CFD analysis of tracer response technique under cake-enhanced osmotic pressure

CFD analysis of tracer response technique under cake-enhanced osmotic pressure G. A. Fimbres Weihs, D. E. Wiley* School of Chemical Engineering The University of New South Wales, Sydney, NSW, 2052, Australia *

Corresponding author. Tel.: +61 2 9385 4755. E-mail: [email protected]

Keywords: Cake enhanced osmotic pressure, Computational Fluid Dynamics, Particulate fouling, Concentration polarization, Sodium chloride tracer response Abstract A cake-layer mass transfer model applicable for RO, that incorporates the cakeenhanced osmotic pressure (CEOP) effect of a particulate fouling layer, is presented. This model includes the effect of a variable dissolved solute concentration on the specific cake resistance and porosity of the cake layer. The model is based on one-dimensional diffusion of the dissolved solute through the cake layer, and uses the solute concentration at the cake surface and the cake mass per unit area to calculate the solute concentration at the membrane surface and the trans-membrane osmotic pressure. The cake-layer mass transfer model is incorporated into a commercial Computational Fluid Dynamics (CFD) software package. Simulations are validated against experimental data, and the model predictions are within ± 7 % for permeate fluxes and within ± 14 % for measured concentration polarisation. The model is used to interpret and assess tracer response test results for estimating concentration polarisation and fouling resistance. Model predictions confirm the assumption for the tracer experiment that the average concentration polarisation along the membrane wall does not change significantly after a step change in the feed concentration of the tracer solute. However, it was found that the tracer experiment over-estimates the concentration polarisation index and under-estimates the fouling resistance, particularly under fouled conditions. The sources of error are discussed and a multiple tracer response test is proposed to minimise estimation error.

NOTICE: This is the author’s version of a work that was accepted for publication in the Journal of Membrane Science. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Membrane Science 449 (2014) 38–49. DOI: 10.1016/j.memsci.2013.08.015

1. Introduction Concentration polarisation and fouling represent two of the biggest challenges faced by membrane separation operations such as Reverse Osmosis (RO). Concentration polarisation reduces the separation performance of membrane systems by adding an extra layer of resistance [1, 2] and/or reducing the driving force in the region near the membrane [3]. Moreover, an increased concentration of solutes near the membrane will also increase the likelihood of precipitation and fouling. A fouled membrane will present a higher resistance to the passage of solvent, or even prevent it altogether, further decreasing the performance of the membrane system. Since concentration polarisation occurs due to solute rejection, it is inevitable and inherent to membrane separation processes. For these reasons, various mathematical models have been proposed for predicting the extent of concentration polarisation [4-8]. However, most of these models do not take into account the effect of a fouling layer, which hinders back diffusion and increases concentration polarisation, an effect referred to as “Cake-Enhanced Osmotic Pressure” (CEOP) [9-11] (see Figure 1). Neglecting to take CEOP into consideration results in an over-estimation of the fouling resistance.

ωb

Feed Bulk Flow

Cake layer

Concentration Profile

ωw,noCEOP Membrane Permeate

ωc

ωw,CEOP

ωp

Figure 1. Schematic of solute concentration (ω) profile with and without the effect of Cake-Enhanced Osmotic Pressure (CEOP).

Incorporation of the CEOP effect into models for the performance of membrane systems can potentially lead to better agreement with experimental data [12]. However, the development of fouling models that take CEOP into consideration also face validation difficulties. This is because fouling occurs within the boundary layer at the membrane surface. Many experimental techniques are too “coarse” and lack the sensitivity required to measure flow variables inside this thin layer, resulting in experimental measurement errors

2

[13]. In addition, these techniques are generally more suited for laboratory scale measurements than for routine use in an industrial plant. Chong et al. [14] proposed a simple technique for assessing the effect of fouling on concentration polarisation, by measuring the response to the introduction of a sodium chloride tracer in a RO system. This tracer response technique can be used for online fouling monitoring. The objective of this paper is to develop a cake-layer mass transfer model that takes CEOP into account and to incorporate this model into a commercial CFD software package. The CFD model is then used to interpret data obtained from the tracer technique. Simulation predictions are compared against experimental data and the accuracy of the tracer response test is assessed. 2. Background The tracer response technique proposed by Chong et al. [14] employs sodium chloride as a tracer. The tracer is injected into the feed stream of the membrane separation unit as an extended pulse, while monitoring permeate flux and trans-membrane pressure (TMP), as well as the concentration of solute in the permeate. This information is then used to calculate the degree of fouling and concentration polarisation. Concentration polarisation refers to a concentration profile characterised by a higher concentration of solute at the membrane surface than in the bulk flow. This phenomenon can be quantified by the use of a concentration polarisation index or modulus (CP). Although there are many forms of the CP index used in the literature, Chong et al. [14] utilise a form derived from the solution of the one-dimensional mass balance differential equation within the boundary layer over the membrane surface [15]: = CPB

ωw − ω p J  = exp  V  ωb − ω p  kmt 

(1)

where CPB is the local concentration polarisation index based on the local solute bulk concentration, JV is the volumetric permeate flux and kmt is the mass transfer coefficient. This concentration polarisation index can also be calculated by using the inlet bulk concentration, and it is expressed as:

CPB 0 =

ωw − ω p ωb 0 − ω p

(2)

3

One of the properties of this form of the concentration polarisation index is that, for one-dimensional diffusion, it only depends on the values for volumetric flux, diffusivity and boundary layer thickness. Therefore, for a constant flux and Reynolds number, the local CPB0 should remain constant. However, the assumption of one-dimensional diffusion from which CPB0 is derived is only an approximation for cross-flow membrane separation, particularly if spacers are present. Spacers promote the formation of vortices that disrupt the boundary layer [16, 17] which results in regions of relatively high and low local CPB0 at the locations of boundary layer separation and reattachment respectively. Nevertheless, the average value of the CPB0 index over the membrane surface is still a useful approximation for predicting the permeate flux through a membrane. 2.1

Other concentration polarisation indices The local permeate flux is usually calculated following the approach of Kedem and

Katchalsky [18] and Merten [19] which, assuming a linear dependence between solute concentration and osmotic pressure, yields the following expression:

JV =

TMP − σϕ (ωw − ω p )

(3)

µ ( Rm + R f )

where the trans-membrane pressure is defined as the pressure difference between the surface of the fouling layer and the permeate (TMP = pc – pp). Substituting the concentration polarisation index from equation (2) into equation (3) yields:

JV =

TMP − σϕ CPB 0 (ωb 0 − ω p )

µ ( Rm + R f )

(4)

