Thin-shell approximation

The film-blowing process is used industrially to manufacture plastic films that are biaxially oriented. Many attempts have been made to predict and model this complex but important process, which continues to mystify rheologists and polymer processing engineers worldwide. A constitutive equation, able to predict well the polymer melt in all forms of deformation, is required to model the process, together with the standard conservation equations of continuity, momentum, and energy. Pearson and Petrie [125,126] were the first to predict the forces within the blown film by the use of the thin-shell approximation, force balances, and the Newtonian constitutive equation. The use of the thin-shell approximation and force balances is standard in any attempt to model the film-blowing process, and it has been used in the vast majority of subsequent studies. 

There have been numerous studies on the film-blowing process. Since the initial thin-shell approximation proposed by Pearson and Petrie [125, 126] with the Newtonian model assumed for deformation, various rheological models have been incorporated in simulations, such as the power-law model [127,128], a crystallization model [129], the Maxwell model [130-133], the Leonov model [133], a viscoplasti-c-elastic model [134], the K-BKZ/PSM model [135-137], and a nonisothermal viscosity model [138]. A complete set of experimental data was reported by Gupta [139] for the Styron 666 polystyrene and by Tas [140] for three different grades of LDPE. 

M. Tobias, A Thin-shell Approximation for Two-group, Two-region Spherical Reactor Calculations, USAEC Report CF-54-6-135, Oak Ridge National Laboratory, June 1954. 

For commodities that soHdify at temperatures commonly encountered during shipping, tank cars are equipped with internal or external heating coils. In some cases, cars are insulated with both sides of the insulation protected by thin steel shells. Approximately 15% of the tank cars in the United States are constmcted for the transportation of pressuri2ed commodities, such as anhydrous ammonia and propane. 

Supporting evidence was given (7,17) for restricting the count to scales sufficiently separated from neighboring scales to respond as independent individuals to the applied oil film. However, oil entered and was retained by scales too near each other for them to be used for mortality counts. If the number and density of these scales were variable, some influence may have been exerted on the relation of the oil film to the individuals coimted. To enhance e independence of each scale coimted, fertile female scales were hand-thinned to approximately 0.5 cm. apart. The adult male insect emerges from the shell covering at an age of about 21 days only the cast shells of the males remain on the fruit at 36 days. These are loose and may be removed easily with an air stream. Any unfertile females are removed at the same time. 

Following the principles of the Petrie model, and recalling that the film thickness <5 is much smaller than the radius S/R thin-film approximation, which implies that field equations are averaged over the thickness and that there are no shear stresses and moments in the film. The film is regarded, in fact, as a thin shell in tension, which is supported by the longitudinal force Fz in the bubble and by the pressure difference between the inner and outer surfaces, AP. We further assume steady state, a clearly defined sharp freeze line above which no more deformation takes place and an axisymmetric bubble. Bubble properties can therefore be expressed in terms of a single independent spatial variable, the (upward) axial position from the die exit,2 z. The object... 

Hence, it is not possible to redefine the characteristic length such that the critical value of the intrapellet Damkohler number is the same for all catalyst geometries when the kinetics can be described by a zeroth-order rate law. However, if the characteristic length scale is chosen to be V cataiyst/ extemai, then the effectiveness factor is approximately A for any catalyst shape and rate law in the diffusion-limited regime (A oo). This claim is based on the fact that reactants don t penetrate very deeply into the catalytic pores at large intrapellet Damkohler numbers and the mass transfer/chemical reaction problem is well described by a boundary layer solution in a very thin region near the external surface. Curvature is not important when reactants exist only in a thin shell near T] = I, and consequently, a locally flat description of the problem is appropriate for any geometry. These comments apply equally well to other types of kinetic rate laws. 

The phototubes are approximated as a smooth spherical shell of radioactive material 2.5 m from the fiducial volume. Contributions from surrounding rock are built up by summing the thin shells out to 1 m. To obtain a rate, a differential flux through a patch on the fiducial surface is obtained by numerically integrating the quantity ... 

