Boundary layer resistance model

Concentration polarisation phenomena lead to an increase Of the solute concentration at the membrane surface. If the solute molecules are completely retained by the membrane, at steady-state conditions the convective flow of the solute molecules towards the membrane surface will be equal to the diffusive flow back to the bulk of the feed. Hence, at 100% rejection the average velocity of the solute molecules in the boundary layer will be zero. Because of the increased concentration, the boundary layer exens a hydrodynamic resistance on the permeating solvent molecules. The solvent flux can then be represented by a resistance model in which both the boundary layer resistance (R, ) and the membrane resistance (Rm) appear (assuming that no gelation occurs ). A schematic drawing of this resistance model is given in figure VII - 18. 

Because both the above resistances operate in series, the solvent flux is given by eq.VII - 32  

This latter equation is the basic equation of the boundary layer resistance model [17-19]. The boundary layer can be considered as a concentrated solution through which solvent molecules permeate, with the permeability of this stagnant layer depending very much on the concentration and the molecular weight of the solute. The resistance exerted by thislayer is far much greater for macromolecular solutes (ultrafiltration) relative to for low molecular weight. solutes (reverse osmosis). Because there is a concentration profile in the boundary layer, the permeability P of the solvent may be written as a function of the distance coordinate x w ith the boundaries x = 0 and x = 6. 

The permeability or permeability coefficient appears in the phenomenological Darcy equation [20], and because the osmotic gradient is the driving force for solvent flow in the boundary layer, the volume flux can be written as [17]  

In order to estimate the boundary layer resistance R),. it is necessary to determine the permeability R This can be done by sedimentation measurements since a correlation exists between the permeation of a solvent through a (stagnant) polymer solution and the. sediihentation of polymer molecules (or molecules as small as sucrose) through a solvent. This is shown schematically in figure VII - 19. According to Mijnlieffet al. [21] the permeability is related to the sedimentation coefficient via 

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Figure VII - 18. Schematic representation of the boundary layer resistance model.

It should be noted that the boundary layer resistance model is equivalent to the osmotic pressure model [16] ... 

All of the many biological transfer processes combine to determine a net surface resistance to transfer. Empirical relationships can be used to infer stomatal resistance from data on photosynthetically active radiation, water stress, temperature, atmospheric humidity and carbon dioxide levels. The resulting net surface resistance has been coupled with mathematical descriptions of aerodynamic and boundary-layer resistances in a "big leaf" model derived on the basis of agricultural and forest meteorology literature (4). At present, the big-leaf model is relatively coarse, permitting application only to areas dominated by maize, soybeans, grass, deciduous trees, and conifers. 

Discrepancies between experimentally obtained and theoretically calculated data for cadmium concentration in the strip phase are 10-150 times at feed or strip flow rate variations. These differences between the experimental and simulated data have the following explanation. According to the model, mass transfer of cadmium from the feed through the carrier to the strip solutions is dependent on the diffusion resistances boundary layer resistances on the feed and strip sides, resistances of the free carrier and cadmium-carrier complex through the carrier solution boundary layers, including those in the pores of the membrane, and resistances due to interfacial reactions at the feed- and strip-side interfaces. In the model equations we took into consideration only mass-transfer relations, motivated by internal driving force (forward... 

These differences between the experimental and the simulated data have the following explanation. According to the model, mass transfer of titanium from the feed through the carrier to the strip solutions is dependent on the resistances boundary layer resistances on the feed and strip sides, resistances... 

Perhaps the most realistic model is the random pore model of Bhatia and Perlmutter (1980 1981a, b 1983), which assumes that the actual reaction surface of the reacting solid B is the result of the random overlapping of a set of cylindrical pores. Surface development as envisaged in this model is illustrated in Figure 11.12. The first step in model development is therefore the calculation of the actual reaction surface, based on which the conversion-time relationship is established in terms of the intrinsic structural properties of the solid. In the absence of intraparticle and boundary layer resistances, the following relationship is obtained ... 

Meyers, T. P. (1987) The sensitivity of modelled S02 fluxes and profiles to stomatal and boundary layer resistance, Water Air Soil Pollut. 35, 261-278. 

The Cake Filtration Model describes the filtration of particles which are much larger than the pores and will be retained, without entering the pores. The particles deposit on the membrane surface contributing to the boundary layer resistance. Included in this model is deposition due to concentration polarisation. 

In VMD, the boundary layer resistance in the permeate side and the contribution of the heat transported by conduction through the membrane are negligible (Lawson and Lloyd, 1996a Bandini et al., 1997 Lawson and Lloyd, 1997). This makes VMD of pure water useful to determine the temperature of the feed solution at the membrane surface (T ), and therefore the boundary layer heat transfer coefficients in the membrane module can be evaluated (Mengual et al., 2004). This helps in selecting the adequate empirical heat transfer correlation of a given MD system, which is a complex task when developing theoretical models to determine the temperature polarization coefficients. 

Permeability-pH profiles, log Pe - pH curves in arhficial membrane models (log Pjpp - pH in cehular models), generally have sigmoidal shape, similar to that of log Dod - pH cf. Fig. 3.1). However, one feature is unique to permeabihty profiles the upper horizontal part of the sigmoidal curves may be verhcally depressed, due to the drug transport resistance arising from the aqueous boundary layer (ABL) adjacent to the two sides of the membrane barrier. Hence, the true membrane contribution to transport may be obscured when water is the rate-limiting resistance to transport. This is especially true if sparingly soluble molecules are considered and if the solutions on either or both sides of the membrane barrier are poorly stirred (often a problem with 96-well microhter plate formats). 

