Constant pressure filtration model
Abstract
One of the major obstacles preventing a more widespread application of membrane technology is the fact that the filtration performance inevitably decreases with filtration time. The understanding of causes of flux decline and the ability of predicting the flux performance is crucial for more extensive membrane applications. Therefore, this review focused on the analysis of the interrelated dynamics of membrane fouling and mass transport in the filtration/separation process. Furthermore, findings from various studies undertaken by many researchers to investigate various aspects of these complex phenomena, were analyzed and discussed. These previous studies form the foundation essential for the alleviation of adverse effect of membrane fouling and emphasize the fact that the development of a complex overlapping approach is needed. Investigations of the mechanisms of mass transfer in low and high pressure membranes, analyses of diffusion mechanisms in the membrane pore together with identifications of transport resistances resulting from membrane fouling will allow for a comprehensive understanding and indepth knowledge of the fouling phenomenon and applicable mass transfer mechanisms.
Keywords
 mass transfer
 modeling
 diffusion
 boundary layer
 transport resistances
 membrane fouling
1. Introduction
Today membrane technology is considered an important tool for sustainable growth in the industry due to its simplicity in concept and operation, modularity for an easy scaleup, potential to reclaim both permeate and retentate for recycling purposes, and reduced energy demand, all of which make it attractive for a more rational utilization of raw materials and waste minimization. Various membrane operations are currently available for a wide spectrum of industrial applications, where most of them can be used as unit operations in pressure driven processes, such as: microfiltration (MF), ultrafiltation (UF), nanofiltration (NF), reverse osmosis (RO), and membrane distillation (MD). Membrane operations show potential in molecular separation, clarification, fractionation, and concentration in liquid and gas phases. Nevertheless, one of the major obstacles preventing a more widespread application of membrane technology is the fact that the filtration performance inevitably decreases with filtration time. This phenomenon is known as membrane fouling and is considered to be the most serious problem affecting system performance and the primary challenge in membrane development. As such, membrane fouling leads to lower mass transfer rate, higher operating cost, higher energy demand, reduced membrane life time, and increased cleaning frequency.
The reduction of membrane flux below that of the corresponding pure water flux, or more generally, pure solvent flux can be divided into two separate components. First, the concentration polarization, which is a natural consequence of the membrane’s selectivity [1]. This phenomenon leads to an accumulation of particles or solutes in a mass transfer boundary layer adjacent to the membrane surface. The dissolved molecules accumulating at the surface hinder the solvent transport, which in turn lowers the solvent flow through the membrane. Second, due to the blockage of membrane pores during the filtration process by the combination of sieving and adsorption of particulates and compounds onto the membrane internal surface or within the membrane pores. During this process additional hydraulic resistance is added to the permeate flux. Fouling leads to an increase in resistance, resulting in less flux for a given transmembrane pressure (TMP), or a higher TMP if flux is kept invariant [2]. Understanding the concentration polarization resistance is important when attempting to distinguish a reduction in the driving force across the membrane from the increase in resistance due to the fouling of the membrane [3, 4]. As the material passes through the membrane, it has to be transfered to the surface, diffused through the membrane, and finally transfered from the other side of the membrane to the bulk phase. When the material dissolves within the membrane, then the components are in equilibrium at the surfaces corresponding to their solubility in the membrane. The solute mass transfer through the membranes is controlled by diffusion as a result of the concentration gradient across the membrane surface. In our previously developed mechanistic model for membrane fouling, the mass of fouling attached to the pore wall was a function of the mass transfer coefficient, which, in turn, was dependent on the diffusion coefficient of the foulants in water, particle sizes, and membrane pore sizes [5, 6].
Researchers have invested a lot of effort in seeking new accurate models for the prediction of membrane performance. In most applications, the system equations were derived based on simplifications and assumptions, which resulted in limited practical application of these models.
Nevertheless, these models provide sufficient basis for an incorporation of more parameters for more realistic scenarios. While so far no universally accepted model exists for describing diffusioncontrolled membrane processes, understanding the multiple factors affecting membrane mass transfer would certainly help to develop better predictive models. There are many different theories and models describing mass transfer in diffusioncontrolled membrane processes, however, several fundamental principles or theories are used as the basis for the majority of these models: convection, diffusion, film theory, and electroneutrality [7, 8]. Most mass transport models assume constant mass transfer coefficients (MTCs) based on a homogeneous membrane surface. Recent innovative studies evaluated mass transfer processes by incorporating membrane surface morphology into a diffusionbased model while assuming that MTCs are dependent on the thickness variation of the membrane’s active layer [9]. A nonhomogeneous diffusion model (NHDM) was then developed in order to account for the surface variations in the active layer [10]. Notably, concentration polarization (CP) is also affected by this nonhomogeneous surface property. As a consequence, the NHDM was modified by incorporating the CP factor. Recent studies have likewise shown that the membrane surface morphology influences colloidal fouling behavior. Therefore, the main objectives of this chapter are to investigate the mechanisms of mass transfer, to critically analyze the different diffusion mechanisms in membrane pores, to identify different transport resistances resulting from membrane fouling, and to discuss various models of mass transfer, in order to obtain a comprehensive understanding of the factors affecting mass transfer phenomenon in membrane and help to develop a comprehensive model predicting the membrane performance.
