Ind. Eng. Chem. Fundam. 1986, 25, 217-228
217
Hollow Fiber Gas Permeator with Countercurrent or Cocurrent Flow: Series Solutions Noureddlne BoucH, Amltava Sengupta, and Kamalesh K. Slrkar Department of Chemistry and Chemical Engineering, Stevens Institute of Technology, Hoboken, New Jersey 07030
The problems in the numerical solution of boundary value problems encountered in hollow fiber binary gas permeatbrs having cocurrent or countercurrent permeate flow with axial pressure drop inside the fiber bore are discussed. A series solution technique is developed to express each product composition and the pressure ratio as a power series in terms of a dimensionless membrane area for shell side constant-pressure feed flow. Cocurrent or countercurrent rating or design problems require the solution of elther one algebraic equation, two coupled
algebraic equations, or three coupled algebraic equations to determine the unknown quantities of interest. Calculated product and reject compositions, stage cuts, and closed-end pressure ratios compare quite well with those from numerical solutions for practically useful low to moderate cuts over a wide range of parameters. Cocurrent rating calculations requiring solution of only one algebraic equation are recommended for shortcut procedures since flow pattern effects are significant but not large.
Introduction Gas separation by selective permeation through a nonporous polymeric membrane is now being commercially achieved by either a hollow fiber or a spirally wound permeator. Among the two membrane configurations, the hollow fiber geometry is utilized in the two most commercially successful devices, e.g., the PRISM separator of Monsanto (Maclean and Graham, 1980) and the cellulose ester hollow fiber permeators of Dow. These devices are being increasingly utilized for H2 recovery, acid gas removal, C 0 2recovery for enhanced oil recovery applications, etc., in preference to conventional gas separation processes. Consequently, there exists a need for rapid and accurate hollow fiber permeator design procedures since exact numerical solutions not only consume considerable computer time but are highly prone to convergence problems (Antonson et al., 1977). The need for analytical or semianalytical solutions is further appreciated when computer simulation of a number of interconnected process units is required for preliminary process design purposes. Commercial hollow fiber gas permeators process fluid mixtures at moderate to high pressure. The mixture usually flows outside the hollow fibers, permeate pressure buildup occurring inside the fiber bore. As indicated by Pan and Habgood (1978), the boundary conditions at either end of the fiber will not be completely known a priori. A trial and error method was adopted by them to solve the boundary value problem numerically for permeation separation of a binary feed mixture in cocurrent or countercurrent flow on the shell side of the rigid hollow fiber module. This paper develops an approximate solution for the same boundary value problem arising out of either countercurrent or cocurrent flow patterns. The only difference between our problem and that treated in Pan and Habgood (1978) lies in our neglect of the fiber lengths embedded in the tube sheets. The binary gas separation by a hollow fiber permeator considered here assumes constant species permeability as in Pan and Habgood (1978). We note that Chern et al. (1984) have numerically studied concentration-dependent species permeability by using a dual-mode model. We are also not concerned here with the effect of fiber deformation in a hollow fiber permeator, the latter having been studied experimentally by Stern et al. (1977) for externally pressurized silicone rubber capillaries. Binary gas separation 01 96-43 1318611025-0217$01.50/ 0
with internally pressurized silicone rubber capillaries has been experimentally and numerically investigated by Thorman et al. (1975). Although hollow fiber deformation is significant in the last two studies using a rubbery polymer, their effect is expected to be minor for hollow fibers of rigid glassy polymers used in commercial permeators. In this paper, a series solution in terms of a nondimensional membrane area is developed for both the permeate and reject compositions, as well as for the pressure in the fiber bore. The method has been applied to both countercurrent as well as cocurrent flow pattern. This study is an extension of the series solution technique used earlier by Boucif et al. (1984) for a binary gas permeator with no axial pressure drop in either stream. As in Boucif et al. (1984), the series solutions obtained herein have been compared with the numerical solutions of the governing equations and the range of validity of the approximate solutions has been identified. The approximate solution technique proposed here is general in nature. Concentration-dependent permeabilities, sweep gases, asymmetric membranes, etc., can be incorporated if necessary. The solution method may also be extended to the rapid approximate design of gas permeators separating a ternary feed mixture. Governing Equations for Cocurrent and Countercurrent Hollow Fiber Gas Permeator Consider Figure 1, a and b, illustrating the hollow fiber permeators under consideration for cocurrent and countercurrent flow patterns, respectively. For simplicity, only a single hollow fiber has been shown in each permeator. High-pressure feed gas of more permeable component 1 mole fraction xf and molar flow rate Lf enters on the shell side outside the fibers at a pressure Pp A high-pressure reject stream having a more permeable component 1mole fraction x, and molar flow rate L, leaves the shell at the other end. The hollow fiber has an active section of length 1, through which permeation takes place. The inactive sections of the hollow fiber are embedded in the tube sheets at two ends. No permeation takes place through these lengths. The permeated stream with a more permeable component mole fraction y, leaves the hollow fiber bore with a molar flow rate of V, and a pressure of pw for the cocurrent scheme of Figure la. For the coun0 1986 American Chemical Society
218
Ind. Eng. Chem. Fundam., Vol. 25, No. 2, 1986
assumption 8 does ignore the cross-flow effect that exists within the porous substrate of an asymmetric fiber (Pan, 1983). Following Pan and Habgood (1978) with the only difference that species 1 is the more permeable component and species 2 is the less permeable, we can obtain the following dimensionless equations for the cocurrent hollow fiber permeator (Figure la): dx _ -
fsHELL FEED Lf,Xf,P FIBER
R+=o
w Rfr
(A) WLLOW FIBER PEFWATOR \ I I M UICLRRENT FLOW
dR,
( 5 ) w - x ) ( x - yy) - x [ ( l - x ) - y ( l - y ) ] }(1) ,,SHELL
SIDE
-(
dY2
(B) HOLLOW FIBER PERMEATOR WITH COWERCURRENT FLOW Figure 1. Schematic of a hollow fiber gas permeator having feed on the shell side and permeate inside the fiber bore in (a) cocurrent flow or (b) countercurrent flow.
