Ind. Eng. Chem. Fundem. 1984, 23, 470-480
470 A
Literature Cited
0 G4and Vy layer depth progresses a s reaction w i t h n.butane continues
( = J
equilibrium
\
unreduced core
G4
stabilized in thebulk by excess phosphorus
Figure 5. Schematic illustration of VPO catalyst structures which can arise during n-butane oxidation in (A) the absence and presence of gas-phase oxygen.
(B)t h e
gration by excess phosphorus for high P:V ratios. This type of structure, illustrated in Figure 5B, tended to favor enhanced activity over the corresponding fully oxidized catalyst: increased selectivity was observed for low P:V ratios. Since reduction did not influence the selectivity of catalysts with P:V ratios 21, it is concluded that it is the value of this ratio and not the V4+content which is important in determining selectivity. On the contrary, the selectivity of catalysts with low P:V ratios seemed more influenced by the V4+content. Acknowledgment One of us (B.K.H.) wishes to thank the Services de Programmation de la Politique Scientifique (Belgium)for a fellowship.
Ai, M. Bull. Chem. SOC.Jpn. 1970, 43, 3490. Ai, M.; Boutry, P.; Montarnal, R. Bull. SOC.Chim. Fr. 1970, 2775. Blelanski, A.; Haber, J. Catal. Rev. Sci. Eng. 1979, 19, 1. Bordes, E. Thesis, Unlversit6 de Technologle de Compl6gne, Compiegne, France, 1979. Bordes, E.; Courtlne, P. J . Catal. 1979, 57, 236. Brkic, D.; Trlfiro, F. Ind. Eng. Chem. Rod’.Res. D e v . 1979, 78. 333. Callaghan, J. L.; Grasselll, R. K. AIChE J . 1983. 9 , 755. Centl, G.; Trlflro, F.; Vaccari, A.; Pajonk, G. M.; Teichner, S. J. Bull. SOC. Chim. Fr. 1981, 1-290. De Maio, D. A. Chem. Eng.May 1980, 104. Fricke. R.; Jerschkewitz. H. G.;Lischke, G.; Ohlmann, G. 2. Anora. - Ala. Chem. 1979, 23, 448. Gopal, R.; Calvo, C. J . Solid State Chem. 1972, 5 , 432. Grasselll, R. K.; Burrington, J. D. A&. Catal. 1981, 30, 133. Grasselli, R. K.; Burrington, J. D.; Brazdll, J. F. Faraday Dlscoss. Chem. Soc. 1981, 72, 203. Grasselll, R. K.; Suresk, D. B. J . Catal. 1972, 25, 273. Haber, J. “Proceedlngs, 3rd International Conference on Chemistry and Uses of Molybybdenum”; Bany, H. F.; Mitchell, P. C. H., Ed.: Ann Arbor, MI, 1979; p 114. Hodnett, B. K.; Permanne, Ph.; Delmon, B. Appl. &tal. 1983, 6 , 231. Hodnett, B. K.; Delmon, B. Bull. Soc. Chim. 8e/ge 1983, 92, 695. Hodnett, B. K.; Delmon, B. Appl. Catal. 1984, 9 , 203. Hodnett, B. K.; Delmon, B. J . Catal. 1984, 88, 43. I.C.I. Belgian Patent 867 189, 1978. Jordan, B.; Cahro, C. Can. J . Chem. 1973, 57, 2621. Morselli, L.; Trlfiro, F.; Urbain, C. J . Catel. 1982, 75, 112. Mount, R. A.; Rafelson, H. US. Patent B 1975. Nakamura, M.; Kawal, K.; Fujiwara, Y. J . Catal. 1974, 3 4 , 345. Poll, G.; Resta, I.; Ruggeri, 0.; Trlflro. F. Appl. Catal. 1981. 7 , 395. Poll, G.; Ruggeri, 0.; Trifiro, F. 9th International Symposium on the Reactivity of Solids, Cracow, 1980; p 512. Schneider, R. A. US. Patent 3864280, 1975. Sunderland, P. Ind. Eng. Chem. Prod. Res. D e v . 1978, 75, 90. Van den Berg, J.; Braus-Brabant, J. H. L. M.; Van Dillen, J. C.; Flach, J. C.; b u s , J. W. Ber. Bunsenges fhys. Chem. 1982, 86, 43. Varma, R. L.; Saraf, D. N. J . Catal. 1978, 55, 90. Varma, R. L.; Saraf, D. N. Indian Chem. Ing. 1978, 20, T44. Wohlfahrt, K.; Emig, G. Hydrocarbon Process. June 1980, 83.
Received f o r review July 5 , 1983 Revised manuscript received February 2, 1984 Accepted M a r c h 19, 1984
Registry No. Vz05,1314-62-1; H3P04, 7664-38-2; MAA, 108-31-6; butane, 106-97-8.
Series Solutions for a Gas Permeator with Countercurrent and Cocurrent Flow Noureddine Bouclf, Sudlpto MaJumdar,and Kamalesh K. Slrkar Department of Chemlstry and Chemical Englneerlng, Stevens Institute of Technology, Hoboken, New Jersey 07030
The difficuitles In the numerical solution of differential equations for countercurrent binary gas permeators are discussed. For a permeator with countercurrent or cocurrent flow of a binary gas mixture without axial pressure drops, a series solution technique is introduced to express each product composition as a power series in terms of a dimensionless membrane area. For countercurrent flow over a wide range of parameters, computed product compositions and fractions of the feed mixture permeating the membranes compare very well with those from a numerical solution of the governing equations for small to moderate permeate fractions in the range considered practical. For ideal separation factors of 2,5, 10, 25,and 80,significant deviations occur at fractlons larger than 0.7,0.5,0.33,0.25,and 0.15,respectively, at a pressure ratio of around 0.1 and a more-permeable-component feed mole fraction around 0.2. Cocurrent flow predictions are excellent at somewhat lower permeate fractions. Preliminary countercurrent permeator design or rating problems will require solution of only one or two cubic equations.
