Equivalent cylinder sphere

In this equation S, and Vp are the external surface area and the volume of the particle and S, is the surface of the equivalent volume sphere = 1 for spheres, 0.874 for cylinders with height equal to the diameter, 0.39 for Raschig-rings, 0.37 for Berl saddles). extends the correlation to particles of arbitrary shape. The product l> dp is sometimes written as a diameter dj, ... 

Figure 9.16, (a) Friction force measurements on crossed polylethylene tercphthalaic) fibers showing agreement with Equation (9.16) with friction coefficient = 0..T.3 and work of adhesion W = 0.01 m . (b) Contact between two crossed cylinders, equivalent to sphere on flat, showing the forces acting. 

Our primary interest in this calculation is the determination of the critical mass of the hot clean reactor and the radial distribution of the fast and thermal flux throughout the core and reflector. An accurate analysis of this system must necessarily take into account the completely reflected cylindrical geometry shown in Fig. 8.216. However, since this would entail a somewhat involved calculation, we will approximate the actual configuration by an equivalent reflected sphere of the same composition. This will reduce our computation appreciably and yet not obscure any of the essential steps in the application of the two-group model. A study of the effect of the corners in the completely reflected cylinder will be deferred until the next section. 

To observe such a pattern from the bcc lattice, the incident X-r beam should be parallel to the [111] direction of the bcc lattice as schematically shown in Figure 4b. At this point we can conclude that the cylinders are broken into a series of spheres with the cylindrical axes parallel to [111] direction of the bcc lattice without changing the grain structure. As shown previously this scheme thermoreversibly occurred le., when bcc-spheres are transformed into hex-cylinders, spheres are connected into cylinders along the [111] direction of bcc-sphere. We should note here that there are 4 possible ways in connecting the spheres into the cylinders, because the [111], [T11], [] il], and [III] directions are physically equivalent. However nature selects only one of those directions and recovers an original orientation of the cylinders. This means some memory of the lattice orientation exists, which will be discussed later. 

This allows for the equivalence between crossed cylinders and the particle on a plane problem. Likewise, the mechanics of two spheres can be described by an equivalently radiused particle-on-a-plane problem. The combination of moduli and the use of an effective radius greatly simplifies the computational representation and allows all the cases to be represented by the same formula. On the other hand, it opens the possibility of factors of two errors if the formula are used without realizing that such combinations have been made. Readers are cautioned to be aware of these issues in the formulae that follow. 

The equivalent particle diameter appearing in these dimensionless groups is the diameter of a sphere having the same external surface area as the particle in question. Thus for a cylinder of length Lc and radius rc, the equivalent particle diameter is given by... 

Several important assumptions have been implicitly incorporated in Eqs. (15) and (16). First, these equations describe the release of a drug from a carrier of a thin planar geometry, equivalent equations for release from thick slabs, cylinders, and spheres have been derived (Crank and Park, 1968). It should also be emphasized that in the above written form of Fick s law, the diffusion coefficient is assumed to be independent of concentration. This assumption, while not conceptually correct, has been... 

Figure 1 shows the DSC cooling scan of iPP in the bulk after self-nucleation at a self-seeding temperature Ts of 162 °C (in domain II). The self-nucleation process provides a dramatic increase in the number of nuclei, such that bulk iPP now crystallizes at 136.2 °C after the self-nucleation process this means with an increase of 28 °C in its peak crystallization temperature. In order to produce an equivalent self-nucleation of the iPP component in the 80/20 PS/iPP blend a Ts of 161 °C had to be employed. After the treatment at Ts, the cooling from Ts shows clearly in Fig. 1 that almost every iPP droplet can now crystallize at much higher temperatures, i.e., at 134.5 °C. Even though the fractionated crystallization has disappeared after self-nucleation, it should also be noted that the crystallization temperature in the blend case is nearly 2 °C lower than when the iPP is in the bulk this indicates that when the polymer is in droplets the process of self-nucleation is slightly more difficult than when it is in the bulk. In the case of block copolymers when the crystallization is confined in nanoscopic spheres or cylinders it will be shown that self-nucleation is so difficult that domain II disappears. 

Bowen and Masliyah examined the axial resistance of cylinders with flat, hemispherical and conical ends, and of double-headed cones and cones with hemispherical caps, together with the established results for spheroids. Widely used shape factors (including sphericity) did not give good correlations, while Eqs. (4-26) and (4-27) were found to be inapplicable to particles other than cylinders and spheroids. The best correlation was provided by the perimeter-equivalent factor Yj defined in Chapter 2. With this parameter, the equivalent sphere has the same perimeter as the particle viewed normal to the axis. Based on their numerical results, Bowen and Masliyah obtained the correlation... 

In this expression, the parameter ad is equal to 1, if the bottom is adiabatic, and zero in other cases. The radius of a sphere that is thermally equivalent to a cylinder is... 

Solution To think specifically, consider a sphere of radius R and a flat (or its Derjaguin-approximation equivalent, two perpendicular cylinders of radius R), and neglect all but nonretarded van der Waals forces. The force is the negative derivative of the free energy — ( Ham / 6) (R/ Z),... 

