Diffusion Simplified solution

The diffusion layer theory, illustrated in Fig. 15B, is the most useful and best-known model for transport-controlled dissolution. The dissolution rate here is controlled by the rate of diffusion of solute molecules across a diffusion layer of thickness h, so that kT kR in Eq. (40), which simplifies to kx = kT. With increasing distance, x, from the surface of the solid, the concentration, c, decreases from cs at x = 0 to cb at x = h. In general, c is a nonlinear function of x, and the concentration gradient dddx becomes less steep as x increases. The hyrodynamics of the dissolution process has been fully discussed by Levich [104]. In a stirred solution, the flow velocity of the liquid dissolution medium increases from zero at x = 0 to the bulk value at x = h. 

Consider two simple cases of extraction processes in which kinetics are controlled by interfacial film diffusion (the solutions are always considered stirred). The two cases are treated with the simplifying assumptions introduced in section 2 (i.e., steady-state and linear concentration gradients throughout the diffusional films). 

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... 

The use of isotopic models in the literature—practical limits of usage As mentioned above, simplified solutions are employed in ion exchange for the estimation of diffusion coefficients. For example, the equations of Vermeulen and Patterson, derived from isotopic exchange systems, have been successfully used, even in processes that are not isotopic. Inglezakis and Grigoropoulou (2001) conducted an extended review of the literature on the use of isotopic models for ion-exchange systems. 

This problem is a good example of the importance of formulating a complex diffusion problem in terms of the equations of change. Hence the simplified treatment given here is discussed in terms of the simplified solutions to the three basic equations. 

The Debye-Huckel theory was developed to extend the capacitor model and is based on a simplified solution of the Poisson equation. It assumes that the double layer is really a diffuse cloud in which the potential is not a discontinuous function. Again, the interest is in deriving an expression for the electrical potential function. This model states that there is an exponential relationship between the charge and the potential. The distribution of the potential is ... 

While numerical methods come into question for solutions involving variable D, D can be assumed to be constant or practically constant for most cases of practical interest. In addition, simplified solutions for diffusion along the x-axis can be used instead of the general solution, except for some particular cases which will be pointed out later. This greatly simplifies presentation of the problem and the resulting equation for diffusion is ... 

A useful literature relating to polypeptide and protein adsorption kinetics and equilibrium behavior in finite bath systems for both affinity and ion-ex-change HPLC sorbents is now available160,169,171-174,228,234 319 323 402"405 and various mathematical models have been developed, incorporating data on the adsorption behavior of proteins in a finite bath.8,160 167-169 171-174 400 403-405 406 One such model, the so-called combined-batch adsorption model (BAMcomb), initially developed for nonporous particles, takes into account the dynamic adsorption behavior of polypeptides and proteins in a finite bath. Due to the absence of pore diffusion, analytical solutions for nonporous HPLC sorbents can be readily developed using this model and its two simplified cases, and the effects of both surface interaction and film mass transfer can be independently addressed. Based on this knowledge, extension of the BAMcomb approach to porous sorbents in bath systems, and subsequently to packed-, expanded-, and fluidized-bed systems, can then be achieved. 

Helfferich [2,3,30] states that in addition to the mutual interference of substances i and j, characterized by the phenomenological cross coefficients of the type L,j, one should take into account the presence of a coion in the ion exchanger as well. As a result, the simplified solution is inappropriate, even to the problem of ordinary IE. By use of only one diffusion mass-transfer equation, as in this case, account for the presence of co-ion has been neglected. It is, as a consequence, necessary to consider the Nemst-Planck relation for the co-ion also. 