Introduction Gas separation by membranes has recently achieved commercial success (Maclean and Graham, 1980). Increasing attention is also being paid to the development of better membranes and/or better separation schemes (Matson et al., 1983). The permeator in such separations is usually operated in countercurrent flow. Prediction of
separation performance of such permeators generally requires numerical iterative solution of three coupled highly nonlinear ordinary differential equations using backward integration from the reject end to the feed (Antonson et al., 1977). Convergence problems are common. Additionally, there exists a need for rapid and accurate design procedures for permeators, especially when computer 0 1984 American Chemlcal Society
Ind. Eng. Chem. Fundam., Vol. 23,
simulation of a number of interconnected process units is required. This is so since replacement or consideration of replacement of conventional gas separation processes such as absorption etc. by membrane processes is an increasing reality. Reviews of analytic studies of the separation of binary and multicomponent gas mixtures by a single permeator are available in Stern and Walawender (1969) and Stern (1976). These studies are restricted to permeators with perfect mixing or cross-flow and provide analytic expressions for results easily evaluated in a computer. Conditions of perfect mixing or cross-flow are, however, usually absent in permeators in practice. Consequently, permeators with countercurrent flow have been studied extensively. Walawender and Stern (1972) have presented a numerical analysis of countercurrent and cocurrent permeators having nonporous homogeneous membranes without axial pressure drop or purge stream. Hwang and Kammermeyer (1975) have also presented the results of Kammermeyer and co-workers obtained from such an analysis. Pan and Habgood (1974) have numerically analyzed countercurrent and cocurrent permeators with a purge stream but no axial pressure drop. Thorman, Rhim, and Hwang (1975) have numerically modeled countercurrent silicone capillary permeators that have axial pressure drops inside silicone capillaries. Antonson et al. (1977) have numerically modeled a commercial scale hollow fiber permeator with a nonzero permeate pressure drop inside the fibers. Pan and Habgood (1978) have numerically studied the same problem with the added complication of a purge stream on the permeate side. No analytical studies of permeators with countercurrent and cocurrent flow exist. The objective of this paper is to develop simple approximate solutions for countercurrent and cocurrent permeators separating a binary gas mixture. A series solution in terms of a nondimensional membrane area is provided for each of the two flow configurations for the case of no axial pressure drop on either side of the membrane. The analysis is restricted to membranes and permeators when constant values of the species permeability coefficient can be assumed and when the membrane thickness is a constant. The results are compared with numerical solutions of the governing equations and the range of validity of the proposed series solutions is delineated. Governing Equations for Cocurrent and Countercurrent Binary Gas Permeators We first present the governing equations for a cocurrent binary gas permeator and then those for a countercurrent binary gas permeator. For steady-state binary gas permeation through an essentially homogeneous membrane having constant species permeability coefficient, governing nondimensionalized equations for a cocurrent permeator (Figure la) with no axial pressure drop and no axial diffusion on either side of the membrane can be written as (Walawender and Stern, 1972; Pan and Habgood, 1974; Hwang and Kammermeyer, 1975)
The nondimensional cumulative membrane area Rf is based on the molar high pressure feed flow rate Lfand the
FEED L x
+
P,L,x
4
1
S
StdS
*I
f, f
s=o
(a)
rn
+
I
REJECT L x w, w
ST
PERMEATOR W I T H COCURRENT FLOW
'
FEED L X
No. 4, 1984 471
P+'
REJECT
* - L
Lw,xw
f, f
(b)
PERMEATOR W I T H COUNTERCURRENT FLOW
Figure 1. Schematic of a gas permeator with (a) cocurrent flow or (b) countercurrent flow.
permeability coefficient Q2of the less permeable species 2
Rf =
(&s Q2 P
(3)
The cumulative membrane area S increases from the high-pressure feed inlet onward. The pressure ratio y (I 1) and the ideal separation factor a are defined by
81
P
y=F;
OL=-
Qz
(4)
with Q1being the constant permeability coefficient of the more permeable species 1 through the membrane of thickness 6. The boundary conditions are at S = 0, Rf= 0, L = Lf, x = xf, V = 0, y = yf (5) Actually, y = yf is not specified in the problem and is obtained naturally as part of the solution procedure. For design problems, the condition of one of the streams at the reject stream end is specified. For example at S = ST,R f = Rm, x = X, (6) with the total membrane area S T to be determined. Or, the value y = y, at Rf = Rm is specified and S T is unknown. For rating problems, the total membrane area S T is known and the reject and permeate stream conditions have to be determined. For a countercurrent permeator as shown in Figure lb, the nondimensionalized governing equations corresponding to eq 1and 2 can be written as (Walawender and Stern, 1972; Pan and Habgood, 1974; Hwang and Kammermeyer, 1975)
Here the nondimensional cumulative membrane area R, is based on the molar high-pressure reject flow rate L, with the membrane area S increasing from the high-pressure
472
Ind. Eng. Chem. Fundam., Vol. 23, No. 4, 1984
reject outlet toward high-pressure feed inlet
Rw =
(&s Q2
P
two product streams in terms of the total nondimensional membrane area Rm
The given boundary conditions is that, at high pressure feed inlet R, = R f l , s = ST,X = Xf, L = L f (10) Note that even if ST is given, R f l may not be available since L, is an unknown in most problems. The solution is developed on the basis of R, = 0, S = 0, x = x,, L = L,, V = 0 (11)
For a design problem, x, (or yf) is specified and R f l is to be determined for given xf, CY, and y. For a rating problem, ST is specified for a given xf,a, and y with reject and permeate stream conditions to be determined. Note that we have not so far indicated the basic balance and transport equations which lead to eq 1and 2, and eq 7 and 8, for cocurrent and countercurrent flows, respectively. We indicate here only the overall and more permeable component mass balances over an envelope extending from one end of the permeator to an arbitrary location where the cumulative membrane area is S. For cocurrent flow (Figure la), these are L,=V+L LfXf = v y + Lx For countercurrent flow (Figure lb), L=V+L, Lx = v y + L&, The fraction of the feed stream that goes through the membrane is identified as the “cut fraction” 0. For the countercurrent permeator it is defined as
