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Journal of Membrane Scrence, 6’7 (1992) 15-28 Elsevler Science Publishers B V . Amsterdam

The use of recycle permeator systems for gas mixture separation N I Laguntsov”, E B. Gruzdev”, E.V Kosykh” and V.Y. Kozhevnickovb “Moscow Engrneermg Physacs Instztute, Moscow (Russaa) bKurchutov Atomic Energy Znstltute, Moscow (RussLa) (Received March 31,1989, accepted m revised form September 10,199l)

Abstract There exist various recycle systems for carrying out membrane gas separation a conventional permeator with recycling, a two-unit series-type separation cell, and different modified contmuous membrane columns Their apphcatlon permits much higher degrees of enrichment to be accomplished than does the use of a single-stage permeator without recycling This paper presents a new approach to choosmg the optimal system to carry out a separation process for a specific separation problem The approach IS based on the parameter optmnzation of a generalized single-cycle single-compressor recycle system The basis of the generalized single-cycle single-compressor recycle system IS introduced and the use of the approach suggested m this paper 1s gwen Possible parameters of the system optnnization are determined and examples are gwen of the numerical parameter optnmzation for separating the mixtures HzCO and 02-Nz Keywords

gas separation, theory, optimization, generalized recycle system

Introduction The recycling of the mixture being separated 1s one of the methods used to increase the degree of ennchment of a gas mixture and to widen the number of separation problems which can be solved m practice by the methods of membrane technology. The features of gas mixture separation using apparatus with different schemes of connection of permeators and with recycling of the mixture being separated have been studied in Refs [l-4] Correspondence to N I Laguntsov, Moscow Engmeermg Physics Institute, Kashlrskolr sh 31, 115409 Moscow (Russia )

0376-7388/92/$05

Ohno et al. [ 1 ] have analysed the separation process in a single-stage permeator incorporatmg one permeate recycle (Fig la) and in a twounit series-type gas separation cell (Fig lb), and have proved that the degree of mixture enrichment in such systems is much higher than m a conventional permeator. A detailed analysis of the separation process m multistage recycle cascades consisting of conventional permeators (the two-unit series-type gas separation cell 1sa particular case of such a cascade; Fig lb,c) 1spresented m Ref. [ 21. Hwang et al. [3] have examined the separation characterlstlcs of a contmuous membrane column (Fig. Id) which can effect the complete separation of a binary mixture, according to the au-

00 0 1992 Elsevler Science Publishers B V All rights reserved

N I LAGUNTSOV ET AL

16

r-i

P,cf

a

P,cf t

b

Fig 1 Recycle systems smgle-stage permeator mcorporatmg one permeate recycle (a), two-umt series-type separation cells (b,c ), contmuous membrane column (d) , contmuous membrane column with an additional permeator m the ennchment unit (e)

thors. The continuous membrane column with an additional permeator in the enrichment section (Fig. le) has the same property. The choice of the optimal system for the separation process to solve a particular separation problem IS of great importance under these conditions. The authors of Refs. [l-4] have made a comparative analysis of some of the recycle systems mentioned above. However, general conclusions about the advantages of one or the other system presented in [l-4] are based on the mvestigation of membrane gas separation not m all possible cases but Just for some specific separation problems. Thus, one paper [3] has shown the advantages of the continuous membrane column for fractionation of air in comparison with a conventional permeator according to their characteristics At the same

time the results of comparison of capital and operating costs for separating the mixture 02N, m a contmuous membrane column, in a single-stage permeator incorporating one permeate recycle and in a two-unit series-type gas separation cell, given in the Appendix of [ 51, are not in favour of the continuous membrane column. The uncertainty m assessing the advantages of any of the systems becomes more severe with the statement in [ 41 about the advantages of the continuous membrane column with an additional permeator in the enrichment unit m comparison with a countercurrent flow cascade of conventional permeators. These results mdicate the absence of the answer to the question: which of the systems is the best for any problem of gas separation. It seems more practical to speak of the classes of

