Review of Basics Reverse Osmosis Process Modeling: A new · PDF fileBasic RO pressco modeling...

AlbertS.Kim, acceptedinMembraneJournal,08-28-2017

Review of Basics Reverse Osmosis Process Modeling: A new

combined fouling index proposed

Albert S. Kim

Civil and Environmental Engineering, University of Hawaii at Manoa, Honolulu, Hawaii 96822,

USA

Abstract

Seawater desalination is currently considered to be one of the primary technologiesto resolve the global water scarcity problem. A basic understanding of membrane�ltration phenomena is signi�cant not only for further technological development butalso for integrated design, optimal control, and long-term maintenance. In this vein,the present work reviews the major transport and �ltration models, speci�cally re-lated to reverse osmosis phenomena, provides theoretical insights based on statisticalmechanics, and discusses model-based physical meanings as related to their practicalimplications.

Keywords: Reverse Osmosis; Concentration Polarization; Solution-di�usion model;combined fouling index (CFI); Modi�ed Fouling Index (MF); Membrane ProcessModeling

1. Introduction

The �rst commercial reverse osmosis (RO) membrane was developed by two re-searchers, Loeb and Sourirajan in early 1960 in the Department of Engineering, Uni-versity of California, Los Angeles (UCLA). After the pioneering work, RO technologyhas been rapidly developed and widely applied in a variety of separation and �ltration�elds, especially for seawater desalination. Fig. 1(a) shows original photo images ofthe prototype desalination cell using fabricated cellulose acetate membranes [1]. Theirproject entitled �Sea Water Demineralization by Means of Semipermeable Membrane�was carried out under the sponsorship of the Statewide Water Resources Center pro-gram in Sea Water Conservation Research. S. Loeb and S. Sourirajan were listed asproject leaders, and the other four personnel include Lloyd Graham, A. Noeggerath,R. Sayano, and M. Accomazzo. The report was signed by Prof. J. M. English, vice-chairman of research, in the Department of Engineering, UCLA. Fig. 1(b) shows the�life test assembly� which contains the desalination cell and circulating and pressuriz-ing pumps. The life test indicates the �ltration experiment, which operated 24 hoursper day for two months. The feed solution was 5.25 percent of seawater (generatedwithin the system), and the applied pressure was 1500 psi (=103.4 bar). During the

Preprint submitted to Membrane Journal August 28, 2017


�rst seven days (period 1) from the fall of 1959, water �ux and permeate concentra-tion were measured as 6.4 gal/ft2day (=10.87 liter/m2 h, LMH) and 0.042 percent.From the second to the fourth week (period 2), water �ux decreased from 5.6 to 5.26.4 gal/ft2day and the permeate concentration remained as 0.040± 0.003 percent.In period 1 and 2, the rejection ratios were calculated as 99.20 and 99.24 percent,respectively. In period 3 of four weeks, the average �ux was 4.75 gal/ft2day with98.97 percent of rejection. Finally, the total cost was estimated as $0.60 per 1,000gallons, i.e., $0.16/m3, which is cheaper than the present water production rate bySorek plant, in Israel, currently producing 624,000 m³/day (26,000 m³/hour) [2]. Inthe 1960 report, Loeb and Sourirajan's future work includes standardization of �lm-fabricating techniques, fabrication cost estimation, and investigation of separationmechanisms, which have been vigorously conducted by subsequent researchers tilldate. The fabricated membrane is later explicitly called Loeb-Sourirajan membrane,and the more detailed story can be found elsewhere [3, 4].

Various mechanisms and models were suggested to explain the RO phenomena.The sieving mechanism [5] indicates that the separation occurs because of the di�er-ence between molecular sizes of solvent and solutes. The wetted-surface mechanism[6, 7] treats the membrane as very wettable material so that water tends to cling tothe membrane surface. The solution-di�usion model [8, 9] followed by the solution-di�usion-imperfection model [10] assumes that both solvent and solutes dissolve inthe homogeneous nonporous surface layer of the membrane and then di�use with-out signi�cant solvent-solute interactions. The preferential sorption & capillary �owmechanism [5, 11, 12] proposes a critical pore size, twice (or smaller than) the waterlayer thickness on the membrane surface, to allow only solvent transport throughthe membrane. Among these models for RO processes, the solution-di�usion modelwas most widely accepted for explanation and prediction. Transport of solvent andsolutes was universally explained using the transmembrane chemical potential andtransition from the solution-di�usion to the pore �ow was investigated [13]. Later,the solution-di�usion model was reformulated as a pressure-driven di�usion processusing rigorous thermodynamic boundary conditions, which led to nonlinear responsesat high pressure and the coupling between solvent and solutes was considered usingthe Maxwell-Stefan formulation for multi-component di�usion [14].

Although the models as mentioned earlier were used to fundamentally explain theRO phenomena, they mostly dealt with speci�c mass transport mechanisms acrossthe polymer membrane, of which thermodynamic state is assumed to be quite closeto the pure static equilibrium. To the best of my knowledge, non-equilibrium ther-modynamics is still at a nascent stage in theoretical statistical physics. The front-endimprovement is a theory to investigate the thermoelectric phenomena, such as trans-ference phenomena in electrolytes and heat conduction in an anisotropic medium,viewed as coupled, irreversible phenomena [15, 16]. A thermodynamic system wasrelaxed from the pure equilibrium to one where the microscopic reversibility could beassumed. This means that an irreversible system of non-equilibrium can be viewed

2


as a collection of a number of small local subregions, having individual processes, inwhich the time-reversal is guaranteed. The time-reversal indicates that an evolvingsystem from its initial condition returns to the initial state if time t is reversed to −t.In other words, an average rate of an individual process is equal to the average rateof its reverse process. In his work, Onsager described the irreversible process usingthe entropy change rate. A phenomenological driving force was de�ned as a partialderivative of the entropy with respect to speci�c �uxes (of multi-species or heat).

The �rst irreversible transport (IRT) model was developed to explain the transferof non-electrolytes through membranes by applying Onsager's reciprocal theorem tothe membrane separations by Kedem and Katchalsky [17], followed by Spiegler andKedem [18]. These irreversible transport models require empirically determining a fewmodel parameters, which is a practical trade-o� to use more realistic models. Mostmembrane systems are thermodynamically open to the ambient environment. If oneof the systems is in a thermodynamic state that is quite close to a static equilibrium,then the irreversible model parameters often converge to those of limiting valuesof the pure equilibrium. In this case, irreversible thermodynamic �ltration modelsbecome mathematically identical to the solution-di�usion model regarding functionalinterdependences between the solvent and solute �uxes and their relationship withthe e�ective driving force.