whereas the cut fraction 0 for a cocurrent permeator is defined as
In membrane separation literature, cut fraction is usually abbreviated as the “cut”. Correspondingly the relation between e and the compositions of three streams entering or leaving the permeator are countercurrent flow: xf = Oyf + (1 - O)x, (13d) (134 cocurrent flow: xf = Oy, + (1 - O)x, A Series Solution for Cocurrent Flow Binary Gas Permeators The approximate series solution technique utilized in this paper for solving the coupled ordinary differential eq 1and 2 with appropriate initial conditions is described in detail in Appendix I. Basically, each product composition at any given distance from the feed point is expressed as a power series in terms of the cumulative dimensionless membrane area R f for that location. Each series is truncated after the RB term. The unknown coefficients in the series expansions are then explicitly obtained in terms of the ideal separation factor a, the pressure ratio y, and the feed mole fraction xf by using the Frobenius method for both eq 1 and 2. For any given cocurrent flow binary gas permeator, the above approach yields two expressions for x, and y,, the mole fractions of the more permeable component in the
The expression for each of the coefficients a,, u2,a3, bo, bl, b2, and b3 in terms of CY, y, and xfis given in Appendix I. One can now design or rate a binary gas permeator with cocurrent flow when a , y, and xf are known. Consider a cocurrent permeator rating problem with Rm being known for a specified membrane area STand given xf, a, y, L f , and ( Q 2 / b ) . The high-pressure reject composition x, is directly obtained from expression 14. Similarly, the permeate composition yw is explicitly given by expression 14a. The cut fraction 8 is easily determined now from definition 13e. On the other hand, if one wishes to know the value of Rr needed to achieve a given x, or y,, he must solve either the cubic eq 14 or 14a for Rm. Then one uses this Rm in the other of eq 14 or 14a to determine the corresponding yw or x,, The determination of the cut fraction O or flow rates V , and L, can then be easily made using relations 13e, 13c etc. The cumulative membrane area STis found from definition 3 for the R f f determined above. Thus designing a cocurrent permeator using series solutions only requires the solution of one cubic equation. Instead of a regular permeator design or rating problem, sometimes one wants to know y, and x, for any given value of 0 for a cocurrent permeator. The procedure to be followed is as follows. Solve eq 14 as a cubic in R f f in terms of x,, xf, al, a2,and u3. Substitute this expression for R f f on the right-hand side of eq 14a. Express yw on the lefthand side of eq 14a in terms of 0, x,, and xf using eq 13e. An algebraic equation is obtained with x, as the only unknown. Solve for x , by trial and error. When x, if determined, yw is obtained from eq 13e. Thus, if we know a , xf,y, and 0 for a cocurrent permeator, x, and yw may be calculated without our knowing the total nondimensional membrane area Rm. A Series Solution for Countercurrent Flow Binary Gas Permeators The approximate series solution technique adopted here for solving the coupled ordinary differential eq 7 and 8 subject to conditions 10 and 11 is treated in Appendix 11. Essentially, the high-pressure stream composition x and the low pressure stream composition y at any location in the permeator are each expressed as a power series in terms of the dimensionless cumulative membrane area R, for that location. Unlike a cocurrent permeator, the countercurrent permeator has, however, R, = 0 at the highpressure reject end (Figure lb) and R, = R f l , the total membrane area required, at the high-pressure feed end. The unknown coefficients in the series expansions are determined in terms of CY,y, and x,. Here x, may or may not be known for the permeator. For any given countercurrent permeator problem, the net result of this approach is the following two relations Xf
= x,
+ c ~ ( ~ , Y J , ) R , T+ C Z ( ~ , Y J , )+RC~(~,YJ,)RMT’ ~~~ (15)
Yf =
~O(~,Y,X,)
+ di(a,r,x,)R,~+ d2(a7~,xw)Rf12 + d3(a,y,xw)Rf13 (1W
The expression for each of the coefficients cl, c2,c3, do, d,,
Ind. Eng. Chem. Fundam., Vol. 23, No. 4, 1984 473
d2,and d3in terms of a,y, and x, is available in Appendix 11. It is now possible to design or rate a binary gas permeator with countercurrent flow using relations 15 and 15a. Consider a countercurrent permeator design problem with the high-pressure reject composition x , specified. Solve the cubic eq 15 for Ra. Substitute this Ra next in eq 15a to determine yfi Since yf and x , are known, 6 is obtained from eq 13d and L, is obtained from definition 13b. The total membrane area S T required for the separation can next be obtained from definition 9. For a design problem with yf specified, solve eq 15 and 15a simultaneously for unknowns x, and R a . Then the total membrane area ST,the cut fraction 6, and L, may be determined in the manner indicated above. Obviously, the traditional numerical solution of the countercurrent permeator with its attendant convergence problems is very different from the schemes proposed above, involving either solution of one cubic equation or simultaneous solution of two algebraic equations. Naturally, the savings in computer time are considerable. For a countercurrent permeator rating problem with given total membrane area ST,the calculation procedure is as follows. The nondimensional total membrane area R f l incorporates L,. The latter is obtained from eq 13b and 13d as L, = Lf(
-)
Using eq 15a, express yf as
+ KXf - X W ) / 6 1
= 2.5 X
(cm3(STP)cm)/(s cm2 cmHg) 6 = 2.54
(18)
Then there are two unknowns in eq 15 and 15a for any given 6, xf, a,and y: x, and ( Q 2 / 6 ) ( P / L f ) S pSolve the two algebraic equations for these two unknowns to obtain x, and then yf from eq 18. Thus, although the present series solution technique via eq 15 and 15a apparently requires the specification of nondimensional membrane area Ra for determining x, and yf, it is possible to determine x, and yf when an exact value of the “stage cut” 6 is specified only if xf, a,and y are given. Numerical Solutions for Countercurrent and Cocurrent Binary Gas Permeators A numerical solution of the governing eq 1 and 2 for cocurrent flow and eq 7 and 8 for countercurrent flow has been carried out earlier in a thesis by Broad (1981), using a fourth-order Runge-Kutta technique. These programs were utilized here for specified values of a and y at xf = 0.2. Values of a = 80,50, 25, and 10 were chosen and y values of 0.1,0.25, and 0.01 were used. This range covered very low to high pressure ratios. Although the value of y varies widely in practice, y = 0.1 is common while y = 0.01 is rarely encountered, except in high-vacuum systems.