RECYCLE PERMEATOR SYSTEMS FOR GAS MIXTURE SEPARATION

separation problem with the optimal system for each one of them. The determination of such classes according to numerical calculations for lfferent values of membrane selectivity and permeability, mixture pressure and initial flow parameters, taking mto account the variety of limitations of the qualitative and quantitative product composition, is of course soundly based but extremely laborious way to choose the optimal system for gas separation. The authors of this paper prefer another approach to choosing the optimal recycle permeator system for a concrete separation problem. In order to understand the main idea, it is necessary to introduce the notion of a single-cycle single-compressor recycle system (Fig. 2), including all possible recycle systems with a single point of compression (Fig. 1) as particular cases. One should then determine the optimization parameters of the generalized system and find their optimal values by numerical experiments. The result of the optimization is the degeneration of the generalized recycle system (Fig. 2) into one of the particular cases (Fig. 1). In this way the choice of the optimal system for the separation process for a specific problem can be solved by calculation of the optimal parameter values for a single-cycle smgle-compressor recycle system, instead of a series of optimization calculations for all known recycle systems and a comparative analysis of the optimal values

17

The aim of our paper is to prove the legnimacy of the suggested approach and to determme the possible optimization parameters for gas separation in a recycle permeator system. Theory The notion of a single-cycle single-compressor recycle permeator system given below means such systems for membrane gas separation as those where a part of the mixture flow goes around a closed loop and in which there is one point of compression of the gas mixture flow. The scheme of the single-cycle smgle-compressor recycle system is shown in Fig. 2 The high pressure flow of the gas mixture being separated goes from the compressor into the high pressure cavity m the conventional permeator 3. The flow which has penetrated through the selectivity permeable membrane of the permeator 3 owing to the pressure gradient (the permeate) is removed as the product P, which is enriched in the most permeable components. The high pressure non-permeate is directed into the high pressure cavity of the conventional permeator 2. The part of this flow not permeated through the membrane of the permeator 2 is fed into the high pressure cavity of the permeator 1. The material that does not permeate through the membrane of permeator 1 forms the product W, which IS enriched m the least permeable components. The permeates from the conventional permeators 1 and 2 are compressed and directed into the high pressure cavity of the conventional permeator 3. The flow of the uutial mixture F goes into the high pressure cavity of one of the conventional permeators; thus:

.S=l

s#t

s,t= 1,2,3

Fig 2 Scheme system

of angle-cycle

smgle-compressor

recycle

The recycle system scheme presented in Fig 2 has one point of compression and a single recycle circuit. the input flow into permeator 3,

N I LAGUNTSOV ET AL

18

the input flow mto permeator 2, the permeate from permeators 1 and 2, and the input flow mto the permeator 3 (the recycle circuit is clearly indicated in Fig. 2 ) . The generalized single-cycle single-compressor recycle system thus introduced contams all the recycle systems shown m Fig. 1 as particular cases. Indeed, the recycle system becomes the continuous membrane column with an additional permeator m the enrichment unit if the nutial mixture flow is fed into the conventional permeator 1 and there is a countercurrent flow m the conventional permeators 1 and 2. If at the same time the membrane selectivity of permeator 3 aTJ3’ =Qa/Q,“, t,~=1,2...N, where Qf is the permeability of the t-th component of an N-component mixture, is lowered to 1, then the recycle system degenerates into a two-unit continuous membrane column. A decrease of the membrane area of permeator 2 to zero while feeding the initial mixture into the permeator 1 or 2 with any flow pattern in the cavities corresponds to the transformation of the recycle system into a two-unit series-type separation cell If at the same time a;y’ -+ 1 the system is transformed into a smgle-stage permeator incorporating one permeate recycle. However, the possibility of universal graphical representation of all the recycle systems of Fig. 1 m the form of Fig. 2 is not enough to consider the systems to be members of a common class - the single-cycle single-compressor recycle system family. The generalization of recycle systems is valid only on the general basis of the mam laws for the separation process. Hwang et al. [3] have considered the possibihty of the complete separation of a binary mixture as being one of the main advantages of the contmuous membrane column. The present authors consider that the possibihty of the complete separation of a binary mixture is common to all recycle systems presented m Fig 1 The proof of this statement follows. Let cF, cp and cw be the concentrations of

the most permeable component m a bmary gas mixture with the feed flow F and the product flows P and W correspondingly m some separation device. The conditions in the device should provide ~‘41. Then m the limit at cp+l the balance equation for the most permeable component m the device will be written like this:

FcF=P+ Wcw

(1)

and, m the hmit at cw-+O, P/F=cF. IfP/F

FcF=PcP

(2)

andif P/F>cF thencP

19

RECYCLE PERMEATOR SYSTEMS FOR GAS MIXTURE SEPARATION

n F;F,=O S#t s,t=1,2,3 does not restrict the common character of the condition 1. The conditions for the separation process in a device providing either cp+ 1 or c w+O were mentioned above. This complex of comhtions is defined by the peculiarities of the membrane gas separation process itself and is formed by the process parameters m a conventional permeator The decrease m the concentration of the most permeable component in the non-permeate with an increase in the cut coefficient 13, is a well known fact [ 11. Moreover, the limit cw-+O at &-+l is true, as is statedm [7], when the gas stream behaves like a “plug flow” (the longitudmal gas velocity and the length of the membrane channel are so large that convective transport in this drection takes precedence over molecular diffusron [8] ) . Thus, the conditions necessary to obtain almost pure the least permeable component in the product W of any permeator system are: (2) to ensure the conditions for an ideal plug flow, to ensure the possibility of unlimited increase of the cut coefficient in the permeator, provldmg the use of the non-permeate as the product W. The product P, which is enriched in the most permeable component, is either the permeate or a part of this flow. It is known that the ratio of the concentration of the most permeable component in the permeate to its input flow concentration is limited from above by the membrane selectivity and by the mixture pressure ratio across the membrane [ 81. Thus, to obtain the most permeable component almost pure m the product P at any membrane select1v1ty ckq;’ and at any ratio of the mixture pressure m the permeate and m the Input flow y=p’/p,, 1sfeasible only m the cases where there is a way to increase the concentration of the

most permeable component m the input flow up to values close to 1. The demonstration of such a posslblhty in the recycle system made for a two-unit seriestype gas separation cell (Fig. lc) follows below. Using the enrichment functions of conventional permeators, which connect the concentrations of the most permeable component in the input flow with the concentrations of the same component in the permeate and nonpermeate: s: (c,,&) =c;

-c,

(3)

6,(cS,eS)=c,-c,,s=1,3 and

taking

Cl =c3

- 6;)

into account the equality the balance equation for the

component WcW+PcP=FcF

(4)

can be rewritten in the followmg way: c3 =cF+&

+a,

-P/F(d,

+&- +S,‘>

(5)

orC3 =f2

(~3 ,PIF,o3)

(6)

where fi(c3J’IF,83)

=CF+&

+a,

(C3,e3)

+s,-

(C3,e33)

[C3 -&(C,,$),hl

-wvr +d;

k3 (C3,e3u

-6:

k3,e33),~,l

(7)

P/F - e, ” = (l-B,)P/F Equation (6) for the unknown c3 has two mdependent parameters S3 and P/F. The location of the solution c3 on the line segment (0,l) depends highly on the values of these parameters Figure 3 presents the function plots fi = c3 and fi The point of intersection of the curves fi (c3) and f2 (c3) is the solution of eqn (6) (points ci, ci, ci and cF m Fig 3 demonstrate the possible location of this solution at various

N I LAGUNTSOV ET AL

20

permeable component being fed mto the last permeator is limited from above by the sum of the concentration of most permeable component in the feed flow and the final enrichments in the preceding permeators. Obviously m this case it is of no use to speak of an increase m concentration of the most permeable component being fed into the last permeator up to values close to unity at any selectivity of the membranes used and at any pressure ratio across the membrane.

1

CF

0

f,=cF+c,

Finally, the conditions for obtaining the most permeable component almost pure in the product P can be formulated as follows: (3) to ensure the incorporation of one permeate recycle of the mixture flow being separated at P/F -SK1 and t9,+0.

values of P/F and 13~;see Appendix, Section (Al) ) It is clear that at P/F<< 1, 19,+0 the concentration c3 will be as close to 1 as possible. In these conditions 6J1-+ 1, G3-+m an increase of the concentration c3 is attained owing to the increase of the recycle flows BIG1and G3. The incorporation of one permeate recycle &G, provides multiple gas mixture separation, separation effect multiplication in the permeate, and an increase of the concentration c3 in the continuous output of a substantial part of the least permeable component from the product IV. This demonstration, with some msignificant changes, can be fulfilled for any of the recycle systems shown m Fig. 1. The necessity of incorporating one permeate recycle to achieve concentrations of the input most permeable component close to 1, mvolvmg the use of the permeate as the product P, is proved by the following fact. In a simple cascade of conventional permeators connected in series, where the permeate from every permeator except the last is fed into the next permeator, recyclmg does not take place. According to this scheme, the concentration of the most

