Separation factors in permeation stages

Journal of Membrane Science, 19 (1984) 101-113 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

101

SEPARATION FACTORS IN PERMEATION STAGES F.P. McCANDLESS Montana State University, Department of Chemical Engineering, Bozeman, MT 59717 (U.S.A.) (Received June 4,1983; accepted in revised form November 23, 1983)

Summary A comparison is made of separation factors commonly used to characterize separations in single-entry barrier-permeation stages. These include the overall, heads, tails and Sandell’s separation factors together with Rony’s extent of separation. Equations are developed which show how the various separation indexes are related to stage cut, recovery and losa terms, and to each other. Calculations were made to compare the various separation indexes to determine which are suitable to characterize the “goodness” of separation in ideal countercurrent, cross-flow, cocurrent, and perfect mixed permeation stages. When compared at the same stage cut all of the separation indexes can be used as a qualitative measure of separation. However, only Rony’s extent of separation appears to be generally applicable when comparing separations that are based on different stage cuts, or product stream rates. In all cases the extent of separation index gives a qualitative, and presumably a quantitative, measure of the goodness of separation.

Introduction and background

In a previous paper [l] Rony’s extent of separation, tP, was used to compare the “goodness” of separation in countercurrent, cross-flow, cocurrent, [Z--4].

[S] , in his pioneering article on the extent of separation, compared various separation factors for two component-two region separators in general and showed that none of the separation factors in common use at that time were consistent in characterizing separations. He defined a new separation index, the extent of separation, which he showed to be superior to the other separation factors. However, many different separation indexes are still commonly used to characterize separations in permeation stages and it is interesting to compare them to determine their usefulness for this purpose. The present paper shows how ,$r is related to stage cut, recovery and loss terms, and to the overall, heads, tails, and Sandell’s separation factors. In addition, judgments are made as to how well the various indexes describe the separations that take place in permeation stages with different ideal flow patterns. 03’76-7388/84/$03.00

0 1984 Elsevier Science Publishers B.V.

102

The extent of separation for single permeation stages For a binary mixture, the general equation for the extent of separation is formulated in terms of segregation fractions: Yij = &j/n!

(1)

For the ideal permeation stages shown in Fig. 1, nij is the molar rate of component i leaving in flow region i (i.e., one of the product streams), while rzp is the molar feed rate of component i to the stage. Thus, Yij represents the fraction of component i present in the feed stream that leaves (or is recovered) in one of the product streams (flow region i). Yij “recovery” is v or a “loss”, i edepending w on how e it d .

(01 COUNTERCURRENT

(bl

(c)

(dl PERFEE;

CROSS-FLOW NO MIXING

COCURREN’I

M IIXING

Fig, 1. Single ideal permeation stages for different flow patterns.

The extent of separation is defined as the absolute value of the determinant of the binary separation matrix written in terms of the four segregation fractions [6, 71: tp s

abs det

y11

YlZ

Y,,

y22

(2)

This index was proposed by Rony as a universal separation index after an extensive investigation of candidate indexes. This was the only index investigated that met very stringent criteria established for a meaningful separation index [ 71. Assuming that compound i = 1 is the more permeable species (i.e., the

103

species concentrated in the low-pressure product stream), it is convenient to take j = 1 to designate the low-pressure product stream (the “heads”), and j = 2 the high-pressure product stream (the “tails”). This is not absolutely necessary, but it is convenient because it makes the calculated value of C;, positive when i = 1 is the more permeable species. The molar flow rate of the feed and product streams to and from the stages shown in Fig. 1 are: n: = LHI (XI)

(3)

n: = LHI(1 - X I )

(4)

nil = LLO (YO)

(5)

nzl = LLO (1 - YO)

(6)

n12 = LHO (X0)

(7)

nz2 = LHO (1 -X0)

(8) where LHI, LLO and LHO are the flow rates of the feed, low-pressure product, and high-pressure streams, respectively, and XI, YO and X0 are the corresponding mole fractions of the more permeable species (i = 1) in the feed and product streams. Other definitions and relationships that are frequently used in the characterization of stage separations are: Stage cut, 0 LLO X I -x0 (j=-= LHI YO-x0 LHO YO -XI 1-~B--.----_ L H I Y O -X0

(9) (10)

Overall separation factor, a0 YO(1 -X0) (yo = X0(1 - YO)

(11)

Heads separation factor, p 1 Pl”

YO(1 - X I ) XI(1 - YO)

(12)

Tails separation factor, p2 XI(1 -X0) 02- X0(1 - X I ) Sandell’s separation factors, S

(13)

(14) S i/2=

W2,l

Distribution coefficients, Ki yi2 J$c = -

nil

(15)

Yil

Recovery of i, Ri = Yij,j=l

=LLo(yo)=~

R

’ LHO (XI) R

($)=I-(1-8)(g)