During ultrasonic irradiation of aqueous solutions, OH radicals are produced from dissociation of water vapor upon collapse of cavitation bubbles. A fraction of these radicals that are initially formed in the gas phase diffuse into solution. Cavitation is a dynamic phenomenon, and the number and location of bursting bubbles at any time cannot be predicted a priori. Nevertheless, the time scale for bubble collapse and rebound is orders of magnitude smaller than the time scale for the macroscopic effects of sonication on chemicals (2) (i.e., nanoseconds to microseconds versus minutes to hours). Therefore, a simplified approach for modeling the liquid-phase chemistry resulting from sonication of a well-mixed solution is to view the OH input into the aqueous phase as continuous and uniform. The implicit assumption in this approach is that the kinetics of the aqueous-phase chemistry are not controlled by diffusion limitations of the substrates reacting with OH. 

The gas flow may be caused by different processes, the major one of which is diffusion. The solution may be simplified by taking into account the results from ref. [41 ] where the expression for the density of the gas flow into a cavitating bubble is given. 

Reverse osmosis is simply the application of pressure on a solution in excess of the osmotic pressure to create a driving force that reverses the direction of osmotic transfer of the solvent, usually water. The transport behavior can be analyzed elegantly by using general theories of irreversible thermodynamics however, a simplified solution-diffusion model accounts quite well for the actual details and mechanism in most reverse osmosis systems. Most successful membranes for this purpose sorb approximately 5 to 15% water at equilibrium. A thermodynamic analysis shows that the application of a pressure difference, Ap, to the water on the two sides of the membrane induces a differential concentration of water within the membrane at its two faces in accordance with the following (31) ... 

Since the solubility of various gases in ILs varies widely, they may be uniquely suited for use as solvents for gas separations [97]. Since they are non-volatile, they cannot evaporate to cause contamination of the gas stream. This is important when selective solvents are used in conventional absorbers, or when they are used in supported liquid membranes. For conventional absorbers, the ability to separate one gas from another depends entirely on the relative solubilities (ratio of Henry s law constants) of the gases. In addition, ILs are particularly promising for supported liquid membranes because they have the potential to be incredibly stable. Supported liquid membranes that incorporate conventional liquids eventually deteriorate because the liquid slowly evaporates. Moreover, this finite evaporation rate limits how thin one can make the membrane. This means that the net flux through the membrane is decreased. These problems could be eliminated with a non-volatile liquid. In the absence of facilitated transport (e.g., complexation of CO2 with amines to form carbamates), the permeability of gases through supported liquid membranes depends on both their solubility and diffusivity. The flux of one gas relative to the other can be estimated using a simplified solution-diffusion model ... 

The general form of the revised embrittlement correlation equation, widely known as the EONY model, is similar to the previous models. Instead of simplifying the existing models, some new key models were introduced into the revision of the models. The use of effective fluence, which had already been suggested by Odette and others, is one such modification. The idea is based on the experimental evidence on enhanced embrittlement, particularly in Cu-containing materials irradiated at low fluxes. This is attributed to the enhanced diffusion of solute atoms, and is modeled by the effective fluence, te, as ... 

Numerous researchers have developed their own simplified solutions to the radiation transfer equation. The first solution were Schuster s equations (3), in which, for simplification, the radiation field was divided into two opposing radiation fluxes (+z and -z directions). The radiation flux in the +z direction, perpendicular to the plane, is represented by /, and the radiation flux in the -z direction, resulting from scattering, is represented by J. The same approximation was used by Kubelka and Munk in their equations, in the exponential (4) as well as in the hyperbolic solution (5). In the exponential solution by Kubelka-Munk, a flat layer of thickness z, which scatters and absorbs radiation, is irradiated in the -z direction with monochromatic diffuse radiation of flux I. In an infinitesimal layer of thickness dz, the radiation fluxes are going in the + direction J and in the -direction I. The average absorption in layer on path length dz is named K the scattering coefficient is S. Two fundamental equations follow directly ... 

The simplified solution of the general problem of simultaneous reaction and diffusion presented in section 5.4.1 was based on the assumption that reactant B is present in a large excess throughout the reaction phase. However, in the case of... 

