Transient thermal stresses and displacements in a transversely isotropic semi-infinite circular cylinder subjected to an arbitrary surface heat generation and a convective heat loss

The transient thermal stresses and displacements in a traction-free, transversely isotropic, semi-infinite circular cylinder subjected to a convective heat loss on the end surface is analyzed by means of a set of stress functions and a generalized Fourier-integral representation for exponential function.The purpose of this paper is to present the results of theoretical analysis which considers the effects of the thermal and elastic anisotropes of the material properties on the thermal stresses and displacements in the transversely isotropic, semi-infinite circular cylinder due to an arbitrary cylindrical surface heat generation.A detailed numerical result of the thermal stresses and displacements is given for various values of the ratios of the thermal conductivity coefficients, Young's moduli and linear thermal expansion coefficients in the axial and radial directions and it is demonstrated that the sensivity of the maximum stress and displacement to the anisotropy of the thermal conductivity coefficient is significant.     

Transient thermal stresses in an orthotropic finite rectangular plate due to arbitrary surface heat-generations

The transient thermal stresses in an orthotropic finite rectangular plate due to arbitrary surface heat-generations on two edges are studied by means of the Airy stress function.The purposes of this paper are to present a method of determining the transient thermal stresses in an orthotropic rectangular plate with four edges of distinct thermal boundary condition of the third kind which exactly satisfy the traction-free conditions of shear stress over all boundaries including four corners of the plate, and to consider the effects of the anisotropies of material properties and the convective heat transfer on the upper and lower surfaces on the thermal stress distribution.     

Three-dimensional transient thermal stresses of reactor graphite subjected to internal heat generation and asymmetric surface heating

The present paper is concerned with three-dimensional transient thermal stresses of graphite in a nuclear reactor. In analyzing this problem, reactor graphite may be approximated by a transversely isotropic finite circular cylinder subjected to internal heat generation and asymmetric heating on an end surface. Thermal stresses are analyzed by means of the transversely isotropic potential functions method proposed by Takeuti and Noda. Numerical calculations were carried out for a special type of heating conditions, and time variations of temperature and thermal stresses of graphite are shown in figures.     

Unsteady thermal stresses in fuel elements under heat generation

In designing some types of fuel elements, it is necessary to find the thermal stresses in a continuous medium performated by cylindrical channels of circular cross-section. In order to set up the problem, assume that the strength and heat transfer calculation are based on the phenomena that occur in a typical unit lattice cell, and that the heat generation is uniform over the whole section: Therefore, the problems reduce to solve the temperature and thermal stresees in a polygonal prism with a central circular hole under uniform heat generation and for insulated outer boundary. In out previous papers we obtained the steady thermal stress distribution in such formed regions by use of the five elementary functions method.

In starting up or shutting down of the reactor, however, the problem becomes one of the unsteady state. It is felt that no analysis has been given to the transient thermoelastic problem of the multiply-connected region. The problems considered here are concerned with unsteady thermal stresses in a polygonal prism with a circular hole under a uniform heat generation. The solving process is divided into two basic steps. First, the heat conduction equation is solved by the technique of Laplace transform. Secondly, the plane thermoelastic problem is solved by use of the five elementary functions method. Throughout the treatment of both problems, in order to satisfy the boundary conditions, we use the point-matching technique in conjunction with the least square method.     

Void fraction distributions of inert gas jets across a single cylinder with non-wetting surface in liquid sodium

Little work on the void fraction behaviors along structural materials with poor-wettability for liquid metals has been performed. In the present study, void fraction behaviors around a single cylinder with non-wetting surface condition were quantitatively discussed by using a gas jet–cylinder system where the impinging jet flow, the boundary layer flow, the separation flow, and the wake flow appear. One cylinder with a non-wetting surface and two cylinders with a wetting surface were used to vary the wettability for liquid sodium, and void fraction distributions were measured around the cylinders. In the case of wetting condition, void fraction distributions around the cylinder decrease clearly in the backward region of the cylinder, and liquid-rich region is formed due to bubble separation from the cylinder surface. On the other hand, under non-wetting condition, because of two-phase flow without bubble separation on the cylinder surface, void fraction distributions show almost steady values around the cylinder compared to those with wetting surface. The void behaviors on a non-wetting surface were also confirmed by a visualization experiment conducted in water. The observed differences can be basically attributed to the work of adhesion required for liquid–solid interfacial separation.     