tercurrent scheme of Figure lb, the permeated stream of mole fraction yf and molar flow rate of Vf leaves the hollow fiber bore at a pressure of pp In each flow pattern, the local pressure p and the local more permeable component 1 mole fraction y of the permeate stream inside the fiber bore vary from one end to the other. Normally the exit pressure in each flow pattern is specified, i.e., p f in countercurrent flow and pw in cocurrent flow. In a real permeator, these pressures will correspond to those at the exit end of the inactive section of fiber embedded in the tube sheet. In our analysis, we neglect the pressure drop in the flow of the permeate gas through the inactive section of the fiber. Consequently, the specified exit pressures will correspond to those at the end of the active section of the hollow fiber. These and other assumptions used here are as follows: 1. The feed gas pressure on the shell side is constant at the value Pp 2. The permeability of each gas species is constant. 3. Laminar flow exists inside the fiber, and the Poiseuille equation may be used for pressure drop estimation along with ideal gas behavior. 4. No axial or transverse diffusion occurs in the gas phases. 5. No purge stream is being used. 6. The viscosity of the gas mixture is assumed to be independent of pressure and composition. 7. There is no permeate stream pressure drop in the inactive sections of the fiber bore. 8. For membrane surface area and thickness (6) in the governing equations, the outside diameter Do is to be replaced by the logarithmic mean diameter DI, = (Do Di)/ln (Do/Di)for a symmetric fiber. For an asymmetric fiber, aDois to be used for membrane surface area per unit length with 6 being the membrane thickness. All these assumptions were utilized by Pan and Habgood (1978) except assumption 7 . In addition, instead of assumption 6, they assumed the permeate viscosity to vary linearly with the permeate mole fraction. The effects of these two additional assumptions are expected to be of a minor nature. We note in passing that the Pan and Habgood (1978) analysis for an asymmetric fiber based on
dRf
(")
256pRTL:
a2 -
6
DhDtPf3 (3)
The first two equations are identical with those in Boucif et al. (1984))while the last one not used by them expresses the gas pressure drop inside the rigid hollow fiber. If one has an asymmetric fiber, replace D,, in eq 3 by Do, the outside hollow fiber diameter. The relevant dimensionless quantities in the above equations are defined by
Y = P/Pf; a = Q1/Q2
P=
256pR TLf2
(
(4)
(4c)
$a2DlmD:p:
where the pressure ratio y (S1)varies along the membrane length due to a variation in p. Further, Rf is the nondimensional area for cocurrent flow measured with the origin at high-pressure feed entry point up to any location of active fiber length 1 (Figure la). If the total active length of the fiber is l,, the corresponding nondimensional membrane area is Rm for cocurrent flow. The initial and boundary conditions for the three governing equations are at 1 = 0: Rf = 0, x = xf,y = yf, y = yf, V = 0 (5) at 1 = 1,: Rf = Rm,y = y,
x = x,, V = V,, y = y ,
(6)
Note y = yf and y = yf are not specified in the problem. Usually x = xf and y = yw are known. If the problem involves designing a permeator, then one of the stream conditions at the reject end, say, x = x , (or y = y,), will be specified with Rm to be determined. For rating problems, the total membrane area and therefore Rm is known with the reject and permeate stream conditions to be determined. To complete the cocurrent flow problem statement, we indicate below the overall and more permeable component mass balances over an envelope extending from the feed
Ind. Eng. Chem. Fundam., Vol. 25, No. 2, 1986 219
end to any arbitrary location where the hollow fiber length
is I: L,=V+L LfXf = v y Lx
+
(7) (8)
The cut 6 for the cocurrent permeator is defined as
Using eq 8 for the whole permeator, one obtains Xf = ey, + (1- e)x,
a permeator design problem, one of the product stream conditions, say, x = x, (or y = yf), will be specified with R a (and therefore I,) to be determined. For rating problems, the totalmembrane length 1, is known and reject and permeate stream conditions are to be determined. The countercurrent flow problem statement is completed by including the overall and more permeable component mass balances over an envelope extending from the high-pressure reject end to any point at a distance 1 from the latter end: L=L,+V
(10)
from which the cut 0 is obtained as
Lx = L&,
(19)
+ vy
(20)
The cut 0 for the countercurrent permeator is defined as
Lf - L w Lf Lf The relation corresponding to eq 10 is
e = - =V-f
For the countercurrent flow hollow fiber permeator (Figure lb), the nondimensionalized governing equations corresponding to eq 1-3 may be written as -dx- -
a,
-dY-
a,
-
(21)
from which
A Series Solution for Cocurrent Flow Hollow Fiber Permeator To solve eq 1-3, subject to initial condition 5 and the condition y = 7, at Rf = Rm of condition 6, we assume the following series solutions for x , y, and y in terms of Rf:
+ alRf + a2Rf2+ a3RB + ... y = bo + b,Rf + b2Rf2+ b3Rt + ... y = co + clRf + c2Rf2+ c3Rf3+ ... x = a.
