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A comparison of some recycle permeators for gas separations
Journal of Membrane Science, 24 (1985) 15-28 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
A COMPARISON SEPARATIONS
OF SOME RECYCLE PERMEATORS
15
FOR GAS
F.P. McCANDLESS Department (U.S.A.)
of Chemical Engineering,
Montana
State
University,
Bozeman,
MT 59717
(Received June 1, 1984; accepted in revised form March 4, 1985)
Summary The performances of several different recycle permeators are compared by calculating the enrichment, extend of separation, and area requirements for different overall stage cuts and recycle ratios, using the separation of air as an example. Membrane modules investigated included a two-unit series type permeator with permeate recycle, a continuous membrane column with high-pressure feed, and two single-unit countercurrent permeators with permeate recycle (both high-pressure and low-pressure feed). The use of Rony’s extent of separation in the analysis greatly aids in the visualization of limiting permeator performance at different recycle ratios. The composition of 0, in the highpressure product (reject) stream becomes zero at specific values of the stage cut which depend on recycle ratio for all but the low-pressure feed module. This would require an infinite permeator area, hence stage cuts greater than this limiting value are not possible. For the case of the low-pressure feed unit, the reject stream composition approaches a minimum finite value that depends on pressure ratio in the permeator. This limiting minimum composition would also be reached at specific values of stage cut that depend on recycle ratio and hence a stage cut greater than this limiting value cannot be obtained. Other permeator types (co-current, perfect mixed models) do not behave in this manner. It is concluded that the two-unit series type permeator gives the best separation and requires smaller membrane areas at high recycle ratios. Performance of the four permeators is not greatly different at lower recycle ratios.
Introduction and background
It is well known that the enrichment that can be achieved in a permeator can be greatly increased by recycling a part of the concentrated permeate stream back to the membrane module. However, there are many possible configurations for recycle permeators, and the effects of recycle ratio and stage cut on the performance of each permeator module will be different. This paper considers several recycle permeators that have been discussed by other authors [ 1,2] and compares their performances using Rony’s extent of separation, in addition to the traditional criteria of enrichment and membrane area requirements. The use of the extent of separation index in the analysis greatly aids in the visualization and understanding of limiting permeator performance at different recycle ratios. 0376-1388/85/$03.30
o 1985
Elsevier Science Publishers B.V.
16
The extent of separation in permeators Rony’s extent of separation for single permeator units (without recycle) has been discussed in detail and compared with other “separation factors” commonly used to characterize the separation that is achieved [ 3,4]. The application of this separation index to the overall separation achieved in a recycle permeator is no different than that for a single unit. However, for clarity and continuity in the development that follows, a brief discussion of this separation index is presented here. The general expression for the extent of separation is formulated in terms of the four segragation fractions Yij = nij/nP
(1)
where nij is the molar rate of component i leaving a permeator module in flow region j (i.e., one of the product streams), while $ is the molar rate of component i in the fresh feed to the permeator. The extent of separation, &,, is defined [ 51 as the absolute value of the binary separation matrix written in terms of the four segregation fractions .&,
s abs det
Some of the alternate equivalent forms for .$, in terms of stage cut, and the feed and product compositions are
6 =e(l-e) =
e-
YO
Lyl;;)]
( 1 XI
+(1-e)
=1-(1-e)($)
=eC
YO-XI
X1(1-X1)
(3)
l-X0 -
(
1 - XI
-e
(i-Z)
>
-1
(5)
1 (7)
In these equations, the stage cut, 0, is defined as the fraction of the fresh feed that is recovered as product concentrated in the more permeable species, while X1, X0 and YO are the mole fractions of the more permeable compound in the feed and product streams as shown in Fig. 1. In this paper the extent of separation refers to .$, while YO is termed “enrichment”. & varies between 0 (no separation) and 1 (perfect separation).
17
/al
TWO-UN/T
SEfffES
TYPE
PERMEATOR
1bJ CONT/NUOUS
MEMYRANE
COLUMN
I
1uo=(e)un
1 +;=ieIcwI)
YO
/c/ ONE-UN/T
Fig. 1.
RECYCLE,
HIGH
PRESSURE
FEfQ
cdl ONE-UhYT RECYCLE, /CMC ENR/CHER)
LOW PRfSSlJRE
FEED
Recycle membrane configurations investigated.
