- Email: [email protected]
Optimized design of a reverse osmosis system with a recycle
Desalination 230 (2008) 128–139
Optimized design of a reverse osmosis system with a recycle Payel Sarkar*, D. Goswami, S. Prabhakar, P.K. Tewari Desalination Division, BARC, Mumbai 400085, India Tel. +91 (22) 2559-4736; Fax: +91 (22) 2550-5151; email: [email protected] Received 22 January 2007; Accepted 15 November 2007
Abstract Small scale brackish water desalination units are used in remote areas and their sustenance depend on the twin factors of consistency of product water quality and availability of raw water resources. A mathematical model has been developed based on the input parameters of feed salinity, basic membrane characteristics, feed temperature, desired product quality and quantity to predict the operating recycle ratio at constant operating parameters. This ensures the consistent quality of product water at the same time minimizing the raw water consumption. The algorithm used include calculation of recovery, determination of membrane configuration and operating pressure within the guidelines of manufacturer’s specified hydrodynamic parameters. The model has a range with respect to reference value of design. The model can estimate the feed quality variation +10% to !70% and feed temperature variation ±5EC based on the design reference value. Keywords: Reverse osmosis; Brackish water; Resource conservation; Reject recycle
1. Introduction Providing safe drinking water has been the major objective in this UN decade of drinking water. Brackish water desalination has a critical role to play in achieving the target, particularly in water scarce remote areas on the Indian subcontinent. The technological constraints include rapid depletion of ground water levels, increase in salinity etc., particularly in drought prone areas. *Corresponding author.
Our experience of installing and operation of brackish water reverse osmosis plants (BWRO) has indicated variations in feed salinity and temperature over the year, besides depleting yield of the sources. The acceptability of the water by the population requires consistent maintenance of the product water quality. Normally the design of reverse osmosis systems are carried out with the conservative analysis of water using higher feed salinity. When the reverse osmosis plant is operated under reduced salinity without change in the operating pressure, the hydrodynamics is not
0011-9164/08/$– See front matter © 2008 Published by Elsevier B.V. doi:10.1016/j.desal.2007.11.021
P. Sarkar / Desalination 230 (2008) 128–139
favorable. All these features call for a design and operational strategy by which the resources could be conserved to the maximum extent possible besides maintaining a consistent level of product quality to ensure acceptability by the local people. In this paper a mathematical model has been proposed involving partial recycle of reject stream and considers all the parameters such astemperature, pressure, feed salinity, recovery relevant to the performance of BWRO system. On the basis of this mathematical model, the design methodology of BWRO system is presented to provide consistent quality and quantity of product water in spite of variation of feed quality and temperature.
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an optimally design RO plant due to seasonal variations of feed quality and temperature lead to significant fluctuations in product quality and recovery ratio. The former becomes a critical factor for acceptability and the latter affects the scarce ground water potential. This calls for a flexible design concept wherein the quality is maintained reasonably constant and the recovery is matched with feed salinity. The proposed mathematical model incorporates the concept of optimization of recycle ratio keeping the operating pressure constant such that the feed flow and feed quality to membrane remains always constant. This concept maintains the hydrodynamics and module recovery (RM) nearly the same. Hence it is reasonable to presume that concentration polarization is constant and hence it does not figure explicitly in the model.
2. Design approach Small brackish water reverse osmosis plants are normally located to provide potable water to a limited population. Off design performance of
2.1. Description of methodology The schematic of reverse osmosis process with recycle provision is shown in Fig. 1. The process
Fig. 1. Schematic diagram of reverse osmosis system with a reject recycle. Ne, total no. of elements; y, no. of elements in series in one pressure vessel.
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P. Sarkar / Desalination 230 (2008) 128–139
envisaged pumping of feed (Qf) at concentration Cf and sent to a series of elements. The number of elements has been chosen to achieve design recovery. The design recovery and design recycle ratio are to be selected from mathematical model taking care of the scaling potential of feed, maximum permissible pressure limit and required product quality. The design needs input data with respect to feed analysis, permeate flow, and permeate quality and the membrane characteristics. Initially the configuration of membrane and operating pressure are obtained without recycle, then the recycle is introduced into the system using the same membrane area but adjusting the operating pressure subject to the satisfaction of hydrodynamic parameters and scaling potential barrier. The design provides operating pressure, recovery and membrane configuration for the reverse osmosis brackish water system. For the sake of design we consider two recoveries: module recovery and system (absolute) recovery. Module recovery is defined as the ratio of permeate flow to the feed flow to the RO module while the system recovery is defined as the ratio of the permeate flow to the flow to the system from the source. Module recovery remains constant whenever any changes occur in the system but the system recovery is a function of recycle flow. When there is a variation of feed quality (Cf) and feed temperature, the model proposes an optimized recycle ratio to get consistent quality and quantity of product water, with all the other parameters remaining constant. 2.2. Basic features of model The basic membrane characteristics (Aw, Bs) and the constraints in membrane hydrodynamics, i.e. maximum feed flow rate, maximum permeate flow rate, and minimum reject flow rate per element are taken from standard literature data [1] or calculatd based on a few experimental data for
other membranes [2]. The scaling potential is assumed to be only due to calcium sulphate, carbonate, fluoride, barium sulphate, strontium sulphate and silica. The design code is developed as follows. A. Equations involved in SCALING function: 1. Determination of ionic strength [1] C Feed:
I f = 1/ 2∑ ( mi × Z i 2 )
(1)
C Concentrate: Ic =
1 1/ 2∑ ( mi × Z i 2 ) 1 − Rd
(2)
2. Determination of solubility product [1]
KspCaSO 4 = 0.0014 I c + 0.0002 KspC
BaSO 4
= 0.7 × 10−7 I c + 0.2 × 10−9
(3) (4)
KspCaF2 = 0.6 × 10−10 log I c + 0.3 × 10−9
(5)
KspSrSO 4 = 0.4 × 10−5 log I c + 0.2 × 10−4
(6)
1 sSiO2 f × pH Corrected × Tcorrected 1 − Rd
(7)
sSiO2 c =
3. Determination of ionic product [1]:
IpCaSO 4 = IpBaSO 4 =
1
(1 − Rd )
2
CCa × CSO4
(8)
2
CBa × CSO4
(9)
1
(1 − Rd )
P. Sarkar / Desalination 230 (2008) 128–139
IpCaF2 =
1
(1 − Rd )
IpSrSO 4 =
sSiO2 c =
CCa × CF2 2
(10)
CSr × CSO4 2
(11)
3
1
(1 − Rd )
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2. Dependence of diffusivity coefficient on temperature — D(T) is approximated by the Wilke–Chang equation [5]: 1
1 sSio × pH Corrected × Tcorrected (12) 1− R d 2 f
D ( T ) = 7.410 × 10
−8
( φM ) 2 T μV
A
0.6
(19)
Diffusivity at the reference temperature (Dto) gives [3]
T μ to × T0 μ t
4. Determination of LSI (for feed TDS <10,000 ppm) and S and DSI (for feed TDS >10,000 ppm) [1]:
D ( T ) = Dt 0 ×
lsic = pH c − pH s
(13)
pH s = pCa + palk + U
(14)
This temperature dependency of the diffusivity coefficient is accommodated in solute the permeability constant.