with the nondimensional membrane area R, being based on the fiber length 1 increasing from the high-pressure reject end and the molar high-pressure reject flow L,:
The quantity p is defined as before by eq 4c. For a total fiber length of l,, the nondimensional membrane area is Ra where
Since only one fiber has been shown in Figure la,b, the flow rates Lf, L,, V, and Vf are on a single fiber basis. For the usual multifiber information, single fiber flow rates are obtained simply by dividing it by the number of fibers. The three governing equations (12) to (14) are to be solved by using the following conditions: (i) At highpressure feed inlet R, = R a , 1 = l,, x = Xf, L = Lf, v = Vf, y = Yf, = Yf (17) (ii) At high-pressure reject stream location R,= 0, 1 = 0, x = x,, L = L,, V = 0, y = Y,, = Y, (18) Note that y = yw and y = 7, are not specified in the problem. But x = xf, L = Lf, and y = yf are known. For
(24) (25) (26)
The infinite series for each of x , y, and y will be truncated here after the Rf3 term. Substitute these expressions into eq 1-3 and collect terms of order RP, Rfl, Rf2, and Rf3. These lead to a total of 1 2 algebraic equations for the 1 2 unknowns ao, bo, CO, al, bl, c1, a2, bp, c2, a3, b3, and c3. Imposing the initial conditions 5, i.e., Rf = 0, x = xf, onto eq 24 yields a, = X f
(27)
Further, at Rf = 0, y = yf (which is unknown) leads to co =
(28)
Yf
from eq 26. We use these two results now in 2 of the 1 2 algebraic equations relating a,, bo, co, al, and bl. Fortunately, the coefficient of bl is xf - ao,Le., 0, so that we have two simple algebraic equations for al and bo to be expressed in terms of a. (= xf) and co (= yf). The equation for bo is a cubic (bo - xf) X (axf - bo + (1 - a)xfbo + (1 - a)boTf +
(01
- l)bo2yf[= 0 (29)
Since bo is obviously greater than xf, we find {(Yxf- bo + (1- a)xfbo + (1- a)bOTf + ((Y- l)bo2Yf)= 0 (30) from which the negative root (being the physically meaningful one) is bo = ([I - (1- a ) ( X f (Xf
+ yf)l - ([(I -
X
+ yf) - 11’ - 4 ( ( -~ l ) ( ~ x f l f ) ” ~ ) / [ 2-( (1)yfl ~ (31)
Ind. Eng. Chem. Fundam., Vol. 25, No. 2, 1986
220
and reject streams. With a design problem where y, is specified, solve eq 41 and 42 for unknowns yf and Rm.
Correspondingly a, = -((a - 1)Xf
+ (1 - a)$ + X f y f - aboyf + ( a - 1)xfbOyf) (32)
Note here that bo, given by eq 31, is yf for cocurrent flow. It depends on yf which is an unknown a t this stage but will be known after the whole procedure is carried out. Thus, a, and bo depend on CY, xf, and yf. When the remaining algebraic equations are solved, the following expressions for c,, b,, a2,b,, c2, a,, b,, and c3 are obtained:
(bo - xf)laal + (1- a)a,bol (34) (bo - X f N l - (1 - a)(xf + Yf - 2b07,)) - a,
+
+
+
l)a, 2(1 - a)xp, yf(a, - ab,) ( a - l)(albo + xfbl)rf)+ a12/[2(xf - bo)] (35)
a2 = -yz{(a-
+ dlR, + d2RW2+ d3Rw3+ ... y = eo + elR, + e2RW2 + e3RW3+ ... y = j o + j,R, + j2RW2 + j3RW3+ ...
x = do
(33)
c, = 0
b, =
A Series Solution for Countercurrent Flow Hollow Fiber Permeator To solve eq 12-14 subject to the specified conditions, we assume the following series solutions for x , y, and y in terms of R,:
b, = [(bo- x f ) ~ 2 b + , (bo - xf)2{aa2+ (1- a) X (a&" + albl + bocz - h 2 Y f - bo2cz)l - ( b , - al)albll/[(bo - xfI211- (1 - a)(yf + xf - 2borf))- 2ai(bo - xf)] (36) (37)
(43) (44) (45)
Each infinite series is truncated after the RW3term. The truncated expansions are substituted for x, y, and y in eq 12, 13, and 14, and terms of order Rwo,RW1,Rw2,and RW3 are collected. Twelve algebraic equations in terms of 12 unknown coefficients, do,dl, d,, d3, eo, el, e,, e3, jo,ji,j z , and j 3 ,are obtained. Using condition 18 at high-pressure reject end as x = x, and y = y, for R, = 0, we get do = x,
(46)
fo
(47)
=
Yw
Using these results in eq 13 and 14 containing unknowns d, and eo, we obtain dl = ( a - l ) x w + (I - a)xW2 xwyw - aeoyw+ ( a - l)x,eoyw (48)
+
and
+
b3 = [(bo - ~ ~ ) ~ (+a(1 a -, a ) ( ~ 3 b o a2bl + a1b2 + blcz + boc3 - 2blbzyf - 2bob1~2- bo2cJ) + (2azbz + ~3b,)(bo- xf)2+ ( b , - ~ ~ 1 ) ~ a-l b i [(2albz + azbl)(bl- a,) + (b2- a 3 a l b l l ( b ~- q ) l / ((bo - .fP[(bO - X f M - (1 - d(Xf + Yf- 2b07f)J- 3Ull) (39) c3
=-
a2P + 4 y d a l - bl) 6yf(xf- bo)
(394
All the 12 coefficients in truncated series expansions 24-26 are now explicitly available in terms of xf, a, and yf, among which yf is still unknown. For Rf = Rff corresponding to reject end location, truncated expansions 24, 25, and 26 yield respectively from condition 6
eo = ([I - (1- ~ H x ,+ 7,)l - ([(I - a ) x ( x w + 7,) - 112 - 4(a - l)aX,r,11'2)/[2(a - l h w l (49)
eo = Y w
In obtaining the last result, we rejected the physically meaningless solution of eo = .x, Further, in the resulting quadratic, the negative root, being the physically meaningful one, was accepted. This quantity eo represents the value of y at the high-pressure reject stream end where y, is specified i.e., y,. At this time note that yf may be represented as a result of eq 49a as Yf = Yw + elRwT+ e 2 R w ~+2
xf + al(Xf,a,Yf)Rm+ aZ(Xf,a,Yf)Rm2+ a3(rf7a,~f)Rt-p3 and (40)
Y, = yf + Cz(Xf,a,Yf)Rm2+ ~ 3 ( ~ f , a , ~ f ) R t(42) -r~