2. Mass transfer and modeling of the filtration process in the absence of fouling
For an indepth analysis of the transport resistances in the filtration/separation process due to membrane fouling, it is necessary to understand the mass transfer occurring through membranes in the absence of fouling. This type of transport in membranes can be generally classified into two types of diffusion: first is the diffusion through a dense membrane, which follows the Fick’s law and does not depend primarily on the structure of the solid membrane; and second is the diffusion in porous membrane where the porous structure and channel type are important. This study will briefly consider both types of this diffusion.
2.1. Solution diffusion theory through dense membranes
The solutiondiffusion process, as illustrated in Figure 1, is the most common mechanism for mass transport through nonporous membranes [11]. It is generally suggested that the component permeation through a homogeneous membrane consists of five fundamental mutually dependent processes: (1) the solute molecules must first be transported or diffused through the liquid film (or gas film; in the later case the mass transfer resistance of the boundary layer is negligible causing this step to be omitted) of the feed phase on the feed side of the membrane, as shown in Figure 1 (B); (2) the solution of the solute molecule in the upstream surface of the membrane; (3) diffusion of the dissolved species across the membrane matrix occurs; (4) desorption of the solute molecules in the downstream side (permeate side) of the membrane; and (5) diffusion through the boundary layer of the permeate phase. The first and the fifth steps can be omitted when there are insignificant mass transfer resistances between the liquid and membrane phases. This assumption is valid for gas permeation or liquid permeation with high flow rates in the fluid phases on both sides of the membranes.
Fick's first law relates the diffusive flux to the concentration under the assumption of steady state. It postulates that the flux goes from regions of high concentration to regions of low concentration, with a magnitude that is proportional to the concentration gradient [12]. Fick’s first law of diffusion is mathematically expressed in Equation (1):
where
2.2. Convective transport through a porous membrane layer
As the basis of the pore flow model, pressuredriven convective flow is most commonly used to describe the flow in capillary tubes. The key mechanism at the core of both microfiltration and ultrafiltration is the sieving process, which effectively rejects molecules with the size greater than that of the membrane’s pores. Commonly, two approaches are utilized as a way of describing permeability through porous membranes. In the first case, when the membrane resembles an arrangement of nearspherical particles, then the Carman–Kozeny equation can be applied [12], as illustrated in Equation (2):
where
In the second approach, the laminar fluid flow through the capillaries can be described by means of the differential momentum balance equation, Newton’s law of viscosity [13, 14]. After integration, the wellknown HagenPoiseuille equation for the average convective velocity through the porous membrane was obtained to calculate the permeate flux, as described by Equation (3):
where τ is the tortuosity of the capillaries and
where
3. Mass transfer boundary layer and membrane fouling
The rejection of contaminated particles results in a higher concentration in the mass transfer boundary layer. The process of mass accumulation in the mass transfer boundary region is called concentration polarization (CP). CP is a natural consequence of membrane selectivity, which leads to a reduction in the permeate flux because of the gel layer formation [15]. As a result of the high concentration of the rejected component, a backdiffusion occurs from the thin layer adjacent to the membrane back into the bulk [16], as shown in Figure 2. The reduction of membrane performance can be represented as a reduction in the effective TMP driving force due to an osmotic pressure difference between the filtrate and the feed solution immediately adjacent to the membrane surface [11]. As shown in Figure (2),
Thus, in general the overall mass balance equation of a component
where
The subscripts
where K is Boltzmann constant,
The curvature of the concentration profile depends on the flux and so does the relationship between the mass transfer coefficients [19]. The mass transfer coefficient may be obtained from experiments using Equation (8), which is rearranged from Equation (6) with the assumption that the solute concentration in the permeate is very small compared to that of the feed and that at the membrane surface.
The mass transfer boundary layer thickness is equal to
