Conditional diffusion models

In transported PDF methods (Pope 2000), the closure model for A, V, ip) will be a known function26 ofV. Thus, (U,Aj) will be closed and will depend on the moments of U and their spatial derivatives.27 Moreover, Reynolds-stress models derived from the PDF transport equation are guaranteed to be realizable (Pope 1994b), and the corresponding consistent scalar flux model can easily be found. We shall return to this subject after looking at typical conditional acceleration and conditional diffusion models. 

Though a porous medium may be described adequately under non-reactive conditions by a smooth field type of diffusion model, such as one of the Feng and Stewart models, it does not necessarily follow that this will still be the case when a chemical reaction is catalysed at the solid surface. In these circumstances the smooth field assumption may not lead to appropriate expressions for concentration gradients, particularly in the smaller pores. Though the reason for this is quite simple, it appears to have been largely overlooked,... 

The conditions for model experiments can be explained in the following way. The governing equations are made nondimensional in the full scale and in the reduced scale used in the model experiments. For example, the velocity in the room is divided by the diffuser velocity in the room, and the velocity in the model is divided by the supply velocity in the model in order to normalize all velocities. The two sets of equations are identical and they describe the same solution provided that requirements f, 2, and 3 mentioned at the beginning of this section have been met. 

It should be noted in conclusion that, by virtue of the impact condition (t,/ > tc) and adiabaticity of the m-diffusion model (cotc > 1), the latter relates to the limit alternative to the perturbation theory ... 

S Neervannan, LS Dias, MZ Southard, VJ Stella. A convective-diffusion model for dissolution of two non-interacting drug mixtures from co-compressed slabs under laminar hydrodynamic conditions. Pharm Res 11 1228-1295, 1994. 

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 ... 

One conclusion from these results is that the axial diffusion model begins to fail as Pe, - small, when an open boundary condition is used at the outlet. The case Pe, - small means increasing backmixing, or that the diffusive flux becomes increasingly significant compared with the convective flux. For an open boundary condition, it is also questionable whether the actual response C(e) can be identified with E(B). Furthermore, regardless of the boundary conditions chosen, it is difficult to envisage that cA... 

A small step rotational diffusion model has been used to describe molecular rotations (MR) of rigid molecules in the presence of a potential of mean torque.118 120,151 t0 calculate the orientation correlation functions, the rotational diffusion equation must be solved to give the conditional probability for the molecule in a certain orientation at time t given that it has a different orientation at t = 0, and the equilibrium probability for finding the molecule with a certain orientation. These orientation correlation functions were found as a sum of decaying exponentials.120 In the notation of Tarroni and Zannoni,123 the spectral denisities (m = 0, 1, 2) for a deuteron fixed on a reorienting symmetric top molecule are ... 

The conditional velocity also appears in the inhomogeneous transport equation for x. / ), and is usually closed by a simple gradient-diffusion model. Given the mixture-fraction PDF, (5.316) can be closed in this manner by first decomposing the velocity into its mean and fluctuating components ... 

In this manner, the non-relaxing property of the IEM model is avoided, and a(f, f) can be chosen such that the limiting mixture-fraction PDF is Gaussian. Indeed, from DNS it is known that the conditional diffusion for the mixture fraction has a non-linear form that varies with time (see Fig. 6.2). In the GIEM model, this behavior is modeled by... 

Using the gradient-diffusion model, (6.29), the term involving the conditional velocity fluctuations can be written as133... 

Some of the models for the conditional diffusion presented in Section 6.6 can be used directly to close the right-hand side of (6.173). For example, the IEM model in (6.84) yields the Lagrangian IEM (LIEM) model. With the LIEM, the drift and diffusion coefficients... 

The Diffusion Model. The uptake of a solute by a sorbent can be analyzed by a diffusion model, which has been used successfully to model adsorption rates onto activated carbon (74, 75), ion exchangers (72), heterogeneous catalysts (76), and soil columns (77). For the purpose of illustration, we can consider the diffusion of a compound into a spherical sorbent grain under conditions of linear sorption and no exterior mass transfer limitations (73), which is described by... 

Pignatello and Xing [107] used two models, the organic matter diffusion model (OMD) and the sorption-retarded pore diffusion model (SRPD), in order to understand better the meaning of slow sorption/desorption observations and mechanisms and to explore the most likely causes of such slow process in natural solid particles. These authors reported that both OMD and SRPD mechanisms operate in the environment, often probably together in the same particle. OMD may predominate in soils that are high in natural OM and low in aggregation, while SRPD may predominate in soils where the opposite conditions exist. 

The simplest practicable approach considers the membrane as a continuous, nonporous phase in which water of hydration is dissolved.In such a scenario, which is based on concentrated solution theory, the sole thermodynamic variable for specifying the local state of the membrane is the water activity the relevant mechanism of water back-transport is diffusion in an activity gradient. However, pure diffusion models provide an incomplete description of the membrane response to changing external operation conditions, as explained in Section 6.6.2. They cannot predict the net water flux across a saturated membrane that results from applying a difference in total gas pressures between cathodic and anodic gas compartments. 

A solution of the diffusion equation for an electrode reaction for repetitive stepwise changes in potential can be obtained by numerical integration [44]. For a stationary planar diffusion model of a simple, fast, and reversible electrode reaction (1.1), the following differential equations and boundary conditions can be formulated ... 

Equation (9.15) was written for macro-pore diffusion. Recognize that the diffusion of sorbates in the zeoHte crystals has a similar or even identical form. The substitution of an appropriate diffusion model can be made at either the macropore, the micro-pore or at both scales. The analytical solutions that can be derived can become so complex that they yield Httle understanding of the underlying phenomena. In a seminal work that sought to bridge the gap between tractabUity and clarity, the work of Haynes and Sarma [10] stands out They took the approach of formulating the equations of continuity for the column, the macro-pores of the sorbent and the specific sorption sites in the sorbent. Each formulation was a pde with its appropriate initial and boundary conditions. They used the method of moments to derive the contributions of the three distinct mass transfer mechanisms to the overall mass transfer coefficient. The method of moments employs the solutions to all relevant pde s by use of a Laplace transform. While the solutions in Laplace domain are actually easy to obtain, those same solutions cannot be readily inverted to obtain a complete description of the system. The moments of the solutions in the Laplace domain can however be derived with relative ease. 

Because the Adler model is time dependent, it allows prediction of the impedance as well as the corresponding gaseous and solid-state concentration profiles within the electrode as a function of time. Under zero-bias conditions, the model predicts that the measured impedance can be expressed as a sum of electrolyte resistance (Aeiectroiyte), electrochemical kinetic impedances at the current collector and electrolyte interfaces (Zinterfaces), and a chemical impedance (Zchem) which is a convolution of contributions from chemical processes including oxygen absorption. solid-state diffusion, and gas-phase diffusion inside and outside the electrode. 

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