If now Rn or 1, is specified for a rating problem, solve the algebraic eq 42 numerically for the unknown yf for known xf, a, and y,. Substitute this value of yf in eq 40 and 41 to determine x, and y,. On the other hand, for a design problem, if, say, x, is specified (with xf, a , and y, being known), then solve eq 40 and 42 simultaneously for two unknowns, yf and Rm The value of y, is obtained from eq 41 by just substituting the values of yf and Rm determined above. Knowledge of x, and y,, for example, allows calculation of 0 from eq 11. Similarly, other mass balance relations may be used for determining flow rates of product
(49b)
For a permeator with given feed conditions and exit pressure ratio, yf, define
x, =
yw = bO(Xf,a,Yf)+ bl(xf,a,yf)Rm+ b2(Xf,a,Yf)Rm2+ b 3 ( ~ f , a , ~ f ) R(41) m~
(494
x, - y, - elRe - ezRwT2- e3Re3 = A
(49c)
xf - yw- elRwT- e2RwT2 - e3RwT3= B
(49d)
By solving the remaining algebraic equations, we obtain the following explicit expressions for the remaining coefficients in the series expansions 43, 44, and 45: fl
=0
(50)
d, = 1/[(a- l ) d l + 2(1 - a)x,d, + d , ~ ,- 2e,yw + ( a - l)(dleo + x,el)ywl - d12/[2(eo- x,)l (52)
Ind. Eng. Chem. Fundam., Vol. 25, No.
f3
=
-4fof2(el - dl)A2 - B2dzP 6A2r,(x, - eo)3
(55)
e3 = ((eo- ad^ + (1- a)(d3eo + d2e1+ d1e2 + elYw+ 4 3 - 2e1e2yW- 2e0eJ2 - eo2f3)I + (2e2d2+ e,d3)(eo - Y , ) ~ + [(el - d1)(2e2dl+ (ez - dz)eldll(eo- x d - eldl(e1 - dJ21/ - (eo - do)((l- a)(do + Y, - Z~OY,) - 111) ((eo(57) Thus, the 12 coefficients in expansions 43, 44, and 45 are expressed explicitly in terms of xf,a,x,, y,, and R a . From definition 16, Ra may be expressed
For a rating problem with 1, known, Ra is then a function of yf and x,. For a design problem with yf specified, Ra is a simple function of I, and x,. For a design problem with x, specified, Rfl is a function of 1, and yf. With that perspective, at R, = Ra, corresponding to feed end location, conditions 17 applied to truncated expansions 43, 44, and 45 yield respectively
xf = x,
2, 1986 221
For a rating problem, 1, or membrane area is specified. With R a given by eq 58, we see that Rfl may be replaced in terms of unknowns yf and x,. Further, with A and B expressed by eq 61a, we have three algebraic equations, eq 59,60, and 61, for three unknowns, x,, yf, and y,. These are to be solved simultaneously in a computer to yield values of x,, yf, and y,. Although the countercurrent permeator problem based on series solutions is numerically more demanding, it is far simpler than solving a boundary value problem with three coupled differential equations and guessing the conditions at the reject end with backward integrations. Numerical Solutions for Countercurrent and Cocurrent Hollow Fiber Permeators It is necessary to recognize first that since there is a pressure drop inside the fiber luman and since the permeate pressure can be specified only at the exit end of the fiber, both cocurrent and countercurrent flow patterns become boundary value problems here. This situation is unlike that considered in Boucif et al. (1984), where the no pressure drop cocurrent permeator was an initial value problem. For either flow pattern, before one can solve the system of differential equations, one needs to know the value of y at the closed end of the fiber. This is done by using the cross-flow criterion at that point (Pan and Habgood, 1978). For cocurrent flow, it means that, at 1 = 0, y = yf is obtained from Y f / ( l - Yf) =
-
rfiYf)I/(l - xf - dl - Yf)J
(62)
One has to select the right root in the above quadratic. The value of dy/dRf at the closed end of the fiber is indeterminate and the L'Hospital rule has to be used. For cocurrent flow, we have
+ dl(a,xf,X,,YW)Ra + dz(at~ft~w,~w)Rp/r~ + d3(a,Xf,Xw,Yw,RwT)R,' (59)
Yf = Y
+
e2(a,Xf,Xw,Yw)Ra2
Yf
+ + e3b,Xf,Xw,Yw3a)R%.T3 (60)
~ ( ~ , ~ f , ~ ~el(a,xf,x,,yw)Ra , ~ w )
= Yw
+ f2(a,Xf,Xw,Yw,RwT)Rf12+ f 3 ( a , X f , X w t r w & T ) R w T
3
(61) Consider permeator design problems now. Suppose x, is specified for known xf,a,and yp Then in eq 59 and 61, there are two unknowns, yw and Ra. Solve these two algebraic equations simultaneously and numerically for yw and Rfl Substitute these values in eq 60 to determine yp The value of 1, is next obtained when yf and x, are substituted in eq 58. Obtain 0 now from eq 23 and L, from eq 21. If in a permeator design problem yf is specified for known xf, a,and yf, the unknowns are x,, y,, and Ran. Note here that from eq 49c and 49d
A = x, - yf and B = xf - yf
For countercurrent flow, equations similar to (62) and (63) have to be used to obtain the value of y at 1 = 0, i.e., y, and its derivative with respect to nondimensional area. However, replace yf by y,, yf by y,, and xf by x,. For actual numerical solution, the countercurrent problem, as defined by eq 12-14, was reformulated in a different way with Rfwas the independent variable instead of R,. This was done because L,, needed to compute Rfl, is not known a priori. At this time, it is advantageous to incorporate V and L as two extra variables in the set of ordinary differential equations:
(614
Therefore, the Coefficients d3and e3 (which have R a dependence due to their dependence on f i and f 3 which in turn depend on R, due to eq 49c and 49d) depend simply on x, and known yr. Further, fi and f3 do not depend on R,; instead they depend on x, and known yP Thus,there are three algebraic equations, eq 59,60, and 61, with three unknowns, x,, y,, and Rat with Ra never appearing with a power greater than 3. Solve these three simultaneously to obtain x,, y,, and Rfl From eq 58,1, is known for the value of Re obtained. The rest of the procedure is straightforward.
