Liquids and solids, diffusion

Fick s law of diffusion is also used for problems involving liquid and solid diffusion, and the main difficulty is one of determining the value of the diffusion coefficient for the particular liquid or solid. Unfortunately, only approximate theories are available for predicting diffusion coefficients in these systems. Bird, Stewart, and Lightfoot [9] discuss the calculation of diffusion in liquids, and Jost [6] gives a discussion of the various theories which have been employed to predict values of the diffusion coefficient. The reader is referred to these books for more information on diffusion in liquids and solids. 

In general, diffusivity depends on pressure, temperature, and composition. With respect to the mobility of molecules, the diffusion coefficients are generally higher for gases and lower for solids. The diffusivities of gases at low densities are almost independent of concentration, increase with temperature, and vary inversely with pressure. Liquid and solid diffusivities are strongly concentration dependent and generally increase with temperature. Tables 2.7 and 2.8 show some of the experimental binary diffusivities for gas and liquid systems. 

Before closing this brief discussion on mass transfer fundamentals, further mention should be made of the diffusion coefficient. Equations for predicting gas diffusivities are given by Fuller and are also given in Perry s Handbook The orders of magnitude of the diffusivities for gases. liquids. and soiids and the manner in which the) vary w-iih temperature and pressure are given in Table 11-2. We note that the Knudsen. liquid, and solid diffusivities are independent of total pressure. 

Fig. 1 Schematic representation of the reversed-flow GC technique for measuring diffusion coefficients, (a) Injection point for gaseous diffusivities (b) injection point for liquid and solid diffusivities.

Here U = T — T )Cp/L is the appropriately rescaled temperature field T measured from the imposed temperature of the undercooled melt far away from the interface. The indices L and 5 refer to the liquid and solid, respectively, and the specific heat Cp and the thermal diffusion constant D are considered to be the same in both phases. L is the latent heat, and n is the normal to the interface. In terms of these parameters,... 

The techniques used for handling various materials depend on their physical states as well as their chemical properties. While it is comparatively easy to handle liquids and solids, it is not as convenient to measure out a quantity of a gas. Fortunately, except under rather extreme conditions, all gases have similar physical properties, and the chemical identity of the substance does not influence those properties. For example, all gases expand when they are heated in a nonrigid container and contract when they are cooled or subjected to increased pressure. They readily diffuse through other gases. Any quantity of gas will occupy the entire volume of its container, regardless of the size of the container. 

Diffusion of particles in the polymer matrix occurs much more slowly than in liquids. Since the rate constant of a diffusionally controlled bimolecular reaction depends on the viscosity, the rate constants of such reactions depend on the molecular mobility of a polymer matrix (see monographs [1-4]). These rapid reactions occur in the polymer matrix much more slowly than in the liquid. For example, recombination and disproportionation reactions of free radicals occur rapidly, and their rate is limited by the rate of the reactant encounter. The reaction with sufficient activation energy is not limited by diffusion. Hence, one can expect that the rate constant of such a reaction will be the same in the liquid and solid polymer matrix. Indeed, the process of a bimolecular reaction in the liquid or solid phase occurs in accordance with the following general scheme [4,5] ... 

Rapid bimolecular reactions are limited by diffusion of reactants in the liquid and solid phases. Diffusion occurs in polymers much more slowly than in liquids. Hence, such rapid reactions as recombination of free radicals occurs in polymers with rate constants of a few order of magnitude more slowly than in solution. For example, the reaction of sterically hindered phenoxyl with the peroxyl radical... 

In this chapter, we consider multiphase (noncatalytic) systems in which substances in different phases react. This is a vast field, since the systems may involve two or three (or more) phases gas, liquid, and solid. We restrict our attention here to the case of two-phase systems to illustrate how the various types of possible rate processes (reaction, diffusion, and mass and heat transfer) are taken into account in a reaction model, although for the most part we treat isothermal situations. 

It is theoretically possible that equilibrium between liquid and resin will be maintained at all points of contact. Liquid and solid concentrations are then related by the sorption isotherm. It is usual, however, that pellet or film diffusion will dominate or control the rate of exchange. It is also possible that control will be mixed, or will change as the ion exchange proceeds. In the latter case, the initial film-diffusion control will give way to pellet-diffusion control at a later stage. 

When diffusion is assumed to be controlled by the boundary him, by implication, all other resistances to diffusion are negligible. Therefore, concentrations are uniform through the solid and local equilibrium exists between huid and solid. The whole of the concentration difference between bulk liquid and solid is conhned to the him. The rate of transfer into a spherical pellet may then be expressed as ... 

Sorption/desorption is the key property for estimating the mobility of organic pollutants in solid phases. There is a real need to predict such mobility at different aqueous-solid phase interfaces. Solid phase sorption influences the extent of pollutant volatilization from the solid phase surface, its lateral or vertical transport, and biotic or abiotic processes (e.g., biodegradation, bioavailability, hydrolysis, and photolysis). For instance, transport through a soil phase includes several processes such as bulk flow, dispersive flow, diffusion through macropores, and molecular diffusion. The transport rate of an organic pollutant depends mainly on the partitioning between the vapor, liquid, and solid phase of an aqueous-solid phase system. 

Filters collect liquid and solid particles by mechanisms including diffusion, impaction, interception, electrostatic attraction, and sedimentation onto the filter while allowing the gas to pass through. The types commonly used in atmospheric particulate collection are membranes, fibrous mats, or porous sheets. Different filter materials are used depending on the particular type of measurement being carried out, including Teflon, quartz fiber, nylon, silver, cellulose filters, glass fibers, and polycarbonate. The characteristics of each are summarized by Chow (1995). 

Infinite fluid volume and solid diffusion control Practically, infinite solution volume condition (w 1) amounts to constant liquid-phase concentration. For a constant diffu-sivity and an infinite fluid volume, the solution of the diffusion equations is (Helfferich, 1962 Ruthven, 1984)... 

In this equation, ,/ and mi/2 are the masses of the two isotopes making up RZ1, and the terms are condensation coefficients for the two isotopes, which are determined experimentally and are typically close to 1. Equation (7.2.1) is valid if a is independent of the evolving composition of the evaporating liquid, and the diffusive transport rate is fast enough to keep the liquid homogeneous. The last condition is violated in solids, where diffusion is very slow relative to the evaporation rate, so solids do not undergo Rayleigh distillation. 

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