Ind. Eng. Chem. Process Des. Dev.
300
sequence two. This result shows that sequence one transfers more energy than sequence two. Also, for a pairwise interchange in a sequence, the preferred order is to match the cold tank with a higher inlet temperature before the alternative cold tank. This result holds for any two consecutive tanks in the sequence. Thus, by a series of pairwise interchanges, the cold tank with the highest inlet temperature will be the first match, the second warmest cold tank, the second match, etc. Therefore, the optimal sequence of cold tanks for any 1hot tank/n cold tank problem is to order the cold tanks from highest initial temperature to lowest. is incorporated easily into this analysis. Replacing y: with yg + (AT),, in the equations for x? will account for the minimum difference in approach temperatures (k = 1, 2, ..., a). Also, xik-') 1 (yg + (AOmin) for a match between hot tank 1and cold tank k to occur. The result, eq 56, remains unchanged by these additions. Heat exchange between a hot tank and a cold tank should continue until no more heat can be transferred. We prove this statement by contradiction. Suppose a match is interrupted before the maximum heat is exchanged. In this case, the result from eq 56 shows that the hottest feasible cold tank should match with the hot tank. However, the hottest feasible cold tank is the cold tank whose match has been terminated early. Thus, once a match has begun, obviously, it should be continued until the maximum amount of heat has been transferred.
(Anmi,
1980,25, 366-375
Literature Cited Bowman, R. A.; Muelier, A. C.; Nagle, W. M. Trans. ASME 1940, 62, 283-294. Chaddock, R. E.; Sanders, M. T. Trans. AIChE 1944, 40 (2). 203-210. Dantzig, G. B. Technical Report 80-18, Stanford University Systems Optimization Laboratory, Stanford, CA, June 1980. Fernandez-Seco, M. Chem. Eng. 1953, 60 (12), 172-173. Fisher, R. C. Ind. Eng. Chem. 1944, 36(10),939-942. Grossmann, I.E.; Sargent, R. W. H. Ind. Eng. Chem. Process Des. Dev. 1979, 18 (2), 343-348. Kern, Donald Q. "Process Heat Transfer"; McGraw-Hili: New York, 1950. Knopf, F. C.; Okos, M. R.; Reklaitis, G. V. Ind. Eng. Chem. Process Des. Dev. 1982, 21 (I), 79-86. Linnhoff, B.; Townsend, D. W.; Boland, D.; Hewitt, G. F.; Thomas, B. E. A.; Guy, A. R.; Marsland, R. H. User Guide on Process Integration for the Efficient Use of Energy"; The Institution of Chemical Engineers: Rugby, England, 1983. Loonkar, Y. R.; Robinson, J. D. Ind. Eng. Chem. Process Des. Dev. 1970, 9 (4), 625-629. Lundberg, W. L.; Christenson, J. A. Final Report EC-77-C-01-5002, Westinghouse Electric Corporation: Pittsburgh, PA, 1979. Nishda, N.; Stephanopoulos, G.; Westerberg, A. AIChE J. May 1981, 27 (3), 321-351. Pho. T. K.; Lapidus. L. AICh€J. Nov 1973, 19 (6), 1182-1 189. Rippin, D. W. T. Comput. Chem. Eng. 1983, 7(3), 137-156. Schrage, L. "User's Manual for LINDO";The Scientific Press: Paio Alto, CA, 1981. Sparrow, R. E.; Forder, G. J.; Rippin, D. W. T. Ind. Eng. Chem. Process Des. Dev. 1975, 14 (3), 197-203. Mah, R. S . H. Ind. Eng. Chem. Process Des. Dev. 1982, 27 ( l ) , Suhami. I.; 94-100. Troupe, R. A. Chem. Eng. 1952, 59(9), 128-131. Unger. T. A. Chem. Eng. 1962, 69(15). 119-122. Vaselenak, J. Ph. D. Thesis, Carnegie-Mellon University, Pittsburgh, PA, 1985.
Received for review September 26, 1984 Revised manuscript received May 15, 1985 Accepted May 23, 1985
Improved Graphical-Analytical Method for the Design of Reverse-Osmosis Plants Franco Evangellsta Dipartlmento di Chimica, Ingegneria Chimica e Materiali, Universits degli Studi dell'Aquila, 67 100 L 'Aquila, Italy
A McCabe-Thiele-type method has been developed for the design of reverse-osmosis plants with rejections of between 80 and 100% . Given the total concentration factor and knowing plant operating conditions and membranes characteristics, the present method predicts the number of parallel lines of modules M p ,the number of series modules in one line M,,the permeate flow rate Q,, and the permeate average concentration c p by simple explicit equations for both straight-through and tapered flow plants irrespective of the geometrical configuration of their modules. The calculation results have been compared with those obtained with rigorous computerized methods. Equal values for M, and Q,, -1 to -5% errors for E , and -5 to +7% errors for M,,have been found for alt practical operating conditions. Greater accuracy can be obtained if the dimensionless parameter A is chosen between 0.02 and 0.1. The same procedure can also be applied to other pressuredriven membrane processes.
The reverse-osmosis technique has received considerable attention in recent years as a substitute for more expensive conventional processes such as evaporation, distillation, extraction, etc. This technique is attractive for its simplicity, and its growth is due both to the development of new and better membranes and to the possibility of tailoring them for specific separations. Because of the low intrinsic permeability of commercial polymeric materials, a large membrane area must be provided in industrial plants in order to achieve practical solvent recovery or concentration factors. So membranes have been fitted into modules of different configurations with the standard membrane area.
Three types of modules are more frequently marketed: tubular, spiral-wound, and hollow fiber. Furthermore these modules can be connected together in "straight-through flow" or "tapered flow" arrangements. There are available computerized mathematical methods for the design of industrial plants either in straight-through or tapered flow arrangements equipped with tubular (McCutchan and Goel, 1974; Harris et al., 1976), spiralwound (Chiolle et al., 1978), and hollow fiber (Dandavati et al., 1975) modules. While the first two can be applied, as far as the transport mechanism allows, to membranes showing different rejecting characteristics, the latter, due to several simplifying assumptions, is restricted to high rejecting fibers only.