S & DSI = pH c − pH s
(15)
pH s = pCa + palk + V
(16)
pCa = −0.4351 × ln ( CCa ) + 4.9948
(16a)
palk = −0.4351 × ln ( C f ) + 4.9948
(16b)
B. Equations involving incorporation of the temperature effect: 1. Dependence of viscosity on temperature — the dependence of viscosity on temperature μ(T) is approximated by the Guzman–Andrade equation [3]:
⎛b⎞ μ(T ) = α exp ⎜ ⎟ ⎝T ⎠
(17)
This α and b values are calculated from [4]. This temperature dependency of viscosity is accommodated in solvent permeability constant.
Aw (T ) = AWT 0
μ ( To ) μ (T )
(18)
Bs ( T ) = Bst 0 ×
(20)
T μ to × T0 μ t
(21)
3. Dependence of osmotic pressure on temperature [3]:
π( c, T ) = icRT
(22)
C. Equations involving evaluation of the configuration: Assumption: (a) Each pressure vessel can contain maximum six elements. (b) Recovery of a single element is constant from element to element connected in series. (c) Membrane characteristics remain constant for a reasonable period. 1. Optimization of flux w.r.t. required permeate quality (Cpr) [6]
⎛ C J opt = Bs ( T ) × CWP ⎜ 1 − f ⎜ C pr ⎝
⎞ ⎟⎟ ⎠
(23)
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P. Sarkar / Desalination 230 (2008) 128–139
Permeate concentration is equal to material balance around the membrane [6]
CPr × J opt = Cwp × J S (T )
where Jopt gives [Cp!CPr] <5 will be the operating flux. 2. Total no. of elements required:
QPr J opt × Sa
(24)
3. No. of elements in series to achieve the design recovery:
ys =
Rd re
(24a)
N P [ j − 1] × Qre Q f max
(25)
Given Np [0] =1.0, when j = 1.0, Qre = Qf. D. Equation evaluating operating pressure:
Pf =
Q f × Rd
N e × Sa × AW ( T )
and
(28b)
y
∑Q
px
(29)
Flow balance equations [6]:
Q f = Q p + (1 − a ) Qr
(30)
Q fm = Q f + aQr
(31)
C Element mass balance equation:
Qr ( x −1)Cr ( x −1) = QrxCrx + Q pxC px
(32)
Qr ( x −1) = Qrx + Q px
(33)
J sx (T ) = Bs (T ) ⎡⎣Cr ( x −1) − C px ⎤⎦
(34)
{
J wx (T ) = Aw (T ) ⎡⎣ Pf − ( x − 1) × 0.5⎤⎦ −π ( T )r ( x −1) + π (T ) px C Element permeate equations:
flow
} mass
(35)
balance
C px × J wx ( T ) = J sx (T ) × Cwp
(36)
Q px (T ) = J wx (T ) × N e × Sa
(37)
y
x =1
× C px
(27)
where
Q p = ∑ Q px
px
Q fmC fm = Q f C f + aQr Cr
(26)
E. Evaluation of basic model parameters: C Mass balance equations [6]
Q f C f = Q p C p + (1 − a )Qr Cr
x =1
C Element flow balance equations:
4. Total no. of pressure vessels in array:
NP =
Cp =
∑Q
x =1
J s (T ) = Bs ( T ) ⎡⎣C f − C pr ⎤⎦
Ne =
y
(28a)
Rd =
Q pr Qf
(38)
P. Sarkar / Desalination 230 (2008) 128–139
2.3. Design algorithm of reverse osmosis system with a reject recycle (Fig. 2) Steps: 1. Assume recycle ratio, initially a = 0.0, recovery Rd, provided user’s input Qpr, Cpr, Cf with ionic analysis. 2. Find Qf according to Eq. (38). 3. Calculate Cfm according to Eqs. (29) and (31). 4. Call function SCALING. 5. Check Ip/Ksp of CaSO4, BaSO4, CaF2, SrSO4, SiO2 and LSI or S and DSI <1. C If Step 5 is YES and a =0.0, follow steps 6–13 for evaluating configuration. C If Step 5 is YES and a … 0.0, follow steps 25–26. C If Step 5 is NO for a = 0.0, Rd should be decreased by 5% and go back to step 2. C If Step 5 is NO for a … 0.0, further recirculation is not possible. 6. Calculate Qfm from Eq. (31). 7. Provided designer’s input (membrane, membrane area, flux range, membrane characteristics, Aw, Bs, membrane hydrodynamics, i.e. max. feed flow, max permeate flow, min reject flow per element), optimization of element flux with respect to Cpr [Eq. (23)]. 8. Obtain no. of elements required [Eq. (24)]. 9. Obtain no. of elements in series [Eq. (24a). 10. Obtain no. of pressure vessels in first array [Eq. (25)]. 11. Obtain no.of elements in each pr. vessel (putting restriction to a maximum of six elements that can present in each pr. vessel). 12. Obtain no. of pressure vessels in arrays [Eq. (25)]. 13. Calculate Pf [Eq. (26)]. Evaluation of basic model parameters, steps 14– 19: 14. Calculate Cpx for each element [Eq. (36)]. 15. Calculate Qpx for each element [Eq. (37)]. 16. Calculate Qp [Eq. (28a)]. 17. Calculate Cp [Eq. (28b)].
