Znd. Eng. Chem. Res. 1987,26, 1109-1116
1109
Approximate Design Method for Reverse Osmosis Plants Equipped with Imperfectly Rejecting Membraned Franco Evangelista Dipartimento di Chimica, Zngegneria Chimica e Materiali, Uniuersitci degli Studi dell'Aquila, 67100 L'Aquila, Italy
An approximate method, analogous t o stagewise calculations of unit operations, has been developed for the design of straight-through and tapered reverse osmosis plants irrespective of the geometrical configuration of their modules. By use of a three-parameter membrane model, explicit equations have been worked out for the whole range of rejections between 0% and 100%. Given the total concentration factor and knowing plant operating conditions and membrane characteristics, the proposed method predicts the number of parallel lines of modules M p ,the number of series modules M,, and the permeate average concentration Cp. The calculation results have been compared with those obtained by rigorous methods. Equal values for M p , errors never exceeding 6% for M,, and errors around 10% for have been found for all practical operating conditions. Slightly lower accuracy has been obtained for Cp in the case of spiral wound modules equipped with high-rejecting membranes.
cp
For many years the design of industrial reverse osmosis plants has been carried out by iterative computerized methods. These have been developed, for different module configurations and different plant arrangements, as soon as they became technologically reliable and commercially available. Three types of modules are more frequently utilized, and correspondingly three design methods have been worked out, respectively, for tubular (McCutchan and Goel, 1974; Harris et al., 1976), spiral wound (Chiolle et al., 1978; Rautenbach and Dahm, 1984), and hollow fiber (Tweddle et al., 1980; Kabadi et al., 1979) modules. All these methods use different membrane models and recursive material balance equations along with appropriate discretizing procedures. The resulting systems of equations are to be solved numerically and the associated algorithms are highly iterative and, therefore, clumsy in terms of computing time and storage. Usually, optimization procedures and technical feasibility checks are also included in the package. Recently, the need to have more practical design schemes has promoted the growth of simple and accurate procedures. While these can be usefully applied to get quick insights and initial estimates of the process, the aforementioned sophisticated procedures should be used for the fine assessment of operating conditions and performances. Sirkar and Rao (1981) have developed simple analytical equations for tubular modules in turbulent flow conditions, equipped with high-rejecting membranes. Sirkar et al. (1982) have extended the above design procedure to spiral wound modules under both turbulent and laminar flow conditions. Dandavati et al. (1975) have worked out explicit and implicit equations for radial flow hollow fiber modules with high-rejecting membranes, and Soltanieh and Gill (1982), too, have developed a simple model for hollow fiber modules assuming complete mixing on the shell side. All these methods start with different membrane models and make several simplifying assumptions, throughout the derivation, according the geometrical configuration of the 'Presented at the 188th ACS National Meeting, Philadelphia, 1984.
0888-5885/87/2626-1109$01.50/0
module, membrane characteristics, and operating conditions. More recently, Evangelista (1985), starting with a twoparameter membrane model, has developed a short-cut design method irrespective of the geometrical configuration of the modules. The method is like the graphical stepwise construction of ideal stages in chemical engineering unit operations. Analytical explicit relationships for the number of series modules and for the permeate average concentration are also given. However, this method is restricted to high-rejecting membranes and to turbulent flow conditions or negligible polarization on the high-pressure side. Later the procedure has been improved taking into account the permeate concentration in design calculations (Evangelista, 1986a); this has made it applicable to less rejecting membranes but still only in those cases where the average rejections are between 80% and 100%. Since the two-parameter model adopted does not account for the convective solute transport through the membrane, it was not possible further to extend the above procedure to low-rejecting membranes. Reverse osmosis technology has confined itself up to now to the processing of dilute or moderately concentrated solutions. However, it is becoming a well-established method of separation and concentration in the chemical industry even for high osmotic pressure solutions, such as sea water for the recovery of salts (Sambrailo et al., 1983) or of sucrose solutions to be fed to crystallizers (Murakami and Igarashi, 1981). The main problem encountered in such processes is the high pressure to be applied in order to get an acceptable solvent flux. Practical and low-cost operations need to operate in a range of pressures and temperatures less than the maximum values that a given membrane can withstand in order to minimize energy consumption and to be competitive with other processes. One way of overcoming this problem is to arrange the plant with a combination of low- and high-rejecting membranes (Van Wijk et al., 1984). For each application, suitable membranes with different rejecting characteristics between 0% and 100% must be used. Permeability, on the other hand, should be always as high as possible. From the designer's point of view, it would be very desirable to have simple and accurate tools to compare the 0 1987 American Chemical Society
1110 Ind. Eng. Chem. Res., Vol. 26, No. 6, 1987
performance of different membranes and to find the best operating conditions of the process. Therefore, the objective of the present work is to develop an approximate, but still accurate, design procedure valid over the whole range of rejections. The analogy with the previous graphical analytical procedures and their simplicity and accuracy will be maintained.
