Maxwell’s transport equation

Using a two-sided Maxwellian velocity distribution functions and Maxwell s transport equation, Asaeda et al. (1974) obtained the following equation for the permeability coefficient in a packed column containing spherical particles ... 

The goal of the presentation in this section is to pose a generalized model comprising all the types of nonequilibrium detectors presented in the literature until now, and applicable to potential novel devices. When deriving the model we start from the semiconductor equations in their general form (Maxwell s equations and Boltzmann s transport equation.). 

The methods of determination of the reaction matrix [AT] are considered in Refs. 167, 181, 183, 184 and 186. Another important matrix parameter entering into the linearized film mass transport equation is the multicomponent diffusion matrix /). The latter results from the transformation of the Maxwell-Stefan Eqs. (1) to the form of the generalized Fick s law (83). Matrix [D] is generally a function of... 

The applicability of Maxwell s equation is limited in describing particle growth or depletion by mass transfer. Strictly speaking, mass transfer to a small droplet cannot be a steady process because the radius changes, causing a change in the transfer rate. However, when the difference between vapor concentration far from the droplet and at the droplet surface is small, the transport rate given by Maxwell s equation holds at any instant. That is, the diffusional transport process proceeds as a quasi-stationary process. 

The droplet current / calculated by nucleation models represents a limit of initial new phase production. The initiation of condensed phase takes place rapidly once a critical supersaturation is achieved in a vapor. The phase change occurs in seconds or less, normally limited only by vapor diffusion to the surface. In many circumstances, we are concerned with the evolution of the particle size distribution well after the formation of new particles or the addition of new condensate to nuclei. When the growth or evaporation of particles is limited by vapor diffusion or molecular transport, the growth law is expressed in terms of vapor flux equation, given by Maxwell s theory, or... 

For a gas mixture at rest, the velocity distribution function is given by the Maxwell-Boltzmann distribution function obtained from an equilibrium statistical mechanism. For nonequilibrium systems in the vicinity of equilibrium, the Maxwell-Boltzmann distribution function is multiplied by a correction factor, and the transport equations are represented as a linear function of forces, such as the concentration, velocity, and temperature gradients. Transport equations yield the flows representing the molecular transport of momentum, energy, and mass with the transport coefficients of the kinematic viscosity, v, the thermal diffirsivity, a, and Fick s diffusivity, Dip respectively. 

The definitions of effective diffusivity tensors are key parameters in the solution of the transport equations above. For an isotropic medium, the effective diffusivity is insensitive to the detailed geometric structure, and the volume fraction of the phases A and B influences the effective diffusivity. When the resistance to mass transfer across the cell membrane is negligible, the isotropic effective diffusivity, Ds e = Dg eI may be obtained from Maxwell s equation... 

For the very simplified situation that the sphere behaves electrically as a pure capacitor, and the solution as a pure resistance, the relaxation can be described by a Maxwell-Wagner mechanism, with T = e e/K, see (1.6.6.321. Although some success has been claimed by Watillon s group J to apply this mechanism for a model, consisting of shells with different values of e and K, generally a more detailed double layer picture is needed. In fact, this Implies stealing from the transport equations of secs. 4.6a and b. generedizing these to the case of a.c. fields. 

In any event, we hope it is now well understood that mass transfer in multicomponent systems is described better by the full set of Maxwell-Stefan or generalized Fick s law equations than by a pseudobinary method. A pseudobinary method cannot be capable of superior predictions of efficiency. For a simpler method to provide consistently better predictions of efficiency than a more rigorous method could mean that an inappropriate model of point or tray efficiency is being employed. In addition, uncertainties in the estimation of the necessary transport and thermodynamic properties could easily mask more subtle diffusional interaction effects in the estimation of multicomponent tray efficiencies. It should also be borne in mind that a pseudobinary approach to the prediction of efficiency requires the a priori selection of the pair of components that are representative of the... 

The conductor can thus be represented as an anisotropic medium with interrelated nonlinear magnetic and electrical properties. The current pattern that results in the conductor from the application of time-dependent external magnetic fields and excitation (transport) currents can be determined in principle, from Maxwell s equations. An integration of pJ and the /-dependent magnetic hysteresis over the entire conductor cross section can then be used to calculate the loss. 

The law of gravitation illustrates the principal goal of science to identify, organize, codify, and compress a large amount of information into a concise statement. Another example is Maxwell s proposal that electricity and magnetism can be described by the same set of differential equations. Still another example occurs in linear transport the-... 

The boundary conditions for the carrier transport equation in the magnetoconcentration detector are posed for the top and bottom surface, assuming that their surface recombination rates are described by the Shocldey-Read model, i.e., that S = 1/tno = 1/tpo- The second assumption often met in literature (e.g., [401]) is that the trap distribution function at the surface is always of Maxwell-Boltzmaim type (nondegenerate semiconductor). In that case Shockley-Read expression for recombination rate (1.70) becomes... 

Depending on the time and length scales, different transport laws can be used. When the objects have comparable size to the wavelength of energy carrier, wave phenomena, that is, reflection, refraction, diffraction, etc., dominate the energy transport mechanism. When the time scale of interest (t) is of the order of collision time scale (t ), time-dependent wave mechanics must be used. Schrodinger s equation must be used for electrons and phonons. Maxwell s equation must be used for photons ... 

Continuum mechanics is the mechanical analog of classical electrodynamics, in which a set of field equations (Maxwell s equations) describe the dynamics of the relevant variables of the electrical and magnetic fields. Whereas Maxwell s equations are linear unless the constitutive behavior is nonlinear, the equations of continuum mechanics are nonlinear, regardless of the constitutive behavior of the materials of interest. The inherent nonlinearity of the conservation equations, which is due to convective transport of momentum, energy, and... 

The theory on the level of the electrode and on the electrochemical cell is sufficiently advanced [4-7]. In this connection, it is necessary to mention the works of J.Newman and R.White s group [8-12], In the majority of publications, the macroscopical approach is used. The authors take into account the transport process and material balance within the system in a proper way. The analysis of the flows in the porous matrix or in the cell takes generally into consideration the diffusion, migration and convection processes. While computing transport processes in the concentrated electrolytes the Stefan-Maxwell equations are used. To calculate electron transfer in a solid phase the Ohm s law in its differential form is used. The electrochemical transformations within the electrodes are described by the Batler-Volmer equation. The internal surface of the electrode, where electrochemical process runs, is frequently presented as a certain function of the porosity or as a certain state of the reagents transformation. To describe this function, various modeling or empirical equations are offered, and they... 

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