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18. Calculate Qr [Eq. (30)]. 19. Calculate Cr [Eq. (27)]. 20. Check |Cp!Cpr| <20 ppm and |Qp!Qpr| <0.05 m3/h. If Step 20 is YES for both a = 0.0 and a … 0.0, follow steps 21–26 to incorporate recycle ratio for a = 0.0 and to increase recycle ratio for a … 0.0, If Step 20 is NO for a = 0.0, Rd should be decreased by 5% and go back to step 2. If Step 20 is NO for a … 0.0, follow steps 25 and 26. 21. Check that hydrodynamics are satisfied (max. permeate flow of first element, min. reject flow of last element in a pr. vessel.) 22. Check Pf is in permissible limits. If Steps 21 and 22 are YES, then follow step 23. If Steps 21 and 22 are NO, then design parameter will be the parameters obtained by taking previous recycle ratio (a) value. 23. Increase recycle ratio by 10%. 24. Go back to step 3 and follow to step 5. 25. Increase pressure Pf by 0.1 bar. 26. Follow steps 14–20 with new Jwx(T) until design parameters are reached (with changing pressure Jwx(T) value will also change). 2.4. Design algorithm of optimization of reject recycle ratio (Fig. 3) Steps: 1. Obtain inputs constant inputs (designer, design code), variable inputs (T, Cf). 2. Initialize with design recycle ratio. 3. Calculate Qf, Cfm [Eqs. (31) and (29)]. 4. Call function SCALING. 5. Check Ip/Ksp of CaSO4, BaSO4, CaF2, SrSO4, SiO2 and LSI or S and DSI <1. If Step 5 is YES, follow steps 6–10. If step 5 is NO, variation in feed or temperature could not be accommodated. Further recycling is not possible. 6. Calculate Cpx [Eq. (36)]
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P. Sarkar / Desalination 230 (2008) 128–139
Fig. 2. Basic inputs and outputs of model.
Fig. 3. Model with variable inputs.
7. Calculate Qpx [Eq. (37)]. 8. Calculate Cp [Eq. (28)]. 9. Calculate Qp [Eq. (28)]. 10. Calculate Cr [Eq. (27)]. 11. Calculate Qr [Eq. (30)]. 12. Check the following parameters: a. |Qp!Qpd| <0.05 m3/h. b. |Cfm!Cfd| <100. c. |Cp!Cpd| <20. d. Membrane hydrodynamics (assume first element permeate flow, Qp[x]) is satisfied. If all steps of 12 are YES, then the system is steady, and present parameters are the design parameters. If any steps of 12 are NO, follow steps 13 and 14.
If 12a and 12b are YES and 12c or 12d are NO, then come out from the program and take the parameters at previous a value. 13. If the system attains steady state by taking the previous a value, the following steps should be followed: a. If CfTd, increase recycle ratio by 10% and go back to step 3. b. If Cf >Cfd and/or T
P. Sarkar / Desalination 230 (2008) 128–139 Table 1 Water Analysis considered for design Ions
Low brackish
High brackish
NH4 K Na Mg Ca Sr Ba CO3 HCO3 NO3 Cl F SO4 SiO2 CO2 TDS pH
0.1 14.0 1375.1 106.0 265.0 10.0 0.03 2.85 350.0 23.0 2300.0 0.4 540.0 4.1 8.04 4990.57 7.6
0.0 0.0 3284.01 283.0 757.67 15.0 0.0 0.41 163.0 40.0 6620.0 1.0 680 18.0 18.88 11862.09 6.8
analysis data, required product quantity and required product quality data to give the design pressure, design recovery, design recycle ratio and design configuration, the following user’s inputs are given to start the model with. For design demonstration purpose, BW-30-4040 Film Tech membrane elements are chosen and the membrane characteristiscs data and hydrodynamics data all are taken from the manufacturer’s guideline [1]. The typical analysis considered for this purpose is given in Table 1. 2.6. Reference design of a brackish water reverse osmosis system with a recycle Case study: design criteria — C 30 m3/d capacity plant. C Product quality less than 100 ppm. User input — C Feed analysis data (Table 1) C Feed temperature: 30EC Designer’s input: C Membrane chosen: BW-30-4040
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C Membrane characteristics: 1. Single element recovery: 0.11 2. Membrane area: 7.6 m2 3. Solvent permeability: 0.002 m3/h.bar 4. Solute permeability: 0.00035×10!6 m3/h. ppm C Membrane hydrodynamics: 1. Maximum feed flow: 3.63 m3/h 2. Maximum product flow: 0.41 m3/h 3. Minimum reject flow: 0.91 m3/h 4. Maximum feed pressure: 41.37 bar C C C C C C C C
Design: Feed flow rate to membrane (Qfm): 2.37 m3/h Recycle flow: 0.126 m3/h Permeate flow (Qp): 1.25 m3/h Recovery: 52.7% Operating pressure (Pf): 12.5 bar Permeate concentration (Cp): 95 ppm Reject concentration (Cr): 10766.52 ppm Configuration: single stage (1×6)
3. Effect of variation of feed temperature and feed salinity on reverse osmosis system performance In the field both temperature and feed salinity would vary depending on time and season. It is well documented with increase in feed temperature, product rate increases with deterioration of product quality, and with increase in feed salinity product rate decreases and product quality also deteriorates. Hence the recycle ratio is optimized to yield designed product quality and quantity maintaining the membrane hydrodynamics. Recycle ratio as a function of feed salinity and feed temperature: India is a tropical country. Only during the monsoons does the aquifer get filled up, and this water is utilized throughout the year. In the summer the actual water problem arises when the water level gets depleted and feed salinity increases. Fig. 4b shows the recycle ratio as a function of feed temperature. With an increase in feed
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P. Sarkar / Desalination 230 (2008) 128–139
power cost, labor, maintenance and membrane
Fig. 4a. Recycle ratio decreases with increasing TDS.
Fig. 5. Specific energy consumption vs. feed TDS.
Fig. 4b. Recycle ratio increases with increasing temperature.
temperature reject recycle has to be increased to get a constant product rate. An increase of about 1.5% in the temperature correspondingly increases the recycle ratio. The conclusions are in consonance with the literature data on increase in flux with temperature [7]. Fig 4a indicates that with increase in feed salinity recycle ratio decreases. In the summer, the tem-perature is higher and normally ground water salinity goes up. These two factors neutralize to give nearly the constant product quantity but results in increase product salinity. In order to maintain the product salinity the recycle ratio may have to be optimized. 4. Impact on energy and recovery on optimization of the recycle ratio The ideal design of a RO plant should be to minimise unit water cost, which is controlled by
Fig. 6. Recovery vs. feed TDS.
life, out of which the effect of maintenance and membrane life can be minimized by carefully selecting operating parameters. Since the ground water resources are limited, there is a need to minimize excess extraction. In rural areas, one may have to compromise between the cost of water and reliability of supply. This results in increase of specific energy consumption in rural RO plant. Fig. 5 shows a comparative analysis of specific energy consumption with reject recycle and without reject recycle. It can be observed that with increasing feed salinity specific energy consumption increases with out recycle. On the other hand specific energy consumption remains almost constant with recycle.
