Forward Osmosis Processes in the Limit of Osmotic Equilibrium

Article pubs.acs.org/IECR

Forward Osmosis Processes in the Limit of Osmotic Equilibrium Sherwood Benavides, Alex S. Oloriz, and William A. Phillip* Department of Chemical and Biomolecular Engineering, University of Notre Dame, South Bend, Indiana 46556, United States S Supporting Information *

ABSTRACT: Forward osmosis processes are an emerging set of technologies that show promise in the treatment of complex and impaired water streams (e.g., those encountered in industrial wastewater treatment and the extraction of unconventional oil resources). The effective operation of these systems requires that the operating conditions be chosen wisely based on the membrane to be used and the streams to be treated. In this work, to aid in the design of these systems, an analytical model was developed that describes the module-level performance (i.e., water recovery rate and separation factors of the feed and draw solutes) for cocurrent and countercurrent forward osmosis systems in the thermodynamic limit of osmotic equilibrium. In the limit of osmotic equilibrium, the model expresses the recovery rate and separation factors in terms of the operating conditions (e.g., the flow rates and concentrations of the feed and draw solutions) and the characteristic membrane transport properties (e.g., the hydraulic and solute permeability coefficients). The model was validated by comparing its predictions with numerical simulations of the full system of governing equations: strong agreement between the model predictions and numerical simulations was observed. Analysis of the model demonstrates that the reverse flux selectivity, the ratio of the forward water flux to the reverse draw solute flux, is a key parameter in the design of forward osmosis systems that controls the maximum solute rejection that the systems can achieve at osmotic equilibrium. Further analysis shows that the flow ratio, the ratio of the inlet flow rate of the draw solution to the inlet flow rate of the feed solution, is an important design parameter. Specifically, in countercurrent operation, a critical value of the flow ratio that maximizes the recovery rate was identified. processes such as reverse osmosis and ultrafiltration.21−23 To date, several efforts have begun to explore the effects of membrane design, module design, channel length, and osmotic equilibrium on the performance of FO systems through mathematical models.24−28 These efforts have relied primarily on numerical algorithms to provide approximate solutions to the set of nonlinear differential equations that govern the performance of FO processes. Although the methods used are accurate, it can be difficult to extract meaningful physical insights and design rules from the results of numerical simulations. Therefore, it would be advantageous to develop analytical solutions that described the behavior of FO systems in physically relevant limits. Osmotic equilibrium is one physical limit of significant and fundamental import to FO processes. In an FO system, the high-concentration draw solution extracts water from the lowconcentration feed solution until the osmotic pressures of the feed solution and the draw solution are equivalent. At this point, the flux of water ceases, and the system is said to be in osmotic equilibrium. Therefore, osmotic equilibrium is a fundamental thermodynamic constraint on FO processes that limits the volume of water that can be recovered from the feed solution by a particular draw solution.24 The constraint of osmotic equilibrium also has implications on the final concentrations of solute in the draw and feed solutions that exit the membrane module. These exiting concentrations play a significant role in determining the design of downstream

1. INTRODUCTION Forward osmosis (FO) has garnered interest recently because of its potential applications in water treatment.1 In particular, FO has demonstrated promise in the treatment of complex and highly impaired feed streams such as wastewater,2−5 highsalinity brines,6−8 and produced water from the oil and gas industry.9−12 Instead of relying on an applied hydraulic pressure to drive water permeation, FO processes rely on an osmotic pressure difference, which is generated when a semipermeable membrane separates a high-concentration draw solution and a low-concentration feed solution, to drive the permeation of water.13,14 The low hydraulic pressures used in FO processes result in several benefits that make these processes well-suited for treating difficult feed streams. For example, reverse osmosis requires the use of an applied pressure that is at least equal to the osmotic pressure of the feed solution.15 For high-salinity brines, which can have osmotic pressures in excess of 80 bar, this is usually not feasible because of the large mechanical stresses placed on the membranes and high-pressure vessels.9,11,12 Alternatively, FO processes do not require the use of expensive high-pressure vessels, and the salinity of feed streams that can be treated is limited only by the concentration of the draw solution. Additionally, in some cases, it has been demonstrated that the lack of an applied pressure in FO processes might mitigate the deleterious effects of irreversible fouling.2,16−20 As opportunities for FO processes are identified and the technology moves from the laboratory scale to the pilot and field scales, the rational design of systems is necessary to maximize the performance of FO processes. Phenomenological models, as a method for guiding the systematic design of unit operations, have helped to advance more established membrane © XXXX American Chemical Society

Received: September 30, 2014 Revised: December 9, 2014 Accepted: December 15, 2014

A

DOI: 10.1021/ie5038787 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research

osmotic pressure gradient that is generated by the unequal total concentration of dissolved solutes.13,33 2.1. Overall Mass Balances and Rate Equations. To describe the cocurrent and countercurrent FO systems, steadystate mass balances are written for the solvent and solutes in differential volumes of the feed and draw streams

processes for draw-solution regeneration, as well as the suitability of the product water for its intended use. In this study, the performance of FO processes in the limit of osmotic equilibrium is examined. First, by solving the governing mass balances and rate equations for cocurrent and countercurrent processes at osmotic equilibrium, we develop analytical expressions that describe the performance of FO modules in terms of the water recovery rate and the separation factors of the feed and draw solutes. Then, using these expressions, we examine the influence of membrane properties (e.g., relative magnitude of the permeability of water to the permeabilities of solutes) and system operating conditions (e.g., ratio of the flow rate of the draw solution to that of the feed solution) on the recovery rate and the solute separation factors. In addition, the effects of cocurrent versus countercurrent operation on the recovery rate and separation factors are studied.

