Gas Separation by Permeation*
Part I, Calculation Methods and Parametric Analysis
C.-Y. PAN mtd H . W . HABGOOD
Alberta Research Council, 11315
- 87th
Calculation methods for single- and multi-stage permeation
of a multi-component mixture are presented. The use of the
local permeate concentration in the formulation results in a
relatively simple form of solution for the binary system in the
cross-flow pattern. A parametric analysis of the permeation
cascade is given, For a given permeability of the more permeable component, there exists a membrane selectivity which
gives rise to a minimum membrane area requirement in the
cascade. It is shown that the compression power requirement
is dependent only on the operating pressure ratios, irrespective
of actual pressure levels in each stage. A study on the effect of
interstage mixing loss shows that the ideal cascade (with no interstage mixing loss) is not necessarily always the tliermodynamically most efficient one; some non-ideal cascades may have
higher efficiencies due t o more effective permeation steps.
Exarnples given relate to He/CH, and Oz/Nz separations.
s
elective permeation through a membrane i s an
increasingly important separation technique for
gas mixtures. The technology of hollow fine fibers and
asymmetric (or composite) membranes recently developed for reverse osmosis“*’ may soon make this
process a standard chemical engineering unit operation. The development of asymmetric membranes in
which the separation takes place in an ultra-thin skin
over a porous supporting layer permits the use of
highly selective polymers with inherently rather poor
permeabilities. The dream membrane of high selectivity and high permeation flux may soon be an
easily achievable reality. The permeation process may
be considered as an extension of the gaseous diffusion
lx-ocess f o r uranium enrichment. But the well developed theory of isotopes ~ e p a r a t i o n ‘ ~ ”is~ ’based on
the assumption of low membrane selectivity, and is
not directly applicable to permeation processes with
highly selective membranes. For permeation systems
with all possible selectivities, only the single-stage
permeation of a binary mixture has been studied
extensi~ely“’-*~’.
The purpose of this study, which is
part of a research program for the recovery of
helium from natural gas, is to develop calculation
methods f o r multicomponent systems in single-stage
and cascade operations, and to analyse the process.
The permeate enrichment in a permeation stage is
generally limited by t h e membrane selectivity and
feed/permeate pressure ratio‘’1’. Thus most permeation processes may require a multistage cascade
to carry out a given separation. The simplest cascade
arrangement is to compress the permeate of one stage
so that it becomes the feed of the succeeding stage.
The various stage residues could be discarded but,
to increase recovery, t h e cascade may be made recycling by having the residue from each stage recycled to the feed of the previous stage as shown in
Figure 1. We then have the concept of ideal cascade,
‘Alberta Research Council Contribution No. 876
Avenue, Edmonton, Alberta T4G 2C2
On presente des mkthodes de calcul dans le cas de la
permkation mouo&ag6e ou multietagh d u n mblange de plusieurs composants. L’emploi, dans la formule de la concentration locale de l’agent de permhtion, permet d’obtenir une
forme relativement simple de solution pour le systLme binaire
en ecoulement transversal. On prksente une analyse paramktrique de la cascade de permkation. Dans le cas d u n e permkabilitt. donnee du composant le plus permkable, il y a
une sklectivitC de la membrane qui donne lieu Q une exigence
minimale relativement Q l’aire de cette membrane dans la
cascade. On constate que l’exigence relative 21 1’Cnergie de
compression ne depend que des rapports de pression opCratoire, quels que soient les degrks rCels de pression dans chaque
&age. Une etude de l’effet de la perte due au mklange entre
les &ages indique que la cascade idkale (dam laquelle il n’y
a pas de perte due au melange entre les ktages) n’est pas
nkcessairement toujours la plus efficace au point de vue thermodynamique; quelques cascades non idkales peuvent fournir des
rendenlent supkrieurs par suite d & a p a de permeation plus
efficaces. On donne des exemples qui ont trait aux skparations
de 1’hClium du mkthane e t de l’oxyghne de l’azote.
as described by Hwang and Kammermeyer‘22’25’,in
which the cascade design ensures that the composition
of each residue is identical with that of the feed
stream with which i t is mixed. In other words, a n
ideal cascade is one with no interstage mixing losses.
For a permeation system with more than two permeable components, however, it is impossible to match
the concentrations of all components, and we define
a multi-component ideal cascade as one having
identical concentrations of t h e desired component in
the residue of a stage and the feed of the preceding
stage. This concept was f i r s t introduced by De La
Garza et al‘za)in multicomponent isotope separation.
The characteristics of the ideal cascade and the
effect of interstage mixing loss in the non-ideal cascade a r e analyzed in this paper. We note in passing,
however, t h a t even with the ideal cascade there a r e
mixing losses within each stage of a cascade because
of the variation of local permeate concentration. The
significance of this is briefly discussed in a following
section on stripping stages, and a detailed analysis
will be given in a separate paper.
Stage calculation
Calculation problems for the single stage permeation of a binary gas mixture have been studied extenSi~elY~11-14,1?-~)
and an iterative computation method
for the multicomponent system has also been described by Stern e t al””. I n the following a calculation method for the permeation of a multicomponent mixture in t h e cross-flow pattern shown in
Figure 2 is presented. It is basically an extension and
partially a revision of what is now a classic treatment of Weller and Steiner“1*’2’for the binary system. The treatment here is simplified by using the
The Canadian journal of Chemical Engineering, Vol. 56, April, 1978
197
I-
1-
STRIPPING SECTION
2
1
ENRICHING SECTION
1
-1-
2
-
Figure 2 Permeation stage in cross-flow pattern. y, is local
permeate concentration and <-is average concentration of the
permeate fluxes.
FE€D
Figure 1 -Permeation cascade. L, is an imaginary stream
(shown as a dotted line) entering the feed to the last enriching
stage representing the unbalanced flows of the non-key components in the cascade during iterative calculations. A portion
of the product is recycled to the feed of the last enriching
stage.
local permeate concentration as a parameter of the
solution. This results in a relatively simple form
of solution for the binary system.
Multicomponent sy&em
Permeation for the cross-flow pattern shown in
Figure 2, may be described by the following equations
-d(Lxi)
- - ( Q i / d ) (Pxi - pyij
ds
two member equations of Equation (1) becomes
where
=
1, n . . . . . . . . . ( 1)
....................................
( ~=i Q i / Q i . .
y
=
P / P . ,......................................
. (6)
.(7)
Throughout this paper, the component 1 (Q,# 0) is considered to be the base component to which the selectivity
ai is referred (a1= 1).Solving for yj from Equation (5)
and then substituting into Equation (4) yields
n
i
!
j z
= 1 aipini
+ Y ( a j -Yail; Yi . = 1
a,xj
i
=
2 , n . . . . . . . . (8)
Eimilarly, solving for xj from Equation ( 5 ) and then
substituting into Equation (3) yields
n
2:
1
n
z
xi
=
1 . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
yi = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi =
,(4)
yo =
1
+ Y ( a i Yo - 1)
aipi
Yi
Yo
.
Yi
pi
i = 2 , n . . . . . . . . . . (9)
n
;
Y j . . . . . . . . . . . . . . . . . . (10)
n
2 yj/aj= 1 -
L
1
where yi is the local permeate concentration and the
product concentration is the average of 34 over the
entire membrane. Equation (2) is the characteristic
of the cross-flow pattern where the permeate flow
may be in any direction in the geometrical plane
perpendicular to the feed flow. Such a flow pattern
is closely approximated by the spiral-wound module
and the hollow fiber module with a porous central
feed distribution tube. It is also possible that the
cross-flow pattern may describe the permeation in
a permeator module of any configuration using asymmetric-type membrane, if the porous supporting layer
of the membrane prevents the mixing of the permeate
fluxes ow the surface of the skin layer. The above
equations are subject to the following assumptions :
(1) negligible g a s - p h w s concentration gradients
in the permeation direction :
(2) negligible pressure drop of the feed and permeate stream;
(3) diffusion along the flow path is insignificant
compared to the bulk flow.