Recycle modules investigated Figure 1 shows schematic diagrams of the recycle permeators investigated. These include a two-unit series type permeation module (TUS), a continuous membrane column with high-pressure feed (CMC), and two single-unit countercurrent permeators with recycle of the permeate stream, both with highpressure feed (OUR) and Iow-pressure feed (ENRC). The latter configuration is sometimes called a “continuous membrane column enriching section”. With no recycle (a), (b) and (c) become simple countercurrent units. Simulation methods All models utilized the countercurrent, “plug-flow” equations developed by either Stern and co-workers or Blaisdell and Kammermeyer [6--81 and the reader should consult these excellent articles for the development of the differential equations needed to model the permeator modules. In general, the model equations consists of three coupled nonlinear ordinary differential equations which relate stream compositions and their rates to membrane area. These equations must be solved simultaneously using numerical methods. All countercurrent permeators require trial and error solution using backward integration starting at the high-pressure product (reject) end. Boundary conditions are different for each case, and the models for the twounit permeators require that the boundary conditions be adjusted between permeator units. For this study a comparison of $.,, YO, X0, and area was desired for the different permeators for different stage cuts, 13, and recycle ratios, RR. For all the permeators the recycle ratio is defined as
18
RR=
Rate of permeate Permeate
product
recycle rate
In addition to 0 and RR the properties of the membrane and operating conditions for the permeator must also be specified. These include: (a) the pressure difference across the membrane, PH - PL, and the pressure ratio, PH /PL ; (b) the ratio of the permeabilities for the different gases in the membrane material, P1/P2= cc* ; (c)the thickness of the membrane material, 6. With all of the above parameters specified the following procedure is used to calculate values of X0 and YO. (1) Assume a value of X0, the high pressure product (reject) composition. (2) Solve the differential equations to yield rates and compositions of all streams. When all material balances are satisfied within a specified small value, the assumed value of X0 is the correct value, and the other compositions, rates and parameters may then be calculated, including values for the areas. The equations were solved on a digital computer using a 4th-order Runge-Kutta numerical integration routine. A Wegstein routine was used to estimate new values of X0 to try in the calculations based on previous trials. The example chosen was Stern’s problem [ 1,7] on the separation of air using the following parameters: XI = 0.209
LHI = 1.0X lo6 cm3 (STP)/sec PH = 380 cmHg R=PH/PL=~ PO2 = 5.0 x 10-g (y* = p**/p% 6 = 2.54X
cm3 (STP)-cm set-cm2 -cmHg
= 10 10m3 cm
These calculations yielded values of X0, YO, stream rates, !+,, and areas as a function of 0 for various assumed values of RR.
Results and discussion The results of the calculations are summarized and discussed below. However, before presenting the results and discussion it should be pointed out that, for each membrane configuration, every RR and 0 combination would result in a different permeator requiring a specific membrane area. Thus, we are not discussing merely 4 permeators of different design, but rather, many permeators of each configuration, each designed with the correct membrane area to result in a specific stage cut for the assumed values
19 of RR, 8, feed rate and other operating comparing the performance of a “family” families of other design.
conditions shown above. We are of permeators of one design with
Enrichment Figure 2 presents the enrichment, YO (permeate product composition) and X0 (reject product composition) as functions of 0 for different recycle ratios. Plots of this nature were first presented by Stern and co-workers [ 21. As can be seen from Fig. 2, YO -+ 1 at values of 0 < 0.3 for all permeators when RR is made large. YO goes through a maximum for the TUS, OUR, and ENRC configurations while X0 decreases steadily for all cases. YO does not exhibit a maximum for the CMC, however, but rather decreases from the maximum at 0 = 0. Interestingly, for the TUS, CMC, and OUR configurations, at higher RR, YO is quite high (> 0.9) and does not vary much over a relatively wide range of 6. Since the permeate product stream is directly proportional to f3, operation at the higher 8s would probably be preferred under these conditions. It is evident that the single permeator unit with low-pressure feed (ENRC) cannot operate without recycle. Also, because of material balance considerations, there is a minimum value of 0 (which depends on RR) below which
0
2
e
4
lb)
(Cl
e
Fig. 2. X0 and YO as functions series; (b) continuous membrane low-pressure
feed.
Cd)
of stage column;
e
8
cut for different recycle (c) one-unit high-pressure
ratios; feed;
(a) two-unit (d) one-unit
20
the permeator cannot operate. In addition, for the feed, membrane, and operating conditions used for this study (XI = 0.209, (Y* = 10, &J /& = 5) the absolute minimum value of X0 that can be obtained is X0 = 0.0456, otherwise internal material balances cannot be satisfied. If the reject stream composition were below this value, permeation would be in the other direction for this configuration. The numerical calculations indicate that this minimum of X0 (X0 = 0.0456) is reached at specific values of 6 which depend on RR and so this value of 0 represents the maximum stage cut that can be attained for that RR. The other three membrane configurations do not have this limitation on X0, and XOmi, = 0 at all values of RR for the TUS, CMC and OUR modules. However, the numerical calculations indicate that X0 would become zero at specific values of B which depend on RR. The attainment of X0 = 0 would require an infinite area and thus, for a given RR it is impossible to obtain a stage cut greater than this B,,. 8,, = 0.871 when RR = 0 for the TUS, CMC, and OUR modules and decreases, approaching emax = 0.209 as RR becomes large. Because of the limitation on the ENRC configuration noted above, 1.0 > eMaX> 0.1712. However, the behavior of the ENRC is different since XO,h is reached at finite areas. Figure 3 shows possible values of X0 and YO vs. 8 combinations for the different membrane configurations. Depending on RR and 0, values of YO will lie within the area bounded by the lines representing YO when RR = 0 (ENRC cannot function when RR = 0); YO when X0 = XO,h, and YO = 1 (YO + 1 as RR -+ =), between 0 = 0 and 6 = 0.209 (0 = O-O.1712 for ENRC). I,
TUS, .8 - YO WILL
X0 W/U
CMC, OUR
YO WHEN X0 = 0
YO AS p,/p,
A.5 /NCREASED
BE NV
(a)
Fig. 3. X0 and YO as functions of stage cut showing regions where the product compositions are possible for the different permeators; (a) two-unit series; continuous membrane column, one-unit high-pressure feed; (b) one-unit low-pressure feed.