P. Sarkar / Desalination 230 (2008) 128–139
Although for low feed salinities reject recycle is not favorable with respect to specific energy consumption, at higher feed salinities the specific energy consumption decreases with the reject recycle. Fig. 6 shows with increasing feed salinity absolute recovery decreases. So, if one wants to maintain the specific energy consumption constant, the absolute recovery is compromised. 5. Conclusions The mathematical model developed can predict the operating recycle ratio, which when followed in practice can yield consistent quality and quantity of product water irrespective of seasonal feed salinity and temperature variation. This paper indicates with respect to reference design feed salinity and temperature data (5000 ppm, 30EC), the model can accommodate the feed quality variation +10% to !70% and feed temperature variation ±5EC .For new reference design data, the model should be reset and it will give a new operating recycle ratio and a new range of accommodation of feed salinity and temperature variations. 6. Symbols a Aw (T)
— —
Aw(T0) b
— —
BS (T)
—
BS (To) c
— —
C
—
Cb
—
Recycle ratio defined as % Temperature dependent solvent permeability, m/h.bar Solvent permeability at 298K Constant of Guzman–Andrade equation Temperature dependent solute permeability, m/h.ppm Solute permeability at 298K Concentration of solute in Vant Hoff equation Solute concentration in boundary layer at a distance y from membrane surface, ppm Bulk concentration, ppm
Cf Cfd Cm Cp Cpd Cpr Cpx
— — — — — — —
Cr Crx
— —
Cr(x!1)
—
Cwp Cfm
— —
CCa CBa CSr CSO4 CF2 D(T)
— — — — — —
Dt0 i Ic If Ip CaSO4 Ip BaSO4 Ip CaF2 Ip SrSO4 j Jopt Js Jsx Jv H2O
— — — — — — — — — — — — —
Jv salt
—
Jwx Ksp BASO4 Ksp CaSO4 Ksp CaF2 Ksp SrSO4
— — — — —
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Feed concentration, ppm Design feed concentration, ppm Membrane concentration, ppm Permeate concentration, ppm Design prdt. conc., ppm Permeate conc. required, ppm xth element permeate concentration, ppm Reject concentration, ppm xth element reject concentration, ppm (x!1)st element reject concentration, ppm Water concentration, ppm Concentration of feed to membrane, ppm Concentration of Ca in feed Concentration of Ba in feed Concentration of Sr in feed Concentration of SO4 in feed Concentration of F2 in feed Temperature dependent diffusivity, m2/s Diffusivity at 298K No. of species Ionic strength of concentrate Ionic strength of feed Ionic strength of CaSO4 Ionic strength of BaSO4 Ionic strength of CaF2 Ionic strength of SrSO4 Array number Optimized permeate flux, m/h Solute flux, m/h Solute flux of xth element, m/h Permeate flux in salt free solution, m/h Permeate flux in saline solution, m/h Permeate flux of xth element, m/h Solubility product, BaSO4 Solubility product, CaSO4 Solubility product, CaF2 Solubility product, SrSO4
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P. Sarkar / Desalination 230 (2008) 128–139
LSI LSIc
— —
M mi Ne NP Pf ΔP pHc pHcorrected pHs
— — — — — — — — —
pHCa pHalk Qf Qfm Qf max
— — — — —
Qp Qpd Qpr Qpx
— — — —
Qr Qre
— —
Qr(x!1)
—
Qrx
—
R
—
Ra Rd
— —
re
—
RM Sa S and DSI SSiO2
— —
Langlier Saturation Index Langlier Saturation Index, concentrate Molecular weight Molecular weight of ith species No. of elements No. of pressure vessels Feed pressure, bar Operating pressure, bar pH of concentrate pH corrected for silica [1] pH at which CaCO3 becomes saturated Negative log of Ca concentration Negative log of alkalinity Feed flow rate, m3/h Feed flow to membrane, m3/h Maximum feed flow to single element Product flow rate, m3/h Design product flow rate, m3/h Product flow required, m3/h Product flow rate of xth element, m3/h Reject flow rate, m3/h Reject flow rate for last element in a pr. vessel, m3/h Reject flow rate of (x!1)th element, m3/h Reject flow rate of xth element, m3/h Universal gas constant, L.atm/g mole.K System (absolute) recovery Assumed recovery/designed recovery Single element recovery (from manufacturer’s guideline) Module recovery Area of membrane element
— —
Stiff and Devis Stability Index Solubility of silica
T — Tcorrected — U — V
—
VA x y
— — —
ys
—
Zi
—
Temperature, K Corrected temperature, K LSI constant dependent on temperature and TDS D and SDI constant dependent on temperature and ionic strength Solute molal volume cc/gmmole Dummy variable Total no. of elements in series in one pressure vessel Total no. of elements in series required to achieve the design recovery Valency of the ith element
Greek α δ μ(T) μ(T0) π(C,T) πm πp πpx πr(x!1) Δπ φ
— Constant of Guzman–Andrade equation — Membrane thickness, mm — Temperature dependent viscosity, cp — Viscosity at 298K — Osmotic pressure, bar — Osmotic pressure at membrane surface, bar — Osmotic pressure of permeate, bar — Osmotic pressure of permeate of xth element, bar — Osmotic pressure of reject of (x!1)th element, bar — Osmotic pressure difference across the membrane, bar — Association factor for solvent
References [1] http://www.dow.com. [2] M. Noronha, V. Mavrov and H. Chemiel, Efficient design and optimization of two stage NF processes by simplified process simulation, Desalination, 145 (2002) 207–215.
P. Sarkar / Desalination 230 (2008) 128–139 [3] S.P. Agashichev, Reverse osmosis at elevated temperatures; influence of temperature on degree of concentration polarization and transmembrane flux, Desalination, 179 (2005) 61–72. [4] R.H. Perry and D.W. Green, eds., Perry’s Chemical Engineer’s Handbook, 6th ed., McGraw-Hill, New York, 1984.
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[5] R.E.Treybal, Mass Transfer Operation, 3rd ed, Mc-Graw Hill, NewYork, 1981. [6] K. Jamal, M.A. Khan and M. Kamil, Mathematical modeling of reverse osmosis systems, Desalination, 160 (2004) 29–42. [5] Z. Amjad, ed., Reverse Osmosis Membrane Technology, Water Chemistry and Industrial Application, Van Nostrand Reinhold, New York, 1993.