X
cm
L f = 1 X lo6 cm3(STP)/s P = 380 cmHg p = 76 cmHg (where a = 2 and y = 0.2) = 1X
(cm3(STP)cm)/(s cm2 cmHg) 6 = 2.54
- xw
Thus Ra in eq 15 and 15a may be expressed in terms of unknowns yf and x, all other quantities (Le., ST, P, L f , (Q2/8),and xf) being known. There are now two unknowns, x, and yf, in the two algebraic eq 15 and 15a. Solve the equations and obtain x, and yf for the given ST, feed and operating conditions. An alternate type of calculation is often necessary if one wants to know yf and xw for a specified value of 6 in a countercurrent permeator. Rewrite Ra using eq 13b as
Yf = x ,
Q2
Q2
Yf - Xf
Yf
Numerical solutions obtained are reported in the results section in terms of the two separated stream compositions and the cut fraction, 6, plotted as a function of the appropriate dimensionless total membrane area Ra or Rm A few numerical computations were also carried out for xf = 0.05 with a = 80 and y = 0.01 to determine the effect of feed composition, if any. For lower ideal separation factors, a = 5 and 2, numerical results for xf = 0.209 are available in Walawender and Stern (1972) for the product and residue compositions as well as membrane area against the cut fraction 6. These results were utilized to determine the total nondimensional membrane areas Ra and Rff needed for our solutions. For this purpose, the following values were chosen since the results in Walawender and Stern (1972) are in cm2of actual membrane area
X
cm
L f = 1 X lo6 cm3(STP)/s
P = 380 cmHg p = 76 cmHg
(where a = 5 and y = 0.2). For any of the membrane areas given in Walawender and Stern (1972), Ra and Rm values are generated first for given exact values of 6, x,, and yf (or y,). These Ra and Rff are then used in the series solutions for predicting x, and yf (or y,). Knowing x, and yf (or y,) leads to an estimate of 6 from eq 13d and 13e. Comparison of Series Solutions with Numerical Solutions A. Countercurrent Permeators. The series solutions developed in this paper contain terms only up to the cubic term in nondimensional area. It is therefore expected that beyond certain ranges of nondimensional area the series solutions will diverge from the numerical solutions. Concurrently, the series solutions will fail to be accurate beyond certain values of the cut fraction. It is of interest to know, however, the regions of divergence to specify the useful range for these truncated series. Note that the usual ranges of cut fractions encountered in commercial permeators vary from around 0.05 to around 0.45-0.5. For a binary feed mixture having a more permeable mole fraction of 0.2 (= xf), computations have been carried out for values of a = 10,25,50, and 80. A number of pressure ratios (y = 0.01, y = 0.1, and y = 0.25) have been used. We first show in Figure 2 the values of yf and x , as a function of R f l for a countercurrent permeator with a y = 0.1. Each Ra stands for a separate countercurrent permeator problem. We notice that, as a increases, deviations in yf from the numerical solutions appear at lower values of R f l . However, there is much less deviation in the value of x, with Ra over the same range of nondimensional areas. Calculations in each case for x , vs. R a were stopped when the analytical solution started deviating significantly from the numerical solution.
474
Ind. Eng. Chem. Fundam., Vol. 23, No. 4, 1984
I c
PERMEATE COMPOSITIONS
X
/-------
= 0.2: f o
k?-
+
A 0 _.NUMERICAL .
3-
Y =
x
0.1
y = 0.25
----NUMERICAL SOLUTION ----SERIES SOLUTION
SOLUTIONS
COUNTERCURRENT FLOW
- SERIES SOLUTIONS
- A
= 0.2,
f
on=lO
IdlI
a - 8 0 a 1 5 0 a = 2 5 a = l O
I : COUNTERCURRENT FLOW
-.___-
[REJECT
30
0.e0
0.16
COMPOSITIONS
.00
0.32
0’. 24
R -> wT
RwT-’
Figure 2. Comparison of series and numerical solutions for product composition against membrane area in countercurrent flow: xf = 0.2, y = 0.1, and a = 80, 50, 25, 10.
x
f
= 0.2;
o +
A
0 . . . .NUMERICAL
Figure 4. Comparison of series and numerical solutions for permeator cut fraction against membrane area in countercurrent flow: xf = 0.2, y = 0.25, and a = 10.
y = 0.1
x
a -80 a = 5 0 a : 2 5 a = l O
f
= 0.2; y 0
a - I O
. . . .NUMERICAL
SOLUTIONS
--SERIES
-SERIES SOLUTIONS
0.25
SOLUTION
SOLUTION
t
COUNTERCURRENT FLOW
Yf
-
REJECT COMPOSITION
X
10 RwT-’
W
36
0.20
0.40
I
0.60
0.80
RwT->
Figure 3. Comparison of series and numerical solutions for permeator cut fraction against membrane area in countercurrent flow: xf = 0.2, y = 0.1, and a = 80, 50, 25, 10.
Figure 5. Comparison of series and numerical solutions for product compositions against membrane area in countercurrent flow: xf = 0.2, y = 0.25, a = 10.
In Figure 3, the cut fraction 8 for the countercurrent permeator is plotted against R f l for y = 0.1 with a = 80, 50,25, and 10. For the lowest value of a = 10, the series solution starts deviating at a value of 8 = 0.30. For a = 25 the deviation starts around 8 E 0.22. Deviations start earlier for higher values of a. For the highest a = 80, deviation s w around 8 = 0.15. We should recognize that xf = 0.20 in terms of the more permeable species. Therefore, for a = 80, a very high value, a cut fraction of 0.15 is quite reasonable since 8 around 0.20 would imply recovery of almost all of species 1 from feed into the permeate. Thus the predictive capacity of the series solutions i s excellent. Since operation at a larger y is tantamount to operating at a lower a , one would expect the range of 8 over which
the series solution is valid to be greater when y is greater. We notice this feature in Figure 4 where results for the cut fraction are shown for y = 0.25 and a = 10. The predictions from the series solution start deviating from the numerical solution around 8 zz 0.35 at Ra values around 0.40. The corresponding values of yf and x , are shown in Figure 5 for a = 10 and y = 0.25. Figures 6 and 7 show, respectively, the behavior of the cut fraction 8 and the product compositions yf and x , as functions of R f l for a very low pressure ratio, y = 0.01, for a = 80,50, and 25. As expected, the deviations begin at lower values of Re The predicted 8 values deviate from the numerical solutions around 8 = 0.20 to 0.25. Walawender and Stern (1972) have provided numerical solutions for low values of cy and xf = 0.209 for O2-N2
Ind. Eng. Chem. Fundam., Vol. 23, No. 4, 1984 475 8
x
f
xf = 0 . 2 0 9 ; y = 0 . 2
COUNTERCURRENT FLOW
= 0 . 2 ; y = 0.01
o a . 8 0
A
+ a = 50 A a = 2 5
t Yf
a = 2. 0 - 5 .
