Simplified diffusion equations

Equations (2), (4), and (5) can be combined with the deviatoric elasticity equation and the equilibrium equations to form a set of field equations consisting of a Navier-type equation and two coupled diffusion equations. For the class of problems characterized by an irrotational displacement field with chemical and hydraulic loadings only, the two coupled diffusion equations simplify to... 

J. Kostrowicki and H.A. Scheraga, Application of the diffusion equation method for global optimization to oligopeptides, J. Phys. Chem. 96 (1992), 7442-7449. M. Levitt, A simplified representation of protein confomations for rapid simulation of protein folding, J. Mol. Biol. 104 (1976), 59-107. 

In addition to the fact that MPC dynamics is both simple and efficient to simulate, one of its main advantages is that the transport properties that characterize the behavior of the macroscopic laws may be computed. Furthermore, the macroscopic evolution equations can be derived from the full phase space Markov chain formulation. Such derivations have been carried out to obtain the full set of hydrodynamic equations for a one-component fluid [15, 18] and the reaction-diffusion equation for a reacting mixture [17]. In order to simplify the presentation and yet illustrate the methods that are used to carry out such derivations, we restrict our considerations to the simpler case of the derivation of the diffusion equation for a test particle in the fluid. The methods used to derive this equation and obtain the autocorrelation function expression for the diffusion coefficient are easily generalized to the full set of hydrodynamic equations. 

In the general case, the initial concentration of the oxidized component equals Cqx and that of the reduced component cRed. If the appropriate differential equations are used for transport of the two electroactive forms (see Eqs 2.5.3 and 2.7.16) with the corresponding diffusion coefficients, then the relationship between the concentrations of the oxidized and reduced forms at the surface of the electrode (for linear diffusion and simplified convective diffusion to a growing sphere) is given in the form... 

Under these simplifying assumptions, the system of diffusion equations becomes... 

In order to make the transport model adaptable to measurement results some simplifications are used. Vertical and lateral components of wind are neglible, the mean transport velocity U in x-direction is steady the pollutant transfer by advection in the drift direction is greater than by turbulent diffusion at the ground total reflection is assumed. For the case that the concentration at any point in space is independent of t and that the diffusivities are independent of x, y and z the simplified diffusion equation of the K-therory /8/ becomes... 

As in the case of burning gaseous fuel jets, the diffusion equations are combined readily by assuming Dp = (Alcp), that is, Le = 1. The same procedure can be followed in combining the boundary conditions for the three droplet burning equations to determine the appropriate b variables to simplify the solution for the mass consumption rate. 

The above equation can be used to estimate the half-time to reach equilibrium. Because the coefficients of 0.953872 and of 1.099056 are not much different from 1, for many applications and colloquial referencing, the mid-diffusion distance is often simplified as and the mid-diffusion time simplified... 

For three-dimensional diffusion in an anisotropic medium, theoretically it is possible to transform the diffusion equation to a form similar to that in an isotropic system. However, in practice, the transformed equation is rarely used, and diffusion is often simplified to be along the fastest diffusion direction. 

Because OH diffusion can be ignored compared to molecular H2O diffusion, Equation 3-78 may be simplified to... 

The general treatment for multicomponent diffusion results in linear systems of diffusion equations. A linear transformation of the concentrations produces a simplified system of uncoupled linear diffusion equations for which general solutions can be obtained by methods presented in Chapter 5. 

Generally, a set of coupled diffusion equations arises for multiple-component diffusion when N > 3. The least complicated case is for ternary (N = 3) systems that have two independent concentrations (or fluxes) and a 2 x 2 matrix of interdiffusivities. A matrix and vector notation simplifies the general case. Below, the equations are developed for the ternary case along with a parallel development using compact notation for the more extended general case. Many characteristic features of general multicomponent diffusion can be illustrated through specific solutions of the ternary case. 

The moisture content distribution is determined from the diffusion equation. Its very simplified form for the cylindical sample reads... 

The above analytical solution was expanded to three dimensions. In such a way, the reactor geometry or the channel can be designed. An appropriate simplified model, given in [38], can be derived from the diffusion equation. Appropriate boundary conditions at the channel walls account for the heterogeneous wall reaction. The concentration of a species A which reacts at the channel wall irreversibly to a species B was given as a function of the lateral channel dimensions y and z and the axial channel dimension xv For an inert gas and for y and z equal to zero (coordinate center indicated in Figure 3.94), Eq. (3.13) reduces to the solution of a non-reactive fluid given above ... 

The collective diffusion coefficient is thus relevant for the mass-transport at surfaces in systems, which are not in thermodynamic equilibrium. It generally depends on coverage. The above diffusion equation is widely employed to determine D, since the adsorbate concentration is a measurable quantity. In practice, frequently the decay of an adjusted coverage gradient is analyzed and diffusion equation is solved numerically or analytically for a given geometry. This task is considerably simplified when diffusion coefficients independent of coverage exist or may be assumed and ... 

In this equation A is the molecular diffusion coefficient expressed as m2/sec, and is the number of liquid moles of component i. This equation can be simplified using approximate analytical solutions for the transient diffusion equation in the vertical direction to ... 

Furthermore, asB in the previously developed theory (7), because the thermal diffusivity is so much greater than the mass diffusivities Equation (2) simplifies to... 

Despite the large number of analytical solutions available for the diffusion equation, their usefulness is restricted to simple geometries and constant diffusion coefficients. However, there are many cases of practical interest where the simplifying assumptions introduced when deriving analytical solutions are unacceptable. Such a case, for example, is the diffusion in polymer systems characterized by concentration-dependent diffusion coefficients.This chapter gives an overview of the most powerful numerical methods used at present for solutions of the diffusion equation. Indeed the application of these methods in practice needs the use of adequate computer programs (software). 

In Section 5.9, we show how to solve the convective-diffusion equation for the rotating disc electrode in order to calculate the diffusion-limited current. When the forced convection is constant, then dc/dt = 0, which simplifies the mathematical solution. 

Hydrodynamic electrodes1 are electrodes which function in a regime of forced convection. The advantage of these electrodes is increased transport of electroactive species to the electrode, leading to higher currents and thence a greater sensitivity and reproducibility. Most of the applications of these electrodes are in steady-state conditions, i.e. constant forced convection and constant applied potential or current. In this case dc/dt = 0, which simplifies the solution of the convective diffusion equation (Section 5.6)... 

If the two species diffusion coefficients are assumed equal (d = 1), the above equations simplify in an obvious way. In fact, then the problem is mathematically equivalent to the simple Cottrell case. Cottrell pointed out [181] that then, initially the concentrations at the electrode of the two species will instantly change to their Nernstian values and remain there after that. 

Under the simplifying assumptions of the Krogh-Erlang model, the steady state oxygen distribution in the tissue at any position z is governed by the steady state diffusion equation in radial coordinates... 

Equation (12-31) assumes that D< is constant however, Dt is rarely constant but varies with moisture content, temperature, and humidity. For long drying times, Eq. (12-31) simplifies to a limiting form of the diffusion equation as... 

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