Furthermore, fouling may be due to adsorption, pore blockage, deposition, and gel layer formation. The adsorption occurs whenever there are interactions between the membrane and the solute or particles. If the degree of adsorption is concentrationdependent, then the CP exacerbates the amount of adsorption creating an extra transport resistance [1]. Notably, the monolayer of particles and solute leads to an additional hydraulic resistance. The membrane surface hydrophilicity is the primary factor affecting particletomembrane attachments [21–23]. The deposit of particles can grow thicker by additional layers on top of the membrane’s surface, leading to a significant hydraulic resistance, often referred to as a cake resistance. For certain macromolecules, CP may lead to gel formation in the immediate vicinity of the membrane surface. Furthermore, the size of contaminated particles plays a major role in the membrane pore blockage, which leads to a reduction in flux due to the partial closure of pores [5, 6]. In general terms, membrane fouling models are concerned with the nondissolved materials that are either deposited on the membrane surface, or deposited in the membrane pores.
4. Modeling of porous membrane filtration in the presence of fouling
Membrane fouling provides additional transport resistance during the separation process. As a consequence, permeate flux and TMP are the best indicators of membrane fouling phenomenon [2]. The transport resistance created by the membrane fouling leads to a significant increase in hydraulic resistance, manifested as a permeate flux decline or TMP increase when the process is operated under constantTMP or constantflux conditions. The flux through most fouled membranes cannot be described by the idealized equation of HagenPoiseuille. It should also be noted that if a solute is present, then there will be CP. The following equation is used to describe the flux in the absence of any fouling cake layer:
where
It was illustrated by Wijmans et al. (1985) that the two expressions are thermodynamically equivalent with the concentration boundary layer impeding the flow of the solvent and thus “consuming” part of the overall driving force [3]. While the value of
The first of the additional hydraulic resistances,
When considering these fouling mechanisms, the critical flux, at which fouling is first observed for a given feed concentration and given crossflow velocity, should be a design consideration for all pressuredriven processes [27]. A model of critical flux,
For low
where
where
Several filtration models have been developed to describe the dynamics that frame the fouling process at a constant TMP. These models relate the permeate flow (Q), peremrate volume (V), the time (t) with the filtration constants for each model (K_{bc}, K_{ic}, K_{sc,} K_{dc}), and the initial permeate flow (Q_{0}). The mathematical expressions of these models and their assumptions are shown in Table 1.
Model  Equation  Assumption 
Complete blocking filtration  
Particles are not superimposed on one another; the blocked surface area is proportional to the permeate volume. 
Intermediate blocking filtration  
Particles can overlap each other; not every deposited particle block the pores. 
Standard blocking filtration  
Particles are small enough to enter the pores; the decrease of pore volume is proportional to the permeate volume. 
Cake filtration  
Particles are big enough to not enter the pores; and therefore forms a cake layer on the surface 
Applications of these models were examined in numerous recent publications [29–33]. Lim investigated the fouling behavior of microfiltration membranes in an activated sludge system [29]. The results show that the main types of membrane fouling in this case were attributed to initial pore blocking (standard blocking filtration model) followed by cake formation (cake filtration model) [29]. Bolton (2006) compared these four models in application to microfiltration and ultrafiltration of biological fluids, the conclusion is the combined cake filtration model and the complete blocking model resulted in the best fit of experimental data [30]. In a cross flow ultrafiltration experiment conducted by Tarabara (2004), cake formation was investigated under variable particle size and solution ionic strength. The results revealed that in all cases, there is a dense layer of the colloidal deposit adjacent to the membrane with an abrupt transition to a much more porous layer near the membranesuspension interface [31]. This observation implies that different models should be considered at different phases of fouling.
5. Diffusion models in the presence of fouling
There is general consensus among research scholars that the solute mass transfer through RO and NF membranes is controlled by diffusion as a result of the concentration gradient across the membrane surface [34–38]. Researchers have invested extensive efforts in seeking new, comprehensive and accurate diffusion models capable of predicting membrane performance in cases involving membrane fouling. In most cases, the models were derived based on simplifications and assumptions, rendering limitation of the application of the models to practice. However, the diffusion models provide the basis and opportunity for incorporation of more complicated and realistic scenarios. Thus far, there is no universally accepted model for describing diffusioncontrolled membrane processes, but understanding the factors affecting membrane mass transfer will certainly help with developing a better predictive model.
5.1. Homogeneous solution diffusion model (HSDM)
The homogeneous solution diffusion model (HSDM) is one of the basic models describing mass transfer as a function of concentration gradient and the pressure difference across the membrane’s film. The HSDM assumes a homogeneous feed stream and homogeneous membrane surface that are linearly correlated to the average feed concentration and system recovery [39, 40]. This model is utilized to predict the permeate concentration, with a given solvent and solute mass transfer coefficient (MTC), overall recovery, transmembrane pressure, and feed concentration over a single element. The HSDM final formulation is given in Equation (16) [41, 42]:
where C_{p} is the permeate concentration, Fw is the water permeate flux, C_{f} is the feed concentration, k_{g} is solute MTC, and Rc is the overall recovery, which is the ratio of permeate flow rate to feed flow rate. The solute MTCs were found to be related to membrane feed water qualities and membrane properties. They can be theoretically determined using Fick’s first law described in Equation (1) assuming a steady state within a thin film.
Duranceau (2011) developed a model correlating the solute MTCs to the solute charge and molecular weight with a normal distribution [41, 42]. Furthermore, Zhao (2005) investigated the influence of membrane surface properties and feed water qualities on solute MTCs. It was found that the MTC increased as the surface roughness and contact angle were increased. Also, the amount of natural organic matter (NOM) in the feed water appeared to positively affect the MTC [35]. The HSDM has been modified to incorporate the osmotic pressure in an integrated incremental model by Zhao [38]. The model considered the osmotic pressure increase as a result of enhanced salt concentration in the membrane channel. Duranceau (2009) has modified the HSDM to predict multistage membrane systems [37]. Absar (2010) modified the diffusion model by comparing the effects of cocurrent and countercurrent permeate flow with respect to the feed stream. It was concluded that the countercurrent flow pattern, especially during the concentrating process, has a better overall efficiency [43].
5.2. Nonhomogeneous diffusion model development
Previous diffusion models had typically assumed a constant mass transfer rate across a nonporous, smooth, flat membrane surface. Nevertheless, the membrane surface characterization has been shown to affect the membrane’s performance, as well as the overall fouling process [32, 33]. Therefore, nonhomogeneous mass transfer model introduced an additional variable affecting mass transport that can be quantified by considering the nonuniform structure of the membrane surface. Research conducted by Mendret (2010) demonstrated that random distribution models may be used to effectively describe the heterogeneity of surfaces [28].
The HSDM can be modified by incorporating the effects of a nonuniform membrane surfaces on the solute permeability so as to obtain a nonhomogeneous diffusion model. In addition, Sung (1993) has incorporated the filmtheory factors accounting for the CP to develop the nonhomogeneous diffusion model (NHDM), as presented in Equation (17)[39]. Notably, the k_{s}′ is the back mass transfer coefficient was also incorporated:
where
where
where
Mendret (2010) developed a model that considered two dimensional flow of a particle suspension in the membrane’s channel in order to investigate the growth of a cake on a nonuniform permeability membrane [28]. The overall mass transfer coefficient (
where
The NHDM and HSDM were verified using pilot plant data from a brackish groundwater plant located in Florida. It was found that the NHDM predictions were in close agreement with actual permeate concentrations. The results likewise indicated that the NHDM provides a more accurate prediction of solute concentration with a high concentration in the feed stream. This can be explained by the role of CP effects on the mass transfer with different feed solute concentration, since the CP appears to be more significant in determining the salt passage in instances when the solute feed concentration is higher. Although it was found that the fouling behavior was related to the degree of surface roughness [2, 44–46], incorporating the surface roughness into the diffusion models and evaluating its impact has not been comprehensively studied yet.
6. Dimensionless mass transfer coefficient models
The mass transfer coefficient (
where d_{h} is the hydraulic diameter of the system;
Sirshendu and Bhattacharya [9] proposed new Sherwood number relationships that incorporate the effects of suction of a membrane for laminar flow in rectangular, radial, and tubular geometries. However, in this research study, the variations in physical properties brought about by concentration, and which become dominant under highly polarized conditions in a UF process, were not taken into consideration during the analysis. The large number of experimental variables and corrections that are associated with the mass transfer relations provide a valid reason for directly determining the mass transfer coefficient experimentally [49]. Based on the data already available in the literature, the variables that have the capacity to influence the value of the mass transfer coefficient can be expected to include: the applied transmembrane pressure, the crossflow velocity, the flux
The commonly used Sherwood number correlation is that from Deissler, as shown in Equation (26). For solutes such as BSA and Dextrans (SC= 13,00022,000), the mass transfer coefficient can be calculated from Harriott and Hamilton’s equation [44].
6.1. The velocity variation method
The Sherwood relations for
where
The velocity variation method is not absolutely reliable when it comes to determining the mass transfer coefficient, primarily because its sensitivity to the experimental parameter values. The best experimental method for determining the mass transfer coefficient remains that of evaluation of the observed retention at varying velocities. Therefore, the use of normal mass transfer relations can be equally reliable and much easier than the velocity variation method [44]. The velocity variation method can be recommended for practice when one or more of the parameters in the conventional mass transfer coefficient relationships are unknown.
6.2. The modified Sherwood relation
The modified Sherwood number relationships, including the effects of suction during vacuum filtration and variations in properties brought about by concentration, may be a more adequate representation of concentration polarization phenomena during ultrafiltration of macromolecules [44]. Such a relationship is formulated for laminar flow in a rectangular geometry. The developed model is then employed to predict permeate flux of the ultrafiltration process of BSA and dextran. The value of the exponent in Equation (25) is found to be 0.01 for all operating conditions of BSA, and 0.013 for dextran at 5 kgm^{−3} and 0.026 at 10 kg m^{−3}. The present model is able to accommodate incomplete solute retention by the membrane and can be effectively applied during the simultaneous prediction of permeate flux and observed retention.