The basic assumption for a mass transport limited model is that diffusion of water vapor thorugh air provides the major resistance to moisture sorption on hygroscopic materials. The boundary conditions for the mass transport limited sorption model are that at the surface of the condensed film the partial pressure of water is given by the vapor pressure above a saturated solution of the salt (Ps) and at the edge of the diffusion boundary layer the vapor pressure is experimentally fixed to be Pc. The problem involves setting up a mass balance and solving the differential equation according to the boundary conditions (see Fig. 10). 

The One-Dimensional Pseudo Homogeneous Model of Fixed Bed Reactors. The design of tubular fixed bed catalytic reactors has generally been based on a one-dimensional model that assumes that species concentrations and fluid temperature vary only in the axial direction. Heat transfer between the reacting fluid and the reactor walls is considered by presuming that all of the resistance is contained within a very thin boundary layer next to the wall and by using a heat transfer coefficient based on the temperature difference between the fluid and the wall. Per unit area of the tube... 

The second approach assigns thermal resistance to a gaseous boundary layer at the heat transfer surface. The enhancement of heat transfer found in fluidized beds is then attributed to the scouring action of solid particles on the gas film, decreasing the effective film thickness. The early works of Leva et al. (1949), Dow and Jacob (1951), and Levenspiel and Walton (1954) utilized this approach. Models following this approach generally attempt to correlate a heat transfer Nusselt number in terms of the fluid Prandtl number and a modified Reynolds number with either the particle diameter or the tube diameter as the characteristic length scale. Examples are ... 

It was postulated that the aqueous pores are available to all molecular species, both ionic and non-ionic, while the lipoidal pathway is accessible only to un-ionised species. In addition, Ho and co-workers introduced the concept of the aqueous boundary layer (ABL) [9, 10], The ABL is considered a stagnant water layer adjacent to the apical membrane surface that is created by incomplete mixing of luminal contents near the intestinal cell surface. The influence of drug structure on permeability in these domains will be different for example ABL permeability (Paq) is inversely related to solute size, whereas membrane permeability (Pm) is dependent on both size and charge. Using this model, the apparent permeability coefficient (Papp) through the biomembrane may therefore be expressed as a function of the resistance of the ABL and... 

Reactions carried in aqueous multiphase catalysis are accompanied by mass transport steps at the L/L- as well as at the G/L-interface followed by chemical reaction, presumably within the bulk of the catalyst phase. Therefore an evaluation of mass transport rates in relation to the reaction rate is an essential task in order to gain a realistic mathematic expression for the overall reaction rate. Since the volume hold-ups of the liquid phases are the same and water exhibits a higher surface tension, it is obvious that the organic and gas phases are dispersed in the aqueous phase. In terms of the film model there are laminar boundary layers on both sides of an interphase where transport of the substrates takes place due to concentration gradients by diffusion. The overall transport coefficient /cl can then be calculated based on the resistances on both sides of the interphase (Eq. 1) ... 

Bennett etal. have presented a model for gaseous pollution sorption by plants. The model includes all the known factors that might have a significant effect on pollution sorption by plant leaves, including gas concentration (ambient air and internal leaf), gas fluxes (external and internal), resistance to flow (leaf boundary layer, stomatal, and internal), nature of leaf surfaces (stomatal presence, cutin, and surface properties), importance of gas solubility and thus solute concentration within the leaf, and ability of the plant to metabolize pollutants (decontaminate itself). They mentioned the reactivity of ozone as another factor to consider. They believe that surface sorption may be important, at least over short periods. They presented a possible mathematical representation of these factors, which they suggested is equivalent to the mathematical statement of Ohm s law. This material is well int ated in the review by Bennett and Hill. ... 

Using this model and the assumption that concentration polarization occurs only on the feed side of the membrane, the flux across the combined resistances of the feed side boundary layer and the membrane can be written as... 

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... 

The formation of concentration gradients caused by the flow of ions through a single cationic membrane is shown in Figure 10.8. As in the treatment of concentration polarization in other membrane processes, the resistance of the aqueous solution is modeled as a thin boundary layer of unstirred solution separating the... 

In a well-fluidized gas-solid system, the bulk of the bed can be approximated to be isothermal and hence to have negligible thermal resistance. This approximation indicates that the thermal resistance limiting the rate of heat transfer between the bed and the heating surface lies within a narrow gas layer at the heating surface. The film model for the fluidized bed heat transfer assumes that the heat is transferred only by conduction through the thin gas film or gas boundary layer adjacent to the heating surface. The effect of particles is to erode the film and reduce its resistive effect, as shown by Fig. 12.3. The heat transfer coefficient in the film model can be expressed as... 

An important example of the system with an ideally permeable external interface is the diffusion of an electroactive species across the boundary layer in solution near the solid electrode surface, described within the framework of the Nernst diffusion layer model. Mathematically, an equivalent problem appears for the diffusion of a solute electroactive species to the electrode surface across a passive membrane layer. The non-stationary distribution of this species inside the layer corresponds to a finite - diffusion problem. Its solution for the film with an ideally permeable external boundary and with the concentration modulation at the electrode film contact in the course of the passage of an alternating current results in one of two expressions for finite-Warburg impedance for the contribution of the layer Ziayer = H(0) tanh(icard)1/2/(iwrd)1/2 containing the characteristic - diffusion time, Td = L2/D (L, layer thickness, D, - diffusion coefficient), and the low-frequency resistance of the layer, R(0) = dE/dl, this derivative corresponding to -> direct current conditions. 

Assuming that the reaction proceeds by a shrinking core model and that the boundary layer does not present an important rate resistance,... 

The effects of dry deposition are included as a flux boundary condition in the vertical diffusion equation. Dry deposition velocities are calculated from a big leaf multiple resistance model (Wesely 1989 Zhang et al. 2002) with aerodynamic, quasi-laminar layer, and surface resistances acting in series. The process assumes 15 land-use types and takes snow cover into account. 

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