Design studies of membrane permeator processes for gas separation

UTTERWORTH El N E M A

Gas. Sep. Pur~/: Vol. 9. No. 3, pp. 151 169, 1995 N

Copyright ( 1 9 9 5 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0950 4214/95 $10.00 + 0.00

N

Design studies of membrane permeator processes for gas separation T. P e t t e r s e n and K . M . Lien Department of Chemical Engineering--NTH, The University of Trondheim, N-7034 Trondheim, Norway The intrinsic behaviour of several single-stage and multi-stage permeator systems has been studied using a recently developed algebraic design model. Upper and lower bounds with respect to product purity and recovery in single-stage systems are presented. It is shown that single-stage permeators without recycle can exhibit maxima in permeate purity as a function of the pressure ratio across the membrane. It is illustrated why bypass configurations may be economically profitable in single-stage systems. The effect of product recycle in single-stage systems has been studied. It is shown that permeate recycle can reduce the compressor load in single-stage permeator systems. Similar benefits of retentate recycle have not been identified• The characteristic behaviour of multi-stage systems has been studied. Based on general design criteria, various module configurations have been classified as suitable for recovery of either the slowest or the fastest permeating component. The effect of using different membrane materials at each stage of a multi-stage permeator cascade has been studied. It is shown that improvements with respect to product recovery can be achieved.

Keywords: membrane permeator processes; design studies; gas separation

Nomenclature A J g

1'*

M e m b r a n e area (m 2) Dimensionless flow rate along feed side of m e m b r a n e Pressure ratio across membrane.

~ = Q1/Q2

Direction of separation in Equation (39) Module cut rate, 0 np/nf Extent of separation in Equation (37) SSP Single-stage p e r m e a t o r SSP-PR SSP with permeate recycle SSP-RR SSP with retentate recyle 2/3-Str 2/3-stage stripper cascade 2/3-Enr 2/3-stage enricher cascade 3StrEnr 3-stage S t r + E n r cascade SSP2Enr Combined SSP and 2Enr cascade /3

g = Pf/pp

Ka, Kc

NCD n P Q R

Cost coefficients in Equation (35), relating R and In g to the annualized cost of m e m b r a n e area and compression, respectively Normalized compressor duty Flow rate [m3(STP) h I] Total pressure (bar) Permeance [m3(STP) m 2 bar I h 1] Dimensionless m e m b r a n e area. R - Q2PfA/nr

rp

Recovery of c o m p o n e n t I in permeate stream,

rr s z,x,y

• To w h o m

Mole fraction of c o m p o n e n t 1 in closed end of the hollow fibres Ideal m e m b r a n e selectivity,

rp = Oy/2

Subscripts

Recovery of c o m p o n e n t 2 in retentate stream, rr - ( 1 - 0 ) ( 1 - . v ) / ( l - - ) Fraction of product stream which is recycled Mole fraction of c o m p o n e n t 1 in the feed. retentate and permeate streams, respectively

f i

p r

c o r r e s p o n d e n c e s h o u l d be a d d r e s s e d

151

Feed stream Components: i = 1 and i - 2 denote the fastest and slowest permeating component, respectively Permeate stream Retentate stream

152

Design of membrane permeator processes." T. Pettersen and K.M. Lien

see F i g u r e 1). Counter-current flow was found to be the best flow pattern for this kind of permeator module, and the use of permeate sweep gas* was found to increase the driving force for permeation significantly. Pan and Habgood 4'5 also presented a similar study on the performance of multi-stage permeator cascades. Stern et al. 6 studied the performance of SSP systems with permeate recycle (SSP-PR). A comparison was made between the SSP-PR configuration and the continuous membrane column proposed by Pfefferle 7. It was found that the continuous membrane column configuration yields higher permeate purity than the SSP-PR configuration only when operated with high reflux ratios and low cut rates. Stern et al. 6 also studied the behaviour of two-membrane systems. Ohno e l al. s proposed a permeator module containing two different membrane materials, which selectively permeates one of the components in the feed stream. The studies presented by Stern et al. 6 involve the effect of flow patterns in a two-membrane permeator and the performance of various two-stage, two-membrane systems, where each module contains only one type of membrane material. With respect to case studies on specific separation problems, the work done by Bhide and Stern on production of oxygen-enriched air 9'1° and on purifica• tlon of natural gas 1 1 1"2 represents some of the more extensive and detailed case studies which have been presented.

Introduction Gas separation is gradually becoming a more important separation technology in areas such as air separation, hydrogen recovery and natural gas purification. Membrane permeation offers several advantages compared to conventional technologies such as absorption, adsorption and cryogenic distillation, not only as a competing technology, but also as a building block in hybrid separation processes. Hybrid separation processes offer the opportunity to exploit the characteristics of different technologies in an efficient way, as illustrated by Ray et al. I One of the advantages of membrane permeation is the inherent simplicity of the process. The separation is accomplished due to the fact that species in a gas mixture permeate at different rates through a polymeric membrane. The driving force for the permeation is a chemical potential gradient across the membrane surface, which is maintained by a partial pressure difference from the feed side to the permeate side of the membrane. Although membrane pcrmeation is a conceptually simple process, there are many design decisions with respect to selection of operating conditions, module configurations and suitable membrane materials. Since Weller and Steiner: presented the first theoretical calculations, the design aspects of gas permeation processes have been addressed by a number of authors. These design studies can be divided into general parametric studies and case studies related to specific separation problems. Pan and Habgood -~ presented an analysis of a single-stage permeator system (the SSP configuration:

* A fraction of the retentate stream is used as the s w e e p gas on the permeate side of the membrane

SSP.RR SSP-PR

SSP

I

Pem~eate Z/3S~ t

' . . . . . . . . . . I

__, t

2/3Enr

~___

i

', . . . . . .

',

i t

I_

r-

2

"

i 3StrEnr

SSP2Enr .j"

Id

Figure 1

1

Various permeator configurations (see the symbol list for complete names)

-2

L .........

i

,

Design of membrane permeator processes." 17Pettersen and K.M. Lien

A more complete overview of theoretical and practical studies on gas permeation processes can be found in some of the review papers which have been published during recent years (see, for example, the work of Lonsdale 13, Matson et al. 14, Fritzsche and Narayan 15, Spillman 16, Kovvali et al. 17. and Koros and FlemminglS). Despite the large number of studies which have been performed, there are still few general guidelines currently available for the design of membrane permeation processes, as pointed out recently by Laguntsov et al. 19 This paper presents the results from a theoretical ~tudy on the behaviour of some commonly used permeator systems for separating a binary mixture. ]{'he systems which have been studied are single- and multi-stage permeator cascades with up to two recompressions (see Figure I). The aim has been to identify general trends which can serve as tools and guidelines for design of membrane-based gas separation processes. M o d e l l i n g of m e m b r a n e

permeator

modules

Figure 2 shows a simplified picture of a membrane

X~Z

Feed

,

f2- IR_ 0

dR f, x

X~X

Retentate

P

P~rmeate

"0, 9

y0-0.

0fl -=y 0 Figure 2

153

Counter-current f l o w in a membrane permeator module

permeating through the membrane is assumed to be proportional to the logarithmic mean of the driving force (partial pressure difference across the membrane) at the feed entrance and at the retentate exit Oyi

(4)

~,RAPi

In order to avoid potential numerical problems associated with the logarithmic mean, the Paterson approximation :1 to the logarithmic mean is used

I (--i - ;',/g) ~ (xi - )';/g) ~ 2

permeator module with a counter-current flow pattern. Fhe mass balance equations for a binary mixture [:lowing along the high-pressure side of the membrane :':an be written in a dimensionless form

APi

,4f_

(1)

The composition in the closed end of the hollow fibres is determined from the Weller and Steiner x equation

(2)

.V;+ 1

[,~(x- y/g) + (l -_\-)

:tR (Ix

_

1

:tR

[<.,-

, , / g ) +_,-

f L

{l

.~,)/g]

I -y*

-

(,(x

-

.rT/g)

(5)

dl] dR

]'he composition in the closed end of the hollow fibres s determined by the well-known equation derived by Weller and Steiner 2 <

+ :3 V/(:, - )',Le)(-\i

_v*/g)

[l - .v - ( 1 - 3'* )/g]

(3)

For a membrane permeator with counter-current flow, Equations (1) and (2) represent a boundary value :woblem which requires an iterative solution algorithm. Unfortunately, solution of the boundary value problem n a y quickly become impractically time-consuming, ,:specially when performing design optimization. In :,rder to avoid solution of the original mass balance equations, several simplified design models have been ,ieveloped for membrane permeators. The review paper -,y Kovvali et al. 17 presents an overview of the models and solution strategies which have been developed for -nembrane permeators. Most of the design models ,vhich have been developed are valid only for separa.ion of binary mixtures. In this work, a recently developed 2° design model for 'nembrane permeators has been used. This algebraic design model is based on an analogy with countercurrent heat exchangers, and is valid also for multicomponent separation problems. The design model can ;',e extended to take into account the pressure drop nside the hollow fibres, but in this work the pressure along either side of the membrane surface is assumed to :',e constant. With constant pressure on both sides of "he membrane, the total amount of component i

