2.01 Fundamentals in Reverse Osmosis
G Jonsson, Technical University of Denmark, Lyngby, Denmark F Macedonio, University of Calabria, Arcavacata di Rende CS, Italy a2010 Elsevier B.V. All rights reserved.
2.01.1 Introduction 2
2.01.2 Phenomenological Transport Models 3
2.01.2.1 Irreversible Thermodynamics-Phenomenological Transport Model 3
2.01.2.2 IT-Kedem?Spiegler Model 4
2.01.3 Nonporous Transport Models 4
2.01.3.1 Solution?Diffusion Model 4
2.01.3.2 Extended Solution?Diffusion Model 5
2.01.3.3 Solution?Diffusion-Imperfection Model 5
2.01.4 Porous Transport Models 5
2.01.4.1 Friction Model 5
2.01.4.2 Finely Porous Model 6
2.01.5 Comparison and Summary of Membrane Transport Models 7
2.01.6 Influence from Operating Conditions on Transport 7
2.01.6.1 Effect of Pressure 7
2.01.6.2 Effect of Concentration 8
2.01.6.3 Effect of Feed Flow 8
2.01.6.4 Effect of Temperature 8
2.01.6.5 Effect of pH 9
2.01.7 Experimental Verification of Solute Transport 10
2.01.7.1 Single-Salt Solutions 12
2.01.7.2 Mixed-Salt Solutions 13
2.01.7.3 Organic Solutes and Nonaqueous Solutions 16
2.01.7.4 Mixed Organic Solutes 18
2.01.7.5 Membrane Charge 18
2.01.7.6 Membrane Fouling and Concentration Polarization Phenomena: Limits of Membrane
Processes 19
References 20
2 Reverse Osmosis and Nanofiltration
2.01.1 Introduction
Several models on reverse osmosis RO transport mechanisms and models have been developed to describe solute and solvent fluxes through RO mem-branes. The general purpose of a membrane mass transfer model is to relate the fluxes to the operating conditions. The power of a transfer model is its ability to predict the performance of the membrane over a wide range of operating conditions. To realize this objective, the model has to be integrated with some transport coefficients often determined based on some experimental results.
When theories are proposed to describe mem-brane transport, either the membrane can be treated as a black box in which a purely thermodynamic description is used, or a physical model of the mem-brane can be introduced. The general description obtained in the first case gives no information on the flow and separation mechanisms. On the other hand, the correctness of data on the flow and separation mechanisms obtained in the second case depends on the chosen model.
The transport models can be divided into three categories:
1.
phenomenological transport models which are inde-pendent of the mechanism of transport and are based on the theory of irreversible thermodynamics irreversible thermodynamics ? phenomenological transport and irreversible thermodynamics ? Kedem?Spiegler models;
2.
nonporous transport models, in which the membrane is supposed to be nonporous or homogeneous solution?diffusion, extended solution?diffusion, and solution?diffusion-imperfection models SDIMs; and
3.
porous transport models, in which the membrane is supposed to be porous preferential sorption-capillary flow, Kimura?Sourirajan analysis, finely porous and surface force-pore flow, and friction models.
Fundamentals in Reverse Osmosis 3
Most models for RO membranes assume diffusion or pore flow through the membrane while charged-membrane theories include electrostatic effects. For example, Donnan exclusion models can be used to determine solute fluxes in the often negatively charged nanofiltration membranes.
Moreover, RO membranes have, in general, an asymmetric or a thin-film composite structure where a porous and thin top layer acts as selective layer and determines the resistance to transport. Macroscopically, these membranes are homoge-neous. However, on the microscopic level, they are systems with two phases in which the transport of water and solutes takes place. Figure 1 provides a schematic presentation of a thin-film composite membrane structure with 1 the highly selective skin layer which acts as a barrier, 2 the intermediate porous layer where the selectivity decreases to zero, and 3 the nonselective porous sublayer.
The porous sublayer influences the total hydrau-lic permeability Lp from Reference 1:
111 1
Lp .Lp sl tLp il tLp pl e1T
But it has almost no influence on the solute rejection properties of the membranes. Therefore, most trans-port and rejection models of RO membranes have been derived for single-layer membranes focusing almost only on the surface thin layer.
