Cylindrical Media

Examples of diffusion problems in wires are fairly common. Of practical importance is the outgassing of metal filaments, and the converse process of sorption in wires. It is necessary to know how much gas remains in the filaments, or how much has diffused away under vacuum conditions. The diffusion of metals such as thorium into or out of tungsten filaments has a profound effect upon the thermionic and photoelectric emission of the filament, and this in its turn is of importance in various types of valve. The quantitative expression for the flow of thorium can be obtained by integration of the differential equation for flow in a cylinder, using appropriate boundary conditions. Sometimes a supply of a solute may be 

Case 1. A circular cylinder of radius r = a is the diffusion medium, at its surface a constant concentration Ci is maintained, and the medium is initially free of solute. In this example the solution may be given in terms of Bessel s function of the first kind and of zero order Jq(x) and its differential Jq(x). The solution is 

These roots give four exponential terms in an infinite series, and it will be found as a rule that these terms are adequate to express the diffusion process. Indeed, for larger values of the 

For most purposes it will be more important to know the mean concentration in the cylinder, or alternatively the quantity Q which has diffused into the cylinder, per unit length. C, the mean concentration in the cylinder, is given by 

The integration of (107) in which (104) has been substituted leads to the equations 

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S2 and S4 Models for Cylindrical Geometries. For the solution of the RTE in cylindrical media, the formulations for the S2 and S4 discrete ordinates (DO) approximations (based on Ref. 69) will be presented here. Note that, in a more recent study, Jendoubi et al. [75] used a similar DO approximation in cylindrical geometry and evaluated the effect of anisotropic phase function on the accuracy of the model. 

Weigh and dissolve the fish oil in the xylenes and add to a mill jar that will accommodate the alumina batch desired. The mill should be one-third filled with grinding media of the appropriate size for the mill for example, 1.27 cm cylindrical media for a one-liter mill. As mentioned previously, the media and mill composition should be selected based upon the material being processed an alumina mill with alumina media would be used for this formulation. 

Case 3. If in the cylindrical medium of radius a the boundary conditions are... 

The first stage moisture sorption rate of fibers R can be obtained by considering that the sorption process is a radial diffusion of moisture in a cylindrical medium, governed by the following relationship [33] ... 

Snyder, A. W. (1971) Mode propagation in a nonuniform cylindrical medium. I.E.E.E. Trans. Microwaves Theory Tech. 19, 402-3. 

The reaction vessel is situated inside a metal of high themial conductivity having a cylindrical, spherical, or other shape which serves as the calorimetric medium. Silver is the most suitable material because of its high themial conductivity, but copper is most frequently used. 

Geankoplis [54] fabricated a porous medium for which the values of K, and are known a priori. This was accomplished by sealing a bundle of identical parallel cylindrical capillary tubes between the two chambers of a Wlcke-Kallenbach apparatus. Then the relevant flux relations are those which apply to a single cylindrical capillary, rather than a porous medium, and these are obtained by setting... 

Nutsche Filter. The nutsche filter (Fig. 8) is simply an industrial-scale equivalent of the laboratory Buckner funnel. Nutsche filters consist of cylindrical or rectangular tanks divided into two compartments of roughly the same size by a horizontal medium supported by a filter plate. Vacuum is apphed to the lower compartment, into which the filtrate is collected. It is customary to use the term nutsche only for filters that have sufficient capacity to hold the filtrate from one complete charge. The cake is removed manually or sometimes by reslurrying. 

The American version of the dynamic filter, known as the Artisan continuous filter (Fig. 30), uses such nonfiltering rotors in the form of turbine-type elements. The cylindrical vessel is divided into a series of disk-type compartments, each housing one rotor, and the stationary surfaces are covered with filter cloth. The feed is pumped in at one end of the vessel, forced to pass through the compartments in series, and discharged as a thick paste at the other end. At low rotor speeds the cake thickness is controlled by the clearance between the scraper and the filter medium on the stationary plate, while at higher speeds part of the cake is swept away and only a thin layer remains and acts as the actual medium. 

Rotary Drum Filters The rotaiy drum filter is the most widely used of the continuous filters. There are many design variations, including operation as either a pressure filter or a vacuum filter. The major difference between designs is in the technique for cake discharge, to be discussed later. All the alternatives are characterized by a horizontal-axis drum covered on the cylindrical portion by filter medium over a grid support structure to allow drainage to manifolds. Basic materials of construc tion may be metals or plastics. Sizes (in terms of filter areas) range from 0.37 to 186 m (4 to 2000 ft ). 

A microscopic description characterizes the structure of the pores. The objective of a pore-structure analysis is to provide a description that relates to the macroscopic or bulk flow properties. The major bulk properties that need to be correlated with pore description or characterization are the four basic parameters porosity, permeability, tortuosity and connectivity. In studying different samples of the same medium, it becomes apparent that the number of pore sizes, shapes, orientations and interconnections are enormous. Due to this complexity, pore-structure description is most often a statistical distribution of apparent pore sizes. This distribution is apparent because to convert measurements to pore sizes one must resort to models that provide average or model pore sizes. A common approach to defining a characteristic pore size distribution is to model the porous medium as a bundle of straight cylindrical or rectangular capillaries (refer to Figure 2). The diameters of the model capillaries are defined on the basis of a convenient distribution function. 

The porous materials that offer the narrowest possible pore size distribution are those that have cylindrical pores of uniform diameter penetrating the entire medium without branching. Branching gives polymer molecules in the junctions extra conformational entropy. An agglomerate of tiny pieces of these porous materials, interlaced with larger voids (much larger than the pore size), should also be chosen. 

Traditional Apparatus. The above reference need only be expanded to Include the common use of cylindrical culture vessels (1), in place of the spherical reaction flasks, for shear sensitive materials and simple boiling water type baths (2) as their heating medium. 

The drum filter consists of a 1 to 5 m diameter cylindrical drum that rotates while remaining partially submerged in an open-feed slurry tank. The drum surface is wrapped around tightly with the filter medium. The drum shell is divided into compartments, and drain lines are connected to the central valve system that permits either vacuum or pres-... 

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