Diffusion mathematics

ANALYTIC SOLUTIONS OF COAL PARTICLE GAS DIFFUSING MATHEMATICAL EQUATION... 

To solve diffusion equation, that is set up and dragged into formula (1), then the coal particle gas diffusion mathematical model can be turned as follows ... 

Coal particle gas diffusion mathematical model was set up basing on the third boundary condition and taking into account gas mass transmission characteristics on borderline. The applying scope includes gas diffusion model under the first boundary condition, thus it is not only more scientific and reasonable, but also is provided with application more widely. 

The analytical solution of coal particle gas diffusion mathematical model was obtained by separate variableness method. The result indicate that we can get out coal particle gas concentration, gas cumulation diffusing capacity at any time and terminal diffusing capacity when t —> o . So, more research on shearing fall coal particles of working face can be carried out thoroughly and the research conclusion will provide theory basis for gas prevention and control. 

Thomas Graham determined that the rates of diffusion and effusion of gases are inversely proportional to the square roots of their molecular or atomic weights. This is Graham s Law. In general, it says that the lighter the gas, the faster it will effuse (or diffuse). Mathematically, Graham s Law looks like this ... 

Guzhev,D.S., Kalitkin,N.N., Seifert,P., Shirkov,P.D. (1992) Numerical methods for problems of chemical kinetics with diffusion. Mathematical Modelling 4, 98-110 (Russian). 

At first we tried to explain the phenomenon on the base of the existence of the difference between the saturated vapor pressures above two menisci in dead-end capillary [12]. It results in the evaporation of a liquid from the meniscus of smaller curvature ( classical capillary imbibition) and the condensation of its vapor upon the meniscus of larger curvature originally existed due to capillary condensation. We worked out the mathematical description of both gas-vapor diffusion and evaporation-condensation processes in cone s channel. Solving the system of differential equations for evaporation-condensation processes, we ve derived the formula for the dependence of top s (or inner) liquid column growth on time. But the calculated curves for the kinetics of inner column s length are 1-2 orders of magnitude smaller than the experimental ones [12]. 

From what has been said, it is clear that both physical and mathematical aspects of the limiting processes require more careful examination, and we will scare this by examining the relative values of the various diffusion coefficients and the permeability, paying particular attention to their depec dence on pore diamater and pressure. 

Rate equations 28 and 30 combine the advantages of concentration-independent mass transfer coefficients, even in situations of multicomponent diffusion, and a familiar mathematical form involving concentration driving forces. The main inconvenience is the use of an effective diffusivity which may itself depend somewhat on the mixture composition and in certain cases even on the diffusion rates. This advantage can be eliminated by working with a different form of the MaxweU-Stefan equation (30—32). One thus obtains a set of rate equations of an unconventional form having concentration-independent mass transfer coefficients that are defined for each binary pair directiy based on the MaxweU-Stefan diffusivities. 

Diffusion and Mass Transfer During Leaching. Rates of extraction from individual particles are difficult to assess because it is impossible to define the shapes of the pores or channels through which mass transfer (qv) has to take place. However, the nature of the diffusional process in a porous soHd could be illustrated by considering the diffusion of solute through a pore. This is described mathematically by the diffusion equation, the solutions of which indicate that the concentration in the pore would be expected to decrease according to an exponential decay function. 

R. R. Dixon, L. E. Edleman, and D. K. McLain,/ Cell Plast. 6(1), 44 (1970) C. F. Sheffield, Description and Applications of a Diffusion Modelfor BJgid Closed-CellFoams, ORNL Symposium on Mathematical Modeling of Roofs, Oak Ridge National Laboratory, Oak Ridge, Term., Sept. 15,1988. 

The mathematical model most widely used for steady-state behavior of a reactor is diffusion theory, a simplification of transport theory which in turn is an adaptation of Boltzmann s kinetic theory of gases. By solving a differential equation, the flux distribution in space and time is found or the conditions on materials and geometry that give a steady-state system are determined. 

Electrokinetics. The first mathematical description of electrophoresis balanced the electrical body force on the charge in the diffuse layer with the viscous forces in the diffuse layer that work against motion (6). Using this force balance, an equation for the velocity, U, of a particle in an electric field... 

An overview of some basic mathematical techniques for data correlation is to be found herein together with background on several types of physical property correlating techniques and a road map for the use of selected methods. Methods are presented for the correlation of observed experimental data to physical properties such as critical properties, normal boiling point, molar volume, vapor pressure, heats of vaporization and fusion, heat capacity, surface tension, viscosity, thermal conductivity, acentric factor, flammability limits, enthalpy of formation, Gibbs energy, entropy, activity coefficients, Henry s constant, octanol—water partition coefficients, diffusion coefficients, virial coefficients, chemical reactivity, and toxicological parameters. 

Aris, R. The Mathematical Theoiy of Diffusion and Reaction in Feimeahle Catalysis, vols. 1 and 2, Oxford University Press, Oxford (1975). 

Equations (12-31), (12-32), and (12-33) hold only for a slab-sheet solid whose thickness is small relative to the other two dimensions. For other shapes, reference should be made to Crank The Mathematics of Diffusion, Oxford, London, 1956). 

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