Transport models can help in identifying the most important membrane structural parameters and showing how membrane performance can be improved by varying some specific parameters. One of the main membrane intrinsic parameters is the reflection coefficient, s, introduced by Staverman
[2] and defined as
? lp P
X lp .Jv .0 e2T
Figure 1 Schematic presentation of thin-film composite membrane structure.
where s describes the effect of the pressure driving force on the flux of solute and represents the relative permeability of the membrane to the solute:
1.
s .1 for a high-separation membrane and
2.
s .0 for a low-separation membrane in which the solute is significantly carried through the mem-brane by solvent flux.
In RO, the intrinsic retention Rmax is related to sand normally sRmax as reported in Reference 3. Pusch
[4] derived the following relationship between Rmax csmax
and :Rmax 1 ?1 ?
.eT? c9
s
where csmax is the mean salt concentration at infinite Jv.
2.01.2 Phenomenological Transport Models
2.01.2.1 Irreversible Thermodynamics-Phenomenological Transport Model
The membrane is treated as a black box when nothing on the transport mechanism and membrane structure is known. In this case, the thermodynamics of irreversible thermodynamics IT processes can be applied to membrane systems. According to the IT theory, the flow of each component in a solution is related to the flows of other components. Then, different relation-ships between the flux through the membrane and the forces acting on the system can be formulated.
Onsager [5] suggests that fluxes Ji are related to the forces Fj through the phenomenological coeffi-cient Lij:
X
Ji .LiiFi tLijFj for i .1;...;n e3Ti.j
6
For systems close to equilibrium, the cross-coefficients are equal:
Lij .Lji for i .6j e4T
Kedem and Katchalsky [6] used the linear phe-nomenological relationships Equations 3 and 4 to derive the phenomenological transport:
Jv P ?
.lpeTe5TJs .!te1 ? TJvecsTln e6T
where parameters lp, !, and sare simple functions of the original phenomenological coefficient Lij.
Usually RO systems are far from equilibrium; therefore, Equation 4 may not be correct. Moreover,
4 Reverse Osmosis and Nanofiltration
phenomenological transport equations 5 and 6 have been rarely applied for describing RO membrane transport both because the often large concentration difference across the membranes invalidates the linear laws and because this analysis does not give much information regarding the transport mechanism.
2.01.2.2 IT-Kedem?Spiegler Model
Spiegler and Kedem [13] bypassed the problem of linearity by rewriting the original IT equations for solvent and solute flux in differential form:
dP d
Jv .pv ?e7T
dx dx
dcs
Js1 ?csJv
.p dx teTe8Twhere pv is the water permeability, x the coordinate direction perpendicular to the membrane, and ps the solute permeability. Integrating Equations 7 and 8 over the thickness of the membrane by assuming pv, ps, and sconstant, the following equations for the solvent flux Jv and retention R are achieved:
pv
Jv .x eP ?Te9T
f1 ? exp.?Jve1 ? Tx=psge10TR .1 ?exp.?Jve1 ? Tx=ps
where x is the membrane thickness. Equation 10 can be rearranged as follows:
c9s1 x
c0s .1 ? ? 1 ? exp ?Jve1 ? Tps e11T
However, similar to phenomenological transport equations, Spiegler and Kedem relationships also do not give information on the membrane transport mechanism.
2.01.3 Nonporous Transport Models
2.01.3.1 Solution?Diffusion Model
The solution?diffusion model assumes that 1 mem-brane surface layer is homogenous and nonporous and 2 both solute and solvent dissolve in the surface layer and then they diffuse across it independently. Water and solute fluxes are proportional to their chemical potential gradient. The latter is expressed as the pressure and concentration difference across the membrane for the solvent, whereas it is assumed to be equal to the solute concentration difference across the membrane for the solute:
Jv .AeP ?Te12T
DvcvVv
A .e13T
R T x
Js .Bces -?cs0Te14T
Dsk
B .e15T
x
where A is the hydraulic permeability constant lp, B is the salt permeability constant, cs -and cs0are, respec-tively, the salt concentrations on the feed and permeate sides of the membrane. Dv and Ds are the diffusivities of the solvent and the solute in the membrane, respectively; cv is the concentration of water in the membrane; Vv is the partial molar volume of water; R is the universal gas constant; T is the temperature; k is the partition or distribution coefficient of solute defined as follows:
kg solute m ? 3 membrane K .kg solute m ? 3 solution e16T
k measures the solute affinity to k 1 or repulsion from k 8:0 10
c
0s
Including pressure, particularly for organic-water
where the ratio k3 =k1 is
1 t
k1
a measure of the relative systems, the solute flux is given by contribution of pore flow compared to diffusive flow.
Dsk
0-
s ? sTtlsp
where lsp is the pressure-induced transport parameter.
