Resistance-in-series model

Using resistance-in-series model (Bosanquet formula), the diffusivity in pore... 

Since the overall concentration drop (cib — cip) is the sum of the concentration drops across the boundary layer and the membrane, a simple restatement of the resistances-in-series model using the terms of Equations (4.1-4.3) is... 

For relatively porous nanofiltration membranes, simple pore flow models based on convective flow will be adapted to incorporate the influence of the parameters mentioned above. The Hagen-Poiseuille model and the Jonsson and Boesen model, which are commonly used for aqueous systems permeating through porous media, such as microfiltration and ultrafiltration membranes, take no interaction parameters into account, and the viscosity as the only solvent parameter. It is expected that these equations will be insufficient to describe the performance of solvent resistant nanofiltration membranes. Machado et al. [62] developed a resistance-in-series model based on convective transport of the solvent for the permeation of pure solvents and solvent mixtures ... 

Generally, applying the resistance-in-series model, the following expression can be given [45] ... 

Here, Kq is an overall mass transfer coefficient. The equation for Kq is analogous to the resistance in series model for electrical circuits. PI is the value of the solute pressure that... 

Under the assumption of steady state aed a linear equilibrium expression y., = mxA i, the well-kenwn resistance in series model cen he derived. 

Table 3.1 illustrates that the separation between the different processes is not precise, as the processes overlap. Therefore, filtration and separation models are generally applicable to mote than one process. Often several phenomena are operative simultaneously and which one dominates depends on the membrane and the solute or particle in question. Concepts such as the resistance-in-series model, the osmotic pressure model or concentration polarisation are principles which are applicable to any membrane operation. These wiU be described in the MF section. 

The Resistance in Series Model describes the flux of a fouled membrane. This is given in equation (3.4). The resistances Ra>, Ri> and Rc denote the additional resistances which result from the exposure of the membrane to a solution containing particles or solute. Rcp is the resistance due to concentration polarisation, Ri> the internal pore fouling resistance, and Rc the resistance due to external deposition or cake formation. These resistances are usually negligible in RO, where the osmotic pressure effects become more important (Fane (1997)). However, the osmotic pressure can also be incorporated into Rcp. 

The osmotic pressure difference can usually be neglected in MF and UF, since the rejected solutes are large and their osmotic pressure small. However, even polymeric solutes can develop a significant osmotic pressure at boundary layer concentrations (Ho and Sirkar (1992)). This naturally implies that the resistance in series model (equation (3.4)) would be more appropriate in MF, while the osmotic pressure model (equation (3.6)) may be more useful in NF and RO. Both models have been applied to UF. 

The effect of the KCl concentradon on the cake resistance after filtradon of 2.8 L feed suspension is presented in Figure 6.21. The calculation of the resistance is based on the resistance in series model presented in equation (6.4) which is further described in Chapter 3. 

The sum involved in equation (7-96) reminds us that we are dealing with a resistance-in-series model, so that we can identify various relationships for corresponding to different controlling resistances. 

Various parameters in Equation 31.17 have been defined earlier. Danesi et al. [92] described a simple correlation between permeability coefficient in FSSLM and HFSLM configuration. At very large values of ( ) (as compared to 1), Equation 31.17 is transformed into the one used for FSSLM by Danesi et al. [92]. Hence, the smaller the value of ( ), the higher will be the negative value of the left-hand side of Equation 31.17, which suggests the higher rate of mass transfer. Later on, D Elia et al. [93] considered the resistance in series model where they have studied the mass transport across hollow-fiber contactors in NDSX mode. They showed that the overall mass transfer resistance is equal to the sum of individual mass transfer resistances across the aqueous boundary layer and membrane phase. Mathematically, it can be written as follows ... 

Finally, Machado et al. [21] developed a resistances-in-series model and proposed that solvent transport through the MPF membrane consists of three main steps (1) transfer of the solvent into the top active layer, which is characterized by surface resistance (2) viscous flow through NF pores and (3) viscous flow through support layer pores, all expressed by viscous resistances, i.e. 

Resistance model for transport in composite hollow fibre membranes based on polysul-fone with siloxane coating has been described in a classical work by Henis and Tripody [51], The resistance in series model assumes that the gas molecules encounter constrictions at certain positions throughout the pore which control the rate of diffusion [20,27,33]. For this scenario the total permeability is inversely related to the total resistance, thus... 

Evaporation is generally described by a resistance in series model, commonly used to describe mass-transfer between phases. The overall transfer coefficient depends on the rate... 

The phenomenon of fouling is very complex and difficult to describe theoretically. Even for a given solution, fouling will depend on physical and chemical parameters such as concentration, temperature, pH, ionic strength and specific interactions (hydrogen bonding, dipole-dipole interactions). However, reliable values of flux decline are necessary for process design. The flux may also be described by a resistances-in-series model, in which a resistance of a cake layer is in series with the membrane resistance. The flux can be described by... 

The MFI is based upon cake filtration theory that particles are retained on the membrane surface during filtration. According to the resistance in series model, the reduction in flux due to the presence of cake layer and the additional resistance from the membrane under constant operating filtration can be described as ... 

Tansel, B., Bao, W. Y, and Tansel, 1. N. (2000). Characterization of fouling kinetics in ultrafiltration systems by resistances in series model. Desalination 129, 7-14. 

Mass Ti ansfer The rate of mass transfer in a liquid-liquid extraction system implemented in a nondispersive membrane contactor is analyzed in the manner followed in conventional contactor analysis. The overall organic phase-based mass transfer coefficient Kio of a species i being extracted (or back extracted) from an aqueous solution into a solvent wetting the pores of a hydrophobic membrane is described via the resistances-in-series model ... 

The resistance offered by the membrane with liquid-fiUed pores wiU be different (generally higher) from the gas-tilled pores, due to the different effective diffusion coefficients. There are two absorption processes operating it is either by physical or chemical absorption. To predict the overall mass transfer coefficient the membrane, liqnid mass transfer coefficient, and Henry s law constant must be known. The mass transfer coefficient is a function of the systan geometry, fluid properties, and flow velocity [1]. It can be expressed by a resistance in series model [9] ... 

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