It is important to note that equations (3) and (4) are valid locally at every point on the membrane surface. They do not refer to the average permeate flux and permeate concentration measured by experiments. This means equations (3) and (4) will not necessarily fit experimental data that has been averaged over the membrane surface and are, therefore, difficult to validate experimentally. The permeate flux typically measured by experiments is equal to the permeate flux given by equations (3) and (4) averaged over the whole membrane area: = JV

x=L TMP − σϕ (ωw − ω p ) 1 = J dx V L x∫=0 µ Rm + R f

(

)

(5)

4

From equation (5), it is important to note that the relevant fouling resistance when considering the area averaged permeate flux is the permeate-flux-averaged fouling resistance, R f . This variable is defined mathematically as the permeate flux weighted average of the

local hydraulic resistance, such that the value of R f is biased towards the values of the local fouling resistance at locations where the permeate flux is larger (e.g. the channel inlet and places with relatively less local fouling resistance). In this sense, R f is representative of the average fouling resistance over the whole membrane surface and, as opposed to the simple area average, it can be used in equation (5) to calculate the average permeate flux for the whole membrane. An alternative expression for the concentration polarisation can be defined by using an analogy to equation (4) for the area averaged permeate flux:

JV =

TMP − σϕ CP B 0 (ωb 0 − ω p ) µ R + R

(

m

f

(6)

)

The concentration polarisation index ( CP B 0 ) from equation (6) depends on areaaveraged concentrations, and is given by:

CP B 0 =

ωw − ω p ωb 0 − ω p

(7)

This form of the concentration polarisation index is not to be confused with the area averaged CPB0, which is given by:  ωw − ω p 1 1 = CPB 0 dx  ∫ ∫ L x 0= L x 0  ωb 0 − ω p =

x L= x L =

= CP B 0

  dx 

(8)

Generally speaking CP B 0 ≠ CP B 0 , although under some practical operating conditions they may approach each other. Although the permeate flux given by equation (6) can easily be measured experimentally, the same is not true for the area-averaged TMP and permeate concentration. Area-averaged

concentrations

generally

cannot

be

determined

from

experimental

measurements other than for some particular variables such as flux. A form of the area averaged flux equation based on experimental variables that are easy to measure is given

5

when the inlet TMP (TMP0) and permeate flux-averaged permeate concentration ( ω p ) are used. This yields the following expression: JV =

TMP0 − σϕ CPM (ωb 0 − ω p ) µ R + R

(

m

f

)

(9)

Equation (9) defines the CPM concentration polarisation index (where the subscript M stands for “measureable”). Comparing equations (5) and (9) it can be seen that the CPM index can be expressed as follows: = CPM

ωw − ω p TMP0 − TMP + ωb 0 − ω p σϕ (ωb 0 − ω p )

(10)

Thus, the CPM index is not a simple ratio of solute concentrations in its own right. However, it can be related to CP B 0 through the following:  ω 0 − ωp CPM  b =  ω − ω p  b0

 TMP0 − TMP  CP B 0 + σϕ (ωb 0 − ω p ) 

(11)

Equation (11) shows that there are two main sources of deviation between the values of CPM and CP B 0 . The first one is caused by the use of the permeate flow averaged permeate concentration ( ω p ) rather than the area averaged value ( ω p ). Given that the local permeate concentration (ωp) is proportional to the local membrane surface concentration (ωw), it is therefore inversely proportional to the local flux. This means membrane regions with higher flux will generally present lower ωp than regions with lower flux, thus ω p < ω p . For this reason, this deviation will cause CP B 0 to be larger than CPM. The second source of deviation is related to the use of the inlet TMP instead of the area averaged value. Given that the difference between the area averaged and inlet TMP would depend on the pressure loss characteristics of the flow channel as well as on the length of the channel, it would be more convenient to utilise an average TMP for any measurement test calculations, rather than the inlet TMP. Equation (9) is used in the tracer response test proposed by Chong et al. [14] for determining the concentration polarisation index. The tracer test is carried out at constant flux by measuring the inlet TMP0 and permeate concentration ( ω p ) at two different inlet

6

concentrations (ωb0,1 and ωb0,2) and these values are then used to estimate CPM. Because this test is carried out at a constant cross-flow velocity, it assumes that kmt remains constant and that equation (1) applies to the values averaged over the whole channel. Hence, it assumes that CPM also remains constant despite the change in inlet concentration. This leads to the following equation for estimating CPM: CPT ,12 =

TMP0,2 − TMP0,1

σϕ (ωb 0,2 − ωb 0,1 + ω p ,1 − ω p ,2 )

(12)

where CPT refers to the estimate of CPM obtained from the tracer response test. If the membrane resistance (Rm) is known, an estimate of the fouling resistance (RfT) can be calculated using the CPT value from the tracer response test: = R fT

TMP0 − σϕ CPT (ωb 0 − ω p )

µ JV

− Rm

(13)

A constant TMP0 tracer response test that assumes a constant kmt is also described by Chong et al. [14]. Both versions of the tracer response test require that the relevant kinematic viscosity (µ/ρ) is approximately independent of concentration, a reasonable assumption for NaCl given that its kinematic viscosity only increases around 0.1 % for each 1 000 ppm increase in concentration [20]. They also assume that the fouling resistance ( R f ) will remain constant despite a change of inlet solute concentration. This is because the tracer test only requires a relatively short amount of time to be performed (< 30 minutes) and, therefore, changes in cake layer mass during that interval can be neglected. 3. Cake-layer mass transfer model Several distinct types of fouling affect membrane separation operations, and they are typically classified as scaling, particulate/colloidal, biological and organic [21]. The different types of fouling may occur concurrently, but differ in the mechanisms by which they cause flux decline. Particulate fouling has been extensively studied [22-26] and, as opposed to other types of fouling, its effect on permeate flux is relatively easy to model due to the simplicity of the mechanisms involved. This section presents the development of a permeation mass transfer model applicable to a particulate cake layer, incorporating the effect of cakeenhanced osmotic pressure.

7

A number of assumptions are used here in order to further simplify the calculations and reduce the computation time, whilst keeping the main characteristics of the problem. The major assumptions used in this paper are as follows: •

Osmotic pressure is assumed to vary linearly with solute concentration.

•

Membrane resistance (Rm) and intrinsic rejection are constant (Rslt).

•

Fouling is simulated as a cake layer consisting of uniform spherical particles with the same diameter (dpart) and density (ρpart). Dissolution of the particles is not considered.

•

The mass of foulant per unit of membrane area ( mc′′ ) is known at every location on the membrane surface. In this paper we assume the foulant particles are spread over the surface of the membrane such that the mass of foulant per unit surface is uniform.

