Rectangular parallelepipeds

Early injection lasers were small rectangular parallelepipeds made by cutting a wafer of GaAs. Feedback was provided by mirrors polished on two edges or by cleaving. The wafer had ap—n junction incorporated into it and broad area or stripe contacts were provided. Laser stmctures have since evolved to satisfy a wide range of appHcation specific requirements. 

We demonstrate the procedure with an experiment conducted on a Bentheimer sandstone sample. For simplicity, we use a relatively thin sample and resolve only the two in-plane spatial coordinates. The sample is a rectangular parallelepiped shape having a length of 50 mm extending in the z direction, width 25 mm along the z2 direction and thickness 5 mm in the z3 direction. The sample was sealed laterally with epoxy and mounted in Plexiglass end-plates with O-rings and tube... 

They were evaluated from our analysis of the primary nucleation and lateral growth rates and that of the l dependence to the melting temperature Tm using the Gibbs-Thomson equation. Insertion of the parameters given by Eq. 20 into Eq. 6 shows that the shape of a nucleus is a long thin rectangular parallelepiped with the ratio of... 

From the experimental fact that the lamellar thickening growth rate (17) of ECSC of PE is independent of l [8], we can regard ve on the end surface of the nucleus as constant and independent of l. For simplicity, we assume that the shape of ECSC is rectangular parallelepiped (Fig. 28). ve is given by the ratio of the volume of the crystal to the surface area of the crystal as shown below,... 

We then wish to discover how tj depends on reaction and particle characteristics in order to use equation 8.5-5 as a rate law in operational terms. To do this, we first consider the relatively simple particle shape of a rectangular parallelepiped (flat plate) and simple kinetics. 

Analytic results for cylinders comparable to those discussed for spheroids are not available. However, Heiss and Coull (H4) reported accurate experimental determinations for cylinders, spheroids, and rectangular parallelepipeds, and developed a general correlation for settling factors. In terms of the volume drag ratio,, their results may be written ... 

The only orthotropic particles for which comprehensive experimental results are available are square bars, rectangular parallelepipeds with one pair of square faces. Symmetry then shows that the two principal resistances corresponding to translation with square faces parallel to the direction of motion are equal. These resistances will be denoted by c 2, while the resistance for translation normal to the square faces will be called cy. Consider such a particle in arbitrary translation at velocity U. Figure 4.11 shows a section of the particle parallel to the square faces (72 is the component of U in this plane, and the angle between U2 and principal axis 2 is 0. From Eq. (4-5), the drag components are as shown in Fig. 4.11. Hence the drag component parallel to U2 is... 

Fig. 4.11 Rectangular parallelepiped with square section in steady translation.

Fig. 4.12 Drag ratios for rectangular parallelepipeds with square section.

Heiss and Coull (H4) measured the drag on rectangular parallelepipeds. Results were correlated by Eqs. (4-26) and (4-27). For a particle with dimensions I X I X El... 

Notable progress in analyzing nonspherical particles has been made by Fuchs (1975), who calculated absorption by cubes in the electrostatics approximation and applied the results to experimental data for MgO and NaCl. We shall discuss Fuchs s results at the end of Section 12.3. Langbein (1976) also did calculations for rectangular parallelepipeds, including cubes, which give valuable insights into nonspherical shape effects. Because the cube is a common shape of microcrystals, such as MgO and the alkali halides, these theoretical predictions have been used several times to interpret experimental data. We shall do the same for MgO. Our theoretical treatment of nonsphericity, however, is based on ellipsoids. Despite its simplicity, this method predicts correctly many of the nonspherical effects. 

The calculation was similar to that made for the problem of three-dimensional heat transmission and uniform heat generation in a rectangular parallelepiped (3) with the surfaces maintained at ambient temperature. By solving the parallelepiped equation with appropriate values, it can be shown that the temperature distribution over the sample cross section is approximately parabolic, with the maximum temperature at the center (7). (Figure 17 gives a graphical presentation of this solution.) This leads to the conclusion that the average temperature is approximately 63.5% of the maximum temperature. 