s
i-
(Walawender 6 Stern (1972))
----NUMERICAL SOLUTION - - - S E R I E S SOLUTION
R
COUNTERCURRENT FLOW
a.
6-
WI REJECT COMPOSITIOF
t
xw
6.03
6.86 R
I
d.09
0. 12
15
i
-1 8
m 0.00
0.16
->
0
wT
Figure 6. Comparison of series and numerical solutions for permeator cut fraction against membrane area in countercurrent flow: 80, 50, 25. lf= 0.2, y = 0.01, and CY
1
0'.32
,
BB
0.64
0.48 4
Figure 8. Comparison of series and numerical solutions for product compositions against permeator cut fraction in countercurrent flow: rf = 0.209, y = 0.2, and a = 2, 5.
%
?, -
0; x
Yf
---
f
= 0.2; y 0 a + a A a
= 0.01
x
= 0.209; y
- 8 0
o A
=SO = 2 5
a a
=
0.2 2
=
5
-
~
I
NUMERICAL SOLUTION
-SERIES SOLUTION COUNTERCURRENT FLOW
REJECT COMPOSITIONS
a 0.88
I
0.03
0.06
0'.09
0: 12
0'. I5
%>-
SI hJ.OO
0.16
0.32
8'.48
0.64
d.50
Figure 7. Comparison of series and numerical solutions for product compositions against membrane area in countercurrent flow: xf = 0.2, y = 0.01, and a = 80, 50, 25.
0 -> Figure 9. Comparison of series and numerical solutions for membrane area against permeator cut fraction in countercurrent flow: xf = 0.209, y = 0.2, and a = 2, 5.
separation through various hypothetical membranes. Here we compare their numerical results for a = 2 and 5 at a pressure ratio y = 0.2 (their r = 5) with values from the series solutions. In Figure 8, we have compared the behavior of yf and x, vs. 0 for these parametric conditions. The series solution predictions agree very well with the numerical solution over a large range of 8. For example, for a = 2, the solutions agree below 8 = 0.70, beyond which calculations were not carried out. For a = 5, the series solutions start deviating slowly from the numerical solution beyond 8 = 0.50. In Figure 9, the logarithm of actual membrane area in cm2required for a given cut fraction is compared for a = 2, 5 and y = 0.2. One notices that for a = 2, the series solution agrees with the numerical solution of Walawender and Stern (1972) for all the values of 8 plotted. The series
solution for a = 5 starts deviating at a 0 = 0.5 as in Figure 8. It is apparent that the series solutions developed in this paper can effectively predict the countercurrent permeator behavior for moderate to large cut fractions as long as the ideal separation factor a is not too large. For large values of a, the series solutions are useful only for low to moderate cut fractions. For a less than 10, we may assume the useful range of 8 to be less than 0.50. For a less than or equal to 25, the effective range is less than 0.30; for a less than or equal to 80,it is less than 0.15-0.20. Obviously, however, such estimates are likely to be influenced by the feed composition. To check the effect of feed composition, if any, on the predictive capabilities of the series solutions, we compare the behavior for xf = 0.05 with a = 80 and y = 0.01. Figure
Ind. Eng. Chem. Fundam., Vol. 23, No. 4, 1984
476
8
6) -?
= 0 . 0 5 ; a = 8 0 ; y = 0.01
x
NUMERICAL SOLUTIONS
NUMERICAL SOLUTIONS
SERIES SOLUTIONS
SERIES SOLUTIONS
o COCURRENT FLOW A COUNTERCURRENT FLO'
COCURRENT FLOW A COUNTERCURRENT FLOW
o
t B
i-
P 6-
Lc Rwfr
,
I
R f T (X10c3).
Figure 10. Comparison of series and numerical solutions for cut fraction against membrane area in countercurrent and cocurrent flow: xf = 0.05, y = 0.01, and a = 80.
8
Figure 12. Comparison of series and numerical solutions for permeate composition against membrane area for countercurrent and cocurrent flow:xf = 0.05, y = 0.01, and a = 80.
CQCURRENT FLOW xf = 0 . 0 5 ;
a =
eo.,
x
f
= 0.209
,
y:
0.2
y= 0 . 0 1 A a = 5 .
---
NUMERICAL SOLUTION
-SERIES
SOLUTION
o COCURRENT FLOW ACOUNTERCURRENTFLOW
PERMEATE
Bs Rnor
Rw
(X~I'~)
->
Figure 11. Comparison of series and numerical solutions for reject composition against membrane area in countercurrent and cocurrent flow: zf = 0.05, y = 0.01, and a = 80.
10 shows the cut fraction f3 against membrane area R a for the countercurrent flow (along with cocurrent flow to be discussed later). The predicted cut fraction starts deviating from the numerical solution at 0 = 0.04. Up to a cut fraction of 0.05, the errors &e very small, since a is very large and the fact that xf = 0.05 indicates excellent predictive ability. Figure 11 shows the value of x, against membrane area under the same conditions. Significant deviations start only when x , is around 0.01, a fourfold reduction in the more permeable species composition in the high-pressure stream. Figure 12 shows the quality of permeate composition prediction showing deviations beginning at membrane areas where 0 and x, started deviating.
eFigure 13. Comparison of series and numerical solutions for product compositions against permeator cut fraction for cocurrent flow: xf = 0.209, = 0.2, and a = 2, 5.