(6)

Ozd÷l (X/+I --1'7+l/g)

Together with a total mass balance around the membrane permeator and the summation of mole fractions in each product stream and in the closed end of the hollow fibres, Equations (4), (5) and (6) represent 3c+ 1 independent algebraic equations which have to be solved simultaneously to determine the 3c+1 variables describing the product compositions and flow rates (xi, Yi, 3'7 and 0). Compared to a numerical solution of the original differential mass balance equations (the model presented by Shindo et al. :2 has been used as the reference model), it has been shown that the above design model predicts permeate purity within 2% accuracy and module cut rate within 5% accuracy 2°. Single-stage

permeator

systems

The single-stage permeator (SSP) system is probably the most common module configuration used in gas separation applications. The reason is obviously its simplicity, and the lack of expensive recompressions of recycle streams. It is therefore important for the process designer to understand and be able to evaluate the performance of SSP systems in an efficient way. The performance of an SSP system separating a binary mixture is determined by four key variables, provided that the pressure drop along either side of the membrane is neglected. The four variables are the feed composition z, the dimensionless membrane area R, the ideal membrane selectivity ~ and the membrane pressure ratio g. Here we will illustrate the behaviour

154

Design of membrane permeator processes." 77Pettersen and K.M. Lien 1

1

0.8

.- "

~'~0.6

/1

0.8

/

~ 0 .6

" ",

0

0.4 ~

~'%0.4

"%

" ..

/ /

0.2

//

0.2

/

°0

o'.2

o14

o'.6

°0

o.8

Dim.less m e m b r a n e area, R

o.2

0.4

o.8

Dim.less m e m b r a n e area, R

F i g u r e :3 Product purity ( - - - - ) and recovery ( - - - ) v e r s u s d i m e n s i o n l e s s m e m b r a n e area. Conditions: z = 0 . 2 , ~ = 5 , g = 5 . ( . . . . . . ) r e p r e s e n t the reference model (Shindo etoL 22)

of SSP systems through particular, we will focus on 1

parametric

studies.

In

The general performance of SSP systems as a function of the four variables, z, R, (~ and g. U p p e r and lower bounds with respect to product purity and recovery are presented. The economic trade-off between the cost of m e m b r a n e area and the cost of compression. It is illustrated why bypass configurations (over-purification of the permeate product, followed by mixing with the feed stream) may be economically profitable for SPP systems. The effect of recycle streams in SSP systems. It is shown that permeate recycle may reduce the compressor duty in SSP systems, while retentate recycle seems to be detrimental for both product purity and recovery.

General characteristics of SSP systems

The performance of a m e m b r a n e permeator system can be described by the product purity and the recovery of the key components in the two products streams c o m p o n e n t 1 in the permeate stream and c o m p o n e n t 2 in the retentate stream. Figures 3 7 serve as illustrations of how product purity and recovery in the two product streams are affected by the four variables _-, R, c~ and g. The base case conditions which have been selected are typical for the separation of air: z - 0.2, c~. - 5, g = 5 and R = 0.40. The system performance has been predicted using the previously presented design model 2°. To veri~' the algebraic design model, numerical solutions* of the model presented by Shindo et al. 22 are also given, indicated by the dotted lines in Figures 3 7. Dimensionless membrane area. F(~ure 3 shows the performance of an SSP system as a function of the dimensionless m e m b r a n e area R. As the dimensionless m e m b r a n e area is increased, the a m o u n t of the feed stream which permeates into the permeate stream increases. Thus, permeate recovery and retentate purity increase, while permeate purity and retentate recovery decrease as R is increased. Note that the deviation between the algebraic design model and the more rigorous DAE-based model is negligible. *The ordinary differential equations were integrated using a 4th/5th-order Runge Kutta algorithm, and the boundary value problem was solved using a N e w t o n - R a p h s o n algorithm

Dotted lines

An upper bound with respect to permeate purity is obtained as R approaches zero. This upper bound for the permeate purity can be calculated from the equation derived by Weller and Steiner 2 [see Equation

(3)1 lira .v

1 + ( , ~ - 1)(1/g + :)

R ~

¢[1 +(c~

1)(1/g + z)] 2 - 4 ( ( ~ - 1)f~z/g

2 ( o - 1)/g

(7)

Equation (7) provides the upper bound for permeate purity as a function of the feed composition, the m e m b r a n e selectivity and the m e m b r a n e pressure ratio. Ideal membrane selectivity Figure 4 shows product purity and recovery versus the ideal m e m b r a n e selectivity (x. Notice that ~ is defined as the ratio between the permeance of c o m p o n e n t 1 (Q1) and the permeance of c o m p o n e n t 2 (Q2). Q2 is part of the definition of the dimensionless m e m b r a n e area R, which is kept constant in the parametric study shown in Figure 4. A change in c~ therefore reflects a change in Q1. Figure 4 shows that both the permeate and the retentate purity can be increased by increasing the ideal m e m b r a n e selectivity. The permeate purity, however, approaches an upper bound. This is consistent with Pan and H a b g o o d ' s analysis 3 which shows that there is little to be gained with respect to permeate purity by using a m e m b r a n e with m e m b r a n e selectivity ct>>g if g < l/z. Also with respect to product recovery, lower and upper bounds exist as o approaches either 1 or infinity. The retentate recovery in particular seems to be almost invariant to the ideal m e m b r a n e selectivity. The limiting case where c~ approaches infinity corresponds to an infinitely large value of the permeance of c o m p o n e n t 1. This implies that c o m p o n e n t 1 permeates completely into the permeate stream as it enters the m e m b r a n e permeator, leaving pure c o m p o n e n t 2 on the feed side of the m e m b r a n e surface. Since c o m p o n e n t 1 is completely transferred to the permeate stream, the permeate recovery is equal to 1 lira rp

=

1;

lira _x= 0JR>0

(8)

Because c o m p o n e n t 1 permeates completely into the permeate stream as it enters the m e m b r a n e module, the

Design of membrane permeator processes." T. Pettersen and K.M. Lien

155

1

1 / ./ /

0.8

0.8

/ / ./

~'~o.6

~. 0.6

/

0

/

? ~0.4

/ /

0.4

0.2

0.2

%0

101

102

00°

101 102 Membrane selectivity, c~

Membrane selectivity, c~

Figure 4 Product purity ( ) and recovery ( - - - ) versus ideal membrane selectivity. Conditions: z=0.2, R=0.40, g = 5 . Dotted lines ( . . . . . ) represent the reference model (Shindo eta/. 22)

dimensionless m e m b r a n e area R

gas which flows along the m e m b r a n e surface contains only c o m p o n e n t 2, except at the feed entrance and at lhe permeate exit, where c o m p o n e n t 1 is present. Based on this description [Equation (8)] of an infinitely .,;elective membrane, Equation (1) can be solved analytically to provide an upper bound for the module cut rate lira 0 = z + R(I - l/g)

(1 - z ) ( 1

(9)

lira rr = 1 ~c,

2

( 1O)

l/g)

R(l

-

l/g)

(11)

--

1

r h e other limiting case ((~ 1) represents a nonselective membrane, therefore no change in composition takes place along the m e m b r a n e surface l i m y = z:

lim.v = z

~ 1

n~l

(12)

Again, Equation (1) is soh'ed analytically to yield the following lower bound for the module cut rate lim0=R(l-

l/g)

lira rp = R ( I - l/R) I-R(I

(14)

1/'gl

(15)

<_t ~ I

F r o m the above analysis it is clear that Equations ( 11 ) and (15) provide the upper and lower bounds for the retentate recovery as a function of - R and g 1

R(I 1 --Z

1/g)

1 --

rr

~
(17)

R= (l-rr)(2-z)+ 2(1 - l / g )

z(1-r,)

(18)

--2(1-l/g)

Note that the relative accuracy of Equation (18), which may be expressed as z / [ 2 ( 1 - z ) ] % , is only dependent on the feed composition to the SSP system. This implies that the m e m b r a n e area required to obtain any retentate recovery in an SSP system is predicted within 20% accuracy provided that z~<0.28. As an example, for the separation of air, the m e m b r a n e area required for a given retentate recovery is predicted within 12.5% accuracy for any retentate recovery! If the main product is the fastest permeating component, permeate purity and recovery become more i m p o r t a n t than the retentate recovery. O f course, for a binary separation, the retentate recovery is expressed as a function of permeate purity and recovery

(13)

]'he bounds with respect to product recovery are obtained directly from the definitions of rp and rr

limrr

rr)

Inequality (17) provides an estimate of the m e m b r a n e area needed to obtain a given retentate recovery. I f we use the arithmetic mean of the upper and lower bound [Equation (17)] as the estimator for R, the m e m b r a n e area is predicted by

The upper bound for the permeate purity y and the lower bound for the retentate recovery r~ are obtained from the definitions of rp and r~, respectively lira y = ,-,~c --~R(I

-

1 - 1/g

~rr~l

-

R(1

1/g)

(161 '

'

F r o m a design point of view it is even more interesting that Equation (16) can be solved with respect to the

r,.