In the following formulation, we seek the differential relationships of the concentration variables
without involving the flow rate and membrane area,
so that the method of solution can be simplified by
calculating the concentrations, flow rate, and membrane area separately. To do this, we first derive
the relation between the feed side concentrations (x)
and the permeate-side concentrations (y)
With the aid of Equation (2), the ratio of any
198
1
.(3j
=
piy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
=
1 -
................................
l/O!i..
(11)
. (12)
Note that Y I = 0 since PI = 0. Equation (8) expresses each Yr in terms of 5s;
without involving
other Yifs and can be transformed into a polynomial
equation of iY4. Equation (9) expresses each st in
terms of Y,'s (this relation can also be obtained from
Equation (8), but i t requires the solution of (n-1)
simultaneous algebraic equations).
Equation (2) can be written in the form
dln(1 - 0) =
dxi
~
Yi
- Xi
i
=
2, n . . . . . . . . . . . . . . . . ..(13)
where
0
=
1
- L/L, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
..(i4)
L, i s a known flow rate usually given a t the feed
or residue end of the permeator. 0 is the fraction
of feed permeated (i.e. stage cut) if Lf is the feed
flow; if Lt is the residue flow, then (1- 0) is the
ratio of feed-side flow to residue flow. Note that
Equation (13) with i = 1 is not independent since i t
can be derived from other member equations. Equation (13) is identical to the Rayleigh equation for
batch distillation and gives ( n - 2) differential
relations among the concentration variables. These
relations can be transformed into the relations
among permeate-side concentrations Y,'s by the sub-
The Canadian Journal of Chemical Engineering, Vol. 56, A p i l , 1978
~
stitution of zr from Equation (9) or the relations
among feed-side concentrations xr)~by the substitution of Y i from Equation (8). The resulting n-2 differential equations can be integrated to obtain the
functional relation among Y l s or s l s (i =2, n).
Generally, i t is more convenient to work with the
variable Yt than xl, because no analytical form of
solution for Y , in terms of xi's can be obtained from
Equation 8 (except for systems with only two permeable components). Consequently, the following
equations are formulated in terms of Y4. Substituting
Equation (9) into Equation (13) yields
i = 2.92..
Dividing Equation (1) with Equation (2) and then
taking the sum of the resulting ratios for each component yields
dS = Yod o , . . . . . . , . . . . . , , . , . . , . . . . . . . . . , . . . . . . . , (16)
where
S = (Qi/d) ( P / L j ) (1 - 7 ) s . . , . . . . . . , . . , . . . . . . . . . .(17)
Equations (15) and (16) may be written in the
following form for convenient solution
+
(anyo- 1) (1 - y
yaiY,) (d Y i/Y i) - (&Yo - 1)
(1 - Y W ~ Y , )( d Y J Y d
i = 2 , n - 1 . . . . (18)
-(1 -'y)(afi-ai)dY,=O
+
parameter used to relate the variable Yi's, it also has
some physical meaning, $Yocan be considered as the
concentration of an imaginary component according
to Equation ( l o ) , and is also equal to the ratio of
an imaginary flux that could have been generated by
a feed stream containing only the base component to
the actual flux produced by the mixed gas permeation
[i.e. ( Q l l d ) ( P - p ) d s l (-dL)]according to Equations (16) - (17).
Alternatively, Equation (18) may be solved for
dY,ldY, in terms of Ytfs after the substitution of Y o
from Equation (10). (The variable Y , is preferably
the most or the least permeable component, because
dlY, of the intermediately permeable component may,
in some instances, vanish). I n this case, there are
only ( n - 2 ) equations for dYddY, (i = 2 , n - 1) to
be integrated, instead of the (12-1) equations for
dY,IdY,, ( i = 2, n ) . But generally it is much more
complicated to solve for dY,/dY, from Equation (18)
than for dYl/dYo,except for the binary and ternary
systems (see below).
Systems with non-permeating components
The above equations are also applicable t o systems
with non-permeating component (a*= 0) in the feed,
However, i t must be recognized that as (Yk approaches 0,
y k becomes 0, but V k / ( Y k and Y h approach a limiting
value which is dependent on the feed-side concentration of the non-permeating component xk. Thus Yk
does not drop out of the above equations. This is
shown as follows. Solving for V k / ( Y k from Equation
(9) with i = k, and then letting f f k + 0 we obtain
Yk
= - [rk/(l
-y -
xk)]
n
(1 - I: Y , ) as ak-+ 0 (21)
2
j f k
This equation may also be written a s
exp
fy'
Y,,,
s
=
y,e -
d Y<
(1 - Y) (a,
Y o - 1) Y ,
s
BdY, .
xk =
. (20)
YO,/
Equation (18) is obtained by dividing Equation (15)
by its member equation with i = n, and Equations
(19) and (20) are the integral forms of Equations
and (16), respectively.
Equations (8) - (9) and Equations (18) - (20) are
the equations required for calculating the multi-component permeation. The composition, flows and membrane area can be ca!culated by Equations (18),
( 19) and (20), respectively, and sequentially. The
average permeate concentrations can then be calculated by material balances from the compositions
and flows of the feed and residue. The initial conditions (Y,'s) for Equation (18) are calculated from
Equation (8) from the concentrations given a t the
feed o r residue end. The feed-side concentrations a t
any point on the membrane can be calculated by
Equation (9) from the corresponding Y,'s obtained
from the solution of Equation (18). The solution
of Equation (18) can best be obtained by first
solving for d Y J d Y . (i = 2, n) in terms of Y o
and Y,'s from Equation (18) (using all n-2 member equations) and Equation (10) (in differential
form), and then integrating the resulting ( n - 1 )
equations for dY,ldY, to obtain Y,'s as functions of
Y,. Here it may be noted that although Y o is merely a
( 15)
- (1 - 7) (yk/yo)
as ark + 0
(22)
The calculation method for systems with non-permeating components is entirely the same as that described above, except that the initial value of Y k
for the non-permeating components is calculated by
Equation (21), after other initial Yi's have been obtained from Equation (8). The feed-side concentration Xk a t any point on the membrane is calculated
by Equation (22) using the values of Y,'s obtained
from Equation (18). In addition, Equation (19) can
be reduced to the following simple form
which can also be obtained by integrating Equation
(13) with Y k = 0.
Binary system
For this system, Equation (18) is not necessary,
and Equations (19) and (20) yield
s = (1
The Canadian Journal of Chemical Engineering, Vol. 56, April, 1978
- y2)e
+
s
yz
Y2,r
e d y z . . . . . . . . . . . .. . .
,
(25)
199
where
YazPz - ffz
..........................
azpz (1 - Y)
b =
. . . . . (27)
Equation (8 ) , which relates xz to Yz can be written
as
yz
- 1/11
=
11
+
+ (at
-
(a2
- 1) (Y
1) (y
+
+
XZ)
X Z ) ] ~-
4yn2
(a2 -
~ ) x z ] / ~ Y o(28)
~z
This equation can be used to calculate the initial
value Y z , ffrom the given feed or residue concentration. The above form of solution, expressed in terms
of the local permeate concentration, is considerably
simpler than that expressed in terms of the feedside
concentration as obtained by Weller and Steiner'").
In addition, Equations (24) and (25) readily reduce
to even simpler forms f o r a2 >> 1 (@ = 1 and b
-1) :
e
=
1 -
.
( ~ 2 / ~ 2 , 1 ~ .
