed membrane area. On the contrary, in many circumstances, membrane area can be reduced. The second way to improve thermodynamic efficiency involved optimal selection of membrane. This paper showed that a membrane with a higher selectivity (and lower permeance as a consequence) should be used for separation of mixtures with higher concentration of the more permeable component and vice versa. An example was given to demonstrate the substantial improvement in efficiency and the modest improvement in membrane area that could be made by using the right membrane at the right place. The concept of separation ability was introduced as a tool to relate membrane area to thermodynamic efficiency. The separation ability is the minimum work of separation achieved in a unit membrane area divided by the universal gas constant and the ambient temperature. It is proportional to the thermodynamic efficiency, the logarithm of the pressure ratio, and the sum of permeation rates. It is a function of the pressure ratio, the mole fractions on the two sides of the membrane, and the membrane selectivity and is proportional to the permeance of the membrane, as well as the pressure on the higher pressure side if the permeation rate of each component is proportional to the difference between the partial pressures on the two sides of the membrane. The concept of separation ability was used to explain why a membrane-separator scheme with less power consumption typically also needs less membrane area, when the pressures on the two sides of the membrane are fixed and only one type of membrane is used throughout the membrane separator and there is no mixing loss. In addition, mixing losses of various kinds and their effects on separator efficiency, objective function for optimization, and backdiffusion and its effects, mass-transfer resistances, and cross flow were also briefly discussed. The purpose of doing so was to identify the proper place for and the limits of thermodynamic efficiency analysis in the design of membrane separators and to highlight the need to address some of the other important but not well-studied aspects of the subject.
Acknowledgment I am grateful to Dr. Rakesh Agrawal of Air Products and Chemicals and Dr. Keith Wilson, a former Air Products employee, for reading the manuscript and providing valuable suggestions.
3138 Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997
in which
Nomenclature A ) membrane area F ) flow rate of the feed stream Fih, Fil ) flow rate of component i on the higher pressure side and lower pressure side, respectively fi ) normalized rate of permeation, defined by eq 2 K ) number of stages in a membrane cascade Mj ) moles of gas in product stream j or its flow rate N ) number of components in the gas mixture n ) number of molecules of gas permeated through the membrane (or its rate) R ) universal gas constant ri ) rate of permeation of component i S ) local separation ability, defined by eq 7 Sstage ) stage-separation ability, defined by eq 6 T ) absolute temperature T0 ) absolute ambient temperature W ) pressure exergy consumed in permeation process or its rate Wi ) work consumed in membrane stage i or its rate (power) Wstage ) pressure exergy consumed in a membrane stage or its rate Wmixing ) exergy loss due to mixing of streams with different compositions or its rate x1l/x1h ) ratio of the mole fraction of the more permeable component on the higher pressure side to that on the lower pressure side xi ) mole fraction of component i xih, xil ) mole fraction of component i on the higher pressure side and lower pressure side, respectively
dni ) ri dA
with N being the number of components in the system and ri the rate of permeation of component i. We will define the normalized rate of permeation, fi, consistent with the paper
fi )
N
dn )
∑ i)1
dni
(A1)
(2)
N
The mole fraction of component i at position A + dA, xih,A+dA, is
Fih
dni -
N
xih,A+dA )
)
) N
xih +
)
N
N
Fjh,A+dA ∑Fjh - dn ∑ j)1 j)1 xih
N
Fjh ∑Fjh ∑ j)1 j)1
Fih - dni
Fih,A+dA
Fjh ∑ j)1
1 - dn/
dni
dn N
N
Fjh ∑Fjh ∑ j)1 j)1 N
∑ j)1
1 - dn/
) xih +
Fjh
[ ] xih - fi N
∑ j)1
1 - dn/
Fjh
dn N
Fjh ∑ j)1 (A3)
Neglecting the second-order term of dn gives
[( ) ] [( ) ]
xih,A+dA ≈ xih 1 + 1 -
fi
dn
xih
Fjh ∑ j)1
fi
dn
N
(A4)
Similarly, it can be derived that
Appendix A Suppose in a membrane element with an infinitely small area dA, dn moles of ideal gas (per unit time) permeate through the membrane from the high-pressure side with pressure of ph and mole fraction of xih for component i to the lower pressure side with pressure of pl and mole fraction of xil for component i; see Figure 15. Notice that the symbols in parentheses in the figure are the same as the subscripts in the text. For example, x(ih) and F(ih,A+dA) in the figure are the same as xih and Fih,A+dA in the text. Notice also that subscript A is dropped to simplify notation. The total moles of gas permeated (per unit time) is the sum of those of each component