) ' ( l - 2) - 2 r p ( l - ) ' ) )'(1 - -)

(19)

Although Equations (19) and (18) m a k e it possible to calculate the m e m b r a n e area required to obtain a specified permeate purity and recovery, they do not indicate whether an SSP system with a finite ideal m e m b r a n e selectivity is able to delivery the specified permeate purity and recovery simultaneously. Membrane pressure ratio. Figure 5 shows product purity and recovery versus the m e m b r a n e pressure ratio g. The feed pressure is part of the dimensionless m e m b r a n e area R, and therefore it is the permeate pressure which varies according to g. As the m e m b r a n e pressure ratio increases, both permeate recovery and retentate purity approach an upper bound, whereas the retentate

156

Design of membrane permeator processes." T. Pettersen and K.M. Lien 1 / /

/

~'~0.6

\

0.6

/ /

0.4 0.2

0.8

/

0.8

/

~0.4

i

,.& /

0.2

/

101

°o0

102

000

M e m b r a n e pressure ratio, g

Figure 5 Produce purity ( ) and recovery ( represent the reference model (Shindo etaL 22)

_r)J

[:,_, + (l

c~- 1 [(~ - 1 ).,-: - (, ./'

(20) 1)_,]

(211

Dividing Equation (20) by Equation (21), and solving the resulting differential equation analytically, we obtain the following relation between the cut rate 0 and the retentate composition .v limO

1

exp((~l

[lnV-- (~ln )l~- x]

(22)

By substituting Equation (22) into Equation (21) (f 1 - 0 ) , an analytical solution of Equation (21) is obtained

Equation (25) provides the upper bound for permeate recovery, and Equation (26) the lower bound with respect to retentate recovery. Equation (24) shows the permeate purity as the pressure ratio approaches infinity. However, this permeate purity is not necessarily the maximum permeate purity which can be obtained in an SSP system. For gas permeation it seems to be generally accepted that permeate purity increases with an increasing pressure ratio across the membrane. Figure 6 shows the results of a parametric study on the relation between pressure ratio and permeate purity in an SSP configuration. Solid lines indicate constant (dimensionless) membrane area and dashed lines indicate constant recovery of component I in the permeate s t r e a m (rp). Since the partial pressure difference across the membrane is the driving force for gas permeation, it is expected that the recovery of components in the permeate stream will increase with an increasing prcssure ratio across the membrane. This is also the case in Figure 6. In terms of permeate purity, the situation is different. For low values of the dimensionless membrane area, the permeate purity increases with increasing pressure ratio over the whole range of pressures displayed. For larger membrane areas, however, permeate purity as a function of pressure ratio passes through a maximum. This behaviour can .

0.5--

l i m R = ( (~

1x -

1) exp ( ( ~ 1

102

- ) versus membrane pressure ratio. Conditions: z = 0.2, R = 0.40, c~.= 5. Dotted lines ( . . . . . )

recovery approaches a lower bound. The permeate purity shows similar behaviour to the permeate recovery but, as will be shown shortly, lime~ ~_ 3, does not necessarily represent the maximum permeate purity for given values of R, : and ~t. The limits for product purity and recovery are useful for evaluating the effect of an increase in the pressure ratio across a membrane with a given selectivity. As the pressure ratio g approaches infinity (the permeate pressure approach zero), the mass balance equations (1) and (2) reduce to the following form

df ]ira g - = dR dx lira e - ~ dR

101 M e m b r a n e pressure ratio, g

.

.

.

.

.

.

.

r

/ /

Crp--0.4

: : :

With this analytical solution of the mass balance equations, the limits with respect to product purity and recovery can be obtained from the total mass balance around the permeator module

: - (1

g~oc

0)x

(24)

=

~

///~"

.

.

s

~ ~

~

-

-

-

~ ~

R=0.20

.. "wo.8

0.35

"'~"

-=

0.3

-

I

rp

=

lira rr =

g~c

.

0

0V liln g.~,~

.

[ In ~- -(~In ~ ] ) 0.45

lim y

.

~

(25)

--

(1

1/: 0.2~. 0

/

i/

.

.

-

.

.

.

.

.

.

i 10 ~

.

.

.

.

.

.

.

. 102

Membrane pressure ratio, g

-

0)(1 J -- 2

.v)

(26)

Figure 6 Permeate purity vs. membrane pressure ratio, z=0.2, c~=5. ( - - ) : Constant dimensionless membrane area R. ( - - - ) : Constant permeate recovery rp

Design of membrane permeator processes." 1~Pettersen and K.M. Lien 1

1

0.8

0.8

~0.6

157

~0.6 O

? 0.4

\

.c. 0 . 4

•

N .

0.2 0

N

0.2

0

0

0.2 0.4 0.6 0.8 Feed composition, z

Figure 7 Product purity ( ) and recovery ( - - - ) ( . . . . . ) represent the reference model (Shindo e t 0/'.22)

0.2 0.4 0.6 0.8 Feed composition, z

versus membrane feed composition. Conditions: R=0.40, (x=5, g = 5 . Dotted lines

be explained by considering the effect that the pressure ratio has on the driving force for permeation. If the pressure ratio approaches 1. the driving force for separation approaches zero, and the composition in the permeate stream approaches the composition in the feed stream. An increase in the pressure ratio g has two effects: 1, it improves the degree of enrichment [see Equation (3)], which leads to an increase in permeate purity; and 2, it increases the driving force for permeation and, as shown previously in Figure 5, the rate of permeation of component 2 increases as g increases. For sufficiently large membrane areas, this increase in the permeation of component 2 may eventually lead to a reduction of the permeate purity as g increases beyond some critical value. This explains ~he behaviour observed in Figure 6 for SSP systems with R > 0 . 3 0 . The interesting thing to note is that. according to Figure 6, this can also occur at modest pressure ratios. Pan and H a b g o o d ~ also analysed the effect of the pressure ratio on the permeate purity in permeators with cross flow on the permeate side of the membrane. t h e y reached the conclusion that there is little to be gained with respect to permeate purity by maintaining :~,~>>~,. The results presented here are consistent with this conclusion. In fact. permeate purity is reduced if the pressure ratio is increased beyond some critical value. For perlneator systems with recycle, e.g. the SSP-PR. :t has already been reported in the literature that such a maximum in permeate purity can bc found s. To our knowledge i t has not previously been published that a +.dmilar maximum phenomenon can occur in simple permeators without recycle.

Feed composition. Figure 7 shows tile influence of the I'eed composition on product purity and recovery in an SSP system. As expected, both pertneate purity and recovery increase as the feed composition _- increases. With respect to the retentate stream, two things can be noted. Firstly, the algebraic design model fails to predict the retentate conditions for values of_- larger than ~0.5. [n particular, the prediction of the retentate purity is qualitatively wrong, compared to the reference model ">2 ~ . According to the reference model, the retentate purity ( l - x ) goes through a minimum as _increases from zero to ~-0.8.

Secondly, the retentate recovery decreases as the amount of component 1 in the feed increases. This behaviour is not expected, since the driving force for permeation of component 2 should decrease as the partial pressure of component 2 decreases in the feed stream. In order to explain these results it is necessary to consider the effect the feed composition has on the local permeation flux along the membrane surface. The permeation flux of component 1 is defined as cKx-y/g) and ik~r component 2 as (1 - x ) - ( 1 -y)/g. The product recoveries are related to the permeation fluxes in the following manner

'L"

rp

~.

#'r

1

~('~

1

y/g) dR

l _jll /"R[(l

.-<)

(27)

(i

)')/gldR

(28)

Figure 8a shows the permeation flux for two different values of feed composition : (the results are obtained from a numerical solution of the reference model presented by Shindo et al.~-2). As the feed composition changes from : - 0.2 to z - 0.8, the permeation flux of component 1 (solid line) is increased, while the permeation flux of component 2 (dashed line) is reduced. However, as ~- increases, the relative a m o u n t of component 2 present in the feed stream is also reduced. As the feed composition changes from z - 0.2 to 0.8, the amount of component 2 present in the feed stream is reduced by a factor of four, while the integral . t ~ [ ( l - x ) - ( 1 - y ) / g ] d R (in Figure 8a) is reduced only by a factor of two. This is illustrated in Figure ,~b which shows the relative permeation flux. The relative permeation flux is defined as ct(x-y/g)/z for component 1 (solid line), and [ ( 1 - x ) - ( 1 - y ) / g ] / ( l - z ) for component 2 (dashed line). Here it is seen that the relative permeation flux for both components increases as the feed composition increases from z = 0.2 to z 0.8. This explains why the retentate recovery decreases as z increases in Figure 7. The deviation between the design model and the reference model is also explained by the permeation flux. As the feed composition changes from z = 0.2 to -0.8, the non-linearities in the permeation flux become more pronounced. Due to the form of the curves describing the permeation fluxes for z = 0.8, a