More rigorous theoretical investigation of RO processes using the non-equilibriumthermodynamics or simply steady-state thermodynamics is necessary to develop next-generation membrane technology. Currently, there are a number of excellent articlesthat provide well summarized technical information and future perspectives of ROtechnology [19�27]. Continuing in this vein, the current work will deal with in-depthand detailed analysis of the solution-di�usion model in various aspects as applied toprocess simulations with a limited literature review and provide theoretical derivationsfor the fouling phenomena on the RO membrane. Fundamental studies and reviewson the future membrane technologies in various aspects can be found elsewhere [28�32]. This paper aims to provide a clear picture of the RO membrane as a platform ofcoupled thermo- and �uid dynamic phenomena and contribute to a solid curriculumfor membrane engineering.

2. Theory and Simulation Review

2.1. Basic RO process modeling

2.1.1. Mass balance equations

Fig. 2 shows the RO schematic, consisting of ten thermodynamic and �uid dy-namic variables. Hydraulic pressure ∆P is applied to the feed stream of concentrationCf and results in feed �ow rate Qf . A portion of the feed stream passes through theRO membrane characterized by solvent permeability A, solute permeability B, andsurface area Am. This product stream is called permeate stream having concentra-tion Cp (usually much lower than Cf ) and out�ow rate Qp. The concentrate (often

3


called retentate) stream has concentration Cc (higher than Cf due to the solventpermeation) �owing with its out�ow rate Qc. The study objective of this basic ROmodeling is to calculate output concentrations and �ow rates in terms of input andoperating conditions. Here we de�ne two representative parameters used to evaluatethe performance of RO membranes: rejection ratio (which we will later call observedrejection)

R = 1− CpCf

(1)

and recovery ratio

Y =Qp

Qf

(2)

which express the quality and quantity of the solvent product, respectively.For both solvent and solute mass transport, the input rate is equal to a sum of

two output rates:

Qf = Qp +Qc (3)

CfQf = CpQp + CcQc (4)

Solvent �ux [m/sec], i.e., the collected volume of water transported through the mem-brane per unit time per unit membrane surface area, is described as

Jw =Qp

Am= A (∆P −∆π) (5)

where ∆π = πf − πp is the osmotic pressure di�erence between the feed (πf ) andthe permeate (πp) streams. Solute �ux, i.e., the solvent �ux multiplied by permeateconcentration [mg/l·m/sec] is expressed as

Js = B (Cf − Cp) (6)

= CpJw (7)

Substituting Eq. (2) in (4) allows us to express the retentate concentration using feedand permeate concentrations and recovery ratio:

Cc =Cf − CpY

1− Y(8)

The permeate concentration of Eq. (7) is rewritten as

Cp =JsJw

=Cf

1 + AB

(∆P −∆π)(9)

and �ow rates of the permeate and retentate streams are then represented using Qf

and Y :

Qp = QfY (10)

Qc = Qf (1− Y ) (11)

4


Note that we initially had total ten variables (shown in Fig. (2)), of which subsetconsists of six knowns: {∆P,A,B,Am, Cf , Qf}. The four balance equations (3)�(8)of solvent and solute transfer rates make the RO process modeling mathematicallysolvable.

2.1.2. Analytic solutions with van't Ho�-type osmotic pressure

If the osmotic pressure is linearly proportional to the solute concentration, thenits transmembrane di�erence is

∆π = b (Cf − Cp) (12)

where b is a proportionality. In van't Ho�'s equation, we have b = RT , where R isthe gas constant and T is the absolute temperature of the membrane system. (J. H.van't Ho� was recognized by the Nobel Prize committee for his discovery of �the lawsof chemical dynamics and osmotic pressure in solutions� and received the �rst NobelPrize in Chemistry in 1901.) Substitution of (12) into (9) gives

Cp = Cα

[√1 + C2

β/C2α − 1

](13)

where

Cα =BA−1 + ∆P − bCf

2b(14)

and

Cβ =

√BCfAb

(15)

Note that Cαand Cβ have a unit of solute concentration. For simplicity, let's setψ = Cβ/Cα. If ψ � 1, then one can approximate the terms in the parenthesis of Eq.(13) using the Taylor expansion as

√1 + ψ2 − 1 =

[1 +

1

2ψ2 + · · ·+O

(ψ4)]− 1 ≈ 1

2ψ2 =

1

2

(CβCα

)2

(16)

where O (ψ4) indicates the remaining terms of ψ4 and higher, and simplify the func-tional form of Cp as

Cp ≈C2β

2Cα=

Cf

1 + AB

(∆P − πf )(17)

which is equivalent to Eq. (9) with a condition of Cp � Cf , and hence indicating∆π ≈ πf . However, note that Eq. (9) is only an implicit solution for Cp, because ∆πincludes Cp itself. For an accurate calculation, Cp of Eq. (17) needs to be used tocalculate ∆π in Eq. (12). An iterative method should continue until Cp in Eq. (9)converges to a speci�c constant value. The accurate calculation of Cp is important notonly for RO but also for NF, in which Cp is comparable with Cf . In this case, higher-order terms in Eq. (16) must be important for such low-rejection �ltration processes,

5


in which the straightforward approximation of Cp � Cf in RO is questionable. Asthe exact solution of Eq. (13) has a perfect closure form, calculations of Cc, Qp (orJw), and Qc are straightforward using Cp of Eq. (13) without any approximation ornumerical iterations.

2.1.3. Causes and e�ects

Fig. 3 shows how output variables Cp, Qp and Cc change with respect to inputvariables of ∆P , Qf and Cf , while A, B and Am are assumed to be invariant duringoperations. Calculation of Qc is straightforward using the solvent mass balance.While one of the three input variables changes with the other two remaining �xed,variations of output variables with respect to the solely changing input variable areanalyzed as follows using Eqs. (8)�(10). Fundamental aspects of the solution-di�usionmodel will be discussed in the later sections.

E�ect of pressure ∆P . First, we let Cf and Qf remain constant and increase onlyapplied pressure ∆P . This type of analysis is mathematically equivalent to calculatingpartial derivatives of Cp, Qp and Cc with respect to ∆P .