0196-4305/86/ 1 125-0366$01.50/0 0 1986 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 2, 1986 367
All these methods utilize different membrane models for the prediction of local solute and solvent fluxes. They involve the application of mass transport equations to small membrane elements in such a way as to neglect any variation of operating conditions as well as changes of solution and membrane properties inside such elements. However this procedure is not so simple, because of the rising of the concentration polarization phenomenon which is a function of the solvent and solute fluxes through an implicit transcendental relationship (Brian, 1966; Sourirajan, 1970). So the local performance of a membrane, that is, solvent flux and solute separation, emerges only from a rather cumbersome trial and error procedure. Introducing these values into recursive solvent and solute material balance equations, applied to a series of elements, allows the designer of industrial plants to calculate the total membrane area required for a given separation together with permeate and reject concentrations and flow rates. One factor affecting the length of these calculations is the choice of the membrane area of the elements. The bigger the area is, the simpler the calculations but the less the accuracy obtained. Another factor is the number of iterations needed in the aforementioned trial and error procedure. Both of them are in turn functions of the membrane properties and operating conditions. Byproducts of such calculations are the overall pressure drop of the brine stream and the wall solute or solutes concentrations in the last element. While the former should be kept as low as possible throughout the plant in order to make the most of pressure as a driving force, the latter may be useful to check the free scaling operation of the plant. Several simple design procedures, valid in cases of high rejecting membranes only, have been also worked out. Dandavati et al. (1975) found an implicit equation for the recovery fraction and an explicit equation for the product concentration of a hollow fiber module. Recently, Sirkar and Rao (1981) derived two explicit equations useful in the design of desalination plants with tubular modules in turbulent flow conditions: one for the prediction of the number of membrane tubes required for a given recovery fraction; the other for the permeate concentration evaluation. Afterwards, Sirkar et al. (1982) extended the method to desalination plants with spiral-wound modules under both laminar and turbulent flow conditions. Later, Sirkar and Dang (1982) improved the design procedure for spiral-wound modules, incorporating in it an implicit equation accounting for the permeate concentration formerly neglected. More recently, Evangelista (1985) has developed a unified design method of straight-through and tapered reverse-osmosis plants regardless of the geometrical configuration of their modules. The method is like the graphical staircase calculation of ideal stages in the McCabe-Thiele-type diagram. An analytical version of the graphical procedure and a generally valid explicit relationship for the permeate concentration have also been given. Its validity is restricted to high rejecting membranes and to turbulent flow conditions or negligible polarization on the high-pressure side. However, when the brine concentration increases, when we are dealing with less rejecting membranes or, in short, in all cases in which the permeate concentration becomes relevant, the above design methods fail to be accurate. So, introducing the permeate concentration into the whole design scheme would make the aforementioned procedures more realistic. This work is therefore a logical extension of the former analysis applied to less rejecting membranes and to all
module geometrical configurations. It retains the simplicity and accuracy necessary for quick engineering design purposes.
Theory The simplest models of mass transport in the membranes under reverse-osmosis conditions are those of Kimura and Sourirajan (1967) and Lonsdale et al. (1965). The first model, applied to one solute solution, characterizes the solvent and the solute passage through the membrane using the two parameters A and (D2,/KS). So the solvent flux is evaluated by N1 = A(Pb - P p - IT, IIp) (1)
+
and the solute flux by
Nz = (&M/K@(C, - C)
(2)
These equations are generally valid, regardless of the module configuration, and therefore can be applied to tubular and spiral-wound as well as hollow fiber membranes anywhere throughout the device. In (1)and (2), the solvent flux and the solute flux are directly proportional to the effective driving forces generated across the membrane, the pressure difference and the concentration difference, respectively. No coupling effects are taken into account. Since the concentration of the permeate is determined by the relative amounts of Nl and N2 fluxes, c may be evaluated with
+ V2N2)
c = N2/(N1
E
N2/N1
(3)
This is strictly valid if the volume flow is given essentially by the solvent flow, and this is the situation encountered in all salts-solvent systems even with very low rejecting membranes. Many authors (Sirkar and Rao, 1981; Sirkar et al., 1982; Evangelista, 1985), dealing with almost perfect rejecting membranes, have neglected c in (2) and likewise IIp in (l), causing no appreciable errors in the evaluation of N, and Nz fluxes. Here, since we are also dealing with less rejecting membranes, the above terms cannot be left out of the respective equations without of being too conservative. However, the validity of (1) and (2) cannot be extended further to very poor rejecting membranes because they do not take into account the convective solute flux which can be of comparable magnitude to the diffusive flux in such cases (Pusch, 1977). When the film model is applied to the boundary layer on the high-pressure side and (3) is utilized the following relationship interrelating C, and C can be derived (Brian, 1966; Sourirajan, 1970): C , = c + (C - c) exp(N,/k,) (4) Knowing lzl, A, and DzM/K6,and a relationship between the osmotic pressure and concentration, the local solvent and solute fluxes, and hence the permeate concentration, can be calculated by solving the nonlinear system of equations (l), (2), (3), and (4). An alternative and easier procedure can be developed by replacing the exponential term of (4)by the first two terms of its series expansion (Sirkar et al., 1982; Sirkar and Rao, 1981). Equation 4 then becomes c, = c + (C - c)(l + N , / k , ) (5) Now we call AP the difference between the brine and the permeate pressure, and we define the concentration factor as
