Ind. Eng. Chem. Process Des. D ~ v1981, . 20, 116-127
116
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Kunii, D.; Levensplel. 0. ”Fluldlzation Engineering”, Wtley: New York. 1969; Chapters 3 and 9. Leva, M. “Fluldlzation”, McGrawHRI: New York, 1959; Chapter 8. ) ” Eng. 1957, 35,71. Leva, M., Can. J. 0 Lese, H. K.; Kermode, R. I.Can. J. Chem. Eng. 1972, 50, 44. Maskaev, V. K.; Baskakov, A. P. J. Eng. phys. 1973, 29,589. Natush, H. J.; Blenke, H. Chem. Ing. Tech. 1973, 7 , 293. Neukiichen, B.; Blenke, H. Chem. Ing. Tech. 1973, 45, 307. Noack. R. Chem. Ing. Tech. 1970. 42, 371. Petrle, J. C.; Freeby, W. A.; Buckam, J. A. Chem. Eng. frog. 1988, 64, 45. Sarklts, V. 8. Candidate Dissertation (in Russian), Leningrad, 1959 (cited in Zabrodsky, 1966). Saxena, S. C.; Grewai, N. S.; Gabor, J. D.; Zabrodsky, S. S.; Galershtlen; D. M. in Adv. Heat Transfer, 1978, 14. 149. Saxena, S. C.; Vogei, G. 9. Trans. Inst. Chem. Eng. 1977, 55, 184. Staub, F. W.; Canada, G. S. In “Fluldizatbn”, Davidson, J. F., Keairns, D. L.. Ed. Cambridge Unhrersity Press: London, 1978; p 339. Ternovskaya, A. N.; Korenberg, Yu. G. “Pyrite Kilning in a Fluidized Bed”, (in Russian), Izd. Khimlya: Moscow, 1971. Tishchenko, A. T.; Khvastukhin, Yu. I.“Fiuidlzed Bed Furnaces and Heat Exchangers”, (in Russian), Neukova Dumka: Kiev, 1973. Tabs, 0. M.; Bondareva. A. K.; Geblker, A. 0. “Dokledy, Ab. 5- 18 ne 20m Vsesowzn, Soveshch, PO Teopb-i Massoobmenu”, Mlnsk (1964) (clted in Ainshtein and Gelperln, 1966). Traber, D. G.; Pomerantsev, V. M.; Mukhlenov, I. P.; Sarklts, V. B. Zh. R k i . Khim. 1982, 35,2386 (cited in Ainshtein and Gelperin, 1966). Vakhrushev, 1. A.; Botnikov, Ya. A.; Zenchenkov, N. G. Int. Chem. Eng. 1983, 3,207. Varygln, N. N.; Marlyushin, I.G. Khim. Mash. 1959, No. 5. 6 (cited in Ainshtein and Gelperin, 1966). Vreedenberg, H. A. Chem. Eng. Sci. 1958, 9 , 52. Zabrodsky, S. S. ”Hydrodynamics and Heat Transfer in Fluldized Beds”, MIT Press: CambrMge, Mass., 1968 Chapter 10. Zabrodsky, S. S. Inzh. F/z. Zhurn 1958, 1, 22. Zabrodsky, S. S.; Antonishin, N. V.; Pamas, A. L. Can. J. Chem. Eng. 1978, 54, 52. Zabrodsky, S. S.; Antonishin, N. V.; Vasillev, G. M.; Paranas, A. L. Vest/ Akad. Nauk BSSR, ser Flz. - Energ. Nauk, 1974, No. 4, 103. Zabrodsky, S. S.; Epanov, Yu. G.; Galershtein, D. M. in ”Fiuldizatlon”, DavM son, J. F., Keairns, D. L. Ed., Cambridge University Press: Cambridge, 1978; p 362. Zenz, F. A.; Othmer, D. F. ”Fluldizath and Fluld-Particle Systems”, Van Nostrand-Reinhold: Princeton, N.J., 1960; Chapter 13. Ziegler, E. N. “Heat and Mass Transfer to a Gas-Fluldlzed Bed of Sdid Particles”, Argonne National Laboratory Report No. 6807, 1963. Ziegler, E. N.; Koppel, L. 6.; Brazeiton, W. T. Ind. Eng. Chem. Fundem. 1984, 3,324.