•

The distance between the particles and, hence, the cake layer porosity (εc) is affected by the ionic strength of the solution [27]. This means that the solute concentration (ω) affects the cake layer porosity.

•

Cake porosity can vary within the cake layer both in the direction of the bulk feed flow (x) and normal to the membrane (y); hence, cake porosity is a continuous variable within the cake layer [28].

•

For simplicity, the cake is assumed to be incompressible in the sense that the specific cake resistance does not depend on compressive drag within the cake layer [29]. This effect could be incorporated at a later stage.

•

Fluid flow within the cake layer is normal to the membrane surface. Cross flow velocity is zero within the cake layer [28].

•

Solute diffusion through the cake layer is hindered by the porosity and tortuosity (τc) of the cake. In this paper, we assume that tortuosity is constant within the cake layer.

Most of these assumptions can be relaxed if desired. In particular, the assumptions about osmotic pressure, constant membrane properties, cake layer tortuosity and uniform mass of foulant can easily be relaxed to accommodate for non-uniform variables, timedependent variables or variables that depend on other simulation parameters. 3.1

Model development A particulate fouling layer reduces the trans-membrane pressure and increases the

concentration of the solute at the membrane surface. According to CEOP theory [9], these two 8

effects are separately accounted for by the fouling resistance (Rf) and the CP index respectively. For these reasons, a more appropriate form of the flux equation (3) for CEOP modelling purposes is one using the conditions at the membrane surface: JV =

pw − p p − σϕ (ωw − ω p )

µ Rm

(14)

In equation (14) the effect of the fouling resistance is taken into account by the value of the membrane surface pressure (pw), and concentration polarisation is accounted for by using the solute mass fraction at the membrane surface (ωw). The values of these variables can be obtained by modelling the effect of the fouling layer on pressure drop and solute transport within the cake layer. The effect of the cake layer on pressure drop can be modelled with the aid of the Carman-Kozeny equation. The original form of the Carman-Kozeny equation relates the pressure drop through a porous bed to the viscosity and velocity of the fluid through this bed, as well as the porosity and surface area per unit volume of the bed. Applying this equation to a cake that consists of spherical particles with uniform diameter on a membrane surface yields the following: dp 180 µ JV (1 − ε c ) = 2 ε c3 dy d part

2

(15)

The specific cake resistance (αc) can be defined as [30].

αc =

180 (1 − ε c ) 2 ρ part d part ε c3

(16)

It should be noted that the cake porosity (and hence specific cake resistance) is not assumed to depend directly on the TMP. However, the TMP has an indirect effect on the hydraulic resistance of the cake through equations (14) and (15); i.e. a higher TMP will lead to a larger flux, thus increasing the pressure drop per unit of cake thickness. Chong [31] carried out dead-end filtration tests of colloidal silica (dpart = 23.8 nm,

ρpart = 2 200 kg/m3) for different concentrations of sodium chloride and determined the resulting specific cake resistance. The experiments were carried out using an UF membrane that fully rejected the silica but did not reject the solute (there was no concentration

9

polarisation). He used the experimental data to develop the following empirical correlation between solute concentration and specific cake resistance, and therefore cake porosity: C2 = α c C= 1ω

180 (1 − ε c ) 2 ρ part d part ε c3

(17)

where, for colloidal silica fouling in a NaCl solution, the values of the constants are:

= C1 7.643755 ×1015 m kg C2 = 0.3384

(18)

Equation (17) implicitly establishes a dependence of cake porosity on solute concentration. The mechanism by which cake porosity is expected to vary with solute concentration is related to the electric double layer that forms around each particle [32, 33]. There is significant evidence that inter-particle interactions are mainly controlled by the ionic strength of the fluid around the particles [28], especially for particle diameters below 50 nm [27]. Given that sharp changes in solute concentration are expected within the cake layer due to concentration polarisation and CEOP, it is reasonable to expect that the ionic strength will vary within the cake layer, as well as the repulsive forces between particles. The cake-layer mass transfer model presented in this paper takes these effects into consideration. It is important to note that although the values for the constants C1 and C2 presented in equation (18) are not geometry dependent, they are only applicable for uniform silica particles in a sodium chloride solution, with the same diameter as that used by Chong [31]. In order to model other types of particles, the values for the constants C1 and C2 for the system under consideration would need to be determined. Given that the dependence of cake porosity on the ionic strength of the solution is affected by the size of the particles [27], it is expected that C2 will be smaller for larger particles. It is also possible to assume a constant specific cake resistance equal to C1 by setting C2 = 0. Assuming that cross-flow velocity is zero at the cake-fluid interphase (i.e. using a noslip boundary condition), solute transport within the cake layer can be modelled as onedimensional solute diffusion in the direction normal to the membrane, with an effective diffusivity affected by the porosity and tortuosity of the cake layer [34]. Therefore:  Dε JV ω +  c  τc

 dω JV ω p =   dy

(19)

10

Tortuosity in the cake layer is typically in the range of 2-3 [34], while typical cake porosity values are in the range of 0.3 to 0.7 [24, 27]. This means that the effective diffusivity is expected to be 65 to 90 % lower than the unhindered solute diffusivity. Making use of equations (17) and (19) as well as the definition of cake porosity, integration of equation (15) results in the following expression for the pressure at the membrane surface:

p= pc + w

180 µ D 2 C2 d part τc

ε c ,c

∫ ε

(1 − ε c )( 2ε c − 3)

1 C2   ε c3 (1 − ε c , w )   c ,w 3 ε c 1 + ( Rslt − 1)  3     ε c , w (1 − ε c )  

dε c

(20)

where pc is the pressure at the cake-fluid interphase. Cake layer porosity at the cake-fluid interphase (εc,c) can be related to the solute mass fraction at that location (ωc) by making use of equation (17). Moreover, the solute mass fraction and cake porosity at the membrane wall (ωw and εc,w) can be obtained by modelling solute transport within the cake layer. Solute transport within the cake layer is described by equation (19). After some manipulation, integration of equation (19) yields: ε

c ,c C2τ c mc′′JV + ∫ ρ part D ε c ,w

( 2ε c − 3)

 ε c3 (1 − ε c , w )  1 + ( Rslt − 1)  3   ε c , w (1 − ε c ) 

1 C2

(21) 0 dε c =

If the solution flux across the membrane (JV) and the solute mass fraction at the cakefluid interphase (ωc) are known, equation (21) can be solved for wall cake porosity (εc,w). This value can then be used to obtain the membrane wall solute mass fraction (ωw) via equation (17). Finally, permeate transport across the membrane can be calculated by making use of the membrane intrinsic rejection: = ω p ωw (1 − Rslt )