0=−

dF − Jw am dz

(1)

0=±

dD + Jw am dz

(2)

0=− 0=±

2. THEORY Schematics of an FO system operating in a cocurrent configuration, where the inlet streams for the feed solution and the draw solution are located at the same spatial point (z = 0), and an FO system operating in a countercurrent configuration, where the inlet streams for the feed solution and draw solution are located at opposite sides of the contactor, are shown in panels a and b, respectively, of Figure 1. The inlet

d(yF ) i

+ Jsi am

(3)

d(xiD) − Jsi am dz

(4)

dz

where F is the total volumetric flux on the feed side, D is the total volumetric flux on the draw side, z is the position along the length of the contactor, xi is the molar concentration of solute i in the draw solution, yi is the molar concentration of solute i in the feed solution, Jw is the flux of water through the membrane, Jsi is the flux of solute i through the membrane, and am is the ratio of the membrane area to the volume of the contactor. The differential equations that correspond to the mass balances on the feed stream (eqs 1 and 3) are identical for the cocurrent and countercurrent configurations. However, the differential terms in the equations that describe the mass balances on the draw stream (eqs 2 and 4) have a negative sign for the cocurrent configuration and a positive sign for the countercurrent configuration. The flux of water through the membrane is expressed as the product of the water permeability coefficient, A, and the local osmotic pressure difference across the membrane, Δπ. Based on the assumption that the feed and draw solutions are ideal and dilute, the van’t Hoff equation can be used to express the osmotic pressure in terms of the number of dissolved species generated by a solute, n; the ideal gas constant, Rg; the absolute temperature, T; and the total molar concentrations of solutes in the feed stream, ∑yi, and in the draw stream, ∑xi

Figure 1. Schematics of (a) cocurrent and (b) countercurrent forward osmosis systems. The feed stream, F, and draw stream, D, exchange solutes and solvent along the length of the contactor. The volumetric inward flux of the feed, Fin, contains only feed solute at a concentration of y1in, and the volumetric inward flux of the draw, Din, contains only draw solute at a concentration of x2in. The volumetric flux of feed out of the contactor, Fout, does not equal Fin because solvent (i.e., water) moves from the feed stream into the draw stream as a result of osmosis. Because of solute permeation through the membrane, Fout and Dout contain both draw and feed solutes at concentrations of y1out, y2out, x1out, and x2out..

Jw = AΔπ ≈ AnR gT (∑ xi −

∑ yi ) = Ã (∑ xi − ∑ yi ) (5)

For simplicity, we have assumed that the feed and draw solutes produce equal numbers of species when dissolved in water, n1 = n2 = n. This assumption can be relaxed easily at the expense of more complicated algebra. Furthermore, the goal of the current effort is to describe the behavior of systems in the limit of osmotic equilibrium, where the water flux is zero. Therefore, the effects of concentration polarization can be ignored in this thermodynamic limit.13,32,33 The flux of solute i through the membrane is written as the product of the solute permeability coefficient, Bi, and the local molar concentration difference of species i across the membrane

stream on the feed side contains only feed solute at a concentration of y1in, where the symbol y represents the molar concentration of a solute in the feed solution and the index 1 represents the feed solute. The inlet stream of the draw solution contains draw solute at a concentration of x2in, where the symbol x represents the molar concentration of a solute in the draw solution and the index 2 represents the draw solute. Over the length of the module, the feed and draw streams exchange solutes, as the species move down their respective concentration gradients,29−32 and solvent, because of the

Jsi = Bi (xi − yi )

(6) 27,29,34

Here, we have ignored any electrostatic coupling or multicomponent interactions.30,35−37 2.2. Recovery Rate Expressed as a Function of Separation Factors. The recovery rate, R, is defined as the B


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Industrial & Engineering Chemistry Research net amount of solvent transferred from the feed stream to the draw stream over the length of the contactor normalized by the amount of feed solution introduced into the contactor R=

Fin − Fout F = 1 − out Fin Fin

coc R OsmEq

(7)

Finally, the flow ratio, β, is defined as the ratio of the entering flux of the draw solution to the entering flux of the feed solution (9)

S Dx 2in B2

−

y1in S F B1

S Fy1in β B1

+

y1 − x1 =

x 2inS D B2

(10)

y2 − x 2 =

The same equation for R, eq 10, is obtained for both the countercurrent (Supporting Information, section S1) and cocurrent (Supporting Information, section S2) systems. 2.3. Recovery Rate in the Limit of Osmotic Equilibrium for Cocurrent Systems. For a long cocurrent contactor, where long is defined relative to the height of a transfer unit for water (i.e., z/HTUw ≥ 1), the system will approach osmotic equilibrium38 HTUw =

)S y F

1in

1in

⎤ + (x 2in − y1in )⎥ ⎦

− [Ã βx 2in + Aỹ 1in + (1 + β)B2 ]

F AnR gTx 2inam

y1in Fin − x1outDout + x1(D − F ) F y2out Fout − x 2inDin + x 2(D − F ) (16)

F

D SOsmEq (z → L )

(11)

=

⎡ Ã S Fy1in ⎢ B 1 + β − ⎣ 1

) + ⎤⎥⎦ − Ã (1 + β)( − ) + y (

S Fy1in

y1in

β x 2in

x 2in

1 B1

y1in x 2in

Ã B2 1in

1 B2

+1

+1 (17)

Furthermore, eq 17 can be substituted back into eq 10 to solve for the recovery rate in the limit of osmotic equilibrium as a function of the feed separation factor