Theory Several models for the prediction of the mass transfer through synthetic membranes under reverse osmosis conditions have been developed. These models characterize the membrane by means of two or three parameters. In the two-parameter models (Lonsdale et al., 1965; Kimura and Sourirajan, 1967), one parameter is for the prediction of solvent flux and the other for the solute flux. Both fluxes are considered to be independent from each other and proportional to their respective driving forces. In the three-parameter models (Spiegler and Kedem, 1966; Sherwood et al., 1967; Jonsson and Boesen, 1975; Pusch, 1977a), a third parameter is added in order to account for the coupling between the two fluxes. The major difference between these two types of models is that in the latter the solute flux is divided into a diffusive part and a convective part, while in the former it is considered only diffusive. Since the convective solute flux becomes relevant as the permeability increases, the validity of a two-parameter model is restricted to high-rejecting and low-permeable membraries. On the other hand, a three-parameter model is suitable also for low-rejecting membranes. In some cases, the prediction has even been attempted taking into account the convective flux only, and reasonable accuracy has been achieved (Kabadi et al., 1979). The three-parameter model is preferable, however, even with high-rejecting membranes having high permeability, such as present commercial desalination membranes. The higher accuracy and the wider validity, however, means both greater mathematical difficulties and a larger number of experiments needed to get the parameters. Among the three-parameter models, Pusch’s model (Pusch, 1977a), derived from the thermodynamics of irreversible processes with some simplifying assumptions, is preferred here since its parameters are more constant with concentration and pressure (Pusch, 1977b). Starting with this model, quite accurate explicit relationships for the prediction of the local solvent flux and the permeate concentration have been recently derived (Evangelista, 198613). However, these results cannot easily be employed in the derivation of simple relationships useful in plant design. Further reasonable simplifying assumptions must be made here. The three parameters of Pusch’s model are hydrodynamic permeability 1,, osmotic permeability l,, and the asymptotic rejection coefficient r,. If the applied pressure is at least twice the difference of the osmotic pressures of the solutions facing the membrane (Pusch, 1977a), the following relationships can be written for all module geometrical configurations: J , = l,[AP - rm(IIw- nP)]a
Cp = (1 - rmKW + (1, - r,21,)Cw(IIw- IIp)/J, C, = Cp + ( c b - Cp)eJJkl where
cy
N
with
(1) (2)
(3)
= 1 for tubular and spiral wound modules and
= q/[l
+ (161,prOl1,q/1.0133 X 1O6r?)]
(4)
-
and k , m for hollow fiber modules. While for tubular and spiral wound modules the above quantities are local evaluations, for hollow fiber modules they are considered averaged along the fiber axis. Moreover, in this case the concentration polarization can be neglected. Relationships 4 and 5 have been derived following the procedure of Dandavati et al. (1975). They derived these relationships by assuming the permeate concentration to be negligible and the shell-side concentration constant along the fiber. The same relationships are obtained if only the concentration difference across the membrane is assumed to be constant along the fiber. This assumption is safer for low-rejecting membranes than for high-rejecting membranes. In fact, as confirmed by numerical calculations, the more rejecting the membrane is, the higher the shell-side concentration gradient along the fiber will be. Moreover, this assumption has been checked by a numerical algorithm also developed by the author. Permeate flux and concentration have been averaged along the fibers and compared with the corresponding quantities calculated from averaged transmembrane pressure difference and shell-side concentration. Their discrepancies never exceeded 5% with higher values for higher rejecting membranes. Usually the system of equations (1143) is solved by iterative procedures. Now a simple and approximate explicit solution will be derived. First we define the constant B as and the concentration factor as Then we substitute the exponential term of eq 3 with its expansion to power series. An economized Tchebichev expansion, truncated to the first term, is suitable (Evangelista, 1986b). So eq 3 is replaced by The best value of a, which is always greater than 1, may be estimated a priori according to the expected extent of polarization. This can be evaluated by calculating the mass-transfer coefficient from dimensionless relationships and the volume flow assuming no polarization. Moreover, ita value can be kept constant throughout the plant since, under turbulent flow conditions, the ratio J v / k ,is almost constant. In order to get simple relationships which may be easily handled mathematically, an approximate solvent flux expression, to be substituted in eq 2 and 8 instead of eq 1, is adopted: J , = I,APa(l - rm2Bf)
(9)
This expression has been worked out from 1, 2, 6, and 7 assuming no polarization and a linear dependence of the osmotic pressure from concentration. The permeate concentration has been evaluated as a first approximation from eq 2 neglecting the diffusive part of the solute flux. It should be noted that no polarization has been assumed for the evaluation of the concentration polarization itself in tubular and spiral wound modules. Therefore, the approximate expression (9) has a secondary effect on the evaluation of the local volume flow and permeate concentration since it will be used only in eq 8 to evaluate the
Ind. Eng. Chem. Res., Vol. 26, No. 6, 1987 1111
I,
I
, /
I
and the actual inlet flow rate per module FF is recalculated. The number of series modules in one line can be calculated by a graphical-analytical procedure like the stagewise calculations of ideal stages in chemical engineering unit operations. A solute and solvent material balance over a differential membrane element gives
i
d F / F = dCb/(Cb - cp)
(18)
Now if the recovery fraction is defined as
4 =P/FF
Permeate
Figure 1. Straight-through flow arrangement of a reverse osmosis
plant.