Thus, the simultaneous fulfilment of all three conditions l-3 formulated above results m the complete separation of a binary mixture m any recycle system, irrespective of the mode connection of the conventional permeators, their number and the position of the feeding point It is very important to stress the following point. Do not consider the material given in the above account in this paper as a recommendation by the authors to use the recycle permeator system for the complete separation of binary mixtures m practice. With the increase in the concentration of the valuable component in the product flows, the increase of the recirculation flow itself will result in the escalation of operating costs That is why today membrane separation to obtain highly enriched products in the product flows becomes unprofitable in comparison with the alternative separation methods, as mentioned in Ref. [ 51 The conclusion about the possible complete separation of a binary mixture in any of the recycle systems presented in Fig. 1 is, m the view of the authors, a theoretical demonstration of the communality of the separating properties of such systems and a basis for the mtroduction

c:cfc:

c;i

c3

Fig 3 Scheme of graphd solution of eqn (6) 1 - ji =c3, 2 - f,=f,(c,,B,,P/F), 2(l) PIF=const, 0,+0 2(2) P/F=l, @,-+O fi=cF-63+, f,=cF+c3(P/F)?, 2(3) PIF=const, t&P/F f2=cF, 2(4) PIFc1, &+O

RECYCLE PERMEATOR SYSTEMS FOR GAS MIXTURE SEPARATION

of a single-cycle single-compressor recycle system as a generalized notion. The above conclusion can be applied to the case of separations of multicomponent mixtures Considering the multicomponent mixture to be a binary one (for example, the most permeable component and the other or the least permeable component and the other), and using the right P/F ratio, the almost pure most permeable component m the product P or the least permeable component in the product W can be obtained in any particular case of a single-cycle single-compressor recycle system As for the intermediate component k according to its permeability (mixture components are numbered m the order of permeability coefficrent decrease Qf , z= l,...N) its limiting concentration 61 m the product P 1s expressed by the equation:

whrch is true when P/F= i c;

(10)

I=1

Almost pure intermediate component can be obtained if the two recycle systems are connected m series and P/F values are properly chosen. This follows from the idea that there are no components in the product flow P of the first recycle system that are less permeable than component k. Thus means that the component k becqmes the least permeable one for the second recycle system. The countercurrent cascade has analogous properties, as stated in [ 21. Results and discussion The common character of regularities of the gas mixture separation process in recycle permeator systems having one recycle crrcuit and one point of compression and the legitimacy of

21

the introduction of the generahzed notion of a single-cycle single-compressor recycle system formulated above were used by the authors to present a new approach to choosing the optimal system for membrane gas separation to solve a specific separation problem. The chorce of the optimal system for membrane gas separation m any particular case should, it is recommended, be made according to the results of the parameter optrmization for a single-cycle single-compressor recycle system. The economic efficiency function of the separation process, includmg membrane, capital and operating (gas compression) costs as the main components, can probably be suggested as an optlmlzation criterion. All possible optrmization parameters can be divided into three groups. The first two groups mclude parameters that determine the efficiency of the separatron process in a conventional permeator: -membrane selectivity in respect to the components of the mixture being separated a!$’ = @/QJ and the ratio of the pressure of the mixture in a low pressure cavity to the pressure m a high pressure cavrty y (or the gas compression degree in a compressor n = l/y) ; -conventional permeator parameters determining the extent of influence of dissipative phenomena in the separating channels (the change in mixture pressure due to the viscous nature of flow, diffusive transfer of mixture components m a channel etc.) on the separation process; for example, the radius, the width and the length of a hollow fibre, the number of fibres in a bundle and the way in which they are packed m the bundle of the fibre module. The thud group mcludes parameters which characterrze the system itself for a given separation problem, i.e. the parameter values are different for different particular cases of a single-cycle single-compressor recycle system: -the feeding pomt location, the cut coefficients 0, and the value P/F (taking eqn. (8) urto ac-