Calculations of cases a and b in figure 5.12 can only be done by numerical methods, described for the first time in the pioneering contribution of Van Krevelen and Hoftijzer (1948). There is, however, a simplified solution for case c, where the reaction can be considered to be instantaneous. The chemical reaction takes place in a very thin zone (at some distance from the interface) where the concentrations of A and B are very small. We can simplify this by assuming the reaction to take place at zero concentrations in a plane at x = X. hi that case the reaction rate is entirely determined by diffusion. 

For describing the combined effect of diffusion and reaction in a porous catalyst particle, we can now use the calculations of section 5.4.1. We shall here again consider the simplified solution that applies for a steady state, with a large excess of reactant B, For this situation, eqs. (5.38a) and (5.38b) are applicable, when is replaced by and 0 by 0... 

Using this simplified model, CP simulations can be performed easily as a function of solution and such operating variables as pressure, temperature, and flow rate, usiag software packages such as Mathcad. Solution of the CP equation (eq. 8) along with the solution—diffusion transport equations (eqs. 5 and 6) allow the prediction of CP, rejection, and permeate flux as a function of the Reynolds number, Ke. To faciUtate these calculations, the foUowiag data and correlations can be used (/) for mass-transfer correlation, the Sherwood number, Sb, is defined as Sh = 0.04 S c , where Sc is the Schmidt... 

Various mathematical concepts and techniques have been used to derive the functions that describe the different types of dispersion and to simplify further development of the rate theory two of these procedures will be discussed in some detail. The two processes are, firstly, the Random Walk Concept [1] which was introduced to the rate theory by Giddings [2] and, secondly, the mathematics of diffusion which is both critical in the study of dispersion due to longitudinal diffusion and that due to solute mass transfer between the two phases. The random walk model allows the relatively simple derivation of the variance contributions from two of the dispersion processes that occur in the column and, so, this model will be the first to be discussed. 

Miyauchi and Vermeulen (M7, M8) have presented a mathematical analysis of the effect upon equipment performance of axial mixing in two-phase continuous flow operations, such as absorption and extraction. Their solutions are based, in one case, upon a simplified diffusion model that assumes a mean axial dispersion coefficient and a mean flow velocity for... 

The overall set of partial differential equations that can be considered as a mathematical characterization of the processing system of gas-liquid dispersions should include such environmental parameters as composition, temperature, and velocity, in addition to the equations of bubble-size and residence-time distributions that describe the dependence of bubble nucleation and growth on the bubble environmental factors. A simultaneous solution of this set of differential equations with the appropriate initial and boundary conditions is needed to evaluate the behavior of the system. Subject to the Curie principle, this set of equations should include the possibilities of coupling effects among the various fluxes involved. In dispersions, the possibilities of couplings between fluxes that differ from each other by an odd tensorial rank exist. (An example is the coupling effect between diffusion of surfactants and the hydrodynamics of bubble velocity as treated in Section III.) As yet no analytical solution of the complete set of equations has been found because of the mathematical difficulties involved. To simplify matters, the pertinent transfer equation is usually solved independently, with some simplifying assumptions. 

The simplified equation (for the general equations, see Section IV, L) in the case of unsteady-state diffusion with a simultaneous chemical reaction in isothermal, incompressible dilute binary solutions with constant p and D and with coupled phenomena neglected is... 

Consider the bimolecular reaction of A and B. The concentration of B is depleted near the still-unreacted A by virtue of the very rapid reaction. This creates a concentration gradient. We shall assume that the reaction occurs at a critical distance tab- At distances r tab. [B] = 0. Beyond this distance, at r > rAB, [B] = [B]°, the bulk concentration of B at r = °°. We shall examine a simplified, two-dimensional derivation the solution in three dimensions must incorporate the mutual diffusion of A and B, requiring vector calculus, and is not presented here. 

Solutions for a number of typical cases are reported below. To simplify our task we use the assumption that reactant migration is not observed (a large excess of foreign electrolyte), that the diffusion coefficients Dj do not depend on concentration, and that for the reactant v = 1. (The subscript j is dropped in what follows.)... 

---