Finite cylinder calculations

Most calculations of stress in fuel pins are done on the assumption of plane strain with the radial direction as the only spatial variable. Different axial positions are then examined separately to obtain the variation of behavior in the z direction. This appears to be a sensible method of simplifying a system where the length to diameter ratio is typically of the order of 1 200 or more. It is important, however, to check the validity of such assumptions. In the first part of this section the possibility of axial extrusion is explored. Then finite cylinder calculations are examined and the results applied to fuel pellet shape, to cracking induced by thermal stresses, and to swelling. 

In the present study the particle effectiveness factor was estimated with the Aris method presented by Ref. 18. In the estimation, the particles were modeled as normal cylindrical catalyst particles, i.e., finite cylinders with no inert kernel. This was necessary because approximative methods to estimate an effectiveness factor cannot account for an inert kernel in a catalyst particle such as the metal wire in the axial hole of the BSR beads. The cylinder diameter used in the calculations was defined in Fig. 18 (the... 

An alternative approach based on a mass spring model (MSM) has been proposed by Williams [ 1 ]. The test system is represented by a lumped mass model with the contact stifBiess between the striker and specimen being ki, this acts on an equivalent mass m which is 17/35 of the specimen mass, and the specimen stiffness is k2- A vital factor is the contact stiffness ki, which controls the dynamics of the system. In reality, this factor is not linear but for a case of the contact of a finite cylinder on plane used here, it can be approximated as linear. From this it is possible to calculate the natural frequency of the system o) where... 

The ability to model can be seen in Fig. 6.6.1 where comparisons are given between finite element calculations and experimental results for the electro-osmotic purging of a 100 mol m acetic acid solution initially saturating a kaolin clay sample. The distributions shown are after 0.12 of a pore volume of liquid is removed. The purge is a 100 mol m NaCl solution and the sample is compacted in an acrylic cylinder 0.5 m in length and 0.1 m in diameter with... 

Other Shapes. For other shapes of particles and finite cylinders, mathematical difficulties arise due to the presence of the edges and comers [302]. The surface modes in small cubes were calculated by Fuchs [303] and Napper [304]. 

The stress contours for the four components of stress are shown in Fig. 4 for a section through a solid pellet of length to diameter ratio 1. The most important feature is the existence of the largest tensile stress at the pellet rim (Fig. 4b). Both Valentin (22) and Matthews (24) predict this from their calculations. Matthews has found, however, that for a Poisson s ratio of 0.3 in both solid (24) and hollow (26) finite cylinders (with length to diameter ratios exceeding 0.6) the hoop stress at the pellet rim is... 

A fluid with a finite yield. stress is sheared between two concentric cylinders, 50 mm long. The inner cylinder is 30 mm diameter and the gap is 20 mm. The outer cylinder is held stationary while a torque is applied to the inner. The moment required just to produce motion was 0.01 N m. Calculate the force needed to ensure all the fluid is flowing under shear if the plastic viscosity is 0.1 Ns/ni2. 

Several boundary conditions have been used to prescribe the outer limit of an individual rhizosphere, (/ = / /,). For low root densities, it has been assumed that each rhizosphere extends over an infinite volume of. soil in the model //, is. set sufficiently large that the soil concentration at r, is never altered by the activity in the rhizosphere. The majority of models assume that the outer limit is approximated by a fixed value that is calculated as a function of the maximum root density found in the simulation, under the assumption that the roots are uniformly distributed in the soil volume. Each root can then extract nutrients only from this finite. soil cylinder. Hoffland (31) recognized that the outer limit would vary as more roots were formed within the simulated soil volume and periodically recalculated / /, from the current root density. This recalculation thus resulted in existing roots having a reduced //,. New roots were assumed to be formed in soil with an initial solute concentration equal to the average concentration present in the cylindrical shells stripped away from the existing roots. The effective boundary equation for all such assumptions is the same ... 

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)... 

Unfortunately, Maxwell s equations can be solved analytically for only a few simple canonical resonator structures, such as spheres (Stratton, 1997) and infinitely long cylinders of circular cross-sections (Jones, 1964). For arbitrary-shape microresonators, numerical solution is required, even in the 2-D formulation. Most 2-D methods and algorithms for the simulation of microresonator properties rely on the Effective Index (El) method to account for the planar microresonator finite thickness (Chin, 1994). The El method enables reducing the original 3-D problem to a pair of 2-D problems for transverse-electric and transverse-magnetic polarized modes and perform numerical calculations in the plane of the resonator. Here, the effective... 