7. Gas transport through membranes
A number of phenomenological and physiochemical models have been developed for gas transport, each of them suited specifically to particular types of penetrating membrane systems over limited pressure and temperature ranges [12]. Gas transport through membranes has been categorized as transport in homogeneous membranes, porous membranes, asymmetric, and composite membranes. The relationship between the permeation, flux, and operating conditions in general can be described by:
where
The permeability value,
The permeation rate,
In this case, the permeation rate,
The resistances in composite membranes are usually connected in a following manner: series, parallel, and two resistance arms. Two resistances are connected in series whenever two layers of membranes are combined in a series, as shown in Figure 3 (a). Two resistances are connected in parallel whenever the material of the membrane is different and connected along the same layer of the membrane’s surface, as shown in Figure 3 (b). The parallel combination of two resistance models is used when a homogeneous film of relatively high permeability is laminated on top of the membrane, as shown in Figure 3 (c).
The mass transport through a porous layer has been extensively studied. Its specific type depends on the pore size and its comparison to the mean path of the transported molecules. In the case of gas transport, three regions are distinguished regarding the transport [50]: (1) molecular and/or viscous flow, occurring when the mean free path of gas molecules is less than the pore size; (2) Knudsen diffusion, occuring when the pore size is larger than the size of transport species, but less than the mean free path of the gas molecule; and (3) molecular sieving transport (configurational diffusion), which tends to occur when the pore size is the same size as that of the transported species [51].
Molecular diffusion and/or viscous flow are dominated by the interaction between the transported molecules whenever the pore diameter is considerably larger than the mean free path. The viscous flow can be calculated according to HagenPoiseuille (Equation (2)), as illustrated in Equation (29):
where
After integration, it can be obtained for the mass transfer rate:
where subscript
In case of Knudsen diffusion, flux can be calculated as follows:
where J_{i} is the mass transfer flux (kmol/m^{2}s). The partial pressure gradient,
where R is the gas constant,
If the pore diameters approach the range comparable to that of the molecule sizes, then the configurational diffusion regime is reached [53]. The corresponding flux density can be described by [54]:
where
In the regions with very small pores compared with the solute molecular size, the effects of the surface diffusion might be significant. The surface diffusion flux can also be critical when it comes to the mass transport through pores in the Knudsen regime due to the adsorption [53, 54]. The most general approach for multicomponent surface diffusion was proposed by Krishna and Wesselingh [55]. Based on the physical picture of molecules moving on the surface, the generalized Maxwell À Stefan equations were applied so as to describe the interactions mechanistically [54, 55].
8. Conclusion
In membrane operations, there are several key issues that influence membrane separation and transport, which in turn affects the performance and the economy of the overall process. Membrane fouling is the major obstacle that inevitably decreases the membrane performance and cause flux decline due to the presence of external and internal membrane fouling. The understanding of causes of flux decline and the ability of predicting the flux performance is essential for more extensive membrane applications. Therefore, this review focused on the analysis of the interrelated dynamics of membrane fouling, flux decline, and mass transport in the filtration/separation process. In addition, this study illustrates that extensive efforts have been undertaken by many researchers to investigate various aspects of these complex phenomena. These studies form the foundation necessary for the alleviation of adverse effect of membrane fouling and emphasize the fact that the development of a complex overlapping approach is needed. Investigations of the mechanisms of mass transfer in high and low pressure membranes, analyses of diffusion mechanisms in membrane pores together with identifications of transport resistances resulting from membrane fouling will allow for a comprehensive understanding and indepth knowledge of the fouling phenomenon and applicable mass transfer mechanism.
Several mathematical models have been developed over the years in order to describe the solute and solvent mass transport in membrane processes based on a homogeneous membrane surface. In addition, the effect of the morphology of the membrane active layer on solvent and solute mass transport was incorporated. It was determined that solute mass transport is controlled by the nonhomogeneous diffusion in the thinner regions of the active layer. This nonuniform surface also affects the concentration polarization layer, where more solutes tend to accumulate on the valleys rather than on the ridges. The NHDM was developed, taking into account the membrane surface property, so as to describe the nonhomogeneous phenomenon while accommodating the fact that solute and solvent MTCs vary with active layer thickness. Based on the same conceptual the dynamic model, which is a nonhomogeneous diffusion model, was developed with and without the CP effect, and then compared with the HSDM. It was found that the mass transport across the membrane film was affected by the morphology of the membrane surface.
Nomenclature