(29)
Ternary system
'
Substituting Equation (10) into Equation (18)
yields (See Equation 31 below)
Using the initial values of Yz and Y , determined by
Equation (8) from the given feed or residue composition, Equation (31) can be integrated by itself to
yield Y z as a function of Y,. This Y2-Ysrelation can
be used to calculate xz, x3, 8, and S by Equations ( 9 ) )
(19)) and (20). The average permeate concentrations and a r e then calculated by material balances
from the feed-side flows and compositions.
c2
Cascade calculation
A permeation cascade such as shown in Figure 1,
resembles such other multistage separation processes
as distillation and absorption in that the material
balance relations are identical (including the differential material balance of Raleigh batch distillation). But the characteristics of the "distribution
coefficient" of the permeation stage set its methods
of calculation somewhat apart from all others. I n an
equilibrium stage process such as distillation and
absorption, the states of the two streams leaving a
stage can be related by a distribution coefficient,
which in most cases depends only on temperature.
I n a permeation stage, however, the states of the
two streams (permeate and residue) leaving the stage
depend not only on the properties of the permeation
stage (membrane area, permeabilities, and operating
pressures) but also on the composition of the feed or
the relative flow rates of the feed and permeate
streams (i.e. stage cut). This is t r u e f o r a permeation stage in t h e countercurrent, cocurrent or crossflow pattern. Thus the membrane cascade is not easily
amenable to the standard calculation methods f o r distillation and absorption. For example, the McCabe-
200
Thiele graphical method generally cannot be applied
to the binary permeation cascade simply because no
single equilibrium curve can be found to relate the
permeate and residue concentrations throughout the
cascade, except for the special case of constant stage
cut in the c a s ~ a d e " " , ~. ~A,s~ to
) the rigorous calculation methods of distillation and absorption such as
the Thiele-Geddes and the modified Lewis-Matheson
method^'^^,*^', the trial calculation is generally made
by assuming a temperature profile in the column,
which gives the distribution coefficient in each stage
to be used in the stage to stage calculation. If these
methods a r e applied to the permeation cascade, one
would have to assume the composition or flow (e.g.
stage cut) of the feed stream in each stage in order
t o determine the distribution coefficients. In the
distillation and absorption processes, a fairly reasonable temperature profile can be easily assumed
(usually estimated by a short-cut method) to start
the calculation ; in the permeation cascade, however,
it is very difficult to assume the flow and composition of any stream because they can take on a wide
range of values. If the initial assumed values differ
too much from the true solution, convergence would
probably never be obtained,
Although the permeation cascade is similar to the
isotope separation cascade, there is a major difference
in the calculation approach. The methods for an
isotope separation cascade depend on the separation
factor being constant and near unity. For the permeation cascade ,with large membrane selectivity,
however, the stage separation factor generally varies
significantly from stage to stage. Thus a special
method must be used in calculating the permeation
cascade. In the following, calculation methods a r e
presented for the ideal and non-ideal multicomponent
cascades. These methods are applicable to cascades
with feed and permeate in countercurrent, cocurrent,
or cross-flow pattern. The general approach here for
the calculation of a non-ideal cascade assumes t h a t
the membrane areas and operating pressures in each
stage a r e given and that recycle of a portion of t h e
last stage permeate is possible. Such a calculation is
particularly useful for predicting the performance
of an existing cascade under various operating conditions. The procedure for calculating the ideal cascade
differs in that there is no recycle and the membrane
areas are not specified but are the quantities to be
determined according to the requirement of zero
interstage mixing loss. Such a calculation is useful
for preliminary cascade design.
Degrees of freedom in cascade
Because there are so many variables and parameters
involved in the permeation cascade, it is worthwhile
to consider f i r s t how much flexibility the designer
has in specifying the various operating conditions.
In a n n-component, N-stage cascade without product
recycle (with or without stripping section) the total
number of flow and concentration variables minus
the number of material balance relations (for each
componenb at each mixing point and around each
permeation stage) is equal to n ( N 1 ) . I n addition
there a r e 3 N parameters (feed and permeate pressures and membrane area in each stage.) These variables and parameters must satisfy the n N differential
equations describing the cascade (Equations (18)
(20) apply to each stage). Thus, in general, t h e
+
-
The Canadian Journal of Chemical Engineering, Vol. 56, April, 1978
degrees of freedom of a permeation cascade without product recycle are equal to 3N
n. For the
cascade with product recycle the degrees of freedom
are 3N n 1.
+
+ +
Binary ideal cascade
In the case of ideal binary cascade (no product
recycle), an additional ( N - 1) constraints arise from
the condition of no mixing loss when the residue o f
one stage is returned to the feed of t h e preceding
stage. The degrees of freedom then are reduced to
(2N + 3). Normally the feed flow and concentration
to the cascade are specified (2) a s well a s the operating pressures for each stage (2N). There then remains
only one variable free to be chosen by the designer.
Probably the most satisfactory procedure is to specify
the residue concentration from the cascade acwrding
to the desired recovery and to achieve the desired
enrichment by adjusting the number of stages (note
that the change in number of stages does not affect
the number of degrees of freedom). Under such conditions, the calculation of a cascade without stripping
srtages is straight-forward and a n example of it was
first given by Weller and Steiner"'). From the given
feed and residue concentrations in the feed stage,
the corresponding permeate concentration can be
easily determined by material balance using the 0
calculated by Equation (24). Since the feed and residue
concentrations of each of the subsequent stages a r e
equal to the permeate and feed Concentrations of its
preceding stage, respectively, the permeate concentrations of all the subsequent stages can be similarly
calculated. Using the calculated concentrations, all
the flow rates in the cascade can be calculated by
simple material balances. Finally, the membrane area
in each stage is calculated by Equation (25). Here
it may be noted t h a t the cascade calculation can also
be started from the product end as described by
Hwang and Kammermeyer'22*25',if the product concentration and stage cut (i.e. e) of the final stage
a r e specified. But the calculation involves a trial and
error procedure in each stage to determine t h e residue
concentration from the corresponding feed and permeate concentrations using Equation (24) and a material balance relation.
If the cascade contains stripping stages, a trial
and error calculation procedure will have to b e employed in the stripping section. I n this case, the most
satisfactory procedure is to assume the feed concentration of the last stripping stage (the residue
of this stage is the cascade residue which is generally
given), and to proceed with the calculation from this
stage through the entire stripping section to determine the permeate concentration of the stripping
stage next to the feed stage. If this concentration is
equal to the given concentration of the feed to the
cascade, then the assumed feed concentration of the
last stripping stage is correct, and the calculation
is then carried through the enriching section without
any trial and error. Here it should be pointed out
t h a t for given feed and residue concentrations of
the cascade, there is a maximum number of stripping
stages allowable without violating the rule of no interstage mixing loss. This number is dependent upon
the desired concentration of the cascade residue relative to the feed concentration. In particular, if the
local permeate concentration corresponding t o the
cascade residue is higher than the feed concentration
to the cascade, then no stripping stage is possible
without introducing mixing of streams of different
concentrations.
Multicomponent ideal cascade
For a multicomponent cascade with matched concentrations of the key-component (usually the most
permeable component) in the interstage mixing
streams, the number of degrees of freedom is equal
to 2N +n+ 1 (i.e. (3N + n ) - ( N - 1 ) ) . Thus f o r
given feed (flow and composition) and operating pressures (feed and permeate pressures) there remains
only one variable free to be chosen by the designer.