21
For X0, the boundary limits are the lines representing X0 for RR = 0 (XI for ENRC since this permeator cannot operate with zero recycle), X0 as RR + m (where YO = l), and the line representing XO,h. Extent of separation in the recycle permeators It is much easier to visualize the limiting behavior of the different permeators when the separation is characterized in terms of the extent of sep aration, .$,. This is done in Figs. 4 and 5. As can be seen from Fig. 4, &, goes through a relative maximum for all permeators, and this maximum occurs at smaller values of 0 as RR is increased. Because XO,h > 0.0456 for the ENRC, .$, -+ 0.819 (where X0 = 0.0456, YO = 1) at 0 = 0.1712 as RR gets large, but for the other three configurations gP + 1 at 8 = 0.209. For a given RR, tp,max can be viewed as the best compromise (at least from a separation standpoint) between the quality and quantity of the two product streams. In addition, since &$ goes through a relative maximum for each RR, the same & may be obtained at two different values of f3 (subject to the restriction on 8 discussed above). This, of course, can happen because of the nature of Ep, that is, it is invariant to a permutation in subscripts i and
1
Cdl
Fig. 4. Extent of separation as a function of stage cut for different recycle ratios; (a) twounit series; (b) continuous membrane column; (c) one-unit high-pressure feed; (d) oneunit low-pressure feed.
22
in eqn. (1) and different combination of product stream composition and rates results in an equivalent separation. A close examination of the results presented in Fig. 4, together with the equations for ,$, in terms of 0 and stream compositions, shows that .$-, approaches limiting values on either side of emax corresponding to points on two straight lines in the tp--e plane. These limiting lines for ,$, are given by eqns. (6) and (7) when X0 = XO,h and YO = 1. These equations are:
j
for TUS, CMC, and OUR.
,$,=9/XIforYO=landB<0.209
(9)
& = (1 - 8 )/(l - XI) for X0 = 0 and B > 0.209
(16)
for ENRC
$, = e/XI for YO = 1 and I3< 0.1712
&, = (1 - e){(XI
- O.O456)/[XI(l-
(11)
XI)]}
for X0 = 0.0456 and
t9> 0.1712
(12)
Plots of the above equations are shown in Fig. 5. The areas bounded by these lines are the regions of possible fP vs. 0 for the different permeators. As discussed above, the right-hand limit branch represents possible values of .$, vs. 0 when X0 = XO,h everywhere, while the left-hand branch gives &, when YO = 1, with the branches meeting at the absolute maximum possible value of gr, for that membrane configuration (i.e., at X0 = XO,h, YO = 1). The two lines in Fig. 5 represent &, vs. 0 when the limiting compositions of the two product streams are reached, that is, when X0 = XO,h and
Fig. 5. Extent of separation as a function of stage cut showing regions where tp is possible for the different permeators; (a) two-unit series, continuous membrane column, one-unit high-pressure feed; (b) one-unit low-pressure feed.
23
YO = 1. As discussed above, XO,h would be reached at specific values of 0 for each RR and this 8 cannot be exceeded. Although this emax cannot be exceeded, it can be approached as closely as desired by making the membrane area large enough (ENRC requires a finite area to reach XO,h), hence the limiting values of & when 0 is increased are on the righthand limit line where X0 = XO,h everywhere. This value of emax decreases as RR is increased and hence, as RR is increased, the limiting value of 6 “slides up” the right-hand branch and only values of 0 less than emax can be “reached”. On the other hand, as RR is increased at values of B < 0.209 for TUS, CMC and OUR (f3 < 0.1717 for ENRC), the .&, vs. B curve only approaches the limiting left-hand branch, that is .$, -+ 0 /XI where YO = 1. and the absolute maximum Finally, as RR + m, Y0-t landXO=XO,i,
Fig, 6. Total area required as a function of stage cut for the different permeators; (a) twounit series; (b) continuous membrane column; (c) one-unit high-pressure feed; (d) oneunit low-pressure feed.
24
can be obtained at 8 = 0.209 for the TUS, CMC and OUR modules and at 0 = 0.1712 for the ENRC module. [p
Area requirements The total area requirements for the different membrane modules as functions of B and RR are compared in Fig. 6. For the TUS, CMC, and OUR configurations, total areas differ by about 2~37% at the same 0 and RR, and in general, the areas decrease in the order TUS > OUR > CMC at the same 8. For the ENRC module, the area requirements are significantly lower than for the other configurations at RR = 1 and 10, but about the same as the other three at RR = 100 at the same B. Figure 7 shows how the two areas (as shown in Fig. 1) vary with 0 for the
Fig. 7. Area requirements for individual units, two-unit permeators; (b) continuous membrane column.
(a) two-unit series;
25
two-unit modules. Interestingly, area 1 is much less than area 2 for the TUS permeator at high values of RR but the two areas are of the same order of magnitude at lower values of RR. For the CMC permeator area 1 goes through a maximum with increasing B for lower values of RR. It is interesting to compare gP as a function of the required membrane area for the different permeators since the area vs. 0 curves do not indicate permeator performance. This is done in Fig. 8. As can be seen, for the TUS, CMC, and OUR configurations, the areas required to reach ,$,max at each RR, are not very different, especially at lower RR. On the other hand, the area required for OUR is about 29% higher than those for the other two at RR = 100. However, in general t;p,maxis higher for the TUS module and this maximum is reached at smaller areas as RR is increased. As a result, it appears that, in general, the TUS unit gives the best performance in terms of required area. &-, is lower for the ENRC configuration because of material balance considerations.