The associated boundary conditions are atR,c"=O, V = O
(64b)
at R,""= RmCC, L = Lf
(644
This formulation is similar to that adopted by Sengupta and Sirkar (1984). For solving the boundary value problems of cocurrent or countercurrent flow, an available packaged software
222
Ind. Eng. Chem. Fundam., Vol. 25, No. 2, 1986
(IMSL subrouting DVCPR) was used. The routine is based on the PASVA3 program (Pereyra, 1978) with adaptive finite difference approach. To use this routine, it is usually necessary to supply initial estimates of the values of each of the variables at user-selected grid points across the domain, including the boundaries. The initial estimates can be generated in the present case by solving the same problem once with artificially simplified boundary conditions. One can first solve the cocurrent no pressure drop problem, for example, by assuming that yf is the same as y, (which is known for cocurrent problem) and calculating yf from eq 62. Since there is no pressure drop, one needs to solve now only eq 1 and 2 from Rf = 0 to Rf = Rm. This initial value problem can be solved by using, e.g., a fourth-order Runge-Kutta method. The values of each variable at user-selected grid points across the domain are stored in the computer memory. These are used as the starting values by the routine DVCPR. To operate the routine, the user needs to supply the derivatives, the boundary conditions, and the Jacobian matrix for the actual system of differential equations. The subroutine initiates a series of iterations which gradually improve the values of the variables until the actual set of differential equations and boundary conditions are simultaneously satisfied. The solution technique is same for countercurrent flow. For the numerical calculations, two sets of parameters have been used as far as feed pressure, feed flow rate, and permeability of the less permeating component are concerned. For both sets of parameters, however, the temperature, viscosity, and fiber dimensions have the same values. They are as follows: 7' = 25 OC = 298 K; p = 0.015 CP = 1.5 X Pa s; Di = 150 pm = 1.5 X m; Do= 300 pm = 3.0 X m; D1, = (Do - Di)/ln (D,/Di) = 2.164 x m. Calculations have been carried out for CY = 5, 10, 25, 30, and 80. For CY = 5 or 10, the following values of Pf,Lf, and Qz/6 were used: Pf = 25 atm; Lf = 65 cm3 (STP)/min = 4.491 X mol/s; Q 2 / 6 = 8.8 X lo* cm3 (STP)/(s cm2 cmHg) = 2.7362 X lod9mol/(s m2 Pa). For values of (Y of 25, 30, or 80, the following corresponding values were used: Pf = 40 atm; L f = 60 cm3 (STP)/min = 4.145 X mol/s; Q2 6 = 1.82 X lo4 cm3 (STP)/(s cm2 cmHg) = 5.659 X 10- mol/(s m2 Pa).
1
Calculations have also used three different exit values of y, 0.1, 0.25, and 0.05, and three different values of xf, 0.209, 0.05, and 0.50. Only a single fiber has been used in the calculation under conditions such that a very significant pressure drop takes place in the hollow fiber bore. Multifiber calculations can be easily adapted to the present arrangement simply by obtaining the feed flow rate per fiber which is Lf in our calculations. This is possible since shell side flow effects and pressure drops (if any) are of no importance to the problem. Further, for membrane area specification, fiber number and length are independent quantities with the length to be determined, usually for given fiber numbers. The numerical solution of algebraic equations obtained from the series solutions are obtained by using a simple Newton-Raphson technique. The IMSL subroutine ZSPOW was used for solving either one algebraic equation containing one unknown or simultaneously two algebraic equations containing two unknowns, The CPU time required for these calculations was, in general, 8-9 times smaller than the corresponding time required for numerical solutions. For the purpose of comparing cocurrent and countercurrent permeators with given a, xf,and exit values of y and a, countercurrent permeator results from series
Y
:.3
COCURRENT ,, =
I
c I
v
'2.0 i / 0.0 5 0.5 1.0 1.5 2.0 2.5 3.0 3.5 G @
TOTAL PERMEATOR LENGTH ( i w ) , meter
Figure 2. Comparison of series and numerical solutions for cocurrent hollow fiber permeators for xf = 0.209, CY = 5, yw= 0.1,Pf= 25 atm, Lf = 65 cm3 (STP)/min, and Q2/6= 8.8 X lo4 cm3 (STP)/(s cm2 cmHg).