22

count, 0, is not an independent optimization parameter in this case). Optimization calculations using all or some of the parameters of the first two groups and of the third as optimization parameters will necessarily lead to the optimal system to solve a given separation problem by the membrane gas separation method. The opportunity for and the necessity of application of one or more values from the third group as optimization parameters depend on the limitations as to the qualitative and quantitative product composition imposed by the conditions of the specific separation problem The examples of such limitations are: maintaining the prescribed values of the product flow P (or W), the valuable component concentration m the product flows cp (or cw), and the degree of extraction of the component #P=P~P/F~F (or gw= WcW/FcF). The concentration and the degree of extraction in different combinations can be prescribed simultaneously. For example, the optimization parameters of the third group are the values P/F and 0, when the valuable component concentration in the product flow is given A change of 19,is equivalent to a change of the feeding point location between permeators 1 and 2. The only optimization parameter of the third group is 0, if the degree of extraction is also a given value. The present authors have worked out the methods and software for the numerical parameter optimization for a single-cycle singlecompressor recycle system wlthm the limitations of the method described above. The automatic degeneration of the conventional permeator 2 and the transition to a series-type separation cell are permissible, and the degeneration of the conventional permeator 3 is carried out by means of a programme. Various unexpected results are presented below They were obtained with the help of the suggested software, and illustrate the potential of the given method to choose the optimal sys-

N I LAGUNTSOV ET AL

tern for the membrane gas separation process. The total membrane area of conventional permeators and the energy consumed by the compressor as functions of the membrane selectivity are shown in Fig. 4 (the algorithm for drawing the curves IS presented m the Appendix; see Section A II) The flow and the composition of the initial mixture and the degree of compression are fixed. The given values c$~=O.O~ and @To =0.9 serve as limitations. After being optimized, the single-cycle singlecompressor recycle system was degenerated mto a series-type separation cell The strongly marked minimum in the total membrane area is an interesting feature to be pointed out, and operating cost changes are negligible beginning with a definite value of ~#.$jc~. As the total membrane area and operating costs for compression represent the mam components of the economic efficiency function for the membrane gas separation process, these curves agmfy the existence of an optimal membrane selectivity providing the minimum cost for the required production. The recycle system that has degenerated into a series-type separation

zsy

io-: n12

E, KWh/m3

Fig 4 Total membrane area and operatmg costs for compression plotted against membrane selectlvlty for separation of Hz-CO mixture, CL, =0 20, QHz =const , W=const , qiFo=0 9, c$ =0 02, y= l/16

23

RECYCLE PERMEATOR SYSTEMS FOR GAS MIXTURE SEPARATION

cell while being optimized (point 1 in Fig. 4) is much more effective than a conventional permeator (point 2) and does not require the use of highly selective membranes. This fact mdicates the prospects for application of the recycle system In the optimization process it was assumed that the membrane selectivity cr&$)co was mcreased because the decrease of CO permeability at Qj$ was constant. This assumption corresponds to the results of Arbatsky et al [ 91, who varied membrane selectivity by plasma modification. Evidently the optimum location can change If both Q& and Q& are changed. However, the selectivity (Y*is the optimization parameter and an increase in membrane selectivity does not always bring profit; this fact is of great importance - the application of a membrane which has limited permselectivity but high flux or high permeability may be more profitable. The next group of relationships presented m Fig. 5 illustrates the influence of the feeding point location F on the values of total membrane surface area and on the operating costs for compression The concentration of Hz in the

zs, !a*,$

990

0,95 a

product, the product W, the pressure ratio y and the membrane selectivity cy* are given values. The dotted curves correspond to the case of the feed input to the conventional permeator 1; the solid curves correspond to the feed input to permeator 3. The conventional permeator 2 was degenerated at the system bemg optimized. It is interesting that m Fig. 5a,b the dotted curves have a mimmum whereas the solid curves fall monotonically as @z. decreases. It is important to take into account the peculiarities of conventional permeator design which lead to deviations of the separation condition from the idealized one durmg separation process optimization in a recycle system The parameters of the second group alone assign the degree of influence of the real physical process, m particular the change of mixture pressure due to the viscous nature of the flow and component diffusion m the separation channels, on the separation process, not only m the conventional permeator but also in the recycle system. The degree of influence of molecular diffusion along the separation channel in permeator 1 can be so high that, for example, complete mixture separation will not be

