Substantial Changes in the Transport Model of Reverse Osmosis and

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Substantial Changes in the Transport Model of Reverse Osmosis and Nanofiltration by Incorporating Accurate Activity Data of Electrolytes William Mickols* Mickols Consulting LLC, 3318 175th Lane SW, Tenino, Washington 98589, United States ABSTRACT: The transport model for reverse-osmosis membranes and nanofiltration membranes was reformulated to include the large changes in the chemical activity of solutes in solutions employed to test these membranes. The approach presented herein uses a solution-diffusion model for membrane transport and a more accurate representation of the test solution and the partitioning of the solutes into the membrane. This approach corrects the simplifying assumption in the Merten model of highly diluted solutions and predicts the large changes in the passage of solutes due to increased solute concentration. These changes are due to the large changes in the activity coefficients of the solutes outside the very dilute regions. These changes can significantly influence saltand water-transport coefficients (e.g., A- and B-values).

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INTRODUCTION Reverse osmosis, nanofiltration, and now forward osmosis are part of the greater than 150 billion USD membrane-based desalination business that supplies water across the globe. These techniques are used to produce drinking water from many sources, industrial water from waste streams, ultrapure water for the electronic industry, the supply of water for medical use and research, and many other uses. Understanding the physical processes that control the water transport and solute transport has been an ongoing area of study since reverse osmosis was first proposed. Reverse osmosis, nanofiltration, forward osmosis, and electrokinetic desalination methods can generate and use high concentrations of salts and solutes. However, much of our understanding of these processes comes from theories developed for dilute solutions. Unlike previous approaches, the model proposed in this work extensively uses the activity coefficients to calculate the solute partitioning and transport. This approach corrects for a major simplification in the Merten, Lonsdale1,2 theory of these processes and identifies some simple approaches to correct the basic transport measurement of these membrane processes and to estimate the distribution of solutes in these complex membrane systems. In many cases, correcting this oversimplification will substantially change the estimates of the water- and salt-transport parameters of reverse osmosis and nanofiltration membranes. This allows the derivation of the basic transport equations without the dilute-solution approximations previously used to describe reverse osmosis, nanofiltration, and forward osmosis. The previous accepted model by Merten and Lonsdale showed that the water transport depends on the applied pressure and osmotic pressure of the feed solution and permeating solutions. The osmotic pressure is due to the activity of various salts in these solutions. Salt transport, which was previously modeled using standard solution-diffusion models, depended only on the difference in salt concentrations © XXXX American Chemical Society

across the membrane. This approach did not include the large changes in salt activity that are strongly dependent on the concentration. The Merten and Lonsdale1,2 model predicted the water transport by using the classic A-value (water transport coefficient, eq 1) and the difference between the osmotic and applied pressures across the membrane. The Merten and Lonsdale model describes the salt-transport with the phenomenological B-value (eq 2) and ignore differences in activity. The membrane-transport model was then reformulated by Paul3 in more general terms to include the Maxwell−Stefan relationship of solute transport of multicomponent mixtures through polymers. Others have used dilute-solution approximations to obtain equations similar to those of the Mertens model.4 These membranes are currently used for a wide variety of waters to produce ultrapure water from already “pure” water, and drinking water from seawater. In high-recovery seawater desalination systems, the concentration of solutes in the reject stream can approach a value of 90 000 ppm, and in forward osmosis can reach values that are 2−5 times higher than these values. In brackish water separations, the salts may start at less than 1 ppm and can be in excess of 30 000 ppm. The activity of the salts in these solutions is the “apparent” concentration of these solutions irrespective of the actual concentration. The activity is defined as the activity coefficient times the concentration. A 1 molar salt solution with an activity coefficient of 0.5 behaves chemically as if the concentration were 0.5 molar. There is a well-developed theory of the thermodynamics of solutions and electrolytes, which describes the strong nonlinearity in the apparent concentrations of salts and solutes as a function of the actual measured concentrations; Received: August 24, 2016 Revised: September 23, 2016 Accepted: October 5, 2016

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DOI: 10.1021/acs.iecr.6b03248 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research

concentration from the large changes in the activity of the solution irrespective of the membrane. A Simple Derivation of the Merten and Lonsdale Model. This simple and powerful model was derived on the basis of the classic solution-diffusion model, in which the classic diffusion equation was modified by replacing the concentration with the chemical potential. The standard solution-diffusion model was used to describe transport of solutes such as salts. The solute flux can be expressed as follows (eq 3): dC Js = −D dx , where integration across the membrane gives:

the nonlinearity can vary greatly with the solute. The nonlinearity is the result of the solvent (water)−solute and solute−solute interactions; in the case of salts, the ion−ion interactions can be very strong. Looking at the range of 0−5 molal for many salts, the activity coefficients vary dramatically. For aqueous NaCl solutions, the activity coefficient ranges between 1 and 0.65;5 for aqueous CaCl2 solutions, the activity coefficient ranges between 1 and greater than 12, with a minimum value of 0.45.6 The activity coefficient of MgSO4 can be as low as 0.045.5 The nonlinearity in the osmotic pressure is smaller than the activity coefficient and is described by the osmotic coefficient, which ranges between 1 and 0.92 for NaCl, and between 0.05 and 5 for other salts. Nanofiltration membranes pass certain amounts and types of solutes and in many cases use highly charged membranes. These are characterized by a very complex dependence of the flux and salt passage on the presence of certain salts and their ionic strength. These complex relationships are very dependent on the specific structure of the membranes. Electrokineticmembrane separations are also used to separate salts from water and can utilize anionic, cationic, or bipolar membranes, and require the coupling of the electric field and ion currents to the solute and water transport. In these processes, the activity coefficients also play an important role. In all of these studies, more complex interactions of water and salt with membranes have been considered. In contrast, in reverse osmosis, the model for the interactions between water solutions and membranes is very simple, and the large changes in the activity coefficients of water solutions are not included. In the Merten and Lonsdale model,1,2 water and salt fluxes are defined, respectively, as follows:

Jw = A(ΔP − Δπ )

(1)

Js = BΔC

(2)

Js =

D mf KD f (m − m mp) = (m − mp) = BΔm L L

(3)

where K is the solubility of the solute in the membrane, D is the diffusivity in the membrane, L is the thickness of the membrane, m is the concentration of solvent or solute, mmf is the concentration of the solute in the membrane on the surface facing the feed solution, and similarly mmp is the concentration of the solute in the membrane on the surface facing the permeate solution. Using the equilibrium partition coefficient, K, the amounts of the solute found on the feed and permeate side of the membrane are calculated on the basis of the bulk solution concentrations in the feed and in the permeate compartment. These are defined in Table 1. Table 1. Membrane Operation Requires Two Solution Chambers Separated by at Least One Membranea

where ΔP is the difference in pressure across the membrane, Δπ is the difference in osmotic pressure across the membrane, and ΔC is the difference in the concentrations of solute (salt) across the membrane. Because the transport of water through the membrane can be higher than diffusion of the other solutes away from the membrane, the other solutes can build up on the surface. This is called polarization, and the polarization ratio is the surface concentration divided by the bulk concentration. The effect of this polarization ratio on the activity of the solutes is not included in eq 2. The effect of increasing concentration on the activity can be high, and as each solute is concentrated the mixed ion effect as well as other nonlinear effects increase. In the transport equations described in this new derivation, the partitioning of the solvent and solutes into the membrane makes use of the solution-diffusion model but includes the activity of the solutes and solvents in the solutions and does not require a particular membrane model; it only requires a chemical activity difference on either side of the membrane. The transport coefficient proposed in this work extrapolates to the standard coefficient of the Merten and Lonsdale1,2 derivation, under the dilute solution approximation (activity coefficient = 1). Not including these effects can underestimate or overestimate the B-value by orders of magnitude. Also, because of the strong concentration dependence of activity, the use of the B-value to predict membrane performance while using wide concentration ranges is problematic. Without this theoretical foundation, it is impossible to separate performance changes in membranes due to the change in solute

feed solution

membrane feed side

membrane permeate side

reject solution

[solvent] = mf [solute] = mfi activity of: solvent = af solute = afi

[solvent] = mmf [solute] = mmf i activity of: solvent = amf solute = amf i

[solvent] = mmp [solute] = mmp i activity of: solvent = amp solute = amp i

[solvent] = mp [solute] = mpi activity of: solvent = ap solute = api

a

This table shows the definitions of concentrations, and activity in the four compartments defined by permeate, membrane permeate side, membrane feed side, and feed.

The B-value is a phenomenological coefficient that describes a more complex system. Modern reverse-osmosis membranes are multilayer structures with multiple functionalities in each layer. Despite such complexity, these are represented by a single layer and a constant B-value. The transport of the solvent (water) was described using the chemical potential of the water solution. These authors considered the chemical potential across the membrane to be proportional to the concentration of water in the membrane as shown in eq 4: Jw = −D

dC w du ≈−D w dz dz

(4)

The chemical potential of the solvent (l) mixed with a solute i is defined with reference to a standard state uol . ul = ulo + Vp l + RT ln(ai)

(5)

The osmotic pressure across a semipermeable membrane with two solutions with a pressure difference across the membrane is defined by the equilibrium of the chemical potential. One side of the membrane is designated as the feed (f in the superscript), and the second solution is designated as the permeate (p in the superscript): B


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field. Also included in the Appendix is a brief description of how polymer type and concentration can greatly affect the activity of water and salts in the membrane. The activity of the water and solutes including salts can be reduced by factors of 10 or more depending on the polymer type and concentration. Reverse osmosis and nanofiltration membranes are defined by their functionality and not by their chemical structure. These membranes have been made from a wide variety of different materials including liquids, solids, solid polymers, gels, alloys, and condensates to name a few.

f p ulo + Vp + RT ln(aif ) = ulo + Vp + RT ln(aip) l l

The difference in osmotic pressure is defined by rearranging the equation and obtaining: π = pf − p p =

RT [ln(aip) − ln(aif )] Vl

When the solute concentration in the permeate solution is zero, this reduces to the following: π = −RT /Vl ln(aif )

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(6)

METHOD Membrane Transport. Two approaches have been proposed to describe the way solutes partition into the membrane phase: the first is to consider that each component independently partitions into the membrane (activities, concentrations, or mole fractions, can be used to describe the partitioning); the activity coefficient can then be defined using the concentration or the mole fraction. The second method uses full representations of the thermodynamics of the polymer solution and feed and permeate solutions. This includes the effect of concentration and activity of water, solutes, and polymers. Partitioning of solutes and solvents into a membrane gives the pool of solute or solvent in the membrane that can be transferred across the membrane. By definition, at equilibrium, the solvent or solute activity in the membrane is equal to that of the feed solution compartments, which can be written as

where Vl is the partial molar volume of the solvent. Using the definitions given in Table 1, the investigators suggested that the membrane chemical potential is proportional to the solution chemical potential; eq 4 was thus conventionally integrated across the membrane. Using eqs 5 and 6, the following expression was obtained: Jw =

⎡ KD ⎤ KD f (ui − uip) = ⎢ [V (p f − p p ) ⎣ L ⎥⎦ l L

+ RT[ln(aip) − ln(aif )]

(7)

At this point, they used the definition of the osmotic pressure to write Jw = A(Δp − Δπ )

(8)