Optimized design of a reverse osmosis system with a recycle Payel Sarkar*, D. Goswami, S. Prabhakar, P.K. Tewari Desalination Division, BARC, Mumbai 400085, India Tel. +91 (22) 2559-4736; Fax: +91 (22) 2550-5151; email: [email protected] Received 22 January 2007; Accepted 15 November 2007
Abstract Small scale brackish water desalination units are used in remote areas and their sustenance depend on the twin factors of consistency of product water quality and availability of raw water resources. A mathematical model has been developed based on the input parameters of feed salinity, basic membrane characteristics, feed temperature, desired product quality and quantity to predict the operating recycle ratio at constant operating parameters. This ensures the consistent quality of product water at the same time minimizing the raw water consumption. The algorithm used include calculation of recovery, determination of membrane configuration and operating pressure within the guidelines of manufacturer’s specified hydrodynamic parameters. The model has a range with respect to reference value of design. The model can estimate the feed quality variation +10% to !70% and feed temperature variation ±5EC based on the design reference value. Keywords: Reverse osmosis; Brackish water; Resource conservation; Reject recycle
1. Introduction Providing safe drinking water has been the major objective in this UN decade of drinking water. Brackish water desalination has a critical role to play in achieving the target, particularly in water scarce remote areas on the Indian subcontinent. The technological constraints include rapid depletion of ground water levels, increase in salinity etc., particularly in drought prone areas. *Corresponding author.
Our experience of installing and operation of brackish water reverse osmosis plants (BWRO) has indicated variations in feed salinity and temperature over the year, besides depleting yield of the sources. The acceptability of the water by the population requires consistent maintenance of the product water quality. Normally the design of reverse osmosis systems are carried out with the conservative analysis of water using higher feed salinity. When the reverse osmosis plant is operated under reduced salinity without change in the operating pressure, the hydrodynamics is not
0011-9164/08/$– See front matter © 2008 Published by Elsevier B.V. doi:10.1016/j.desal.2007.11.021
P. Sarkar / Desalination 230 (2008) 128–139
favorable. All these features call for a design and operational strategy by which the resources could be conserved to the maximum extent possible besides maintaining a consistent level of product quality to ensure acceptability by the local people. In this paper a mathematical model has been proposed involving partial recycle of reject stream and considers all the parameters such astemperature, pressure, feed salinity, recovery relevant to the performance of BWRO system. On the basis of this mathematical model, the design methodology of BWRO system is presented to provide consistent quality and quantity of product water in spite of variation of feed quality and temperature.
129
an optimally design RO plant due to seasonal variations of feed quality and temperature lead to significant fluctuations in product quality and recovery ratio. The former becomes a critical factor for acceptability and the latter affects the scarce ground water potential. This calls for a flexible design concept wherein the quality is maintained reasonably constant and the recovery is matched with feed salinity. The proposed mathematical model incorporates the concept of optimization of recycle ratio keeping the operating pressure constant such that the feed flow and feed quality to membrane remains always constant. This concept maintains the hydrodynamics and module recovery (RM) nearly the same. Hence it is reasonable to presume that concentration polarization is constant and hence it does not figure explicitly in the model.
2. Design approach Small brackish water reverse osmosis plants are normally located to provide potable water to a limited population. Off design performance of
2.1. Description of methodology The schematic of reverse osmosis process with recycle provision is shown in Fig. 1. The process
Fig. 1. Schematic diagram of reverse osmosis system with a reject recycle. Ne, total no. of elements; y, no. of elements in series in one pressure vessel.
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P. Sarkar / Desalination 230 (2008) 128–139
envisaged pumping of feed (Qf) at concentration Cf and sent to a series of elements. The number of elements has been chosen to achieve design recovery. The design recovery and design recycle ratio are to be selected from mathematical model taking care of the scaling potential of feed, maximum permissible pressure limit and required product quality. The design needs input data with respect to feed analysis, permeate flow, and permeate quality and the membrane characteristics. Initially the configuration of membrane and operating pressure are obtained without recycle, then the recycle is introduced into the system using the same membrane area but adjusting the operating pressure subject to the satisfaction of hydrodynamic parameters and scaling potential barrier. The design provides operating pressure, recovery and membrane configuration for the reverse osmosis brackish water system. For the sake of design we consider two recoveries: module recovery and system (absolute) recovery. Module recovery is defined as the ratio of permeate flow to the feed flow to the RO module while the system recovery is defined as the ratio of the permeate flow to the flow to the system from the source. Module recovery remains constant whenever any changes occur in the system but the system recovery is a function of recycle flow. When there is a variation of feed quality (Cf) and feed temperature, the model proposes an optimized recycle ratio to get consistent quality and quantity of product water, with all the other parameters remaining constant. 2.2. Basic features of model The basic membrane characteristics (Aw, Bs) and the constraints in membrane hydrodynamics, i.e. maximum feed flow rate, maximum permeate flow rate, and minimum reject flow rate per element are taken from standard literature data [1] or calculatd based on a few experimental data for
other membranes [2]. The scaling potential is assumed to be only due to calcium sulphate, carbonate, fluoride, barium sulphate, strontium sulphate and silica. The design code is developed as follows. A. Equations involved in SCALING function: 1. Determination of ionic strength [1] C Feed:
I f = 1/ 2∑ ( mi × Z i 2 )
(1)
C Concentrate: Ic =
1 1/ 2∑ ( mi × Z i 2 ) 1 − Rd
(2)
2. Determination of solubility product [1]
KspCaSO 4 = 0.0014 I c + 0.0002 KspC
BaSO 4
= 0.7 × 10−7 I c + 0.2 × 10−9
(3) (4)
KspCaF2 = 0.6 × 10−10 log I c + 0.3 × 10−9
(5)
KspSrSO 4 = 0.4 × 10−5 log I c + 0.2 × 10−4
(6)
1 sSiO2 f × pH Corrected × Tcorrected 1 − Rd
(7)
sSiO2 c =
3. Determination of ionic product [1]:
IpCaSO 4 = IpBaSO 4 =
1
(1 − Rd )
2
CCa × CSO4
(8)
2
CBa × CSO4
(9)
1
(1 − Rd )
P. Sarkar / Desalination 230 (2008) 128–139
IpCaF2 =
1
(1 − Rd )
IpSrSO 4 =
sSiO2 c =
CCa × CF2 2
(10)
CSr × CSO4 2
(11)
3
1
(1 − Rd )
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2. Dependence of diffusivity coefficient on temperature — D(T) is approximated by the Wilke–Chang equation [5]: 1
1 sSio × pH Corrected × Tcorrected (12) 1− R d 2 f
D ( T ) = 7.410 × 10
−8
( φM ) 2 T μV
A
0.6
(19)
Diffusivity at the reference temperature (Dto) gives [3]
T μ to × T0 μ t
4. Determination of LSI (for feed TDS <10,000 ppm) and S and DSI (for feed TDS >10,000 ppm) [1]:
D ( T ) = Dt 0 ×
lsic = pH c − pH s
(13)
pH s = pCa + palk + U
(14)
This temperature dependency of the diffusivity coefficient is accommodated in solute the permeability constant.