-LHo(l-xo) 2 - LHI(1 -XI)

LOSS Of

L

= 1

L

(16)

(18)

i, Li = Yij,jz# 1

(19)

LHO (X0) =(1-8)($)=1--8(E)

LHI (XI)

_LLO(l-YO)= LHI (1 -XI)

(20) (21)

2-

R,+L,=l,

(17)

R2+L2=1

(22)

Many other relationships are possible, for example: 00

=P1S2

(23)

K2

=a&1

(24)

YO= XI = x0=

81 (XI) QIO (X0) 1+ @l-1)x1 = 1 + (cyo -1)X0 a0

(X0)

010 (X0) + fl1(1 -X0) = YO +

YO P2 (I- w

XI YO YO + OLo (1 - YO) =1+p2(1-xI)

(25)

(26) (27)

The reader is referred to Rony [6, 71, Benedict et al. [8], Sandell [5], and Sirkar [ 91 for further relationships and a discussion of the various separation factors. With these definitions, some of the many possible alternative forms for iP are shown in Table 1. Sirkar [9] applied & to a single-entry barrierseparation stage and showed that it could be re-arranged into the form of a stage capacity factor times an enrichment factor, either (cyo - 1) or (PI - l), such as shown in Table 1, eqns. (28) and (31). However, as shown by Table 1,

-

1 + Y,JYZl

1

= 1 + n,Jn,,

Ic 1 1 + Mn,,

1

I[ 1 1 = -_1 + K, 1 + K,

I

(33)

(32)

(31)

(30)

Note: For the designation of i and j for separation discussed in this paper, Ep is positive. For other component and stream designations it may be necessary to take the absolute value to make i a positive number.

~[l+(~~~)(~)-l+(‘;l’i(~~~~=[::.I”;-’.h,,,]

=[ 1 + Y,,/Y,,

1

Alternate forms for tp

TABLE 1

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many other equivalent forms also can be found by simple re-arrangement of eqn. (2) in terms of the definitions of other separation factors. Other equivalent forms are probably possible, the list is not meant to be exhaustive. The advantages of using individual forms for {r is not known at this time, but may become apparent as further studies are made. Calculations were carried out to compare the various separation factors for the different ideal permeation stages shown in Fig. 1. As in the previous paper [ 11 the example chosen was for the enrichment of O1 in air as reported by Stern and co-workers [2-41. The following parameters were used as defined by Walawender and Stern [3] : XI = 0.209, PH = 380 cmHg, PL = 76 cmHg, (Y* = P"z/PN2 = 10, LHI = 1 X 10" cm3 (STP)/sec, and 6 = 2.54 X 10m3 cm. The equations as presented by Stern and co-workers were programmed on a digital computer using these values of the parameters and solved (where necessary) using a 4th order Runge-Kutta numerical integration routine. These calculations yielded values of LHO, LLO, X0 and YO as a function of 8 for the different ideal permeation stages. These values were then used in the preceding equations to calculate values of the various “separation factors” for comparison. The results of these calculations are presented in Figs. 2-7. Figure 2 shows how the extent of separation, [r, varies with stage cut for the different ideal permeation stages. As can be seen, tp has a value of zero for the two extremes, fl = 0 and B = 1, where there is no separation. It goes through a maximum at an intermediate value of 0 and, from a strict separation point of view, presumably this maximum represents the “best” separation. This optimum separation can be viewed as the best compromise between the quality of the two product streams, and their rates. As can be seen, there is a significant difference in the maxima for the different stage flow patterns with countercurrent > cross-flow > cocurrent > perfect mixing. In Fig. 2 the arrows indicate the value of 0 where the maxima occur which are at different values of r3 for each flow pattern. Figure 3 shows how the overall separation factor, eo, varies with stage cut for this example. As can be seen, the behavior of co is different for the different flow patterns, with e. increasing rapidly at higher values of 0 for the countercurrent and cross-flow stages, while for cocurrent and perfect mixing stages a0 varies little with stage cut and either decreases slightly (perfect mix), or goes through a very shallow relative maximum (cocurrent). The very rapid increase in 01~ for the countercurrent and cocurrent flow is due to the rapid depletion of the more permeable component in the high-pressure stream. In these cases X0 + 0 as 0 -+ 1 and thus czo + m. However, for the cocurrent and perfect mix cases X0 does not approach zero at high values of 0. For example, for the perfect mix case, X0 = 0.062 at B = 0.99999. Figures 4 and 5 show how the heads and tails separation factors vary with stage cut. As can be seen, p1 decreases with increasing 0 while & increases for all stage models. For the countercurrent and cross-flow stages fiz gets very

107 ! 2. 3. 4.