In this model, the nonlinearity of the osmotic pressure of the solutions with concentration is accurately described for water transport. The A-value does use the chemical activities of the salts and water (as reviewed later). The osmotic pressure7,8 difference arises from the difference in the logarithm of the activity. This factor is absent for the solute transport from the feed and permeate compartments. The membrane model described here has four compartments (Table 1): two solvent-solution compartments (with the arbitrary names feed and permeate) and one membrane compartment with one side in contact with the feed and the other side in contact with the permeate. The membrane phase contains the same solvents and solutes and is denoted in a similar manner using the compartment descriptor m. The permeate side and feed compartments can be differentiated using p and f (i.e., concentration of NaCl in the feed compartment = mfNaCl, activity of a solute i on the membrane surface on the permeate side = amp i ). At equilibrium, the partitioning of the feed and permeate components into the membrane can be described using a partition coefficient. It is important to note that the feed and permeate sides are identical at equilibrium. In the model described below, the discussion is focused on the partitioning of the solvent and solutes into the membrane; this approach makes use of the classic solution-diffusion model for membranes and does not require a particular membrane model; it only requires a chemical activity difference on either side of the membrane. The model proposed in this work predicts the standard Merten and Lonsdale1,2 model under the dilute solution approximation (activity coefficient = 1). A brief description of solution electrolyte theory is presented in the Appendix. It reviews the temperature, pressure, and concentration of the osmotic pressure, which is derived from the activity coefficients of monovalent, multivalent, and mixed salt solutions containing multiple ion types. The selection of molal versus molar is also described. In this Article, molal will be used extensively because historically it has been used in this

alf = almf , which can be written as: γlf mlf = γlmf mlmf

(9)

Using eq 9, the traditional operational definition of the partition coefficient (Q) for the solvent in the membrane is given by rearranging eq 9 to give the ratio of the activity coefficients. mlf mlmf

=

γlmf γlf

=Q (10)

This is a very brief description of this activity-based equilibrium constant. A more complete description includes the binding energy of the solutes and solvent to the membrane (using the Gibbs energy of binding, ion exchange capacity, etc.), the temperature dependence, concentration dependence, and the interaction between the solutes in the membrane. Part of that discussion is included in the Appendix. For this discussion, the membrane is assumed to be homogeneous. For more complex solutions and experiments, such as polymer in water or complex solutions in equilibrium with a polymer, the equilibrium condition is more completely described by including the chemical potentials of the solution and the water polymer solution. The chemical potential of a simple solution (like water) is described in eq 5. Because the chemical potential of a volatile solution is pressure dependent, it has different distributions in a two-phase system under different pressures. Water is incompressible; that is, concentration does not change with pressure, but the effective concentration does. The chemical potential of the water− polymer solution is the sum of the chemical potentials of the components. The equilibrium condition across the membrane− feed interface can be written as the sum of the chemical potentials, as follows: C


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Industrial & Engineering Chemistry Research f o mf ulo + Vp l + RT ∑ ln(ai ) = ul + Vp l + RT ∑ ln(aj ) i

j

(11)

This general equation shows that the equilibrium condition is both temperature and pressure dependent. As either the temperature or the pressure increases, the chemical potential increases. In the case of a membrane that takes up only water and excludes all other solutes, the equation can be written as f mf mf Vp l + RT ∑ ln(ai ) = Vp l + RT[ln(al ) + ln(a p )] i

(12)

afm p

where is the activity of the polymer or liquid membrane at the feed/membrane interface. During membrane operation, different concentrations of solute in the feed and permeate compartments are possible. The steady-state assumption affirms that the near-surface membrane is at equilibrium with the solution compartment. The difference in activity between the two compartments is therefore equal to the difference between the activities of the two surfaces of the membrane. Equation 9 is applicable to both sides: aif − aip = aimf − aimp

Figure 1. Activity coefficient of five common salts (NaCl, KCl, CaCl2, Na2SO4, and MgSO4) as a function of concentration (molal). Data from the Handbook of Aqueous Electrolyte Thermodynamics, ref 8.

coefficient, their difference can be reduced by over a factor of 10. This is critical to understand the membrane performances. The salt passage and flux can be experimentally measured; they are typically corrected on the basis of the polarization, which is estimated on the basis of the concentration of salt at the surface of the membrane; this is, however, nontrivial, as discussed in the next section. The A- and B-values are generally measured at different solute concentrations, and then used over a narrow concentration range. Equation 14 shows that the membrane surface concentration is dependent not only on the concentrations of the solution compartments, but also on the activity coefficients, which are concentration and solutionchemistry dependent. Thus, the measured B-values contain the thermodynamic quantities specific to a solution with a certain concentration. If the activity coefficients contained in the measured B-values are not removed, the reported B-values can strongly vary depending on the exact concentrations of all of the solutes used during the test. The models for the partitioning of solutes into the membrane and the transport across the membrane are not required to be linear, and the diffusion coefficients can also be complex functions. However, for this model, the diffusion rate, partition coefficient, and solubility in the membrane are considered to be linear and are best described by eq 3. In this case, the membrane is homogeneous, and the activity coefficients across the membrane are constants. This is in accordance with the Merten Lonsdale1,2 model. Thus, eq 3 can be rewritten as dm Ji = −D dxi , with integration across the membrane, and using eq 9 gives

(13)

This can be rewritten as γi f mif − γi pmip = γi mf mimf − γi mpmimp

(14)

Equation 13 shows that the difference in the concentration of a solute across the membrane (designated by m) is equal to concentration times the activity coefficient in both the permeate and the feed compartments. The chemical activities in both the feed and the permeate compartments and membrane are dependent on the temperature, pressure, and concentration of other solutes. The concentrations used in the primary transport equation (eq 3) are not simply dependent on the concentrations in the feed and permeate compartments; they are also dependent on the activities, which equal bulk concentrations times the activity coefficient. The initial derivation of the solution diffusion equation employed the factor KD to calculate the concentration in the membrane from the solution concentration. This is equivalent to assuming that the activity coefficients are not concentration dependent, and the binding is simply described by a constant. In this modified model presented in this contribution, solution-thermodynamics changes are specifically included. This does not require any change in the membrane model. The large changes in the activity coefficients that this modified model includes are shown in Figure 1, which displays some of the most commonly studied salts in membrane transport. The large changes in activity coefficient that require this modification include the simplest salts. The activity coefficients of five common salts versus concentration are shown in Figure 1. The coefficients have been plotted as a semilog function, because the activity coefficients differ by several orders of magnitude, even though they all have an activity coefficient of 1 at the dilute solution limit. Notably, the activity of MgS04 is typically near 0.1, while that of 1:1 salts is closer to 0.7. MgSO4 has a very low salt passage (often reported as 0.1%), while NaCl has a passage of 0.4%−1% in standard reverse-osmosis membrane testings under similar conditions. When the Bvalues of NaCl and MgSO4 are corrected for the activity