Usually this variable is the product recovery. A
typical design calculation is t o determine the product
concentration and membrane areas required i n the
cascade for a given recovery of the key-component
(note t h a t the desired product enrichment can always be obtained by adjusting the number of stages,
which does not affect the degrees of freedom in the
calculation). This is a problem of solving a system
of simultaneous differential and algebraic equations
(Equations (18) - (20) and material balance relations applied to each stage). Direct integration of
the simultaneous differential equations is a formidable boundary value problem because most of t h e
boundary conditions a r e unknown and vary from
equation to equation. I n the following, we present
an iteration method with stage to stage calculation
using only the assumed flow rates of the non-key
components in the cascade residue. This method applies only to cascades without stripping stages. The
calculation procedure is outlined below.
1. Calculate the flow rate of t h e k e y component
i n the f i r s t stage residue according to the desired recovery from the given feed. As a first
approximation, assume the attendant flow rates
of the non-key components i n t h e f i r s t stage
residue to be equal t o those in the feed to the
cascade. This gives the total flow and composition of the f i r s t stage residue to initiate t h e
following stage to stage calculation.
2. Integrate Equations (18) and (19) from t h e
residue end of t h e feed stage until the calculated
feed-side concentration (Equation (9) ) of t h e
key-component is equal to t h a t of the feed to
the cascade. This gives the flows and compositions of the feed and permeate of the f i r s t stage.
Calculate the states of the second stage residue
(which is recycled to the f i r s t stage feed) by
material balance around the mixing point (Figure 1).
3. Repeat step 2 f o r the subsequent stages, except
that f o r the last stage, the integration of Equations (18) - (19) is terminated when the feedside flow rate of t h e k e y component is equal to
t h a t in the permeate of the preceding stage.
This balances the flows of t h e key component
throughout the cascade. Calculate the unbalanced
flows of t h e non-key components as represented
by the imaginary stream Lo i n Figure 1 by material balance between the last stage feed and the
permeate of the preceding stage (no product
recycle).
4. Subtract these unbalanced component flows in
Lo (retaining the mathematical signs as calculated) from t h e respective component flows in
the f i r s t stage residue, and calculate the corresponding total flow and composition of the
first stage residue. Repeat from step 2 (if,
however, subsequent calculations result in a
negative component flow in the residue of any
stage indicating a n over correction, then only a
The Canadian Journal of Chemical Engineering, Vol. 56, April, 1978
201
portion of the flow of t h a t particular component
in L. should be subtracted from the f i r s t stage
residue).
5. Convergence is obtained when the unbalanced
component flows in the stream Lo a r e negligible
compared to the respective component flows in
the last stage permeate. The membrane area in
each stage is then calculated by Equation (20).
Non-ideal cascade
Most real cascades will probably be non-ideal with
Rome interstage mixing loss f o r various practical
reasons. For example, a cascade designed to be ideal
under specific conditions may become non-ideal due
to changes in membrane properties, feed, or operating
pressures. In addition, the product recycle, which is
essential in providing flexibility f o r the cascade operation, will always introduce interstage mixing losses.
Thus, an important calculation problem f o r the nonideal cascade is to calculate the performance and required recycle of an existing cascade with fixed
membrane areas for given operating pressures, feed
conditions, and recovery. The calculation procedure is
similar to that for the multicomponent ideal cascade
described above, except t h a t the integration of Equations (18) - (20) f o r each stage (step 2) is terminated when the calculated membrane area is equal
to the given value. In addition, the unbalanced flow
rates of the non-key components in the stream Lo
are calculated after the recycle flow is determined
by balancing the key-component among the last stage
fped, preceding stage permeate, and product recycle
(so that the flow of the key-component in Lo is zero).
The calculation method applies also to cascades containing a stripping section; in this case the calculation starts from the last stripping stage.
The above calculation procedure is very similar
to the Lewis-Matheson method for distillation'28',
except that the calculation here is always started
from the residue end of the cascade, not from the
product and residue ends toward the feed stage. If
the calculation were started from the product end,
then it would be necessary to use a tedious trial and
error procedure to solve Equations (18) - (20) for
each successive stage, since neither the feed nor the
residue conditions would be known from the stage
calculated before (but only the permeate which is
insufficient to s t a r t t h e integration).
Other workers have also studied the calculation
problem of the non-ideal cascade. Naylor and
and Hwang and Kammermeyer'22'5" have proposed a
McCabe-Thiele type graphical method for determining
the required number of stages and the interstage concentrations and flows in a binary non-ideal cascade.
This method is based on the existence of a single
equilibrium curve for the entire cascade, which requires the stage cut, pressure ratio, and membrane
selectivity being identical in each stage. I n practice,
however, these parameters will probably vary from
stage to stage and the stage cut will likely be dictated
by minimizing the interstage mixing loss (e.g. ideal
cascade). For the multicomponent system, Blumkin(")
has described a method of solution for a cascade with
specified permeate flow in each stage.
General comments on iteration procedures
The above iteration methods" have been tested for
*Computer programs for binary and ternaw systems are available
from Alberta Helium Ltd.. 826 Eighth Ave. S.W. Calgary. Alberta.
202
a few systems, some of which a r e presented in t h e
analysis section. The following has been observed
during the course of calculation. 1. The ratio of the
unbalanced component flow (in L,j to the flow of
the same component in the product stream can generally be reduced to below 0.005 within 10 iterations.
2. High calculation precision is required f o r cases
with low residue/feed concentration ratio (below
0.001) in any stage of the cascade. But such cases
do not usually have practical importance because of
the low enrichment which almost inevitably will accompany a low residue/feed concentration. 3. The effect of product recycle on the cascade performance
can best be calculated by starting the calculation f o r
a low recovery (high recycle) and then gradually
increasing the recovery until near zero recycle is
reached. This is especialIy important in cases where
the recovery is not very sensitive to the variation of
recycle (see below). The number of iterations required
can also be reduced by using the values of t h e component flows of the non-key components in the f i r s t
stage residue f o r a given recovery as t h e starting
values f o r the calculation of a somewhat higher
recovery.
Cascade analysis
Gas separation by permeation ultimately depends
on the permeability and selectivity of the membrane.
With a given membrane, however, there is a n infinite
number of cascade designs t h a t will perform a given
separation if one includes all of the possibilities of
number of stages, operating pressures, membrane
areas, and product recycle. Thus the knowledge of
the effects of membrane properties and operating
conditions on the plant requirements is of prime importance in determining the optimum design. In t h e
following, some of t h e important characteristics of
the ideal and non-ideal cascades are analyzed.
Ideal cascade
Parametric analysis of single-stage permeation has
been carried out by many ~ o r k e i ~ s ( ~ ~. Although
-'~~''-~~~~~
these parametric effects apply also to the multistage
cascade, they may be somewhat complicated by the
interstage recycle flows. The important effects of
membrane properties and operating conditions on the
membrane area and compression power requirements
of t h e binary ideal cascade a r e given below. Although
these effects a r e analyzed in terms of the binary
system, they are also generally true of the multicomponent ideal cascade.
Membrane Area Requirement. As in the singlestage operation"", the membrane area required i n
the cascade always decreases with decreasing recovery
and permeate/feed pressure ratio, and for a given
pressure ratio, t h e membrane area required is inversely proportional to the feed pressure. The effects
of membrane permeabilities however, are less straightforward in cascade operation. In single-stage permeation, the membrane area required for recovering
a given amount of the more permeable component in the permeate always decreases with increasing permeabilities of both gas components'21).
This is because the rate of permeation of the more
permeable component is proportional not only to its
permeability but also to its pressure differential across
the membrane which can be increased by diluting
the permeate with increased permeation of the less
The Canadian Journal of Chemical Engineering, Vol. 56, April, 1978
I
I
9-
I-
7-
DIMENSIONLESS MEMBRANE AREA [Q,/d)(P/LJr
M E M B R A N E S E L E C T I V I T Y , o( ( ' Q ? / Q l )
Figure 3 - Effects of permeabilities on membrane area requirement in the cascade. The effect of Q1 with a given QZ, and
with a given Ql are shown by the curves for
the effect of
(QuZ/d)(P/L,)s and (Ql/d)(P/Le)S, respectively.