f and the constant B as
=
c/cF
(6)
388
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 2, 1986
B = n~/hp
(7)
When a linear relationship between the osmotic pressure and concentration is assumed and ( 5 ) , (6), and (7) are utilized, the difference in the osmotic pressures of (1) becomes II, - I I p = (IIF/CF)(C, - C) = B W f - c / C F ) ( ~+ Ni/ki) (8) Substitution of (8) in (1) leads to F )
+ Ni/ki)
+ ( Y 2 + 20f + f)’/’]/2
[(D,~/K6)(1/Ahp+ 1 / k 1 )
+ l/kJ
(11)
+ 1]/B
(12)
- 1]/B
(13)
Substituting (11) in eq (9), we get the explicit equation for N,
N, =
AAP[1 - B[f + Y - (u2 + 20f 1
+ (ABAP/k,)[f
+ (16Apr,ll,q/ri4.1.0133
X
lo6)]
(16)
+ f)’/’]/2] + Y - (v2 + 20f + f2)1’2]/2
tanh [(16AprO/r,2-1.0133X 106)1/2(l/rL)] (17) [(16Apr,/ri2.1.0133 X 106)1/2(l/rJ]
II=
and this expression for the solute flux
where o = [(D,M/KS)(l/AAP
y = q/[1
(10)
When (9) is substituted in (10) and then both in (3), a quadratic expression for c is found. Its solution is
V
(15)
where
(9)
Substitution of (5) in (2) leads to
c = CF[f -
N1 = A W [ 1 - B ( f - C/CF)]r
with
A@[1 - B(f - c/cF)I N, = 1 + (ABP/hi)(f - C / ~ N2 = ( D z M / K ~ ) C-F c/cF)(1 (~
effects, the net pressure driving force tends to equalize, further guaranteeing the above assumption. Following a procedure similar to that of Dandavati et al. (1975), we get this expression for the solvent flux
Nz = ( D ~ M / K ~ ) C- Fc/CF) (~
The difference between present results and those of Dandavati et al. (1975) is in that they neglect the permeate concentration. This difference will make the present treatment more accurate than the original one since the averaging procedure along the fiber axis is safer in this case. Now utilizing (3), (15), and (18),we can similarly calculate the permeate concentration. It is given by
c=
c,[f - v + (v2 + 2uf + f2)1/2]/2
(19)
where
(14)
Equations 11 and 14 predict the solvent flux and permeate concentration, given the local numerical values of the parameters involved. They can substitute the system of equations (11, (21, (3), and (41, with the assumptions contained in (3), (5), and (8). In order to extend the validity of (11) and (14) to whole modules or lines of modules, the following further assumptions should be made for tubular and spiral-wound modules: (1) the transmembrane pressure difference is constant everywhere in the device and equal to AP,(2) kl is constant and equal to an averaged value between the feed and the reject ends, and (3) membrane characteristics parameters are constant. A different procedure, with further simplifying assumptions, needs to be developed for hollow fiber modules. Part of the above assumption (1) in hollow fiber membranes no longer holds since the pressure drop in the permeate stream along the fiber axis may become relevant, even as high as 30% of the applied pressure, depending on the permeability, the length, and inside radius of the fibers. On the other hand, a common assumption is that of a negligible concentration polarization since the fibers have a much lower permeability than flat sheet membranes made of comparable polymeric materials. With nonperfect rejecting membranes, the polarization phenomenon is less severe, so the above assumption is even more valid although the membrane permeability may be larger. In order to obtain simple analytical expressions, one more assumption must be made: that the net driving forces can be considered constant along the fiber axis and equal to the averaged values. In radial-flow hollow fiber modules, due to pressure drops in the bore of the fibers, the permeate flux is greater at the open end of the fibers than at the closed end. This causes the brine concentration, and hence the associated osmotic pressure, to rise faster at the open side than at the other as the solvent is progressively removed. So, apart from the initial entrance
(181
v = [(D2M/K6) / A u ? ’ + 11/ B
(20)
= [(~~M/KS)/A -A 1I/B ~
(21)
and o
Substituting (19) into (15), we get the following explicit expression for the permeate flux:
N1 = AAP[l - B [ f + v - (2+ 20f
+ f)1’2]/2]y
(22)
As can be seen, (19) is formally equal to (11) and (20), (21), and (22) have the same structure as (12), (13), and (14), respetively. For this reason, quite general equations for all module configurations can be derived. In fact, the permeate flux equation becomes
N, =
AAPa[l - B[f + Y - (2+ 20f 1
+ f2)’’2]/2]
+ (AAPaB/kI)[f + Y - (2+ 20f + f2)1/2]/2 (23)
where Y
= [(DZM/K6)(l/AAPa
+ l/ki) + 1]/B
(24)
= [(DZM/KS)(l/AAPa
+ l/k1)
(25)
~7
-
1]/B
The permeate concentration is calculated by (11)or (19) with (24) and (25). Equations 23, 24, and 25 are utilized for all module configurations with the understanding that a = 1 for tubular and spiral-wound modules and a = y and k , goes to infinity for hollow fiber modules. More approximate expressions for the permeate concentration and permeate flux will also be derived now. In industrial applications, the applied pressure is much greater than the osmotic pressure of the solutions to be processed. In such cases, as a first approximation, we can evaluate the permeate flux as the permeability times the pressure difference. Under this assumption, (3) and (10) become c = NZ/AAPa
Nz = (D,,/KG)C,(f
- C/CF)(1
(26)
+ AAPa/hl)
(27)
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 2, 1986 369 E
G
1
Permeate
Figure 1. Flow sheet for a straight-through flow arrangement of a reverse-osmosis plant.
Even though the solute and solvent fluxes are overpredicted, the permeate concentration, since it is the ratio of the two quantities, is evaluated with reasonable accuracy. When (27) is substituted into (26), the following expression for the permeate concentration is obtained: (D2M/KG)(k1 + AMs) (28) c = CFfAAPakl (D,M/KG)(kl AAPa)
+
+
In general, (28) underpredicts the permeate concentration. Since all quantities of the right-hand fraction are constant, (28) can be considered as derived from a constant rejection model. When (28) is introduced into (9), the following expression for the solvent flux is derived: I r N , = A A P a 1-Bf 1-
I1
11-
1-
0.0
1 / f h
1.0
l/f
l/fMs
Figure 2. Graphical construction for calculating the number of series modules of a reverse-osmosis plant arranged in straightthrough flow.
