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Effectiveness–mass transfer units (–MTU) model of an ideal pressure retarded osmosis membrane mass exchanger The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Sharqawy, Mostafa H., Leonardo D. Banchik, and John H. Lienhard. “Effectiveness–mass Transfer Units (–MTU) Model of an Ideal Pressure Retarded Osmosis Membrane Mass Exchanger.” Journal of Membrane Science 445 (October 2013): 211–19. As Published http://dx.doi.org/10.1016/j.memsci.2013.06.027 Publisher Elsevier Version Author's final manuscript Accessed Thu May 26 21:39:26 EDT 2016 Citable Link http://hdl.handle.net/1721.1/102433 Terms of Use Creative Commons Attribution-NonCommercial-NoDerivs License Detailed Terms http://creativecommons.org/licenses/by-nc-nd/4.0/ Effectiveness-Mass Transfer Units (ε-MTU) Model of an Ideal Pressure Retarded Osmosis Membrane Mass Exchanger Mostafa H. Sharqawy1, Leonardo D. Banchik2, John H. Lienhard V*, 2 1 Department of Mechanical Engineering, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia 2 Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA ______________________________________________________________________________ Abstract Many membrane-based systems, such as reverse osmosis (RO), forward osmosis (FO) and pressure retarded osmosis (PRO), are being used in desalination, water treatment, and energy production. These systems work on the basis of mass transfer through a semipermeable membrane which allows for the permeation of water while rejecting salts and other substances. The membrane-based devices are essentially mass exchangers which are analogous to heat exchangers. The driving potentials in these mass exchangers are the concentration and pressure differences, whereas in heat exchangers the driving potential is the temperature difference. Closed form solutions of the permeation rate through an ideal PRO mass exchanger are obtained for parallel and counter flow configurations. The recovery ratio (RR) is obtained as a function of dimensionless parameters such as the number of mass transfer units (MTU), mass flow rate ratio (MR), and osmotic pressure ratio (SR). The resulting mathematical expressions form an effectiveness-NTU model for osmotic mass exchangers. These expressions are analogous to those for heat exchangers and can be used as an initial design for PRO membrane based mass exchange devices. Keywords Membrane mass exchanger, pressure retarded osmosis, effectiveness, mass transfer units * Corresponding author Email address: lienhard@mit.edu (John H. Lienhard V), Tel. +1 617-253-3790 1 Nomenclature A Am B C M m P R w Greek symbols water permeability coefficient total membrane surface area molality – moles of solutes per kg of solvent modified van ’t Hoff coefficient molar weight mass flow rate hydraulic pressure ideal gas constant salinity - grams of solutes per kg of solution kg m-2 s-1 kPa-1 m2 mol kg-1 kPa kg g-1 kg mol-1 kg s-1 kPa kJ mol-1 K-1 g kg-1 osmotic pressure density osmotic coefficient kPa kg m-3 per unit area m-2 Superscripts ’’ Subscripts d f i, in j o, out p pure recipe s solutes Abbreviations CF ECP ICP MR MTU NTU PRO RO RR SR draw feed inlet jth solute in a solution outlet permeate refers to pure water corresponds to a reference of seawater constituents salt solutes concentration factor external concentration polarization internal concentration polarization mass flow rate ratio mass transfer units number of transfer units pressure retarded osmosis reverse osmosis recovery ratio osmotic pressure ratio 2 1. Introduction Pressure retarded osmosis (PRO) systems are mass exchangers which are currently receiving great attention for their capabilities to produce renewable power from two streams of different salinities. In a PRO system, the higher salinity solution is called the draw stream and the lower salinity solution is called the feed stream. For power production, seawater and river water are typically used as the draw and feed streams respectively. These streams enter a mass exchanger where a semi-permeable membrane allows water to pass through, but not salts. The difference in osmotic pressure drives pure water (permeate) from the feed stream, through the membrane, and into the pressurized draw stream to dilute it. The pressurized diluted draw stream can be depressurized through a turbine to produce power. In this paper, we investigate the performance of the PRO exchanger relative to the amount of permeate and develop a sizing methodology similar to that commonly used for heat exchangers. The concept of PRO was first proposed by Loeb [1]. Since