⎛ γ fm f γ pmip ⎞ i Ji = D/L(mimf − mimp) = B⎜⎜ i mf − i mp ⎟⎟ γi ⎠ ⎝ γi

(15)

If the partition coefficients (the ratio of activities, eq 10) are the same on both sides of the membrane, and the activity coefficients are 1, then a simple K and the classic B-value can be used. In the case of differences in the activity coefficients between the two solution compartments, and of unknown activity coefficient within the membrane (which is then assumed to be constant), then a modified B-value designated Ba can be used: D


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Industrial & Engineering Chemistry Research Ji =

D (γ f mif − γi pmip) = Ba(γi f mif − γi pmip) Lγi m i

For an osmotically driven process such as forward-osmosis, they use draw solutions that significantly exceed these concentrations. Similarly, for the concentration of acids and bases by diffusion dialysis, the concentrations can exceed these figures. The concentration of mineral salts using electrodialysis also operates at these concentrations. Figure 2 shows how the osmotic coefficient and the activity coefficient of NaCl change, depending on its concentration.

(16)

The presence of other solutes or polymers can affect the activity coefficients of the solutes described here; however, this dependence is included in the activity coefficient. For highly rejecting membranes, the solute concentration in the permeate compartment is small, and the solute activity coefficient may be approximated as 1. This new derivation allows for the calculation of the relative B (eq 17) as the ratio of the experimentally measured B at a given concentration to the B measured under standard conditions. The relative B then varies as a function of the large deviations in the measured B value due to the change in the activity coefficient, which depends on the solute concentration. In the limit of very high rejection, the solute concentration in the permeate solution is very low, and it does not significantly change the solute transport; therefore, the relative water and salt transport (relative A- and B-values) can be estimated for most common test conditions: relative B =

B(γi f ) at concentration mif B(γi f ) at standard concentrations

(17)

The relative B- and relative A-values (which are similarly derived) are used to estimate the maximum size of the error commonly observed in membrane testing and operation. This will be described in more detail in the next section. The relative A-value reduces to the ratio of osmotic coefficients at the experimental concentration divided by the coefficient at the standard test condition. The relative B-value reduces to the ratio of activity coefficients at the experimental concentration divided by the coefficient at the standard test condition.

Figure 2. Osmotic and activity coefficients of NaCl at 25 °C as a function of concentration, from 0 to 2.5 M. Osmotic and activity coefficients are from ref 5.

Figure 2 covers the range of interest for desalination using pressure-driven separation processes. Both the activity coefficient and the osmotic coefficient show minima; however, the positions of the minima differ, partly due to the definition of the osmotic coefficient, which includes the density of the solution. NaCl behaves as if the concentration were 35% lower than the actual concentration at 1.05 m. This value is in the expected range of seawater desalination. For commercial FT30-type membranes and disulfonated polyarylethers (Xie et al.),9 1 M solutions have been used as test solutions. If the Bvalue is measured at 1.05 M, it needs to be multiplied by 1/ activity coefficient (i.e., 1.52). Therefore, the activity of NaCl substantially affects the interpretation of the membrane performance and the value of the measured salt and water transport in most of this range. The large difference between the osmotic coefficient (Figure 2) and the activity coefficients clearly shows that estimating polarization without taking the activity into account will cause large errors. This can significantly affect modeling water and salt flux using the membrane A-value and B-value. When the polarization is determined, the surface concentration is estimated on the basis of either the changes in the measured flux as a function of solute concentration or pressure or the change in salt transport as a function of solute concentration and pressure. Because the Merten Lonsdale1,2 model does not include the activity coefficient, the effective salt concentration can vary by the activity coefficient. In NaCl solutions, this can be off by 50%. The relative B-value for NaCl (Figure 3A) is calculated using well-established standard tests in the reverse osmosis industry for both standard brackish water and seawater membranes. The “standard” brackish water test used here is 2000 ppm of NaCl (0.0365 M), at an applied pressure of 225 psi (1.55 MPa). For reverse osmosis of seawater, the standard seawater test condition is 32 000 ppm of NaCl (0.602 M, 0.630 m), and

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RESULTS AND DISCUSSION To quantify the impact of not correcting the measured B-value for the solution properties, the relative B-value can be calculated across the operational concentration range of the membranes. This effect partly explains why different research groups have obtained different A- and B-values as well as different projected salt passage and flux values at different concentration of salts. To demonstrate the importance of this point for single salts and for complex waters, a brief review of the methods used to process waters is needed. Because natural waters contain many different ions, the overall total concentration is often defined as the total amount of dissolved solids, or in terms of ppm (mg/kg). The reported fluxes and salt passage for brackish-water reverse-osmosis membranes are often defined at salt concentrations ranging between 1000 and 2000 ppm of NaCl (0.0365 M); the corresponding value for seawater membranes is 32 000 ppm (0.602 M, 0.630 m). Both membrane types have recommended feed-concentration ranges. Within the available membranes for brackish water and seawater, some types are intended for higher ionic-strength and some for lower ionic-strength waters, but their performance is defined at a few specified concentrations. In general, the brackish water membranes are used in waters with salt concentrations ranging between 10 and 30 000 ppm. This range includes the recommended feed values as well as the concentrations of the reject solution in higher recovery processes. For seawater membranes, the starting waters begin at around 10 000 ppm (waters found in estuaries), and the final reject stream may have concentrations as high as 90 000 ppm. E