Received for review June 2, 1980 Accepted October 6 , 1980
Approximate Design Equations and Alternate Design Methodologies for Tubular Reverse Osmosis Desalination Kamalesh K. Slrkar’ Department of Chemistw and Chemical Engineerlng, Stevens Instnute of Technokgy, Hoboken, New Jersey 07030
Goruganthu Hanumantha Raa Department of Chemical Engineering, Indian Instnute of T e c h n o w , Kanpur, 2080 16, UP., India
Simple analytical equations have been developed to determine the number of membrane tubes required for a given reverse osmosis desalination plant capacity with tubular modules In “straight-through” arrangement and turbulent flow inside tubes. The average solute concentration in permeate may also be estimated for such reverse osmosis plants by means of simple equations. The predictive accuracies of such approximate analytical design equations have been checked against the standard numerical design procedure of McCutchan and Goel for a single stage reverse osmosis seawater desalination plant. The design method based on breaking up the plant in two or more sections yields a better design. A simplified alternate numerical design procedure having the accuracy of the standard procedure has also been developed.
Introduction Remarkable advances in the reverse osmosis (RO) membranes and processes in the past decade have not only resulted in significant cost advantage for seawater desa0196-4305/81/1120-0116$01.00/0
lination by RO over that by distillation (Channabasappa, 1975) but also established the reverse osmosis process as a versatile and economic method for removal of small inorganic and organic solutes from aqueous solutions. This 0 1980 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 20, No. 1, 1981
study is concerned with the “straight-through tubular reverse osmosis module design even though three module configurations for RO equipment, namely, hollow fiber, spiral wound, and tubular, are commercially available. A number of design studies of the tubular module to predict the number of tubular membranes of specified lengths required for obtaining a given desalinized water production rate from a particular brine feed flow rate are available (McCutchan and Goel, 1974; Scherer and Chan, 1976; Harris et al., 1976; McCutchan, 1977). All such studies are based on an iterative solution for the desalinized water output from a single membrane tube since the solvent flux expression at a given location is not of the explicit type due to the presence of the high pressure side wall solute concentration which is a function amongst others of the solvent flux itself. These studies then proceed from membrane tube to membrane tube with the help of a set of recursive equations to determine the brine flow rate, concentration, and pressure for the feed to each succeeding tube till the total desalinized water flow rate from all the tubes equals the desired plant capacity. Regardless of whether the arrangement of tubes in the reverse osmosis plant is of the “straight-through or “tapered-flow cross section” type (Harris et al., 1976), the existing design procedures are quite complex. There exists a need for simpler computational schemes with sufficient accuracy. Furthermore, for a quick economic evaluation of a number of competing processes for a given separation, one would like to have simple analytical design equations for tubular reverse ostnosis separations especially since RO is being increasingly explored for many conventional separations (Channabasappa, 1975). Rao and Sirkar (1978) have recently developed explicit flux expressions in terms of intrinsic membrane parameters and the given operating variables, namely the applied pressure difference, feed concentration and the feed side mass transfer coefficient for practical tubular reverse osmosis desalination. Such expressions developed for solution diffusion as well as other models of membrane transport are valid, in general, for tubular reverse osmosis processes with turbulent flow. In this paper, we develop simple approximate design equations for tubular RO processes with the help of such explicit flux expressions and other suitable assumptions. The total number of membrane tubes needed, for example, in the “straightthrough” arrangement for a given desalination problem and the average salt concentration of the permeate thereof can be obtained easily from such design equations. The principal utility of such equations is in the rapid preliminary process design calculations at the cost of a limited loss of accuracy. The extent of loss of accuracy has been determined by carrying out a complete numerical iteration procedure such as that of McCutchan and Goel (1974). By means of such explicit flux expressions of Rao and Sirkar (1978), a considerably simplified numerical procedure having the accuracy of McCutchan and Goel (1974) has also been developed. Although these design methodologies are based on a desalination example for which reasonable data are available for comparison, they can be utilized in practical tubular reverse osmosis process design for the removal of small solutes from dilute solutions in general.
A n Approximate Design Equation The tubular reverse osmosis module under consideration is basically a number of tubular membranes of fixed length 1 and diameter 2R connected in series through suitable fittings (Figure 1). Feed brine of molar salt concentration Czlf enters the first membrane tube with an average velocity of Oxf and at a gauge pressure APfi The total number
117
F
ox, BRINE FEE
t
V P cp “ E T WT FREE PWE4TE I
Figure 1. Tubular straight-through reverse osmosis module-schematic for use with eq 11and 14.