then, numerous mathematical models have been developed and experiments implemented to determine the work and permeate flux performance of PRO membranes. Mehta and Loeb [2] were early to recognize the significance of internal concentration polarization, which acts as a resistance to mass transfer inside the support layer of the membrane. Lee et al. [3] more rigorously investigated concentration polarization in PRO membranes and developed equations to determine the maximum power density and flux for zero-dimensional ideal membranes and membranes with internal concentration polarization (ICP). Two decades later, McCutcheon and Elimelech [4] developed a model which incorporates internal and 3 external concentration polarization (ECP) for a zero-dimensional FO and PRO membrane. Most recently, Tan and Ng [5] revisited the FO and PRO transport equations to include ECP and ICP resistances, which consider the interaction of the solute and the membrane and consider additional draw solutes. Currently, PRO related research is being conducted to determine the effect of fouling on flux performance [6-8] and the viability of novel materials for new membranes and hollow fibers [9-11]. The performance of oneand two-dimensional forward and pressure retarded osmosis exchangers have also been numerically investigated very recently [12-16]. The operation of an osmotic mass exchanger looks very similar to that of a heat exchanger when the systems are compared. In the heat exchanger schematic drawing shown in Fig. 1a, the temperature difference between a hot and cold fluid is the driving potential for heat transfer. The resistance to heat flow per unit area is the reciprocal of the overall heat transfer coefficient, U. In the osmotic mass exchanger shown in Fig. 1b, the osmotic pressure difference between a concentrated and a diluted stream, or draw and feed stream, is the driving potential for mass transfer. The resistance to mass transfer per unit membrane area for an ideal membrane is the reciprocal of the water permeability coefficient, A. For PRO systems, the driving potential, and thus the amount of mass transfer, is retarded by the hydraulic pressure difference, Pd - Pf which is greater than zero and less than the maximum osmotic pressure difference in the exchanger. For FO systems, this hydraulic pressure difference is equal to zero. 4 For heat exchangers, the effectiveness - number of transfer units (ε-NTU) method developed by Kays and London [17] is a well-known design technique that determines the required surface area of a heat exchanger for a fixed effectiveness and inlet conditions. The method uses three dimensionless groups: the effectiveness, which is the ratio of actual heat transfer to the maximum heat exchange possible; a heat capacity ratio, which is the lower heat capacity divided by the higher heat capacity; and the number of transfer units, which is an effective size of the heat exchanger. This paper proposes an -MTU method for a PRO mass exchanger. The local transport equation for permeate flow in a zero-dimensional exchanger is combined with conservation of mass and a linearized equation for osmotic pressure to develop dimensionless expressions for parallel-flow and counterflow PRO exchangers. The expressions are closed-form solutions which relate dimensionless performance parameters of the exchanger to the effective size and input stream properties. The effectiveness of the PRO exchanger is defined as a novel performance parameter. The dimensionless groups used in the closed form solutions are discussed, and an analogy to the heat exchanger groups is made. A numerical model which uses a nonlinear osmotic pressure function is also implemented in order to assess the errors associated with linearizing the osmotic pressure function. 5 2. PRO Mass Exchanger Model Figure 2 is a schematic drawing of a membrane-based mass exchanger device. A feed solution with low salt concentration (low salinity) flows through a channel where one side of the channel has a semi-permeable membrane. On the other side of the membrane , a draw solution with higher salt concentration flows in the same direction (Fig. 2a, parallel-flow configuration) or in the opposite direction (Fig. 2b counterflow configuration). The inlet and outlet conditions of both feed and draw streams are given as the mass flow rate, osmotic pressure, and pressure as indicated in Fig. 2. The membrane characteristics are given as the total membrane area (Am) and water permeability coefficient (A) of the membrane material. The model makes the following assumptions: Concentration polarization effects are neglected and the salt concentration near the membrane surface is equal to the bulk concentration of the flow stream. Pressure drop through the flow channel is negligible on both the feed and the draw side. Hence the hydraulic pressure difference between the draw side and feed side (P) is fixed over the length of the exchanger. The salt rejection is 100% and only pure water flows through the membrane. Within the operating salinity range of the PRO exchanger, the osmotic pressure follows van ’t Hoff’s law so that it is linearly proportional to the stream salinity (see Appendix Section A.1; the constant of proportionality may differ under different operating conditions). 