(a) In most RO cases of a high rejection ratio close to 1.0, B � Jw must be a goodapproximation in RO processes. (Note that B and Jw have the same unit ofvelocity [m/s].) As Jw is proportional to ∆P , the permeate concentration Cpof Eq. (9) decreases with ∆P :

CpCf

=1

1 + Jw/B≈ B/A

(∆P −∆π)(18)

If concentration polarization is negligible above the membrane surface, one canapproximate ∆π ≈ πf , Cp � Cf , and hence Cp ∝ 1/∆P . One may apply ahigher pressure to decrease the permeate concentration Cp, to be obtained byincreasing water �ux Jw through the membrane. If the feed concentration isclose to the seawater concentration having the osmotic pressure about 400 � 500psi, then a decrease in Cp with respect to ∆P is not as much as that of brackishwater of a few thousand mg/l. As one increases ∆P , the enhanced pressurepushes more water to the membrane to have a higher permeate �ux, Jw. In thiscase, the convective solute transport (roughly equal to CfJw) increases at themembrane surface. As the membrane rejects solute ions , Cm increases fromCf , providing a higher osmotic pressure di�erence between the feed side and thepermeate side of membrane surface, i.e., ∆πm (= πm − πp) > ∆πf (= πf − πp).The increase in ∆πm is a partial feedback from increased ∆P , so that a decreasein Cp is lessened by the Cp phenomena, which is fundamentally inevitable.

(b) The permeate �ow rate is conceptually equivalent to the permeate �ux, becausethe available membrane surface area Am is usually �xed. (See Eq. (5).) Thisindicates that the variation of Qp with respect to ∆P is the same as that ofJw. Here, we assume for simplicity that ∆π is insensitive to ∆P , and Jw is not

6


meaningful for ∆P ≤ ∆π. As indicated in Fig. 3(b), the onset of non-zero Qp

occurs when ∆P exceeds ∆π. After that, Qp monotonously increases with ∆Pand the slope is equal to A · Am from Eq. (5). In reality, measured Qp residesbelow the linear line, because the Cp increases the osmotic pressure di�erenceand therefore decreases the e�ective pressure, ∆Peff = ∆P −∆πm.

(c) As Cp decreases with respect to ∆P , more solutes are rejected by the mem-brane. Overall, the amount of solutes retained per unit volume, i.e., retentateconcentration Cc, increases with ∆P . For high rejection of Cp � Cf , we canneglect CpY = Cf (1−R)Y in the numerator of Eq. 8:

Cc ≈Cf

1− Y(19)

Note that Y is proportional to Qp ∝ Jw ∝ ∆Peff . Cc is therefore linearlyproportional to ∆P if and only if the recovery ratio is small (i.e., Y � 1). Tovalidate this, one can use Taylor's series of Cc with respect to Y :

Cc ≈ Cf(1 + Y +O

(Y 2))

(20)

Otherwise, the higher-order terms become signi�cant and Cc must non-linearlyincrease with ∆P .

E�ect of feed �ow rate Qf .

(d) The feed �ow rate Qf usually does not signi�cantly change the characteristics ofthe permeate stream, unless ∆P depends on Qf . The permeate concentrationis pseudo-independent of Qf .

(e) In the same vein, the permeate �ow rate is indi�erent to the feed �ow ratebecause Qp primarily depends on the applied pressure ∆P . The amount ofwater that passes in the longitudinal direction (tangentially to the membranesurface) in the feed stream does not noticeably change the permeate �ux Jw orpermeate �ow rate Qp.

(f) Because Qf = Qp +Qc, for a constant Qp, Qc increases with Qf . Eq. (19) canbe then rewritten as

Cc = CfQf

Qc

= Cf

(1 +

Qp

Qf −Qp

)(21)

to show Cc gradually decreases with Qc or Qf

E�ect of feed concentration Cf .

(g) When the applied pressure is much higher than the feed osmotic pressure, Cp islinearly proportional to Cf ; in other words, Cp/Cf = constant. When the feedapplied pressure is comparable with the osmotic pressure, Cp versus Cf curveshows a non-linearly increasing trend, which is above the linear line. Eq. 9

7


indicates that the increase in Cf secondarily contributes to ∆πm, increases Cm,and �nally reduces ∆Peff and Jw. As a consequence, the solvent and solute�uxes decrease and increase, respectively, with respect to Cf , and therefore thepermeate concentration increases.

(h) When ∆P and Qf are �xed, Qp linearly decreases with respect to the feedconcentration:

Qp = AmJw ∝ ∆P −∆πm = ∆P + bCp − bCm (22)

The permeate �ow rate vanishes when πm = ∆P + πp ≈ ∆P .

(i) When recovery is small, i.e., Y � 1 or Qp � Qf , the retentate concentrationCc does not change signi�cantly from the feed concentration level, i.e., Cc ∼ Cf .As Cf increases when ∆P is �nitely higher than πf , Y decreases because Qp

monotonously decreases with Cf . Therefore, the slope of Cc versus Cf plottedfrom Eq. (8) also decrease with Cf : 1/ (1− Y ) ≈ 1 +Y +Y 2 +Y 3 + · · · . Notethat (1− Y )−1 > 1 indicates Cc is unconditionally higher than Cf , except forthe zero-recovery case. The slope of Fig. 3 can be calculated as

∂Cc∂Cf

=1− Y0

[1− Y0 + Y1Cf ]2 (23)

where Y0 = AmA∆P/Qf and Y1 = AmAb/Qf . Using parameter values in Fig. 3gives the Y0 value higher than 1.0, which provides a negative value of ∂Cc

∂Cf, and

therefore Fig. 3(i) shows the gradually decreasing behavior of Cc with respectto Cf .

2.2. Concentration Polarization

2.2.1. Phenomena

Fig. 4 shows a schematic of di�usive and convective transport of solutes near themembrane surface. During the RO/NF �ltration process, pressurized feed stream ofconcentration Cf �ows in a tangential (x-) direction to the membrane with velocity u.Solutes are rejected by the membrane, whereas solvent (water) molecules pass throughthe membrane. The permeate concentration Cp is therefore much lower than the feedconcentration Cf in proper operations. The hydraulic pressure gradient between thebulk and permeate stream generates the solvent �ow across the membrane, whichis de�ned as the permeate �ux Jw, i.e., the volume of solvent passing through themembrane per unit membrane surface area: a unit of Jw is [m3/m2 s], equivalent to[m/s] or [µm/s]. As the transverse solvent �ow brings solutes down to the membranesurface, solutes are retained on the membrane surface where concentration Cm ishigher than Cf . This phenomenon of the uneven or biased concentration distributionnear the membrane surface is called the concentration polarization (CP), and theregion where the CP occurs is called the CP layer. δp denotes the thickness of theCP layer above which the concentration remains Cf .