A surface is closed if it has no boundary curves. By this definition surfaces of a sphere and a torus are closed, whilst the surfaces of a hollow cylinder and of a disc are open. Boundary curves of two-sided surfaces are curves which separate one side of the surface from the other, for example the edges of a piece of thin paper. A completely open cylinder has two boundary curves. A cylinder which is half-open has only one boundary curve, and is continuously deformable into, and therefore topologically equivalent to a disc. Similarly, the removal of a disc from the surface of a sphere leaves an... 

For simple-shaped electrodes (plane, cylinder, and sphere), the problem can be solved analytically [1-9]. For more complex-shaped electrodes, numerical methods are used. For the numerical solution of the set of equations (11), both iteration methods (Ref. 36, for example) and the method of variation of inequalities are used. The latter allows one to reduce the initial problem to an equivalent problem in the region with fixed boundaries [37, 38]. 

This case is easily solved analytically for both the spherical and slab geometries. However, for cylindrical particles, the analytical solution is not as simple as for the cases of slab and sphere. However, this does not present any real difficulty since Aris (1957) has proved that all shapes can be transferred into the equivalent slab with the same specific surface as the original shape. Figure 5.46 shows the effectiveness factor versus Thiele modulus profiles for the cases of slab, cylinder and sphere geometries. 

For cubes and cylinders for which the length L equals the diameter, the equivalent diameter is greater than L and found from the equivalent diameter would be 0.81 for cubes and 0.87 for cylinders. It is more convenient to use the nominal diameter L for these shapes since the surface-to-volume ratio is 6/i)p, the same as for a sphere, and this makes equal to 1.0. For column packings such as rings and saddles the nominal size is also used in defining... 

The dimensionless shape factor for the right circular cylinder is in very close agreement with the values for the oblate spheroid in the range 0 < Lid < 1 and with the values for the prolate spheroid in the range 1 < Lid < 8. The difference when Ud = 1 is less than 1 percent. This shows that the results for the sphere and a finite circular cylinder of unit aspect ratio are very close. The simple expression obtained from the Smythe solution can be used to estimate the shape factors of circular disks, oblate spheroids, and prolate spheroids in the range 0 < Ltd < 8. For Ud > 8, the prolate spheroid asymptotic result can be used to provide accurate results for long circular cylinders and other equivalent bodies. 

Other 3D Enclosures With Interior Solids. Warrington and Powe [278] showed that so far as the heat transfer is concerned, cubes and stubby cylinders behave similarly to equivalent spheres of the same volume. This appears to be the case for both the inner and outer body shape. So Eqs, 4.121,4.124, and 4.128 appear to be applicable to other inner and outer body shapes as well, it being understood that D0 = (6V0/7t)l/3 and D, = (6 Vz/Jt)1 3, where Va and U, are the inner and outer body volumes, respectively. Sparrow and Charmichi [258], using stubby cylinders for the inner and outer body shapes, confirmed the conduction layer model prediction that the heat transfer is independent of eccentricity E when Ra (based on inner cylinder diameter) is greater than about 1500. 

If the continuum treatment is to be employed, we must first identify the elements that make up the system. The choice of elements might be obvious (as in the case of a packed bed of spheres) or some simplifying assumptions might have to be made. Common simplifying assumptions are assuming the system to be made up of cylinders of infinite length (for fibrous media) or assuming arbitrary convex-surfaced particles to be spheres of equivalent cross section or volume. Then the properties of an individual particle can be determined. If the system cannot be broken down into elements, then we have no choice but to determine its radiative properties experimentally. 

It is of interest to note that, by judicious definition of the characteristic diameter of nonspherical bodies, good agreement with the equations for spherical solids was obtained. A diameter defined by the total surface area of the body, divided by the perimeter normal to flow, was successfully used for spheres, hemispheres, cubes, prisms, and cylinders (PI), yielding a = 0 b - 0.692 m = 0.514 and n = [Eq. (4)]. Similar results were obtained for spheroids (S14), namely a = 0 6 = 0.74 w = 0.5 and n =. The commonly used equivalent diameter of a sphere of the same volume as the body yields transfer coefficients increasing with eccentricity (SI4). 

Figure J,. Scattering matrix elements for randomly oriented circular cylinders with diameter-to-length ratio 1, surface-equivalent sphere size parameter 180, and refractive index 1.311. Thin curves show T-Matrix computations, thick curves represent ray-tracing results.

This expression is an adaptation of the I(Q) calculation for a sphere and the required integration can be performed numerically. Particles that do not have spherical or near-spherical symmetry do not exhibit the minima and maxima noted above, and the scattering curve I(Q) declines more uniformly as Q increases. Other analytical expressions exist for the calculation of I(Q) for ellipsoids, prisms and cylinders and their hollow equivalents [55]. It should be noted, however, that I(Q) for ellipsoids, prisms and cylinders do not differ greatly. For simple models, a first indication of the macromolecular shape in terms of a triaxial body can be extracted by curve-fitting of the calculated scattering to the experimental curve at low Q. 

P = design internal pressure, psi P = allowable external pressure, psi Px = design external pressure, psi R = outside radius of spheres and hemispheres, crown radius of torispherical heads, in, t = thickness of cylinder, head or conical section, in. t, = equivalent thickness of cone, in. oc = half apex angle of cone, degrees... 

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