6 2.1 Parallel-flow configuration PRO model The differential permeate flow rate in a PRO parallel-flow configuration is given by dmp = A ´ ( Dp - DP) ´ dAm (1) where m p is the permeate mass flow rate through the membrane in kg/s, A is the modified1 water permeability coefficient of the membrane in kg/m2 s kPa, is the osmotic pressure difference between the draw and feed (d – f) in kPa, P is the hydraulic pressure difference between the draw and feed (Pd – Pf) in kPa, and Am is the membrane surface area in m2. Using van ’t Hoff’s law for osmotic pressure Dp = p d - p f = C ( wd - w f ) (2) where w is the salt concentration or salinity (mass of solutes per total mass of solution) in g/kg and C is a modified van ’t Hoff coefficient, it follows that dmp = A éëC ( wd - w f ) - DPùû dAm (3) Under the assumed condition of 100% salt rejection, only pure water permeates though the membrane; hence, the salinity of the permeate is zero. Applying conservation of solutes to the feed stream between the inlet and any arbitrary location along the flow channel yields ms, f = m f ,in ´ w f ,in = m f ´ w f (4) 1 It is important to note that the water permeability coefficient (A) is often given in units of m/s-bar, or m/satm, or L/m2-hr-bar [3, 4], which is the permeate water volume flux per unit pressure difference; however, for the present model, we express this coefficient on a mass basis (equivalent to multiplying it by the density of pure water and some SI conversion factors). 7 For the same arbitrary control volume, conservation of total mass requires that m f ,in = m f + mp (5) Substitution of Eq. (5) into Eq. (4) yields wf = m f ,in ´ w f ,in m f ,in - m p (6) Similarly applying conservation of solutes and total mass on the draw side for a parallel configuration yields wd = md,in ´ wd,in md,in + m p (7) Substituting Eq. (6) and (7) into Eq. (3) yields é æm ´w ù m f ,in ´ w f ,in ö d,in d,in ÷ - DPú dAm dmp = A êC çç m f ,in - mp ÷ø êë è md,in + m p úû (8) We now proceed to cast Eq. (8) in a dimensionless form. Four dimensionless parameters are introduced for this purpose. Recovery ratio, RR RR º mp m f ,in (9) The recovery ratio is a primary performance metric of the PRO mass exchanger as it represents the amount of pure water recovered from the feed stream. In so far as the inlet mass flow rate is greater than the maximum amount of permeate that can be recovered, the recovery ratio should not be confused with the effectiveness which will be described in Section 3. 8 Mass flow rate ratio, MR MR º md,in m f ,in (10) The mass flow rate ratio is the ratio of the mass flow rate of the draw solution to that of the feed solution at the inlet of the PRO mass exchanger. Osmotic pressure ratio, SR For the draw side: SRd º p d,in DP (11) For the feed side: SR f º p f ,in DP (12) The osmotic pressure ratio is the ratio of the osmotic pressure at the draw or feed inlet to the hydraulic pressure difference. For PRO exchanger operation, SRd will always be greater than SRf. Mass Transfer Units, MTU MTU º Am A m f ,in DP (13) The number of mass transfer units (MTU) is a dimensionless parameter for a membrane mass exchanger similar to the number of transfer units (NTU) used in heat exchanger design. The total membrane area, Am, is analogous to the total heat exchanger surface area and A is the overall water permeability coefficient, which is analogous to the overall heat transfer coefficient in heat exchangers. Therefore, the MTU in the membrane-based 9 mass exchanger will play the same role that NTU plays in the -NTU analysis of heat exchangers. Substituting Eqs. (10 – 13) into Eq. (8) yields, æ MR ´ SRd SR f ö dRR = ç -1÷ dMTU è MR + RR 1- RR ø (14) Equation (14) can be integrated as follows: RR ò MR ´ SR 0 1 SR f -1 MR + RR 1- RR dRR = MTU (15) d Equation (15) can be simplified into RR ( MR + RR) (1- RR) dRR = MTU ò ( RR - a ) ( RR - b ) (16) 0 where 2 ù 1é a = ê(1+ MR ( SRd -1) + SR f ) - ( -1- MR ( SRd -1) - SR f ) - 4MR ( SRd - SR f -1) ú (17) û 2ë 2 ù 1é b = ê(1+ MR ( SRd -1) + SR f ) + ( -1- MR ( SRd -1) - SR f ) - 4MR ( SRd - SR f -1) ú (18) û 2ë Therefore the integration of Eq. (16) will be MTU = ( b -1) ( MR + b ) ln æ b - RR ö - (a -1) ( MR + a ) ln æ a - RR ö - RR ç ÷ ç ÷ è a ø è b ø (a - b ) (a - b ) (19) Therefore, Eq. (19) can be used in the design of a membrane mass exchanger where the required mass transfer units (hence the membrane area) is given as an explicit relation. An additional dimensionless parameter which may be useful to a designer is the concentration factor. The concentration factor is the ratio of the outlet salinity of a stream 10 to the inlet salinity. If a designer is limited to output brine below a certain salinity from the PRO exchanger, the concentration factor can be useful in determining the maximum MTU allowable. By considering a pure permeate and applying conservation of solution and solutes to the draw and feed streams separately, the expressions for the draw and feed concentration factors can be determined, as given by Eqs. (20) and (21). For the draw side: CFd º wd,out MR = wd,in MR + RR (20) For the feed side: CFf º w f ,out 1 = w f ,in 1- RR (21) Figures 3 and 4 show the variation of the recovery ratio (RR) and the concentration factor (CF) with the mass transfer units (MTU), respectively, at different mass flow rate ratios for the parallel-flow configuration. 2.2 Counterflow configuration PRO model The transport model Eq. (1) and the van ’t Hoff osmotic pressure model Eq. (2) will again be used to describe the permeate flow rate in the counterflow configuration shown in Fig. 2b. The differential permeate flow rate is given by Eq. (3) and the conservation of solution and solute for the feed side between the inlet and any arbitrary location along the flow channel leads to Eq. (6) of the feed salinity. Applying a similar conservation of solution and solute on the draw side for the counterflow configuration in the same arbitrary direction as taken for the feed stream gives a result that differs from that for the parallel-flow configuration (Eq. 7). 11 wd = md,out ´ wd,out md,out - m p (22) Substituting Eqs. (6) and (22) into Eq. (3), é m ´p ù m ´p dmp = A ê d,out d,out - f ,in f ,in - DPú dAm m f ,in - mp êë md,out - m p úû (23) Using the same dimensionless parameters as used for the parallel configuration (i.e. RR, MR, SRf, and MTU), Eq. (23) can be rewritten in a dimensionless form. However, two additional dimensionless parameters are also required; they are defined as follows. Outlet mass flow rate ratio, MRo MRo º md,out m f ,in (24) Osmotic pressure ratio at draw outlet, SRd,o SRd,o º p d,out DP (25) Since we are interested in expressing the recovery ratio (RR) as a function of the inlet flow conditions, we proceed to develop relations between the outlet dimensionless groups defined by Eqs. (24 – 25) and the inlet dimensionless groups defined by Eqs. (10 – 12). The mass flow rate ratio defined in Eq. (10) can be written as a function of the outlet mass flow rate ratio defined by Eq. (24) and the recovery ratio defined by Eq. (9) as follows. MR = md,in md,out - m p = = MRo - RR m f ,in m f ,in (26) 12 Therefore, MRo = MR + RR (27) Similarly, the osmotic pressure ratio of the outlet draw stream can be written as a function of the osmotic pressure ratio of the inlet draw stream, the mass flow rate ratio, and the recovery ratio as follows. SRd,o = SRd MR MR + RR (28) Using these dimensionless groups, Eq. (23) can be rewritten in a dimensionless form: æ MR ´ SRd,o SR f ö dRR = ç o -1÷ dMTU è MRo - RR 1- RR ø (29) This result can be integrated RR ò MR ´ SR 0 1 SR f -1 MRo - RR 1- RR o dRR = MTU (30) d,o Equation (30) can be simplified into RR ( MRo - RR) (1- RR) dRR = -MTU ò ( RR - a¢) ( RR - b ¢) (31) 0 where 2 ù 1é a¢ = ê(1+ MRo (1- SRd,o ) + SR f ) - ( -1+ MRo ( SRd,o -1) - SR f ) - 4MRo (1- SRd,o + SR f ) ú û 2ë (32) 2 ù 1é b ¢ = ê(1+ MRo (1- SRd,o ) + SR f ) + ( -1+ MRo ( SRd,o -1) - SR f ) - 4MRo (1- SRd,o + SR f ) ú û 2ë (33) Therefore the integration of Eq. (31) will be 13 MTU = ( b ¢ -1) ( b ¢ - MRo ) ln æ b ¢ - RR ö - (a¢ -1) (a¢ - MRo ) ln æ a¢ - RR ö - RR ç ÷ ç ÷ è a¢ ø è b¢ ø (a¢ - b ¢) (a¢ - b ¢) (34) Therefore, Eq. (34) combined with Eqs. (27) and (28) can be used in the design of a membrane mass exchanger where the required mass transfer units, effectively the membrane area, is given as an explicit relation of the form MTU = f ( RR, MR, SR f , SRd ) (35) Figures 5 and 6 show the variation of the recovery ratio (RR) and the concentration factor (CF) with the mass transfer units (MTU) respectively at different mass flow rate ratios (MR) for the counterflow configuration. The concentration factors for the feed and draw stream as defined in Eqs. (20) and (21) are applicable to the counterflow configuration as well. It can be seen by comparing Figs. 3 and 5 that, for each contour of MR, the recovery ratio is higher for the counterflow case than that of parallel-flow case. This is an expected result which is found in PRO literature [11, 12, 15] and is analogous to similar results for heat exchangers. 