Equilibrium Flux Models

The form of Darcy s law which is usually applied in membrane filtration is  

Mcrofihration usually provides complete retmtion of the particle phase on or in the membrane, but for ultrafiltration this may not be the case and the retention (i ) on the membrane is defined as the fi action of retained edes by the membrane  

Cake Filtration model. This model treats both the membrane resistance and deposit resistances in a very similar way to that described in Chapter 2. Equation (10.10) is the starting point for fiirther development which is concerned initially with the in ease in cake resistance to some equilibnum value. This is analogous to accounting for the increasing cake d th in convoitional dead-end filtration, see Section 2.5. In membrane filtration a mass balance over the deposit layer is performed  

The deposit concentration on the membrane N is equal to the product of the true solid density and the solid volume firaction  

At the veiy start of the filtration the membrane resistance alone opposes permeate flow and Equation (10.10) can be written as  

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It is obvious from the conditions defined above that the rate-based model equations and variables are more numerous and complex than those in the equilibrium stage model described in Chapter 13. Other features of the rate-based model are that the exiting liquid and vapor from a stage can be at different temperatures since separate balance equations are written for each phase. Each phase on a stage can have a different externally transferred heat duty. The exiting phases in general are not at equilibrium the liquid may be subcooled and the vapor may be superheated. In a rate-based model the phase interface must be defined. The variables defining the interface include the liquid and vapor compositions and the temperature at the interface, and the molar flux across the interface. 

The tube models a concrete folding mechanism by following the center of the tube and the fluctuations near that center. This concrete mechanism can be examined with respect to the overall funnel picture. For example, if the rate in the tube is exceptionally slow compared to the experimental rate, it is unlikely that the sampled tube is important. Computationally and conceptually, we should partition the funnel to tubes and analyze them one by one. We note that all the tubes meet at the folded state. Therefore the weight of each tube can be measured by overlapping the equilibrium flux into the reactant with the flux from a tube. The calculation of weights of tubes is a topic for future research. 

As with the equilibrium solvation models introduced earlier, it is also possible to incorporate quantum mechanical effects into the non-equilibrium transport model. Our motivation is to account for non-equilibrium ion fluxes and induced response in the electronic structure of the solute or membrane protein. To this end, we combine our DG-based DFT model with our DG-based PNP model as illustrated in Fig. 12.4 to develop a free energy functional and derive the associated governing equations. 

These models consider either the thermodynamic or mechanical non-equilibrium between the phases. The number of conservation equations in this case are either four or five. One of the most popular models which considers the mechanical non-equilibrium is the drift flux model. If thermal non-equilibrium between the phases is considered, constitutive laws for interfacial area and evaporation/condensation at the interface must be included. In this case, the number of conservation equations is five, and if thermodynamic equilibrium is assumed the number of equations can be four. Well-assessed models for drift velocity and distribution parameter depending on the flow regimes are required for this model in addition to the heat transfer and pressure drop relationships. The main advantage of the drift flux model is that it simplifies the numerical computation of the momentum equation in comparison to the multi-fluid models. Computer codes based on the four or five equation models are still used for safety and accident analyses in many countries. These models are also found to be useful in the analysis of the stability behaviour of BWRs belonging to both forced and natural circulation type. 

FIG. 26-69 Ratio of mass flux for inclined pipe flow to that for orifice discharge for flashing liquids by the homogeneous equilibrium model. Leung, J. of Loss Prev. Process Ind. 3 pp. 27-32, with kind peimission of Elsevier Science, Ltd, The Boulevard, Langford Lane, Kidlington, 0X5 IGB U.K., 1990.)... 

Jaquet and Miller [1985] have studied the transfer of hydrogen atom between neighbouring equilibrium positions on the (100) face of W by using a model two-dimensional chemosorption PES [McGreery and Wolken 1975]. In that calculation, performed for fairly high temperatures (T> rj the flux-flux formalism along with the vibrationally adiabatic approximation (section 3.6) were used. It has been noted that the increase of the coupling to the lattice vibrations and decrease of the frequency of the latter increase the transition probability. 

Bowers and Mudawar (1994a) performed an experimental smdy of boiling flow within mini-channel (2.54 mm) and micro-channel d = 510 pm) heat sink and demonstrated that high values of heat flux can be achieved. Bowers and Mudawar (1994b) also modeled the pressure drop in the micro-channels and minichannels, using the Collier (1981) and Wallis (1969) homogenous equilibrium model, which assumes the liquid and vapor phases form a homogenous mixture with equal and uniform velocity, and properties were assumed to be uniform within each phase. 

Solute Flux Solute partitioning between the upstream polarization layer and the solvent-filled membrane pores can be modeled by considering a spherical solute and a cylindrical pore. The equilibrium partition coefficient 0 (pore/bulk concentration ratio) for steric exclusion (no long-range ionic or other interactions) can be written as... 

Assuming that the target interface can be modeled as a quiescent, sharp boundary, with Eq. (30) initially at equilibrium there is zero net flux of species Red across the interface and each phase has a uniform composition of Red, CRed, (where the integer i = 1 or 2). The initial condition is identical to Eqs. (11) and (12). 

This section describes the continuous flux melting model used in Bourdon et al. (2003) and has many similarities with the model of Thomas et al. (2002). A significant difference is that the model described here keeps track of the composition of the slab as it dehydrates. This model is based on mass balance equations for both the mantle wedge and the slab. We assume secular equilibrium in the U-series decay chain initially ... 

Torii, H., and M. Tasumi. 1993. Infrared Intensities of Vibrational Modes of an a-helical Polypeptide Calculations Based on the Equilibrium Charge/Charge Flux (ECCF) Model. J. Mol. Struct. 300,171-179. 

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