This constraint on the total solute concentration is used to calculate the draw separation factor at osmotic equilibrium as a function of the feed separation factor (Supporting Information, section S3) as follows

counterc R OsmEq (z → L ) =

⎡ ⎤ x A SF(1 + β) B x 2in + 1 + y2in − 1 ⎥ ⎡ β ⎤ ⎢ 1 1in ⎢ ⎥ =⎢ ⎥× ⎛ Ã B2 x 2in ⎞ ⎥ ⎣ 1 + β ⎦ ⎢ F Ã ⎜ ⎟ + β − + + S x (1 ) 1 x ⎢⎣ B2 2in B1 y1in ⎠ ⎥ ⎝ B2 2in ⎦

( (

(15)

2.4.1. Osmotic Equilibrium at z = L. The recovery rate for a countercurrent system that achieves osmotic equilibrium at z = L is calculated by evaluating eq 16 at z = L and substituting the results into eq 15. The resulting expression is combined with eq 10 to solve for the draw separation factor in the limit of osmotic equilibrium as a function of the feed separation factor

In the limit of osmotic equilibrium, the water flux through the membrane is zero because the total concentration of dissolved solutes in the feed stream is equal to the total concentration of dissolved solutes in the draw stream. For a cocurrent system, the total concentrations will be equal at z = L, where both solutions exit the module x1out + x 2out = y1out + y2out (12)

D SOsmEq

B2 B1

F

Equation 3 is added to eq 4. Then, for the feed solute, the sum of eqs 3 and 4 is integrated from z = 0 to an arbitrary position along the module length; for the draw solute, the sum is integrated from z = L to an arbitrary position along the module length. The integrated equations are then solved for the concentration differences of the solutes at an arbitrary position along the contactor

This nomenclature applies for both cocurrent and countercurrent configurations. However, the location of the inlet of the draw solution differs between the two configurations. As detailed in the Supporting Information, the governing equations can be solved to express the recovery rate in terms of the separation factors for the feed and draw solutes

+

(

Ã (1 + β) 1 −

)S y

x1 + x 2 = y1 + y2

Din Fin

1 Ã

=

B2 B1

2.4. Recovery Rate at Osmotic Equilibrium for Countercurrent Systems. For a long countercurrent contactor, the system will tend toward osmotic equilibrium. The countercurrent system is unique because it can approach osmotic equilibrium at z = 0 or at z = L.39 Where the system approaches osmotic equilibrium depends on the inlet solute concentrations and the inlet flow rates. For now, to keep the derivation general, the constraint on the solute concentrations at osmotic equilibrium (i.e., equal total concentrations of solute) is written for an arbitrary position along the module length

The separation factors of the feed solute, S , and draw solute, SD, which quantify the amounts of solute transferred through the membrane over the length of the contactor, are given by the equations y x S D = 2out S F = 1out x 2in y1in (8)

R=

(

(14) F

β=

⎡ − Ã β ⎢(1 + β) 1 − ⎣

)

( S(

)y − 1)y

S Fβ 1 − F B2 B1

B2 B1

1in

1in

+ x 2in − y1in +

B2 Ã

+ x 2in

(18)

2.4.2. Osmotic Equilibrium at z = 0. A similar process can be used in the limit that osmotic equilibrium is approached at z = 0, in which case eq 16 is evaluated at z = 0 and substituted into eq 15. The resulting expression is combined with eq 10 to solve for the draw separation factor as a function of the feed separation factor

)

(13)

Additionally, eqs 13 and 10 are utilized to obtain an expression for the recovery rate at osmotic equilibrium as a function of the feed separation factor C


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method in MATLAB (i.e., ode45).40 This is a one-step solver that calculates values for the independent variables based on their values at the previous step. 3.2.1. Boundary Value Problem, Shooting Method, and Collocation Method. In the countercurrent configuration, the inlet conditions for the feed stream and the inlet conditions for the draw stream are located at opposite ends of the contactor. Two different approaches are used to find a solution that satisfies this system of ODEs and boundary conditions: a shooting method and a collocation method.40 The shooting method is an iterative process whereby the values of the independent variables at z = 0 for the draw stream (Dout, x1out, x2out) are guessed repeatedly until the solution for the independent variables at z = L matches the boundary conditions at z = L (Din, x1in, x2in). For each step in the iteration, the governing equations are solved by an explicit Runge−Kutta method using the known conditions for the feed stream and the initial guesses for the draw stream as boundary conditions. The resulting values for the independent variables at z = L are compared to the values specified by the boundary conditions; if these values do not fall within a 10−3 tolerance, the guessed values at z = 0 are modified. This process is automated by employing a nonlinear constrained optimization that minimizes the following error function

D (z → 0) SOsmEq

(

SFy1in β

=

(

SFy1in Ã (1 + β)

1 B1

1 x 2in

−

+

1 B2

Ã B1

)+

y1in

) + β(1 − ) x 2in

Ã [y (1 B2 1in

+ β) − x 2inβ ] + 1

(19)

and the recovery rate in the limit of osmotic equilibrium as a function of the feed separation factor counterc (z R OsmEq

→ 0) =

(

) + β (x − y ( − 1) + + y

S Fy1in β 1 − S Fy1in

B2 B1

B2 B1

2in

B2 Ã

1in

)

1in

(20)

2.4.3. Critical Flow Ratio. A countercurrent FO system can reach osmotic equilibrium at z = 0 or at z = L. Which situation occurs depends on the operating conditions, namely, the inlet volumetric fluxes and inlet concentrations. The value of the flow ratio that results in a transition from approaching osmotic equilibrium at z = 0 to reaching osmotic equilibrium at z = L can be identified by setting eqs 18 and 20 equal to each other and solving for β, which gives βcrit =

S Fy1in

(

B2 B1

)

−1 + B2 A

+ x 2in

B2 A

+ y1in

E step =

(21)

2 step 2 (Din − DLstep)2 + (x 2in − x 2step L ) + (x1in − x1L )

For β > βcrit, osmotic equilibrium occurs at z = L, and eqs 17 and 18 should be used; for β < βcrit, osmotic equilibrium occurs at z = 0, and eqs 18 and 20 should be used. It can be shown that, if the incorrect equations are used (e.g., if eqs 17 and 18 are used when β < βcrit), a negative driving force results. The equations detailed above provide analytical expressions that enable the prediction of system performance in the limit of osmotic equilibrium based on characteristic membrane properties for FO systems operating in a cocurrent or countercurrent configuration.