3. PRO Mass Exchanger Effectiveness ( – MTU Model) The effectiveness of the PRO system can be defined as the ratio of the permeate flow rate to the maximum permeate flow rate, which occurs when MTU is increased to infinity. The effectiveness can also be defined as the recovery ratio divided by the maximum recovery ratio. Note that the maximum permeate flow rate is not the inlet feed flow rate. The maximum accumulated permeated water occurs when the net driving pressure ( P) to draw water from the feed stream has decreased to zero at one end of the mass 14 exchanger. The following is a derivation of the maximum permeation flow rate and, hence, the effectiveness of the PRO exchanger. 3.1 Parallel-flow PRO Effectiveness Using Eq. (1), the maximum permeate in the case of parallel flow configuration will occur when the hydraulic pressure difference is equal to the osmotic pressure difference at the outlet. Dp out = p d,out - p f ,out = DP (36) Using the van ’t Hoff model C ( wd,out - w f ,out ) = DP (37) Applying conservation of solutes and solution on the draw side, one can find that wd,out = MR wd,in MR + RR (38) Similarly on the feed side, one can find that w f ,out = 1 w f ,in 1- RR (39) Substituting Eq. (38) and (39) into Eq. (37) and using the dimensionless groups defined earlier SR f MR× SRd =1 MR + RR 1- RR (40) Solving Eq. (40) to find the maximum recovery ratio, one can find that there are two solutions RRmax,1 = a (41a) 15 RRmax,2 = b (41b) where and are given by Eqs. (17) and (18), respectively. We notice from Eqs. (17) and (18) that is always less than 1 and is always greater than 1. The recovery ratio must be less than one. Therefore, the maximum recovery ratio is equal to . Now the effectiveness is defined as e= RR RRmax (42) Hence, RR = e RRmax = e a (43) By substituting Eq. (43) into Eq. (19), an expression for MTU as a function of the effectiveness can be obtained: MTU = ( b -1) ( MR + b ) ln æ b - a e ö - (a -1) ( MR + a ) ln 1- e - a e ( ) ç ÷ è b ø (a - b ) (a - b ) (44) Figure 7 shows the effectiveness changing with the mass transfer units for varying mass flow rate ratios. 3.2 Counterflow PRO Effectiveness Using Eq. (1), the maximum permeate in the case of counterflow configuration will occur when the hydraulic pressure difference is equal to the osmotic pressure difference at the right side or the left side of the exchanger schematic shown in Fig. 2b. Therefore, there are two conditions at which the driving potential for permeate flow will become zero. 16 p d,out - p f ,in = DP (45) Using the van ’t Hoff model and applying conservation of solution and solute this condition will lead to RRmax,1 = MR SRd - MR 1+ SR f (46) The other condition will lead to p d,in - p f ,out = DP (47) And RRmax,2 =1- SR f SRd -1 (48) Since there are two solutions for the maximum recovery ratio, we should take the minimum value, hence: RRmax = min RRmax,1, RRmax,2 (49) The effectiveness defined by Eq. (42) and the recovery ratio can be written in terms of the effectiveness and maximum recovery ratio as given by the first equality of Eq. (43). Substituting Eq. (43) into Eq. (34), an expression for MTU as a function of the effectiveness is obtained: MTU = ( b ¢ -1) ( b¢ - MRo ) ln æ b ¢ - e RRmax ö - (a¢ -1) (a¢ - MRo ) ln æ a¢ - e RRmax ö - e RR ç ÷ ç ÷ max è ø b¢ a¢ è ø (a¢ - b ¢) (a¢ - b ¢) (50) Figure 8 shows the effectiveness changing with the mass transfer units for varying mass flow rate ratios. 17 It should be noted that by varying the modified van ’t Hoff coefficient of each stream, the closed-form solutions given in the above sections allow for the two streams entering the PRO exchanger to have different compositions, such as seawater and ammonia-carbon dioxide, sodium chloride and pure water, or flowback water and ammonia-carbon dioxide. 