Design of membrane permeator processes. 7~Pettersen and K.M. Lien

158

logarithmic mean related to the conditions at the feed entrance and at the retentate exit provides an overestimation of the permeation of c o m p o n e n t 2 and an underestimation of the permeation of c o m p o n e n t 1. This is consistent with the results shown in Figure 7 where the algebraic design model underestimates the permeate recovery and overestimates the retentate recovery for large values of z. F r o m Figure 7 it is clear that the retentate purity is not predicted with the desired accuracy for cases where z~>0.5. For cases where the retentate purity is a critical product property and z>~0.5, e.g. removal of carbon dioxide from natural gas, the accuracy can be improved by dividing the calculations into steps, so that in each calculation step the cut rate is less than 0.5. Since both permeate purity and recovery are predicted with reasonable accuracy, this stepwise calculation procedure should not be necessary for cases where the retentate purity is of less importance, e.g. in recovery of hydrogen from purge streams, where the recovery of hydrogen in the permeate stream is the most critical design parameter. In Figure 7 it is seen that the retentate recovery approaches an upper limit and the permeate recovery approaches a lower limit as z approaches zero. For z = 0 the following limits are valid with respect to composition along the m e m b r a n e lim x = 0 ; z~O

limv=0 --~0 ~

=

(1

-

~0

l/g)R

(30)

The upper limit with respect to the retentate recovery can now be obtained from the definition of r~ limr,

=

1 -

(1

l/g)R

-

--~0

1) l-1,/g]]- I

limO )
(32)

/~)c~/[(c~

1)(1

l/g)]]/0

(33)

Solving Equation (32) with respect to x/z and substituting into Equation (33), the following approximation for the permeate recovery is obtained ]in(]rp = 1 - [ 1

R(1 -- l/g)] (r/[a-((~-l)(I-1/g)]

(34)

Equation (34) does not represent a tight lower bound for permeate recovery. C o m p a r e d to the values of rp obtained from the reference model, Equation (34) provides a conservative estimate (underestimation) ot the permeate recovery. The deviation is typically - 5 % c o m p a r e d to the values obtained from the reference model.

(29)

Again, the mass balance equation (1) is solved analytically with respect to the module cut rate lira0

To determine the permeate recovery rp as z approaches zero is more difficult, because both the n u m e r a t o r and the d e n o m i n a t o r in the definition of rp a p p r o a c h zero. However, an a p p r o x i m a t e solution can be obtained. Pan and H a b g o o d 3 showed that the performance of a p e r m e a t o r module with cross flow on the permeate side is described by the following equations, as the feed composition z approaches zero

(31)

a)

Summary of purity and recovery constraints for SSP systems. The parametric studies have illustrated that the performance of an SSP system with respect to product purity and recovery can be described by upper and lower limits. These limits are determined once two or three of the key variables R, c~, g or z are given. Table 1 provides a s u m m a r y of these limits. In order to illustrate the use of these limits, we will present a simple design problem. Problem." Determine whether an SSP system can be used to remove carbon dioxide (CO2) from a stream ot

b)

3.5

3.5

*"" ~z=0.8

t ~

2.5

=

g

2

Q)

~1.5

0.

1

. _ - : : 2 1 _ _ Z_;.~ . . . . . . . . . . . . . . 0.5

0

o.as

o'.,

o.~s

0'.2

o.~s

o'.a

Dimensionless area from the leed entrance,

o.~s

R

o.~

.

0.;5

5

011

03S

~

0'2

0.)5

0'.3

0.;5

Dimensionless membrane area from the feed entrance, R

Figure 8 An SSP system, (~ = 5, g = 5, R = 0.4. (a) Permeation flux. Component 1 ( ): (~(x- y/g). Component 2 ( (b) Relative permeation flux. Component 1 ( - - - - ) : ~(x-y/g)/z. Component 2 ( - - -): ((1 - x ) - (1 -y)/g)/(1 z)

0.4

- -): (1 - x ) - (1

-y)/g.

Design o f m e m b r a n e p e r m e a t o r processes." T. Pettersen a n d K.M. Lien ]Fable 1 Summary of upper and lower bounds with respect to product purity and recovery for SSP systems Given:

Permeate stream

z.{t. ga

y%limR .0 Y

z.R,g

Y<~z.R,1z 1-~,

1

rn~R(1

r~l

1/gl

Retentate stream

RI1

1 'g)~r~ R,1 1 g

,

z.R,(i bc

lim 9 .., y

• ~,,

1

x~lim#.,

rp ~ i~y

ft..% 1

rp ~>limz .OG

r~l

•

x

~t 1 x, f~

R, ~t, gd

R(1

1/gl

limR~ o y is determined from Equation (7)

lim#~• 0 and lim~,_ ~ x are determined from Equations (22) and 123), respectively limg~, y does not necessarily represent an upper bound for the permeate purity r Equation (34) provides an approximation for l i m ~ o r p

natural gas (CH4). The following data are given: ~*(:O,/C'H~ -- 35, QCH~ 0.0073 m ~ (STP) h I bar i m -'. Pr-~ 50 bar, Pp>~l.5 bar, n~.-- 10 000 tn 3 (STP) h 1 : : : c o , - 0 . 3 . Product requirements are: .v(o.40.025, t'r-CH4>/0.95.

The dimensionless membrane area needed to meet lhe recovery constraints is determined from Equation (18): R - 0.044+0.008. Inserting the given values for (_)('H4, Pr and nr, the membrane area determined is ,~ ~ 1200 ± 2 2 0 m 2. For 0.3. the membrane area is determined within 20% accuracy. The next step is to determine whether an SSP system with the given membrane selectivity is able to meet the product requirements. This can be done bx. checking if an SSP system with an infinite pressure ratio across the membrane (an optimistic assumption) rcquires a larger membrane area than the upper bound already obtained (R~<0.052). Inserting values for (~{O/Cn,, .\CO: and :co2 into Equation (23) results in lim~,~, R - 0.064, which shows that the selectivity of the membrane is too low to obtain the specified retentate recovery. This conclusion is reached without solving any implicit :tlgebraic or differential equations. If we do solve the algebraic design model with 'r.CH4 0.95 (and purity unspecified), we obtain R - 0.044, which for this case is exactly the same value as the one estimated from the upper and lower bounds. The retentate purity obtained from this SSP ,,ystem is, however, x c o . 0.09, which does not meet the specified purity. I n - o r d e r to obtain the specified purity (Xco,~<0.025), the membrane area would have to be R =-0.09 which, according to the algebraic :[esign, results in a retentate recovery ~-90%. This illustrates a simple fact about SSP membrane systems: with finite selectivities, purity and recovery constraints :~.re tightly coupled.

and investment costs associated with a membrane permeator system for gas separation. Figure 9 shows how the dimensionless membrane area and the pressure ratio affect the permeate purity (solid lines) and the permeate recovery (dashed lines) in an SSP configuration. The important thing to note in Figure 9 is the trade-off between the membrane area and the pressure ratio. For a given permeate recovery, one can either install a large membrane module and operate the system at low pressure ratio, or increase the pressure ratio across the membrane and install less membrane area. The latter will lead to a permeate product with high purity, whereas the former will lead to low permeate purity. The optimal trade-off between pressure ratio and membrane area will depend on the cost of compression versus membrane area, i.e. the ratio k',, K~.. If we assume that feed is available at the same pressure as the permeate product and a vacuum is applied at the permeate side of the membrane, that the investment cost of the compressor/vacuum-pump is proportional to the compressor duty, and that the feed is free, which are reasonable assumptions e.g. for the separation of air, the annual cost of an SSP system can be related to R and g by the following cost function (see Appendix 1) Costssp - K~. ln(g) + K~R

-I:he membrane area and the compressor duty are the most important factors which determine the operating

(35)

With Equation (35) as the objective function to be minimized, the optimal trade-off between compression and area costs has been studied for various values of K,/K~.. Three such optimal trade-off curves are plotted in Figure 9 (dot and dashed lines) as a function of the permeate recovery rp. Note that as compression becomes more expensive (e.g. K,/K~ decreases from 20 to 5), it becomes profitable to install more membrane area in the SSP system and reduce the compressor duty. In addition to productivity (recovery) requirements, there will also be product purity requirements, stated as ~> inequality constraints. Depending on the ratio K~,/K~, purity may or may not be an active constraint.

I/ I ' I

._o-10

~ I I'

°o',sq @, ,':

~ [ '1

'/

[; K/K ~n

',

'

",KJ~=IO

o

8 13

r~ = 0.2 10~0

Trade-off b e t w e e n m e m b r a n e area a n d :::ompressor d u t y

159

0'1

012

013 014 015 016 0:7 Dimensionless membrane area, R

rp = 0.4 0'8

0'.g

Figure 9 The effect of the pressure ratio g and the dimensionless membrane area R on permeate purity ( ) and permeate recovery ( ). z = 0 . 2 , ~ = 5 . ( . . . . ) Optimal trade-off curves between g and R as a function of rp and the ratio Ka/Kc

160

Design of membrane permeator processes." T. Pettersen and K.M. Lien

b)

a ) 0.65

5

-0.4

0.6[

~

.

/,=0.2 /,=03/,=0,

/

m

', ~

R=0.1

't\ 1 \\%-.
"o=I

0.25 0 2L 0

',,\',X

°4

~--~3.5

\\.\ \ \ "x.--'"\

;

/

/k /\ 1/

"'"-

""

"'.