8


2.2.2. Mass balance

Solutes are transported from the bulk phase toward the membrane by two masstransfer mechanisms, i.e., convection and di�usion, which are balanced as

JwC −(−D∂C

∂y

)= JwCp (24)

Here, JwC is the convective transport of solutes from the bulk phase toward themembrane. Within the concentration polarization layer, 0 < y < δp, the soluteconcentration C (y) decreases with respect to y so that −D ∂C

∂yis positive and indicates

the magnitude of di�usive transport of solutes from the membrane surface to the bulkphase. Therefore, speci�c boundary conditions are:

C (y = 0) = Cm (25)

C (y = δp) = Cf (26)

The CP layer of thickness δp is usually much smaller than the channel height of thefeed �ow. Within the CP layer, it is appropriate to approximate that the permeate�ux Jw is constant with respect to y and the concentration is independent of theaxial position x of the membrane surface. Then, the partial derivative of ∂C/∂y inEq. (24) becomes its ordinary di�erential, i.e., dC/dy. Integration of Eq. (24) withrespect to y using boundary conditions of Eqs. (25) and (26) yields

Cm − CpCf − Cp

= eJw/kf (27)

where kf = D/δp is the mass transfer coe�cient, indicating how quickly solutes back-di�use from the membrane to the bulk phase. (See Appendix A.1 for the detailedderivation of Eq. (27).) Usually, δp (or kf ) is unknown and often estimated usingempirical correlations, because a coupled mass transfer equation using varying cross-�ow velocity is hard to solve. The right-hand side of Eq. (27) is interpreted as theratio of excessive concentrations at the membrane surface to that of the bulk phase.In RO/NF, this ratio is about 2�3.

2.2.3. Rejection ratios

From Eq. (27), the solute concentration on the membrane surface is rewritten as

Cm = Cp + (Cf − Cp) eJw/kf (28)

= Cp + CfRobs eJw/kf (29)

where Robs is the observed rejection ratio, de�ned as

Robs = 1− CpCf

(30)

9


which indicates the fraction of solutes retained by the membrane. The permeateconcentration can be calculated using Robs:

Cp = Cf (1−Robs) (31)

and now we can eliminate Cp in Eq. (29) to derive

CmCf

= 1 +Robs

(eJw/kf − 1

)(32)

For the perfect rejection (Robs = 1.0), Cm reduces to

Cm = Cf eJw/kf (33)

as a product of Cf and the exponential factor. In addition to Robs, the intrinsicrejection is de�ned as

Rint = 1− CpCm

(> Robs) (34)

Substitution of Eq. (28) in (34) derives

CmCf

=eJw/kf

Rint + (1−Rint) eJw/kf> 1.0 (35)

which requires known values of Jw, kf , and Rint. In normal RO processes, measuredJw is about a few µm/s, kf can be estimated using empirical correlations, and Rint

is often close to 1.0. If the intrinsic rejection Rint is close to zero, the right-hand sideconverges to one. No concentration polarization occurs and the concentration hasan even distribution along the y-direction, i.e, Cm w Cf . Similarly, if the membraneresistance is too high (e.g., almost impermeable when deleteriously fouled), the solvent�ux becomes very small, i.e., Jw → 0. Hence, we calculate that:

eJw/kf ≈ 1 +O(Jwkf

)≈ 1.0 (36)

or equivalently Cm w Cf .

2.3. Solution-di�usion model

2.3.1. Governing equations based on Fick's law

Solvent Transport. We assume that water transport through the normal membranesis by di�usion through a single phase and write transport equation of water:

Jw = −Dw∇Cw (37)

where Cw andDw are concentration and di�usivity of water dissolved in the membrane[33]. We accept the Henrian approximation that in an isothermal environment

µw = RT lnCw + µwo (38)

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where µw is the chemical potential of the water and µwo is an isothermal constantindependent of Cw. Substitution of Eq. (38) in (37) gives

Jw =DwCwRT

∇µw ≈DwCwRT

∆µwδm

(39)

which represents the solvent �ux driven by the chemical potential gradient ~∇µw '∆µw/δm, where ∆µw is the chemical potential di�erence across the membrane ofthickness δm. In pressure-driven membrane separation processes, the chemical poten-tial of water may be governed by the applied pressure and water concentration andcan be expanded as

dµw =

(∂µw∂Cw

)P,T

dCw +

(∂µw∂P

)Cw,T

dP (40)

Integration of Eq. (40) across the membrane gives

∆µw =

∫ (∂µw∂Cw

)P,T

dCw +

∫ (∂µw∂P

)Cw,T

dP

=

∫ (∂µw∂Cw

)P,T

dCw + v̄w∆P (41)

If the applied pressure is equal to the osmotic pressure di�erence (∆P = ∆π), thenmass �uxes are zero since the chemical potential has zero gradient. Hence, we obtain∫ (

∂µw∂Cw

)P,T

dCw = −v̄w∆π (42)

and therefore∆µw = v̄w (∆P −∆π) (43)

where V̄w is the molar volume of the solvent. Substitution of Eq. (43) in (39) givesthe solvent �ux

Jw =DwCwv̄wRTδm

(∆P −∆π) = A (∆P −∆π) (44)

where

A =DwCwv̄wRTδm

(45)

is called the solvent permeability having a unit of [m/s·atm], which is often assumed tobe independent of ∆P . Eq. (44) indicates that the water �ux through the membraneis proportional to the e�ective pressure, i.e. the ∆P and ∆π. The origin of thisconclusion is from the thermodynamic relationship:(

∂µw∂P

)Cw,T

= v̄w (46)

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or equivalentlyµw = v̄w∆P + f (Cw, T ) (47)

where f is an arbitrary function of Cw and T . Comparison of Eqs. (47) and (40)gives a self-consistent result in terms of speci�c dependence of µw on ∆P , Cw , andT .

Solute Transport. The transmembrane solute di�usion is also assumed to be Fickian:

Js = −Ds∇Cs ≈Ds

δm∆Cs = B∆Cs (48)

where Js, Ds, and Cs are the mass �ux, di�usivity, and concentration of the solute,respectively, in the membrane. The phenomenological solute transport coe�cient canbe de�ned as

B =Ds

δm(49)

which is called the solute permeability. It is often assumed that Ds is independent ofthe solute concentration, but maybe varies with temperature. In Eq. (48), ∆Cs indi-cates the transmembrane concentration di�erence, measured on the exterior surfacesof the membrane.