Fig. 8. Extent of separation as a function of total area requirements permeators; (a) two-unit series; (b) continuous membrane column; pressure feed; (d) one-unit low-pressure feed.
for the different (c) one-unit high-
General discussion
All of the permeator configurations that were investigated are capable of producing a greatly enriched permeate product stream where YO -+ 1 at B = 0.2 as RR is made large. However, there are subtle differences in the performance of the different permeators as RR and 0 are varied, which are important in overall performance. These differences can best be analyzed and evaluated in terms of gp. The numerical calculations using very small integration increments (dX =
26
lo-‘) in the 4th-order Runge-Kutta numerical integration method, and using the double-precision mode on the computer, indicate that X0 would reach an absolute minimum value at specific values of 0 which depend on membrane configuration and RR. It would be impossible to obtain stage cuts above this emax because this would either require an infinite area in the case of the TUS, CMC and OUR modules, or, in the case of the ENRC unit, imply that material balances cannot be satisfied. This result was surprising because previous investigators (including the present author) apparently tacitly assumed that XO,h could only be reached with 0 + 1. Upon reflection, this behavior should be expected for the countercurrent permeators but it is a characteristic of the countercurrent units only; X.0 will remain finite for cocurrent and perfect mixed modules. This is illustrated in Table 1, which gives X0 for a one-unit cocurrent permeator with recycle. The ENRC permeator cannot operate at some stage cuts and RR because of material balance limitations. XO,h for this configuration depends weakly on 01* but is a strong function of the pressure ratio. The basis for all calculations in this paper was R = PH /PL = 5 and (Y* = 10. XO,h decreases quite rapidly with increasing R but changes very little with (Y* as shown in Table 2. Thus, as the pressure ratio is increased, XO,h + 0 and the limit lines shown in Figs. 3b and 5b will change as shown. At high enough R the ENRC ‘TABLE 1 Reject composition
as a function of RR for a cocurrent permeator
RR
8
x0
0 10 10 10 10
0.9999 0.9 0.99 0.999 0.9999
0.04561 0.014403 0.012969 0.012841 0.012827
TABLE XOmi,
10 10 10 10 10 10 1000
2 as a function of pressure ratio and 01* for ENRC
2 5 10 20 100 5
0.110644 0.045601 0.023015 0.011560 0.002320 0 0.0418
0.111 0.171 0.191 0.200 0.207 0.209 0.1745
0.529 0.819 0.911 0.956 0.991 1.0 0.835
27
configuration will have “available” other recycle permeator modules.
essentially the same tP-O
area as the
List of symbols LHI LHO LLO nij
nf PH
R RR XI x0
xomin yij YO
molar rate of fresh feed to the recycle permeator molar rate of high-pressure (reject) product from the permeator molar rate of low-pressure (permeate) product from the permeator moles of compound i. leaving in flow region j moles of compound i in fresh feed pressure on high-pressure side of membrane pressure on low-pressure side of membrane permeability coefficients of O2 and N2 pressure ratio, = PH /PL recycle ratio = rate of permeate recycle/LLO mole fraction of O2 in fresh feed mole fraction of O2 in the reject product stream minimum value of X0 that can be obtained in a permeator module segregation fraction of compound i in flow region j enrichment, mole fraction of 0, in low-pressure (permeate) product stream
Greek letters a* ideal separation factor, = P”2/f12 membrane thickness 6 extent of separation for a permeator EP e overall stage cut, = LLO/LHI e max maximum value of 0 that can be obtained in a permeator module e min minimum value of 6’ in ENRC; below this 0 material balances cannot be satisfied
References 1 2 3 4 5
S.L. Matson, J. Lopez and J.A. Quinn, Separation of gases with synthetic membranes. Review article No. 13, Chem. Eng. Sci., 38 (1983) 503-524. S.A. Stern, J.E. Perrin and E.J. Naimon, Recycle and multimembrane permeators for gas separations, J. Membrane Sci., 20 (1984) 25-43. F.P. McCandless, The extent of separation in single permeation stages, J. Membrane Sci., 17 (1984) 323-328. F.P. McCandless, Separation factors in permeation stages, J. Membrane Sci., 19 (1984) 101-113. P.R. Rony, The extent of separation: On the unification of the field of chemical separations, Recent Advances in Separation Techniques, AIChE Symposium Series No. 120, 68 (1972) 89-104.
28 6 S.A. Stern and SC. Wang, Countercurrent and cocurrent gas separation in a permeation stage. Comparison of computation methods, J. Membrane Sci., 4 (1978) 141-148. 7 W.P. Walawender and S.A. Stern, Analysis of membrane separation parameters. II. Countercurrent and cocurrent flow in a single permeation stage, Sep. Sci., 7 (1972) 553-584. 8 CT. Blaisdell and K. Kammermeyer, Countercurrent and cocurrent gas separation, Chem. Eng. Sci., 28 (1973) 1249-1255.