(22)

Dstep L ,

xstep 1L ,

xstep 2L

where and are the values for the independent variables D, x1, and x2, respectively, at z = L that result from the simulation. The algorithm of the minimization routine is defined by the built-in MATLAB function fmincon and is constrained by mass balances on the feed and draw streams. For some cases, numerical integration is not stable, and the shooting method is unable to find a solution. In this scenario, a collocation routine based on Simpson’s method is implemented to find an approximate solution to the governing equations. This algorithm finds a polynomial solution that satisfies the boundary conditions and the set of governing differential equations along the length of the contactor. The built-in MATLAB function bvp5c was used to implement this routine.

3. METHODS 3.1. Numerical Simulations. The full governing equations for an FO process in a cocurrent or countercurrent configuration comprise a system of nonlinear, coupled ordinary differential equations (ODEs). There are six independent variables that vary along the contactor in the z direction: y1, y2, x1, x2, F, and D. The numerical solutions of eqs 1−4 show how these variables depend on parameters such as the inlet volumetric fluxes, Fin and Din; the inlet solute concentrations, x2in and y1in; the permeability coefficients of water and the solutes, A, B1, and B2; and the contactor length, L. It is assumed that the water permeability coefficient is of higher magnitude than the permeability coefficients of the solutes. Therefore, the characteristic length for water transport (i.e., HTUw) is used to define the contactor length in all of the simulations. We anticipate that, for values of the distance down the module, z, near the value of HTUw, osmotic equilibrium will be approached. 3.1.1. Explicit Runge−Kutta Method. The inlet feed and draw fluxes are located at z = 0 for the cocurrent configuration. This nonlinear system of differential equations is an initial value problem because values for all independent variables are known at the same location, z = 0. The evolutions of the solute concentration and volumetric fluxes down the length of the module were determined using an explicit Runge−Kutta

4. RESULTS AND DISCUSSION 4.1. Comparison of Asymptotic Solutions and Numerical Simulations. Equation 10 describes the performance of an FO system in terms of the recovery rate R, feed separation factor SF, and draw separation factor SD. When plotted, it generates a three-dimensional surface that defines all possible combinations of solvent and solute exchange that do not violate an overall mass balance. In Figure 2a, the threedimensional surface that is generated for a system utilizing a membrane with water and solute permeabilities of Ã am = 3 L mol−1 s−1, B1am = 0.03 s−1, and B2am = 0.03 s−1 and operating with Fin = 3 m s−1, Din = 3 m s−1, y1in = 1 M, and x2in = 5 M is plotted.41 The concentrations of the draw and feed solutions were selected to mimic those from a recent field study.11 To facilitate comparison between different systems, it is more convenient to plot the three-dimensional surface as a twodimensional projection (i.e., Figure 2b). In this format, values of the draw separation factor are plotted on the vertical axis; values of the feed separation factor are plotted on the horizontal axis; and values of the recovery rate are represented D


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factors for cocurrent and countercurrent FO systems at osmotic equilibrium are studied in the subsequent sections. 4.2. System Performance in the Limit of Equipermeable Solutes. The equations derived above are capable of predicting the performance of cocurrent and countercurrent forward osmosis unit operations at osmotic equilibrium. Furthermore, for equipermeable solutes, namely, B2 = B1, they simplify significantly and result in equations that express a key performance metric for FO processes, namely, the recovery rate, solely as a function of known operating conditions, namely, the inlet concentrations of the solutes, the flow ratio, and the reverse flux selectivity of the membrane (Supporting Information, section S4). 4.2.1. Role of Reverse Flux Selectivity. The reverse flux selectivity for the draw solute, which is defined as Ã /Bi = (AnRgT)/Bi, is an important figure of merit for FO membranes. Physically, it can be regarded as the volume of water produced per mole of solute that permeates through the membrane. In the development of FO membranes, fabricating membranes with high reverse flux selectivity is vital to maximizing water permeation while deterring the exchange of solutes between the feed and draw streams.26,30−32 The recovery rate is plotted as a function of the feed and draw separation factors for three different values of the reverse flux selectivity in Figure 3. In all three cases, the entering feed stream contains only feed solute (y1in = 1 M), the entering draw stream contains only draw solute (x2in = 5 M), and the entering volumetric fluxes are equal (β = 1). The three cases were generated by holding the permeability of the solutes constant at Bam = 0.03 s‑1 and modifying the hydraulic permeability coefficient. Values for the modified hydraulic permeability coefficient of Ã am = 0.3, 3, and 6 L mol−1 s−1 were used to generate the results in panels a−c, respectively. In each panel, the results for the countercurrent configuration are plotted in white, and the results for the cocurrent configuration are plotted in black. The solid lines describe the asymptotic limit where the system is at osmotic equilibrium. The discrete circles describe the results obtained by integrating the system of nonlinear governing equations numerically. Each discrete point represents a contactor of a different length. The maximum length used in each case studied was 5 times (i.e., x2in) HTUw. The results in Figure 3 demonstrate that the cocurrent and countercurrent configurations offer similar solvent recovery rates and solute separation factors for lengths that are short relative to the HTUw. As the lengths of the contactors increase, the trajectories of the two configurations diverge. Although both configurations achieve their maximum recovery rate as they approach osmotic equilibrium, the countercurrent contactors achieve a higher maximum recovery rate and result in a greater extent of solute exchange than cocurrent contactors. These observations result directly from the ability of the countercurrent contactors to maintain a larger osmotic driving force over the length of the membrane module than the cocurrent contactors.24−26 The osmotic driving force decreases over the length of the module because of the fluxes of water and solutes. The flux of water dilutes the draw solution and concentrates the feed solution, whereas the flux of solutes decreases their concentration gradients. For short contactors, little water or solute permeates through the membrane (i.e., the recovery rates and separation factors are small), so there is little change in the osmotic driving force over the length of the module. Therefore, the cocurrent and countercurrent configurations perform comparably.