2.3.2. Solvent and solute �uxes

The solvent �ux Jw is proportional to the e�ective pressure, of which the osmoticpressure di�erence can be more accurately represented as:

∆π = πm − πp (50)

The van't Ho� equation indicates that the (absolute) osmotic pressure is linearlyproportional to the solute concentration, unless the concentration is too high nearthe solubility limit. In this case, the solution-di�usion model is equivalent to theosmotic pressure model and then we have:

∆π = πf

(CmCf− CpCf

)(51)

and hence using Eq. 32 we make a relationship between the thermodynamic variable∆π and the hydrodynamic variable Jw through the mass transfer coe�cient k.

∆π = πfRobseJw/kf (52)

By substituting Eq. (52) in 44, we obtain

Jw = A(∆P − πfRobse

Jw/kf)

(53)

which needs to be solved iteratively for Jw with an estimated value of kf . Phenomeno-logically, the permeate �ux increases if the applied pressure increases. The enhanced

12


permeate �ux contributes additionally to the convective solute transport toward themembrane surface, which brings more solutes on the membrane surface and henceincreases Cm. The osmotic pressure on the membrane surface πm therefore increasesexponentially with Jw (See Eq. 52). As a consequence, the net pressure, the drivingforce of the solute permeation, does not increase as much as ∆P increases, becausethe concentration polarization causes the reduction of the driving force as indicatedin Eq. 52. Since Jw is on both sides of Eq. (53), a nonlinear or iterative solver isrequired to calculate Jw. If the mass transfer coe�cient is larger than the permeate�ux or

Jwkf

< 1 and

(Jwkf

)2

� 1

then we can expand the exponential form in Eq. 52 as

eJw/kf w 1 +Jwkf

+O

(J2w

k2f

)(54)

In this case, we derive an analytic expression for the permeate �ux:

Jw = Aeff (∆P − πfRobs) (55)

under the in�uence of the Cp, where the e�ective solvent permeability Aeff is calcu-lated as

Aeff =A

1 + πfRobsk−1f

(< 1) (56)

Eq. (55) explains the solvent permeation giving a di�erent picture from that of Eq.(53). πfRobs indicates the osmotic pressure di�erence between the feed and permeatestreams. The e�ective solvent permeability Aeff , which is smaller than A, includesthe resistance for solvent permeation from the membrane and the CP layer. Whenthe concentration polarization is negligible and so its thickness is very small, i.e.,δp → 0, then the mass transfer coe�cient kf diverges, because a �nite concentrationdi�erence exists within a CP layer of zero thickness k−1

f → 0. The e�ective solventpermeability Aeff converges to A as the CP layer disappears.

In the solution-di�usion model, the driving force for the solute �ux is the concen-tration di�erence between the membrane surface and the permeate stream. Replacing∆Cs in Eq. (48) by Cm − Cp gives

Js = B (Cm − Cp) (57)

where B has the same unit of Jw [m/s]. We rewrite Eq. (57),using Eq. 29, as

Js = BCfRobseJw/kf (58)

which implies that Js increases exponentially with respect to Jw. Note that the solute�ux must be equal to the permeate concentration multiplied by the permeate �ux:

Js = CpJw = Cf (1−Robs) Jw (59)

13


Rigorously, Js in Eq. (58) indicates the solute �ux through the membrane interiordriven by the external concentration di�erence, Cm − Cp. Eq. (59) is based on theglobal mass balance implying that the solute molecules are uniformly mixed in thepermeate stream after they have passed through the membrane.

2.3.3. Parameter estimation

Solvent permeability. The solvent permeability A is an intrinsic material constant ofa speci�c membrane and so needs to be experimentally measured. When the feedstream of zero concentration (Cf = 0) is �ltered using an RO membrane, we have

A =J0w

∆P(60)

where J0w indicates the permeate �ux of zero feed concentration. Using pure water,

a series of �ltration experiments can be conducted to measure J0w with respect to

∆P as schematically shown in Fig. 5. The slope of the �ux vs. pressure line can becalculated using a simple linear regression method to calculate the must probable A.

Solute permeability. Here, we simply assume that B is also a constant within typicalranges of the solute concentration and applied pressure of normal RO processes. FromEqs. (53), 58 and (59), we obtain:

RobseJw/kf =

A∆P − JwAπf

=(1−Robs) Jw

B(61)

which leads toB

A=

πwJ0w

J0w − Jw

(62)

where πp = πf (1−Robs) is the osmotic pressure of the permeate stream and J0w−Jw

indicates the permeate �ux lost from the pure water �ux J0w due to the concentration

polarization.

Low �ux limit. When the permeate �ux is low due to small e�ective pressure, thefollowing approximations can be made. The intrinsic rejection Rint converges to theobserved rejection, Rint → Robs, because the CP must be negligible on the membranesurface Cm → Cf , and hence πm → πf . The solute �ux is

Js = CpJw = B0 (Cf − Cp) (63)

which gives

B0 = Jw1−Robs

Robs

(64)

where the subscript 0 indicates no or negligible concentration polarization. Then, Bof Eq. (61) can be expressed as:

B = B0e−Jw/kf (65)

If Jw/kf � 1, then B converges to B0 of the dilute limit

14


Mass transfer coe�cient. In system design and performance evaluation of RO pro-cesses, estimation of the mass transfer coe�cient kf is of great importance. A numberof experiments can be conducted and accumulated data can be used to create empir-ical correlations for later use. Here, kf can be represented using the solvent �ux Eq.(53) and the solute �ux Eq. (58), which are

1

kf=

1

JwlnJ0w − JwAπfRobs

(66)

and1

kf=

1

Jwln

(1−Robs) JwBRobs

(67)

respectively. Values of kf estimated using the above two equations should be compa-rable within a tolerable error.

2.3.4. Empirical correlations

When a feed solution is physico-chemically characterized and a module geometryis given, the cross�ow speed v is almost the only controllable parameter to changethe mass transfer coe�cient kf . For dimensionless analysis, Sherwood number isrepresented as a function of Reynolds and Schmidt numbers and the aspect ratio ofthe channel geometry. (See Appendix A.2 for details.) Table 1 shows exponent valuesof a, b, c, and d in Eq. (A.9). For laminar �ow, a of a rectangular channel is slightlyhigher than that of a cylindrical tube, while all other exponents are equally 1/3.In�uences of Re, Sc, and module geometry on the kf must be similar in cylindricaland rectangular channels and a represents the e�ect of the cross-section shape. Forturbulent �ow, a, b, and c are same for the cylindrical and rectangular channels andinterestingly d = 0. Due to the complex nature of the turbulent �ow �eld, the e�ectof hydraulic diameter vanishes, as the wetted surface area in the turbulence fails toprovide a controllable impact on the mass transfer.