solution are expressed in terms of Rm" instead of Re with 1, however, starting from the high-pressure reject end. Since cocurrent results are expressed against Rm,both flow patterns can be compared on an equivalent basis for equal 1, and LP
Comparisons of Series Solutions with Numerical Solutions Cocurrent Hollow Fiber Gas Permeators. We present below the results of calculations in terms of values of exit compositions, closed-end permeate pressures, and stage cuts plotted against I, the tote1 permeator length, for different cocurrent hollow fiber gas permeators. Two types of results are plotted in each figure. The solid lines represent the results obtained by using the series solutions developed here. The dashed lines display the numerical solutions for the same parametric conditions. The juxtaposition of the numerical results on the series solutions will identify the ranges of validity of the series solutions. It is useful to recall here that the usual ranges of cut B encountered in commercial permeators vary from around 0 to about 0.45-0.5. Thus, we will not be concerned with the accuracy of the series solution predictions for high values of cut. Figure 2 compares the cocurrent series solution predictions of y,, x,, pf, and 8 against I, with a numerical solution for xf = 0.209, CY = 5, p w = 2.5 atm, and Q2/6 = 8.8 X lo4. Each value of 1, represents a different permeatur for given operating conditions. The conditions are such that, at the highest membrane length used (I, = 3.8 m), the value of cut 8 is almost 0.5. Further, there is a significant pressure drop such that, for a permeator of 1, = 3.8m, p f is around 5 atm with p , = 2.5 atm. Note that, unlike in Boucif et al. (1984), the curves of y, and x , against I, cannot be used to indicate the composition profiles along a cocurrent permeator. Further, the value of pf for any permeator with a given 1, is only indicated in Figure 2. Thus, pf values, as shown, apparently rise with l,, whereas an actual p profile decreases along the length of a cocurrent permeator from a high value of pf to p,. It is clear from Figure 2 that, for the low value of CY = 5, the predictive accuracy of the series solutions for yw,x,,
Ind. Eng. Chem. Fundam., Vol. 25, No. 2, 1986 223 5rn
5
-
Lc
COCURRENT
LI
u.10
-
1.0
L
c.
a.80
= = Y
p: Y
Y)
z Y
0.8
c Y
sec-cm2-cmHq
L yl
9 5
y 0
su.
D
2 Y
PERMEATE HOLE FRACTION
5
SERIES SOLUTION NUMERICAL SOLTUION
10.0
7.5
I
0 ZZ
m
REJECT MOLE FRACTION (x,)
1 .o
1.5
2.0
n
; Pf = 40 a t m ;
02 6
0.209;
=
1,82
p,
0
z W a
stdcm3 sec-cm2-cmHg
wl Y
s
U
g 5 wl
0.4
6.0
- SERIES
SOLUTION
_ _ _ _ -NUMERICAL -
12.0
SOLUTION
0.2
STAGE CUT
(e)
0.0
'
g.0
1.0 MOLE REJECT F R A C T I O N (x,)
2.5 0.0
TOTAL PERiEATOR LENGTH ( e w ) , meter
0.50
0.25
Figure 3. Comparison of series and numerical solutions for cocurrent hollow fiber permeators for xf = 0.209, a = 10, yw = 0.1, and other parameters same as in Figure 2. E
-Ig 402
1.0
= 4 atm
U
2.5
0.5
xf
z
-
a
5.0
0
= 60
I-
/
0.0
Lf
0.6
YI
s
n c Y
6h -n
PI W
(Y,)
!U
&
Y
a m Lo
PERMEATE MOLE FRACTION
0.8
I. 0 25
1 .oo
0.75
T O T A L PERMEATOR LENGTH ( E,),
meter
Figure 5. Comparison of series and numerical solutions for cocurrent hollow fiber permeators for xf = 0.209, a = 80, y,,, = 0.1, and other parameters same as in Figure 4.
Y-
COCURRENT a.30
n
- SERIES
_____
D LUClll
m, n
; P,
= 40 a t m ;
p,
= 4 atm
COCURRENT a-1 0
SOLUTION
NUMERICAL SOLUTION
i
'p: A
E? !? 0.61 L
'
I/)
-5.0 REJECT MOLE F R A C T ~ ~(x,) N 0.0
I
0
0.5
I
I
1 .o
1.5
TOTAL PERMEATOR LENGTH ,( k,),
2.0
2.5
meter
0 . o y 0.0
,
I
1 .o
'
I
I
2.0
I
I
I
TOTAL PERMEATOR LENGTH (k,),
I
4.0
3.0
.o
meter
Figure 4. Comparison of series and numerical solutions for cocurrent hollow fiber permeators for xf = 0.209, a = 30, yw= 0.1,Pf = 40 atm, Lf = 60 cm3 (STP)/min, and Q 2 / S = 1.82 X 10" cm3 (STP)/(scm2 cmHg).
Figure 6. Comparison of series and numerical solutions for cocurrent hollow fiber permeators for xf = 0.209, a = 10, yw = 0.25, and other parameters same as in Figure 2.
and 6 is very high, even at the highest cut level used. The accuracy in pf prediction at the highest 0 values is slightly lower. However, the effect of this deviation in pf values is highly attenuated insofar as predictions of y,, x,, and 0 are concerned. Since for design or rating purposes one is interested only in y,, x,, and 6 for a particular ,I we conclude that the series solutions are highly efficient for a = 5. Figure 3 explores the comparative behavior when a is increased to 10 while all other conditions are exactly those used in Figure 2. The accuracy of x, predictions appears to be very high. At the highest 1, (= 2.8 m) corresponding to a cut of 0.40, the error in predicting y , is less than 6% while that in predicting 0 is around 8%. The error in
predicting pf is similar to that in Figure 2. When a is increased to 30, the comparative behavior shown in Figure 4 indicates that the series solutions are quite accurate up to a 0 N 0.20 beyond which errors in 0 and x, predictions become considerable. Figure 5, calculated for a = 80, similarly illustrates that, beyond a cut of 0 'v 0.14, considerable errors are encountered if series solutions are used. The behavior is similar to that observed by Boucif et al. (1984). Note that the values of Lf, Pf,pw, and Q z / 6 in Figures 4 and 5 are different from those in Figures 2 and 3. The cases considered so far utilized yw = 0.10. Figure 6 explores the results for a higher yw = 0.25 at CY = 10, zf = 0.209, Pf = 25 atm, and p , = 6.25 atm. Other conditions
224
Ind. Eng. Chem.