When (19) is substituted into (33), the following differential equation if found: dc$/(l - 4) = 2 df/[f + u - (2+ 20f + f2)”2] (34) The left-hand side can be integrated between 0 and 4; correspondinglythe right-hand side is integrated between 1 and f . This gives 1 - $J = exp(-/3/2) (35) where
1
[
/3 = B (f - 1) + (v2 + 2uf
(AAPaBf/kI) 1(D2M/KG)(kl + AMs) AAPakl + (D,,/KG)(k, AAPa)
+
]
(29)
Equation 29 also underpredicts the permeate flux to a small extent and will cause the membrane area to be safely overestimated. Plant Design. Reverse-osmosis plants are made of many modules connected together in different ways. Here we will deal only with the design of the straight-through flow arrangement (Figure 1)since the design of the plants in tapered flow is straightforward (Evangelista, 1985). The number of parallel lines of modules Mp is easily found, given the maximum-allowed inlet flow rate for the modules employed, using the formula Mp = Q F / F M ~ ~ (30) If this value is not an integer, it is rounded up to the nearest integer. The number of series modules M , can be calculated by a graphical-analytical method like the stagewise calculations. A differential solute and solvent balance over a differential membrane element gives dF/F = dC/(C - C) (31) Now defining the recovery fraction as 4 = Po/FF
(32)
and remembering ( 6 ) , (31) becomes d + / ( l - 4) = df/(f - c/CF)
(33)
2u In f
+
f)1/2
- (u2 + 2a
+
+
f + a + (u’ + 2uf + f2)1/2 + u In 1 + u + (v2 + 2a + I)’/’ a f / u + v + (u2 + 2af + f2)’/’ u In u / u + u + (u2 + 2a +
]
(36)
Relation 35 is the equation of the “operating curve” on the (1 - 41, l / f reference frame. It has an upward concavity and crosses the points of coordinates (1,1)and (0, 0) as shown in Figure 2. The straight line between the same points, obtained for the “operating curve” in Evangelista (1985) neglecting the solute leakage through the permeate stream, is here substituted by a curve given by (35). The departure from the diagonal of the square is greater as the solute concentration in the permeate increases. Equation 35 allows us to calculate the concentration factor given the fractional solvent recovery or vice versa. But while the recovery fraction can be calculated directly knowing the concentration factor, the concentration factor corresponding to a given solvent recovery can be found through a Newton-Raphson technique only. Now we need to find the “equilibrium curve” of the operation in terms of the (1- 4) and l / f quantities. The solvent material balance for the generic module of the line is F - P = R = F - SN1 (37) When (6),(32), and (35) are remembered and the solvent and solute material balances for the envelope I1 of Figure
370
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 2, 1986
1 are utilized the second equality of (37) becomes 1 - 4 = exp(-P/2) - SNl/FF
(38)
The flux N , can be calculated at the inlet, outlet, or averaged conditions of the module. While inlet conditions will certainly underpredict the membrane area, the outlet conditions may overpredict it too much. So 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. When it is noted that the recovery fraction or concentration factor of one module is low and it is assumed for a while that the solute leakage is negligible, the outlet concentration factor may be assumed to be 1 / ( 1 - 4). So the averaged concentration factor, needed to calculate the solvent flux,can be set as Lf + 1 / ( 1 - 4)]/2. The fact that the solute leakage is not negligible has little, if any, importance. It would mean that the solvent flux is not calculated at an arithmetically averaged concentration factor but nearby and, still, within the module under consideration. Modifying (29) according to the above assumption and introducing it into (38),we have
than the membrane surface area of one module. As will be shown later, the accuracy of the present calculations slightly depends on the value of A,, and its best value will be established later. The curve representing (39) is also shown in Figure 2 . It crosses the straight line l l f = 1 a t a point G in such a way as that the segment EG represents the recovery fraction of the first module. This curve also crosses the operating line, showing that there is a maximum recovery fraction and a maximum concentration factor. However, in that region, the equations derived above are no longer valid, and the above maximum values should be indicative only. In practice, recovery fractions and concentration factors are very far from the maximum ones. The number of series modules M, is then calculated by drawing the stairlike construction EGHOU-Z between the "operating curve" and the "equilibrium curven as shown in Figure 2. The step EGH corresponds to the first module and so on. The point E represents the feed, the point Z the reject, and points H,U,etc., the intermodule streams. Since the minimum outlet flow rate for a module is nearly half the maximum inlet flow rate, the maximum recovery fraction which can be gotten from one line of modules is 0.4-0.5. From Figure 2, it is easily seen that in the range 0-0.5 of the recovery fraction, both the "operating curven and the "equilibrium curve" are almost linear. Thus, applying a Kremser-like procedure (Treybal, 1980) to the present case, we get an analytical expression for the number of series modules:
where me and m, are the averaged slopes of the equilibrium and operating curves, respectively. They may be given by This is the equation of a practical equilibrium line, not in a thermodynamic sense but in that it allows us to know the concentrations and flow rates of the outlet streams. It depends, as expected, on membrane and module characteristics as well as on operating conditions. Assuming AP, kl,A , and D Z M / K 6to be constant throughout the plant, a quadratic equation in ( 1 - 4) can be worked out. Its solution is 1 - 4 = [A,
+ (AI2+ 4A2)'/2]/2
(40)
where
A , = exp(-P/2) -
A~(2.44- AAPaBkJ)
+ A2AF"a2B
2A4 + A2AP2a2Bf (41)
+ AAPa exp(-P/2)] 2A4 + A2AP2a2Bf
AAPaB[A&, A2 =
A3 = SAAPa/FF
(42)
4M,+1 - 41 me =
l / f O - l/fM8
(47)
Other averaging procedures could have been employed, but (46) and (47) have been chosen in that they will lead to a better estimate of concentrations and flow rates of the outlet streams from one line of modules. If fractional modules are not involved, their values are equal to those obtained by a graphical method. However, fractional series modules usually emerge from the calculations. But since industrial plants are made up of integer numbers of modules, the designer is interested in knowing the actual conditions of the outlet streams. This can be accomplished, unlike in other methods, by using (46) and the explicit relationship