E

I,00

,KW h/m3

0,90

0.95 b

i,oo

Fig 5 Total membrane area (a) and operatmg costs (b) as functions of the extraction degree $& in separation of HZ-CO mixture m a two-unit senes-type separation cell, c&, =0 20, c~&,~o = 10, y= l/16,1, c$ =0 02,2,0 0% 3,O 06 (-) feed input to conventional permeator 3, (-----) feed mput to conventional permeator 1

24

N I LAGUNTSOV ET AL

achieved even in theory under the condition of complete mixing and when the influence of molecular diffusion along the separation channel is at its greatest. The influence of the pressure change in the conventional hollow fibre permeators at the gas input to the fibre is introduced in Fig. 6. There are curves of equal expenditure at the compressor input for the systems described. The membrane area changes along the dotted curves for a single-stage permeator incorporating one permeate recycle, and the membrane area S,, of permeator 3 changes along the curves correspondmg to the two-unit series-type separation cell at the fixed membrane area S,, of the conventional permeator 1 It is also seen m the figure that the system of two conventional permeators is more effective However the concentration cz,oZhas an upper

0.20 Fig 6 Concentration of 02, c& , as a function of the product flow P m the separation of an 02-N2 mixture using fibres made of poly-4-methyl-1-pentene (Y&+ = 3 5, y= l/ 7,r,=125X10-6m,r,=275X10-6m,I=10m (-----) Single-stage permeator mcorporatmgonepermeate recycle, S,=const , 1, G= 1 0,2, 14,3,2 0 (-,- - -) Two-umt senes-type separation cell, feed input to conventionalpermeator 1, (-_) S,, =const ,4, S,, =0 75 S,,,5,060S,,,6,050S,,,7,040S,,,(- -)G1=const, 8,G*=13G,9,17G,10,20G

limit due to the restrictions in the fibre capability. An appreciable rise in the concentration of 0, in the product P can be expected if there is an increase in the internal radius of the fibre. The use of membranes of different selectivity can make recycle systems more effective. The largest contribution to the total membrane area is made by permeators 1 and 3, as was shown above The optimal selectlvities of the membranes used in these permeators are determined according to Fig. 4. A highly selective membrane, although of low permeability, can be used in permeator 3, although the pressure ratio across the membrane may increase owing to pressure decrease behind the membrane; this IS not particularly difficult to do because, usually, P/F << 1 Then the required value of 0, may become less owing to an increase in the separation effect S;, leading to a substantial decrease of the membrane area in permeators 1 and 2 and of the compressor power m the recycle circuit. The results of actual numerical investigations presented m this paper do not pretend to be full and general. The economic efficiency function used as the optimization criterion should include specific economic values, providmg a way to evaluate the competition between the membrane gas separation and alternative technologies. The main idea of this investigation was to develop a new approach to choose the optimal systems for a membrane separation process; the results illustrate the potential of the method recommended by the authors Conclusion

The present investigation led to the following main results. (1) The common character was established of the features of gas mixture separation m a single-stage permeator incorporating one permeate recycle, in a two-unit series-type sepa-

RECYCLE PERMFXI’OR SYSTEMS FOR GAS MIXTURE SEPARATION

ration cell (two-stage recycle cascade), m a continuous membrane column and in a continuous membrane column with an additional conventional permeator. (2) The generalized notion of a single-cycle single-compressor recycle system including all suggested systems as particular cases has been introduced. A new method to choose the optimal system for membrane gas separation by the numerrcal optrmlzation of the parameters of a smgle-cycle single-compressor recycle system has been developed. (3) A new group of optrmizatlon parameters in addition to those already known has been established feeding point location, cut coefficient, and ratio of product flow to feed flow, which characterizes the system Itself for the separation problem and makes particular cases of a single-cycle single-compressor recycle system differ from one another. (4) The methods and the software for numerical optimization of the parameters for a smgle-cycle single-compressor recycle system have been worked out. They provide for the automatic transltlon from a generalized system to the particular cases of one- and two-unit recycle systems during the optimrzatlon process. The practical results of the separation process optrmization to solve specific problems of membrane gas separatron according to the suggested method are of original srgnificance. For example, an increase m membrane selectivity does not always result m an mcrease of the economic efficiency of membrane separation technology. The use of less selective membranes in a recycle system can be more profitable than the use of a conventional permeator with a more selective membrane.