(22)

where the membrane intrinsic rejection (Rslt) can be assumed to be constant for the length of the membrane channel. The value of Rslt can be calculated from experimental data. 3.2

Model limitations One of the main features of the model developed in section 3.1 comes from the

estimation of specific cake resistance and cake porosity based only on solute concentration within the cake, as indicated by equation (17). The estimation does not incorporate any 11

compressive drag effects on the cake porosity due to the viscous forces of the solution flux acting on the cake particles. This means that the values for the constants presented in equation (18) are only applicable for fluxes in the same range as in the experimental data from which the constants were estimated, that is, between 25 and 350 LMH [31]. Other parameters that are assumed to be constant in this model may also vary across wider ranges of concentration or pressures than those considered in this paper. In particular, the osmotic pressure may not follow a linear relationship with solute concentration over wide ranges of concentration, such that the osmotic pressure coefficient (ϕ) may depend on concentration and not be constant. In this paper, the range of solute concentrations considered was between 100 and 10 000 ppm. Further examples of variables that may vary depending on the operating parameters of the membrane system include the intrinsic rejection (Rslt), membrane resistance (Rm) and cake tortuosity (τc). 4. CFD Simulation The commercial CFD code ANSYS CFX-13.0 is used to solve the continuity, momentum and mass transport equations [35] in a rectangular channel. This geometry was chosen for the sake of simplicity, but the CEOP mass-transfer model can be applied to more complex geometries. Previous work [36, 37] has shown that neither gravity nor density variation will have a significant effect on the solutions obtained. Therefore, constant properties are employed and the effect of gravity is excluded. In order to further simplify this system the fluid is assumed to be Newtonian, at steady state (i.e. no time variations), in the laminar flow regime, the flow two-dimensional, and a binary mixture of water and salt is considered, with no sources of salt in the fluid. 4.1

Cake-layer mass transfer model in CFD In the CFD model used in this paper, the cake layer does not form part of the fluid

domain and the boundary of the fluid domain is fixed at the cake-fluid interphase. Solute diffusion in the cake layer is modelled by integration across the cake layer. The boundary condition for the solute mass fraction at the cake-fluid interphase is calculated in the usual manner [38], using the mass balance across the cake-membrane:

 ∂ω  −D   = JV (ωc − ω p )= JV ωc − ωw (1 − Rslt )   ∂y c

(23)

12

The salt mass fraction at the cake-fluid interphase (ωc) is taken from the CFD simulation and εc,c is calculated via equation (17). The remaining unknown variables (εc,w, ωw,

ωp, pw and JV) form a set of non-linear algebraic equations that can be reduced to a single nonlinear equation that is a function of only one variable (εc,w):

C2κ 2ξ1 + κ1ξ 2 +

ρ part Rm ξ3 = 0 mc′′

(24)

where the auxiliary variables are:

κ1 =

180 2 d part

κ2 =

τc µD

 κ1 (1 − ε c , w )  ξ1 = pc − p p − ϕσ Rslt   3  C1 ρ part ε c , w 

1 C2

ξ2 =

ξ3 =

(1 − ε c )( 2ε c − 3)

ε c ,c

∫ ε

1 C2 3     ε ε − 1 c ,w ( )   c c w , ε c3 1 + ( Rslt − 1)  3    ε c , w (1 − ε c )   

ε c ,c

∫

ε c ,w

( 2ε c − 3)

 ε c3 (1 − ε c , w )  1 + ( Rslt − 1)  3   ε c , w (1 − ε c ) 

1 C2

dε c

dε c

(25)

Equation (24) can be solved iteratively using the Newton method. Then, the local value of the solution flux is given by: = JV

κ1ξ 2  1   ξ1 +  µ Rm  κ 2 C2 

(26)

In addition, the local fouling resistance is given by: Rf =

mc′′κ1ξ 2

ρ partξ3

(27)

13

4.2

Problem geometry and boundary conditions Wall Inlet

Outlet

hch Wall Entrance

Membrane L

Wall Exit

Figure 2. Geometry of the channel used for testing the cake-layer mass transfer model with ANSYS CFX, indicating channel height (hch) and membrane length (L).

A 2D rectangular empty channel similar to that used by Chong et al. [14] is used as the fluid domain, and is shown in Figure 2. Simulation parameters used in this paper are summarised in Table 1. Even though this geometry is two-dimensional, the developed CEOP model and user subroutines written are also valid for 3D geometries. As can be seen in Figure 2, entrance and exit lengths separate the membrane wall section from the inlet and outlet respectively. The entrance and exit length dimensions follow the considerations stated by Schwinge et al. [39]. Table 1. Summary of simulation parameters used in this paper Variable Channel height (hch) Channel length (L) Solution density (ρ) Solution viscosity (µ) Solute diffusivity (D) Osmotic pressure coefficient (ϕ) Reflection coefficient (σ) Solute intrinsic rejection (Rslt) Membrane resistance (Rm) Fouling particle diameter (dpart) Fouling particle density (ρpart) C1 C2 Cake tortuosity (τc)

Value 1.4 mm 300 mm 1000 kg/m3 8.94×10-4 Pa s 1.5×10-9 m2/s 80.51 MPa 1 0.974 1.2×1014 m-1 23.8 nm 2 200 kg/m3 7.64×1015 m/kg 0.3384 2

The channel entrance and exit region wall surfaces, as well as the membrane channel top wall, are treated as non-slip walls with no mass transfer, where all velocity components and the mass fraction gradient normal to the boundary are set to zero (u = v = 0 and ∂ωc/∂nw = 0). The boundary condition at the membrane surface is specified by the cake-layer mass transfer model and the permeable wall condition, as described in sections 3.1 and 4.1. The velocity component parallel to the membrane surface is set to zero (uw = 0), while the normal velocity component (vw) and the mass fraction at the wall (ωw) are computed by the cake-layer mass transfer model. A membrane resistance value of 1.2×1014 m-1 is used, which 14

is typical for brackish water membranes [34] such as the one used by Chong et al. [14]. For simplicity, a linear dependence between dissolved solute concentration and osmotic pressure is assumed. This is a reasonable approximation (± 3.5 %) for concentrations of NaCl up to 17 550 ppm [40]. At the inlet of the flow domain, a flat velocity profile with u = uavg, v = 0 and ω = ωb0 is specified. At the outlet, an average reference pressure of zero is specified. A Schmidt number of 600 is used to represent sodium chloride. Mesh independent solutions are obtained using a structured mesh with smaller cells closer to the channel walls and membrane surface. This element size was determined after a series of runs with increasingly finer meshes. The thickness of the first grid element layer is of the order of 1 µm. 5. Results and discussion 5.1

Clean membrane tracer test assessment The performance of the tracer technique for measuring the CP index is first assessed for

a clean membrane ( mc′′ = 0). For this purpose, simulations were conducted at a constant permeate flux of 30 LMH, an inlet average velocity of 0.1 m/s, and three different inlet salt concentrations of 2 000, 2 850 and 4 000 ppm. The results of these simulations are summarised in Figure 3, which shows the local permeate flux and concentration polarisation (CPB0) distributions along the membrane channel. It can be seen in Figure 3 that the concentration polarisation along the membrane wall does not change significantly after a step change in the feed salt concentration.

