Solution diffusion model assumptions

The second assumption concerns the pressure and concentration gradients in the membrane. The solution-diffusion model assumes that when pressure is applied across a dense membrane, the pressure throughout the membrane is constant at the highest value. This assumes, in effect, that solution-diffusion membranes transmit pressure in the same way as liquids. Consequently, the solution-diffusion model assumes that the pressure within a membrane is uniform and that the chemical potential gradient across the membrane is expressed only as a concentration gradient [5,10]. The consequences of these two assumptions are illustrated in Figure 2.5, which shows pressure-driven permeation of a one-component solution through a membrane by the solution-diffusion mechanism. 

The general approach is to use the first assumption of the solution-diffusion model, namely, that the chemical potential of the feed and permeate fluids are... 

There are several models that describe the transport of mass through RO membranes. These are based on different assumptions and have varying degrees of complexity. The solution-diffusion model best describes the performance of "perfect," defect-free membranes and is considered the leading theory on membrane transport.2 Three other theories are presented here for completeness. 

The solution diffusion model is the most widely used model to describe permeation in dense membranes, as is the case for PV [3,32]. This model is based on the following assumptions ... 

Farmer (6) reviewed the various diffusion models for soil and developed solutions for several of these models. An appropriate model for field studies is a nonsteady state model that assumes that material is mixed into the soil to a depth L and then allowed to diffuse both to the surface and more deeply into the soil. Material diffusing to the surface is immediately removed by diffusion and convection in the air above the soil. The effect of this assumption is to make the concentration of a diffusing compound zero at the soil surface. With these boundary conditions the solution to Equation 8 can be converted to the useful form ... 

The second assumption has been effectively invalidated by the discovery of the hydrated electron. However, the effects of LET and solute concentration on molecular yields indicate that some kind of radical diffusion model is indeed required. Kuppermann (1967) and Schwarz (1969) have demonstrated that the hydrated electron can be included in such a model. Schwarz (1964) remarked that Magee s estimate of the distance traveled by the electron at thermalization (on the order of a few nanometers) was correct, but his conjecture about its fate was wrong. On the other hand, Platzman was correct about its fate—namely, solvation—but wrong about the distance traveled (tens of nanometers). 

From a plot of the internalisation flux against the metal concentration in the bulk solution, it is possible to obtain a value of the Michaelis-Menten constant, Am and a maximum value of the internalisation flux, /max (equation (35)). Under the assumption that kd kml for a nonlimiting diffusive flux, the apparent stability constant for the adsorption at sensitive sites, As, can be calculated from the inverse of the Michaelis-Menten constant (i.e. A 1 = As = kf /kd). The use of thermodynamic constants from flux measurements can be problematic due to both practical and theoretical (see Chapter 4) limitations, including a bias in the values due to nonequilibrium conditions, difficulties in separating bound from free solute or the use of incorrect model assumptions [187,188],... 

In another model, Harland and Peppas [159] considered the diffusion of solutes through semicrystalline hydrogel membranes. These types of membranes were assumed to consist of a crosslinked, swollen (amorphous) phase through which solute diffusion occurred and an impermeable, crystalline phase. A simplified form of the model assumes uniform amorphous regions. With this assumption, the diffusion coefficient through a semi-crystalline membrane, Dc, was written as... 

According to their analysis, if c is zero (practically much lower than 1), then the fluid-film diffusion controls the process rate, while if ( is infinite (practically much higher than 1), then the solid diffusion controls the process rate. Essentially, the mechanical parameter represents the ratio of the diffusion resistances (solid and fluid-film). This equation can be used irrespective of the constant pattern assumption and only if safe data exist for the solid diffusion and the fluid mass transfer coefficients. In multicomponent solutions, the use of models is extremely difficult as numerous data are required, one of them being the equilibrium isotherms, which is a time-consuming experimental work. The mathematical complexity and/or the need to know multiparameters from separate experiments in all the diffusion models makes them rather inconvenient for practical use (Juang et al, 2003). 

As mentioned above, the nonequilibrium radiation code NEQAIR is employed for prediction of ultraviolet emission from the DSMC flow field solutions. The modeling of ultraviolet emission with this code is discussed for nitric oxide in Ref. 84 and for atomic oxygen in Ref. 87. A common assumption made in using the NEQAIR code is that a quasisteady state (QSS) exists for the number densities of the electronically excited species. The assumption requires that the time scale of chemical processes is much smaller than the time scales for diffusion and for changes in overall properties. Under these conditions, the local values of temperatures and ground state species number densities obtained from the DSMC computation may be used to compute the populations of the electronically excited states. 