--He-CH4,
feed = .l%o, residue = .04%, product > 90%,
Y = .02
_ _ - - 02-Nz,feed = 21%, residue = 8.4%, product > 90%,
a
Y
= .lo
permeable component. Although such effects are also
present in the cascade, they are complicated by other
factors. Figure 3 is an illustration of the effects
of permeabilities on the membrane area requirement
in the cascade for the Sle-CH4 (natural gas) and
OTN3 (air) systems. For a given permeability of
the less permeable component ( & I > , the membrane
area required always decreases with increasing permeability of the more permeable component (Qz) as
shown by the bottom two lines of Figure 3; the corresponding membrane area in each stage of the
cascade for various a (or Qz) is shown in Figure 4.
For a given Qz, however, the membrane area required
in the cascade, in contrast to the single-stage operation, goes through a minimum with the variation of
(Y
(or l/Qlas plotted) as shown by the top two
lines of Figure 3. The same is true for the membrane
Figure 4-Hfect of permeability of the more permeable
component (Qz) on the membrane area requirement in each
stage with a given permeability of the less permeable component (Q1).
--HeACH,,
feed = .l%,residue = .04%, product > 90%,
Y = .02
----Oz-Nz,feed = 2170, residue = 8.4%, product > 9076,
Y = .lo
area required in each stage which is shown in Figure
5 ; for example in the He-CH4 system, the a of 10 and
60 give the least membrane area requirement in
the first and second stages, respectively. The reason
for this is explained below.
For a given Q2, the membrane area required in
each stage will decrease with increase in the permeation driving force and increase with increase in
the amount of the more permeable component to be
permeated. It is this latter factor which comes into
play in the cascade; the amount of the more permeable component to be permeated per stage relative
to the amount produced from the last stage of the
\
\
\
1.8 - \
\
\
DIMENSONLESS MEMBRANE AREA (02/d)(P/L,)s
STAGE
Figure 5 - Effed of permeability of the less permeable compcment (Q,) on the membrane area requirement in each stage
with a given permeability of the more permeable component
---He-CH,,
Y
= .02
(Qd.
feed = .l%,residue = .04%, product
--- -O&, feed = 219'0, residue = 8.4%,
Y = .lo
product
> 90%,
> 90%,
NUMBER
Figure 6-Effect of membrane selectivity on the volume of
the more permeable component permeated in each stage per
unit volume of the more permeable component recovered
in the final-stage permeate.
---He-CH,,
feed = .l%,residue = .04%, product > 90%,
Y = .02
----Oz-Nz,feed = 21%, residue = 8.4%, product 90%
Y = .lo
The Canadian Journal of Chemical Engineering, VoZ. 56, April, 1978
>
203
cascade is found to increase markedly a t low Q due
to an increased extent of residue recycle. If we
define this ratio as U (volume of more permeable
component permeated in the stage/volume of t h e more
permeable component recovered from t h e cascade),
then it is possible by carrying out a material balance
among the three flow streams in the cascade (the
permeate of the stage concerned, residue of the following stage, and the permeate of the last stage)
to show t h a t
1-stage feed conc./cascade product conc.
U = 1-stage feed conc./stage permeate conc.
' '
.'.'
'
(32)
For a given product concentration, t h e value of U
in each stage generally increases with decreasing
membrane selectivity due to decreased enrichment
ratio (permeate concentration/feed concentration).
This is shown in Figure 6 which gives the values
of U in each stage for the He-CH4 and Oz-N2cascades
with various membrane selectivities. It is seen t h a t
at high a the U is not very sensitive to t h e variation
of a ; consequently, for a given Qz,the membrane
area required in each stage will decrease with decreasing a (increasing Q I ) due to increased permeation
driving force across the membrane. At low a, however, the much increased U due to lower a will tend
to increase the membrane area requirement. Such
effects are clearly shown in Figures 3 and 5. It must
be pointed out, however, that the membrane selectivity
corresponding to t h e minimum membrane area requirement is not necessarily the optimum one because
t h e compression power requirement always increases
with decreasing selectivity. For a given Q2, the optimum a will be determined by the optimum trade
off between the membrane and compression costs.
Compression Power Requirement. In single-stage
operation, the compression power requirement (to
recompress the permeate to the feed pressure) for a
given separation depends only on the membrane
selectivity and permeate/feed pressure ratio, irrespective of the actual pressure levels. We will see
that the same is true for multistage operation.
Specifically, i t will be shown that for given membrane, feed flow, feed and residue concentrations,
the compression power requirement in a n enriching
cascade is independent of the operating pressure
levels provided that the perrneate/feed pressure ratio
in each stage 7 , (i being the stage number) and
the first stage feed/final product pressure ratio
y p are fixed (the following discussion is also t r u e
for a cascade with stripping stages if t h e pressure
ratio of cascade feed/cascade residue i s specified).
First of all, it is clear from the above equations and
calculation method that the flow rates and compositions of all streams in the cascade a r e fixed by the
operating pressure ratios, irrespective of the actual
pressure levels. The compression power required in
the cascade will then depend on the pertinent compression ratios which may be different from l/y,
since each permeate is to be compressed to the feed
pressure level of the succeeding stage. If the feed
pressure i n each stage of the cascade is equal t o the
f i r s t stage feed pressure (uniform feed pressure
mode), the compression ratio for the permeate of
any stage except the last will be l/y,. The compression ratio f o r the final product will be l / ( y q p )
where n is the number of enriching stages i n the
cascade. Since the y,'s and y p a r e fixed at the outset,
the compression power required f o r the cascade will
also be fixed.
204
Under the constraints of fixed yi)a and yp,however,
the feed preseure in each of the second and subsequent enriching stages may still be established at any
intermediate value between the feed and permeate
pressures of the preceding stage (cf Figure 1). This
will, of course, require the addition of residue compressors in t h e residue lines to boost the residue
back to the feed line of t h e preceding stage. The
pertinent permeate and residue compression ratios
will, in this case, be different from the feed/permeate
pressure ratio. It will be shown, however, t h a t such
an operation only redistributes the compression power
requirements in the cascade without affecting t h e
total power requirement.
Let u s f i r s t examine the extreme case of setting
the feed pressure of each stage to be equal to the
permeate pressure of the preceding stage, a case in
which the feed and permeate pressures step down
successively with the succeeding stage. Such a n
operation can be accomplinshed by merely shifting the permeate compressor in each stage except
the last to t h e residue line of the suceeding stage.
The compression ratio of each residue compresor
will still be l/y, (the feed/permeate pressure
ratio of the preceding stage). But the product
compressor will have a compression ratio equal to I/
( 7 : . . . y..