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Figure 4. Relative osmotic coefficient and relative B-value (eq 17) for CaCl2 over a wide range of concentrations. These relative numbers are normalized to the estimated B-value on the basis of standard test conditions (100 ppm of CaCl2, 0.5 mM) versus molar CaCl2.

bicarbonate or chloride (sulfate is less frequent). The main solutes that make desalination necessary are the main hard water components Ca, Mn, Mg, and bicarbonate along with various forms of iron. Because pH adjustments of hard waters with HCl are commonly used to increase the solubility of the various CO2-derived components, even bicarbonate-based waters often contain large amounts of chloride before the desalination step. It is common for surface waters and near-surface well waters to contain 100−300 ppm of total dissolved solids and with high bicarbonate, therefore scaling control is required. If water being processed contains calcium bicarbonate as the main salt and requires a pH adjustment, then this may result in 100−200 ppm of CaCl2 (1−2 mM). Because of high recovery designs, the reject stream may contain a range from 4−60 mM. CaCl2 may also be used as a draw solution in forward osmosis, with the advantage that smaller changes in concentrations produce larger changes in the osmotic pressure (i.e., in some cases, up to 12 times the expected value). Other 2:1 salts show similar nonlinearity and have been suggested for use as draw solutions in forward osmosis. Figure 5 shows the range of CaCl2 concentrations from 0 to 0.06 M. Over this range, the relative B-value changes from 113% to 65%, confirming the difficulty of predicting the transport of CaCl2. However, in this case, the relative osmotic pressure difference is only 14%. High-rejection membranes eliminate most of the calcium, but nanofiltration passes a substantial fraction of CaCl2, and the energy required for the separation would be poorly predicted without using activity coefficients. Thus, the calcium passage may be either underestimated or overestimated, depending on the salt concentrations. Similarly, predicted osmotic coefficients and B-values for other ions (i.e., phosphate, various iron species, Mn, acetate, etc.) may be incorrect. The mixed-ion solutions can have strong ion−ion interactions, which can further complicate the interpretation. Noninclusion of these factors in nanofiltration membrane models would result in a poor prediction of the strong ion−ion interactions in the feed and permeate solutions. Both experiment and electrolyte theory show substantial changes in the activity and osmotic pressure when NaCl and CaCl 2 are mixed. Shatkay10 reported that, for a CaCl2 concentration of 1.52 mM, the CaCl2 activity changes from 0.97 to 0.65 when the NaCl concentration changed from 0.01

Figure 3. Change in relative salt transport (relative B-value, eq 17) (A) and relative osmotic coefficient (B) versus molar concentration of NaCl as a function of the concentration of feed NaCl concentration. The relative value for the brackish water membrane is 100% under standard conditions (2000 ppm of NaCl). The standard seawater membrane has a value of 100% under standard seawater conditions (32 000 ppm of NaCl). Osmotic and activity coefficients are from ref 5.

an applied pressure of 800 psi (5.52 MPa). This includes no correction for concentration polarization. Figure 3 shows the relative osmotic pressure and relative Bvalue for NaCl as a function of the concentration of NaCl across the operation range of both the standard brackish water and the seawater membranes. For membranes tested at 1.0 M NaCl, the actual Ba-value is about 50% higher than reported. The relative B-value for the brackish water membrane varies from 115% to 80% under the expected conditions. This shows that the measured B-value may change by greater than 40% (115/80), depending on the concentration of the salt. The estimated osmotic pressure deviates by 7% from the standard condition (Figure 3B). For seawater membranes, the relative Bvalue varies by 10% with respect to the measured values under the standard conditions. A 7% change in the apparent osmotic pressure over the same range occurs. Even this seemingly small change in osmotic pressure can greatly affect the projected performance of seawater desalination in the high salinity portion of the separation. Figure 4 shows the relative A- and B-values over a wide concentration range for CaCl2; the apparent CaCl2 B-value drops to 50% of the standard test B-value, only to increase by 12 times the expected value at higher concentrations. The apparent osmotic pressure in this range reaches a minimum of 88% and increases to more than 300%. The activity of mixed salt solutions greatly affects the activity of divalent salt solutions. This effect is very pronounced in natural waters. Two main cations, Ca or Na, can be found in brackish waters; the most common anions found are F


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Prediction of Separation Factors and the Calculation of Individual Membrane Separation Factors. Prediction of separation factors for different salts is normally done through modeling transport through a membrane. Substantial salt dependence on rejection and passage is predicted due to the strong dependence of activity of different salts and ionic mixtures. This source of separation factor has not been included in any model until now. Modeling an indiscriminate membrane that shows no dependence of the B-value with salt type shows the difference using a constant B-value (eq 15) versus the corrected Ba-value (eq 16). Considering the three salts NaCl, CaCl2, and MgSO4 (1:1, 2:1, and 2:2 salts) and using a B-value of 0.1, the results of using the correct activity can be compared to calculations without using activity. If the feed solution was a constant 10 mM and with a membrane with a constant B-value of 0.1 (mM based), the membrane would transport 1 mmol of any salt (per unit area and time). When the activity coefficient corrected Ba-value of 0.1 is used, then the amount of salt transported would be the concentration times activity times the Ba-value times the feed concentration. The ratio of the mmoles of salt transported for MgSO4/CaCl2/ NaCl would be 0.2/0.55/0.95 at 25 °C. This assumes the experiment is done in a diffusion cell where the permeate concentration is negligible or if the rejection was very high leading to a negligible permeate concentrate. This predicted separation is without any membrane separation specificity at all. These ratios change with salt concentration, salt type, and mixtures of salts. This order of separation has been the source of speculation for many years. This is the first example of a priori calculation of the order of separation salts due to electrolyte theory. Different salt concentrations and membrane rejections will change these ratios. Mixed ion effects can change these activities by at least another 40%, In Figure 7, the NaCl permeability data of a sulfonated nanofiltration membrane obtained from Xie et al.9 are plotted