At pH 5.2, hollow spheres are formed with a uniform size distribution and an average diameter of approximately 1 fj,m. These silica spheres are thermally very stable and hollow (Fig. 9.17), possessing a relatively thin shell with a very regular thickness of 80-130 nm composed of 7-8 nm thick layers of silica with an interlamellar spacing of 3-4 nml l. ... 

J.L. Sanders, An improved first-approximation theory for thin shells. Technical Report NASA TR R-24, National Aeronautics and Space Administration, Langley Research Center (1959)... 

In optics, one frequently wants to know the deflections of an axially loaded plate. Mirrors are often good approximations to thin plates and it is important to understand the deflections of mirrors when loaded by gravity, wind, or other point loads. Fortunately this is a relatively well studied subject, so many solutions exist in the literature. A particularly valuable general reference book is Theory of Plates and Shells, by Timoshenko and Woinowsky-Krieger. 

Rutherford based his model on a refinement of von Lenard s electron scattering experiment carried out by Geiger and Marsden in 1909. They used u-particles, which were known to be much heavier than electrons (more than 7000 times heavier), instead of electrons as the shells . Using a thin gold foil, they observed that almost all the u-particles went through the foil undeflected, but approximately 1 in 20 000 was reflected back towards the radioactive source. Rutherford, in describing this experiment, is widely quoted as saying It was almost as if you fired a 15 inch shell at a piece of tissue paper and it came back and hit you, but the source of this quote is obscure. 

Diffraction patterns from thin polycrystalline Ge films were measured by the eleetron diffraetometer. After refinement of scale and thermal factors and corrections for the primary extinetion within the two-beam approximation, the parameters k (spherieal deeompression of valence eleetron shell) and multipoles P32- and P40 (anisotropy of electron density) were ealeulated (Table 4). The residual faetor R ealeulated from the experimental and theoretical amplitudes (the latter were ealeulated by the LAPW method, Lu Z.W., et al. Phys.Rev. 1993, B47, 9385) is 2.07% and proofs the high quality of the experimental. 

The common mature coconut fruit weighs more that 1 kg and is ovoid in shape and green or yellow in color. The nut has a smooth epidermis over a fibrous mesocarp (husk) that covers the hard endocarp (shell). Within the shell is the endosperm (kernel, meat) approximately 1-2 cm thick. A thin brown layer called testa separates the kernel from the inner surface of the shell. The cavity within the kernel has an average volume of 300 mL and contains the endosperm liquid (coconut water) (see Figure 2). 

The emulsion fed to the HOE can be preheated to approximately 125 (257 F) by a shell-and-tube heat exchanger (HT-4). The hot emulsion is then fed into the evaporator vessel (V-3), through a spreader, onto a wide, shallow tray. The spreader ensures that a thin uniform coat of oil is deposited onto the tray, which is sloped downward and is heated from below by steam coils. As the emulsion runs down the tray, water, light ends, and any remaining diluent are evaporated. This process is repeated consecutively on two additional trays. 

An AGB star is a red giant with a C-O electron degenerate core surrounded by a thin He shell outside of which is the extensive H-rich convective envelope. The C-O core with the He shell is approximately the size of the Earth but contains several tenths of a solar mass of material. This core is a putative white dwarf. The radius of the AGB star is approximately the size of the Earth s orbit around the Sun. Stellar evolution in the nuclear sense is driven by H-burning in a shell just below the convective envelope and by He-burning in the thin He shell. Evolution is also driven by mass loss off the surface. At the highest luminosities, mass loss controls the evolution of the AGB star. 

The VSEPR theory allows chemists to successfully predict the approximate shapes of molecules it does not, however, say why bonds exist. The quantum mechanical valence bond theory, with its overlap of atomic orbitals, overcomes this difficulty. The resulting hybrid orbitals predict the geometries of molecules. A quantum mechanical graph of radial electron density (the fraction of electron distribution found in each successive thin spherical shell from the nucleus out) versus the distance from the nucleus shows maxima at certain distances from the nucleus—distances at which there are higher probabilities of finding electrons. These maxima correspond to Lewis s idea of shells of electrons. 

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