In the case of selective neutrality—this means that all variants have the same selective values—evolution can be modeled successfully by diffusion models. This approach is based on the analysis of partial differential equations that describe free diffusion in a continuous model of the sequence space. The results obtained thereby and their consequences for molecular evolution were recently reviewed by Kimura [2]. Differences in selective values were found to be prohibitive, at least until now, for an exact solution of the diffusion approach. Needless to say, no exact results are available for value landscapes as complicated as those discussed in Section IV.3. Approximations are available for special cases only. In particular, the assumption of rare mutations has to be made almost in every case, and this contradicts the strategy basic to the quasi-species model. 

In summary, models with a variety of assumptions have been developed to explain diffusion in polymer gels. A recent review of the models for solute diffusion in hydrogels [85] compares data from the literature with a variety of models, including models that attempt to account for reduced diffusion by reduction in free volume (see, e.g., [86]), by the increase in hydrodynamic drag experienced by the solute and created by polymer chains (see, e.g.. Equation 4-32), and by the presence of physical obstructions to solute diffusion (see, e.g.. 

Three-dimensional models offer more realism, at least apparently, but with the cost of greater complexity, a more limited number of simulations, and a higher probability of crucial regional errors in the base solutions, which may compromise direct, quantitative model-data comparisons. Ocean GCM solutions, however, should be exploited to address exactly those problems that are intractable for simpler conceptual and reduced dimensional models. For example, two key assumptions of the 1-D ad-vection-diffusion model presented in Figure 2 are that the upwelling occurs uniformly in the horizontal and vertical and that mid-depth horizontal advection is not significant. Ocean GCMs and tracer data, by contrast, show a rich three-dimensional circulation pattern in the deep Pacific. 

The distinguishing feature of the classical diffusion model of Springer et al. [39] (hereafter SZG) is the consideration of variable conductivity. SZG relied on their own experimental data to determine model parameters, such as water sorption isotherms and membrane conductivity as a function of the water content. Alternative approaches include the use of concentrated solution theory to describe transport in the membrane [45], and invoking simplifying assumptions such as thin membrane with uniform hydration [46]. 

Tinland and Borsali [123] used FRAP and QELS to measure Dp of 433 kDa dextran in aqueous solutions of 310 kDa polyvinylpyrrolidone (PVP) for 0 < c < 120g/L. M ,/Mn was 1.5 for the matrix polymer but ca. 1.9-1.95 for the probes. Dp from the two techniques do not agree. We analyzed Dp from FRAP measurements, because FRAP does not require the detailed model assumptions needed to relate the QELSS spectrum to diffusion coefficients. Tinland and Borsali s data agree well with stretched exponentials in c. 

The above-mentioned mechanistic models of separation were based on the assumption that the time necessary to transport the solute molecules between the phases is much shorter than the time spent by them at a given position inside the column. To overcome the difficulty to explain the differences in K ec measrued rmder static and dynamic conditions, a mechanism of restricted diffusion has been suggested. Unfortimately, the restricted diffusion model is incoherent with the independence of the K ec on the flow rate and temperatiue observed within a wide range of experimental conditions. 

Note that if Z) is a scalar Z>o, then d is real and equal to Dq. Generally, d is positive. This is because the uniform steady solution is stable below criticality by assumption. In Appendix B, the derivation of the Ginzburg-Landau equation (in particular, the explicit calculation of Ai, d, and g) is illustrated for a hypothetical reaction-diffusion model. 

The solution-diffusion theory models the performance of the perfect membrane. In reality, industrial membranes are plagued with imperfections that some argue must be considered when developing a complete theory that models performance. The basis of the Diffusion Imperfection Model is the assumption that slight imperfections in the membrane occur... 

To calculate and we use the experimental value of the diffusion coefficient of Zr impurity in Al, Dzr- = 728 x 10" exp (—2.51 eY/ksT) m. s [33,34]. The kinetic parameters can be deduced from this experimental data by using the five frequency model for solute diffusion in fee lattices [35], if we make the assumption that there is no correlation effect. We check afterwards that such an assumption is valid at T = 500 K the correlation factor is fzr- = 1 and at T = 1000 K fzr- = 0.875. Correlation effects are thus becoming more important at higher temperature but they can be neglected in the range of temperature used in the fitting procedure. 

Although transport processes in aqueous NF systems have been studied for several years and much knowledge has been gained, OSN systems are not yet well understood. While some studies support the use of pore-flow models, others suggest using a solution-diffusion approach. The basic equations of these models are outhned below along with the major simplifying assumptions. 

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