S & DSI = pH c − pH s
(15)
pH s = pCa + palk + V
(16)
pCa = −0.4351 × ln ( CCa ) + 4.9948
(16a)
palk = −0.4351 × ln ( C f ) + 4.9948
(16b)
B. Equations involving incorporation of the temperature effect: 1. Dependence of viscosity on temperature — the dependence of viscosity on temperature μ(T) is approximated by the Guzman–Andrade equation [3]:
⎛b⎞ μ(T ) = α exp ⎜ ⎟ ⎝T ⎠
(17)
This α and b values are calculated from [4]. This temperature dependency of viscosity is accommodated in solvent permeability constant.
Aw (T ) = AWT 0
μ ( To ) μ (T )
(18)
Bs ( T ) = Bst 0 ×
(20)
T μ to × T0 μ t
(21)
3. Dependence of osmotic pressure on temperature [3]:
π( c, T ) = icRT
(22)
C. Equations involving evaluation of the configuration: Assumption: (a) Each pressure vessel can contain maximum six elements. (b) Recovery of a single element is constant from element to element connected in series. (c) Membrane characteristics remain constant for a reasonable period. 1. Optimization of flux w.r.t. required permeate quality (Cpr) [6]
⎛ C J opt = Bs ( T ) × CWP ⎜ 1 − f ⎜ C pr ⎝
⎞ ⎟⎟ ⎠
(23)
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P. Sarkar / Desalination 230 (2008) 128–139
Permeate concentration is equal to material balance around the membrane [6]
CPr × J opt = Cwp × J S (T )
where Jopt gives [Cp!CPr] <5 will be the operating flux. 2. Total no. of elements required:
QPr J opt × Sa
(24)
3. No. of elements in series to achieve the design recovery:
ys =
Rd re
(24a)
N P [ j − 1] × Qre Q f max
(25)
Given Np [0] =1.0, when j = 1.0, Qre = Qf. D. Equation evaluating operating pressure:
Pf =
Q f × Rd
N e × Sa × AW ( T )
and
(28b)
y
∑Q
px
(29)
Flow balance equations [6]:
Q f = Q p + (1 − a ) Qr
(30)
Q fm = Q f + aQr
(31)
C Element mass balance equation:
Qr ( x −1)Cr ( x −1) = QrxCrx + Q pxC px
(32)
Qr ( x −1) = Qrx + Q px
(33)
J sx (T ) = Bs (T ) ⎡⎣Cr ( x −1) − C px ⎤⎦
(34)
{
J wx (T ) = Aw (T ) ⎡⎣ Pf − ( x − 1) × 0.5⎤⎦ −π ( T )r ( x −1) + π (T ) px C Element permeate equations:
flow
} mass
(35)
balance
C px × J wx ( T ) = J sx (T ) × Cwp
(36)
Q px (T ) = J wx (T ) × N e × Sa
(37)
y
x =1
× C px
(27)
where
Q p = ∑ Q px
px
Q fmC fm = Q f C f + aQr Cr
(26)
E. Evaluation of basic model parameters: C Mass balance equations [6]
Q f C f = Q p C p + (1 − a )Qr Cr
x =1
C Element flow balance equations:
4. Total no. of pressure vessels in array:
NP =
Cp =
∑Q
x =1
J s (T ) = Bs ( T ) ⎡⎣C f − C pr ⎤⎦
Ne =
y
(28a)
Rd =
Q pr Qf
(38)
P. Sarkar / Desalination 230 (2008) 128–139
2.3. Design algorithm of reverse osmosis system with a reject recycle (Fig. 2) Steps: 1. Assume recycle ratio, initially a = 0.0, recovery Rd, provided user’s input Qpr, Cpr, Cf with ionic analysis. 2. Find Qf according to Eq. (38). 3. Calculate Cfm according to Eqs. (29) and (31). 4. Call function SCALING. 5. Check Ip/Ksp of CaSO4, BaSO4, CaF2, SrSO4, SiO2 and LSI or S and DSI <1. C If Step 5 is YES and a =0.0, follow steps 6–13 for evaluating configuration. C If Step 5 is YES and a … 0.0, follow steps 25–26. C If Step 5 is NO for a = 0.0, Rd should be decreased by 5% and go back to step 2. C If Step 5 is NO for a … 0.0, further recirculation is not possible. 6. Calculate Qfm from Eq. (31). 7. Provided designer’s input (membrane, membrane area, flux range, membrane characteristics, Aw, Bs, membrane hydrodynamics, i.e. max. feed flow, max permeate flow, min reject flow per element), optimization of element flux with respect to Cpr [Eq. (23)]. 8. Obtain no. of elements required [Eq. (24)]. 9. Obtain no. of elements in series [Eq. (24a). 10. Obtain no. of pressure vessels in first array [Eq. (25)]. 11. Obtain no.of elements in each pr. vessel (putting restriction to a maximum of six elements that can present in each pr. vessel). 12. Obtain no. of pressure vessels in arrays [Eq. (25)]. 13. Calculate Pf [Eq. (26)]. Evaluation of basic model parameters, steps 14– 19: 14. Calculate Cpx for each element [Eq. (36)]. 15. Calculate Qpx for each element [Eq. (37)]. 16. Calculate Qp [Eq. (28a)]. 17. Calculate Cp [Eq. (28b)].