Figure 5. Relative B-value (eq 17) and relative osmotic coefficient over a concentration range typical of easily processed hard waters. The standard condition used to calculate the relative osmotic coefficient and activity coefficient is 100 ppm of CaCl2 or 0.5 mM CaCl2. Osmotic and activity coefficients are from refs 6 and 16.

mM to 0.02 M. A literature survey by Butler11 gives a compendium of different concentrations of NaCl with different concentrations of CaCl2 and the resultant change in activity coefficient of CaCl2. Figure 6 shows the changes in the CaCl2

Figure 6. Mixed-ion effect showing the CaCl2 activity coefficient change as a function of the molar concentration of NaCl. Osmotic and activity coefficients are from ref 17.

activity as the concentrations of CaCl2 and NaCl increase. While the initial effect is small, during the purification process, Ca and Na concentrations increase by 5−20 times (i.e., 80− 90% recovery). The mixed ion effect would give an apparent change in B-value of 30−40%. This effect is only due to the addition of NaCl to the solution. A similar effect is predicted when MgCl2 is used as a draw solution. Wu and Scatchard12,13 measured the activity coefficient of MgCl2 in the presence of MgSO4; the activity of MgCl2 dropped by about a factor of 3 when part of MgCl2 was replaced with MgSO4. The addition of monovalent ions reduces the back transport of MgCl2 when a sufficient quantity of MgSO4 is present in the system. Other multivalent counterions should improve this further; this effect is observed for all 2:1 salts. Along with the advantages in terms of cost and solubility,14 the addition of monovalent and divalent ions may reduce the back transport of the osmolyte.

Figure 7. Data obtained from permeation cell experiments adapted from Xie et al. The B-values were plotted versus NaCl concentration (molar), and the Ba-values were plotted versus the NaCl activity (molar).

versus the molar concentration of NaCl. These data are then corrected for the activity of the NaCl solution and replotted. This corrected permeability (Ba-value) is then plotted versus the molar activity (molar concentration times the activity coefficient). The data were collected in a permeation cell; G


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separation order is predicted by the estimates of activity coefficients in the solution without considering the membrane separation. This also predicts changes in the apparent permeability based on the concentration, solute and solvent type, and mixed ion effects (Figures 1−7). These effects are commonly ascribed to nanofiltration membranes but are found in all membrane systems and predicted with electrolyte theory. This is the first prediction of separation factors based on the solution physical chemistry and electrolyte theory. When moving from a concentration based analysis of salt/ solute and solvent transport to an activity-based model of membrane transport, several important things were found. The relative B-value (eq 16) and relative osmotic coefficient were used to show the error due to not including the activity in the analysis of the B-value (eq 15). For different salts and different common testing conditions (Figures 3−6), some corrections are relatively small, but in some cases there are differences by factors exceeding 10. This correction to the B-value (Ba) also corrects mixed ion effects (Figure 6). In most cases, the correction to the B-value is larger than the correction to the osmotic coefficient. Modern researchers would always correct the osmotic coefficient when studying large ranges of concentrations, so it is surprising that the activity correction has not been included before. This model also allows for the direct calculation of the difference in activity across a membrane (eq 16). In the case of pressure-driven separations, both the pressure driven water transport and the backward permeation of water can be calculated directly. For solvents like water and other solutes, the difference in activity across the membrane differs from the concentration differences by small amounts as well as above 10 times the concentration difference. Developing an activity-based membrane model of transport can be difficult while using the previous models. This modified model allows the researcher to calculate the activity difference across membranes in operation while only using commonly collected rejection and flux data. When combined with static solubility and activity data, an exact experimentally measured activity-based model can be developed. This should allow for much more accurate models of separation in membranes. The development of an activity-based membrane transport model when combined with modern electrolyte theory allows several new important steps to be taken. A new source of concentration, salt type, and mixed ion-dependent separation was calculated on the basis of the activity of the solutions. The expected order of separation of the salts (MgSO4/CaCl2/NaCl) was predicted. This exact order is expected to change with concentration and with mixed ions. This new separation effect should be added to past membrane modeling to improve their accuracy. A method to directly measure the membrane activity difference across a membrane allows for a membrane activitybased model using the theory of polymer/solute activities and transport in polymers. The correction of NaCl permeability of a polyelectrolyte membrane for activity and comparing the activity corrected permeability versus NaCl activity showed a substantial change from the previous concentration-dependent measurements. This model has already shown the importance of using activity coefficients in interpreting the transport of solvents and solutes, in predicting separation factors, and in determining membrane activity differences and back transport of solvents in membrane systems.

therefore, the polarization is minimal and does not need to be corrected. The NaCl permeabilities were corrected for activity by a factor of 1.5 and 1.3 at 1 and 0.2 molar NaCl, respectively. Xie9 reported a change by a factor of 5 in the permeability per molar concentration in the range 0.2−1 M NaCl; when this was corrected for the solution activity, the corrected permeability (Ba-value) per molar activity was 6.8-fold. If the permeability versus activity was linear, then a simple linear partitioning could be proposed. Using Ba-values versus NaCl activity, there was a linear range from 0 to 0.4 m activity. In the modeling of salt and solvent transport through polymers, the permeability, diffusivity, and solubility should be compared to the activity of the salt or solute, rather than the simple concentration. This approach also allows for the estimation of the membrane parameters from the rejection and salt passage measurements. Equation 13 shows the difference in activity of permeate and feed is equivalent to the difference in activity across the membrane. When bulk activities are known and ultrathin membranes are used, a ratio of activities allows for estimates of the activity in the membrane. As an example, the water activity and activity ratio of water across a membrane can be easily done. At 1 molar NaCl, the water activity is 0.995, while the collection vessel contains water with negligible NaCl with an activity of 1. The water activity ratio is 0.974 (0.995*54.3 m/55.5 m), and the difference in activity of water across the membrane is 1.46 molar. This is the actual driving force for transport of water across the membrane. Similarly, the effective concentration (activity) of NaCl in the feed during operation of a reverse osmosis membrane with 99.9% rejection and 1 M NaCl is about 0.63 M (activity), and the concentration in the permeate is about 0.001 M. Therefore, the difference in activity across the membrane is about 0.63 M, and the activity ratio is 0.0159 (0.01 activity/0.63 activity) or 60% higher than the concentration ratio.