(43)
A , = A U a [ k l + ( D Z M / K @+] ( D ~ M / K ~(44) )~I As can be easily worked out from the above, the "equilibrium curven depends mainly on the value of the membrane surface S , the only quantity left to the discretion of the designer. The other quantities are established by the operating conditions of the process and by membrane characteristics. In fact, S can be equal to the membrane area of one module as assumed so far or to a portion of membrane area which can be smaller or larger
Another important quantity which the designer is also concerned with is the average concentration of the permeate. When the solvent and the solute material balances for the envelope I of Figure 1 are utilized, the following simple expression is obtained: FP = cF[l
- (1 -
$M,)fM81/4M,
(49)
Once the recovery fraction or the concentration factor has been chosen, (49) together with (35) can be used to
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 2, 1986
Table 1. Operating Conditions and Input Data for Testing Present Design Methods" low concn and pressures design ouantities feed concentration 103CF,g-mol/cm3 feed osmotic pressure IIF, atm total fractional recovery *T module configuration pure water permeability 105A, cm/(s.atm) max feed flow rate per module FMa., cm3/s min brine outlet flow rate per module FMint Cm3/S av mass-transfer coeff 103k1reported, cm/s
0.034 246 5 (2000 ppm) 1.568 42 0.4 tubular spiral-wound hollow fiber 2.5 2.5 0.4
371
high concn and pressures 0.6 (35000 ppm) 27.4788 0.3 tubular spiral-wound hollow fiber 1.5 1.5 0.3
140
420
460
140
420
460
70
220
210
90
260
300
5.8
2.75
5.8
2.75
(D2M/K6)parameters have been reported in the figures.
check if the membrane used gives the desired performance, before the entire design is completed.
Results and Discussion In this section, the merits and the limitations of this short-cut procedure will be extensively reviewed and commented on. The method can be applied, as will be shown later, to high-rejecting membranes and to medium-rejecting membranes. Since simplier and perhaps more accurate procedures for high-rejecting membranes can be found elsewhere (Evangelista, 1985), we will now focus our attention on medium-rejecting membranes. According to the model adopted, flux and rejecting capabilities of a membrane are quantitatively accounted for by two parameters: A and DZMIK6. While the flow of permeate is mainly determined by A , the rejection is determined by a combination of A and DZM/K6. Different membranes with different values of the two parameters may give the same rejection. In order to explore rejections of between 70 and loo%, one value of A and different values of D,M/K6 have been chosen for a given module configuration. Although these values do not refer to any particular membrane, they give rejection percentages which could conceivablybelong to real membranes. For simplicity, equal values for tubular and spiral-wound modules have been selected. Such commercial membranes in fact show comparable permeability and rejection characteristics. On the other hand, since existing hollow fiber modules have smaller values for the permeability and solute passage, the choice of figures has been made accordingly. These values are all reported in Table I as well as operating conditions. These are selected in such a way as to cover the entire working conditions of reverse-osmosis technology, ranging from low solution concentration and pressures to high concentrations and pressures. Lastly, averaged mass-transfer coefficients for tubular and spiral-wound modules have been reported in the same table. These values have been obtained by an arithmetic mean of the corresponding values calculated at the maximum and minimum brine velocity by dimensionless correlations valid for turbulent flow conditions. These correlations are given in Evangelista (1985) together with the geometrical characteristics of the modules used here for calculations. For hollow fiber modules, the value of K, has been omitted. Any value great enough for the concentration polarization to be negelected is suitable. It should be noted that for these design methods and under normal operating conditions, only the numerical value of k1 takes into account the different geometry of modules: low values for spiral-wound, somewhat greater values for tubular, and high values for hollow fiber modules. Other situations, however, with different values of h, can be also handled without impugning the validity of
the method provided the correctness of ( 5 ) is still maintained. Evangelista (1985) has shown that the accuracy of the method in calculating the number of series modules depends mainly on the value of the dimensionlessparameter AB. This represents, according to (43), the recovery fraction of a module of membrane area S fed with distilled water, keeping the other operating conditions unchanged. For high-rejecting membranes, the value of A3 should be selected in the range 0.02-0.1 in order to keep errors in M, less than 2.5% for all operating conditions and for all module configurations (Evangelists, 1985). Now the sensitivity of the M,value on the A , parameter for medium rejecting membranes will be also explored. Calculated values of M,for different A3 values are compared with those obtained by rigorous methods for all module configurations. The iterative methods of McCutchan and Goel (1974) for tubular modules and that of Chiolle et al. (1978) for spiral-wound modules are utilized for comparison purposes. For hollow fiber module, I have also developed a rigorous method starting from the analysis of Dandavati et al. (1975). Their results are not directly applicable since the permeate concentration is neglected. Here the permeate concentration is accounted for, while as done by Dandavati et al. (1975) the variation of all quantities along the fiber axis on the shell side and of all quantities along the radius in the bore of the fibers is neglected. The result of the foregoing analysis is a system of four partial differential equations, three of the first order and one of the second order. It has been solved by a Crank-Nicolson discretizing procedure choosing suitable intervals along the axis and the radius of the module. A t each step, the total solvent recovery is checked so even fractional modules can be calculated. Average quantities along the fiber axis, such as shell pressure and concentration, bore pressure and concentration, and permeate flux are calculated. In order to check the validity of ( E ) , the permeate flux is calculated from the aforementioned averaged quantities and compared with the averaged flux above. Their discrepancies never exceed 5%. Discrete values of A, have been calculated for all module configurations by changing the membrane surface area. Correspondingly, the values of series modules, for design data of Table I, have been calculated by both graphical and analytical methods. They have been compared with those obtained by rigorous methods for the same operating conditions. Relative errors for the analytical method have been evaluated and reported, as a function of A , in Figures 3, 4, and 5 for tubular, spiral-wound, and hollow fiber modules, respectively. The graphical method would give in general somewhat better estimates. Positive errors mean