\

lr~0-3

/

\

"'\ \

I

s_-o

.6 s=0.O 0.5

~=v.~ 1

Dimensionless membrane area,

1.5 R

0

0.1

0.2

0.3 Permeate

0.4 0.5 recycleratio, s

0.6

Figure 10 The performance of an SSP PR system. Conditions: z=0.2, c~= 5. (a) Permeate purity vs. dimensionless membrane area, g = 10. ( ) Different values of the permeate recycle ratio, (- -) constant permeate recovery. (b) Compressor duty indicator vs. permeate recycle ratio with constant permeate purity y--0.5Q () Constant dimensionless membrane area R. Note that g is here freetovary

If the purity constraint is not active at the optimal trade-off between compression and area, then overpurification of the product is profitable, and a bypass configuration, where parts of the feed are mixed with the overpurified product, reduces the load on both the m e m b r a n e p e r m e a t o r system and the compressor, without compromising purity requirements. The conditions when overpurification of the permeate product and use of bypass configurations are profitable are cases where compression is inexpensive c o m p a r e d to the cost of m e m b r a n e area, and where permeate purity requirements are modest. A typical example is small scale production of low purity O~enriched air (Yo~~>0.30) using an inexpensive compressor solution (vacuum p u m p on the permeate stream). The use of overpurification in such cases may seem counterintuitive and may thus be suspected to be an 'academic artifact'. However, this solution has already been reported in m e m b r a n e literature: Bhide and Stern "~ indicate that this m a y indeed be a cost-efl'ective solution in practice. The above analysis explains why.

Single-stage recycle configurations One extension of the single-stage permeator configuration is obtained by introducing a recycle stream, where a fraction of one of the product streams is mixed with the feed stream before it enters the permeator module. The main effect of a recycle stream in an SSP system is to change the feed composition to the m e m b r a n e permeator. Figure 7 illustrated the effect of the feed composition with respect to product purity and recovery. According to Figure 7, an increase in z leads to an increase in both permeate purity and recovery. An increase in _- can be accomplished using a permeate recycle configuration (SSP-PR). On the other hand. Figure 7 suggests that a reduction in -, which can be accomplished by a retentatc recycle configuration (SSP-RRL may improve both retentate purity and recovery. Both the SSP-PR and the SSP-RR config-

urations have been proposed earlier as potential module configurations 9. Here we will illustrate h o ~ retentate and permeate recycle influences the performance of an SSP system.

Single-stage permeator systems with permeate recycle. F(~ure 10 shows the effect of permeate recycle in an SSP system, again with the production of Oz-enriched air as the example. It is noted that increasing the recycle ratio for a fixed value of the dimensionless m e m b r a n e area R leads to an increase in permeate purity and a decrease in permeate recovery is (and notice that R is defined in terms of the system feed flow rate nr, not in terms of the feed flow rate to the m e m b r a n e module). The reduction of permeate recovery is caused by the increase in the feed flow rate to the m e m b r a n e module. Thus, a permeate recycle stream increases the permeate purity, but reduces the productivity of the system. As the dimensionless m e m b r a n e area R is increased from zero. the permeate purity goes through a m a x i m u m if the recycle ratio is held constant at a value larger than ~ 0.6. Stern et al. 6 have previousb reported such maxima in permeate purity for SSP-PR configurations. Since the permeate recycle stream has to be recompressed, the flow rate through the compressor is larger in an SSP-PR than in an SSP processing the same teed stream. The actual feed flow rate to the m e m b r a n e module in an SSP-PR system is given by the recycle ratio s and the module cut rate Itf.SSP

PR -- l

nr

5;0

(3<

Thus, the compressor duty in an SSP-PR system. where the feed is compressed from permeate pressure to the m e m b r a n e feed pressure, is proportional tc lng/(l -sO), according to Appendix 1. However, the permeate purity increases with increasing permeate

Design of membrane permeator processes: T. Pettersen and K.M. Lien

a)

b)

0.5

1

0.41

0.98

0.4~

0.96

0.44

0.94

_~0.42

~'~-"0.92

® 0.4

go.9 0~

a. 0.38

0.88

0.36

0.86

0.a4

0.84

0.32

0.82

0.3

0.()5

0'.1

0.16

0.2

0.~)5

013

0.35

rr= 0.8

0.8 0

0.4

Dimensionless membrane area, R

o'1

F i g u r e 11 Product purity vs. dimensionless m e m b r a n e area in an SSP-RR system. ( - - ) p r o d u c t recovery. Conditions: z = 0.2, ~ = 5, g = 10

recycle, and therefore an SSP-PR system could be operated at a lower pressure ratio than an SSP system, ,delivering the same permeate purit~ and permeate recovery'. It is therefore worth investigating whether this possible reduction in pressure ratio outweighs the increase in flow rate through the compressor, in terms of the compressor duty'. In Figure lob, the permeate purity is fixed (y = 0.50), and the compressor duty' is plotted versus the permeate recycle ratio s for various ,combinations of the permeate recovery rp and the dimensionless m e m b r a n e area R. It is seen that a reduction in compressor duty is obtained at the expense of an increased m e m b r a n e area. For example, consider ~'p = 0.4. By' increasing R from 0.1 to 0.2 and s from 0.03 to 0.48, a 35°/0 reduction is obtained in the compressor duty'. It should be noted that the possible reduction in compressor duty increases with increasing permeate recovery. In other words, the SSP-PR ~ecomes more profitable compared to an SSP if permeate purity' and recovery requirements are high, and if compressor duty' is expensive relatixe to the cost of m e m b r a n e area.

Single-stage permeator systems with retentate recycle. Figure l l shows product purity; versus the dimensionless m e m b r a n e area for different values of the retentate recycle ratio. Curves for constant product recovery are also given (dashed lines). As expected from the previous analysis, both permeate purity and recovery are reduced by retentate recycle in an SSP system. 'More interesting is that retentate purity and recovery are also reduced by the retentate recycle stream. Thus, retentate recycle seems to be detrimental to both product purity and recovery. Although the SSP-PR has been proposed as a potential configuration, it is not known by the authors whether any case study has found the SSP-PR to be competitive. The results shown in Fi,~,m'e 11 are consistent with this.

161

o'.2

o'3

0'4

o's

Dimensionless membrane area, R

0'6

Different values of the recycle ratio. ( - - - ) C o n s t a n t

Multi-stage and recycle configurations Since the m e m b r a n e selectivity in most commercial m e m b r a n e s is limited (typically in the range 2-100), a single-stage p e r m e a t o r configuration (SSP) is not always sufficient to perform the desired separation task. To overcome this lack of selectivity, various multi-stage recycle configurations have been proposed (see Figure 1). Here we will study the characteristic behaviour of the module configurations shown in F~gure 1 based on a general design criterion. The objective is to: I

2

3

C o m p a r e the performance of SSP systems and multi-stage and recycle cascades. What m a y be gained by using a multi-stage cascade instead of an SSP system? Study' the performance of the various multi-stage configurations. Do general trends exist which can serve as guidelines for designs? Study' the effect of m e m b r a n e properties (selectivity and permeability) in a multi-stage cascade. W h a t may be gained by using different m e m b r a n e materials in each stage of a multi-stage cascade?

Design criteria Since different module configurations have different degrees of freedom in terms of design, a design criterion is needed to compare the system performance in a consistent way. The 'extent of separation', defined by R o w > , is a general measure of the degree of separation which is achieved by a separation system. For a l-input 2-output system separating a binary mixture, the extent of separation is defined as det

Cl_

El C',

det

Ep ' 1

rr

rr F

rp + ,'r

1

Oz( 1

--

Z

z)

(37)

Design of membrane permeator processes. 77Pettersen and K.M. Lien

162

b)

a) 1/ 0.9l

t

.

/

0.8

C~2 " 0.8

0.7

~

>=

0.6

o

0.6

~ 0.5

o

~

rr 0.4

g

0.4 0.3 0.2

C

0.2

0.1

0.2

F i g u r e 12

0.4 0.6 Permeate recovery

0

0.8

0

0.4

0.6

0.8

1

Permeate recovery

G e o m e t r i c i n t e r p r e t a t i o n o f : ( a ) the e x t e n t of separation, C (b) the d i r e c t i o n of separation, ,4

The row vector C1 represents the distribution of c o m p o n e n t 1 between the two product streams, while C2 represents the corresponding distribution of component 2. For a binary separation the extent of separation can be given a geometrical interpretation. It can be shown that ~ is proportional to the length of the vector C1 - (~2 (see Figure 12a). In this work. the extent of separation has been maximized for various values of the total dimensionless m e m b r a n e area, given the feed composition to the system, the m e m b r a n e selectivity and a m a x i m u m pressure ratio across each m e m b r a n e m.odule in the system. This corresponds to solving the following nonlinear optimization problem: Maximize Subject to :

[Model of permeator system] ~

-

constant

g = constant - -- constant

~stage,, Rstage

constant

(38)

In addition to the extent of separation, information on the product distribution is also needed to fully describe the performance of a separation system. Here we introduce the term 'direction of separation' J, which is based on the geometric interpretation of the extent of separation. The direction of separation 54 is defined as the angle between the d i a g o n a l i n the parallelogram formed by two vectors Cl and C:, and the abscissa in Figure 12b. In terms of product recoveries, the direction of separation is defined as l - - rp - ~ r r tan/7

0.2

(39)

--

1 + rp -- rr

This definition implies that a separation system which favours high recovery of the slowest permeating c o m p o n e n t in the retentate s t r e a m ( r r > r p ) is characterized by D/>45 . Likewise, if a system favours high

recovery of the fastest permeating c o m p o n e n t in the permeate s t r e a m (rp > rr), then /3 < 45'.