ri rj ∑ j)1
Greek Letters Ri ) selectivity, defined by permeance of the most permeable component divided by the permeance of component i γ ) ratio of the pressure on the higher pressure side to that on the lower pressure side γ* ) optimal pressure ratio at which the local thermodynamic efficiency is at the maximum µ ) minimum work of separation achieved in the permeation process η ) local thermodynamic efficiency, defined by eq 1 η* ) maximum local thermodynamic efficiency with the composition on the higher pressure side and that on the lower pressure side and membrane selectivity fixed ηsep ) thermodynamic efficiency of the membrane separator ηstage ) thermodynamic efficiency of a membrane stage ξ ) extent of separation, defined by eq 5
(A2)
xil,A+dA ≈ xil 1 + 1 -
xil
N
(A5)
Fjl ∑ j)1
For the stream on the higher pressure side, the change in chemical potential independent of changes in pressure and temperature is N
N
Fih,A+dA ln xih,A+dA - ∑Fih ln xih] ∑ i)1 i)1
(dµ1)T,p ) RT[
(A6) in which R is the universal gas constant and T the absolute temperature. Substituting eq A5 into eq A6 and using the first-order Taylor expansion give
[
Fih,A+dA ln xih +
( ) ] ( ) { [ ( ) ] [( ) ]} [( ) ] fi
-
N
Fjh ∑ j)1
ih
fi
N
ih
RT
Fih 1 ∑ x i)1
N
Fjh ∑ j)1
xih
(dµ)T,p
N
∑ i)1
N
Fjh ∑ j)1
fi ln xih dn (A7)
N
∑ i)1
Fih 1 -
fi
dn
-
xih
N
Fjh ∑ j)1
N
N
fi ln xih dn ) RT∑(xih - fi) dn ∑ i)1 i)1
RT
N
∑ i)1
RT
N
∑ i)1
fi ln xih dn ) RT[
N
xih -
N
∑ i)1
RT
fi] dn ∑ i)1
N
fi ln xih dn ∑ i)1
fi ln xih dn ) -RT
(A8)
Similarly, it can be derived that for the stream on the lower pressure side, the change in chemical potential independent of pressure and temperature is N
N
(xil - fi) dn + RT∑fi ln xil dn ) ∑ i)1 i)1
(dµ2)T,p ≈ -RT N
-RT[
∑ i)1
N
xil -
∑ i)1
η)
dW
N
T0 T
RT0 ≈
∑ i)1
()
fi ln
xil
xih
N
∑ i)1
dn
fi ln
()
) RT0 ln(γ) dn
ln γ
xil
xih
(1)
Literature Cited
Neglecting the second-order term of dn gives
(dµ1)T,p ≈ RT
Multiplying T0/T converts it into a term useful for exergy calculation, the minimum work of separation. On the other hand, in the permeation process, dn moles of gas are reduced in pressure. The loss of pressure exergy due to this pressure reduction is equal to
in which ph and pl are the pressures on the higher pressure side and on the lower pressure side, respectively. The efficiency of the permeation process, which converts pressure exergy into composition exergy, is
fi 1 ∑ i)1
- RT
dn (A10)
dW ) RT0 ln(ph/pl) dn ) RT0 ln(γ) dn (A11)
N
dn
dn2
)
dni ln xih ) ∑ i)1
-
ih
fi
N
fi ln ∑ x i)1
ih
- RT
Fjh ∑ j)1
fi
N
Fih ln xih ∑ i)1
() xil
N
RT
N
dn
(Fih - dni) 1 ∑ x i)1
(dµ)T,p ) (dµ1)T,p + (dµ2)T,p ≈
N
dn
Fih,A+dA 1 ∑ x i)1
RT
brane element dA is the sum of these two terms. It is therefore
N
∑ i)1
(dµ1)T,p ≈ RT
N
Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997 3139
Agrawal, R.; Xu, J. Gas separation membrane cascades II. Twocompressor cascades. J. Membrane Sci. 1996a, 112, 115. Agrawal, R.; Xu, J. Gas separation membrane cascades utilizing limited number of compressors. AIChE J. 1996b, 42 (8), 2141. Hwang, S. T.; Thorman, J. M. The continuous Membrane column. AIChE J. 1980, 25 (4), 558. Laguntsov, N. I.; Gruzdev, E. B.; Kosykh, E. V.; Kozhevnickov, V. Y. The use of recycle permeator systems for gas mixture separation. J. Membrane Sci. 1992, 67, 15. Matson, S. T.; Lopez, J.; Quinn, J. A. Separation of gases with synthetic membranes. Chem. Eng. Sci. 1983, 38 (4), 503. Pfefferle, W. C. Diffusion purification of gases. U.S. Patent 3,144,313, 1964. Robeson, L. M. Correlation of separation factor vs permeance for polymeric membranes. J. Membrane Sci. 1991, 62, 165. Xu, J. Compressed permeate sweep membrane-separation process. U.S. Patent 5,252,219, 1993. Xu, J. High-pressure feed membrane separation process. U.S. Patent 5,282,969, 1994a. Xu, J. Low-pressure feed membrane separation process. U.S. Patent 5,306,427, 1994b. Xu, J.; Agrawal, R. Membrane separation process analysis and design strategies based on thermodynamic efficiency of permeation. Chem. Eng. Sci. 1996a, 51 (3), 365. Xu, J.; Agrawal, R. Gas separation membrane cascades I. Onecompressor cascades with minimal exergy losses due to mixing. J. Membrane Sci. 1996b, 112, 129.
N
fi ln xil dn ) ∑ i)1
Received for review October 2, 1996 Revised manuscript received March 11, 1997 Accepted March 12, 1997X
fi] dn + RT
N
RT
fi ln xil dn ∑ i)1
(A9)
IE960617I
The overall change of chemical potential due to composition change resulting from the permeation in mem-
X Abstract published in Advance ACS Abstracts, June 15, 1997.