. ynyp), instead of the 1/ (y,,yp) for t h e
uniform feed pressure mode. The power requirement
for the residue compressors will be less than that
for the permeate compressors in uniform feed pressure operation, since each residue stream which is
to be compressed is smaller than the permeate stream
of the preceding stage. Specifically since Vz - Lz =
VZ- LB= V , (cf. Figure 1), the total power reduction
from the ( n - 1) residue compressors is exactly equal
to the power required to compress the product stream
V , in ( n - 1 ) steps with compression ratios of
l y , , . . ., l / y , . . ., l/yn., respectively. But the power
required for the product compressor must be increased
for the compression ratio is now ( U Y L .. .y i . . . ynyp>,
instead of 1/ (yIIyp) for the uniform feed pressure
mode. The power increase for the product compressor
is approximately equal to the total power reduction
from the (n - 1) residue compressors, since the power
increment f o r compressing a gas stream with a compression ratio of l / y l . . . y z . . . y Y y p ) over a compression ratio of 1/ (yay,> is approximately equal to the
power required to compress the same gas stream in
( n - 1) steps with compression ratios l / y l... , U y , . . . ,
l/yn.l respectively. Consequently the total power requirement for the cascade with step-down feed pressures is identical to that f o r uniform feed pressure
operation (strictly speaking, the equality of the power
requirements will depend on the method of compression. They a r e exactly identical if the compression
is carried out isothermally, or if under adiabatic conditions, the compression of the product stream is
carried o u t in n distinct steps with compression ratios
of l i Y i . .,
. l / y l . . . , l/y,,+I/ (ynYp) respectively. For
more details on compression, the book by Dodge'J')
may be consulted). The same result can be similarly
obtained for the case of intermediate feed pressure
operation utilizing both permeate and residue compressors. Thus it may be concluded t h a t the total
power requirement for a permeation cascade depends
only on the operating pressure ratios (of permeate
feed in each stage and cascade feed/cascade product)
irrespective of the pressure levels in each stage.
In choosing the mode of operation, i t must be remembered t h a t for given operating pressure ratios,
the membrane area required in each stage is inversely
The Canadian Journal of Chemical Engineering, Vol. 56, April, 1978
-ICOMPRESSOR
21%
CROSS-FLOW PATTERN
-- 1
I
' X
'FEED.21
.02
.04
.06
DIMENSIONLESS MEMBRANE AREA, S
Figure 7 - Concentration profiles on the feed and permeate
sides of the membrane in a single enriching stage (permeate
= 0.24) or, as shown schematically at the
concentration
0.38) plus a
top of the figure, an enriching stage
stripping stage
0.21).
7'
(7"
'
(y=
proportional to its feed pressure
Therefore
the uniform feed pressure mode is generally preferred. The step-down feed pressure mode may be
considered only if the required pressure of the final
product gas is so low that the product compressor
may be eliminated. But the saving in compression
power must be balanced against the increased membrane area requirement.
Stripping Stages. The purpose of employing stripping stages in the cascade is generally to increase
recovery without sacrificing permeate enrichment.
Under certain conditions, this is very important in
cascade design and can be appreciated from the operation of a cascade with one enriching and one stripping stage a s shown in Figure 7. The feed-side concentration profile (z vs. S) is calculated by Equations
( 2 4 ) , (25) and (28). The profile of local permeate
concentration (y vs. S) is related to the feed-side
profile through Equation (28). The average permeate
concentration over a section of the membrane area is
designated by $ and is calculated by material balance
from feed-side flows and concentrations. Here it is
important to note that both the feed and permeate
side concentrations decrease along the flow path; if
the residue concentration is lower than a certain
value, it is possible that the local permeate concentrations near the residue end may be actually lower
than the inlet feed concentration. This is the case
shown in Figure 7. The local permeate concentrations beyond the point A on the y-profile a r e all
lower than the inlet feed concentration. As a result,
if the entire membrane area is used as a single enriching stage (the x and y profiles will remain unchanged) the overall average permeate concentration
will be only 24% due to the dilution by the low concentration permeate fluxes near the residue end.
Clearly this is not an efficient way to enrich a gas
stream. If the membrane area is broken up into
enriching and stripping sections divided a t the point
B as shown in Figure 7, the average permeate concentration from the enriching stage will be 38%. The
permeate from the stripping stage has the same concentration as the inlet feed and can be recycled back
to the feed of the enriching stage without any dilution
effect. However, because of this recycling, the ex-
ternal feed to the enriching stage must be proportionally reduced in order to maintain t h e desired residue concentration of 2.1%; o r equivalently, for a given
feed flow rate, the membrane area (and permeate
compression power) required for the two-stage operation will be larger than t h a t for the single stage
operation. But the thermodynamic efficiency of the
two-stage operation, defined as the ratio of the thermodynamic minimum work required for the separation,
(see below) to the actual permeate compression work,
is much higher due to higher permeate enrichment.
This comparison is summarized as Cases 1 and 2 in
Table 1. Furthermore, since the lowest possible permeate concentration (4.7% a t the residue end of the
y-profile) is still lower than the feed concentration
to the stripping stage ( l 8 % ) , the stripping stage
can be further broken up into two stripping stages
to improve the performance by further reducing t h e
dilution effect of the low concentration permeate
fluxes. This results in a considerable reduction of
compression power and membrane area requirements
a s shown by Case 3 in Table 1. Although the product
concentration is somewhat reduced, the thermodynamic
efficiency is higher due to lower permeate compression power. The effect of stripping stages, however,
generally decreases with increasing number of stages.
Case 4 shows that the employment of an additional
stripping stage yields only a slightly higher product
concentration with virtually identifical compression
power and membrane area requirements.
Here i t may be noted that for Cases 1-4 where
the product concentration, compression power and
membrane area vary considerably with the number
of stripping stages, the thermodynamic efficiency is
probably the only way for measuring the effectiveness
of each case. If a few enriching stages a r e used,
however, the product concentration may not vary
significantly with the number of stripping stages
and t h e increase in thermodynamic efficiency can be
translated into reduction of compression power and
membrane area requirements. This is shown by Cases
5-8 in Table 1.
The above discussion may be summarized as follows: the permeation process can be looked upon as
consisting of two sub-processes : permeation through
the membrane and mixing of permeate fluxes of
various concentrations. In cases where the concentration of the local permeate flux corresponding to the
desired residue concentration is lower than the given
feed concentration, the employment of stripping stages
can decrease the permeate mixing loss with a resultant
increase in product enrichment and/or decrease of
membrane area and compression power requirements.
Generally speaking, the conditions that favor the employment of stripping stages a r e Iow residue concentration, low membrane selectivity and low feed/ permeate
pressure ratio.
Non-ideal cascade
The non-ideal cascade is characterized by the interstage mixing loss which nullifies some of the
separation obtained by permeation. In analyzing the
non-ideal cascade, i t will be useful to have a measure
for gauging the extent of mixing loss relative to the
separation achieved by permeation. The thermodynamic minimum work required for a given separation
is, perhaps, the best measure of the separation work
and mixing loss, and is used in the following analysis.
The minimum power required for separating a binary
gas mixture (flow rate Lf, composition xf) into two
streams of different compositions (residue L , x and
The Canadian Iournal of Chemical Engineering, Vol. 5 6 , April, 1978
205
TABLE 1
EFFECTOF NUMBER
OF STRIPPING STAGES ON CASCADE PERFORMANCE
Membrane Area and Permeate Compression Power Requirements
~~
Stripping
Stages
Enriching
_-Cases*
Stages
Total
1
r-1
--
2
3
____
6370
-
~
_
27.4
4
--
I
214
18300
5790m2
43
3140
30.4
I
2590
_
_
31.0
_
1
_
~
I I
38.1
2.06
_
12880
32.7
2.56
_
33.7
2.75
_
150
12830
50.8
-127
11800
4820
54.9
7290
74.5
~
5050
87.5
~
6370
6640
I
35090
22.6
2980
95.1
438
1590
48.9
4800
56.5
27.8
3500
66.2
28000
4660
258
21440
3750
328
~
%
~~
6
48.5
1
5970 m2
49.4
30000
I
I
~
5
8
5%
-___
150
360
7
1
area
Thermo
Efficiency
71.7 kW
~-
3920
3600
_-___
1
power
51.2
42.1
2850
2630
71.7 kW
Product
conc.