Figure 2. Solvent recovery rate plotted as a function of the draw solute and feed solute separation factors for a cocurrent system. On the left, a three-dimensional plot for a system with B1 = B2, AnRgT/B = 100 L mol−1, y1in = 1 M, and x2in = 5 M is displayed. The colored surface was generated using eq 10. The solid line describes the system in the limit of osmotic equilibrium (eq 14). The dashed line describes the system in the limit of no water permeation. The discrete circles define the path obtained by solving the governing equations numerically. On the right, a two-dimensional plot of the same data is shown. The magnitude of the recovery rate is represented by a color gradient in the two-dimensional plot.

by the color gradient, where a value of R = 0 corresponds to blue and a value of R = 1 corresponds to red. Identifying the trajectories followed by contactors of differing lengths for a specific membrane and set of operating conditions requires that the nonlinear system of governing equations be solved, which is not plausible to do analytically and is tedious to do numerically. Therefore, it is useful to solve the equations in physically relevant limits. Two asymptotic limits are considered here: (1) the limit where no water permeates through the membrane and (2) the limit where the system reaches osmotic equilibrium. In the limit that the permeation of water through the membrane is negligible, eqs 3 and 4 reduce to linear ODEs that can be solved readily using standard techniques.38 Note that, in this limit, the recovery rate is equal to zero for all cases, but the variation in SD with SF can be elucidated. This boundary, for a cocurrent system, is plotted in Figure 2 as a dashed black line. The second asymptotic limit of interest occurs for systems that approach osmotic equilibrium. For cocurrent systems, this second boundary, which is plotted as a solid black line in Figure 2, is defined by eq 14. The limits of no water permeation and osmotic equilibrium bound the solution of the governing equations to a region on the surface defined by eq 10. To compare the asymptotic solutions to the full solution, eqs 1−4 were simulated numerically for monotonically increasing values of L, and the results are plotted as black circles in Figure 2. For small values of L, little water permeates through the membrane, and the solution in the limit of no water permeation describes the performance of the system well. As the value of L increases, the numerical solution deviates from the two asymptotic solutions because an appreciable amount of water has permeated through the membrane but the system is not yet at osmotic equilibrium. With further increases in L, the system comes close to reaching osmotic equilibrium. In this limit, the system performance calculated for the cocurrent system using numerical methods aligns perfectly with the performance predicted by eq 14. This strong agreement confirms that the recovery rate and solute rejection near osmotic equilibrium can be predicted for a cocurrent system using the model developed here. The same is true for countercurrent systems when eqs 17−21 are used (vide infra). The effects of the operating conditions and the water and solute permeabilities on the recovery rate and separation E


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Figure 4. Recovery rate at osmotic equilibrium plotted as a function of the reverse flux selectivity, AnRgT/B. The cocurrent and countercurrent curves were generated using eqs 14 and 18, respectively, in the limit of equipermeable solutes (i.e., B1am = B2am = 0.03 s−1). A flow ratio of β = 1 and entering concentrations of y1in = 1 M and x2in = 5 M were used.

recovery rate at osmotic equilibrium (i.e., the maximum recovery rate) is plotted against the reverse flux selectivity. As the value of the reverse flux selectivity increases, the recovery rate increases for both configurations. Interestingly, for AnRgT/ Bi > 10 L mol−1, the increments in the recovery rate as a result of increments in the reverse flux selectivity asymptotically approach zero. Interactions between the permeating solute and solvent might require the use of a reflection coefficient,37,42 but the point remains that, above a certain value, the primary benefit of increasing the reverse flux selectivity is reducing the amount of solute permeation, not increasing the recovery rate. This observation is consistent with experimental studies that have investigated the use of FO processes to remove a variety of dissolved solutes (e.g., heavy metal ions43 and trace organic compounds44). The systems studied were not operated at the thermodynamic limit of osmotic equilibrium because this would require infinitely long modules. However, the water fluxes and recovery rates reported were little affected by the feed stream composition. The percent rejections reported for the feed solutes, however, varied widely. 4.2.2. Impact of the Flow Ratio. The flow ratio is an easily modified operating parameter for an FO system that determines the volume of feed solution that is processed per unit volume of draw solution used. It is an important operating parameter because selecting the ratio intelligently can be used to achieve high recovery rates while minimizing the exchange of solutes. Furthermore, the flow ratio plays a key role in determining the concentration and volume of the diluted draw solution. Therefore, in system designs that require the diluted draw solution to be processed further (e.g., regenerated), the flow ratio will directly affect the size and energy demand of the downstream unit operations.38,45 In Figure 5, the recovery rate is plotted as a function of the feed and draw separation factors for cocurrent and countercurrent systems operating with different flow ratios. In all cases, the feed solute enters at a concentration of y1in = 1 M, the draw solute enters at a concentration of x2in = 5 M, and the reverse

Figure 3. Recovery rate as a function of draw and feed separation factors for different values of the reverse flux selectivity, AnRgT/B. The reverse flux selectivity is reported in units of L mol−1. The feed and draw solutes are equipermeable (i.e., B1am = B2am = 0.03 s−1). For all cases, the inward fluxes of the feed solution and draw solution are equal to 3 m/s, and entering concentrations of y1in = 1 M and x2in = 5 M were used. Values for the countercurrent configuration are plotted in white, and values for the cocurrent configuration are plotted in black. The solid lines describe the asymptotic limit where the system is at osmotic equilibrium. The discrete circles describe the path obtained by numerically integrating the system of nonlinear governing equations (i.e., eqs 1−4). Note that the scales for SF and SD change for the different values of the reverse flux selectivity.