.6

COUNTERCURRENT CROSS - FLOW COCURREN T PERFECT M/X

.I

.2

.3

.4

.5

.6

.0

.7

.S

8 Fig. 2. Extent of separation as a function of stage cut for the different ideal stages. 50

4 0 -

1. 2. 3. 4.

COUNTERCURRENT CROSS-FLOW COCURRENT PERFECT M/X

3 0 -

a0 2 0 -

10 -

4 .I

.2

.3

.4

5

.6

.7

.8

.S

8 Fig. 3. Overall separation factor as a function of stage cut for the different ideal stages.

108 I COUNTERCURRENT 2. CROSS - FLOW 3. COCURRENT 4. PERFECT M/X

\

L

3

.I

j

.2

4

.3

.4

I

.5

.6

I

.7

I

.6

.s

3

e

Fig. 4. Heads separation factor as a function of stage cut for the different ideal stages.

1 I. COUNTERCURRENT 2. CROSS-FLOW 3. CCWJRRENT 40 - 4. P E R F E C T MIX

.I

.2

.3

.4

.5

.6

.7

.6

.S

1.0

e

Fig. 5. Tails separation factor as a function of stage cut for the different ideal stages.

large as 8 increases for the same reason that a0 gets large, that is, because X0 + 0. /3 1 decreases and approaches 1 as 6 + 1 since at a stage cut of 1 all of the high-pressure stream will have permeated and YO = XI. It is interesting to note that for the cocurrent model the maximum gP occurs when 0 I = pz, but this is not the case for the other ideal models.

109

In Fig. 6 Sandell’s separation factors for the four permeation stages are plotted vs. 0. As can be seen, the values for S,,, and S,,, differ very little for the different ideal stages. For all models S,,, + 00 as 13 + 0 and S,,, -+ 0 as 0 -+ 1 because of the (1 - 0 )/f3 term in the expression. Interestingly, this separation factor varies very little for the different ideal permeation stages at the same 0, indicating there would be little differences in the separation, while .& indicates significant differences. 5.0

.I

.2

.3

.4

.6

6

.7

.6

.S

8

Fig. 6. Sandell’s separation factor as a function of stage cut for the different ideal stages.

.6

R1

CouNrET?cuRRt-Nr

.5

CROSS -FLOW COCURRENT PERfEcr M/X IND/CATES WHERE MAX/MA OCCU? IN THE &, CURVES

.I

.2

.3

.4

.5

.8

.7

.8

.9

Fig. 7. Recovery of 0, as a function of stage cut for the different ideal stages.

110

Finally, Fig. 7 shows how the recovery varies with stage cut for the different stage flow patterns. As can be seen, above a stage cut of about 0.15, R increases more rapidly with increasing 0 for the more efficient countercurrent and cross-flow permeation stages. However, of course, in all cases R -+ 1 as 8 + 1, a necessary consequence of the definition of R. General discussion The purpose of a separation factor should be to aid in answering the question “which separation scheme results in the best separation?” This question is relatively easy to answer when considering the separation that is occurring in the different ideal stages, all at the same stage cut, that is, where the two product streams’ flow rates are the same for each stage. For example, consider the data shown in Table 2A, which gives the product stream compositions and values for the various separation indexes for the different ideal stages, all at the same stage cut, 6 = 0.45. In comparing these data we know intuitively that 0 = 0.45 with YO = 0.413 and RI = 0.889 is a better separation than 13 = 0.45 with YO = 0.335 and RI = 0.723, because in the former case we have recovered a larger percentage of the desired product, at a higher concentration, with the same product stream rates for the two cases. Also, in comparing the values of the various separation factors in Table 2A we can see that all of them have higher values for the cases we intuitively know represent better separations (the exception is &, which decreases for the better separations, but its reciprocal, S1,2, shows the same trends as the others). Thus, when a comparison is made at equal stage cuts, co, pl, pz, S,,z TABLE2 Selected values of the various separation factors for different ideal permeation stages;example ofSternandWalawender[4] Stage flow

e

x0

0.45 0.45 0.45 0.45

0.042 0.060 0.087 0.106

0.15 0.45 0.70

YO

010

81

82

S,,,

16.00 10.14 5.83 4.28

2.66 2.45 2.11 1.91

6.01 4.14 2.76 2.24

0.749 0.781 0.824 0.861

6.26 6.26 0.152 0.530 16.OC 2.66 0.042 0.413 0.001 0.298 316.4 1.61

1.47 6.01 196.9

S,,,

RI

tp

Table 2A Countercurrent Cross-flow Cocurrent Perfect mix

0.413 0.393 0.358 0.335

1.34 1.28 1.21 1.16

0.889 0.841 0.770 0.723

0.555 0.498 0.405 0.344

Table 2B Countercurrent Countercurrent Countercurrent

cv* = 10, r = 5, XI = 0.209.