465
1070
49.3
36850
24.2
3150
1230
49.2
3110
23.8
~
1200
I
352
28290
339
58.2
2.25
71.1
2.69
66.7
3.30
~
~
67.6
27030
3.48
*Feed 18.6 mol/s, 21%; Residue: 2.1%
Cross-flow pattern
a 2 = 4, Ql/d = 0.67 nmol/(m2.s. Pa)
y = .25, fi = 1100 kPA
c),
product V ,
all a t the same temperature and pressure, is given by‘31’.
w,= RT { v [V In? + (1 - ?)ln(l 3
1
+ L[x lnx + (1 , ~ ) l n (-l x ) ]
-
-
-~ , [ ~ ,+
l ~(1~ -, x , ) ln(l
- ,Kf)ll,,
,
,
,
.
,
.
. , (33)
where
v = L,(x,
-
x)./(? - x)
He N CHI
-80
---_____-
THRD STAGE AREA S, mz
206
-
=
L/(Y
-
-
x,)/(y - x ) . . . . . . . . . . . . . . . . . . . . . . . . . .(35)
Equations (34) and (35) a r e obtained by material
balance. The process which performs the separation
is said to have a separative power W., and conversely
the loss due to the mixing of the two streams L and
V is equal to -Ws. Separative power and mixing loss
can be calculated for the overall process and also for
. . (34)
t
100
FEED 01%
L
-24
-
e
Figure 8 - Effect of last-stage membrane
area on the performance of a %age
binary (He-CH,) cascade. Feed flow =
18.6 mol/s, feed concentration = . l %
(He), az(He/CH,) = 60, Q,/d(CH,) =
0.67 nmol/(mz. sec. Pa), P = 6210 kPa,
p = 69, 1’72, 345 kPa (for 1, 2, 3 stage),
sl= 88 m2, s2 = 12.7 m2. The cascade
is ideal if s3 = 1.06 m2.
-’
Figure 9 - Effect of second-stage niembrane area on the performance of a 3stage binary (He-CH,) cascade. Parameters are the same as in Figure 8
except that sg is fixed a t 1.06 nlz and
s2 is variable.
FEED OJ%
9
H. IN CH4
lo-
[
01
/--
--
‘ - t i 2 STAGE W,
CASCADE
01-
0.
2-3
I0
20
SECON)
30
’40
STAGE AREA
53
&I
s2 rnz
The Canadian lournal of Chemical Engineering, V a l . 56, April, 1978
w
1.4
V
z
Q
11.2
a
9
m
E 1.0
W
0
a
xa 0.8
V
Ly
$ 0.6
a
--I
W
a 0.4
0.2'
I
I
1
t
,
1
1
1
1.0
t
I
t
.
1
1
1
,
10
RELATIVE FEED FLOW
PRODUCT RECYCLE X
Figure 10-Effect of product recycle (to third-stage feed) on
the performance of a 3-stage binary (He-CH,) cascade designed
as an ideal cascade at zero recycle. Parameters the same as in
Figure 8 but membrane areas fixed at 88, 12.7 and 1.06 m2.
the individual steps of permeation and mixing. I n
particular, it can be shown that the separative power
of a permeation cascade calculated from the concentrations and flows of the feed, residue and product
of the cascade is equal to the arithmetic sum of the
separative powers and mixing losses of t h e individual
permeation and mixing steps, each calculated from
the appropriate flows and concentrations. This rate
of the additivity is very useful in analyzing a nonideal cascade to determine where the major work of
separation is obtained and the seriousness of any
mixing losses.
For the multicomponent system, if we are only interested in the separation of a key-component, then
the attendant separation or mixing among the nonkey components in each permeation and mixing step
should not be taken into account in calculating the
separative power or mixing loss. Under such conditions, equation 33 can be used for the multicomponent
system by lumping all the non-key components as one
component.
Cascade Without Product Recycle. Any change in
the membrane areas or operating conditions of an
ideal cascade will give rise to some interstage mixing
losses. At the same time, the separative power of
each permeation stage may also be affected due to
t,he change of flows and compositions (This effect
will be analyzed in detail in a separate paper). Thus
the overall efficiency of the cascade will depend on
the combination of these effects. We can now show
that the ideal cascade is not necessarily always the
most efficient one even though it has no interstage
mixing loss. In the following, the effect of interstage
mixing on the cascade performance is analyzed by
varying t h e membrane area in a stage while keeping
the others constant.
Figures 8 and 9 show the effect of membrane areas
on the 3-stage cascade performance for the He-CH?
system which corresponds to the practical problem
of recovering helium from natural gas. It is seen that
Figure 11 - Effect of feed flow rate on the performance of the
He-CH, cascade designed as an ideal cascade for a specific feed
flow rate. The cascade is the same as that described in Figure 10
but with zero recycle. The values of product rate (referring
to the volume of He in the product), product concentration,
compression power, and recovery, are shown relative to those
obtained at ideal cascade conditions (i.e. feed flow = 18.6
mol/s).
the cascade designed without mixing loss is, in this
case, not the thermodynamically most efficient one.
The reason for this can be elicited from the interstage
mixing losses and the separative power of each permeation stage shown in the figures. The effect of
third stage membrane area s3 is discussed first. (Figure 8). With increasing ss from the point of no mixing
loss, the decrease in cascade efficiency, as evidenced
by the decreased product concentration, is mainly due
to the decreased separative power of the third permeation stage W,,s (resulting from the decreased stage
residue concentration which produce low concentration permeate). The third stage will, in fact, become
useless if its membrane area is so large that all
the permeate from the second stage permeates through
it. The interstage mixing loss, due to membrane area
variation, is in this case, largely negligible. With
decreasing s3, the second stage separative power W,,Z
increases significantly, causing the cascade efficiency
to increase to a maximum. The corresponding product
enrichment increases markedly with only small decrease in recovery and small increase in compression
power. Here it is worth noting that the increased
separative power of the second stage with decreasing
s3 results from the processing of a larger and more
concentrated feed stream in the second stage. A t
lower ss, however, the cascade efficiency decreases
with decreasing ss because of (1) increased interstage mixing loss, ( 2 ) increased compression power
due to high permeate flows resulting from the buildup of helium concentration in the cascade (primarily
in the second stage). The decreased separative power
of the cascade is indicated by the low recovery. Here
the effect of reducing ss is very similar to that of
product recycling to be discussed later. In fact, if
ss = 0, the cascade becomes essentially a two-stage
unit with total product recycle. The corresponding
cascade efficiency is zero because the sum of separative powers of all the permeation stages is equal to
the interstage mixing losses ; the permeate compression work is completely wasted as heat of compres-
The Canadian Journal of Chemical Engineering, Vol. 56, April, 1978
207
w
U
Z
I
lx
0,
lx
LL
Y
Y
n
i
Q
U
Ln
.1 0
10
10
10
RELATIVE FEED C O N C E N T R A T I O N
Figure 12 - Effect of feed concentration on the performance
of the He-CH, cascade designed as an ideal cascade for a
specific feed concentration. The cascade is the same as that
described in Figure 10 but with zero recycle. The values of
product rate (referring to the volume of He in the product),
product concentration, compression power, and recovery, are
obtained at a fixed feed flow rate (18.6 mol/sec) and are
shown relative to those obtained at ideal cascade conditions
(i.e. feed concentration = .1% He).
sion. It is significant t h a t the interstage mixing
loss is, in this example, negligible over a wide range
of s3. This offers flexibility for t h e plant design.
The effect of the variation of the second stage
membrane area ( s s ) shown in Figure 9, can be
similarly explained by the mixing losses and the
separative and compression powers in each stage.