wall concentration on the high-pressure side and in the second term of the right-hand side of eq 2. If eq 2 and 6-9 are rearranged, an explicit expression for the permeate concentration is derived: cp
=
Gf(di+ dd
+ d$)
where
d l = lpa(l- rm)(ukl + 1,APa) rm21,a(l- r,)(ak,
+ 21,APa)l
(11)
(12)
d3 = rm2B2APZPa[(l - r,)rm2lPa- (1, - rm21p)] (13) d 4 = lpa[rmkl+ (1 - r,)(akl
+ lPUa)l
(14)
d 5 = B[(l, - rm2Zp)(ukl+ 1,APa) rmzZpa(l- rm)(ukl + 21,APa) - r,3kllpal (15) If eq 1, 6-8, and 10 are rearranged, the following explicit relation is obtained for the volume flow:
d4 + d5f + d 8
J, = 1+
and relation 7 is remembered, eq 18 becomes d$/(l-
$1
= d f / ( f - CP/CF)
(20)
If eq 10 is substituted into eq 20, the following differential equation is found:
(10)
d4 + d5f + d3P
d 2 = B[(l, - rm2lP)(akl+ 1,APa) -
(19)
To integrate eq 21, the following further assumptions must be made: (1) The pressure drop on the brine side is negligible in comparison to the transmembrane pressure difference which is considered constant and equal to AP. If this assumption is not applicable to the whole plant, it is possible to divide the plant into several appropriate sections so that it becomes valid within each of them. In this case the procedure developed later is to be applied several times and the transmembrane pressure difference AP updated accordingly. (2) Turbulent flow on the high-pressure side is fully developed ‘so as to neglect the variations of the masstransfer coefficient with bulk velocity. (3) Membrane characteristic parameters and physical properties of the solutions are constant throughout the plant. From the above assumptions, the constancy of dl-d5 is derived and eq 21 is easily integrated so as to give 1 - $ = ,-6
kl
Regardless of the geometrical configuration of the modules, eq 10 and 16 can be used to evaluate in an explicit way the permeate concentration and the volume flow anywhere in the device, knowing the local concentration factor and the other operating conditions. Plant Design. Usually reverse osmosis plants are composed of many modules connected together in different ways. Depending on the size and overall recovery fraction, plants are arranged in straight-through or tapered flow. Whatever the arrangement is, the basic configuration, from which any plant size and arrangement can be derived, is shown in Figure 1. This scheme is composed of Mp parallel lines of modules and M , series modules in one line. If the feed flow rate and concentration are known, the design of such a plant consists in calculating the number of parallel lines of modules, Mp, the number of series modules, M,, needed to reach a given concentration factor or recovery fraction, and concentrations and flow rates of the outlet streams. The number of parallel lines is easily found knowing the maximum inlet flow rate per module by means of Mp = Q F / F M ~ ~ (17)
If M p is not an integer, it is rounded up to the next integer
(22)
where
d4
d3(d4- d,)
[AL--d5 - dz d4 - dl
(d5 - d2)’
I
Brm2f- 1
In Brm2- 1
(23)
Relation 22 is the equation of the “operating curve” on (1 - $), l / f reference frame. It crosses points ( 1 , l ) and (l/fMa,O), as shown in Figure 2. The value of fMax, the maximum concentration factor, can be calculated from the argument of the last logarithm on the right-hand side of eq 23: fMax
= 1/Brm2
(24)
This value, however, should be indicative only, since it is derived approximately. In fact it could also have been worked out from the approximate relation (9) adopted for the solvent flux. Practical concentration factors are usually very far from the maximum concentration factor evaluated by relationship 24, and their difference is a measure of the correct use of the reverse osmosis technique. Another consequence of the approximations is the nonuniqueness of the operating curve concavity which is upward for low-
1112 Ind. Eng. Chem. Res., Vol. 26, No. 6, 1987 A€
1.0 I
both (1- 4) and f , we will get an average slope. Now if eq 27 is utilized, the outlet concentration factor can be evaluated by
with
and an arithmetically averaged concentration factor as
y
0
0.0
I
0.0
I*
1 / L X
i 1/f%
1
I
I
l/f
I
1 .o
Figure 2. McCahe-Thiele-type diagram to calculate the number of series modules.