Very low cut fractions in the range 0 . 5 are used often in purification applications, whereas higher values are used in recovery applications. Thus, the series solutions proposed here should be very useful in developing preliminary designs of countercurrent permeators having not too large values of tJ over a wide range of a and y. We note that the numerical solutioii required an average CPU time of 1min 40 s on a DEC-10 computer, while the analytical series solutions required around 2.6 s when "ZSCNT" subroutine was used for the solution of the cubic equation in Re B. Cocurrent Permeators. We first compare the results calculated from the series solutions with those available in Walawender and Stern (1972) for a = 2 and 5,y = 0.2, and cocurrent flow. In Figure 13,we observe that, for a = 2, deviations in both the x , and y, start around a cut fraction 0 = 0.50. For a = 5, deviations in
177
Ind. Eng. Chem. Fundam., Vol. 23, No. 4, 1984
x o
a = 2.
A a = 5.
f o +
= 0 . 2 ; y = 0.25
A 0
*--4
a 0 a a
= 8 0 . 5 0 . 2 5 = l O
---NUMERICAL SOLUTIONS -SERIES SOLUTIONS COCU RRENT FLOW
P'
-SERIES
SOLUTIONS
COCURRENT FLOW
I
I
d
I
I
0'.32
0.48
0.64
30
33
0.05
0'.10
0.15
0.20
c
25
Rf T
Figure 14. Comparison of series and numerical solutions for membrane area against permeator cut fraction for cocurrent flow: z, = 0.209, y = 0.2, and a = 2, 5.
Figure 16. Comparison of series and numerical solutions for permeator cut fraction against membrane area for cocurrent flow: xf = 0.2, y = 0.25, and a = 80, 50, 25, 10.
8
8
-I
c
x
f o
= 0 . 2 ; y = 0.25
X( I
a = 80. + a = 50. A a = 25. 0 a = IO. ---NUMERICAL SOLUTIONS -SERIES
t
r'
+
a . 5 0 a = 2 5 0 a.10 --- NUMERICAL SOLUTIONS -SERIES SOLUTIONS
: 0 I
A
m" T
SOLUTIONS
yws ai
COCURRENT FLOW
$1
PERMEATE COMPOS ITlONS
"h 4
$
= 0 . 2 ; y = 0.1 o a = 8 0
COCURRENT FLOW
PERMEATE -IONS]
Ql 0
2 ? ?O0.00
8 0.10
0.05 R
0.15
0.20
!
6
-> fT
6' 0.00
1
0.05
I
0.1Q
0.15
0.20
25
R -> fT
Figure 15. Cornparison of series and numerical solutions for product compositions against membrane area for cocurrent flow: xf = 0.2, y = 0.25, and a = 80, 50, 25, 10.
Figure 17. Comparison of series and numerical solutions for product compositions against membrane area for cocurrent flow: y = 0.1, zf= 0.2, and a = 80, 50, 25, 10.
x, and y, start earlier at cut fraction values around 0.35.
numerical solutions appear earlier. The corresponding values of cut fraction 6 for cocurrent flow are plotted in Figure 16. One notices again that the deviations, for example, for a = 10 start at a cut fraction of 6 = 0.15 which is somewhat lower than that for countercurrent flow (Figure 4). The pressure ratio is now changed to y = 0.1 and one observes from Figures 17 and 18 that deviations start somewhat earlier both with respect to the behavior at y = 0.25 as well as with those for countercurrent flow at the same y = 0.1 (Figures 2 and 3). An aspect of some importance is that the predictions of y, values are somewhat more accurate in cocurrent flow than in countercurrent flow for the same cut fraction. On the other hand, the predictions of x, values are less accurate in cocurrent flow than in countercurrent flow under comparable conditions.
Thus deviations in cocurrent flow start at smaller values of cut fraction than in countercurrent flow (Figure 8). Figure 14 shows again that for cocurrent flow, the deviations in membrane area start at smaller values of cut fraction than in countercurrent flow (Figure 9). However, even up to a cut of 0.35, the deviation in the logarithm of the membrane area is not more than a few percent. But at a given membrane area, the percentage error in cut fraction is somewhat larger. Next we compare the predictions from series solutions at higher a values with the numerical solutions for cocurrent flow. In Figure 15, the values of y, and x, are plotted against R f f for a = 80,50,25, and 10 and y = 0.25. For a = 10, deviations start at around Rm = 0.16. For higher a,deviations between the series solutions and the
478
Ind. Eng. Chem. Fundam., Vol. 23, No. 4, 1984
€4 Xf
o
dl
y = 0 I
= 0.2
o t
a = 5 0 a . 2 5 o a x 1 0 _ _ - - _NUMERICAL SOLUTIONS __SERIES SOLUTIONS COCURRENT FLOW *
R
0.20
0.15
0.10
A
0
,,,,,,''
,/
0.65
f
25 R f T >
~
x
= 0.1
~~~
~~~~
~
y = 0.01
-
o
a = 8 O a = 5 0 A a = 2 5 0 a = IO - - - NUMERICAL SOLUTIOh -SERIES SOLUTIONS COCURRENT FLOW
REJECT COMPOSITIONS
0'.03
0.06
16
->
fl
Figure 18. Comparison of series and numerical solutions for permeator cut fraction against membrane area for cocurrent flow: 0.2, y = 0.1, and a - 8 0 , 50, 25, 10.
30
y = 0.01
a = 80. a = 50.
a = 25. a = IO. - - - NUMERICAL SOLUTIONS -SERIES SOLUTIONS
a
30
= 0.2
Xf
a = B O
0. 10
0.13
Figure 19. Comparison of series and numerical solutions for product compositions against membrane area for cocurrent flow: xf = 0.2, y = 0.01, and a = 80, 50, 25, 10.