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CONCLUSIONS A uniform model for solute and solvent transport in reverseosmosis membranes and related membrane systems is presented for solutes and solvents with nonconstant activity coefficients. The previous models based on the work of Merten1,2 and Londsdale2 only included activity effects in the calculation of osmotic pressure. This work completes the transformation of the transport model to a thermodynamic consistent model. The solute and solvent permeability coefficient (B-value, eqs 2, 15) was modified to incorporate activity coefficients into the solute and solvent permeability coefficient (Ba-value, eq 16). This modification allows for the prediction of the order of separation of different salts based on the activity coefficients. The previously used B-value included the activity of the feed and permeate of the test solution (eq 15). The utilization of the previous B-value has been the basis for many models of reverse osmosis, nanofiltration, forward osmosis, pressure retarded osmosis, etc. Currently, the measured B-values will overestimate or underestimate the Bavalue by 0.04 to over 12. Many of the previous models should be reevaluated on the basis of this current correction to the solution physical chemistry. Furthermore, using the extreme changes in activities (Figures 1−6), the order of separation of different salts/solutes/solvents for an indiscriminate membrane (one with constant permeability vs salt and concentration) can be predicted. This model allows for the prediction of the order of separations commonly found in membrane studies. The 2:2, 2:1, 1:2, and 1:1 salt H


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APPENDIX

second-order corrections (β) to the osmotic coefficient are given by20

Solution Electrolyte Theory

The chemical potential of a chemical species (ui) is defined as a reference to a standard state (uio), as developed by G. N. Lewis (eq 5). The standard state of various electrolyte physical parameters can be defined in terms of molarity and molality. For historical reasons, the activity coefficient, osmotic coefficient, and the Gibbs free energy are typically defined with respect to the ideal case of unit molality at standard temperature and pressure (eqs 5, 18, 19, 20, 21). For this discussion, molality is used, and some differences between molality and molarity are described. The activity coefficient is used to convert from molality to activity, and is defined as

γi = aimo /mi

φ−1=−

(18)

ul = ulo + RT ln(ai) = ulo + RT ln(mi) + RT ln(γi) (19)

In the special case of a solvent or solute that is volatile, then both the pressure and the partial molar volume (Vl) are included. o ul = ulo + Vp l + RT ln(ai) = ul + Vp l + RT ln(mi )

(20)

Multiple solutes in solution can be included by summing over the solutes (designated (i)). For a nonvolatile solute like a salt that dissociates in water, the chemical potential is ui = uio + RT ln(γmi)v

(21)

where v is the stoichiometric number of moles of ions in one mole of salt. The osmotic pressure (eq 6), a result of the activities of the solvent and solute, can be measured and is used to estimate solute activities in the solution. Because the activity of water changes as a function of dilution (by the addition of the solutes), temperature, and pressure, it can be calculated separately from the contributions of the solutes. In this case, the osmotic coefficient includes the changes in the solvent (water) and summing over the solutes (i) in the definition of the osmotic coefficient. Thus, the osmotic coefficient is conventionally defined in any aqueous solution as

φ=−

1000 ln(a w ) 18.0153 ∑i vm i i

(22)

Experimentally, the osmotic pressure (π) is defined using the osmotic coefficient φ: π = vRT

miWl φ 1000Vl

1/2

1 + 1.2m

1/2

+ m[β (0) + β (1) e−2m ]

(24)

Compilations of the β parameter for salts have been tabulated by several researchers, but the most complete set has been compiled using the Pitzer method. One of the earliest complete sets included >100 salts.18 Methods to estimate the activity coefficients, osmotic pressure, and Gibbs free energy of mixed salts of different valence states have been proposed using this model.18,21 The temperature and pressure dependence of the osmotic coefficient is contained in the Debye−Huckel parameter.16 This parameter is commonly used to describe the osmotic coefficient, activity coefficient, and Gibbs free energy of electrolyte solutions, and is linear with respect to the osmotic coefficient. It is, however, dependent on the particular derivation of the parameter. According to the final formation by Pitzer, the Debye−Huckel parameter for the osmotic coefficient of water under standard conditions is 0.391. The Debye−Huckel parameter can be used to estimate the osmotic coefficient difference with respect to temperature and pressure. A change in temperature from 10 to 50 °C results in a variation in the Debye−Huckel parameter from 0.382 to 0.41. This 10% change includes the most important range of temperatures for reverse osmosis. In contrast, the effect of pressure is significantly smaller; that is, the Debye−Huckel parameter at 100 bar (and 25 °C) and 1 bar is 0.390 and 0.391, respectively. Thus, the effects of pressure are not considered in this discussion. Importantly, any volumetric change due to the temperature is solvent/solute dependent and is included separately. Because the activity coefficient and Gibbs free energy of the solutions are similarly derived, these differences lead to slightly different values of the Debye−Huckel parameter, which, however, shows the same temperature and pressure dependence. Because the osmotic coefficient is defined on the basis of the natural log of the chemical activity and is dependent on the density of the solution, it will have different minimum and maximum values from the activity coefficient. For instance, for NaCl solutions, a minimum in the activity coefficient can be found at 1.05 m (0.968 M, 54490 ppm), with the activity coefficient varying from 1 to 0.656. The osmotic coefficient also has a minimum, but only varies from 1 to 0.92. This demonstrates that transport parameters for NaCl that are linearly dependent on the osmotic coefficient differ by ∼8% from the measured values; in contrast, the measured salttransport coefficient (B-value), which depends on the activity coefficient, is ∼35% lower than the actual B-value due to the activity; the correct value is therefore 152% of the previously reported values. In water solutions, different salts can change their activities from 1 to below 0.1 to above 12. This suggests that the actual NaCl B-values for seawater are in fact ∼50% higher than the reported values (1 m). In reverse osmosis and nanofiltration, water transport is dependent on the combination of the osmotic coefficient and the rate of salt transport. By examining the thermodynamics of the electrolyte solutions, the solute and solvent transport through these membranes is expected to be dependent on the activity of salts in both the feed and the permeate. In the Merten1,2 model, water transport is dependent on the differences in both the applied pressure and the osmotic pressure between the feed and permeate. The