and medium-rejecting membranes and downward for high-rejecting membranes. However, as will be shown below, eq 22 is a fair approximation in the practical operating range of the recovery fraction (between 0% and 50%) of a plant in straight-through flow arrangement as shown in Figure 1. If the rejection had been loo%, the operating curve would have been given by the diagonal of the square (Evangelista, 1985). So the departure from the diagonal is a measure of the average rejection: the greater the distance, the lower the rejection obtained. Equation 22 allows us to calculate the concentration factor given the recovery fraction or vice versa. But, while the recovery fraction can be calculated explicitly knowing the concentration factor, the concentration factor corresponding to a given recovery fraction must be calculated through a Newton-Raphson technique. Now we need to find the “equilibrium curven of the operation in terms of (1 - 4) and l / f quantities. The solvent material balance of the generic module of a line is F - P = R = F - SJ, (25)
If eq 7 , 19, and 22 are recalled and the solute and solvent material balances for envelope I1 of Figure 1 are used, the second equality of eq 25 becomes 1 - 4 = e-0 - S J v / F F (26) The flux J , can be calculated at the inlet, outlet, or averaged conditions of a module. While the inlet conditions will certainly underpredict the membrane area, the outlet conditions may overpredict it to too great an extent. Therefore, we choose to evaluate the permeate flux at averaged conditions. A t the generic module i, we know the inlet conditions but the outlet conditions are still unknown. So, in order to evaluate the averaged conditions, we must forecast the outlet conditions. This can be done by evaluating first the average slope of the operating curve from eq 21:
Equation 27 would give the local slope of the operating curve a t the inlet of the module if we replace (1 - 4) by using eq 22 or at the outlet conditions if we substitute for f a value obtained from eq 22 knowing (1- 4). So leaving
If, for simplicity, only the first f s of the numerator and of the denominator of eq 16 are replaced with 7 and then eq 16 is substituted into eq 26, this latter becomes 1 - 4 = e-@- [A, - (A12 - 4A2)1/2]/2
(32)
where A1 = [ar-BfA3(2&!lg2 + gZ? zA3kl2+ glg2) 2e-Bk12+ fA4glg2)1/[2kl2 + g1gJ + fA4(2g,g2 + g22)l (33) A2
=
2e-@A3(g12+ ar,BfglgJ 2k12 + g1g2) + fA4(2g,gz + g,2) A3
= Sl,APcY/FF
A4 = l,hParmB/K1
(34) (35) (36)
This derivation is consistent with a constant rejection model which usually gives quite good estimates of the permeate flux (Soltanieh and Gill, 1982). Moreover, the subscript i off is omitted to indicate that f is the currently independent variable. Relationship 32 is the equation of a practical equilibrium curve, not in a thermodynamic sense but in that it allows us to calculate, together with the operating curve, the concentrations and flow rates of the outlet streams knowing those of the inlet stream of a module. As can be easily worked out from relations 32-36, the “equilibrium curve” depends on the operating conditions and membrane characteristics and even more on the value of the membrane surface S, the only quantity left to the discretion of the designer. In fact, S can be equal to the area of one module or to a portion of membrane area which can be smaller or larger than the membrane area of one module. However, whatever the value of S, and hence of the parameter AB, this method, as will be shown later, always gives good accurate estimates. The curve representing eq 32 is also shown in Figure 2. It crosses the straight line l / f at a point G in such a way that segment EG represents the recovery fraction of the first module and tends to coincide with the operating curve at high concentration factors, confirming again that further concentration is not possible. The number of series modules is then calculated by drawing the stairlike construction EGHOU ...Z between the “operating curve” and the “equilibrium curve” as shown in Figure 2. Segment GH is the reciprocal of the concentration factor of the same module. Hence, step EGH corresponds to the first module and so on. Point E represents the feed, point Z the reject, and points H, U, etc., the intermodule streams. From Figure 2 it is easily seen that in the practical range of recovery fractions, that is, between 0 and 0.5, both the operating curve and the equilibrium curve are almost
Ind. Eng. Chem. Res., Vol. 26, No. 6, 1987 1113 1 .o
1-4
0.5
I /
3
4
0.0
0.0
0.5
0.0
Vf
1 .o
Figure 3. Comparison between different membrane models and calculation methods: (-) three-parameter model, eq 1-3; (- - -) three-parameter model, eq 22; (. .) two-parameter model (Evangelista, 1986a) with eq 37; (- -) two-parameter model (Evangelista, 1986a) with eq 38.