2.3.5. Long membrane modules

Fig. 6 shows a schematic of cross�ow RO �ltration. For a short membrane, theretentate concentration is

Cc =Cf − CpY

1− Y(68)

which implicitly neglects the concentration variation in the longitudinal direction, asshown in Fig. 2. For a long membrane, Eq. (68) holds its validity only if Cp isreplaced by its length-averaged value:

C̄p =1

L

∫ L

0

Cp (x) dx (69)

15


where Cp (x) is a local permeate concentration at x (0 < x < L). A local averageconcentration of Cp(x) is denoted as

C̄ ′p =1

x

∫ x

0

Cp (x′) dx′ (70)

Then, the retentate concentration of Eq. (68) can be rewritten as

Cc =Cf − C̄pY

1− Y(71)

where Y is the (global) recovery ratio de�ned as Y = Qp/Qf .

Mean osmotic pressure in the bulk phase. To apply the solution-di�usion model for along membrane module, a longitudinally mean osmotic pressure π̄ needs to be usedto calculate the mean transmembrane osmotic pressure: ∆π = π̄ − πp. A goodapproximation, especially for membrane array design, can be

π̄ =πf + πc

2(72)

where πc = π (Cc). Note that Eq. (72) does not include the e�ect of the concentra-tion polarization phenomena in the transverse direction. This forceful decoupling ofmass balance equations in the transverse and longitudinal direction allows analyticalsolutions, which are later combined using empirical correlations. The mass trans-fer coe�cient kf , estimated using an empirical correlation, implicitly includes e�ectsof the membrane length and the channel cross-section in addition to the transportmechanisms.

Mulder's theory. Now we apply the same analysis at x = l. For a partial membraneof length x from 0, Cc and Cp of a full membrane are replaced by their partial valuesC ′c and C

′p, respectively, to give

C ′c =Cf − C ′pY ′

1− Y ′(73)

where Y ′ is a local recovery ratio for the partial membrane of length x. To solve this,we need an additional relationship such as

C ′

1−R= 〈C ′c〉 (74)

for which Mulder [34] assumed that

〈C ′c〉 =1

Y ′

∫ Y ′

0

C ′c (Y ) dY (75)

16


Substitution of Eq. (73) and (75) in (74) gives

C ′c · (1− Y ′) = Cf − (1−R)

∫ Y ′

0

C ′cdY (76)

which is an integral equation for C ′c. We di�erentiate Eq. (76) with respect to Y ′ tohave

dC ′cC ′c

= RdY ′

1− Y ′(77)

and integrate it such that ∫ Cc

Cf

dC ′cC ′c

= R

∫ Y

0

dY ′

1− Y ′(78)

to obtainCc = Cf · (1− Y )−R (79)

Substitution of (79) into (71) represents the mean permeate concentration:

C̄p = Cf1− (1− Y )1−R

Y(80)

represented as a function of Cf , R, and Y . In this approach, Y implicitly includesimpacts of A,Am and ∆P , and R contains the rejecting role of B.

Examples in the Appendix A.3indicate that Mulder's theory is valid when therejection ratio is high, such as standard RO processes. The key equations (79) and (80)stem from the partial mass balance Eqs. (74) and (75). The solution-di�usion modeluses speci�c permeability values of A and B, to iteratively calculate the permeateconcentration. As the CP is incorporated into the solution-di�usion model, Cm isconsidered higher than Cf , but does not explicitly include variation of Cp in thelongitudinal direction (from the inlet to the exit of the membrane module). Anempirical correlation for the mass transfer coe�cient kf implicitly includes the length-averaged dimensionless numbers, and perhaps so does Cm. Therefore, combinationof the solution-di�usion model and kf estimated using an empirical correlation isconceptually equivalent to Mulder's intuitive assumption:

C̄ ′p1−R

=1

Y ′

∫ Y ′

0

C ′c (Y ) dY

Usually, vendors provide a rejection ratio for a membrane, measured at a referencecondition, which in this case can be used as an intrinsic constant similar to A or B.Mulder's theory allows us to practically estimate the product permeate concentrationusing R and Y without dealing with speci�c transport models.

In section 2.1, a membrane is characterized using ten variables. Of these, sixvariables of A,B, Am, ∆P , Cf and Qf usually are known. The four remaining ones

17


are calculated using the same number of equations, which are global mass balancesof Eqs. (3) and (4), and solvent and solute �uxes of Eqs. (5) and (6), respectively. InMulder's approach, the membrane is treated as a black box of known R and Y so thatthe variable set includes eight elements {R, Y, Cf,Qf , Cp, Qp, Cc, Qc} when R, Y , Cf ,and Qf are known, then de�nitions of R and Y , and global mass balance equationsof solutes and solvent will be used to calculate the same total number of unknowns,such as {Cp, Qp, Cc, Qc}.

2.4. Coupled Governing Equations

An accurate governing equation without a forceful decoupling between the trans-verse and longitudinal directions is

∂C

∂t=

∂

∂y

(D∂C

∂y

)− u (y)

∂C

∂x− vw (x)

∂C

∂y(81)

where the solute di�usivity D is often assumed to be constant and the longitudinaldi�usion is discarded by indicating ∂2C/∂x2 � ∂2C/∂y2. Within the CP layer, thecross�ow velocity is often represented as a linear shear �ow with respect to y:

u (y) = γy (82)

where

γ =

[∂u

∂y

]y=0

(83)

is a shear rate on the membrane surface. The mathematical rigor of the coupled gov-erning equation is closely related to the exponential dependence of the concentrationnear membrane surface on the permeate �ux (see Eq. (27))[35, 36]. The only nu-merical solution seemed to be available for the 2D convection and di�usion of soluteson the membrane surface. A general solution of Eq. (81) was developed using Airyfunctions, but coe�cients were obtained by numerical integrations [37]. This workdiscovered that an in�ection point of the concentration pro�le exists in the longitu-dinal cross�ow direction. Even if these analytic approaches provide a fundamentalinsight of cross�ow membrane �ltration, they are still restricted to solute migrationon the �at, slip-less surface in the linear shear �eld of Eq. (82). It is formidablydi�cult to develop an analytic solution of the 2D governing equation if one or someof the followings are considered: the presence of spacers, transient hydraulic pressurefor pulsing, curved channels, and parabolic or nonlinear �ow �elds.