Therefore, eq 5 becomes

+ RT ln(γi)

Aφm1/2

(23)

where Wl is the molecular weight of the solvent (water), and Vl is the molar volume of the solvent. Models for the electrolyte activity, osmotic coefficient, and Gibbs free energy have been developed for concentrated salt solutions at temperature and pressure ranging between 0 and 823 K and 0 and 100 MPa, respectively.15−20 For a 1:1 salt, the I


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an intermediate salt partitioning depending on the salt concentration between an uncharged polymer and Donnan exclusion of polyelectrolytes. The partitioning in a neutral polymer system (e.g., cellulose acetate) showed a simple salt partitioning. Furthermore, the processing effects on salt transport and water permeability and the mixed-ion effects on membrane performance are worth mentioning. Xie26 studied a 35%disulfonated polyarylether cast as a potassium salt, cast as an acid, swollen in acid, swollen in high-temperature water, and heat treated. Consequently, the water and salt transport changed dramatically, showing a somewhat predictable response to swelling as measured by water uptake. Sulfonated polysulfone and 20%-random disulfonated polyarylether membranes were examined as reverse-osmosis membranes. The sulfonated polysulfone membrane showed a reasonable NaCl passage of 2−3% at a moderate ionic strength, but with a strong ionic-strength dependence.27 The mixed-ion effect of added calcium significantly affected the NaCl passage. Stevens et al.28 showed that a 20%-random disulfonated polyarylether membrane has an improved NaCl passage (of about 1%), which could be further improved to 0.3% by deswelling in a highosmotic pressure solution at elevated temperatures. This same treatment improved the mixed-ion effect by nearly a factor of 10. This is not simply due to the reduction in water content, because the acid form has the worst salt passage and the lowest water content measured. Thus, the specific relaxation of the polymer in the water solution containing salt is needed. This effect is probably a combination of two factors, that is, the solution region, where the osmotic pressure is above that of the internal polymeric osmotic pressure, and the polymer relaxation in the presence of co- and counter ions.

difference in the osmotic pressure in turn depends on the concentrations and activity coefficients of the solutes in both the feed and the permeate solutions. Salt transport is only dependent on the concentration difference. Polymer Activity and Osmotic Pressure

Only limited data on the activity or osmotic pressure of polyelectrolytes such as poly(carboxylic acid)s, polysulfonic acids, and polyquaternary amines and at high concentrations are available. Lammertz et al.22 measured the water activity and rational osmotic coefficient of 10 different polymers as a function of the NaCl content in polymer solutions, with concentrations up to 40%. They found that the water activity is strongly affected by the charge density and concentration of NaCl and the polymer. The water activity and the osmotic coefficient of the polymer vary from 0.9803 to 0.8748, and from 0.4 to nearly 1, respectively. Horvath and Nagy23 studied the activity of water as a function of the concentration of poly((vinyl alcohol)-co-(vinyl sulfate)) for divalent metals (e.g., Co, Cu, Ni); they found that the activity of water is mostly independent of the nature of the ions, except for the binding, which leads to different concentrations of free divalent ions. The activity of water was strongly dependent on the charge density and polymer concentration. In addition, it was found that the natural log of the activity of water (directly related to the osmotic pressure) with divalent ions is less than one-half that of monovalent ions in the same polymers. Newman and McCormick,24 who performed sodium-23 NMR experiments of five anionic polyelectrolytes in NaCl solutions, confirmed that sodium is preferentially displaced by divalent ions. These results suggested that the limited mobility of sodium near the polymer is increased by the binding of divalent ions; however, the sodium exchange with the calcium-ion condensed region around the polymer is very limited. These findings showed that the water activity within a polymer is significantly lower than that in water and is dependent on both the polymer and the state of salts in the polymer in the membrane. While the solution-polyelectrolyte theory continues to be developed for water and salts in high concentrations of polymers, an alternative approach has been suggested; that is, Flory−Huggins polymer theory is currently being modified according to the ion−polymer interaction parameter23 and water interaction with polymers.25 Unfortunately, equations that can quantitatively describe water and ions in polymer solutions (60−90% polymer) are scarce. Polymers such as sulfonated polyarylethers can phase separate into interconnected regions of high charge or can be randomly dispersed. The packing and extent of interconnectivity of different regions govern the transport. In the case of highly charged partitioned regions, the mobile counterions are expected to form a highly mobile continuum with a few polymer-dependent features. Condensation of the ions in free standing polyelectrolyte films may depend on the ionic strength or the nature of the added divalent ions. These condensations typically occur with the removal of water; however, this does not occur in the case of a rigid polymer system. In at least one rigid polyelectrolyte system, the water content of the membrane showed little change as a function of the applied pressure and salt concentration. In these systems, the water content is expected to only slightly change after calcium ions are bound to the charged polymer; nevertheless, the water and sodium activity and mobility are expected to change substantially. The partitioning of salts was discussed by Geise25 et al.; they found that ion-exchange membranes have

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The author declares no competing financial interest.

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ACKNOWLEDGMENTS

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REFERENCES

I would like to thank Prof. B. Freeman for his suggestions and initial review of the manuscript.

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Substantial Changes in the Transport Model of Reverse Osmosis and

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