Figure 3. CFD simulation results for local volumetric permeate flux (JV) and concentration polarisation (CPB0) obtained under a constant area-averaged flux of 30 LMH, for three different inlet salt concentrations.

15

During the constant area-averaged permeate flux experiment, an increase in inlet bulk concentration causes an increase in TMP that counteracts the increase in osmotic pressure. As is evidenced by Figure 3, CFD simulation results show that although the area-average permeate flux remains constant, the local permeate flux and local CPB0 change at every point along the membrane surface. Both of these values increase near the channel inlet and decrease near the channel outlet when the inlet solute concentration is increased. This phenomenon is related to the developing concentration boundary layer, which results in lower levels of concentration polarisation near the inlet than near the outlet. Because of the lower CP near the channel inlet, the increase in TMP has a stronger effect on flux than near the outlet, where CP is lower. Therefore, in order to attain the same average flux after an increase in bulk concentration, the local flux must decrease near the outlet and increase near the inlet when compared to the initial condition. In addition, equation (1) dictates that CP will increase in the regions where the local flux increases and decrease where the local flux decreases. This explains the results shown in Figure 3. Table 2. CFD clean membrane simulation results of global quantities, for three different inlet concentrations of sodium chloride. Variable Set JV (LMH) TMP0 (MPa) ωb 0 (ppm)

1 30.0 1.21 2 000

2 30.0 1.32 2 850

3 30.0 1.49 4 000

ωw (ppm)

3 937

5 419

7 572

ω p (ppm) ω p (ppm)

113.9

157.2

220.3

113.1

155.6

217.4

CP B 0 , from eq. (8)

1.9419

1.9443

1.9474

CP B 0 , from eq. (7) CPM, from eq. (10) CPT, from eq. (12) CPM,1 relative error (%) CPM,2 relative error (%) CPM,3 relative error (%)

1.9394 1.9392

1.9418 1.9412 1.9463 0.369 0.266 N/A

1.9449 1.9438 1.9487 0.488 N/A 0.251

The results of the clean membrane runs are used to calculate the value of different CP indices. Table 2 summarises the effect of a change in inlet solute concentration on the different forms of the CP index for a typical tracer test. From this data it can be seen that none of the forms of the CP index remain constant after the increase of solute concentration. Again, this is related to the developing concentration boundary layer, as the mass transfer coefficient is not constant along the membrane. Because of this, the tracer experiment (CPT) slightly over-estimates the concentration polarisation index it intends to measure (CPM) by 0.25 % to 0.49 %, where the CPM,i relative error is defined as (CPT – CPM,i) / CPM,i. 16

The main source of error in the tracer test comes from the assumption that the concentration polarisation index that is to be calculated remains constant when operating at constant permeate flux, despite the change in inlet concentration. This assumption is based on the solution of the one-dimensional diffusion equation. Even assuming that this relationship is valid for cross-flow experiments, it would apply only to the local concentration polarisation. However, the tracer test extrapolates this dependency to the relationship between the areaaveraged permeate flux and the calculated CPM. A perturbation approximation can be obtained for the deviation of the CPT index if the assumption of a constant CPM is relaxed and we assume that the observed rejection, R= obs

(ω

b0

− ω p ) ωb 0 , is approximately constant during the tracer response test. For a clean

membrane, this approximation yields:

CPT ,12 ≈ CPM ,1 + ωb 0,2

dCPM dCPM ≈ CPM ,2 + ωb 0,1 d ωb 0 d ωb 0

(28)

where CPT,12 represents the measured CPT index using a change of feed concentration from

ωb0,1 to ωb0,2. From equation (28) it is evident that if CPM increases during the tracer response test, the test (CPT,12) will over-estimate both the initial CPM,1 and the final CPM,2 indices by an amount proportional to the product of the derivative of CPM with feed concentration and either the final or the initial feed concentration, respectively. Given that the feed concentration is a positive number, this means that if CPM increases with an increase in feed concentration (ωb0), CPT,12 will over-estimate both the initial CPM,1 and the final CPM,2 indices. Conversely, if CPM decreases with an increase in ωb0, CPT,12 will under-estimate CPM,1 and CPM,2. Moreover, equation (28) shows that if CPM does not vary with ωb0 during the tracer test, then there will be no error and CPT,12 = CPM. Using equation (28) we can plot the relationship between the deviation of the tracer response for the clean membrane, the feed concentration, the measured CPT,12 index and the variation of CPT index with feed concentration. This relationship is depicted in Figure 4 as a relative error, and shows that a larger variability of CPT index will lead to larger measurement errors. It can also be seen that a lower feed concentration for the tracer test and a higher CPT index will reduce the measurement error.

17

Figure 4. Relationship between tracer response test error for a clean membrane, feed concentration (ωb0) and variation of CPT index with feed concentration.

One of the reasons for the variable concentration polarisation, despite the constant permeate flux, is the entrance effect. As can be seen in Figure 3, over the first 50 mm of the channel both the changes in permeate flux (10 %) and concentration polarisation (60 %) are quite dramatic. It may be possible to obtain better experimental results if the permeate from this region is not used for measuring the permeate concentration. However, this would be more difficult to do when applying the tracer test in an industrial setting. Nevertheless, commercially available membrane modules use spacer meshes to separate the membrane leaves. Spacers limit the entrance effects to within a very small (5-10 mm) distance from the module inlet [17, 41]. Therefore, the tracer test may be more accurate in commercially available membrane modules than in lab-scale membrane cells without spacers. The CPM derivative can be estimated by differentiating equation (9) under the assumption of approximately constant observed rejection, which yields:

= CPM

dCPM 1 dTMP0 − ωb 0 σϕ Robs dωb 0 d ωb 0

(29)

From successive differentiation of equation (29) it can be shown that the kth derivative of CPM is given by: d k CPM d k +1CPM  1  1 d k +1TMP0 = − ω   b0 d ωbk0 k + 1  σϕ Robs d ωbk0+1 d ωbk0+1 

(30)

Equation (30) can be used recursively to estimate the first derivative of CPM, based on increasingly higher order derivatives of TMP0. This value can then be substituted into 18

equation (29) to estimate CPM. If the highest order derivative of CPM is neglected, this approach yields the following Nth order truncation approximation of CPM: 1 CPM ≈ σϕ Robs

N

∑

( −ωb 0 )

k =1

k −1

k!

d k TMP0 d ωbk0

(31)

From the tracer response test, values of TMP0 can be measured at different feed concentrations, and the TMP0 derivatives can be calculated using polynomial interpolation. The tracer test method defined by equation (12) is equivalent to the first order (N = 1) truncation of equation (31) using two data points, and gives a relative error of 0.35 %. If the three data points presented in Table 2 are used, the second order (N = 2) truncation approximation of CPM gives a relative error of 0.06 %, about one order of magnitude smaller than the error from using the first-order two-point tracer response test described by equation (12). Higher order approximations are possible if more data points are used. 5.2

Fouled membrane simulations Table 3. Fouled membrane simulation conditions.