A three-dimensional analog of the one-dimensional particle in a box is a particle in a rectangular parallelepiped with sides a, b, and c we have... 

Another useful quantity is the rate at which molecules hit a unit area of a wall (or of a hole). Placing the x axis perpendicular to the wall, (N/V)vx molecules with the x-component of velocity vx will hit a unit area of the wall in unit time (all of the molecules in the rectangular parallelepiped of height vx and unit cross-sectional area). This quantity has to be averaged over the distribution of vx for positive values of vx (only molecules moving in the positive direction will hit the wall) ... 

Mathematical analysis of the diffusion problem in this case for a rectangular parallelepiped adsorbent leads to an equation for the total moles of adsorptive that have entered the adsorbent pores by the elapsed time t (the time integral of the right side of Eq. 4.72) 39... 

Hosono, E., Fujihara, S., Kakiuchi, K. and Imai, H. (2004). Growth of submicrometer-scale rectangular parallelepiped rutile Ti02 films in aqueous TiCE solutions under hydrothermal conditions. J. Am. Chem. Soc. 126(25), 7790-7791. 

Isothermal rectangular parallelepiped buried in semi-infinite medium having isothermal surface... 

Product solutions for temperatures In multidimensional systems (a) semi-infinite plate (b) infinite rectangular bar (c) semi-infinite rectangular bar (cf) rectangular parallelepiped (e) semi-infinite cylinder (0 short cylinder. 

Example 10 Furnace Simulation via Zoning The furnace chamber depicted in Fig. 5-20 is heated by combustion gases passing through 20 vertical radiant tubes which are backed by refractory sidewalls. The tubes have an outside diameter of D = 5 in (12.7 cm) mounted on C = 12 in (4.72 cm) centers and a gray body emissivity of 0.8. The interior (radiant) portion of the furnace is a 6 x 8 x 10 ft rectangular parallelepiped with a total surface area of 376 ft2 (34.932 m2). A 50-fL (4.645-m2) sink is positioned centrally on the floor of the furnace. The tube and sink temperatures are measured with embedded thermocouples as 1500 and 1200°F, respectively. The gray refractory emissivity may be taken as 0.5. While all other refractories are assumed to be radiatively adia-... 

With homogeneous strain, the deformation is proportionately identical for each volume element of the body and for the body as a whole. Hence, the principal axes, to which the strain may be referred, remain mutually perpendicular during the deformation. Thus, a unit cube (with its edges parallel to the principal strain directions) in the unstrained body becomes a rectangular parallelepiped, or parallelogram, while a circle becomes an ellipse and a unit sphere becomes a triaxial ellipsoid. Homogeneous strain occurs in crystals subjected to small uniform temperature changes and in crystals subjected to hydrostatic pressure. 

Figures 22a, b provides a schematic representation of the pilot scale reactor. Essentially it is a rectangular parallelepiped limited by two parallel windows made of borosilicate glass and operated as a slurry reactor inside the loop of a batch recycling system. Irradiation of one of the reactor faces is obtained using two tubular lamps that were placed along the focal axis of two parabolic reflectors made of specularly finished aluminum (Brandi et al., 1996, 1999, 2002). The specific information concerning the experimental device is presented in Table 9, and more details can be found in Satuf et al. (2007b).

A. We will calculate the total surface area of the palisade mesophyll cells (rectangular parallelepipeds) and spongy mesophyll cells (spheres) and express it per unit area of the leaf surface ... 

We may think of a free-electron gas as having a vanishing potential (or equivalently, a constant potential, since wc can measure energies from that potential level). The Hamiltonian becomes simply -h V Ilm, and the solutions of the time-independent Schroedinger equation, Eq. (1-5), can be written as plane waves, e h Wc must apply suitable boundary conditions, and this is most conveniently done by imagining the crystal to be a rectangular parallelepiped, as shown in Fig. 15-1. Then wc apply periodic boundary conditions on the surface, as wc did following F.q. (2-2). The normalized plane-wave stales may be written as... 

A metal in the shape of a rectangular parallelepiped. Periodic boundary conditions are applied on the surface. The total volume is... 

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