372
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 2, 1986
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that the present method overpredicts the number of series modules, negative errors the contrary. In the same figures, errors obtained by using the analytical method of Evangelista (1985) are also reported. Dashed area and full lines refer to the present work, while dashed lines or line-dot lines refer to the previous work. For both designs presented, the three different values of the DzM/K6parameter are also reported. For low concentration and pressures, these figures have been placed close to lines referring to the previous work, while for high concentration and pressures they have been placed close to lines referring to this work. The dashed area refers to this paper and to a low concentration and pressures. All three curves are
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confined within that area. Of the three dashed lines, only two have been reported because the last, referring to the highest DzMjK6parameter, would have been well out of the coordinates. Corresponding to the D 2 ~ j K parameters, 6 average rejection coefficients have also been reported. From these figures, two main considerations can be made. Choosing the value of the A3 parameter in the range 0.02-0.1, the present method gives errors never exceeding 5 6 % for all operating conditions, all module configurations, and rejection coefficients ranging from 0.8 to 1. The previous analytical method (Evangelista, 1985) gives better estimates at low concentrations but becomes unacceptable at higher concentrations and lower rejections. This is because, at low concentrations, the permeate osmotic pressure is still negligible in comparison with the effective driving force, even though the rejection is low. As the feed concentration increases and the rejection decreases, the osmotic pressure of the permeate becomes relevant and leads to higher overestimates. However, the previous method is still acceptable for all concentrations if the membranes show rejection coefficients between 0.96 and 1. It should be noticed that all figures are dependent on the recovery fraction, and higher accuracy can be obtained by applying the same procedure to lower recovery fractions repeatedly until the total recovery fraction is reached. There is also another point of great interest which needs to be emphasized. All curves obtained by the previous method show a divergent trend as the parameter A3 increases, while all curves obtained by the present method have an upper limit and are barley dependent on AS. This is because in the previous paper, the solvent flux, to be introduced in (38), was calculated at the outlet conditions of a module, while here it has been calculated at averaged conditions. The improvement lies in that it will give good estimates (errors between -5 and +7%) whatever value of the A3 parameter we select, for all module geometrical configurations and rejection coefficients ranging from 0.8 to 1. This makes it possible to use the membrane surface
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 2, 1986 373
Table 11. Operating Conditions and Input Data for the Design of a Tapered Reverse-Osmosis Plant Equipped with Spiral-Wound Modules 7233.8 feed flow rate, cm3/s QF, cm3/s 0.1 feed concn 103C~,g-mol/cm3 40 applied pressure AP,atm 4.58 feed osmotic pressure IIF, atm 0.7 total fractional recovery 2.5 pure water permeability 105A, cm/ (watm) 1.0 solute transport parameter (D2M/K6)io4, cm/s 420 max feed flow rate per module FM-, cm3/s min brine outlet flow rate per module FM~,,,cm3/s 210 0.352 pressure drop per module APM,L-, atm 2.61 av mass-transfer coeff lo%, reported, cm/s 0.1434 A3 parameter @
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area of any module in the calculations. Greater accuracy, however, can also be obtained by dividing the modules in an integer number of portions. The only requirement is that the applied pressure must be at least double the osmotic pressure of the brine. This condition will make (26) and the related constant rejection model more realistic. Constant rejection models have frequently been used (Gill and Bansal, 1973; Saltonstall and Lawrence, 1982) and extensively critized (Bansal and Gill, 1974; Dandavati et al., 1975) when applied to plants with high recovery fractions (greater than 0.5-0.6). But here, this kind of model is applied to one module at a time and, together with the arbitrary averaging procedure adopted for the solvent flux, accounts for the slight overestimates of the number of series modules at higher concentrations and lower rejections. Different considerations must be made for the permeate average concentration. The above constant rejection model has no effect on its estimate since (19) and not (28) is used in the derivation of (35). Moreover, its value does not depend on the value of the parameter A3 and is not sensitive to the actual value of the recovery fraction. Figure 6 shows the errors oc the permeate average concentration as a function of 1 - R. Errors obtained from the previous work (Evangelista, 1985) are also reported. As can be seen, the present work (dashed area) gives quite satisfactory estimates (-1 to -5%) for all module configuration, all operating conditions, and all rejection coefficients. On the other hand, the previous work gives better estimates, when
25
20
15
10
5
0
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0 .o
0.1
0 2 1 - - R
0.3
Figure 6. Percentage deviations on the permeate average concentration.