(ppm)

mc′′

(kg/m2)

Experimental data available

2 000

2 850

0.0127

Yes

0.1

2 000

2 850

0.0217

No

1.57 MPa

0.1

2 000

4 000

0.0122

No

TMP0

1.57 MPa

0.1

2 000

4 000

0.0131

Yes

5

TMP0

1.57 MPa

0.26

2 000

4 000

0.0131

Yes

6

TMP0

1.57 MPa

0.26

2 000

4 000

0.0145

Yes

7

TMP0

1.57 MPa

0.26

2 000

4 000

0.0219

No

8

TMP0

1.57 MPa

0.26

2 000

4 000

0.0242

No

uavg

ω b 0,1

ω b 0,2

(m/s)

(ppm)

30 LMH

0.1

JV

30 LMH

3

TMP0

4

Run

Constant

Value

1

JV

2

The conditions of four experimental tracer test results reported by Chong [31] are simulated using the cake-layer permeable wall model described in section 4.1. Four other simulations are also carried out at similar flow conditions as the experimental data but at different foulant mass ( mc′′ ) conditions. Table 3 reports the model parameters estimated from the data of Chong [31] as well as the conditions of all the simulation runs. Two of these runs (runs 1 and 2) are carried out at constant area-averaged volumetric flux ( JV ), while the other six runs are carried out at constant inlet trans-membrane pressure (TMP0). The values for mc′′

19

reported in Table 3 are based on the R f measurements reported by Chong [31], and assumed to be constant along the membrane surface. The fouled membrane simulation results for runs 1, 4, 5 and 6 are compared against the experimental data of Chong [31] in Figure 5 and Figure 9. There is good agreement between the predicted and measured flux and TMP0 values, as all of the predicted values are within 7 % of the reported measurements. The differences can be attributed to the use of a uniform fouling layer for the simulations (a constant mc′′ along the membrane surface) and, to a lesser extent, to the assumption of constant solution properties (µ, ρ and D). Flux values for the constant pressure run are over-predicted at the lower solution cross-flow rate (run 4). This appears to indicate that the value for the average mc′′ along the membrane surface is underestimated for run 4. On the other hand, the TMP0 results for the constant flux run (run 1) and the flux results for the constant pressure runs at the higher solution cross-flow rate (runs 5 and 6) show an excellent agreement with the experimental results. The maximum error for runs 1, 5 and 6 is under 2.3 %, which is within the measurement uncertainty of ± 3 %.

Figure 5. Comparison of predicted average fluxes (for constant TMP0 runs) and TMP0 values (for constant flux runs) against the measurements reported by Chong [31] for the fouled membrane conditions described in Table 3.

For the constant flux runs (1 and 2) the concentration polarisation along the membrane wall does not change significantly after a step change in the salt concentration. This is evidenced in Figure 6 for run 1, where it can be seen that the increase in salt concentration at constant average flux causes an increase in concentration polarisation in the upstream half of the channel and a decrease in the downstream half of the channel. The average concentration polarisation index ( CP B 0 ) increases by about 0.32 %, which is slightly more than its increase

20

for the clean membrane simulation (0.12 %). Despite the small scale of this increase, the change in the average CPB0 index has the potential to affect the tracer response measurement of the CPT index, as was evidenced for the clean membrane simulation.

Figure 6. Variation of the local concentration polarisation (CPB0) along the membrane surface, for a fouled

6 000

ω (ppm)

Salt mass fraction, ω (ppm)

7 000

Cake layer

5 000

7 000

0.42

6 000

0.41

5 000

0.40

4 000

4 000

Clean

3 000

2 000

0

50

0

4

Run 2

100

150

8

y (µm)

200

12

16

250

0.39

Cake porosity, εc

membrane constant flux simulation (run 1), for two different feed salt concentrations (ωb0).

300

Distance from membrane, y (µm)

Figure 7. Comparison of concentration profiles prior to tracer injection for the clean run and fouled run 2, at a location 290 mm from the channel inlet. Inset shows concentration and porosity profiles within the cake layer for run 2.

The effect of the cake layer on solute diffusion leading to CEOP is evidenced in Figure 7 for run 2 near the channel outlet (chosen for illustrative purposes due to higher extent of fouling than for run 1). Because of the lack of cross-flow and the decrease in effective diffusivity within the cake layer (80 % lower than in the bulk), concentration at the membrane surface is about 50 % higher for run 2 than for the clean run. The increase in concentration at the membrane leads to a higher osmotic pressure and, hence, a higher TMP (by about 30 %) is 21

required to produce the same flux. More than half of the increase in TMP – an increase of about 17 % of the clean run TMP – is due to CEOP, and the rest is due to the hydraulic resistance of the cake layer. This highlights the importance of the CEOP effect for this case. The inset in Figure 7 also shows the porosity profile within the cake layer for run 2 near the channel outlet. Simulation results for porosity are expected to be lower close to the membrane surface due to the change in ionic strength within the cake layer. However, the variation in cake porosity is not very significant (less than 4 %) despite a 50 % change in concentration. In fact, as seen in Figure 8, cake porosity ranges from just over 0.39 to about 0.43 over the whole cake layer for run 2. This suggests that the variation of cake porosity due to ionic strength for particulate fouling does not play a significant role in the CEOP phenomenon under the conditions simulated in this paper.