A COMPARISON SEPARATIONS
OF SOME RECYCLE PERMEATORS
15
FOR GAS
F.P. McCANDLESS Department (U.S.A.)
of Chemical Engineering,
Montana
State
University,
Bozeman,
MT 59717
(Received June 1, 1984; accepted in revised form March 4, 1985)
Summary The performances of several different recycle permeators are compared by calculating the enrichment, extend of separation, and area requirements for different overall stage cuts and recycle ratios, using the separation of air as an example. Membrane modules investigated included a two-unit series type permeator with permeate recycle, a continuous membrane column with high-pressure feed, and two single-unit countercurrent permeators with permeate recycle (both high-pressure and low-pressure feed). The use of Rony’s extent of separation in the analysis greatly aids in the visualization of limiting permeator performance at different recycle ratios. The composition of 0, in the highpressure product (reject) stream becomes zero at specific values of the stage cut which depend on recycle ratio for all but the low-pressure feed module. This would require an infinite permeator area, hence stage cuts greater than this limiting value are not possible. For the case of the low-pressure feed unit, the reject stream composition approaches a minimum finite value that depends on pressure ratio in the permeator. This limiting minimum composition would also be reached at specific values of stage cut that depend on recycle ratio and hence a stage cut greater than this limiting value cannot be obtained. Other permeator types (co-current, perfect mixed models) do not behave in this manner. It is concluded that the two-unit series type permeator gives the best separation and requires smaller membrane areas at high recycle ratios. Performance of the four permeators is not greatly different at lower recycle ratios.
Introduction and background
It is well known that the enrichment that can be achieved in a permeator can be greatly increased by recycling a part of the concentrated permeate stream back to the membrane module. However, there are many possible configurations for recycle permeators, and the effects of recycle ratio and stage cut on the performance of each permeator module will be different. This paper considers several recycle permeators that have been discussed by other authors [ 1,2] and compares their performances using Rony’s extent of separation, in addition to the traditional criteria of enrichment and membrane area requirements. The use of the extent of separation index in the analysis greatly aids in the visualization and understanding of limiting permeator performance at different recycle ratios. 0376-1388/85/$03.30
o 1985
Elsevier Science Publishers B.V.
16
The extent of separation in permeators Rony’s extent of separation for single permeator units (without recycle) has been discussed in detail and compared with other “separation factors” commonly used to characterize the separation that is achieved [ 3,4]. The application of this separation index to the overall separation achieved in a recycle permeator is no different than that for a single unit. However, for clarity and continuity in the development that follows, a brief discussion of this separation index is presented here. The general expression for the extent of separation is formulated in terms of the four segragation fractions Yij = nij/nP
(1)
where nij is the molar rate of component i leaving a permeator module in flow region j (i.e., one of the product streams), while $ is the molar rate of component i in the fresh feed to the permeator. The extent of separation, &,, is defined [ 51 as the absolute value of the binary separation matrix written in terms of the four segregation fractions .&,
s abs det
Some of the alternate equivalent forms for .$, in terms of stage cut, and the feed and product compositions are
6 =e(l-e) =
e-
YO
Lyl;;)]
( 1 XI
+(1-e)
=1-(1-e)($)
=eC
YO-XI
X1(1-X1)
(3)
l-X0 -
(
1 - XI
-e
(i-Z)
>
-1
(5)
1 (7)
In these equations, the stage cut, 0, is defined as the fraction of the fresh feed that is recovered as product concentrated in the more permeable species, while X1, X0 and YO are the mole fractions of the more permeable compound in the feed and product streams as shown in Fig. 1. In this paper the extent of separation refers to .$, while YO is termed “enrichment”. & varies between 0 (no separation) and 1 (perfect separation).
17
/al
TWO-UN/T
SEfffES
TYPE
PERMEATOR
1bJ CONT/NUOUS
MEMYRANE
COLUMN
I
1uo=(e)un
1 +;=ieIcwI)
YO
/c/ ONE-UN/T
Fig. 1.
RECYCLE,
HIGH
PRESSURE
FEfQ
cdl ONE-UhYT RECYCLE, /CMC ENR/CHER)
LOW PRfSSlJRE
FEED
Recycle membrane configurations investigated.