2.5. Fouling indexes and scaling potential

2.5.1. Modi�ed fouling index (MFI) for colloidal fouling

When colloidal particles deposit on the membrane surface (typically, but not lim-ited to, MF or UF membranes), the resistance-in-series model represents the permeate�ux

Jw =∆P

η (Rm +Rc)(84)

18


where Rc is the resistance of the cake layer, i.e., temporarily or permanently built de-posit layer of solid materials such as nano- or colloidal particles, (deformable) macro-molecules or combined forms. In the dead-end �ltration or at the initial stage of thecross�ow �ltration, Rc continuously increases with respect to time. Moreover, oftencausing noticeable declining trends of the permeate �ux, the speci�c cake resistanceis de�ned as

rc = Rc/δc (85)

which is independent on the cake thickness δc if the cake layer has a uniform massdensity. In principle, the speci�c resistance is an inverse of the hydraulic permeabilityκ, i.e., rc = κ−1, which is generally a function of particle size, particle shape, andcake porosity. If particulate materials are perfectly removed by a membrane, theamount of particle mass transported from the bulk (feed) phase to the membranesurface is equal to the particle mass accumulated on the membrane surface, which ismathematically written as

φfV = φcAmδc (86)

where φf and φc are particle volume fractions in the feed solution and of the cakelayer, respectively, and V is the permeate volume, i.e., the solvent volume passedthrough the membrane having the surface area Am. Substitution of (86) in (85) gives

Rc = αV (87)

where α = φfrc/φcAm is the proportionality between the cake resistance and thepermeate volume. Eq. (87) indicates that the cake resistance increases as water is�ltered by the membrane. By de�nition, the permeate �ux is written as

Jw =1

Am

dV

dt(88)

as it is proportional to the volume of produced solvent per unit time , i.e., dv/dt.Substitution of (87) and (88) in (84) provides

1

Am

dV

dt=

∆P

η (Rm + αV )(89)

which is simply the �rst order ordinary di�erential equation of the �ltered volume V ,rewritten as

η (Rm + αV ) dV = ∆PdtAm (90)

in an integrable form. Integration of this equation using the initial condition ofV (t = 0) = 0 gives

t

V= C1 + C2 V (91)

where

C1 =ηRm

Am∆P(92)

19


is the y-intercept of t/V versus V plot and

C2 =ηφfrc

2φcA2m∆P

≡ MFI (93)

is the slope, which is de�ned as modi�ed fouling index (MFI) [38, 39]. This MFIcannot be easily calculated using Eq. (93), because the cake volume fraction φc isneither known nor (easily) measurable and rc is a complex non-linear function of φc.Theoretical calculation of MFI is challenging, because φc strongly depends on inter-particle and particle-membrane interactions and rc is a complex, non-linear functionof φc.

2.5.2. Combined fouling index (CFI)

When the feed solution contains both salt ions of high concentration and colloidalparticles, the permeate �ux may be expressed as a combination of the osmotic pressuremodel and the resistance-in-series model:

Jw =∆P −∆πmη (Rm +Rc)

(94)

where ∆πm (= πm − πf ) is the transmembrane osmotic pressure di�erence in the pres-sure of CP. Consider that the cake layer exists inside the concentration polarizationlayer of salt ions, i.e., δp > δc, where δp is the thickness of the salt concentrationpolarization layer above the membrane surface. Then we de�ne

δs = δp − δc (95)

which is the partial thickness of the concentration polarization layer above the cakelayer, within which the tangential cross �ow velocity is assumed to be negligible.The surface of the cake layer may provide the no-slip boundary condition, which issimilar to the (bare) membrane surface without the particle deposition. Then, themass balance Eq. (24) can be employed using the solute di�usivity changing withrespect to y:

D =

{D0 for δc < y < δp

D0ετ

for 0 < y < δc(96)

where ε (= 1− φc) is the cake porosity and τ is the di�usive tortuosity. In Eq. (24),dy/ (C − Cp) is multiplied on both sides to give∫ Cf

Cm

dC

C − Cp= −

∫ δc

0

Jwτ

D0εdy −

∫ δp−δc

δc

JwD0

dy (97)

which is solved as

lnCf − CpCm − Cp

= −JwD0

(τεδc + δs

)(98)

20


The cake volume fraction φc is often assumed to be a random close packing ratio of0.64 [40, 41], and the di�usive tortuosity τ is in principle greater than 1.0, varyingwith φc and the internal structure of the cake layer. For a thick cake layer, theconcentration polarization above the cake layer does not signi�cantly contribute tothe permeate �ux. So, one can approximate Eq. (98) by removing δs as

Cm − CpCf − Cp

= exp

(JwδcD0ε/τ

)= ej (99)

where we de�ne a dimensionless permeate �ux

j =Jwkc

(100)

and

kc =D0

δcτ/ε(101)

which is interpreted as the di�usive mass-transfer coe�cient of solute ions in the cakelayer of porosity ε. The denominator of Eq. (101), δcτ/ε, can be considered as thee�ective path length of di�using solutes within the cake layer, which is longer thanthat in the void space.

Substitution of Eq. (99) in (94) gives the �nal equation to solve for j:

j =∆P − πfRobse

j

kc η (Rm +Rc)(102)

Similar to the previous case, we assume j < 1 to use

ej = 1 + j +O(j2)

(103)

and substitute Eq. (103) with (102). We then replace j by (kcAm)−1 dV/dt and useEq. (87) to give

η

[Rm +

πfRobs

kcη+ αV

]dV = (∆P − πfRobs)Amdt (104)

Integration of both sides using an initial condition of V = 0 at t = 0 provides

t =

(ηRm

AmdP

)(1 + π̂f

Jw0

kc

1− π̂f

)V +

(ηφfrc

2φcA2m∆P

)1

(1− π̂f )V 2

simpli�ed to

t

V= C1

(1 + π̂f

Jw0

kc

1− π̂f

)+

MFI

(1− π̂f )V (105)

21


where

π̂f =πfRobs

∆P(106)

indicates the ratio of the osmotic pressure of the net concentration πfRobs to theapplied pressure ∆P . Now, we can take the proportional constant of V in Eq. (105)to de�ne the combined fouling index (CFI) as

CFI =MFI

1− π̂f(107)

The absence of the salt ions in the feed stream can be considered by setting π̂f = 0which makes CFI converge to MFI and Eq. (105) equal to (91). In this case, appli-cations are limited to MF or UF processes. The fouling tendency of RO desalinationcan be quanti�ed using the CP factor β, de�ned as

β = Cm/Cf (108)

which can be calculated using the measured permeate �ux Jw and the empirically-determined mass transfer coe�cient kf in Eq. (27). In addition to β, CFI can beused to estimate the combined fouling tendency in the presence of both ionic andparticulate species. Note that CFI is always larger than MFI. For example, if ∆P isset as twice seawater osmotic pressure πsw, then we calculate π̃f = 2πsw/πsw = 2 andCFI = 2 MFI using Eq. (107). Related experimental and modeling studies can befound elsewhere [42�45].