-
.
straight lines. This suggests that a Kremser-like procedure (Treybal, 1980), referred to in the following as analytical method, could be usefully applied without any great loss of accuracy. For detailed equations and their use, see Evangelista (1986a). Once the total concentration factor has been fixed and the corresponding recovery fraction calculated, or vice versa, the concentrations and flow rates of the permeate stream and of the reject stream can easily be found from solute and solvent overall material balances. Both the graphical and analytical methods can straightforwardly be applied to the design of reverse osmosis plants in tapered flow arrangement. Further details are reported in Evangelista (1985). The analytical method is also useful for the design of plants with recirculation stages. Recycle streams are likely to introduce iteration in design procedures. But linearization of the operating curve allows us to work out an explicit relationship between the feed concentration and the reject concentration (Evangelista, 1986~).
Results and Discussion The calculation results of the present method will now be compared with those of computerized iterative procedures. But first, the assumptions made in the preceding sections are verified. I have also developed sophisticated algorithms for the design of reverse osmosis plants, both in straight-through and tapered flow, equipped with tubular, spiral wound, and hollow fiber modules. These methods make use of eq 1-3 for tubular and spiral wound modules and of eq 1and 2 only, with the wall quantities replaced by corresponding bulk quantities, for hollow fiber modules. For tubular and spiral wound modules, eq 1-3, applied to small-membrane elements, are solved by a Newton-Raphson technique; then, the total membrane area is calculated by applying recursive solute and solvent material balances equations from the feed end toward the reject end until the required concentration factor or recovery fraction is reached. Successive substitution procedures might also have been employed to solve the above nonlinear system but the Newton-Raphson technique has been used in all cases since the latter, unlike the former, has always proved convergent. (Evangelista, 1986b). For
0.1
0.2
0.3
0.4 A 3
Figure 4. Errors in the number of series modules for tubular configuration.
hollow fiber modules, a procedure similar to that of Kabadi et al. (1979) has been developed, but adopting Pusch’s model for membrane transport. More details of these design methods are given elsewhere (Evangelista, 1985, 1986a). In Figure 3 a comparison between the present method, eq 22, and the computerized procedure for spiral wound modules is given for four different cases. The values of kl and a of eq 8 have been kept constant and equal to 2.5 X and 1.03, respectively. As can be seen, the agreement is quite good for the whole range of rejections and for the practical range of recovery fractions. The different slope of the operating curve at high concentration factors in case 1is to be ascribed to the several assumptions made. Calculations obtained by a two-parameter model (Evangelista, 1986a) are also reported. For this purpose, 1, and A are considered almost equivalent, while the solute transport parameter (DZM/K6) has been calculated, with the l,, l,, and r , values reported in Figure 3, by
(DZM/KB) = (1, - r m 2 1 p ) n F (DzM/KB) = (1, - rmzLp)nF/rm
(37) (38)
While relation 37, given by Pusch (1977b), accounts for the diffusive solute flux only, relation 38, given by Pusch and Riley (1977), is an attempt to account as diffusive also its convective part. Higher discrepancies are shown for higher concentrations and lower rejections (cases 3 and 4) by this model. It fails also for low concentrations and high permeabilities (case 2). However, all models are adequate for high-rejecting membranes (case 1). Now we have to test the accuracy of the proposed method in estimating the number of series modules, M,, For this purpose, two extreme design cases have been chosen: low concentration and pressures and high concentration and pressures. For simplicity’s sake and to cover the wide range of rejections in both cases, one value of the hydrodynamic permeability and four values of the osmotic permeability and asymptotic rejection coefficient have been selected. For tubular and spiral wound modules, equal values have been chosen, whereas for hollow fiber modules they have been taken considerably lower. Although these parameters do not belong to any real membrane, real membranes, with slightly different combinations of parameters, could give the same performances. All these parameters and other operating conditions are reported in Table I and Figures 4,5, and 6 for tubular, spiral
1114 Ind. Eng. Chem. Res., Vol. 26, No. 6, 1987
Table I. Operating Conditions and Input Data for Accuracy Check of Present Design Methods” low concentration and pressures
design quantities
0.1 4.58 spiral wound 2.5 420 210 0.5 30 2.61 1.033
1 0 3 ~ ~ nF
module config
P 1o3k,
tubular 2.5 110 60 0.4 :30 4.52
n
1.012
1051, FMaX
FM,” %
high concentration and pressures
hollow fiber 0.5 450 220 0.4 30 1.0
tubular 1.5 110
60 0.3 70 4.77 1.014
0.6 27.48 spiral wound 1.5 420 250 0.4 70 2.71 1.03
hollow fiber 0.3 450 250 0.3 60 1.0
1, and r , parameters are reported in Figures 4, 5, and 6, respectively, for tubular, spiral wound, and hollow fiber modules 0.99 0 9 4 4
0:665
AP=70
%
1
-c,= 0 . 6 . 1 6 ~ 0.426
3 QT = 0.4 2 -
0.30 7
0.970
-1
I,.lO’
-
,
I
’’
0.7 61
AP=30
.