the rejection coefficient ranges from 0.96 to 1but becomes unacceptable outside those limits. As an example, design calculations for a tapered reverse-osmosis plant equipped with spiral-wound modules are also reported. Design input data are given in Table 11. The figures refer to low concentration and pressures and to medium-rejecting membranes. In Table 111, the results of the design calculations for four methods have been reported. The first column refers to the present graphical method for the number of series modules and to (49) for the permeate average concentrations. The whole procedure has been carried out by computer, and the corresponding drawings are shown in Figure 7. The use of a computer in plotting the graph, as it involves one trial and error procedure for each step, could
Table 111. Comparison of Design Methods for Tapered Reverse-Osmosis Plants Equipped with Spiral-Wound Modules iterative analytical graphical analytical tapered method method method plant method design quantities section (this work) (this work) (Chiolle et al., 1978) (Evangelista, 1985) no. of parallel nodules Mp I 18 18 18 18 I1 10 10 10 10 I I1
4 4.221
4 4.424
4 4.378
4
permeate flow rate QP,cm3/s
I I1
3413.24 1650.42
3363.04 1700.62
3359.67 1705.60
3191.93 1871.73
permeate av concn 106~,,,g-mol/cm3
I I1
1.77954 3.361 05
1.753 15 3.339 34
1.85690 3.472 86
2.162 12 4.225 94
reject flow rate QR, cm3/s
I I1
3820.55 2170.14
3870.76 2170.14
3874.09 2168.50
4041.87 2170.14
reject concn i04CR,g-mol/cm3
I I1
1.734 41 2.797 83
1.716 51 2.79996
1.706 2.775
1.619 2.6508
reject pressure, atm
I I1
38.59 37.10
38.59 37.04
38.57 37.05
38.59 36.72
115.78 5065.30 2.401 1.73
125.09 5063.66 2.925 0.144
no. of series modules M ,
total no. of modules MT overall permeate flow rate Q cm3/s i, permeate av concn io5&, g-mol/cm3 CPU time, s
Whole Plant 116.24 114.21 5063.66 5063.66 2.286 2.295 0.172 0.581
5.309
374
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Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 2, 1986
l.O
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be more cumbersome than plotting by hand! Manual plotting on reasonable graph paper does not necessarily mean loss of accuracy as the differences are confined to the fractional series module of the last section only. The second column refers to the present analytical method, to (45) for the number of series modules, and to (46), (48), and (49) for the permeate average concentration. The third column refers to the iterative method of Chiolle et al. (1978) and will be used for comparison purposes. The results of the analytical method of the previous work (Evangelista, 1985) have also been reported in the last column. For all the data reported, there is good agreement between both present methods and the iterative method with errors of less than 5% for the permeate average concentrations. The previous analytical method shows higher discrepancies (greater than 20%) both for the number of series modules and for the permeate average concentrations, denoting its inaccuracy in this region of the operating conditions. At the bottom of the table, a “summary” of the plant together with the CPU times elapsed on a UNIVAC 1100/60 computer has also been reported. The latter are reported to show the simplicity of the present design methods and particularly the straightforwardness of the analytical method which can even be implemented on a pocket calculator. Note that if the concentration factor is fixed, the analytical method is completely explicit, while it requires one Newton-Raphson procedure only if the fractional solvent recovery is known. Moreover, its accuracy is quite good since in the range of practical operating conditions, both operating and equilibrium curves are congruent and almost linear. So the lineralization made is unlikely to introduce any relevant errors, as it keeps the effective driving force unchanged and slightly shifts the outlet conditions of the intermodule streams only. Conclusions Starting with a two parameter model, a simple and reasonably accurate method, useful in the design of industrial reverse-osmosis plants, has been worked out. It is based on the graphical stairlike construction as in ideal stage calculations of chemical engineering unit operations. Almost equal approximations are obtained by its analytical version derived from a Kremser-like procedure. The suitability of the graphical method and even more of the analytical method lies in the extremely low varia-
bility of the quantities involved, such as brine and permeate concentrations and solvent flux; corresponding averaged quantities can be substituted for local quantities, giving fairly good approximations. Straight-through flow and tapered flow plants can be easily designed irrespective of the geometrical configuration of their modules. Turbulent flow conditions or negligible concentration polarization on the high-pressure side of the membrane, small pressure drop in comparison to the applied pressure, and an almost constant mass-transfer coefficient in the brine compartment have been assumed. While the previous work (Evangelista, 1985) is valid for high-rejecting membranes, the present methods can be applied, with slightly lower accuracy, to membranes whose average rejection coefficients are between 0.8 and 1. They are able to predict the number of parallel lines of modules, the number of series modules in one line with errors never exceeding 7 % and permeate average concentrations with errors between -1 and -5%. If greater accuracy is to be obtained (less than 5%), the value of the dimensionless parameter A3 should be selected in the range 0.02-0.1. Provided a two-parameter membrane model is still valid, these design methods can also be applied to other pressure-driven membrane processes such as ultrafiltration. Their suitability for inclusion in teaching programs should also be emphasized since chemical engineering students are very familiar with such kinds of calculations. Acknowledgment The financial support of the Italian Minister0 della Pubblica Istruzione is greatly acknowledged. Nomenclature A = pure water permeability constant, cm/(s.atm) A,, A,, A,, A4 = defined by (41), (42), (43), and (44), respectively B = defined by (7) c = molar solute concentration of the permeate, g-mol/cm3 Cp = average solute concentration of the permeate of a straight-through arrangement, g-mol/cm3 (D,M/KS) = solute transport parameter, cm/s exp = exponential j = concentration factor defined by (6) j o = concentration factor at the inlet of the first module, j o )
=1 jMs = concentration factor after the M,th module ji,k= concentrationfactor at the end of the ith module of the
kth section of a tapered plant jT= total concentration factor F = feed flow rate of the generic module, cm3/s F,, = maximum feed flow rate per module, cm3/s FMin= minimum brine outlet flow rate per module, cm3/s k, = solute mass-transfer coefficient on the high-pressure side of the membrane, cm/s 1 = active length of the fibers, cm 1, = seal length of a hollow fiber module, cm m, = average slope of the operating curve, (46) me = average slope of the equilibrium curve, (47) M,, = number of parallel lines of modules in a straight-through
plant M, = number of series modules or portion of the module in a line MT = total number of modules in a reverse-osmosis plant N , = solvent flux, cm/s N , = solute flux, g-mol/(cm2.s) P = permeate flow rate of the generic module, cm3/s PO = current overall permeate flow rate of one line of modules, cm3/s Pk = overall permeate flow rate of one line of modules, cm3/s Pb = brine pressure, atm Pp = permeate pressure, atm AP = pressure difference across the membrane, atm
Ind. Eng. Chem. Process Des. Dev. 1986, 25, 375-381
aPM,Loss = brine average pressure drop in one module or portion of module, atm QF = plant feed flow rate, cm3/s QR = plant reject flow rate, cm3/s r, = outside radius of the fibers, cm ri = inside radius of the fibers, cm R = reject flow rate of the generic-module, cm3/s R = average rejection coefficient, R = 1 - Cp/CF S = membrane surface area of one module or portion of module, cm2 tanh = hyperbolic tangent V = molar volume, cm3/g-mol Greek Letters = applied pressure averaging constant:
a! = 1 for tubular and spiral-wound modules, a! = y for hollow fiber modules /3 = defined by (36) y = defined by (16) 7 = defined by (17) u = defined by (24) u = defined by (25) + = recovery fraction, (32) +o = recovery fraction at the inlet of the first module, +o = a!
