Spheres, and Cylinders

In this rate expression we have lumped C/js into the effective surface rate coefficient by defining k — CC s- AU sohd reactions have reaction steps similar to those in catalytic reactions, and the rate expressions we need to consider are basically Langmuir-Hinshelwood kinetics, which were considered in Chapter 7. Our use of a first-order irreversible rate expression is obviously a simplification of the more complex rate expressions that can arise from these situations. 

We title this chapter the reactions of sohds and we deal mostly with gaseous and sohd reactants and products, but the same ideas and equations apply to gas-hquid and liquid-liquid systems. For example, in fiying of foods, the fluid is obviously a liquid that is transferring heat to the sohd and carrying off products. The same equations also apply to many gas-hquid and liquid-hquid systems. The drops and bubbles change in size as reactions proceed so the same equations we derive here for transformation of sohds wiU also apply to those situations. 

In Chapter 12 we will consider multiphase reactors in which drops or bubbles carry one phase to another continuous fluid phase. In fact, these reactors frequently have a sohd also present as catalyst or reactant or product to create a three-phase reactor. We need the ideas developed in this chapter to discuss these even more complicated reactors. 

The first two profiles are hrniting cases of solid concentration profiles, and most situations may be somewhere between those limits, as shown in the third panel. 

The growing or dissolving of a porous solid is a variant on the porous catalyst, but this situation combines transport through a porous solid with changing solid phases and is too complex to be considered here. 

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L. Other objects, including prisms, cuhes, hemispheres, spheres, and cylinders forced convection... 

Separate regions in the figure account for the scatter of velocities for spheres and cylinders separating into 2, 10 or 100 fragments. The number of fragments must first be chosen, usually on the basis of scaled energy. 

The absorption of reactants (or desorption of products) in trickle-bed operation is a process step identical to that occurring in a packed-bed absorption process unaccompanied by chemical reaction in the liquid phase. The information on mass-transfer rates in such systems that is available in standard texts (N2, S6) is applicable to calculations regarding trickle beds. This information will not be reviewed in this paper, but it should be noted that it has been obtained almost exclusively for the more efficient types of packing material usually employed in absorption columns, such as rings, saddles, and spirals, and that there is an apparent lack of similar information for the particles of the shapes normally used in gas-liquid-particle operations, such as spheres and cylinders. 

SK Friedlander. Mass and heat transfer to single spheres and cylinders at low Reynolds numbers. AIChE J 3 43-48, 1957. 

For more complicated geometries, the computations become more and more involved as it is the case for the ordinary electromagnetic Casimir effect. However, Casimir calculations of a finite number of immersed nonoverlapping spherical voids or rods, i.e. spheres and cylinders in 3 dimensions or disks in 2 dimensions, are still doable. In fact, these calculations simplify because of Krein s trace formula (Krein, 2004 Beth and Uhlenbeck, 1937)... 

The consequences of this normalization are summarized for the various shapes in Table 8.2. In Table 8.2, subscripts FP1 and FP2 refer to a flat plate with 1 and 2 faces permeable, respectively, and subscripts s and c refer to sphere and cylinder, respectively, all as given in Table 8.1. The main consequence is that, if 0 replaces 0 in Figure 8.11, tj for all shapes lies approximately on the one line shown. The results become exactly the same for large values of 0 (tj - T/< , independent of shape). In the transition region between points G and H, the results differ slightly (about 17% at the most). 

Predictions can be made for the initial velocity of fragments provided certain properties of the explosive material are known. The Gurney equation (8), or one of its many variations (9,10), is the most widely accepted method for predicting fragment initial velocity. The equations are slightly different for spheres and cylinders ... 

The second model, proposed by Frank-Kamenetskii [162], applies to cases of solids and unstirred liquids. This model is often used for liquids in storage. Here, it is assumed that heat is lost by conduction through the material to tire walls (at ambient temperature) where the heat loss is infinite compared to the rate of heat conduction through the material. The thermal conductivity of the material is an important factor for calculations using this model. Shape is also important in this model and different factors are used for slabs, spheres, and cylinders. Case B in Figure 3.20 indicates a typical temperature distribution by the Frank-Kamenetskii model, showing a temperature maximum in the center of the material. 

Similar equations may be developed for other geometries such as spheres and cylinders. To complete the mathematical representation of a problem, initial and boundary conditions are specified. 