In a manner similar to that in Section 3.4, where we considered cross sections of finite particles, we can calculate cross sections per unit length of an infinite cylinder by constructing an imaginary closed concentric surface A of length L and radius R (Fig. 8.6). The rate Wa at which energy is absorbed within this surface is... 

The scaled particle approach is exact at the limit of infinite dilution and makes it possible to formulate static solution properties at finite dilutions in an approximate way. In this approach, we first calculate Sex for one hypothetical scaled particle with a size smaller or larger than the real particle and then find Sex of the real size by interpolation. For the wormlike spherocylinder, the scaled particle is assumed to have a cylinder length kL and a hard-core diameter Kd where X and k are scaling factors. The persistence length q of the scaled particle may be chosen rather arbitrarily. Here we do not scale q of the scaled particle but take it to be the same as the real q [17, 18],... 

The diffusion coefficients at infinite dilution (D]0, D 0, and Dr0) for the fuzzy cylinder reduce to those for the wormlike cylinder, which can be calculated as explained in Appendix B. On the other hand, these diffusion coefficients, D, Dx, and Dr, for the fuzzy cylinder at finite concentrations can be formulated by use of the mean-field Green function method and the hole theory, as detailed below. 

Unfortunately, the air flow in the vicinity of these microscopic sensory hairs is extremely difficult to calculate. Cheer and Koehl (1987b) provide a solution for the flow field in the vicinity of two parallel and infinitely long cylinders. Even for this simple geometry, the solution (expressed as a stream function) has enough terms that it takes up most of a printed journal page, and the reader must differentiate the provided stream function with respect to the spatial variables in order to solve for the velocities at different points in space. Finite hairs usually experience less flow between them than predicted assuming infinite length because fluid can go around the tips as well as the sides of an array (Koehl, 2001). 

Two finite-length concentric cylinders are placed in a large room maintained at 20°C. The inner cylinder has a diameter of 5.0 cm and the outer cylinder has a diameter of 10 cm. The length of the cylinders is 10 cm. The inner cylinder is newly turned cast iron and maintained at a temperature of 400°C. The outer cylinder is Monel metal oxidized at 1110°F. Calculate the heat lost by the inner cylinder. 

The temperature distribution in a cylinder of finite length is given by (2.192). In order to explicitly calculate the cooling time tk, we will limit ourselves to the first term in each infinite series and then later we will check whether this simplification was correct. We therefore put... 

X-ray residual stress determination was performed on the surface of the samples prepared by HIP sintering. The measured residual stress was compared with the results calculated by the finite element method (FEM). The electrical resistivity was measured by the four probes method on the slices cut from the cylinder samples. In order to inspect the thermal stability, the samples were annealed at 900 °C for 24 hour in vacuum. The microstructure on the section was observed by scanning electron microscope. 

Solvation energies for other multipoles inside a spherical cavity, including corrections due to salt effects, can be found, for example in Ref. 29. Analytical solutions of the Poisson equation for some other cavities, such as ellipse or cylinder, are also known [2] but are of little use in solvation calculations of biomolecules. For cavities of general shape only numerical solution of the Poisson and Poisson-Boltzmann equations is possible. There are two well-established approaches to the numerical solution of these equations the finite difference and the finite element methods. 

The situation for tumors is more complex. It is known that a tumor may infiltrate the surrounding tissue in a complex geometrical fashion. However, for simple geometries (i.e., cylinder, sphere), it is possible to obtain analytical solutions. For more complex geometries, finite difference or finite element methods are necessary to calculate the temperature field in the tumor and the surrounding normal tissues. 

Finite Square Cylinder. The dimensionless shape factors for finite square cylinders of length L and side dimension W (Fig. 3.4d) can be calculated using the finite circular expression by means of the equivalent aspect ratio... 

Therefore, determination of the magnetic field on the borehole axis when the formation has a finite thickness, consists of calculation of the field in a horizontally layered medium and geometric factors of vertical cylinders with a finite height which are coaxial to the borehole. 

Thus, calculation of the field on the borehole axis consists of determination of the field in the horizontally layered medium when the borehole and the invasion zone are absent and calculation of geometric factors of vertical cylinders of a finite height the axis of which coincides with the borehole one (Table 6.1). 

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