0
recovery fraction after the M,th module 4, = recovery fraction of a tapered reverse-osmosis plant +i,k = recovery fraction at the end of the ith module of the kth section of a tapered plant dT = total recovery fraction of a reverse-osmosis plant p = viscosity, g/(cm.s) n = osmotic pressure of a solution, atm +Ms =
Subscripts
o = refers to the inlet of a module or to outside 1 = refers to the solvent or to the first module or to the first section 2 = refers to the solute b = refers to the brine F = refers to the feed
375
i = refers to the ith series module or to inside j = refers to the jth parallel line of modules k = refers to the kth section of a tapered plant
P = refers to the permeate R = refers to the reject s = refers to a series modules or to the seal of a hollow fiber module T = total w = refers to the membrane wall on the high-pressure side Superscripts O = overall quantities - = averaged or partial molar quantities
L i t e r a t u r e Cited Bansal, 6.; Gill, W. N. AIChE Symp. Ser. 1974, No.
7 0 , 136. Brian, P. L. T. "Desallnatlon by Reverse Osmosis"; Merten, U., Ed.; MIT Press: Cambridge, MA, 1966; pp 178-181. Chiolle, A,; Gianotti, G.; Gramondo, M.; Parrini, G. Desalination 1978, 2 4 , 3. Dandavati, M. S.; Doshi, M. R.;Gill, W. N. Chem. Eng. Sci. 1975, 30, 877. Evangelista, F. Ind. Eng. Chem. Process Des. Dev. 1985, 2 4 , 211. Gill, W. N.; Bansal, B. AIChE J. 1973, 19, 823. Harris, F. L.; Humphreys, G. 6.; Spiegler, K. S. "Membrane Separation Process"; Meares, P. Ed.; Elsevier: Amsterdam, 1976; pp 121-186. Kimura, S.; Sourlrajan, S. AIChE J . 1967, 13, 479. Sci. 1965, 9 , 1341. Lonsdale, H. K.; Merten, U.; Riley, R. L. J. Appl. Po&" McCutchan, J. W.; Goel, V. Desalination 1974, 1 4 , 57. Pusch, W. Ber. Bunsenges. Phys. Chem. 1977, 8 1 , 856. Saltonstall, C. W.; Lawrence, R. W. Desalination 1982, 4 2 , 247. Sirkar. K. K.; Dang, P. T. "Harnessing Theory Practical Application, World Filtration Congress, 3rd; Uplands Press: Croydon, UK, 1982, Vol. 2, pp 564-71. Sirkar, K. K.; Dang, P. T.; Rao, G. H. Ind. Eng. Chem. Process Des. Dev. 1982, 2 1 , 517. Sirkar, K. K.; Rao, G. H. Ind. Eng. Chem. Process Des. Dev. 1981, 2 0 , 116. Sourirajan, S. "Reverse Osmosis"; Logos Press: London, 1970. Treybal, R. E. "Mass Transfer Operations", 3rd ed.; McGraw-Hill: New York, 1980; D 126.
Received for review July 12, 1984 Revised manuscript received May 23, 1985 Accepted July 1, 1985
Performance of a Sieve Plate with Small Holes Canan Ozgen' and Tarlk G. Somer Chemical Engineering Department, Middle East Technical University, Ankara, Turkey
I n this study, a new type of distillation column plate is developed by a special technic which may be used to manufacture industrial plates having small holes. The performance of this plate is investigated in a glass distillation column with an internal diameter of 0.225 m, using air-water, acetic acid-water, and ethanol-water systems. The glass column operating at total reflux contained three plates having 25 mm thickness and a free area of 6.15 % . Two sets of plates were used, having equivalent hole diameters of 0.615 and 1.025 mm. The Murphree plate efficiencies of the plates under study were found to be between 75 % and 95 % , depending on the vapor rate. The plate pressure drop was comparable with those of the sieve plates having the same hole size. The plates used tolerate lower allowable vapor rates before weeping and dumping can occur.
Since the early 19509, the literature contains numerous articles promoting the use of sieve plates in place of the bubble-cap. There are many reasons for this popularity, but, certainly, greatly improved design methods is one of the factors in the acceptance of these plates. Hole size in a sieve plate is one of the important factors on the performance and efficiency of the plate. Small holes, although having certain limitations, are known to provide good vapor-liquid distribution and reduce pressure drop and "weeping", while increasing plate efficiency by
producing small bubbles with large interfacial area per unit volume of vapor, leading to greater mass transfer (Bain and Winkle, 1961; Smith, 1963; Wade et al., 1966; Smith and Upchurch, 1981). Small holes also offer an appreciable capacity advantage (Lemieux and Scotti, 1969) especially with positive surface tension systems (Fell and Pinczewski, 1977). Manufacturing of plates with small holes introduces, however, certain mechanical difficulties; on an ordinary steel plate, for instance, the minimum diameter of a hole
0196-4305/86/1125-0375$01.50/00 1986 American Chemical Society