-2
, 0.316 0.1
’0.8
10.6
-4
0.0 0.1
0.2
0.3
0.4
A3
Figure 5. Errors in the number of series modules for spiral wound configuration.
p’
0.9541
cx
0 -1
-2
0.2
0.3
0.4
0.5 A 3
Figure 6. Errors in the number of series modules for hollow fiber configuration.
wound, and hollow fiber modules, respectively. Different recovery fractions, different pressures, and different mass-transfer coefficients have been explored. The geometrical characteristics of the modules used are reported elsewhere (Evangelista, 1985). In previous papers (Evangelista, 1985, 1986a), i t was shown that the accuracy could somehow depend on the value of the parameter A,. So Figures 4-6 report the error percentages for the number of series modules as function of parameter A3 for tubular, spiral wound, and hollow fiber modules. These errors, defined as 70 = 1OO(M,8- MSi)/Msi have been calculated by comparing the results of the
0.0
........ -.-.
..-..
--- - .--. 0.5
T
\1 1.0
Figure 7. Errors in the permeate average concentration.
analytical method with those of the iterative methods. Usually better estimates are obtained by the graphical procedure since linearization of the operating and equilibrium curves is not involved. However this is not always true since errors can cancel each other out and the analytical method may perform better than the graphical method. Although the aforementioned dependence cannot be neglected, the errors, whatever parameter A,, are always below 6% for all module configurations and all operating conditions. Besides, these figures are within the scattering of performances of commercial modules. The empty circle on the lines indicates the values of the parameter A , calculated from the total membrane area of the modules. The corresponding errors denote the safety of the procedure without dividing the modules into portions as done elsewhere (Evangelista, 1985) for hollow fiber modules. Indeed, lower values (between 0.05 and 0.2) must be preferred for high concentration and higher values (between 0.2 and 0.4) for low concentration. However, proper values of the parameter A,, the most suitable for each design task, can be selected from Figures 4-6. Average rejections have also been reported for each case. While local rejections depend only on membrane parameters and osmotic and applied pressures, average rejections depend also on the recovery fractions. In Figure 7 the errors in the permeate average concen- CPi)/Cpi,are reported tration, defined as % = 100(pPa as a function of the average rejection. Permeate average concentration does not depend on A3 because it is calculated by using only eq 22 and material balance equations. Higher discrepancies are obtained with higher rejections. Although these figures depend on the operating conditions, the errors are confined within 10% with the only exception of spiral wound modules fed with the more concentrated
Ind. Eng. Chem. Res., Vol. 26, No. 6, 1987 1115 Table 11. Operating Conditions and Input Data for the Design of a Tapered Reverse Osmosis Plant Equipped with Hollow Fiber Modules 7233.8 feed flow rate, QF, cm3/s 0.1 x 10-3 feed concen, Cp,g-mol/cm3 applied pressure, AP,atm 30 4.58 feed osmotic pressure, I&, atm total fractional recovery, @T hydrodynamic permeability, I , cm/(s.atm) osmotic permeability, l,, cm/i$.atm) asymptotic rejection coeff, r , max feed flow rate per module, FM,, cm3/s min brine outlet flow rate per module, FMin, cm3/s pressure drop per module, APM,Laa, atm A3 for the graphical method I section
I1 section A , for the analytical method I section I1 section
0.6 0.5
X
10"
LO x 10-5 0.9 450 250 0.432 0.114 0.111 0.456 0.436
Table 111. Comparison of Design Methods for Tapered Reverse Osmosis Plants Equipped with Hollow Fiber Modules method design quantities no. of ~arallellines of modules, Mp no. of series modules in one line, M, permeate flow rate, QP
perme_ate av concn, 105cp reject flow rate, QR
reject concn, 104cR reject pressure
tapered plant section I
I1 I
I1 I I1 I I1 I I1 I I1 I I1
analytigraphical, this work 17 10 1 0.85 2933.88 1406.40 3.9583 6.4028 4299.92 2893.52 1.4122 1.7874 29.57 29.20
Whole Plant total no. of modules, MT 25.5 overall permeate flow rate, Qpo 4340.28 permeate av concn, 1o5CPo 4.7504 CPU time, s 0.792
cal, this work 17 10 1 0.92 2850.62 1489.66 3.9069 6.3449 4383.18 2893.52 1.3963 1.7884 29.57 29.17 26.2 4340.28 4.7437 0.172
0.0
l/fT
l/f,,,
l/f
1.0
Figure 8. Graphical construction for calculating the number of series modules of a reverse osmosis plant in tapered flow arrangement.