For long contactors, the water and solute permeations become substantial. In a cocurrent operation, where the feed and draw solutions are moving in the same direction down the module, the diluted draw solution contacts the concentrated feed solution. This pinches the osmotic driving force to zero quickly. On the other hand, in a countercurrent system, the feed and draw solutions are moving in opposite directions. In this configuration, the diluted draw solution moves toward less concentrated feed solution, which helps to sustain the osmotic driving force and enables higher recovery rates to be achieved. The countercurrent configuration also maintains a larger concentration gradient for the individual solutes, which results in a greater amount of solute permeation and higher separation factors. For both cocurrent and countercurrent systems, the maximum recovery rate increases as the ratio of hydraulic permeability to solute permeability increases. However, increasing the hydraulic permeability indefinitely does not drive the recovery rate to 100% because the systems are limited thermodynamically by osmotic equilibrium. In Figure 4, the F


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The flow ratio also plays a key role in determining the recovery rate that can be achieved at osmotic equilibrium. These findings are illustrated in Figure 6, where the recovery

Figure 6. Recovery rate in the limit of osmotic equilibrium plotted as a function of the flow ratio, Din/Fin. A hydraulic permeability of Ã am = 3 L mol−1 s−1, solvent permeabilities of B1am= B2am = 0.03 s−1, and entering concentrations of y1in = 1 M and x2in = 5 M were used in conjunction with eqs 14, 18, and 20 to calculate the recovery rate in the limit of osmotic equilibrium. A countercurrent system can operate in two regimes: a regime in which osmotic equilibrium is approached at z = L (eq 18) and a regime in which osmotic equilibrium is maintained at z = 0 (eq 20). The transition between the two regimes occurs at a flow ratio of βcrit, defined by eq 21. Equation 14 was used to calculate the recovery rate for the cocurrent system.

Figure 5. Recovery rate as a function of draw and feed separation factors for different values of the flow ratio, β = Din/Fin. The solutes are equipermeable (i.e., B1am = B2am = 0.03 s−1). The water permeability was held constant, Ã am = 3 L mol−1 s−1, and the entering concentrations were y1in = 1 M and x2in = 5 M. Values for the countercurrent configuration are plotted in white, and values for the cocurrent configuration are plotted in black. The solid lines describe the asymptotic limit where the system is at osmotic equilibrium. The discrete circles describe the path obtained by numerically integrating the system of nonlinear governing equations (i.e., eqs 1−4).

rate is plotted as a function of β for cocurrent and countercurrent systems. For countercurrent systems above βcrit, the maximum recovery rate does not change as a function of the flow ratio. In these cases, the system pinches at z = L and is limited by water permeating from the feed solution to the draw solution, which concentrates the solutes in the feed solution. Once the overall concentration of dissolved solutes in the feed solution equals the entering concentration of the draw solution, the permeation of water ceases. Below βcrit, R decreases linearly with β. This is a result of the system pinching at z = 0 where it is limited by the dilution of the draw solution. As β decreases, the volume of feed solution entering the system increases while the volume of water that permeates from the feed to the draw solution remains nearly constant, and as a result R decreases. In many FO processes, the water extracted from the feed solution is the valuable product being recovered.4,46 Therefore, it is better to be limited by concentrating the feed solution rather than diluting the draw solution. The simple criterion for βcrit detailed in eq 21 helps to select operating conditions that ensure that this is the case. For cocurrent systems, osmotic equilibrium is always approached at z = L. For these systems, as β increases and approaches infinity, the recovery rate of the cocurrent system asymptotically approaches the recovery rate of the countercurrent system. In the limit of large β, the volume of water that permeates through the membrane becomes infinitesimally small compared to the volume of the draw solution. Therefore, the draw solution is not diluted as it flows through the contactor, and the performance limitations associated with draw-solution dilution vanish for the cocurrent system. Instead, it is limited by

flux selectivity is held constant at a value of 100 L mol−1. The flow ratio is varied by holding the entering volumetric flux of the draw solution constant at 3 m s−1 and modifying the inlet volumetric flux of the feed solution. In each panel, the results for the countercurrent configuration are plotted in white, and the results for the cocurrent configuration are plotted in black. The solid lines describe the asymptotic limit where the system is at osmotic equilibrium, and the discrete circles represent the results obtained from the numerical solutions of eqs 1−4. Examining how the separation factors vary with the flow ratio allows an important observation to be made. For both cocurrent and countercurrent systems, with all other variables held constant, the value of the feed separation factor when the system first approaches osmotic equilibrium decreases with increasing β. This is a result of dilution. As β increases, the volume of the draw solution relative to the volume of the feed solution increases, which means that a larger volume of draw solution is available to dilute the solute that permeates from the feed solution into the draw solution. This dilution results in lower values of x1,out and SF. The same rationale highlights how the flow ratio can be used to tune the solute concentration and the volume of the draw solution exiting the FO module. G


