Finite difference techniques diffusion modeling

The model equations (Eqs. (60) and (61)) were solved numerically using an implicit finite difference technique. Typical profiles for the solvent volume fraction as a function of position and time in the rubber and the polymer volume fraction in the diffusion boundary layer are shown in Figs. 31 and 32 respectively. Other features of the simulation are the prediction of the temporal evolution of the rubbery-solvent interface and the mass fraction of the polymer dissolved as a function of time (Figs. 33 and 34). The simulations showed that the dissolution could be either disentanglement or diffusion controlled depending on the polymer molecular weight and the thickness of the diffusion boundary layer. 

The finite difference technique is a widely used numerical method for solving the reaction-diffusion problems [7]. When applying this approach, the model equations are transformed into a form such that the differentiation can be performed by numerical calculations. There are several difference schemes that can be used for... 

Pollutants emitted by various sources entered an air parcel moving with the wind in the model proposed by Eschenroeder and Martinez. Finite-difference solutions to the species-mass-balance equations described the pollutant chemical kinetics and the upward spread through a series of vertical cells. The initial chemical mechanism consisted of 7 species participating in 13 reactions based on sm< -chamber observations. Atmospheric dispersion data from the literature were introduced to provide vertical-diffusion coefficients. Initial validity tests were conducted for a static air mass over central Los Angeles on October 23, 1968, and during an episode late in 1%8 while a special mobile laboratory was set up by Scott Research Laboratories. Curves were plotted to illustrate sensitivity to rate and emission values, and the feasibility of this prediction technique was demonstrated. Some problems of the future were ultimately identified by this work, and the method developed has been applied to several environmental impact studies (see, for example, Wayne et al. ). 

Gradient diffusion was assumed in the species-mass-conservation model of Shir and Shieh. Integration was carried out in the space between the ground and the mixing height with zero fluxes assumed at each boundary. A first-order decay of sulfur dioxide was the only chemical reaction, and it was suggested that this reaction is important only under low wind speed. Finite-difference numerical solutions for sulfur dioxide in the St. Louis, Missouri, area were obtained with a second-order central finite-difference scheme for horizontal terms and the Crank-Nicolson technique for the vertical-diffusion terms. The three-dimensional grid had 16,800 points on a 30 x 40 x 14 mesh. 

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