Relations between Shape Factors

Two parallel black plates 0.5 by 1.0 m are spaced 0.5 m apart. One plate is maintained at 1000°C and the other at 500°C. What is the net radiant heat exchange between the two plates  

Some useful relations between shape factors may be obtained by considering the system shown in Fig. 8-19. Suppose that the shape factor for radiation from A3 to the combined area A,.2 is desired. This shape factor must be given very simply as 

Solving for A F -3 from (c), inserting this in (b), and then inserting the resultant expression for A,F, 3.4 in (a) gives 

Notice that all shape factors except F 4 may be determined from Fig. 8-14. 

In the foregoing discussion the tacit assumption has been made that the various bodies do not see themselves, i.e., 

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Fig. 8-19 Sketch showing some relations between shape factors.

The industrial point of view of catalyst characterization is different. Here the main interest is to optimize or produce an active, selective, stable and mechanically robust catalyst. In order to accomplish this, tools are needed which identify those structural properties that discriminate efficient from less efficient catalysts. In principle, all spectroscopic information that helps to achieve this is welcome. Empirical relations between the factors that govern catalyst composition, particle size and shape, and pore dimensions on one side, and catalytic performance on the other can be extremely useful in developing catalysts. 

The hydrauhc diameter method does not work well for laminar flow because the shape affects the flow resistance in a way that cannot be expressed as a function only of the ratio of cross-sectional area to wetted perimeter. For some shapes, the Navier-Stokes equations have been integrated to yield relations between flow rate and pressure drop. These relations may be expressed in terms of equivalent diameters Dg defined to make the relations reduce to the second form of the Hagen-Poiseulle equation, Eq. (6-36) that is, Dg (l2SQ[LL/ KAPy. Equivalent diameters are not the same as hydraulie diameters. Equivalent diameters yield the correct relation between flow rate and pressure drop when substituted into Eq. (6-36), but not Eq. (6-35) because V Q/(tiDe/4). Equivalent diameter Dg is not to be used in the friction factor and Reynolds number ... 

It is only reasonable to suggest that the compressibility of a liquid may depend partly on the shape of the molecules, and that we should, therefore, expect only a rough relation between a- and /S, as we go from one liquid to another. We can eliminate this factor by keeping to the same molecules, for instance, by working with solutions of the same... 

Table IV shows whole data relating to CSD and j, p values, the latters of which were mainly estimated by the relation between Sw-Sn (Figure 2), because that almost the same values were obtained with different methods, as shown in Table III. The data in Table IV are arranged in order of volumetric shape factor of crystals, fy. fv is unity...

Figure 4 Relation between volume shape factor and unit number...

The relationship amongst the X- (here, atomic spectra visualised in Figure 4.9) and Y-blocks (here, [Sb]), can be evaluated by the t-u plof. This plot shows how the different factors (latent variables) account for the relation between X and Y. Hence we can inspect whether a sample stands out because of an anomalous position in the relation. Also, we can visualise whether there is a fairly good straight relationship between the blocks. Whenever the samples follow a clear curved shape, we can reasonably assume that the model has not... 

Transforming a mass distribution to a number distribution, or vice versa, requires a relationship between the measured and desired quantities. The mass of a single crystal, mciys, is related to crystal size by the volume shape factor, feyoi (see Eq. (19)) ... 

Some caution should be used in the interpretation of the activation parameters obtained in many of these studies. Particularly in the earlier work, approximate equations relating line shape to rate parameters often have been used, so many of the rates are basically order of magnitude values. The current use of computers to determine the best agreement between calculated and observed spectra should improve the accuracy of these calculations. Other factors such as changes in the chemical shifts between two resonances with temperature can cause large errors in the activation energy. 

All the preceding discussions have considered radiation exchange between diffuse surfaces. In fact, the radiation shape factors defined by Eq. (8-21) hold only for diffuse radiation because the radiation was assumed to have no preferred direction in the derivation of this relation. In this section we extend the analysis to take into account some simple geometries containing surfaces that may have a specular type of reflection. No real surface is completely diffuse or completely specular. We shall assume, however, that all the surfaces to be considered emit radiation diffusely but that they may reflect radiation partly in a specular manner and partly in a diffuse manner. We therefore take the reflectivity to be the sum of a specular component and a diffuse component ... 

Steric hyperlayer-FFF is well established as a fast separation technique for mi-cron-sized particles, although the hydrodynamic lift forces are not yet well understood. This is worse for the steric elution of non-spherical particles. Despite over thirty years of application of FFF techniques, only very little has been reported about the fractionation of non-spherical particles by any FFF mode. The few available studies so far reported are the investigation of coal particles [423,424], inorganic colloids [462], metal particles [69] and doublets of polystyrene beads, rod-shaped glass fibers, compressed latex discs and quartz particles with complex shape [427]. In the latter paper, systematic studies of particle shape on the retention behavior of non-spherical particles are reported with the result that the qualitative major retention behavior of spheres and other shapes is equal (e.g. response to increase in the field strength, etc.). However, the quantitative differences in the retention behavior were found to depend on numerous factors in a complex way so that no quantitative relation between the hydrodynamic radius and the retention ratio could be established. 

Figure 8-23. Relation between net pho tosynthesis (Jco and transpiration (Jwv) as stomatal conductance is varied. The three curves depict various PPF levels, indicated as low, medium, and high. The circles indicate where the slope dJwv/dJco2 1000 H20/C02 (the slope is at 45° on each curve, as the ordinate is in mmol m-2 s l and the abscissa is in [xmol m-2 s which differ by a factor of 1000). Cu ticular transpiration is ignored. Hie numbers indicate changes in the fluxes, on the medium PPF curve for no net change in transpiration and at the same slope on the high PPF curve for no change in net C02 uptake. Curve shapes indicate that Jwv increases faster than does Jco2as stomata open.

The relation between mass and funnel flows for conical bins is shown in Fig. 21-20. The angle of kinematic friction which is a measure of the friction coefficient between the solid and the material of construction used for the conical-shaped hopper, is measured with the flow-factor tester. The degree of finish of the metal surface can have a large effect in determining whether the vessel will function in mass or funnel flow. Finer degrees of finish are being used more fre-... 

Particle shape is also an important parameter for the shear strength depending on the saturation of the filter cakes. For the fully dried filter cakes, shear strength is independent of the shape factor. The relation between the shear strength and the particle size and the other related parameters for the saturation region S=0% can be written as ... 

The numerical relationships between the various sizes of a particle depend on particle shape, and dimensionless ratios of these are called shape factors the relations between measured sizes and particle volume or surface area are called shape coefficients. 

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