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source (e.g., wastewater) while simultaneously diluting a seawater reverse-osmosis (RO) feed47 and (2) a membrane brine concentrator to enhance the reuse of produced water from oil production.11 These two cases provide extremes in feed solution concentration. The total dissolved solids (TDS) content of wastewater is ∼500 mg L−1, whereas produced water has a TDS of ∼75 g L−1.11,47 It is possible for both of the proposed FO processes to achieve high recovery rates, but the draw solutions necessary to accomplish this are quite different. Pretreated seawater (∼0.6 M NaCl) could be used to recover 95% of the volume from the wastewater feed solution. On the other hand, a much more concentrated engineered draw solution (5.5 M NH4HCO3) is necessary to recover 76% of the produced water volume. A recent pilot-scale membrane brine concentrator using an engineered draw solution and operating conditions similar to those described above was able to achieve a recovery rate of 64%.11 This is less than the recovery rate predicted at osmotic equilibrium, which is expected because operating in the limit of osmotic equilibrium would require infinite membrane area, but it highlights the ability of eq 18 to establish practical limits for the performance of FO modules. 4.3. Impact of Disparate Solute Permeabilities on Recovery Rate. In the preceding sections, the performance of systems with equipermeable solutes was considered. This is a useful, but unrealistic, limit. Therefore, the impact of disparate solute permeability coefficients on solvent and solute exchange in an FO process at osmotic equilibrium is analyzed in this section. In Figure 8, the recovery rate is plotted as a function of the feed and draw separation factors for cocurrent and countercurrent systems operating under identical operating conditions: the entering feed stream contains only feed solute (y1in = 1 M), the entering draw stream contains only draw solute (x2in = 5 M), and the entering volumetric fluxes are equal (β = 1). Three cases are studied, where Ã am is held constant at 3 L mol−1 s−1 and the ratio of the feed solute permeability to the draw solute permeability is altered. Values for the countercurrent configuration are plotted in white, and values for the cocurrent configuration are plotted in black. The solid lines describe the asymptotic limit where the system is at osmotic equilibrium. The discrete circles describe the results obtained from the numerical solutions of eqs 1−4. For equipermeable solutes (Figure 8b), once the system reaches osmotic equilibrium the recovery rate reaches its maximum value, and no further solvent exchange takes place between the streams. Further increases in the length of the module result in the exchange of solutes across the membrane, but solvent is not exchanged because the solutes are equipermeable and, for every molecule of feed solute that enters the draw stream, one molecule of draw solute leaves the draw stream. Therefore, the total number of solute particles present in each stream remains constant and equal. This equal exchange of molecules between the feed and draw streams results in separation factors that vary linearly at osmotic equilibrium (Figure 8b). In a system where the feed and draw solutes have different values for their permeability coefficients (Figure 8a,c), the relationship between the separation factors at osmotic equilibrium is not linear. This asymmetric solute exchange at osmotic equilibrium has an interesting effect. Specifically, it results in an osmotic pressure differential that drives solvent permeation being generated constantly as the length of the contactor is increased. Hence, the recovery rate is not constant

the increasing concentration of solutes in the feed solution, which is similar to a countercurrent system that pinches at z = L. 4.2.3. Impact of the Inlet Feed and Draw Concentrations. The ratio of the inlet concentration of the feed solution to the inlet concentration of the draw solution establishes a thermodynamic limit for the recovery rate that a given process can achieve. In many cases, the concentration of the feed solution is determined by the source (e.g., wastewater or produced water) and cannot be manipulated easily. Therefore, in cases where the concentration of the draw solution is also set by its source (e.g., pretreated seawater), eqs 14, 18, and 20 can be used in conjunction with the membrane properties and flow ratio to calculate the maximum recovery rate possible. Alternatively, if an engineered draw solution (e.g., NH4HCO3) is being implemented and a specific recovery rate is desired, eqs 14, 18, and 20 allow the minimum drawsolution concentration necessary to accomplish the desired separation to be determined. The recovery rate at osmotic equilibrium is plotted against the ratio of inlet concentrations in Figure 7. For both cocurrent

Figure 7. Recovery rate at osmotic equilibrium for different values of the inlet concentration ratio, y1in/x2in. The reverse flux selectivity and entering concentration of the draw solution were kept constant such that B/(Ã x2in) = 0.002. Assuming a flow ratio of Din/Fin = 1, eqs 14, 18, and 20 were evaluated for equipermeable solutes (Supporting Information, section S4).

and countercurrent systems, the recovery rate approaches unity asymptotically as the ratio of initial feed concentration to initial draw concentration goes to zero. On the other hand, the recovery rate drops rapidly toward zero as the ratio of the initial concentrations tends toward 1, which is sensible because the osmotic driving force vanishes in this limit. The benefits of operating an FO module in a countercurrent configuration are apparent for concentration ratios between 1 × 10−2 and 1. Over this range, a countercurrent process results in a higher recovery rate than the cocurrent process for all ratios of inlet concentrations. Equation 18 can be used to examine the recovery rate for two proposed FO systems. A selectivity of Ã /B = 100 L mol−1, which is characteristic of commercial FO membranes,31 and a flow ratio of 1 are assumed. The two processes examined are (1) osmotic dilution to recover water from an impaired water H


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contactor lengths. In Figure 8c, where the permeability coefficient of the draw solute is 10 times that of the feed solute and 1/10 that of the solvent (Ã /B2 = 10 and Ã /B1 = 100 L mol−1), the separation factor of the draw solute is significantly higher than the separation factor of the feed solute. This reduces the osmotic potential of the draw solution over the length of the contactor, which results in a reduced maximum recovery rate that decreases with increasing module length.