Multi-stage v e r s u s single-stage system

Figure 13a shows the m a x i m u m extent of separation plotted versus total dimensionless m e m b r a n e area for the single-stage systems (SSP, SSP-PR) and the twostage recycle cascades (2Enr,2Str). These results have been obtained with the following specifications: - - 0.2, ~ = 5 and g~< 10. It is seen that single-stage systems without recycle (SSP) behave fundamentally differently than the recycle systems. In terms of the extent of separation, the SSP configuration shows a maximum, whereas for the recycle systems, ~ increases monotonically with increasing R. The recycle systems arc able to perform a substantially sharper separation than an SSP configuration with the same m e m b r a n e selectivity and m e m b r a n e pressure ratio. The reason why an SSP configuration shows a m a x i m u m in ~ is rather obvious when considering the definition of the extent of separation [Equation (37)]. At low values of R, only a small a m o u n t of the feed permeates through the membrane. This corresponds to 0 ~ 0, which implies that ~ 0 . Similarly for large values of R: here 0 ~ 1, but in this case no enrichment can be obtained in the permeate stream, and since y - z ~ O , then ( ~ 0 . Thus, for a fixed m e m b r a n e pressure ratio, there is a trade-off between product recovery and product purity which gives rise to the m a x i m u m in observed for the SSP configuration. This trade-off is observed in Figure 13b where the same results are displayed in terms of permeate recovery and permeate purity. The figure also illustrates the differences in performance a m o n g the recycle configurations. In terms of the extent of separation (Figure 13a), the difference between the 2Str and 2Enr configurations does not seem to be large. F r o m Figure 13b it is seen that the 2Enr cascade yields a permeate product with a high, relatively constant purity over the whole range of R, whereas for the 2Str cascade, both permeate purity and recovery increase with increasing R.

Design of membrane permeator processes. T. Pettersen and K.M. Lien

a)

163

b) 1 0.9

:

0.5 . . . . . . .

\ SSP

"

'

'.

!

'

0.95

i

2Enr

0.9

/12,.,

,_.O,SE 0.5 . . . . . . . .

/'/""-

. ~

SSP-PR

~0.~

!0,,I

.. .i.......

...... ... "..'."

.

i 0.7

Q.

0.65 0 2t/

........

' SSP

0.6

// 0"10~__

0.55

......

0

o~

0.5 1 1.5 Dimensionless membrane =tea, R

c)

0'3

0:4

o's

0:5

0:7

Permeate purity, y

o'8

0:9

o16

o19

d) 3 2.8

ev

=*

1 Ill

2.6 t-i 0

: SSP-PR

z 2.4

~

2.2

i 0.

~1,8 ._~ ~1.6 1.4

2Enr

SSP-PR

En 2St~

1.2

03 F i g u r e 13

04

o.s

0:5

o'.7

Permeate purity, y

o'.~

Parametric study of 1-stage and 2-stage

0'9 systems. Maximize

Figure 13c shows the relationship between permeate purity and dimensionless m e m b r a n e area. This plot shows the penalty which has to be paid in terms of increased m e m b r a n e area in order to obtain an increase in permeate purity using recycle configurations. As an illustration, consider the various configurations when the required permeate purity 3. = 0.5. The SSP :onfiguration (a single-stage p e r m e a t o r without re:ycle) can deliver this purity in the limit, but the permeate recovery for an SSP system at y 0.5 ~.pproaches 0. F r o m Figure 13b it is seen that, at 1' = 0.5, the permeate recovery of the SSP-PR config:ration ( r p ~ 0 . 8 5 ) is ~ 5 % larger than for the 2Str ,::onfiguration; but according to Figure 13c the SSP-PR ,:onfiguration requires three times as much m e m b r a n e area as the 2Str configuration at v 0.5 to achieve this !:% better recovery. The compressor duty is another important factor in determining the cost of a p e r m e a t o r system. Compres:;or duty is mainly determined by the flow rate through "he compressors and the pressure ratios across the compressors. Since we are using a dimensionless model ~:[tow rates are incorporated in the dimensionless

~

013

014

015 o16 oi~ Permeate purity, y

(, s u b j e c t to: z = 0.2, c~.= 5, g <~ 10

m e m b r a n e area), a normalized compressor duty ( N C D ) was defined (see Appendix 2). N C D is defined as the ratio between the sum of duties in all compressors (including a feed compressor) and the duty required to compress the system feed from permeate to feed pressure. The N C D for an SSP configuration where the feed is available at permeate pressure is therefore equal to 1. Figure 13d shows N C D versus permeate purity. If again we c o m p a r e the SSP-PR and the 2Str configurations at y = 0.5, now with respect to the required NCD, we find that the 5% better recovery of the SSP-PR requires more than twice the compressor duty. This is due to the fact that the SSP-PR configurations will here require larger recycle streams than the 2Str configurations.

Characteristics of some multi-stage module configurations In the previous section it was noted that recycle configurations have different characteristics in terms of product purity and recovery. Here we will study these trends more closely. Figure 14 illustrates the

164

Design of membrane permeator processes." T. Pettersen and K.M. Lien

a)

b)

.o

1

0.95 55

0.85

E 50

0.8

4s

/3Stl'

//

0.9

0.75

O.7

2

:

i

Enr

//

40

0.65

SSP2Enr

06 2Enr.

055

&

;

0

I'.5

.2

o'a

014

Dimens,onl~s membrane area, B

c)

I

812

0'.8

0.9

!

n

,2 013

07

O'"

O)~

L2

Permeate punty, y

0.3

2S!r

i

04

05 06 07 Permeate purity, y

018

og

Parametric s t u d y of recycle system performance. Maximize (, subject to: z = 0.2, c~ = 5, g <~ I0, R = fixed parameter

performance at the m a x i m u m cxtenl o1" separation for various three-stage systems in addition to the two twostage systems illustrated in F~,~ure 13. It is seen that at the m a x i m u m extent of separation, the three-stage extensions fill in and extend the picture indicated by the two-stage systems. Some fairly obvious trends can be recognized:

2

0;7

2.8 2.6 2.4 2.2 2 1.8 1.0 1.4

IEnr

1

0.6

Permeate purity, y

d

.5

F i g u r e 14

o;s

Performance in terms of the direction of separation shows that the p e r m e a t o r systems may be divided into two categories. The enrichcr cascades (2Enr. 3Enr, 3StrEnr and SSP2Enr) generally favour retentate recovery (,'4>45). Notice that high retentate recovery also leads to high permeate purity. The stripper cascades (2Str,3Str) and SSPPR configurations, on the other hand, favour permeate recovery ( , 4 < 4 5 ) . Introduction of more stages, e.g. using a 3Str instead of a 2Str, makes it possible to obtain the same purities and recoveries by use of less m e m b r a n e area and less compressor duty. (However, since the number of compressors increases, this does not necessarily, imply reduced total costs.)

By using a 3Enr configuration instead of a 2Enr configuration, higher purity can be obtained for the same recovery. In addition, it is observed that: 4

5

At the m a x i m u m extent of separation, increased area and compressor duty do not significantly affect the permeate purity obtained by the enricher cascades. For these systems, the main effect of increasing area and recycle flows is increased permeate recovery. The stripper cascades exhibit both increased recovery and purity when the area and recycle flows are increased, and the same seems to be the case for the "combined' configurations SSP2Enr and 3StrEnr.

All the plots presented up to this point have been based on fixed values for feed composition, m e m b r a n e selectivity and pressure ratio across the membranes. It can therefore be questioned whether the findings reported above are valid only for selected values of these parameters. In an attempt to assess whether this

Design of membrane permeator processes: T. Pettersen and K.M. Lien is the case, parametric studies have been performed for different values of these parameters. The following sets of conditions have been studied: 1, z 0.2, c ~ - 15, g~<10; 2, z 0.2, c~ - 5, g~<5: and 3, z 0.5, (~ - 5, g~< 10. The results obtained are shown in Figure 15, and it is clear that the qualitative picture remains the same. Increased selectivity makes the curves move towards higher purities and they drift farther apart. Decreased pressure ratios push the curves towards lower purities and enriched feeds naturally push the curves towards higher permeate purities. However. the general shape of the curves remains the same, and the relative position of the individual curves remains invariant. This indicates that the qualitative picture provided by these curves is rather general. It may also be questioned whether designing and operating a structure at its m a x i m u m extent of separation is always the best choice, since this measure involves a clear trade-off between purity and recovery [see Equation (37)]. This question cannot be answered in any general way without including considerations of factors such as the cost of compression, m e m b r a n e area and the value of the product being purified. Such considerations are most easily covered in problemspecific case studies.