n of membrane tubes required in the tubular module to generate a certain production rate of desalinized water is unknown. The average velocity oZLwith which the concentrated brine leaves the nth tube after traversing a total membrane length L (= nl) is known since the fractional water recovery is usually specified. Bigger plant capacities are usually achieved by having a number of such tubular modules in parallel. Here we are concerned with only a single tubular module with a “straight-through” arrangement or a number of such modules connected in series such that the brine flow cross sectional area is unchanged with the tube number. The following assumptions are introduced for analyzing such an arrangement: (1)Total molar density C of any brine or permeate is constant along the whole module. (2) The solute leakage through the membrane to the permeate may be neglected for obtaining a solute balance between any two cross sections of the module perpendicular to the mean flow since reverse osmosis membranes as used have high solute rejection. (3) Turbulent flow conditions exist and the explicit flux expressions for solution-diffusiontype membranes and other associated relations obtained by Rao and Sirkar (1978) for tubular RO desalination are valid. (4) Neglect entrance effects in any membrane tube. ( 5 ) Perfect mixing conditions exist in the fittings between consecutive membrane tubes. (6) The assembly of membrane tubes connected in series is such that the bulk solute concentration Czl at any location is continuously varying with distance 3e along any membrane tube and therefore along the whole assembly due to assumption (5). Further, the wall solute concentration CZ2on the high pressure side at any location is related to Czl a t that location through an equation (describing concentration polarization) given by Brian (1966) or Sourirajan (1970) and utilized in arriving at explicit flux expressions by Rao and Sirkar (1978). (7) Neglect the pressure drop due to flow along tubes and through fittings. (8) Assume the tube-side solute mass transfer coefficient to remain unchanged along the module length due to any change in the bulk velocity of high pressure feed in the tube as a result of solvent permeation through the membrane. (9) Assume that the intrinsic membrane properties remain unchanged from tube to tube. Assumption 1 is routine in reverse osmosis of dilute solutions (see Sourirajan, 1970) and leads to cxZ1f= CZU; CXZlL = c21L; CXZZ = c2Z; cxZ3-= C23; CXzl = C21; CV1 z 1 (1) Assumption 2 is quite realistic since practical reverse osmosis is usually carried out with membranes having a high solute rejection. For example, consider recovering 219 cm3/s (5OOO U.S. gal/day) of desalinized water containing 400 ppm of salt from 731 cm3/s (16700 gal/day) of seawater as feed (35 OOO ppm of salt) (McCutchan and Goel, 1974). By assumption 2, a salt balance between the inlet and exit of the tubular assembly would be rR20,&’21f = rR2Ox~CZ1~ (la) In actuality, relation l a should be
118
Ind. Eng. Chem. Process Des. Dev., Vol. 20, No. 1, 1981 rR2ijxfC21f
= rR2ijxL(C21L)actud
Thus the percent error in replacing
+ VpCp
(C21~)actudby
Ob)
CZILis
which is negligible. If instead of seawater as feed a brackish water with 5OOO ppm of salt is used as a feed with, say, a 95% rejection membrane so that the permeate concentration Cp corresponds to a brine having 250 ppm salt, the percentage error increases to only 1.5%. Thus assumption 2 is quite realistic. In fact, McCutehan and Goel (1974) also used this assumption: see their eq 20. The error will, however, increase with increasing fractional water recovery. The solvent and solute fluxes through the membrane for real flow conditions in the tube (i.e., with concentration polarization) may be expressed for a solution-diffusion type of model as N1, = A ( M - bCX22 bCX23,) (2)
membranes in practical RO is then eq 6 for solvent flux. We will consider the corresponding problem of determining the solute flux in high rejection reverse osmosis somewhat later. We now turn to assumptions 4, 5, and 6. These imply that Xll, X21,and N,, are continuous functions of axial distance x from module inlet with eq 6 being valid everywhere. If we make a water balance over a differential length dx of any membrane tube, we obtain (7) Further, assumption 2 is also equivalent to nR2D,fC21f
= rR2ij,C21
(8)
Noting that C = CI1+ CZlat any location, we obtain from relations 1, 6, 8, and eq 7
+
N2r
= (DPM/K6)(CX22 - cx23r)
(3)
See, for example, Lonsdale et al. (1965), Sourirajan (1970), and McCutchan and Goel (1974). The concentration polarization relationship for a film-type model in turbulent flow in a membrane tube as developed by Brian (1966) may be expressed in Sourirajan (1970) nomenclature and formalism as
so that L
n l = L = i dx=
2
Rao and Sirkar (1978) have shown that for tubular reverse osmosis with turbulent flow, this expression may be simplified to
They have further obtained the following expression for Nl, by substituting eq 4a in eq 2
Here the constant b appears since the osmotic pressure r(C2j) of a solution of molar salt concentration at location j is expressed as T(C,j)
=
bC2j = bCX2j
(5a)
For practical reverse osmosis, membranes have a high rejection of solute so that, in general, XBr