133
18. Calculate Qr [Eq. (30)]. 19. Calculate Cr [Eq. (27)]. 20. Check |Cp!Cpr| <20 ppm and |Qp!Qpr| <0.05 m3/h. If Step 20 is YES for both a = 0.0 and a … 0.0, follow steps 21–26 to incorporate recycle ratio for a = 0.0 and to increase recycle ratio for a … 0.0, If Step 20 is NO for a = 0.0, Rd should be decreased by 5% and go back to step 2. If Step 20 is NO for a … 0.0, follow steps 25 and 26. 21. Check that hydrodynamics are satisfied (max. permeate flow of first element, min. reject flow of last element in a pr. vessel.) 22. Check Pf is in permissible limits. If Steps 21 and 22 are YES, then follow step 23. If Steps 21 and 22 are NO, then design parameter will be the parameters obtained by taking previous recycle ratio (a) value. 23. Increase recycle ratio by 10%. 24. Go back to step 3 and follow to step 5. 25. Increase pressure Pf by 0.1 bar. 26. Follow steps 14–20 with new Jwx(T) until design parameters are reached (with changing pressure Jwx(T) value will also change). 2.4. Design algorithm of optimization of reject recycle ratio (Fig. 3) Steps: 1. Obtain inputs constant inputs (designer, design code), variable inputs (T, Cf). 2. Initialize with design recycle ratio. 3. Calculate Qf, Cfm [Eqs. (31) and (29)]. 4. Call function SCALING. 5. Check Ip/Ksp of CaSO4, BaSO4, CaF2, SrSO4, SiO2 and LSI or S and DSI <1. If Step 5 is YES, follow steps 6–10. If step 5 is NO, variation in feed or temperature could not be accommodated. Further recycling is not possible. 6. Calculate Cpx [Eq. (36)]
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P. Sarkar / Desalination 230 (2008) 128–139
Fig. 2. Basic inputs and outputs of model.
Fig. 3. Model with variable inputs.
7. Calculate Qpx [Eq. (37)]. 8. Calculate Cp [Eq. (28)]. 9. Calculate Qp [Eq. (28)]. 10. Calculate Cr [Eq. (27)]. 11. Calculate Qr [Eq. (30)]. 12. Check the following parameters: a. |Qp!Qpd| <0.05 m3/h. b. |Cfm!Cfd| <100. c. |Cp!Cpd| <20. d. Membrane hydrodynamics (assume first element permeate flow, Qp[x]) is satisfied. If all steps of 12 are YES, then the system is steady, and present parameters are the design parameters. If any steps of 12 are NO, follow steps 13 and 14.
If 12a and 12b are YES and 12c or 12d are NO, then come out from the program and take the parameters at previous a value. 13. If the system attains steady state by taking the previous a value, the following steps should be followed: a. If Cf
P. Sarkar / Desalination 230 (2008) 128–139 Table 1 Water Analysis considered for design Ions
Low brackish
High brackish
NH4 K Na Mg Ca Sr Ba CO3 HCO3 NO3 Cl F SO4 SiO2 CO2 TDS pH
0.1 14.0 1375.1 106.0 265.0 10.0 0.03 2.85 350.0 23.0 2300.0 0.4 540.0 4.1 8.04 4990.57 7.6
0.0 0.0 3284.01 283.0 757.67 15.0 0.0 0.41 163.0 40.0 6620.0 1.0 680 18.0 18.88 11862.09 6.8
analysis data, required product quantity and required product quality data to give the design pressure, design recovery, design recycle ratio and design configuration, the following user’s inputs are given to start the model with. For design demonstration purpose, BW-30-4040 Film Tech membrane elements are chosen and the membrane characteristiscs data and hydrodynamics data all are taken from the manufacturer’s guideline [1]. The typical analysis considered for this purpose is given in Table 1. 2.6. Reference design of a brackish water reverse osmosis system with a recycle Case study: design criteria — C 30 m3/d capacity plant. C Product quality less than 100 ppm. User input — C Feed analysis data (Table 1) C Feed temperature: 30EC Designer’s input: C Membrane chosen: BW-30-4040
135
C Membrane characteristics: 1. Single element recovery: 0.11 2. Membrane area: 7.6 m2 3. Solvent permeability: 0.002 m3/h.bar 4. Solute permeability: 0.00035×10!6 m3/h. ppm C Membrane hydrodynamics: 1. Maximum feed flow: 3.63 m3/h 2. Maximum product flow: 0.41 m3/h 3. Minimum reject flow: 0.91 m3/h 4. Maximum feed pressure: 41.37 bar C C C C C C C C
Design: Feed flow rate to membrane (Qfm): 2.37 m3/h Recycle flow: 0.126 m3/h Permeate flow (Qp): 1.25 m3/h Recovery: 52.7% Operating pressure (Pf): 12.5 bar Permeate concentration (Cp): 95 ppm Reject concentration (Cr): 10766.52 ppm Configuration: single stage (1×6)
3. Effect of variation of feed temperature and feed salinity on reverse osmosis system performance In the field both temperature and feed salinity would vary depending on time and season. It is well documented with increase in feed temperature, product rate increases with deterioration of product quality, and with increase in feed salinity product rate decreases and product quality also deteriorates. Hence the recycle ratio is optimized to yield designed product quality and quantity maintaining the membrane hydrodynamics. Recycle ratio as a function of feed salinity and feed temperature: India is a tropical country. Only during the monsoons does the aquifer get filled up, and this water is utilized throughout the year. In the summer the actual water problem arises when the water level gets depleted and feed salinity increases. Fig. 4b shows the recycle ratio as a function of feed temperature. With an increase in feed
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P. Sarkar / Desalination 230 (2008) 128–139
power cost, labor, maintenance and membrane
Fig. 4a. Recycle ratio decreases with increasing TDS.
Fig. 5. Specific energy consumption vs. feed TDS.
Fig. 4b. Recycle ratio increases with increasing temperature.
temperature reject recycle has to be increased to get a constant product rate. An increase of about 1.5% in the temperature correspondingly increases the recycle ratio. The conclusions are in consonance with the literature data on increase in flux with temperature [7]. Fig 4a indicates that with increase in feed salinity recycle ratio decreases. In the summer, the tem-perature is higher and normally ground water salinity goes up. These two factors neutralize to give nearly the constant product quantity but results in increase product salinity. In order to maintain the product salinity the recycle ratio may have to be optimized. 4. Impact on energy and recovery on optimization of the recycle ratio The ideal design of a RO plant should be to minimise unit water cost, which is controlled by
Fig. 6. Recovery vs. feed TDS.
life, out of which the effect of maintenance and membrane life can be minimized by carefully selecting operating parameters. Since the ground water resources are limited, there is a need to minimize excess extraction. In rural areas, one may have to compromise between the cost of water and reliability of supply. This results in increase of specific energy consumption in rural RO plant. Fig. 5 shows a comparative analysis of specific energy consumption with reject recycle and without reject recycle. It can be observed that with increasing feed salinity specific energy consumption increases with out recycle. On the other hand specific energy consumption remains almost constant with recycle.