Membrane properties in multi-stage systems At present, most commercial multi-stage permeator systems use the same m e m b r a n e material in all stages of the cascade (single-membrane systems). Here we wish to study how a multi-membrane system (where different m e m b r a n e materials are used in each stage of the cascade) can improve the performance of a permeator system. Removal of carbon dioxide (CO_,) from natural gas (CH4) with a desired product purity of 1.5 m o l % CO2 (Xco,~<0.015) in the retentate stream is used as an example. T h i s separation problem has been addressed by several authors 0 "1-~ 16 , and ~ith respect to module configuration, the SSP2Enr (see Figure 1} seems to be the best suited system. In the most recent and extensive evaluation of m e m b r a n e processes for treatment of natural gas by Bhide and Stern ]~'~e cellulose acetate is used as m e m b r a n e material in all stages of the SSP2Enr system. The effect of m e m b r a n e selectivity is only considered through a simultaneous variation of the permeability of CO~ for fixed permeability of C H 4 . In reality, however, the m e m b r a n e selectivity is highly connected to permeability. A change in m e m b r a n e selectivity can be accomplished by changing operating conditions such as temperature, or by changing the m e m b r a n e material in the permeatot. In both cases it is likely that the permeabilities of all the components are afl'ected. With respect to membrane material it is often noted that m e m b r a n e materials which provide high m e m b r a n e selectivity tend to have low permeability, and vice versa. Thus. one might expect to find a trade-off between membrane selectivity and m e m b r a n e permeability. Recently, Robeson e4 proposed an upper bound relationship between the permeability one might expect to find in polymeric materials and the ideal m e m b r a n e selectivity for a number of different binary gas mixtures. Based on this correlation and data provided by' Bhide and Stern tl for cellulose acetate membranes, we obtain the following upper bound relationship

165

between the permeance of CH4 and the ideal m e m b r a n e selectivity (where the active layer is assumed to have a thickness of 1000 ,~,) 6264 QCH4<- 2 8 9 3 . 4 ~C6-/'CH4

(40)

The total processing cost is calculated as the sum of investment costs (compressors and membranes), operating costs (compression duties, m e m b r a n e replacement, labour and maintenance) and the value of lost p r o d u c t (CH4). The economic data used for cost estimation are the same as those used by Bhide and Stern it. For each case which is presented, the following non-linear optimization problem has been solved: Minimize

Total processing cost

Subject t o :

[Model of an SSP2Enr system]

[$/10.000 m2(STP) feed] -),~ c,.~ .

3.6264

QCH~ ~ , ~ o . ~ OIC02/,CH4 P r - - 55 [bar] Pp >/1.4

nr

[bar]

40.000 [m3(STP)h 1]

xco~ ~<0.015 z - constant

(41)

Figure 16a shows the total separation cost versus feed composition. Note that for each value of the feed composition, m e m b r a n e properties were constrained by inequality (40). In terms of separation costs, the difference between a single- and a multi-membrane system does not seem to be large. The latter yields only 2 3% lower separation costs. In terms of product recovery, the situation is different. Loss of methane represents 30 40% of the total separation costs, and the loss of methane in a single-membrane system is 15 20% larger than in a multi-membrane system. Thus. for processes where recovery of the product is important, multi-membrane systems might be considered. In Figure 16b. corresponding results for m e m b r a n e selectivities are presented for the two systems. For the multi-membrane systems, high selectivity (ct~,34) is preferred for the stages where CO2 is removed from the system (stage l and 3; see Figure 1). In stage 2 of the SSP2Enr cascade, the optimal m e m b r a n e selectivity is lower ( ~ z 2 4 ) , which, according to inequality (40), implies that the permeance of CO2 is approximately three times larger for this m e m b r a n e material than for the material used in stages 1 and 3. This makes it possible to obtain larger recycle ratios in the 2Enr part of the cascade c o m p a r e d to the single-stage system where a m e m b r a n e material with selectivity o~,~30 is selected. The recycle ratio in the 2Enr part of the cascade is also the reason why the multi-membrane system exhibits higher investment and operating costs, compared to the single-membrane system where the recycle ratio is smaller. Figure 16a also contains another interesting feature with respect to the m e m b r a n e permeator system. The total processing cost has a m a x i m u m , corresponding to a feed composition of 35 tool% carbon dioxide. Thus. from a design point of view, an increase or a reduction of the CO~ content in the feed stream both

166 Design of membrane permeator processes. T. Pettersen and K.M. Lien

a)

b)

3Enr 3StrEnr

0.95 0.g

f

.~ 0.85

e~

....

60

1

,

,

,

,

,

55

,r

\

\

.3Enr

50

0.8

tr

45

c "~ 0.75 uJ

40

0.7 SSP2E~//

~ p

35

065 0.(

01

0.2 0.3 04 Dimensionless membranearea, R

30

0.5

c)

0.1

0.2 0.3 0.4 Dimensionless membrane area, R

0.5

d) 6O 0.9 0.85

55

0.B

~

0.75 c "~ 0.7 ~0.65 "6 E O.6

3Enr

50

;inr

2Enr

2s::

45 3Str _ SSP 40

n

05 0.45 0.4

r

SSP2E/Kt"~S~p 0.2

~

,

SP'PR

,

0.4 0,6 o.a 1 1:2 114 Dimensionless membranearea, R

1:6

e)

SSP-PR

Ii

0.55

0.2

0.4 0.6 0.8 ; 112 114 Dimensionless membrane area, R

%

f)

6o~ ~[ ~ ~ ~ ~

t

09

3StrEnr 0.85

~

0.8 0.75 o.7

(n 0.65 "6 ® 0.6 LU 0.55

2Str

5f

~50

3Enr r

9 I

/-

~3s.

_ _ _ _ - -

0.5 0.45 0.4

Figure15

o2

o4

o:6

o18

;

1:2

1'4

Dimensionless membrane area, R

116

30 I 0

0.2

0.4 0.6 0.8 1 1.2 1.4 Dimensionless membrane area, R

1.6

~ystem p e r f ~ r m a n c e a t v a r i ~ u s d e s i g n c ~ n d i t i ~ n s : ( a ~ b ) z ~ 2 ~ g ~ 1 ~ = 1 5 ; ( c ~ d ) z ~ 2 ~ g ~ < 5 ~ = 5 ; ( e ~ f ) z = ~ 5 ~ g ~ < 1 ~ a ~ = 5

Design of membrane permeator processes. 77.Pettersen and K.M. Lien

a)

167

b) 40

; ~)

7

3

8

8

8

8 38

Total

8

£6 ts

I-o~

1.stage o o 3.stage

36 3 4

Invaslment and operation o o o o+ o+ + + +

o +

0

o

0

0

o

o

0

0 8

O

o o

32

o

o

o o

o +

+

0

+

4-

+

+

+

+

0

0

0

0

0

÷

÷ + +

83 g

o +

+

+

+

o

o

o

+ o

o

o

+

~28

+

o

o

2.stage

26

Value of lost product

0

0

Q)

0

24

1

o +

Multi-membrane systems Single-membrane systems

1

0:2

22

013 014 015 Feed composition, z(C02)

0

o +

28 1

0.6

Multi-membrane systems Single-membrane systems 012

013 014 Feed c o m p o s i t i o n ,

0

1 015

0.6

z(CO2)

F i g u r e 16 Removal of C02 (various feed compositions) from natural gas using an SSP2Enr permeator system, described by inequality (41). (a) Separation cost vs. feed composition. (b) Optimal membrane selectivities vs. feed compositions

a)

b)

8OO0

700(

7

$Q}

- - Stage 1

600(

13_ 1--

Stage 2

~5000

-

Stage 3

400~

r

83

/

g ~2

/ / /

2000

Total sep. cost Membranes

Compression - - Lost product

m

3000

I --

--

e~

1000

.1

Figure stage

17

012

0'a

0'.4

0's

Feed composition, z(C02)

o'.6

0.7

0:2

0'a

0'.4

0'.s

Feed composition, z(C02)

0'.6

Removal of CO 2 from natural gas using an SSP2Enr permeator system. (a) Separation costs.(b) Membrane area in each permeator

lead to a reduction in the total separation costs. The explanation for this behaviour can be found from a more detailed study of the distribution of the separation cost between the different parts of the SSP2Enr process. Figure 17a shows the contributions to total separation costs from the membranes, the compression and the value of the lost product (note that labour and maintenance costs are not shown in this figure). As the C02 content in the feed stream decreases, the cost of compression decreases monotonically, while both the cost of membrane area and the value of lost product go through a maximum. Figure 17b shows corresponding values for the membrane area for each stage in the permeator cascade. As the CO, content in the system feed decreases, the membrane area in the 2Enr part of the cascade (stages 2 and 3) decreases. This means that

as the CO2 content of the feed increases, more o f the separation is performed in the inexpensive SSP part of the cascade, and therefore the cost of compression (the recycle compressor is in the 2Enr part o f the cascade) is reduced. To explain why the membrane area and the recovery of methane (value o f lost product) both go through a m a x i m u m is not easy. This is a result o f the trade-off between the three major cost factors in the process. However, it should be noted that the optimal value for the total membrane area is sensitive to the CO2 content in the feed stream. As Zco, increases from 0.15 to 0.20, the total membrane area increases by 15% from 6000 m 2 to ~7000 m 2. This indicates that variations in the feed composition should also be addressed in the design of membrane permeator systems for gas separation.