4. Numerical PRO mass exchanger model The closed form solutions derived in the previous sections required the assumption that osmotic pressure is a linear function of salinity. While this assumption is valid for relatively dilute solutions, the variation becomes increasingly nonlinear as the mixture salinity increases. In this section, a numerical model of a one-dimensional PRO mass exchanger is developed using a nonlinear function for the osmotic pressure of seawater, so as to determine the accuracy of the approximate analytical results given earlier. The model applies a discretized version of the transport equation, Eq. (1), and conservation of solution and solutes to N membrane elements in series. The number of elements was increased to 50 at which point the results were grid independent. The total amount of permeate is calculated by numerically integrating the permeate mass flow rate produced by all elements. The equations comprised by the numerical model were solved using Engineering Equation Solver [18], a simultaneous equation solver which iterates to find a solution to sets of coupled nonlinear algebraic equations. The details of the nonlinear osmotic pressure function used in this numerical model is given in Banchik et al. [19] whereas the modified van ’t Hoff coefficient are given in the Appendix. 18 The numerical model is used to estimate the error in the analytical solutions that results from using a linearized osmotic pressure function (the van ’t Hoff equation). All other assumptions made for the analytical model are also made for the numerical model. An additional assumption is that the PRO membranes can withstand arbitrary net driving pressures. Two representative uses of a PRO system are for power production at a river delta [20] and for recovering the chemical energy which exists between the rejected brine of a desalination system and the available seawater [21-23]. Therefore, two numerical cases are considered: (1) a power production case with seawater and river water, wd,in = 35 g/kg and w f ,in =1.5g/kg; (2) and an energy recovery case with brine and seawater, wd,in = 70 g/kg and w f ,in = 35g/kg. The inputs for the numerical model are given in Table 1. The water permeability coefficient used is representative of a typical spiral wound forward osmosis membrane [24]. The percent error of the analytical model is given by Eq. (51): error = 1- analytical ´100% numerical (51) The inlet salinities given in Table 1 are used to calculate the inlet osmotic pressures using the nonlinear osmotic pressure function and the modified van ’t Hoff coefficient. For both cases, the hydraulic pressure difference is set to equal one-half of the maximum osmotic pressure difference. This assumption requires that SRd - SR f = 2 . To improve the accuracy of the linear case, we use different values of the modified van ’t Hoff coefficient to cover ranges of interest. Table 2 lists modified van ’t Hoff coefficients for three ranges with the coefficient of determination given in the third column. The first and second 19 value of C from Table 2 will be used in the linear model to compare the two numerical cases. We first consider the seawater-river water stream combination with a counterflow exchanger configuration. Figure 9 shows the recovery ratio versus mass transfer units for contours of the mass flow rate ratio, MR, and for fixed inlet draw and feed salinities. The inlet salinities are used to calculate the nonlinear and linear osmotic pressure for the draw and feed stream and the hydraulic pressure difference is determined by assuming that SRd - SR f = 2 as described in the previous section. It can be seen in Fig. 9 that the errors associated with linearization are highest for low mass flow rate ratios and for high values of MTU. For MR = 0.5 and for MTU > 2, the largest error associated with linearization of the osmotic pressure function is 4.62%. Figures 10 and 11 show the concentration factor and effectiveness plotted versus MTU for the same contours of mass flow rate ratios and inlet salinity. The maximum error for the draw and feed stream concentration factors is 2.12% and 20.3%, respectively. The relatively high error of 20.3% results from the contour of MR = 1 and corresponds to an outlet salinity difference between the numerical model and the analytical solution of about 1.9 g/kg. The errors are higher for the feed stream concentration factor because the feed stream salinities are lower, which lends to a smaller denominator in the error calculation even though the overall difference in outlet feed salinities may be low. The maximum error for the effectiveness is 4.41%. Figures 12-14 display the same dimensionless groups as Figs. 9-11 except that the inlet salinities are now representative of brine and seawater as defined by the energy 20 recovery case 2. The maximum error between models for the recovery ratio versus MTU of Fig. 12 is 5.37% and corresponds to the lowest mass flow rate ratio of MR = 0.5 at MTU = 5. For the case 2 concentration factor and effectiveness versus MTU shown in Figs. 13 and 14, the maximum error is less than 2% and 5%, respectively. Table 3 summarizes the maximum error incurred for each dimensionless variable and for each case and configuration. 