struction is reported in Figure 8. iterative 17 10 1 0.89 2906.16 1455.11 3.9068 6.2475 4327.64 2872.53 1.4092 1.8065 29.57 29.13 25.9 4361.27 4.6877 58.75
solution. All these errors and the maxima and minima shown in Figure 7, are a consequence of the substitution of the approximate eq 9 in eq 2 and 8. Equation 9 overestimates the permeate flux, and, hence, eq 2 underestimates its concentration. Though approximations only are given on the diffusive part of the solute flux, such effects are more pronounced for high permeability and high-rejecting membranes and lower mass-transfer coefficients (spiral wound modules). Where the present method shows higher errors, that is, between 80% and 100% rejections, other approximate procedures with higher accuracy have already been developed (Evangelista, 1985, 1986a). Finally, a design example of a tapered reverse osmosis plant equipped with hollow fiber modules is reported. The input data and operating conditions are given in Table 11, and the calculation results are reported in Table 111. The design has been carried out with both the graphical and analytical methods and by a sophisticated computerized algorithm. The agreement between the three methods is quite satisfactory. Parameter A3 has been calculated from the geometrical membrane area of the module for the analytical method and from its forth part for the graphical method. As can be worked out, discrepancies on the number of series modules agree very well with the errors reported in Figure 6. The corresponding graphical con-
Conclusions If we start with a three-parameter membrane model and take into account the nonperfect rejecting characteristics of the membranes, a graphical-analytical method, useful in the design of reverse osmosis plants, is developed. Straight-through and tapered flow plants can be designed easily irrespective of the geometrical configuration of their modules. Given the total concentration factor, the proposed method predicts the number of series modules, M,(errors not exceeding 6%), and the permeate average concentration, Cp (errors around l o % ) , by simple explicit equations. Slightly higher errors are obtained for the permeate average concentration in the case of high-rejecting membranes. Since more accurate and simple procedures exist for this case (Evangelista, 1985) and for medium-rejecting membranes (Evangelista, 1986a), this method has its best range of applicability with low-rejecting membranes. So, while previous methods cover the range of rejections between 80% and loo%, the proposed one is suitable for average rejections between 0% and 80%, with some overlapping. Acknowledgment
I am grateful to the Italian Minister0 della Pubblica Istruzione for financial support. Nomenclature a = constant introduced in eq 8 A = pure water permeability, cm/(s.atm) A,-A4 = defined by eq 33-36, respectively B = constant defined by eq 6 C = solution concentration, g-mol/cm3 d,-ds = defined by eq 11-15, respectively (D2M/K6)= solute transport parameter, cm/s e = Naperian logarithm base f = concentration factor, defined by eq 7 fMs = concentration factor after the M,th module f,,k = concentration factor at the end of the ith module of the kth section of a tapered plant
F = feed flow rate of the generic module, cm3/s g,, g, = defined by eq 29 and 30, respectively J, = volume flow, cm/s k l = mass-transfer coefficient on the high-pressure side of the membrane, cm/s