168 Design of membrane permeator processes." T. Pettersen and K.M. Lien Conclusions The behaviour o f several single- and multi-stage permeator systems has been studied using a recently developed algebraic design model. Upper and lower bounds with respect to product purity and recovery in single-stage systems are presented. It is s h o w n that single-stage permeators without recycle can exhibit maxima in permeate purity as a function of the pressure ratio across the membrane. It is illustrated why bypass configurations may be economically profitable in single-stage systems. The effect o f product recycle in single-stage systems has been studied. It is shown that permeate recycle can reduce the compressor load in single-stage permeator systems. Similar benefits o f retentate recycle have not been identified. The characteristic behaviour of multi-stage systems has been studied. Based on some general design criteria. various module configurations have been classified as suitable for recovery of either the slowest permeating c o m p o n e n t or recovery of the fastest permeating component. The effect of using different membrane materials in each stage o f a multi-stage permeator cascade has been studied. It is shown that improvements with respect to product recovery can be achieved.

13

Lonsdale H.K. The growth of membrane technology J Membram Sci(1982) 10 81 181 14 Matson S.L., Lopez J. and Quinn J.A. Separation of gases with synthetic membranes Chem Eng Sei (1983) 38(4) 503 524 15 Fritzsche A.K. and Narayan R.S. Gas separation by membrane systems ('hem Eeon Eng Rev (1987) 19 19 31 16 Spillman R.W. Economics of gas separation membranes Chem Eng Prog (1989) 85 41 62 17 Knw, ali A.S., Vemury S., Krovvidi K.R. and Khan A.A. Models and analysis of membrane gas permeators J Membrane Sei (19921 73 I 23 18 Kurus W.J. and Flemming G.K. Membrane-based gas separation J Membrane Sei (19931 83 1 80 19 Laguntsov N.I., Gruzdev E.B., gosykh E.V. and Kozheniekov V.Y. The use of recycle permeator systems for gas mixture separation J Membrane Sei (1992) 67 15 28 2(I Pettersen T. and Lien K.M. A new robust design model for gas separating membrane modules, based on analogy with countercurrent heat exchangers Comput Chem Eng (1994) 18(5) 4 2 7 4 3 9 21 Paterson W.R. A replacement for the logarithmic mean Chem Eng Sei (19841 39(111 1635-1636 22 Shindo Y., Hakuta T., Yoshitome H. and Inoue H. Calculation methods for multicomponent gas separation by permeation S~7~aration Sci Teehnol (19851 20(5,6) 445 M-59 23 Rony P.R. The extent of separation: on the unification of the field of chemical separations A I C h E Syrup Series (1972) 68(1201 89 104 24 Rubeson L.M. Correlation of separation factor versus permeability' for polymeric membranes J Membrane Sei (1991) 62 165 185 25 Douglas J.M. Conceptual Design (!~'Chemical Processes Chemical Engineering Series. McGraw Hill, New York, USA (19881

Acknowledgement This work has been financed in part by, the Nordic Energy Research Program, by Norsk Hydro and Statoil and by the Norwegian Science Foundation (NFR).

References l

Ray R., ~:ytcherley R.~'., Newbold D., McCra) S,, Friesen D. and Brose D. Synergistics, membrane-based hybrid separation systems .I Membrane Sci (19911 62 347 369 2 Weller S. and Steiner W.A. Engineering aspects of separation of gases. Fractional permeation through membranes ('hem Eng Prog (1950) 46(I l) 585 590 3 Pan C.Y. and Habgood H.~,~,'. An analysis of the single-stage gaseous permeation process lint l:'ng (Tw,1 Fumlam (19741 13(4! 323 331 4 Pan C.Y. and Habgood H.W. Gas ~,cparation by permeation. Parl I: Calculation methods and parametric analysis ('an J Chem End, (19781 56 197 209 5 Pan C.Y. and Habgood H.~'. (}as separation by permeation Part II: EflZ'ct of permeate pressure drop and choice of permeate pressure ('an J Chem En~ (I 9781 56 21(I 217 6 Stern S.A., Perrin d.E. and Naimon E.J. Recycle and multimembrane permeators for gas separations J 3h,mhrane Sci (I 984) 20 25 43 7 Pfefferle V~'.C. US Patent No. 3 144313 iAugust 1964) 8 Ohno M., Morisue T., Ozaki O., Heki H. and Miyauehi T. Separation of rare gases by membranes Radiochem Radioanal LeH (1976} 27 299 9 Bhide B.D. and Stern S.A. A nc~ exaluation of membrane processes for the oxygen enrichment of a i r I Identilication of optimum operatmg conditions and process conliguration J Membrane Sei (1991) 62 13 33 10 Bhide B.D. and Stern S.A. A new evaluation of membrane processes for the oxygen enrichment of a i r I I Effects of economic parameters and membrane properties J Ilemhrane Sci {19911 62 37 58 11 Bhide B.D. and Stern S.A. Membrane processes for the removal of acid gases from natural gab. I. Process configurations and optimization of operating conditions J !,lemhram, Sci (19931 81 209 237 12 Bhide B.D. and Stern S.A. Membrane processes for the removal of acid gases from natural gas. 11. Etlects of operating conditions. economic parameters, and membrane properties J 34emhrane S,i (1993) 81 239 252

Appendix 1 The cost of compression can be written as the sum of operating and investment costs. Both the operating and the investment costs can be related to the compressor duty, which for a single-stage compressor can be written as H' - n,.

RT

1

V0 ta~'l]ad

(g~

11

(42)

where g is the ratio between the exit pressure and the inlet pressure to the compressor. For a multi-stage compressor with interstage cooling to inlet temperature, the minimum compression work is obtained with equal compression ratios across each stage e5. For a multi-stage compressor, the compressor duty becomes H

nr

RT

ns

V0 h;?]ad

(gs - 1)

(43)

The compression ratio across each stage is determined as g~ gl/,,~. In practice an upper limit exists for the pressure ratio across each stage, g .... . The minimum number of stages in the compressor is then the smallest integer which is larger than the ratio l n ( g ) / l n ( g ..... ). Here we will treat the number o f stages in the compressor as a continuous variable

ns

-

lng In gmax

(44)

By doing so, the compressor duty in a multi-stage compressor with interstage cooling to inlet temperature is estimated as g ....

nr

R T g~m~x -

1

v0 ~lngmax

In g

(45)

Design of membrane permeator processes." T. Pettersen and K.M. Lien C o m p a r i s o n o f E q u a t i o n (43) and E q u a t i o n (45) shows 1:hat the simplified model o f the c o m p r e s s o r duty [Equation (45)] overestimates the c o m p r e s s o r duty. The largest deviations are observed for small values o f ~; and for large values o f g . . . . . The deviation is usually within 10%, which is acceptable for design purposes. If the investment costs and operating costs are p r o p o r t i o n a l to the c o m p r e s s o r duty, the annualized c o m p r e s s o r cost becomes CoStComp = (Kcomp,anv -I- Kcomp.Opr) D" = Kc l n g

,

tlf

(49)

1 - sot

NCD2str

02(1 - 01)lng2

1~ [1

NCD3str = l +

(50)

0 2 ( 1 - 01)]lngt

02(1 - 0])1ng2+03(1

02)(1 - 01)lng3

[1 - (1 - 0 ] ) 0 2 - (1 - 02)03]1ngl

(46)

Fhe annualized cost o f the m e m b r a n e system is related t:o the actual m e m b r a n e area in the system. F r o m the definition o f the dimensionless m e m b r a n e area. we obtain the following cost function COStMem = (KMem,lnv "1" KMem,Opr) PfQ2 R = KaR

sol

NCDssP-PR = 1 . 1 . -

(47)

(51) NCD2Enr

1 ~-

01 lngj

NCD3]~nr = 1 +

- 02(1 - 03)] lngj + 0201 lng2 [1 - 0](1 - 02) - 02(1 - 03)]lngl

0j[l

(53)

The normalized c o m p r e s s o r duty ( N C D ) is defined as the ratio between the sum o f duties in all c o m p r e s s o r s ca feed c o m p r e s s o r ) and the duty o f the feed compressor, which compresses the system feed stream from permeate pressure to the feed pressure for the first p e r m e a t o r m o d u l e in the cascade. The N C D for the module configurations in Figure 1 is defined as follows 1

(52)

[1 - 01(1 - 02)]1ng2

Appendix 2

NCDssp-

169

(48)

NfD3strEnr

03(1 - Oj)lng3+Ollngl 1 q [1 ,1,1,03(1 01) 01(1 - 02)]1ng2

NCDssP2Enr = 1 -I-

02(1 - 0 1 ) l n g 2 [1 - 0 2 ( 1 - 03)]lng]

(54)

(55)

The indices 1, 2 and 3 c o r r e s p o n d to stage n u m b e r s according to Figure 1.