25

point of vrew, and look forward to further drscusslons on the use of recycle permeator systems List of symbols C

.?,,,c+ c-*,a 8,1,

CF I, cpI, cw L

c;, c& c:

c:

E

F

relative molar concentrations of the mrxture component z in the input, permeate and nonpermeate flows respectively of the conventional permeator s (G, c,’ and c: are the relative molar concentrations of the most permeable component for the case of a bmary gas mixture) (-) relatrve molar concentrations of the mixture component Lin the feed flow F and m the product P and W flows correspondmgly ( cF, cp and cw are the relatrve molar concentratrons of the most permeable component for the case of a bmary gas mixture ) (- ) possible values of the single solution c3 of eqn. (6); see Fig. 3 (-) limiting molar concentration of the mixture component k in the product flow P (-) operatmg costs for the compressron of the mixture being separated ( kWhr/m3 (STP)) nntial mixture

flow

(feed

flow), F= i F, (m3 (STP)/ .S=l hr)

Acknowledgement The authors are grateful to the members of the edrtonal board of the Journal of Membrane Science for the opportunity to express their

F.3

fi9fi

feed flow being fed to the conventional permeator s (m3 (STP)/hr) the functions fi (c3)= c3,f2 defined by correlation (7) (-)

N I LAGUNTSOV ET AL

26

1 L

N

QT

P

Ph, P’

(P/F) (O) R r0, rl

S m*

T W

input flow of the mixture to the conventional permeator s (m3 (STP)/hr) non-dimensional quantity Gs/ G, (-) fibre length (m ) local value of mixture flow in high pressure cavity of permeator s ( m3 (STP ) /hr ) number of components of the mixture being separated (-) permeability coefficient of the mixture component z through the membrane of permeator s ( m-m3 ( STP)/m2-hr-atm) non-dimensional quantity L,/ G, (-) product flow enriched with the most permeable component (m3 (STP)/hr) mixture pressures in the input flow to the conventional permeator and in the permeate (atm) initial (approximate) value of P/F parameter (-) universal gas constant (J/ mol-K) inner and outer radii of the hollow fibre respectively (m) membrane surface area of the conventional permeator s (m”) mixture temperature (K) product flow enriched with the least permeable component (m” (STP)/hr) local values of relative molar concentrations of the mixture component z in the high and low pressure cavities respectively of permeator s (-)

Greek letters a:#’

=Q:/

Y’P’

hh

n=l/y cs=

; s,,

SC1

Q;

membrane selectlvlty factor of the conventional permeator s relative to the components z and] (-) mixture pressure ratio across the membrane (-) functions of mixture ennchment m the conventional permeator s according to relationship (3) (-) degree of mixture compression in a compressor (- ) total membrane area ( m2) cut coefficient in the conventional permeator s, equal to the ratio of the permeate to the input flow to the permeator

0,

ego, P,W @I

Subscrzpts z, I, k s,t

(-) initial (approximate ) value of 0, parameter (-) extraction degree of the component z m product P or W flows (-) number (symbol) of mixture component numbers of permeator

References M Ohno, H Hekl, 0 Ozakl and T Mlyauchl, Radloactive rare gas separation performance of a two-unit series-type separation cell, J Nucl Scl Technol, 15 (1978) 668 E B Gruzdev, N I Laguntsov, B I Nlkolaev and G A Sulabendze, Investlgatlon of the separation characteristics of the countercurrent cascades without mlxmg of flows with different relative concentrations for arbltrary ennchments on the stage, At Energ ,61 (1986) 454 (m Russian) S -T Hwang, K H Yuen and J M Thorman, Gas separatlon by a contmuous membrane column, Sep Scl Technol ,15 (1980) 1069