5. CONCLUDING REMARKS This study investigated the performance of cocurrent and countercurrent FO processes in the limit of osmotic equilibrium. We developed a simple mathematical model for systems at osmotic equilibrium that can be used to calculate the maximum recovery rate and characteristic separation factors of the feed and draw solutes. This model was then used to elucidate the effects that operating conditions and membrane properties have on the performance of FO processes in this fundamental thermodynamic limit. Specifically, we highlight the roles the reverse flux selectivity, the flow ratio, and the solute permeability play in determining the amounts of water extracted and solutes exchanged in FO modules. This study suggests that developing membranes that are more selective than those currently available will help to reduce the extent of solute exchange (i.e., separation factors) in FO modules, but likely will have less impact on the recovery rates that are achieved. We also report that, when the exchange of solutes through the membrane is asymmetric, a pseudo-osmotic equilibrium state can result. Furthermore, we demonstrate the importance of the flow ratio. In particular, for a countercurrent configuration, a critical flow ratio is identified that indicates whether diluting the draw solution or concentrating the feed solution limits module performance. The recovery rate is maximized in cases where concentrating the feed solution limits performance. The expressions presented in this work provide a simple but accurate estimate for the water recovery and solute leakage as functions of operating parameters and known membrane properties. Previously, this information, which is essential for evaluating the feasibility and viability of an FO process, could be obtained only by solving a complex system of nonlinear differential equations. As such, the equations presented here can be used to quickly evaluate the feasibility of a proposed process (e.g., whether the proposed process can achieve the desired recovery rate). Furthermore, the equations can be used to predict the concentration and volume of the draw solution exiting an FO module, which determine the size and energy demand of downstream unit operations used for draw-solute recovery. This information is essential when critically evaluating the viability of hybrid FO processes that include an FO module in tandem with a downstream unit operation used for drawsolution regeneration. Using the expressions developed herein, future studies will explore how varying operating conditions and membrane properties impact the overall design of hybrid FO processes.

Figure 8. Recovery rate as a function of draw and feed separation factors for different ratios of the solute permeabilities, B1/B2. The same inlet flux (β = 1) and water permeability (Ã am = 3 L mol−1 s−1) were used in all cases. Entering concentrations of y1in = 1 M and x2in = 5 M were used. Values for the countercurrent configuration are plotted in white, and values for the cocurrent configuration are plotted in black. The solid lines describe the asymptotic limit where the system is at osmotic equilibrium. The discrete circles describe the path obtained by numerically integrating the system of nonlinear governing equations (i.e., eqs 1−4). Note that the scales for SF and SD change for the different ratios of solute permeabilities.

near osmotic equilibrium. This counterintuitive result occurs because the ratios Ã /B1 and Ã /B2 are large (at least equal to 10 L mol−1) for all of the cases studied, which produces a pseudoosmotic equilibrium state. Solvent permeation drives the system toward osmotic equilibrium, and once the system is close to osmotic equilibrium, the solvent permeation is able to respond rapidly to the small osmotic pressure differences generated by asymmetric solute permeation. Therefore, the behavior of the system is well-predicted by the results for osmotic equilibrium derived above. In Figure 8a, the system is driven to osmotic equilibrium primarily by the exchange of solvent because the hydraulic permeability is 2 orders of magnitude higher than the draw solute permeability (Ã /B2 = 100 L mol−1) and 1 order of magnitude higher than the feed solute permeability (Ã /B1 = 10 L mol−1). Therefore, the recovery rate at osmotic equilibrium is comparable to the case of equipermeable solutes. Because feed solute permeates from the feed solution into the draw solution more rapidly, the feed separation factor is higher than the draw separation factor (i.e., the rejection of the feed solute is reduced). This produces a larger osmotic potential in the draw solution, and the recovery rate increases with increasing

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ASSOCIATED CONTENT

* Supporting Information S

Detailed derivations of recovery rate as a function of separation factors for both countercurrent and cocurrent systems, recovery rate at osmotic equilibrium for cocurrent systems, and recovery rate in the limit of osmotic equilibrium for equipermeable I


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solutes. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS A.S.O. gratefully acknowledges support for this project from the Vincent P. Slatt Fellowship for Undergraduate Research in Energy Systems and Processes at the University of Notre Dame.

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NOMENCLATURE A = hydraulic permeability coefficient (m s−1 Pa−1) Ã = modified hydraulic permeability coefficient (A = nRgT) (m L mol−1 s−1) am = specific membrane area (m−1) Bi = permeability coefficient of solute i (m s−1) D = volumetric flux of the draw solution (m s−1) F = volumetric flux of the feed solution (m s−1) HTUw = height of a transfer unit for water (m) Js = solute flux (mol m−2 s−1) Jw = water flux (m s−1) L = contactor length (m) n = number of dissolved species generated by a solute R = recovery rate Rg = ideal gas constant (L Pa mol−1 K−1) SD = draw separation factor SF = feed separation factor T = temperature (K) x = molar concentration of a solute in the draw solution (mol L−1) y = molar concentration of a solute in the feed solution (mol L−1) z = distance along the contactor (m)

Greek Letters

β = ratio of the inlet draw flux to the inlet feed flux βcrit = flux ratio that defines the transition from approaching osmotic equilibrium at z = 0 to approaching osmotic equilibrium at z = L for a countercurrent system Δπ = osmotic pressure differential (Pa) Subscripts

0 = evaluated at z = 0 1 = feed solute 2 = draw solute in = inlet L = evaluated at z = L OsmEq = evaluated at osmotic equilibrium out = outlet Superscripts coc

= cocurrent configuration = countercurrent configuration

counterc

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Forward Osmosis Processes in the Limit of Osmotic Equilibrium

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