P. Sarkar / Desalination 230 (2008) 128–139
Although for low feed salinities reject recycle is not favorable with respect to specific energy consumption, at higher feed salinities the specific energy consumption decreases with the reject recycle. Fig. 6 shows with increasing feed salinity absolute recovery decreases. So, if one wants to maintain the specific energy consumption constant, the absolute recovery is compromised. 5. Conclusions The mathematical model developed can predict the operating recycle ratio, which when followed in practice can yield consistent quality and quantity of product water irrespective of seasonal feed salinity and temperature variation. This paper indicates with respect to reference design feed salinity and temperature data (5000 ppm, 30EC), the model can accommodate the feed quality variation +10% to !70% and feed temperature variation ±5EC .For new reference design data, the model should be reset and it will give a new operating recycle ratio and a new range of accommodation of feed salinity and temperature variations. 6. Symbols a Aw (T)
— —
Aw(T0) b
— —
BS (T)
—
BS (To) c
— —
C
—
Cb
—
Recycle ratio defined as % Temperature dependent solvent permeability, m/h.bar Solvent permeability at 298K Constant of Guzman–Andrade equation Temperature dependent solute permeability, m/h.ppm Solute permeability at 298K Concentration of solute in Vant Hoff equation Solute concentration in boundary layer at a distance y from membrane surface, ppm Bulk concentration, ppm
Cf Cfd Cm Cp Cpd Cpr Cpx
— — — — — — —
Cr Crx
— —
Cr(x!1)
—
Cwp Cfm
— —
CCa CBa CSr CSO4 CF2 D(T)
— — — — — —
Dt0 i Ic If Ip CaSO4 Ip BaSO4 Ip CaF2 Ip SrSO4 j Jopt Js Jsx Jv H2O
— — — — — — — — — — — — —
Jv salt
—
Jwx Ksp BASO4 Ksp CaSO4 Ksp CaF2 Ksp SrSO4
— — — — —
137
Feed concentration, ppm Design feed concentration, ppm Membrane concentration, ppm Permeate concentration, ppm Design prdt. conc., ppm Permeate conc. required, ppm xth element permeate concentration, ppm Reject concentration, ppm xth element reject concentration, ppm (x!1)st element reject concentration, ppm Water concentration, ppm Concentration of feed to membrane, ppm Concentration of Ca in feed Concentration of Ba in feed Concentration of Sr in feed Concentration of SO4 in feed Concentration of F2 in feed Temperature dependent diffusivity, m2/s Diffusivity at 298K No. of species Ionic strength of concentrate Ionic strength of feed Ionic strength of CaSO4 Ionic strength of BaSO4 Ionic strength of CaF2 Ionic strength of SrSO4 Array number Optimized permeate flux, m/h Solute flux, m/h Solute flux of xth element, m/h Permeate flux in salt free solution, m/h Permeate flux in saline solution, m/h Permeate flux of xth element, m/h Solubility product, BaSO4 Solubility product, CaSO4 Solubility product, CaF2 Solubility product, SrSO4
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P. Sarkar / Desalination 230 (2008) 128–139
LSI LSIc
— —
M mi Ne NP Pf ΔP pHc pHcorrected pHs
— — — — — — — — —
pHCa pHalk Qf Qfm Qf max
— — — — —
Qp Qpd Qpr Qpx
— — — —
Qr Qre
— —
Qr(x!1)
—
Qrx
—
R
—
Ra Rd
— —
re
—
RM Sa S and DSI SSiO2
— —
Langlier Saturation Index Langlier Saturation Index, concentrate Molecular weight Molecular weight of ith species No. of elements No. of pressure vessels Feed pressure, bar Operating pressure, bar pH of concentrate pH corrected for silica [1] pH at which CaCO3 becomes saturated Negative log of Ca concentration Negative log of alkalinity Feed flow rate, m3/h Feed flow to membrane, m3/h Maximum feed flow to single element Product flow rate, m3/h Design product flow rate, m3/h Product flow required, m3/h Product flow rate of xth element, m3/h Reject flow rate, m3/h Reject flow rate for last element in a pr. vessel, m3/h Reject flow rate of (x!1)th element, m3/h Reject flow rate of xth element, m3/h Universal gas constant, L.atm/g mole.K System (absolute) recovery Assumed recovery/designed recovery Single element recovery (from manufacturer’s guideline) Module recovery Area of membrane element
— —
Stiff and Devis Stability Index Solubility of silica
T — Tcorrected — U — V
—
VA x y
— — —
ys
—
Zi
—
Temperature, K Corrected temperature, K LSI constant dependent on temperature and TDS D and SDI constant dependent on temperature and ionic strength Solute molal volume cc/gmmole Dummy variable Total no. of elements in series in one pressure vessel Total no. of elements in series required to achieve the design recovery Valency of the ith element
Greek α δ μ(T) μ(T0) π(C,T) πm πp πpx πr(x!1) Δπ φ
— Constant of Guzman–Andrade equation — Membrane thickness, mm — Temperature dependent viscosity, cp — Viscosity at 298K — Osmotic pressure, bar — Osmotic pressure at membrane surface, bar — Osmotic pressure of permeate, bar — Osmotic pressure of permeate of xth element, bar — Osmotic pressure of reject of (x!1)th element, bar — Osmotic pressure difference across the membrane, bar — Association factor for solvent
References [1] http://www.dow.com. [2] M. Noronha, V. Mavrov and H. Chemiel, Efficient design and optimization of two stage NF processes by simplified process simulation, Desalination, 145 (2002) 207–215.
P. Sarkar / Desalination 230 (2008) 128–139 [3] S.P. Agashichev, Reverse osmosis at elevated temperatures; influence of temperature on degree of concentration polarization and transmembrane flux, Desalination, 179 (2005) 61–72. [4] R.H. Perry and D.W. Green, eds., Perry’s Chemical Engineer’s Handbook, 6th ed., McGraw-Hill, New York, 1984.
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[5] R.E.Treybal, Mass Transfer Operation, 3rd ed, Mc-Graw Hill, NewYork, 1981. [6] K. Jamal, M.A. Khan and M. Kamil, Mathematical modeling of reverse osmosis systems, Desalination, 160 (2004) 29–42. [5] Z. Amjad, ed., Reverse Osmosis Membrane Technology, Water Chemistry and Industrial Application, Van Nostrand Reinhold, New York, 1993.