Recycle modules investigated Figure 1 shows schematic diagrams of the recycle permeators investigated. These include a two-unit series type permeation module (TUS), a continuous membrane column with high-pressure feed (CMC), and two single-unit countercurrent permeators with recycle of the permeate stream, both with highpressure feed (OUR) and Iow-pressure feed (ENRC). The latter configuration is sometimes called a “continuous membrane column enriching section”. With no recycle (a), (b) and (c) become simple countercurrent units. Simulation methods All models utilized the countercurrent, “plug-flow” equations developed by either Stern and co-workers or Blaisdell and Kammermeyer [6--81 and the reader should consult these excellent articles for the development of the differential equations needed to model the permeator modules. In general, the model equations consists of three coupled nonlinear ordinary differential equations which relate stream compositions and their rates to membrane area. These equations must be solved simultaneously using numerical methods. All countercurrent permeators require trial and error solution using backward integration starting at the high-pressure product (reject) end. Boundary conditions are different for each case, and the models for the twounit permeators require that the boundary conditions be adjusted between permeator units. For this study a comparison of $.,, YO, X0, and area was desired for the different permeators for different stage cuts, 13, and recycle ratios, RR. For all the permeators the recycle ratio is defined as
18
RR=
Rate of permeate Permeate
product
recycle rate
In addition to 0 and RR the properties of the membrane and operating conditions for the permeator must also be specified. These include: (a) the pressure difference across the membrane, PH - PL, and the pressure ratio, PH /PL ; (b) the ratio of the permeabilities for the different gases in the membrane material, P1/P2= cc* ; (c)the thickness of the membrane material, 6. With all of the above parameters specified the following procedure is used to calculate values of X0 and YO. (1) Assume a value of X0, the high pressure product (reject) composition. (2) Solve the differential equations to yield rates and compositions of all streams. When all material balances are satisfied within a specified small value, the assumed value of X0 is the correct value, and the other compositions, rates and parameters may then be calculated, including values for the areas. The equations were solved on a digital computer using a 4th-order Runge-Kutta numerical integration routine. A Wegstein routine was used to estimate new values of X0 to try in the calculations based on previous trials. The example chosen was Stern’s problem [ 1,7] on the separation of air using the following parameters: XI = 0.209
LHI = 1.0X lo6 cm3 (STP)/sec PH = 380 cmHg R=PH/PL=~ PO2 = 5.0 x 10-g (y* = p**/p% 6 = 2.54X
cm3 (STP)-cm set-cm2 -cmHg
= 10 10m3 cm
These calculations yielded values of X0, YO, stream rates, !+,, and areas as a function of 0 for various assumed values of RR.
Results and discussion The results of the calculations are summarized and discussed below. However, before presenting the results and discussion it should be pointed out that, for each membrane configuration, every RR and 0 combination would result in a different permeator requiring a specific membrane area. Thus, we are not discussing merely 4 permeators of different design, but rather, many permeators of each configuration, each designed with the correct membrane area to result in a specific stage cut for the assumed values
19 of RR, 8, feed rate and other operating comparing the performance of a “family” families of other design.
conditions shown above. We are of permeators of one design with
Enrichment Figure 2 presents the enrichment, YO (permeate product composition) and X0 (reject product composition) as functions of 0 for different recycle ratios. Plots of this nature were first presented by Stern and co-workers [ 21. As can be seen from Fig. 2, YO -+ 1 at values of 0 < 0.3 for all permeators when RR is made large. YO goes through a maximum for the TUS, OUR, and ENRC configurations while X0 decreases steadily for all cases. YO does not exhibit a maximum for the CMC, however, but rather decreases from the maximum at 0 = 0. Interestingly, for the TUS, CMC, and OUR configurations, at higher RR, YO is quite high (> 0.9) and does not vary much over a relatively wide range of 6. Since the permeate product stream is directly proportional to f3, operation at the higher 8s would probably be preferred under these conditions. It is evident that the single permeator unit with low-pressure feed (ENRC) cannot operate without recycle. Also, because of material balance considerations, there is a minimum value of 0 (which depends on RR) below which
0
2
e
4
lb)
(Cl
e
Fig. 2. X0 and YO as functions series; (b) continuous membrane low-pressure
feed.
Cd)
of stage column;
e
8
cut for different recycle (c) one-unit high-pressure
ratios; feed;
(a) two-unit (d) one-unit
20
the permeator cannot operate. In addition, for the feed, membrane, and operating conditions used for this study (XI = 0.209, (Y* = 10, &J /& = 5) the absolute minimum value of X0 that can be obtained is X0 = 0.0456, otherwise internal material balances cannot be satisfied. If the reject stream composition were below this value, permeation would be in the other direction for this configuration. The numerical calculations indicate that this minimum of X0 (X0 = 0.0456) is reached at specific values of 6 which depend on RR and so this value of 0 represents the maximum stage cut that can be attained for that RR. The other three membrane configurations do not have this limitation on X0, and XOmi, = 0 at all values of RR for the TUS, CMC and OUR modules. However, the numerical calculations indicate that X0 would become zero at specific values of B which depend on RR. The attainment of X0 = 0 would require an infinite area and thus, for a given RR it is impossible to obtain a stage cut greater than this B,,. 8,, = 0.871 when RR = 0 for the TUS, CMC, and OUR modules and decreases, approaching emax = 0.209 as RR becomes large. Because of the limitation on the ENRC configuration noted above, 1.0 > eMaX> 0.1712. However, the behavior of the ENRC is different since XO,h is reached at finite areas. Figure 3 shows possible values of X0 and YO vs. 8 combinations for the different membrane configurations. Depending on RR and 0, values of YO will lie within the area bounded by the lines representing YO when RR = 0 (ENRC cannot function when RR = 0); YO when X0 = XO,h, and YO = 1 (YO + 1 as RR -+ =), between 0 = 0 and 6 = 0.209 (0 = O-O.1712 for ENRC). I,
TUS, .8 - YO WILL
X0 W/U
CMC, OUR
YO WHEN X0 = 0
YO AS p,/p,
A.5 /NCREASED
BE NV
(a)
Fig. 3. X0 and YO as functions of stage cut showing regions where the product compositions are possible for the different permeators; (a) two-unit series; continuous membrane column, one-unit high-pressure feed; (b) one-unit low-pressure feed.