Fundam., Vol. 25, No. 2,
1986 c w m
i
1
.
Lc
a
I
i
rn 3 LT
LT 4
w a
a rn
stdcm3 if = 60 min ; Pf = 4 0 a t m ; p,
Y
c 4 4 c
ti a n Y z
n Y
SERIES SOLUTION hUMER!CAL SOLUIION
in
0
u _J
- 10.0 STAhE CUT
0
,
ol/ 0.5
c
4 atm
Iii
'o'6t x
u 4 c:
= 0.50;
f
-S E R I E S
0 I:
I
----
9
4,
stdcm'
(2) = 1 82 x 5
Ba Y ser-:mi-cmH
SOLUTION
/
NUMERICAL SOLUTION
w
v)
"
0
i
-
16.0
- 5.c
-
!2.0
-
-
8.0
- 7.5
REJECT, MOLE FRACTION (x,) 2.0 2.5
1.5
1 0
=
2.5 0.0
- 4.0
3 0
TOTAL PERMEATOR LENGTH ( e w ) , m e t e r
0.0
I
I
1
I
0
Figure 7. Comparison of series and numerical solutions for cocurrent hollow fiber permeators for xf = 0.209, a = 10, yw = 0.05, and other parameters same as in Figure 2.
PERMEATE MOLE F R A C T I O N ( y , )
410 1
i
7.5
0.2
45 3
p -
.__
- - - NUMEQICAI
Y
t
z
a
SERIES SOLUTION
I
W
c LT
SOLUTION
J
a
0.2-
- ,? . 0
0 L.
Q
6.0
"?
0.0
iL_ ,
I. 2
t
0.5
I
I
1 .o
TOTAL PERMEATOR LENGTH
1
1.5
I
0.0 2 0
0.1
Pf
r--
4.0
peter
( r w ~ ,
Figure 8. Comparison of series and numerical solutions for cocurrent hollow fiber permeators for xf = 0.50, a = 10, yw= 0.1, and other parameters same as in Figure 2.
are the same as those in Figure 2. To the extent operation at a higher y is tantamount to operating at a lower a,we notice that the series solution predictive accuracy is somewhat higher than those shown in Figure 3 at comparable cuts. In fact, one can go &s far as a cut of 0 = 0.45 with little error. Reduction of the yw to 0.05 and therefore of pw to 1.25 atm, on the other hand, is equivalent to increasing a. Thus, we see in Figure 7 somewhat higher errors at the highest B N 0.40 than those observed in Figure 3, where yw = 0.1, i.e. pw = 2.5 atm, was used. The feed mole fraction was, so far, fixed at xf = 0.209. When the feed composition is increased to xf = 0.50 in Figure 8 for a = 10, all other conditions being similar to those in Figure 3, we observe, as in Figure 3, that the predictive accuracy is quite good up to a cut of B N 0.40. Thus, changing xf from 0.209 to 0.5 did not materially affect the range over which the series solutions are accurate. This is further reinforced if we observe the behavior
T O T A L PERMEATOR LENGTH ( O w ) ,
peter
Figure 10. Comparison of series and numerical solutions for cocurrent hollow fiber permeators for xf = 0.05, a = 80, y.+, = 0.1, and other parameters same as in Figure 4.
for a = 25, xf = 0.50, and pw = 4 atm (yw = 0.1) in Figure 9 where the series solutions are quite efficient up to a cut of about 0.35. However, when a xf = 0.05 is chosen for a = 80 and pw= 4 atm (yw = O.l), we find from Figure 10 that beyond a cut of 8 N 0.05 accuracy is low, unlike that in Figure 5 where accuracy is reasonable up t o 0 = 0.14. This is somewhat expected since a cut of 0 0.05 for xf = 0.05, a = 80, and yw = 0.1 is quite high. It is to be noted that all conditions in Figures 9 and 10 except xfand cy are exactly the same as those in Figures 4 and 5. Countercurrent Hollow Fiber Gas Permeators. We next present the results for countercurrent hollow fiber gas permeators using essentially the format followed for co-
Ind. Eng. Chem. Fundam., Vol. 25, No. 2, 1986 225 E Q m
1 .o
d
COUNTERCURRENT a=5 L f = 65
-$$$;
0.8 Xf
10
0.209 ;
T
0
Pf
42
=
25 a t m ;
i
8.8 x 10-6
pf = 2.5
v
COUNTEXCURREE a=30
2
-':!
0
v
atm
Stdcm3 sec-cm2-crnHq
0 c
__ SERIES SOLTUION
3
5
t
'
NUMERICAL SOLUTION
0
-----
w
o'6-
xf
x
=
0.209 ;
9
(2) = 1.82 x
s e c - c m -cmHq
6
W 0
m
Ln
2 0
c a u
w
SERIES SOLUTION
.
_ _ - - _ _NUMERICAL
w
-16.0
SOLUTION
5
v)
-12.0 0.2
4.0 0.5
0.0
2.5 3.0 TOTAL PERMEATOR LENGTH ( Pw), m e t e r
1.0
1.5
3.5
2.0
4.0
Figure 11. Comparison of series and numerical solutions for countercurrent hollow fiber gas permeatore for xf = 0.209, cy = 5, Tf = 0.1, and other parameters same as in Figure 2.