5. Conclusions The major conclusions of this paper are as follows: Closed form analytical solutions for a one-dimensional pressure retarded osmosis mass exchanger using an ideal membrane were developed. The equations express the recovery ratio of the membrane as a function of the configuration of the membrane and four dimensionless groups: two osmotic pressure ratios, a mass flow rate ratio, and the number of mass transfer units. A robust analogy exists between heat exchangers and osmotic mass exchangers in which the effectiveness can be expressed by four dimensionless groups. The new -MTU model developed for the osmotic mass exchanger can be used as a design tool for PRO systems using a linearized osmotic pressure function and ideal membrane characteristics. Combinations of stream compositions can be analyzed by selecting the appropriate modified van ’t Hoff coefficient to be used in the dimensionless osmotic pressure ratio. In order to develop closed-form analytical solutions of the PRO exchanger performance, osmotic pressure must be approximated as a linear function of 21 salinity. Modest errors are incurred in doing so. Two cases were modeled to determine the errors associated with linearization: a power production case with inlet seawater and river water, and a desalination energy recovery case with inlet brine and seawater. For both cases the hydraulic pressure was set to equal half of the maximum osmotic pressure difference. Also for each case, the modified van ’t Hoff coefficient is varied to more closely approximate the nonlinear osmotic pressure for the range of salinity. For the power production case, the maximum error is less than 5% for the recovery ratio and effectiveness of a counterflow PRO exchanger. For the energy recovery case, the maximum error is less than 5.5%. Acknowledgments The authors would like to thank King Fahd University of Petroleum and Minerals in Dhahran, Saudi Arabia, for funding the research reported in this paper through the Center for Clean Water and Clean Energy at MIT and KFUPM. 22 Appendix A.1 The Modified van ’t Hoff Coefficient A nonlinear seawater osmotic pressure function is developed using the relation with the osmotic coefficient of seawater. The osmotic coefficient of seawater is given by a piecewise function which uses Brønsted’s equation for 0 ≤ w < 10 g/kg and an empirical correlation given by Sharqawy et al. [25] for 10 ≤ w < 120 g/kg. For details regarding the development of the nonlinear seawater osmotic coefficient function, please refer to the Appendix of Banchik et al. [19]. The nonlinear osmotic pressure of seawater as a function of salinity in g/kg is shown in Fig. (A.1). The modified van ’t Hoff coefficient (C) can be determined as the slope of the nonlinear osmotic pressure between specific ranges of interest. The osmotic pressure can be approximated as the product of the modified van ’t Hoff coefficient and the salinity of the solution using Eq. (A.1). p » Cw (A.1) Using a best-fit least squares method between the range of 0 and 35 g/kg, the modified van ’t Hoff coefficient is determined to be 73.07 kPa-kg/g at a temperature of 25oC. This linear model represented by Eq. (A.1) can be used for a salinity range of 0 to 35 g/kg, which is the typical range for power production at a river delta with a PRO exchanger. For salinities between 35 and 70 g/kg, the modified van ’t Hoff coefficient is determined to be 76.76 kPa-kg/g which is the typical range for power production using seawater and disposed brine from a seawater desalination plant. 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Lienhard V, Thermodynamic analysis of a reverse osmosis desalination system using forward osmosis for energy recovery, in: International Mechanical Engineering Congress and Exposition IMECE-2012, ASME, Houston, TX, USA, 2012. [23] W.A. Waqas, M.H. Sharqawy, J.H. Lienhard V, Energy utilization from disposed brine of msf desalination plant using pressure retarded osmosis, in: IDA World Congress, Tianjin, China, under review 2013. [24] J.R. McCutcheon, M. Elimelech, Modeling water flux in forward osmosis: implications for improved membrane design, AIChE J, 53 (2007) 1736-1744. [25] M.H. Sharqawy, J.H. Lienhard V, S.M. Zubair, Thermophysical Properties of Sea Water: A Review of Existing Correlations and Data, Desalination and Water Treatment, 16 (2010) 354 – 380. 