Crosssectional area to flow [m^{2}] 

Constant depending on flow patterns [dimensionless] 

concentration of diffusing substances [kg/m^{3}] 

Concentration of solute 

Concentration of solute 

Concentration of solute 
C_{f}  Feed concentration [kg/m^{3}] 

Diffusivity of component 

Back diffusivity [m^{2}/s ] 
D^{∞}  Bulk diffusivity [m^{2}/s ] 

Diameter of the capillaries [m] 
Diameter of retained particles [m]  
Hydraulic diameter of the membrane channel [m]  

Activation energy of diffusion [kJ/kmol] 

Total thickness of the porous subdomain [m] 

Deposit thickness [m] 

Membrane thickness [m] 
Water permeate flux [m^{3}/m^{2}s]  

Permeate flux [m^{3}/m^{2}s] 

Permeate flux of pure water [m^{3}/m^{2}s] 

Resistance coefficient for membrane 

Resistance coefficient for cake layer 

Resistance coefficient for pore constriction 
Kozeny coefficient (typically 5) [dimensionless]  

Permeability constant [m^{2}] 

Mass transfer coefficient [m/s] 
Deposit permeability [m^{2}]  
Solute mass transfer coefficient [m/s]  
Back mass transfer coefficient [m/s]  
Filtration constant of complete blocking model [dimensionless]  
Filtration constant of intermediate blocking model [dimensionless]  
Filtration constant of standard blocking model [dimensionless]  
Filtration constant of cake model [dimensionless]  
Boltzman constant [dimensionless]  

Membrane thickness [m] 

Thickness of the porous layer [m] 

Thickness of the mass transfer boundary layer [m] 

Dynamic viscosity of the permeate [kg/(m·s) or Pa·s] 

Pressure drop [psi] 
( 
Pressure drop [psi] 
P  Permeability [m^{2}] 

Total discharge [m³/s] 
Initial permeate flow [m³/s]  
q  Constant depending on flow patterns [dimensionless] 
Reynolds number [dimensionless] 

r  Constant depending on flow patterns [dimensionless] 
Particle radius [m]  

Membrane resistance [m^{1}] 

Resistance of the concentration polarization layer [m^{1}] 

Hydraulic resistances, due to surface or pore adsorption [m^{1}] 

Irreversible resistance due to irreversible deposition [m^{1}] 
Overall recovery [%]  

Observed retention coefficient [dimensionless] 

Intrinsic retention coefficient [dimensionless] 

Resistance to flow 
R  Gas constant [J/mol. K] 
S  Specific area per unit volume [m^{2}/m^{3}] 
Schmidt number [dimensionless] 

Sherwood number [dimensionless] 

T  Absolute temperature [K] 
τ  Tortuosity of the capillaries [dimensionless] 
t  Filtration time [seconds] 

Permeate volume [m^{3}] 

Crossflow velocity [m/s] 

Space coordinate measured normal to the membrane [dimensionless] 

Osmotic pressure [atm] 
Δπ_{m}*  Osmotic pressure difference [atm] 
ε  Surface porosity [dimensionless] 
Thickness of cake layer [m] 
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