RECYCLE PERMEATOR SYSTEMS FOR GAS MIXTURE SEPARATION

E B Gruzdev, V K Ezhov, E V Kosykh, N I Laguntsov, B I Nlkolaev and G A Sulabendze, On the separation characterlstlcs of the contmuous membrane column with a conventional permeator on top of the enricher, Cbem Eng Fundam, 23 (1989) 195 (m Russian) S L Matson, J Lopez and J A Quinn, Separation of gases with synthetic membranes, Chem Eng Scl , 38 (1983) 503 V P Mmenko, The hmlt enrichment of mtermedlate isotopes m the product from the ends of the cascade, At Energ, 33 (1972) 703 (m Russian) N A Kolokoltaov, N I Laguntsov and GA Sulaberadze, Calculation and estimation of the integral charactenstlcs of the Ideal two-component cascades with arbitrary ennchmenta on the stage, At Energ, 34 (1973) 259 (m Russian) S -T Hwang and K Kammermeyer, Membranes m Separations, Wiley, New York, NY, 1975 A E Arbataky, A K Vakar, A V Golubev, E G Krashenmmkov, V V Lmentaov, S 0 Marcheret, V D Rusanov and A A Fndman, Polymer gas separation membranes modified m oxygen-contamed plasma Theory and comparison with expenmenta, Kblm Vys Energ, 24 (1990) 256 (m Russian)

Sectwn AI

Equation (6) is an algebraic one, which can be written in common form as (Al)

where x=c3, fi(x)=r and fi(x) is defined by correlation (7) The root of eqn (Al) can be found graphically as the point of intersection of the curves fi (x) and fi (x) . The procedure of graphical solution of eqn. (6) is shown in Fig. 3 Location of the root of eqn. (6) depends on the lsposition of the curve fi ( c3), but the disposition of curve fi ( c3), in its turn, depends on the values of P/F and 0,. That is why there are four characteristic variants of location of the single solution c3 m Fig. 3 (ci, cF, c3 and cE ) and four corresponding variants of the disposition of curve fi ( c3) (curves 2 ( 1) -2 (4 ) m Fig. 3) for various values of P/F and 0,: P/F= const, f3,+O

In this case ST -+O, 0,+ 1 (according to eqn. (8)), 6, -cl, c13c3 and correlation (7) can be rewritten as: f2(c3,P/F,83)

-cF+c3

-P/Fcp

Disposition of this curve depends on the value P/F: 0 forP/F-+l

(curve2(2)),fi-+cF-83+,values f2(c3) and solution ci are less than cF, l for P/F<<0 (curve 2(4)),f2-+CFfC3, values fi (c3) and eqn. (6) solution (c: ) are greater than cF and moreover, in this case, the solution of eqn. (6) can be as close to 1 as possible; l for 0

Appendix

f1(x)=fz(r)

27

(i-e,)~,

(A2)

andf,zc* (curve2(3)). So eqn. (6) has the sole solution c3 and its location depends on the values of P/F and $ Sectwn AII

Membrane area S,, (s= 1,2,3) was found by solving the well-known system of differential equations (see e g., Ref. [ 4 ] ) : ’

1

(A3)

with boundary conditions (?*=I.

x,,,=c1. s,=o

(A4)

as S,, = S, (q= 1 - 0,). Here qs= LJG,; L, 1sthe

N I LAGUNTSOV ET AL

28

local value of mixture flow in the high pressure cavity of permeator s; 1 < qs < 1; x,,,, xi,, are local values of relative molar concentrations of the mixture component z in the hrgh and low pressure cavrties of permeator s respectively. The system (A3), (A4) is correct only for the case of “plug flow”, when the longitudinal gas velocity and the length of the membrane channel are so large that the convectrve transport in thus duection dominates over molecular diffusion The following algorithm was used to draw the curves in Fig. 4: (1) FIX the value of membrane selectivity a*(s) 111* (2) Define all cuts 0, and mixture flows G, for the given CV>~),the uutial values t9$‘) and (P/F) (O)are chosen instead of unknown 0, and P/F, 0, is defined from eqn. (8), G, = IV/ (l-o,), Ga=GJ(l-($3) (3) Solve the system of differential equations (A3) with boundary conditions (A4); concentrations x& are defined at every step of

the procedure of numerical integratron from a system of algebraic equations which, for example, in the case of “cross plug flow” can be wrltten as [8].

Define (4) S,, = S, ( qs = 1 - 0,))

membrane total membrane

area area

3

CS, = C S, s=l

and approximate operating costs

(energy consumption) E = EG Inn r 3

(A6)

where q is compression efficiency, (5 ) Correct the untial values O$O),(P/F) (O), define 01, G1, G3again and repeat the procedure of point 3 and point 4 until grven values of cw and @w can be obtained. (6) Select another value of a:/(l”) and repeat the algorithm from point 2 to point 5.