Diffusion equation general formulation

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. 

The original theory of Brownian motion by Einstein was based on the diffusion equation and was valid for long times. Later, a more general formulism including short times also, has been developed. Instead of the diffusion equation, the telegrapher s equation enters. Again, an indeterminacy relation results, which, for short times, gives determinacy as a limit. Physically, this simply means that a Brownian particle s... 

We propose the balance principles for an immiscible mixture of continua with microstructure in presence of phenomena of chemical reactions, adsorption and diffusion by generalizing previous multiphase mixture [9] and use a new formulation for the balance of rotational momentum. New terms are also included in the energy equations corresponding to work done by respective terms in the micromomentum balances. 

Rieckmann and Keil (1997) introduced a model of a 3D network of interconnected cylindrical pores with predefined distribution of pore radii and connectivity and with a volume fraction of pores equal to the porosity. The pore size distribution can be estimated from experimental characteristics obtained, e.g., from nitrogen sorption or mercury porosimetry measurements. Local heterogeneities, e.g., spatial variation in the mean pore size, or the non-uniform distribution of catalytic active centers may be taken into account in pore-network models. In each individual pore of a cylindrical or general shape, the spatially ID reaction-transport model is formulated, and the continuity equations are formulated at the nodes (i.e., connections of cylindrical capillaries) of the pore space. The transport in each individual pore is governed by the Max-well-Stefan multicomponent diffusion and convection model. Any common type of reaction kinetics taking place at the pore wall can be implemented. 

The j° term denotes the ordinary concentration diffusion (i.e., multi-component mass diffusion). In general, the concentration diffusion contribution to the mass flux depends on the concentration gradients of all the substances present. However, in most reactor systems, containing a solvent and one or only a few solutes having relatively low concentrations, the binary form of Pick s law is considered a sufficient approximation of the diffusive fluxes. Nevertheless, for many reactive systems of interest there are situations where a multi-component closure (e.g., a Stefan-Maxwell equation formulated in terms... 

A general model for hydrogen diffusion with reversible trapping was formulated by McNabb and Foster [103]. The general diffusion equation can be written as... 

The general prineiples of the development of nonlinear models of mass transfer in elastieally deformed materials were developed in studies. The general formulation of eon-stitutive equations and the use ofnon-traditional thermodynamie parameters sueh as partial stress tensors and diffusion forees lead to signitieant diftieulties in attempts to apply the theory to the deseription of speeitie objeets. Probably, beeause of this, the theory is little used for the solution of applied problems. 

In general, solubility and diffusivity are concentration-dependent. A number of mathematical equations for mass transport have been formulated on the basis of Pick s diffusion equation using different empirical expressions of concentration dependency of solubility and/or diffusivity. However, these equations cannot be taken for granted unless they are used within the experimentally established range for which the relationships expressed for diffusion and thermodynamic equilibria are applicable. 

If condition 20 is not satisfied, the term r cannot be ne glected in Eq.l7. Yet that term depends not only a, but also on the concentration bj of the non-volatile components hence in principle Eq.l7 is coupled with the diffusion equations of all other liquid phase components. The problem formulated in such a general form is very difficult to solve. 

This model in which the neutron source is taken to be (which implies that neutrons from fission appear at the one velocity of diffusion) is not expected to apply directly to operating reactors however, the techniques to be used later are well illustrated by this formulation of the problem, and the results are useful, if properly adapted, to more general situations. We take then as our neutron-balance relation the time-dependent diffusion equation (5.21) along with the assumed source term... 

It is again supposed that translational diffusion motions of molecules can be described by a diffusion equation. The theory of spin relaxation by translational diffusion can, in principle, be formulated [7.16]. The review by Kruger [7.57] provides an exhaustive description and interpretation of the behavior of mass diffusion in different thermotropic mesophases. The mass diffusion is anisotropic in mesophases and, in general, will be given by a second-rank tensor D, the symmetry of which is related to the symmetry of the mesophase under consideration. For a uniaxial system with the z axis along the director, the translational diffusion tensor is... 

Cahn s kinetic theory is a formulation of a generalized diffusion equation for an inho-mogenous system, which accounts for the phenomena described above.The flux of matter is phenomenologically related to the gradient in chemical potential... 

This chapter provides analytical solutions to mass transfer problems in situations commonly encountered in the pharmaceutical sciences. It deals with diffusion, convection, and generalized mass balance equations that are presented in typical coordinate systems to permit a wide range of problems to be formulated and solved. Typical pharmaceutical problems such as membrane diffusion, drug particle dissolution, and intrinsic dissolution evaluation by rotating disks are used as examples to illustrate the uses of mass transfer equations. 

Generalized local Darcy s model of Teorell s oscillations (PDEs) [12]. In this section we formulate and study a local analogue of Teorell s model discussed previously. The main difference between the model to be discussed and the original one is the replacement of the ad hoc resistance relaxation equation (6.1.5) or (6.2.5) by a set of one-dimensional Nernst-Planck equations for locally electro-neutral convective electro-diffusion of ions across the filter (membrane). This filter is viewed as a homogenized aqueous porous medium, lacking any fixed charge and characterized... 

It is worth noting that the derivation outlined above is more generally applicable since, for many types of diffusional mass transfer (spherical, cylindrical, bounded, etc.), it is possible to rewrite the original differential equations in the time domain in terms of new variables in such a way that the second diffusion law is of the same form as eqn. (19b), with appropriate formulation of the boundary conditions [22, 75]. However, in finding the inverse transforms, difficulties may arise because of the more complex meaning of the time domain variables. 

Fick s second law states the conservation of the diffusing species i no i is produced (or annihilated) in the diffusion zone by chemical reaction. If, however, production (annihilation) occurs, we have to add a (local) reaction term r, to the generalized version of Fick s second law c, = —Vjj + fj. In Section 1.3.1, we introduced the kinetics of point defect production if regular SE s are thermally activated to become irregular SE s (i.e., point defects). These concepts and rate equations can immediately be used to formulate electron-hole formation and annihilation... 

---