3. Concluding Remarks

In this study, I brie�y reviewed the fundamentals of reverse osmosis processes,based on the solution-di�usion model. Speci�c variations of output variables such asconcentrations and out�ow rates of the permeate and brine streams are characterizedwith respect to the input and operating parameters. Transverse variations of thesolute concentration are reviewed by solving the decoupled convection-di�usion equa-tion. Mulder's theory is discussed to explain the longitudinal variations of permeate�ux, which primarily controls the rejection and recovery ratios. The solution-di�usionmodel was also reviewed using principles and concepts of statistical mechanics. Fi-nally, the degree of combined fouling (by both ionic solutes and particulate materials)is quanti�ed using a novel combined fouling index (CFI) as an extension of the mod-i�ed fouling index (MFI).

In environmental engineering, which is the discipline closest to mother nature, aholistic understanding of transport phenomena at the basic level of thermodynamics,statistical mechanics, and �uid mechanics is as important as practically dealing withdesigning, optimizing, and maintaining speci�c processes. Hopefully, this incompletemanuscript can be a stepping stone for future membrane engineers, who may resolvethe impending global water shortage.

22


(a) (b)

Figure 1. (a) Assembling �lm packages in prototype desalination cell and (b) life test assembly(Figure 3 and 8 of Ref. [1], respectively. Reprinted.)

Figure 2. A schematic of the RO process.

23


Figure 3. Theoretical causes and e�ects in basic RO modeling. Parameters used are T = 25◦C,Am = 150 m2, A = 1.36 × 10−7m/s psi, B = 5.0 × 10−8 m/s, Qf = 864 m3/day, Cf = 35 g/l,∆P = 800 psi. Three variables of ∆P , Qf and Cf are selected as independent variable in the �rst,second and third rows, respectively. Variations of Cp, Qp and Cc are calculated with respect to oneindependent variable while the rest two are kept constant.

24


Figure 4. Schematic of concentration pro�le across the membrane in the cross�ow.

Figure 5. Schematic of the pure water �ux with respect to the applied pressure.

25


Figure 6. RO schematic with local balance.

�ow geometry a b c d

laminar tube 1.62 1/3 1/3 1/3laminar rectangular 1.85 1/3 1/3 1/3turbulent tube 0.44 3/4 1/3 0turbulent rectangular 0.44 3/4 1/3 0

Table 1. Exponents of the mass transfer coe�cient in terms of channel geometry and �ow regions

26


Appendix A. Appendix

Appendix A.1. Proof of Eq. (27)

The permeate �ux Jw, the permeate concentration Cp, and the solute di�usivityD are constant. Eq. (24) is rewritten, using the net or excessive concentrationC ′ = C − Cp, as

DdC ′

dy= −JwC ′ (A.1)

Because only C ′ is a sole function of y, one rewrites Eq. (A.1) as

dC ′

C ′= −Jw

Ddy (A.2)

Integration of Eq. (A.2) gives

lnC ′ = −JwDy + constant (A.3)

of which the constant is determined using the boundary condition of Eq. (25):

ln [Cm − Cp] = constant (A.4)

The boundary condition of Eq. (26) on the top of the CP layer provides

ln [Cf − Cp] = −JwDδp + constant (A.5)

Substitution of Eq. (A.4) in (A.5) generates

Cm − CpCf − Cp

= exp

[+

JwD/δp

](A.6)

in which the solute di�usivity per CP layer thickness,D/δm, refers to the mass transfercoe�cient kf of the same dimension of the permeate �ux Jw. Substitution of Eq. (A.4)in (A.3) gives the solute concentration C (y) as a function of the distance from themembrane surface y:

C (y)− CpCm − Cp

= exp

[−JwyD

]for 0 < y < δp (A.7)

The concentration exponentially increases from Cf at the CP layer boundary to Cmon the membrane surface.

27


Appendix A.2. Dimensionless number analysis

The performance of a membrane is typically estimated using the recovery andrejection ratios, which are primarily determined by A and B, respectively, which areintrinsic material properties of the membrane. On the other hand, the mass transfercoe�cient kf is strongly dependent on �uid dynamics and module geometry. Inengineering and applied sciences, dimensionless numbers are often used to representcorrelations between representative physical quantities. The Sherwood number (Sh)includes the mass transfer coe�cient such as

Sh =kfdhD

=convective mass transfe rate

diffusive mass transfer rate(A.8)

implying the signi�cance of the convection over the di�usion of solutes, where dh isthe hydraulic diameter. The Sherwood number is often represented as a function ofReynolds (Re) and Schmidt (Sc) numbers:

Sh = aReb Scc(dhL

)d(A.9)

where L is the membrane (or channel) length, and a−d are constants. Here, dh/L canbe considered as the aspect ratio of the �ow channel. The Reynolds number measuresa ratio of inertial to viscous forces for given �ow conditions, which is de�ned as

Re =vdhν

=vρdhη

(A.10)

where v is the �ow speed, ρ is the �uid density, and η and ν are the absolute andkinematic viscosities, respectively. In the case of �ow through a straight pipe with acircular cross-section, �uid motion will be laminar at Re < 2000, whereas at Re >4000, the �ow is turbulent. The Schmidt number represents a ratio of convective todi�usive mass transport:

Sc =ν

D(A.11)

Appendix A.3. Application of Mulder's theory to Seawater Desalination

An RO desalination process has a feed water of 35,000 ppm and the rejectionratio of the selected membrane is reported by a vendor as 99.00%. If the process isoperated for 60% recovery, then the permeate concentration is predicted as

C̄p = Cf1− (1− 0.6)1−0.99

0.6= Cf × 0.01520 = 532.06 ppm (A.12)

If so, one calculates

1− C̄pCf

= 0.9848 (A.13)

28


which is close enough to 0.9900. If we replace the rejection by 40% (such as that ofnano�ltration), then we calculate

C̄p = Cf1− (1− 0.6)1−0.40

0.6= Cf × 0.7048 = 24670.33 ppm (A.14)

and

1− C̄pCf

= 0.2951 (A.15)

which is di�erent from 40%.

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29


[12] E. Glueckauf, �The distribution of electrolytes between cellulose acetate mem-branes and aqueous solutions�, Desalination 18 (2) (1976) 155�172,

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