Diffusivity: and temperature

An extension of this one-dimensional heterogeneous model is to consider intraparticle diffusion and temperature gradients, for which the lumped equations for the solid are replaced by second-order diffu-sion/conduction differential equations. Effectiveness factors can be used as applicable and discussed in previous parts of this section and in Sec. 7 of this Handbook (see also Froment and Bischoff, Chemical Reactor Analysis and Design, Wiley, 1990). 

As shown by the equations in Section 12.1, column plate height is affected by many parameters, including flow velocity, particle diameter, packing nonuniformity, diffusivities, degree of retention, stationary phase structure, temperature, pressure drop, and pressure. Some of these parameters are interdependent, such as diffusivity and temperature also velocity and pressure drop. Finding a minimum with respect to all of these parameters is an extended task we shall not attempt here. However, we can readily uncover some simple rules for optimizing a few of the major parameters. First we choose flow velocity. 

Pmi, Pm2 6 the partial pressures of vapor (water) at the membrane surfaces on the feed and permeate sides, respectively Kiji is the membrane coefficient that is a function of membrane properties (pore size, thickness, porosity, and tortuosity), properties of the vapor transported across the membrane (molecular weight and diffusivity) and temperature gradient... 

Tang et al. (2004) presented a cure model that captures transient, thermal and chemical effects that are ignored in typical threshold-based models. This new model incorporates as inputs photoinitiation rates, reaction rates, diffusion and temperature distributions, and is able to determine the spatial and temporal distributions of monomer and polymer concentrations, molar masses, crosslink densities and degree of cure. 

Gorb, Kholyavenko, Diffusion and temperature effects in porous (56)... 

All materials tend to be somewhat permeable to chemical molecules, but the permeability rate of some elastomers tends to be an order of magnitude greater than that of metals. Though permeation is a factor closely related to absorption, factors that influence the permeation rate are diffusion and temperature rather than concentration and temperature. Permeation can pose a serious problem in elastomer-lined equipment. When the corrodent permeates the elastomer, it comes into contact with the metal substrate that is then subject to chemical attack. 

The data in Table II apply to the solutes benzene for Al and water for Ac at 20°C, and corrections should be applied for other solutes by using the diffusivity ratio. There are strong theoretical reasons for suggesting that the best correcting ratio for both solute diffusivity and temperature is to use the Schmidt number to the power —0.5 for the liquid and -0.67 for gas phases. The dimensionless Schmidt number is the ratio [viscosity/(density x diffusivity)] in consistent units (33). 

Figure 7.1.2 shows the effect of temperature on the diffusivity of four solvents. The relationship between diffusivity and temperature is essentially linear. Only solvents having the smallest molecules (methanol and acetone) depart slightly from a linear relationship due to the contribution of the energy term. The diffusivity of the solvent decreases as temperature decreases. Several other solvents show a similar relationship. ... 

Dependence of particle growth rate on the activation energy for diffusion and temperature... 

Do we expect this model to be accurate for a dynamics dictated by Tsallis statistics A jump diffusion process that randomly samples the equilibrium canonical Tsallis distribution has been shown to lead to anomalous diffusion and Levy flights in the 5/3 < q < 3 regime. [3] Due to the delocalized nature of the equilibrium distributions, we might find that the microstates of our master equation are not well defined. Even at low temperatures, it may be difficult to identify distinct microstates of the system. The same delocalization can lead to large transition probabilities for states that are not adjacent ill configuration space. This would be a violation of the assumptions of the transition state theory - that once the system crosses the transition state from the reactant microstate it will be deactivated and equilibrated in the product state. Concerted transitions between spatially far-separated states may be common. This would lead to a highly connected master equation where each state is connected to a significant fraction of all other microstates of the system. [9, 10]... 

Micropore Diffusion. In very small pores in which the pore diameter is not much greater than the molecular diameter the diffusing molecule never escapes from the force field of the pore wall. Under these conditions steric effects and the effects of nonuniformity in the potential field become dominant and the Knudsen mechanism no longer appHes. Diffusion occurs by an activated process involving jumps from site to site, just as in surface diffusion, and the diffusivity becomes strongly dependent on both temperature and concentration. 

Dielectric Film Deposition. Dielectric films are found in all VLSI circuits to provide insulation between conducting layers, as diffusion and ion implantation (qv) masks, for diffusion from doped oxides, to cap doped films to prevent outdiffusion, and for passivating devices as a measure of protection against external contamination, moisture, and scratches. Properties that define the nature and function of dielectric films are the dielectric constant, the process temperature, and specific fabrication characteristics such as step coverage, gap-filling capabihties, density stress, contamination, thickness uniformity, deposition rate, and moisture resistance (2). Several processes are used to deposit dielectric films including atmospheric pressure CVD (APCVD), low pressure CVD (LPCVD), or plasma-enhanced CVD (PECVD) (see Plasma technology). 

Sintering consists of heating a mixture of fine materials to an elevated temperature without complete fusion. Surface diffusion and some incipient fusion cause the soHd particles in contact with one another to adhere and form larger aggregates. In the processing of hematite, Fe202, or magnetite,... 

Ceramic—metal interfaces are generally formed at high temperatures. Diffusion and chemical reaction kinetics are faster at elevated temperatures. Knowledge of the chemical reaction products and, if possible, their properties are needed. It is therefore imperative to understand the thermodynamics and kinetics of reactions such that processing can be controlled and optimum properties obtained. 

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