Figure 8. Contours of porosity within the cake layer for run 2, prior to tracer injection

5.3

Fouled membrane tracer test assessment Values for CPT are obtained following the tracer response test of Chong et al. [14],

using both the experimental and simulated data. These values are reported in Figure 9. The simulated runs at constant TMP0 (runs 4, 5 and 6) correctly predict a decrease in concentration polarisation when the inlet salt concentration is increased. However, despite the good agreement between simulated and predicted flux and TMP0 values, lesser agreement is found for the CPT predictions using simulation data, with relative errors ranging from 7 % to 14 %, and a RMS relative error of 11 %. This can be considered a relatively good agreement given the uncertainties involved (e.g. fouling layer distribution along the membrane surface and estimation of parameters involved in the simulation, such as Rslt and mc′′ ) and the assumptions of constant properties and linear osmotic pressure dependency. Moreover, the data presented

22

in Figure 9 further supports the idea that the value for the average mc′′ along the membrane surface was under-estimated for run 4 (hence the under-prediction of CPT by the simulation). In addition, it indicates that the CPT index is more sensitive than the flux or TMP0 to variations in fouling layer mass.

Figure 9. Comparison of concentration polarisation index calculated by the tracer test (CPT) from the experimental data of Chong [31] and from the simulation results for the fouled membrane conditions described in Table 3.

It is important to note that the tracer response test assumes that the fouling resistance remains constant after the change in inlet salt concentration. However, as was observed by Chong [31], a change in feed concentration has an effect on the porosity of the cake layer, and consequently on the fouling resistance. The cake-layer mass transfer model used in the simulations presented here takes into account this effect and assumes a constant fouling layer mass, instead of constant fouling layer resistance. However, it is unclear whether the effect of concentration on cake porosity observed by Chong [31] would be valid for a consolidated cake or for other types of fouling. In Figure 10 and Figure 11, the CPT indices calculated from the simulation data using the tracer response test are compared against the CPM indices calculated from equation (10), also using simulation data. As was the case for the clean membrane runs, the CPT overestimates CPM. However, the relative error for the tracer test in the fouled runs is in the range of 9 % to 23 %, much higher than the 0.35 % relative error observed for the clean membrane runs. The magnitude of the error appears to be directly proportional to the fouling layer mass. In addition, as predicted by equation (28), a higher feed concentration leads to a lower CPM estimation error. As explained for the clean membrane runs, the error in the constant pressure tracer response test is related to the assumption of a constant CPM index. For the fouled

23

membrane runs the increased error is also due to a variation in the fouling resistance, which in turn is related to the effect of salt concentration on fouling layer porosity and specific cake resistance, as described by equation (17).

Figure 10. Comparison of concentration polarisation index estimated by the tracer test (CPT) and the “measureable” concentration polarisation index (CPM), calculated from simulation results.

Figure 11. Relationship between CPM relative error and mass of fouling layer ( mc′′ ). Feed concentration (ωb0) is indicated for the constant pressure runs.

The main purpose of the tracer response test is to provide accurate estimates of CP and fouling resistance under CEOP conditions. Hence, fouling resistance estimates calculated from the simulated tracer response data using equation (13) are compared against actual fouling resistances in Figure 12. The fouling resistance estimates significantly under-predict the actual fouling resistance, with relative errors in the range of 35 % to 53 %. Higher relative errors are found for higher feed concentration. Although the absolute CP error and fouling

24

resistance error are related through equation (13), there does not appear to be a correlation between fouling layer mass or CP relative error and fouling resistance relative error. Nevertheless, the fouling resistance relative errors are much higher than the errors found for the CP estimates. This raises concerns about the usefulness of the tracer response test for estimating fouling resistance under the CEOP conditions reported in this paper. Moreover, it should be noted that the estimation errors for the tracer response test are not specific to particulate fouling, but are related to the deviations in CP and fouling resistance during the tracer test, regardless of the type of fouling present.

Figure 12. Comparison of fouling resistance estimated by the tracer test (RfT) and the permeate flux averaged fouling resistance ( R f ), calculated from simulation results.

In order to estimate the error for the tracer response test under fouled conditions, it is possible to define a pressure-loss fouling factor R′f = µ JV R f such that equation (9) becomes:

JV =

TMP0 − σϕ CPM (ωb 0 − ω p ) − R′f

µ Rm

(32)

If the assumptions of constant CPM and R f are relaxed, we can obtain a perturbation approximation for the deviation of the CPT index. Assuming that the observed rejection, R= obs

(ω

b0

− ω p ) ωb 0 , is approximately constant during the tracer response test, this

approximation yields:

CPT ,12 ≈ CPM ,1 + ωb 0,2

dR′f dCPM 1 + d ωb 0 σϕ Robs d ωb 0

(33)

25

A comparison of equation (33) to equation (28) shows that an increase in fouling resistance during the tracer test increases the over-estimation error of the CPM index by the tracer response test. For the case of particulate silicate fouling in sodium chloride, Chong [31] found that the specific cake resistance increases with salt concentration. Therefore, for the constant flux fouling test run simulated in this work, R f and R′f may increase with concentration and cause a larger over-estimation of the CPM index than if no fouling layer was present. Given that equation (17) from the model used in this paper is based on the results of Chong [31], this explains the higher relative error for the CPM index using the tracer test for fouled run 1 than for the clean membrane run, as seen in Figure 10. As opposed to the clean membrane cases, for fouled runs it is not possible to estimate the changes in CPM and R′f based on the tracer test data. This is because in the tracer test equation (9) is used to calculate the estimated fouling resistance (RfT) using CPT as the estimate for CPM, and it is also used to calculate CPT. For this reason, the derivatives of RfT and CPT with respect to concentration cancel each other and do not provide any further information about the actual changes CPM and R′f :

dCPT 1 dR′fT = ωb 0 + 0 d ωb 0 σϕ Robs dωb 0

(34)

Nevertheless, it is unclear whether the hydraulic fouling resistance of a consolidated cake layer would respond to a change in feed concentration during the tracer test. If the fouling resistance does not change during a constant flux tracer test, equation (33) is reduced to equation (28), and the error estimation techniques proposed in section 5.1 can be used. Another significant source of error during a tracer response test is measurement error. The uncertainty in CPT can be related to the standard deviation in the measurements of TMP0,

ωb0 and ω p through the following relationship:

σ CP

= CPT ,12 T ,12

2 2 σ TMP + σ TMP

(TMP

0,1

0,2

0,2 − TMP0,1 )

2

+

σ ω2

b 0,1

+ σ ω2 p ,1 + σ ω2b 0,2 + σ ω2 p ,2

(ωb0,2 − ωb0,1 − ω p,2 + ω p,1 )

2

(35)

From equation (35) it can be concluded that small changes in feed concentration (which would therefore lead to small changes in TMP0) would result in larger uncertainties for CPT. For the experimental data reported by Chong [31], the uncertainty in CPT can be as high as ± 39 %. Therefore, small ωb0 changes would not be recommended for the tracer response test.