21
For X0, the boundary limits are the lines representing X0 for RR = 0 (XI for ENRC since this permeator cannot operate with zero recycle), X0 as RR + m (where YO = l), and the line representing XO,h. Extent of separation in the recycle permeators It is much easier to visualize the limiting behavior of the different permeators when the separation is characterized in terms of the extent of sep aration, .$,. This is done in Figs. 4 and 5. As can be seen from Fig. 4, &, goes through a relative maximum for all permeators, and this maximum occurs at smaller values of 0 as RR is increased. Because XO,h > 0.0456 for the ENRC, .$, -+ 0.819 (where X0 = 0.0456, YO = 1) at 0 = 0.1712 as RR gets large, but for the other three configurations gP + 1 at 8 = 0.209. For a given RR, tp,max can be viewed as the best compromise (at least from a separation standpoint) between the quality and quantity of the two product streams. In addition, since &$ goes through a relative maximum for each RR, the same & may be obtained at two different values of f3 (subject to the restriction on 8 discussed above). This, of course, can happen because of the nature of Ep, that is, it is invariant to a permutation in subscripts i and
1
Cdl
Fig. 4. Extent of separation as a function of stage cut for different recycle ratios; (a) twounit series; (b) continuous membrane column; (c) one-unit high-pressure feed; (d) oneunit low-pressure feed.
22
in eqn. (1) and different combination of product stream composition and rates results in an equivalent separation. A close examination of the results presented in Fig. 4, together with the equations for ,$, in terms of 0 and stream compositions, shows that .$-, approaches limiting values on either side of emax corresponding to points on two straight lines in the tp--e plane. These limiting lines for ,$, are given by eqns. (6) and (7) when X0 = XO,h and YO = 1. These equations are:
j
for TUS, CMC, and OUR.
,$,=9/XIforYO=landB<0.209
(9)
& = (1 - 8 )/(l - XI) for X0 = 0 and B > 0.209
(16)
for ENRC
$, = e/XI for YO = 1 and I3< 0.1712
&, = (1 - e){(XI
- O.O456)/[XI(l-
(11)
XI)]}
for X0 = 0.0456 and
t9> 0.1712
(12)
Plots of the above equations are shown in Fig. 5. The areas bounded by these lines are the regions of possible fP vs. 0 for the different permeators. As discussed above, the right-hand limit branch represents possible values of .$, vs. 0 when X0 = XO,h everywhere, while the left-hand branch gives &, when YO = 1, with the branches meeting at the absolute maximum possible value of gr, for that membrane configuration (i.e., at X0 = XO,h, YO = 1). The two lines in Fig. 5 represent &, vs. 0 when the limiting compositions of the two product streams are reached, that is, when X0 = XO,h and
Fig. 5. Extent of separation as a function of stage cut showing regions where tp is possible for the different permeators; (a) two-unit series, continuous membrane column, one-unit high-pressure feed; (b) one-unit low-pressure feed.
23
YO = 1. As discussed above, XO,h would be reached at specific values of 0 for each RR and this 8 cannot be exceeded. Although this emax cannot be exceeded, it can be approached as closely as desired by making the membrane area large enough (ENRC requires a finite area to reach XO,h), hence the limiting values of & when 0 is increased are on the righthand limit line where X0 = XO,h everywhere. This value of emax decreases as RR is increased and hence, as RR is increased, the limiting value of 6 “slides up” the right-hand branch and only values of 0 less than emax can be “reached”. On the other hand, as RR is increased at values of B < 0.209 for TUS, CMC and OUR (f3 < 0.1717 for ENRC), the .&, vs. B curve only approaches the limiting left-hand branch, that is .$, -+ 0 /XI where YO = 1. and the absolute maximum Finally, as RR + m, Y0-t landXO=XO,i,
Fig, 6. Total area required as a function of stage cut for the different permeators; (a) twounit series; (b) continuous membrane column; (c) one-unit high-pressure feed; (d) oneunit low-pressure feed.
24
can be obtained at 8 = 0.209 for the TUS, CMC and OUR modules and at 0 = 0.1712 for the ENRC module. [p
Area requirements The total area requirements for the different membrane modules as functions of B and RR are compared in Fig. 6. For the TUS, CMC, and OUR configurations, total areas differ by about 2~37% at the same 0 and RR, and in general, the areas decrease in the order TUS > OUR > CMC at the same 8. For the ENRC module, the area requirements are significantly lower than for the other configurations at RR = 1 and 10, but about the same as the other three at RR = 100 at the same B. Figure 7 shows how the two areas (as shown in Fig. 1) vary with 0 for the
Fig. 7. Area requirements for individual units, two-unit permeators; (b) continuous membrane column.
(a) two-unit series;
25
two-unit modules. Interestingly, area 1 is much less than area 2 for the TUS permeator at high values of RR but the two areas are of the same order of magnitude at lower values of RR. For the CMC permeator area 1 goes through a maximum with increasing B for lower values of RR. It is interesting to compare gP as a function of the required membrane area for the different permeators since the area vs. 0 curves do not indicate permeator performance. This is done in Fig. 8. As can be seen, for the TUS, CMC, and OUR configurations, the areas required to reach ,$,max at each RR, are not very different, especially at lower RR. On the other hand, the area required for OUR is about 29% higher than those for the other two at RR = 100. However, in general t;p,maxis higher for the TUS module and this maximum is reached at smaller areas as RR is increased. As a result, it appears that, in general, the TUS unit gives the best performance in terms of required area. &-, is lower for the ENRC configuration because of material balance considerations.