COUNTERCURRENT a-1 0
Lf
=
0.0
0.0 0 TOTAL PERMEATOR LENGTH
(aw), m e t e r
Figure 13. Comparison of series and numerical solutions for countercurrent hollow fiber permeators for xf = 0.209, 01 = 30, -yf = 0.1, and other parameters same as in Figure 4.
-
E
-3
Y rn
CI
cz Y
65 stdcp3. P = 25 a t m : p f = 2 . 5 a t m min ' f
3
I!
PERMEATE MOLE F R A C T I O N ( y f )
- -----_
I
7.5
0.4
I-
------
i
W
VI
0
u -J
NEMERICAL SOLLlTION
16.0
12.0
2.5
STAGE CUIJB&-
0.0
0 5
1 0
1 5
2 0
TOTAL PERMEATOR LENGTH (Lw), m e t e r
2 5
I?
n
SERIES SOLUTION
5.0
0.0 0 0
= 3 yi VI
--------
0.8
10.0
4
6.0
3 0
Figure 12. Comparison of series and numerical solutions for countercurrent hollow fiber permeators for xt = 0.209, a = 10, yr = 0.1, and other parameters same as in Figure 2.
current permeators. Note, however, that 1, for the following figures starts from the high-pressure reject end. Further, the closed-end permeate pressure p , is plotted here instead of p p As before, solid lines represent series solutions while the dashed lines are obtained from a numerical solution. The curves depict the behavior of yf, x,, p,, and 8 against 1, with each 1, representing a different permeator if Lfis fixed. Figure 11obtained for CY = 5 , xf = 0.209, and pf = 2.5 atm (all other conditions being similar to those of Figure 2) shows that the countercurrent permeators display a behavior very similar to that in cocurrent permeators. The percent errors in yf and 0 at the highest cut of 8 = 0.47 at 1, = 3.8 m are around 4 % and 8%, respectively. Note that the error in countercurrent flow series solution predictions is somewhat higher than in cocurrent flow. An additional feature worth noting is that the countercurrent permeator is producing a richer permeate, a
4.0 0.0 0.0
0.0 0.25
0.50
0.75
1 .oo
1.25
TOTAL PERMEATOR LENGTH ( tw), m e t e r
Figure 14. Comparison of series and numerical solutions for countercurrent hollow fiber permeators for q = 0.209, a = 80, -yr = 0.1, and other parameters same as in Figure 4.
leaner reject (lower x,), and a higher cut at the same value of the hollow fiber length, 1, if the two numerical solutions are compared. Further, any cocurrent permeator has a higher pressure generated at the closed end than that in a countercurrent permeator with the same fiber length, other conditions remaining the same. This feature will be observed in all of our calculations and will be discussed more fully elsewhere. The efficiency of the series solutions is investigated next in Figure 12 when CY is changed to 10, all other conditions remaining the same as in Figure 11. Again, the behavior of the series solution vis-8-visthe numerical solution is very similar to that with cocurrent flow permeators (Figure 3). In fact, the percent errors for the largest permeator (1, =
226
Ind. Eng. Chem. Fundam., Vol. 25, No. 2, 1986 45
m
1 . r -
__^_
1 COUNTERCIJARENT ,,=lo
-I ,-I
213
i , 25
? . 5'1 T e
JTB? 'EPME;TC"
C.!i .L'IGTI [ ,
---------Db
1.3: 'I,
'.25
i
?.'
1.53
meter
Figure 15. Comparison of series and numerical solutions for countercurrent hollow fiber permeators for xf = 0.209, a = 10, yf = 0.25, and other parameters same as in Figure 2.
Figure 16. Comparison of series and numerical solutions for countercurrent hollow fiber permeators for xf = 0.5, a = 10, yf = 0.1, and other parameters same as in Figure 2.
2.8 m) in yf and 8 are around 7% and 9% with 8 around 0.40. Increasing a to 30 for countercurrent permeators (Figure 13) indicates that the series solution accuracy is quite good up to 8 N 0.24. Series solution accuracies decrease beyond 0 N 0.16 if cy is increased to 80 for countercurrent permeators (Figure 14). Changing the pressure ratio yf from 0.1 (Figure 12) to 0.25 and therefore pf from 2.5 atm to 6.25 atm for CY = 10 leads to higher predictive accuracies as shown in Figure 15. Thus, the behavior of the series solution for countercurrent flow is essentially similar to that for cocurrent flow (Figure 6) when pressure ratio is increased. Although we are not showing any results here for yf = 0.05, the behavior again is essentially similar to that in cocurrent flow; i.e., for the same conditions, accuracy at a given cut is reduced by decreasing y. The effect of a change in feed composition from xf = 0.209 to xf = 0.50 for a = 10 and pf = 2.5 atm (yf = 0.10) is shown in Figure 16. As observed in Figure 8 for cocurrent flow, predictive accuracy is quite good up to 8 = 0.45. Although we are not showing here any results for countercurrent flow for a low xi = 0.05, the behavior is essentially similar to that observed in cocurrent flow (Figure 10).
not too large. Thus, such solutions are useful for preliminary process design or performance analysis since very high cuts are never encountered in commercial permeators. An additional aspect needs to be stressed. The structure of a numerical design program for a hollow fiber gas permeator has to be quite different from that for a numerical rating program. No such distinction is necessary for the approximate series solutions developed here. The difference between the various problems lies only in the number of algebraic equations to be solved simultaneously. Our choice of a and xfvalues reflects conditions likely to be encountered in industrial membrane gas processing. For example, 02-N2 separation factors through cellulose acetate (CAI or PRISM membranes are around 5-6 with air as feed. Acid gas purification of natural gas (a C02-CH4 system, say) encounters a separation factor of around 25-30 through the same membranes (Mazur and Chan, 1982). Recovery of H2from H2