25 Fig. 1a Schematic drawing of a counterflow heat exchanger. A hot and cold stream enter the exchanger where a temperature difference drives heat transfer. 26 Fig. 1b Schematic drawing of a counterflow osmotic mass exchanger. A concentrated and dilute stream, draw and feed, enter the exchanger where a salinity difference drives mass transfer. 27 Fig. 2a Schematic diagram of a PRO mass exchanger in parallel-flow configuration 28 Fig. 2b Schematic diagram of a PRO mass exchanger in counterflow configuration 29 Fig. 3 Recovery ratio vs. mass transfer units at different mass flow rate ratios for a parallel-flow configuration 30 Fig. 4 Concentration factor of the feed (dashed curves) and draw (solid curves) stream vs. mass transfer units at different mass flow rate ratios for a parallel-flow configuration 31 Fig. 5 Recovery ratio vs. mass transfer units at different mass flow rate ratios for a counterflow configuration 32 Fig. 6 Concentration factor of the feed (dashed curves) and draw (solid curves) stream vs. mass transfer units at different mass flow rate ratios for a counterflow configuration 33 Fig. 7 Effectiveness vs. mass transfer units at different mass flow rate ratios for a parallel-flow configuration 34 Fig. 8 Effectiveness vs. mass transfer units at different mass ratios for a counterflow configuration 35 Fig. 9 Recovery ratio vs. mass transfer units for varying mass flow rate ratios, counterflow configuration, and fixed inlet salinities representative of seawater and river water. Lines are from analytical Eq. (34), and the points are the results of a numerical model using a nonlinear osmotic pressure function. 36 Fig. 10 Concentration factor vs. mass transfer units for varying mass flow rate ratios, counterflow configuration, and fixed inlet salinities representative of seawater and river water. Solid and dashed lines are from analytical Eqs. (20) and (21), respectively, and the points are the results of a numerical model using a nonlinear osmotic pressure function. 37 Fig. 11 Effectiveness vs. mass transfer units for varying mass flow rate ratios, counterflow configuration, and fixed inlet salinities representative of seawater and river water. Lines are from analytical Eq. (50), and the points are the results of a numerical model using a nonlinear osmotic pressure function. 38 Fig. 12 Recovery ratio vs. mass transfer units for varying mass flow rate ratios, counterflow configuration, and fixed inlet salinities representative of brine and seawater. Lines are from analytical Eq. (34), and the points are the results of a numerical model using a nonlinear osmotic pressure function. 39 Fig. 13 Concentration factor vs. mass transfer units for varying mass flow rate ratios, counterflow configuration, and fixed inlet salinities representative of brine and seawater. Solid and dashed lines are from analytical Eqs. (20) and (21), respectively, and the points are the results of a numerical model using a nonlinear osmotic pressure function. 40 Fig. 14 Effectiveness vs. mass transfer units for varying mass flow rate ratios, counterflow configuration, and fixed inlet salinities representative of brine and seawater. Lines are from analytical Eq. (50), and the points are the results of a numerical model using a nonlinear osmotic pressure function. 41 Fig. A.1 Seawater osmotic pressure versus salinity for T = 25 °C shown as a solid curve. The points indicate the linear osmotic pressure approximation using varying modified van ’t Hoff coefficients. 42 Table 1 Data input for numerical model Input Value/Range Ambient temperature, To 25 °C 2 Modified water permeability coefficient, A 3.07´10-6 kg/m -s-kPa Feed mass flow rate, m f ,in 1 kg/s Inlet draw salinity, wd ,in 70 g/kg and 35 g/kg Inlet feed salinity, w f ,in Mass flow rate ratio, MR 35 g/kg and 1.5 g/kg 0.5, 0.75, 1, 2, 3 Hydraulic pressure, P (p d,in - p f ,in ) / 2 Trans-membrane pressure difference, P Membrane area, Am 1.24 – 1.47 MPa 0 – 9.45´105 m2 43 Table 2 Modified van ’t Hoff coefficients over three ranges for determining osmotic pressure as a function of salinity at T=25°C C [kPa-kg/g] 73.07 76.76 82.65 Range [g/kg] 0 - 35 35 - 70 70 - 105 R2 0.9997 0.9926 0.9691 44 Table 3 Maximum error resulting from linearized osmotic pressure Case 1: Seawater + River Water RR CFd CFf ε Parallel-flow Counterflow Case 2: Brine + Seawater Parallel-flow Counterflow 3.69% 4.62% RR 2.34% 5.37% 1.61% 2.12% CFd 0.43% 1.26% 3.41% 20.03% CFf 0.45% 1.68% 3.36% 4.41% ε 1.71% 4.95% 45