26

On the other hand, larger ωb0 changes can lead to significant changes in CPM and R′f , which introduce error to the tracer test as predicted by equations (28), (29) and (33). Nevertheless, if the change in R′f is minimal or can be neglected, multiple changes in tracer concentration coupled with the use of equation (31) can reduce this source of error and provide a better estimate of CPM. For the multiple tracer response test, a larger range of ωb0 is therefore recommended. 6. Conclusions A cake-layer mass transfer model applicable for RO, which incorporates the cakeenhanced osmotic pressure (CEOP) effect of a particulate fouling layer, was developed. This model includes the effect of a variable dissolved solute concentration on the specific cake resistance and porosity of the cake layer. The model is based on one-dimensional diffusion of the dissolved solute through the cake layer, and uses the solute concentration on the membrane surface to calculate the trans-membrane osmotic pressure. For simplicity, a linear dependence between dissolved solute concentration and osmotic pressure was assumed. The model is expected to produce accurate results for fluxes within 25 to 350 LMH and NaCl concentrations up to 10 000 ppm. The cake-layer mass transfer model was incorporated into the commercial Computational Fluid Dynamics (CFD) software package ANSYS CFX-13.0. Subsequent simulations have been validated against experimental data, and the model predictions were satisfactory (± 7 % for predicted permeate flux, and around ± 14 % for the CPT index). These predictions may be improved if variable properties and a fouling layer mass distribution are used instead of constant values. Despite the fact that the model assumes that the cake porosity is affected by changes in concentration, it is unclear whether the effect of concentration on cake porosity would be valid for a consolidated cake. In any case, it is possible to assume a constant cake layer porosity by setting C2 = 0 in the cake-layer model. In addition, it is also possible to use the cake-layer mass transfer model presented in this paper to predict the time dependence of flux if the rate of foulant deposition per unit of membrane area is known. The CFD model was used to help interpret and assess the tracer response technique. Model predictions confirmed the assumption that the average concentration polarisation along the membrane wall does not change significantly after a step change in the feed concentration of the tracer solute. However, it was found that the tracer experiment slightly over-estimates the concentration polarisation index and significantly under-estimates the fouling resistance. It is important to note that the estimation errors for the tracer response test are not specific to

27

particulate fouling, but are related to the deviations in CP and fouling resistance, regardless of the type of fouling present. These deviations are mainly due to entrance effects, which give rise to small variations in concentration polarisation and fouling resistance during the tracer tests. Moreover, the magnitude of this over-estimation is greater for fouled membrane conditions, increases with fouling layer mass and decreases with tracer concentration. Small measurement uncertainty was also found to be a significant source of error. A multiple data point tracer response test is proposed to take into account slight variations in concentration polarisation during the tracer tests. This proposed test reduces the error in the estimation of the concentration polarisation by one order of magnitude when using three data points (double step), compared to the single step tracer response test for a clean membrane. As with the original tracer response test, the proposed test is not limited to particulate fouling, but is valid for estimating concentration polarisation and fouling resistance for any type of fouling. It may be possible to further improve the accuracy of the estimation of the concentration polarisation index by means of the tracer response test if the permeate from the entrance region is not used for measuring the permeate concentration, or if spacers are introduced into the feed channel. Both of these approaches would reduce entrance effects on the measured concentration values. Acknowledgements The authors would like to acknowledge the Australian Department of Education, Science and Training for funding this project through the International Science Linkage grant established under the Australian Government’s innovation program ‘Backing Australia’s Ability’, in association with the “Membrane-Based Desalination: An Integrated Approach” (MEDINA) project as part of the European Union 6th Framework Program (EU FP6). Nomenclature

C1

Auxiliary constant, (m/kg)

C2

Auxiliary constant

CPB

Theoretical local concentration polarisation index based on the local bulk solute concentration

CPB 0

Theoretical local concentration polarisation index based on the inlet bulk solute concentration

CP B 0

Area averaged concentration polarisation index defined by equation (8)

CP B 0

Alternative averaged concentration polarisation index defined by equation (7)

28

CPM

“Measureable” concentration polarisation index defined by equation (9)

CPT

Concentration polarisation index obtained by tracer experiment equation (12)

D

Diffusion coefficient, (m2/s)

d part

Particle diameter, (m)

hch

Channel height, (m)

JV

Volumetric flux of solution through the membrane, (LMH, L m-2 h-1)

kmt

Mass transfer coefficient, (m/s)

L

Membrane channel length, (m)

mc′′

Mass of fouling layer per unit area of membrane, (kg/m2)

n

Distance in the direction normal to a surface, (m)

N

Order of truncation approximation for CPM

p

Pressure, (Pa)

Rf

Fouling layer hydraulic resistance, (1/m)

R′f

Pressure loss fouling factor, (kg m-1 s-2)

R fT

Fouling layer hydraulic resistance calculated from tracer experiment data using equation (13), (1/m)

R′fT

Pressure loss fouling factor from tracer experiment data, (kg m-1 s-2)

Rm

Membrane hydraulic resistance, (1/m)

Robs

Solute observed rejection by the membrane

Rslt

Solute intrinsic rejection by the membrane

Sc =

µ ρD

Schmidt number

TMP

Trans-membrane pressure, (Pa)

u

Velocity in x direction, (m/s)

v

Velocity in y direction, (m/s)

x

Distance in bulk flow direction, (m)

y

Distance in direction perpendicular to bulk flow, (m)

Greek letters

εc κ1 =

Cake layer porosity 180 d p2

Auxiliary constant, (1/m2)

29

κ2 =

τc µD

Auxiliary constant, (s/m2)

µ

Dynamic viscosity of the fluid, (kg m-1 s-1)

π

Osmotic pressure, (Pa)

ρ part

Particle density, (kg/m3)

σ

Reflection coefficient

σi

Standard deviation or measurement uncertainty for variable i

τc

Cake layer tortuosity

ξ1

Auxiliary variable, (Pa)

ξ2

Auxiliary variable

ξ3

Auxiliary variable

ϕ

Osmotic pressure coefficient, (Pa)

ω

Salt mass fraction, (ppm)

Subscripts

0

Value at the domain inlet or feed

1

Value before tracer response test

2, 3

Values during tracer response test

avg

Average value integrated over channel cross-section

b

Mass flow average or bulk flow value

c

Value for cake layer, or at cake-fluid interphase for flow variables

p

Value for the permeate

w

Value at the membrane wall, or at cake-membrane interphase for flow variables

Over-bars x=L

1 φ dx L x∫=0

φ

Area averaged value, such that: φ =

φ

1 JV φ dx Permeate flux averaged value, such that: φ = JV L x∫=0

x=L

30

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[2]

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[3]

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[4]

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[5]

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[6]

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[9]

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