1116 Ind. Eng. Chem. Res., Vol. 26, No. 6, 1987 1 = active length of the fibers, cm In = natural logarithm I , = hydrodynamic permeability, cm/ (s-atm) 1, = seal length of a hollow fiber modules, cm 1, = osmotic permeability, cm/(s.atm) m, = slope of the operating curve Mp= number of parallel lines of modules in a straight-through plant M , = number of series modules or portion of module in one line MT = total number of modules in a plant P = permeate flow rate of the generic module, cm3/s P = current overall permeate flow rate of one line of modules, cm3/s Ppo= overall permeate flow rate of one line of modules, cm3/s hp = pressure difference across the membrane, atm QF = plant feed flow rate, cm3/s QR = plant reject flow rate, cm3/s r = rejection coefficient ri = inside radius of the fibers, cm ro = outside radius of the fibers, cm r, = asymptotic rejection coefficient R = reject flow rate of the generic module, cm3/s S = membrane surface area of one module or portion of module, cm2 tanh = hyperbolic tangent Greek Symbols
a = applied pressure averaging constant p = defined by eq 23 7 = defined by eq 5
4 = recovery fraction, defined by eq 19 4M8= recovery fraction after the M,th module @,,k = recovery fraction at the end of the ith module of the kth section of a tapered plant 4T = total recovery fraction p = viscosity, g/(cms) n = osmotic pressure, atm Subscripts b = refers to the brine
F = refers to the feed i = refers to the ith series module
i = refers to inlet j = refers to the jth parallel line of modules k = refers to the kth section of a tapered plant o = refers to operating curve or to outlet P = refers to permeate p = refers to parallel R = refers to reject w = refers to the membrane wall on the high-pressure side
Superscripts a = analytical i = iterative
o = overall quantities -
= averaged quantities
Literature Cited Chiolle, G.; Gianotti, G.; Gramondo, M.; Parrini, G. Desalination 1978,24,3. Dandavati, M. S.; Doshi, M. R.; Gill, W. N. Chem. Eng. Sci. 1975, 30, 877. Evangelista, F.Ind. Eng. Chem. Process Des. Deu. 1985,24, 211. Evangelista, F. Ind. Eng. Chem. Process Des. Deu. 1986a,25,366. Evangelista, F.Chem. Eng. Sci. 1986b,41, 1913. Evangelista, F. Proceedings of the 5th Yugoslavian-Austria-Italian Chemical Engineering Conference, Portoroz, Yugoslavia, Sept 1986~;pp 251-258. Harris, F. L.; Humphreys, G. B.; Spiegler, K. S.Membrane Separation Processes; Meares, P., Ed.; Elsevier: Amsterdam, 1976; pp 121-186. Jonsson, G.; Boesen, C. E. Desalination 1975,17, 145. Kabadi, V. N.; Doshi, M. R.; Gill, W. N. Chem. Eng. Commun. 1979, 3, 339. Kimura, S.; Sourirajan, S. AIChE J. 1967,13, 497. Lonsdale, H. K.; Merten, U.; Riley, R. L. J . Appl. Polym. Sci. 1965, 9, 1341. McCutchan, J. W.;Goel, V. Desalination 1974,14, 57. Murakami, H.; Igarashi, N. Ind. Eng. Chem. Prod. Res. Deu. 1981, 20, 501. Pusch, W. Ber. Bunsenges. Phys. Chem. 1977a,81, 269. Pusch, W. Ber. Bunsenges. Phys. Chem. 1977b,81, 854. Pusch, W.; Riley, R. Desalination 1977,22,191. Rautenbach, R.; Dahm, W. Europe-Japan Congress on Membrane and Membrane Processes, Stresa, Italy, June 1984. Sambrailo, D.; Kunst, B.; Ivekovic, H.Desalination 1983,46, 125. Sherwood, T. K.; Brian, P. L. T.; Fisher, R. E. Ind. Eng. Chem. Fundam. 1967,6,2. Sirkar, K. K.; Dang, P. T.; Rao, G. H. Ind. Eng. Chem. Process Des. Deu. 1982,21,517. Sirkar, K. K.; Rao, G. H. Ind. Eng. Chem. Process Des. Deu. 1981, 20, 116. Soltanieh, M.;Gill, W. N. Chem. Eng. Commun. 1982,18, 311. Spiegler, K. S.; Kedem, 0. Desalination 1966,1 , 311. Treybal, R. E. Mass Transfer Operations, 3rd ed.; McGraw-Hill: New York, 1980; p 128. Tweddle, T. A.; Thayer, W. C.; Matsuura, T.; Hsieh, F.; Sourirajan, S. Desalination 1980,32,181. Van Wijk, H. F.; Jansen, A. E.; Creusen, R. J. M. Desalination 1984, 51, 103. Received for review March 17, 1986 Accepted January 29, 1987