Equation 22 has been proved to be accurate for different organic solutes with cellulose acetate mem-branes [8].
2.01.3.3 Solution?Diffusion-Imperfection Model
The solution?diffusion model is one of the most referred membrane models. It presupposes that the membrane surface is homogenousnonporous and it has the limitation that the intrinsic value of retention is always unity.
The SDIM developed by Sherwood et al. [9] considers that small imperfections exist on the mem-brane surface due to the membrane-making process,
P e22T
Js
.x e
cc
This model has been successfully applied for the
performance description of several solutes and
membranes [10], particularly it is proper for those membranes exhibiting lower separation than that cal-culated from solubility and diffusivity measurements.
2.01.4 Porous Transport Models
Among the transport models in which the membrane is supposed to be porous, friction and finely porous models are described in this section.
2.01.4.1 Friction Model
Friction model considers that the transport through porous membrane occurs both by viscous and diffu-sion flow. Therefore, the pore sizes are considered so
6 Reverse Osmosis and Nanofiltration
small that the solutes cannot pass freely through the pores but friction between solute-pore wall and sol-vent-pore wall and solvent?solute occurs. The frictional force F is linearly proportional to the velo-city difference through a proportional factor X called friction coefficient indicating the interaction between solute and pore wall:
F23 .?X23 eu2 ?u3T.?X23u2 e27T
F13 .?X13 eu1 ?u3T.?X13u1 e28T
F21 .?X21eu2 ?u1Te29T
F12 .?X12eu1 ?u2Te30T
Equations 27?30 are derived considering the membrane as reference u3 .0. Considering that the frictional force per mole of solute F23 is given by
J2p
F23 .?X23u2 .?X23 e31T
c2p
Equation 27 can be written as
J2p
F23 .?X23 e32T
c2p
Jonsson and Boesen [10] have presented a detailed description of this model and they have shown that, as F21 is the effective force for diffusion of solute in the center of mass system, the solute flux per unit pore area J2p is given by
1 J2p .X21 c2pe?F21Ttc2p ?u e33T
A balance of applied and frictional forces is equal to
F2 .? eF21 tF23Te34TNeglecting the pressure term and in the case of dilute solution behavior, F2 is equal to
RT dc2p
F2 .? c2p dx e35T
Defining b as the term that relates the frictional coefficients X23 between solute and membrane and X21 between solute and water
b .X21 tX23 e36TX21
and inserting in Equations 29, 32, and 34?36, J2p can be written as
RT dc2p c2p?u J2p .? X21?b dx tb e37T
The coefficient for distribution K of solute between bulk solution and pore fluid is given by
K .c2p =c2 e38T
with Jv ."?u, Ji .J2?", and .?x, using the pro-duct condition
c02 .Ju 2p e39T
and integrating Equation 37 with the boundary conditions x .0: c2p .Kc92; x .?: c2p .Kc0 the following equation for the ratio c2 9 =c2 0 is obtained:
c9b exp u"?X21 ? 1
X21
c022 .1 tk exp u""?"RTRT e40T
In this derivation K, b, and X21 are assumed to be independent from the solute concentration.
2.01.4.2 Finely Porous Model
The finely porous model was developed by Merten
[11] using a balance of applied and frictional forces proposed by Spiegler [12]. It is a combination between viscous flow and frictional model presented in detail by Jonsson and Boesen [10]. The premise of the model is to describe, reasonably, the transport of water and solutes in the intermediate region between solution?diffusion model and Poiseuille flow:
1.
the first is reasonable when applied to very dense membranes and solutes which are almost totally rejected, whereas
2.
Poiseuille flow can be used to describe the trans-port through porous membranes consisting of parallel pores.
Jonsson and Boesen [10] showed that the following equation can be used to determine Rmax from RO experiments:
c92 bb ?Jv
c02 .K t1 ? K exp ? "? D2 e41T
where D2 is the solute diffusion coefficient. From Equation 41, the maximum rejection Rmax at Jv !1 is given by K 1
Rmax ..1 ? b .1 ? K e42T
X231 tX21
Equation 42 shows how rejection is related to a kinetic term the friction factor b and to a thermo-dynamic equilibrium term K. Spiegler and Kedem
[13] derived the following corresponding expression:
1 X23u2
.1 ? K 1 tX21 1 tX21u1 e43T
X23
Equations42and43areidenticalapartthecorrection term X13u2 =X21u1 whichismuchsmaller than 1for
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