Hollow cylindrical catalyst pellets are sometimes employed in commercial chemical reactors in order to avoid excessive pressure drops across a packed bed of catalyst. A more complex expression for the effectiveness factor is obtained for such geometry. This case was first discussed by Gunn [4]. Figure 2 illustrates the effectiveness factor curves obtained for the slab, sphere and cylinder. 

For top spin at higher rotation rates, Sh first decreases and then increases with increasing rotational Reynolds number, Qc/ /v, at constant translational Reynolds number, Re. The overall change in Sh is generally relatively small except for very rapid rotation. Analyses have been carried out (F4, PI 3) for spheres and cylinders with Re 1 and Pe = ReQ Sc 1 in simple shear. At low Re there are closed streamlines around the body at high Pe these streamlines are also lines of constant composition so that Sh (or Nu) becomes independent of Pe. For a sphere, Sh = 8.9 for Pec oo (PI 3). More complex velocity fields have also been considered at low Re. ... 

However, bed voidage depends on dji) (Dixon, 1988). It can be proved that for dji) values lower than 0.1, the bed voidage can be considered practically constant. Indeed, it is common practice to use ratios of dJD lower than 0.1, and therefore, the bed voidage can be actually considered the same for common fixed beds. For instance, a bed consisting of particles with 2 mm diameter should have a diameter greater that 2 cm, which leads to a dji) ratio with a value of 0.1 maximum. Consequently, the bed voidage is critical only in laboratory experiments. In Figure 3.39, Dixon s correlations are presented for spheres and cylinders, in the case of dji) < 0.4 (Dixon, 1988). For cylinders, dp is equal to the diameter of a sphere of equal volume. 

It was found that surfaces of grains of proplnt exhibited contamination by micro-organisms which ranged in shape from spheres and cylinders to rods and squares. Penetration of these bacteria beyond the surface was also apparent. The contamination probably resulted from bacteria encountered in water used during manuf of M8 by the slurry method Refsi l)T.Bokorny, ChemZtg 20,985-6(1896) 2)B.Malenkovic,MitteilGegenstande-Artil-Genie-wesen 1907,599-615 CA 1,2411(1907) 3)B. 

Nonplanar electrodes such as spheres and cylinders exhibit an increase in it1/2 with increasing t. However, planar diffusion can be closely approximated... 

One possible structure would consist of irregular forms among which spheres and cylinders with alternatively hydrophobic and hydrophilic outer surfaces are formed according to Figure 8. This suggestion has an immediate attraction its features are similar to critical phenomena [cf., two-dimensional Ising s model (40)]. The resemblance between micellar associations and the fast fluctuation aggregates before phase separations... 

Thus qmin as well as the average q are proportional to p for a constant value of 8. If in the case of spherical aggregates 1.2 qmin, which is always smaller than 1, becomes smaller than 0.5, the two-block polymers tend to avoid this strong deformation of the butadiene sequences by forming cylindrical aggregates. Both particle shapes are observed in cases when qmin is calculated to be approximately 0.5 to 0.7 both for spheres and cylinders. 

Note that at uniformly accessible electrodes, spheres and cylinders, the mass flux is identical at all the points of its surface, whereas at non-uniformly accessible ones, discs and bands, the mass flux varies through the radius and the width, respectively. Therefore, in these cases, the surface gradient should be calculated by integrating the flux over the electrode surface such that the current is given by (see Scheme 2.5) ... 

Spheres and Cylinders. The drag force F exerted by a fluid on a body would be expected to be a function of the body size (area), relative velocity with fluid, fluid density and viscosity, and body shape (see Figure 17). Thus F = f (A, u, p, p, shape). 

This relation applies generally and specifically to spheres and cylinders. Empirical relations have been developed to convert the measured lifetime (x) to this mean free path ( ). In case of small pores (<2 nm) an earlier version is adequate [43, 44], The model is based on the assumptions of spherical potential wells with infinite depth and radius r that traps the positronium. 

M. Other objects, including prisms, cubes, hemispheres, spheres, and cylinders forced convection jD=omzN ,NMr = -M- Terms same as in 5-20-J. [E] Used with arithmetic concentration difference. Agrees with cylinder and oblate spheroid results, 15%. Assumes molecular diffusion and natural convection are negligible. 500 < Nfo p < 5000. Turbulent. [88] p. 115 [141] p. 285 [111][112]... 

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