Fig. 8. Extent of separation as a function of total area requirements permeators; (a) two-unit series; (b) continuous membrane column; pressure feed; (d) one-unit low-pressure feed.
for the different (c) one-unit high-
General discussion
All of the permeator configurations that were investigated are capable of producing a greatly enriched permeate product stream where YO -+ 1 at B = 0.2 as RR is made large. However, there are subtle differences in the performance of the different permeators as RR and 0 are varied, which are important in overall performance. These differences can best be analyzed and evaluated in terms of gp. The numerical calculations using very small integration increments (dX =
26
lo-‘) in the 4th-order Runge-Kutta numerical integration method, and using the double-precision mode on the computer, indicate that X0 would reach an absolute minimum value at specific values of 0 which depend on membrane configuration and RR. It would be impossible to obtain stage cuts above this emax because this would either require an infinite area in the case of the TUS, CMC and OUR modules, or, in the case of the ENRC unit, imply that material balances cannot be satisfied. This result was surprising because previous investigators (including the present author) apparently tacitly assumed that XO,h could only be reached with 0 + 1. Upon reflection, this behavior should be expected for the countercurrent permeators but it is a characteristic of the countercurrent units only; X.0 will remain finite for cocurrent and perfect mixed modules. This is illustrated in Table 1, which gives X0 for a one-unit cocurrent permeator with recycle. The ENRC permeator cannot operate at some stage cuts and RR because of material balance limitations. XO,h for this configuration depends weakly on 01* but is a strong function of the pressure ratio. The basis for all calculations in this paper was R = PH /PL = 5 and (Y* = 10. XO,h decreases quite rapidly with increasing R but changes very little with (Y* as shown in Table 2. Thus, as the pressure ratio is increased, XO,h + 0 and the limit lines shown in Figs. 3b and 5b will change as shown. At high enough R the ENRC ‘TABLE 1 Reject composition
as a function of RR for a cocurrent permeator
RR
8
x0
0 10 10 10 10
0.9999 0.9 0.99 0.999 0.9999
0.04561 0.014403 0.012969 0.012841 0.012827
TABLE XOmi,
10 10 10 10 10 10 1000
2 as a function of pressure ratio and 01* for ENRC
2 5 10 20 100 5
0.110644 0.045601 0.023015 0.011560 0.002320 0 0.0418
0.111 0.171 0.191 0.200 0.207 0.209 0.1745
0.529 0.819 0.911 0.956 0.991 1.0 0.835
27
configuration will have “available” other recycle permeator modules.
essentially the same tP-O
area as the
List of symbols LHI LHO LLO nij
nf PH
R RR XI x0
xomin yij YO
molar rate of fresh feed to the recycle permeator molar rate of high-pressure (reject) product from the permeator molar rate of low-pressure (permeate) product from the permeator moles of compound i. leaving in flow region j moles of compound i in fresh feed pressure on high-pressure side of membrane pressure on low-pressure side of membrane permeability coefficients of O2 and N2 pressure ratio, = PH /PL recycle ratio = rate of permeate recycle/LLO mole fraction of O2 in fresh feed mole fraction of O2 in the reject product stream minimum value of X0 that can be obtained in a permeator module segregation fraction of compound i in flow region j enrichment, mole fraction of 0, in low-pressure (permeate) product stream
Greek letters a* ideal separation factor, = P”2/f12 membrane thickness 6 extent of separation for a permeator EP e overall stage cut, = LLO/LHI e max maximum value of 0 that can be obtained in a permeator module e min minimum value of 6’ in ENRC; below this 0 material balances cannot be satisfied
References 1 2 3 4 5
S.L. Matson, J. Lopez and J.A. Quinn, Separation of gases with synthetic membranes. Review article No. 13, Chem. Eng. Sci., 38 (1983) 503-524. S.A. Stern, J.E. Perrin and E.J. Naimon, Recycle and multimembrane permeators for gas separations, J. Membrane Sci., 20 (1984) 25-43. F.P. McCandless, The extent of separation in single permeation stages, J. Membrane Sci., 17 (1984) 323-328. F.P. McCandless, Separation factors in permeation stages, J. Membrane Sci., 19 (1984) 101-113. P.R. Rony, The extent of separation: On the unification of the field of chemical separations, Recent Advances in Separation Techniques, AIChE Symposium Series No. 120, 68 (1972) 89-104.
28 6 S.A. Stern and SC. Wang, Countercurrent and cocurrent gas separation in a permeation stage. Comparison of computation methods, J. Membrane Sci., 4 (1978) 141-148. 7 W.P. Walawender and S.A. Stern, Analysis of membrane separation parameters. II. Countercurrent and cocurrent flow in a single permeation stage, Sep. Sci., 7 (1972) 553-584. 8 CT. Blaisdell and K. Kammermeyer, Countercurrent and cocurrent gas separation, Chem. Eng. Sci., 28 (1973) 1249-1255.