With increasing sz from the point of no mixing
loss, the separative power of the cascade remains
nearly constant ; the increased separative powers of
the second and third stages are just about equal to
the interstage mixing losses. The cascade efficiency
drops because of high compression power requirement,
resulting primarily from the increased second stage
permeate flow. With decreasing s2, the cascade efficiency increases slightly and then decreases sharply
due to mixing losses and decreased separative powers
of the second and third stages. Both Figures 8 and
9 show that the efficiency of the ideal cascade can
be improved by somewhat reducing the second and
third stage membrane areas. Generally, the cascade
which yields a high ratio of final product concentration/feed concentration can be expected to show the
same behavior as the He-CH4 cascade described above.
Cascade W i t h Product Recycle. The recycling of
a portion of the product to the feed of the last permeation stage is done to increase the concentration
of the product, but it affects all the concentrations
and flow rates in the cascade. Compression power,
separative power and interstage mixing loss a t each
stage generally increase with increasing recycle. The
corresponding cascade efficiency will depend on the
relative magnitude of these increases. Figure 10
shows the effect of product recycle on the performance of the cascade designed with no mixing loss a t zero recycle for the He-CHa system. The
cascade separative power and efficiency increase with
recycle up to about 8070 recycle; the product concentration increases markedly without significantly
affecting the recovery and permeate compression
208
Figure 1 3 - Effect of product recycle on the Performance of
a 3-stage ternary (He-C02-CH,) permeation cascade designed
with matched He concentration for interstage mixing streams
at zero recycle. Thermodynamic efficiency rl is based on equation 33 using helium 2oncentrations. Feed flow = 18.6 mol/s,
feed composition = .1% He and .594 CO,, a,(He/CH,) =
60, a2(CQ2/CF14)= 20, Ql/d(CH4) = 0.559 nmol/(m2. S.
Pa), P = 6210 kPa, p = 69, 172, 345 kPa, (for 1, 2, 3 stage),
s, = 103 m2, s2 ~ 1 1 . 8m2, ss = 372 mz.
power requirement. But after 90% recycle, the cascade efficiency drops rapidly due to increased mixing
loss and compression power requirements. The increased compression power results from the recycling
of high flows of helium in the cascade.
E f f e c t o f Feed Conditions. The effect of feed flow
rate and concentration on the performance of a cascade with fixed membrane areas is of practical importance in the operation of a permeation plant. Figure 11 shows the effect of feed flow rate on the performance of a He-CH4 cascade designed as an ideal
cascade for a specific feed flow rate. The values of
product rate (referring to the volume of the more
permeable component), product concentration, recovery, compression power, and feed flow, a r e shown
relative to those obtained a t ideal cascade conditions.
The product rate and concentration increase with increasing feed f l ~ wrate at the expense of recovery
(i.e. fraction of the more permeable component in the
feed recovered). Figure 12 shows the effect of feed
concentration on the performance of the same He-CH4
cascade with a fixed feed flow rate. Here it is important to note that the recovery remains nearly
constant but the product rate is almost linearly proportional to the feed concentration. Consequently the
helium cost is approximately inversely proportional t o
the feed concentration.
Ternary System. Most multicomponent systems a r e
worked for the separation of a single preferred component. As an example of a tenary system of some
potential interest we consider the He-C02-CH4system.
Such a system may be encountered in the recovery of
helium from natural gas containing CO,. Figure 13
shows the effect of product recycle on the performance
of the cascade designed with matched helium concentrations for the interstage mixing streams at zero recycle. The thermodynamic efficiency shown in the fig-
The Canadian Journal of Chemical Engineering, Vol. 56, April, 1978
ure is calculated according to Equation (33) by lumping
COz and C H I as one component; t h a t is, the minimum
theoretical work is calculated by ignoring t h e separation of C 0 2 / C H 4occurring. It is interesting that the
C 0 2 concentrations of the second stage permeate G,Z
and, to a lesser degree, the first stage permeate
go through a maximum with increasing product recycle ; a t low recycles both H e and COz concentrations
increase a t the expense of the C H , concentration;
but a t high recycles helium is enriched at the expense of both CO, and C H , concentrations. It is also
worth noting that at a sufficiently high recycle, the
maximum CO, concentration in the cascade exists in
the permeate of an intermediate stage. Of particular
importance is the effect of the presence of COz on the
performance of the permeation cascade for helium
recovery (cf. Figures 10 and 13). With CO, present
in the feed, much greater compression power is required for a given helium enrichment. For the case
shown in Figure 13, it is probably more practical
to remove CO, from the first stage permeate by other
processes.
A cknowledgment
This work was done under arrangement with and finnncial suppurt
from Alberta Helium Ltd., Calgary. Alberta. AHL is owned by the
Government of Alberta, Trans-Canada Pipe Lines Limited, and Alberta
and Southern Gas Company.
Nomenclature
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
dimensionless constant defined by Equation (26).
dimensionless constant defined by Equation (27).
membrane thickness, m.
feed-side flow rate, mol/s.
feed flow to a permeation cascade, mol/s.
feed-side flow rate a t the feed or residue end of a permeation stage, mol/s.
imaginary flow stream representing the unbalanced
flows of the non-key components in the cascade
during iterative calculations, mol/s.
total number of component in the feed stream.
feed-side pressure, Pa (abs).
permeate-side rressure, Fa (abs).
permeability of component i, mol/(m.s.Pa).
uermeability of the less permeable component in a
binary gas mixture, mol/(m. s. Pa).
uermeability of the more permeable component in a
binary gas mixture, mol/(m.s.Pa).
gas constant.
dimensionless membrane area defined by Equation
(17).
membrane area, mz.
membrane area of the j‘* permeation stage, m*.
temperature, K.
volume of the more permeable component in a binary
mixture permeated in a stage of a cascade per unit
volume of the same component recovered in the finalstage permeate.
= permeate flow rate, mol/s.
= permeate flow rate of thejfhpermeation stage, mol/s.
= product flow rate of a permeation cascade, mol/s.
= compression power required for the total cascade
(calculated by assuming adiabatic compression with
90% efficiency and using a minimum number of compression stages with the maximum uermissible compression ratio in each stage limited to 5), W.
= compression power required for the permeate of the
f h stage, W.
= interstage mixing loss (calculated by Equation (33)
with a negative sign), W.
= separative power of a permeation cascade (calculated
by Equation (33)).W.
= separative power of the jrapermeation stage (calculated by Equation (33)), W.
= mole fraction of component i in the feed-side stream.
= mole fraction of component i in the feed-side stream a t
the feed or residue end.
= & Yi, permeate-side concentration variable.
= value of Yi a t the feed or residue end of a permeation stage.
= local permeate concentration of component i, mole
fraction.
-
Y
=
yi,$
=
QI,
=
B,
=
Y
YJ
=
=
yp
4,
=
=
7
=
0
=
average permeate concentration, mole fraction.
average concentration of component i in the permeate
of thejdk stage, mole fraction.
Q,/Q1, selectivity of component i to the base component 1.
1 - l / a , ,dimensionless constant related to the membrane selectivity.
ratio of permeate pressure/feed pressure.
ratio of permeate pressure/feed pressure of the j t h
permeation stage.
ratio of first stage feed pressure/final product pressure.
percent of component i in the feed recovered in the
product stream, %.
WJW,, thermodynamic efficiency defined as the ratio
of the minimum to the actual (compression) work
required for the separation, %.
l-L/L,, fraction of feed permeated (i.e. stage cut if
L, is the feed) or the negative ratio of permeate flow/
residue flow (if L f is the residue).
Su bscrifits
=
f
=
=
p,, ,
referring to conditions at the feed or residue end of a
permeation stage.
referring to the product stream.
component identification number or stage number.
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Manuscript received August 3, 1976 : revised manuscript received
November 9. 1Y77 : accepted for publication, December 6 , 1977.
The Canadian Iournal of Chemical Engineering, Vol. 56, April, 1978
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