2.40 0.417 0.380 0.291 0.889 0.550 0.749 1.34 0.998 0.377 0.379 2.64

111

and iP all give an indication as to which separation is best, at least qualitatively, because they follow the trends our intuition tells us they should, that is, they are larger for the better separations. However, the significance of the magnitude of the values for the different separation factors is not apparent, and, for example, we do not know how much better the separation is for ~1~ = 16.00 than for 01~ = 4.28. Of the separation indexes considered, only tP is normalized, that is, varies between 0 (no separation) and 1 (perfect separation). Apparently tP is also a quantitative measure of how good the separation is. However, the same significance cannot be placed on the trends in the values of cro, pl, pz, S1,2 and RI when they are not based on the same product flow rates. This is clearly shown in Table 2B which shows the values of the various separation indexes at different stage cuts, all for the countercurrent flow model. In this case our interaction fails us because we do not know which of the following represents the best separation: 0 = 0.15 with YO = 0.530 and RI = 0.380, or 8 = 0.45 with YO = 0.413 and RI = 0.889, or 8 = 0.70 with YO = 0.298 and RI = 0.998. In comparing these data we might reason that 8 = 0.70 does not represent the best separation because the product concentration is too low, having only increased in concentration from XI = 0.209 to YO = 0.298. We also might reason for the csse of B = 0.15 that the recovery is too low because we have only recovered 38% of the desired product. However, these are only guesses and from the above data we do not know with certainty which represents the best separation. Inspection of the values for (Ye, pl, Dz and S,,, does not aid in solving our dilemma. For example, the magnitude of CY~ conflicts with our reasoning above which indicates that 0 = 0.70 probably is not the best separation, although a0 = 316.4 for this case and is much higher than for the other two cases. pz and S,,z show the same trends as (Ye, while the trend for bi is opposite. Only tP is consistent for all of the data presented in Table 2. In comparing the separations at the sense stage cut, any of the separation factors in this study appears to qualitatively describe how good the separation is. However, for unequal 0, only gP gives consistent results. Thus, it can be concluded that, in general, of the separation indexes investigated, only .$r is a valid separation factor. In order for a comparison of different separation schemes to be valid, the comparison must be made when the different separation parameters are onn the same basis. Table 1 shows some of the many equivalent ways that .$, can be expressed. Since [r is defined in terms of normalized segregation fractions, the definition assures that all stage separations are compared on the same basis regardless of stage cuts, and gives a meaningful measure of the goodness of separation.

112

List of symbols Distribution ratio for compound i (Ki = niz/ni 1). Fraction of compound i “lost” in a product stream. Molar rate of feed stream to a stage inlet. LHI LHO Molar rate of high-pressure product stream from a stage. LLO Molar rate of low-pressure product stream from a stage. Moles of compound i leaving in flow region j. ni* d Moles of compound i in the feed to a stage. “i Pressure on high-pressure side of permeation stage. PH Pressure on low-pressure side of permeation stage. PL pO,pN, Permeability coefficients of O2 and Nz. Fraction of compound i “recovered” in a product stream. Ri Sandell’s separation factor using compound 1 as the reference. s 2/l S l/2 Sandell’s separation factor using compound 2 as the reference. Mole fraction of compound 1 in the feed stream. XI x0 Mole fraction of compound 1 in the high-pressure product stream. Segregation fraction of compound i into flow region j (note that Li = yij Yii,j+ i, Ri = Yij,i_ i in this paper). YO Mole fraction of compound 1 in the low-pressure product stream.

Ki Li

Greek letters Overall separation factor. ;: Ideal separation factor. Heads separation factor. 01 Tails separation factor. 02 6 Membrane thickness. Extent of separation. Extent of separation for an ideal permeation stage. :, Stage cut, i.e., the fraction of the feed allowed to permeate through 6 the membrane. References 1 F.P. McCandIess, The extent of separation in single permeation stages, J. Membrane Sci., 17 (1984) 323-328. 2 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. 3 W.P. WaIawender 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. 4 S.A. Stern and W.P. Walawender, Analysis of membrane separation parameters, Sep. Sci., 4 (1969) 129-159. 5 E.B. SandeIl, Meaning of the term “separation factor”, Anal. Chem., 40 (1968) 834835. 6 P.R. Rony, The extent of separation: On the unification of the field of chemical separations, in: Recent Advances in Separation Techniques, AIChE Symp. Ser. No. 120, 68 (1972) 89-104.

113 7 P.R. Rony, The extent of separation: A universal separation index, Sep. Sci., 3 (1968) 239-248. 8 M. Benedict, T.H. Pigford and H.W. Levi, Nuclear Chemical Engineering, 2nd edn., McGraw-Hill, New York, N.Y., 1981, pp. 647-648. 9 K.K. Sirkar, On the composite nature of the extent of separation, Sep. Sci., 12 (1977) 211-229.