Since the error in prediction of cut fraction 0 is much more influenced by the error in x, prediction-0 = ( x f - x,)/Cyf - x,) in countercurrent flow and 0 = (xf- x,)/(y, - x,) in cocurrent flow-we find the error in 0 in cocurrent flow to be significantly greater. The behavior for y = 0.01 and a = 80,50,25, and 10 is similar and shown in Figures 19 and 20. The effect of feed composition, if any, on the predictions in cocurrent flow is checked in Figures 10, 11, and 12, where results of calculations for xf = 0.05, a = 80, and y = 0.01 are shown. Figure 10 shows that the predicted cut fraction 0 starts deviating from the numerical value at around 0 = 0.03 i.e., somewhat earlier than in countercurrent flow. Similarly the x, prediction shown in Figure 11starts deviating significantly earlier around x, = 0.03, unlike that in countercurrent flow. However, as we have
Figure 20. Comparison of series and numerical solutions for permeator cut fraction against membrane area for cocurrent flow: xf = 0.2, y = 0.01, and a = 80, 50, 25, 10.
seen before, the yw prediction is much better in cocurrent flow (Figure 12). For the cocurrent flow the numerical solution consumed an average 13.2 s of CPU time while the analytical series solution required about 0.6 s. The CPU time required for the numerical solution for cocurrent flow is a couple of times lower then that in countercurrent flow since the latter requires a number of iterations along the permeator with different guesses at the reject stream end until convergence is achieved. Discussion It is apparent that the analytical solutions developed here for a binary gas permeator with countercurrent flow can be used to compute the product stream compositions for a given membrane area accurately as long as the cut fraction is not too large. The usefulness of these solutions for preliminary process design or permeators is therefore apparent since very high cuts are rarely used in practice. The accuracies of the series solutions for a binary gas permeator with cocurrent flow are somewhat poorer than those with countercurrent flow, especially as the cut fraction is increased. Perhaps this can be traced to the nondimensional areas used for a particular flow pattern. For given operating conditions with a given membrane area, Re is greater than Rm If one considers the cubic expression merely from the point of view of representing a fiiite difference, say (xf- x,), then the cubic containing Re will have a better accuracy than that with Rm. Countercurrent flow is used almost always in gas permeators, industrial or otherwise. Thus, the better predictive capabilities of the series solutions for countercurrent flow pattern should make them useful. The predictions for cocurrent flow are of course useful at lower values of cut fraction. Acknowledgment One of us (N.B.) acknowledges with thanks the financial support of Sonatrach DRD/END, 16 Rue Sahara, Hydra, Algiers, Algeria, through its scholarship award. Appendix I Series Solution Technique for Cocurrent Gas Permeators. The governing equations to be solved for
Ind. Eng. Chem. Fundam., Vol. 23,No. 4, 1984 479
a cocurrent binary gas permeator are eq 1 and 2 subject to initial conditions 5. The solution will express the high-pressure gas composition, x , and low-pressure gas composition,y , in a permeator at any location as separate functions of the nondimensional, cumulative membrane area Rf at that location. CY, x f , and y are the parameters for the problem. Assume the following series solutions for x and y
+ alRf + ~2Rf2+ a,&? + ... = bo + blRf + b2Rf2 + b,&$ + ...
x =
(AI)
y
(A2)
We truncate each infinite series after the R? term. Substitute these expressions for x and y in eq 1 and 2 and collect terms of order RP, Rfl, Rf2, and R?. These lead to a total of 8 algebraic equations in terms of 8 unknowns ao, al, a2,a3,bo,bl, b2,and bB. Imposing the initial conditions 5 onto eq A1 leads to a0 = X f
(A31
Using this result in two of the algebraic equations containing either a,, bo, and bl or ao,bo, and al, we find that a1
bo =
+ Cxf)+ bo(B - 0 x 3 (A41 + Cxf)- [ ( E+ Cxf)2- 4 a D ~ f ] l / ~
= -xAA
-(E
20
(A5)
The quantities A , B , C, D , and E are defined by A=a+y-l
(A61
B = ay
(A64
C=l-a
(A6b)
D = ~ ( - a1)
(A64
E=y-l-CYy
(-464
After we reject the physically meaningless solutions of bo = x f , eq A5 is obtained from a cubic in b,. This leads to a quadratic equation in whose solution only the negative root is physically meaningful (Walawender and Stern, 1972). Note that bo in eq A5 is equal to y f for cocurrent flow and is exactly equal to the expression obtained by Walawender and Stern (1972) and Pan and Habgood (1974), who used the criterion that, at S = 0, crossflow conditions exist. Solution of the remaining algebraic equations leads to the following explicit expressions for bl, a2,b2,a3,and b,.
b3 =
- bo),[CYU3 + 2Dblb2 + C(a3bo + ~ 2 b 1+ albz)] (a0 - b0)~(2azbz + a3bi) + [(ai - b1)(2aib2 + a2b1) + ((a0
(a2 - b&ibil(ao - bo) - biai(a1 - bJ2)/ - b0)~[3al- (a0 - bO)(E +2Dbo C U ~ ) (]A) l l )
+
((a0
Since all the coefficients in the truncated series expansions A I and A2 are known, x and y are known in terms of Rf. For the full permeator, the value of R f at the reject end is R f f and the corresponding values of x and y are x , and y,, respectively. Therefore X , = xf + alRW + a2Rpr2 + a3Rpr3 (A12)
yw = bo + blRff + b2Rff2+ b3Rff3
(A12a)
Appendix I1 Series Solution Technique for Countercurrent Gas Permeators. For the solution of eq 7 and 8 with condition 10 specified, assume the following series solutions for x and y in terms of R , x = co + clRw + c2Rw2+ c3RW3+ (A13)
...
y = do + dlR,
+ d2RW2+ d3R$ + ...
(A13a)
Again we truncate both the infinite series after the RW3 term. Substitute these truncated expressions for x and y into eq 7 and 8 and collect terms of order RWo,RW1,Rw2, and RW3. Eight algebraic equations are obtained in terms of the eight unknowns co, cl, c2, c3, do, d l , d,, and d,. If we utilize condition 11in the series expansion A13 at the reject end of the permeator, we get co = x, (A14 since R , is zero. Now x, may be specified if a permeator design is to be developed, or x, may have to be determined, given ST, for a rating or performance analysis problem using condition 10. But the solution will be developed in terms of x,. Using co = x, in two of the algebraic equations containing either only co, do, and c1 or co, do, and d l , we get ~1 = x,[A + CX,] + do[Dx, - B ] (A15)
-(E do = .
+ CX,) - [ ( E+ C X , ) ~- ~ c Y D x , ] ' / ~
(A161 20 The quantities A, B , C, D , and E have already been defined by relations A6 to A6d. The result, eq A16, is obtained from a cubic in do by rejecting the physically meaningless solution, do = x,. In the resulting quadratic expression, the negative root is accepted since it is physically meaningful (Walawender and Stern, 1972). Note that do is the value of y at the high-pressure reject stream end. It has been obtained by Walanwender and Stern (1972) and Pan and Habgood (1974) using a crossflow condition to start their numerical integration for any particular guess of x,. Solution of the remaining algebraic equations lead to the following explicit expressions for the coefficients d l , c,, d2, c3, and d3 in the series expansions A13 and A13a
c2
=
Ind. Eng. Chem. Fundam. 1904, 23,400-404
480
Qz= permeability coefficient of more permeable species 1 and the less permeable species 2, respectively Rf = nondimensional membrane area for cocurrent permeator defined by eq 3 R, = nondimensional membrane area for countercurrent Q1,
d3 = ((do - x,)~[(Yc~ + 2Ddid2 + C(~3do+ ~ 2 +4ciddl (do - ~,)~[2czdz + c@il + (do - xw)[(2cidz+ c2dJ(di - ci) + (4- ~ J c i d i l- cidi(di - cJ21/ ((do - X,)2[3C1 - (do - x,)(E +2Ddo + C X ~ ) (A21) ])
Thus the coefficients cl, c2, c3, do, dl, d2, and d3 are known in terms of the quantities x,(= co), a, and y. Representations A13 and A13a for x and y at any location in the countercurrent permeator may now be used to represent conditions a t the feed end X f = X , + CiRfl + c2Re2 + c3Re3 (A22) yf = do + dlRfl
+ d2Re2 + d3Rf13
(A22a)
Nomenclature A = constant defined by eq A6 a,,, al,a2,a3 = coefficients in expansions 14 and A 1 defined by eq A3, A4, AS, and A10, respectively B = constant defined by eq A6a bo, bl, b2, b3 = coefficients in expansions 14a and A2 defined by eq A5, AI, A9 and All, respectively C = constant defined by eq A6b co, cl, c2,c3 = coefficients in expansions 15 and A13 defined by eq A14, A15, A18 and A20, respectively D = constant defined by eq A6c do, dl, d2,d3 = coefficients in expansions 15a and A13a defined by eq A16, A17, A19, and A21, respectively E = constant defined by eq A6d L = molar flow rate of high pressure gas at a given permeator location P = pressure of high-pressure gas p = pressure of low-pressure gas
permeator defined by eq 9 S, ST = membrane area at a given permeator location and the total membrane area, respectively V = molar flow of low pressure gas at a given permeator location x , xf, x , = mole fraction of more permeable species 1 in high-pressuregas at any permeator location, feed entry, and residue exit locations, respectively y , yf, yw = mole fraction of more permeable species 1in lowpressure permeate gas at any permeator location, feed entry, and residue exit locations, respectively Greek Letters (Y = ideal separation factor, defined by eq 4 y = pressure ratio, defined by eq 4 6 = thickness of nonporous membrane 0 = permeator cut fraction defined by eq 13b or 13c Subscripts f = pertaining to the permeator axial location where high pressure feed enters T = total value, used for membrane area w = pertaining to the permeator axial location where high pressure feed exits Literature Cited Antonson, C. R.; Gardner, R. J.; King, C. F.; KO, D. Y. Ind. Eng. Chem. Process Des. Dev. 1977, 16, 473. Broad, R.; M.Eng. (Chemical) Thesis, Stevens Institute of Technology, Hoboken, NJ, 1981. Hwang, S. T.; Kammermeyer, K. "Membranes in Separations"; Wiley: New York, 1975. Maciean, D. L.; Graham, T. E. Chem. Eng. Feb 1980, 25, 54. Matson, S. L.; Lopez, J.; Quinn, J. A. Chem. Eng. Scl. 1983, 38(4), 503. Pan, C . Y.; Habgood, H. W. Ind. Eng. Chem. Fundam. 1974, 13, 323. Pan, C . Y.; Habgood, H. W. Can. J. Chem. Eng. 1978, 56, 210. Stern, S. A. I n "Membrane Separation Processes",Meares, P.. Ed.; Eisevier: New York, 1976; Chapter 8. Stern, S. A.; Walawender, W. P. Sep. Scl. 1989, 4, 129. Thorman, J. M.; Rhim, H.; Hwang, S. T. Chem. Eng. Sci. 1975, 3 0 , 751. Waiawender, W. P.; Stern, S. A. Sep. Sci. 1972, 7 ( 5 ) ,553.
Received for review July 11, 1983 Revised manuscript received February 13, 1984 Accepted March 20, 1984
Liquid Extraction of Furfural from Aqueous Solution John R. Croker' and Ron 0. Bowrey School of Food Technology, University of New South Wales, Kensington, N.S. W., 2033, Australia
Three solvents were evaluated for the liquid-liquid extraction of furfural from aqueous solution. Complete ternary diagrams were prepared for each of the systems water-furfural-methyi isobutyl ketone, water-furfural-toluene, and water-furfurfal-isobutyl acetate by use of data obtained at 30 ' C . The data were analyzed to give equations for the equilibrium lines for each system and the conslstency of the tie-line data was confirmed using an Othmer-Toblas plot. The data indicate that toluene is the most effective solvent of those tried for the removal of furfural from aqueous solution, although isobutyl acetate may be preferred because of its low toxicity.
Introduction When furfural is formed at high temperature and pressure (180-200 OC, 8-10 atm) and separated from aqueous solution by azeotropic distillation, as occurs in
* CSR Limited, Sugar Division, 55 Clarence St., Sydney, NSW
2000, Australia.
0196-4313/04/ 1023-0400$01.50/0
present furfural proce3ses, considerable product loss occurs due to thermal degradation of the furfural. Commercial processes operate with a yield of 30-35% of theoretical furfural (Jaeggle, 1975). As an alternative, a process has been suggested by the authors in which furfural is made at high temperature and pressure in a plug flow reactor with fast cooling of the contents at the point of maximum $3 1904 American Chemical Society