Transient thermal stresses and displacements in a transversely isotropic, heat-generating circular cylinder

The transient thermal stress field in a traction-free, transversely isotropic, heat-generating circular cylinder is studied by means of a set of stress functions developed by Singh. The purpose of this paper is to present the results of theoretical analysis which consider the effects of the thermo-elastic anisotropies of the material on the thermal stress field in the transversely isotropic, circular cylinder due to an abrupt axial change in the internal heat-generation rate. A detailed qualitative picture of the thermal stress state for two values of the Biot number and various values of the ratio of the thermal conductivity coefficients in the axial and radial directions is given and it is demonstrated that the magnitude of the maximum thermal stress is significantly sensitive to the anisotropies of the thermal conductivity coefficient and the Young's modulus.     

Length effects of unsteady thermal stresses in a finite short circular cylinder with an arbitrary heat supply

An unsteady thermoelastic study, taking consideration of the length effects, was made of a finite short circular cylinder under an arbitrary heat supply along its cylindrical surface in the longitudinal direction. The analysis was treated by the thermoelastic potential method using a general form of the Love's displacement function which has not so far been used. The results obtained were compared with those of an approximate solution for the finite cylinder using Saint-Venant's principle. From the results the effects of the length-to-diameter ratio evidently appear in the transient thermal stress distributions for the finite short cylinder.     

Asymmetrical transient thermoelastic problems in a composite hollow circular cylinder

The present paper is concerned with the transient thermal stresses in a bonded composite hollow circular cylinder under an arbitrary asymmetrical heat supply. Numerical work is carried out using Laplace transforms and is given for the composite cylinder made of the different materials under three, band heat sources on its outer surface.     

Natural convection heat transfer from a horizontal cylinder in liquid sodium: Part 2: generalized correlation for laminar natural convection heat transfer

Rigorous numerical solution of natural convection heat transfer, from a horizontal cylinder with uniform surface heat flux or with uniform surface temperature, to liquid sodium was derived by solving the fundamental equations for laminar natural convection heat transfer without the boundary layer approximation. It was made clear that the local and average Nusselt numbers experimentally obtained and reported in part 1 of this paper were described well by the numerical solutions for uniform surface heat fluxes, but that those for uniform surface temperatures could not describe the angular distribution of the local Nusselt numbers and about 10% underpredicted the average Nusselt numbers. Generalized correlation for natural convection heat transfer from a horizontal cylinder with a uniform surface heat flux in liquid metals was presented based on the rigorous theoretical values for a wide range of Rayleigh numbers. It was confirmed that the correlation can describe the authors’ and other workers’ experimental data on horizontal cylinders in various kinds of liquid metals for a wide range of Rayleigh numbers. Another correlation for a horizontal cylinder with a uniform surface temperature in liquid metals, which may be applicable for special cases such as natural convection heat transfer in a sodium-to-sodium heat exchanger etc. was also presented based on the rigorous theoretical values for a wide range of Rayleigh numbers. These correlations can also describe the rigorous numerical solutions for non-metallic liquids and gases for the Prandtl numbers up to 10.     

Finite integral transform method to solve asymmetric heat conduction in a multilayer annulus with time-dependent boundary conditions

The separation of variables (SOV) method has recently been applied to solve time-dependent heat conduction problem in a multilayer annulus. It is observed that the transverse (radial) eigenvalues for the solution in polar (r-θ) coordinate system are always real numbers (unlike in the case of similar multidimensional Cartesian problems where they may be imaginary) allowing one to obtain the solution analytically. However, the SOV method cannot be applied when the boundary conditions and/or the source terms are time-dependent, for example, in a nuclear fuel rod subjected to time-dependent boundaries or heat sources. In this paper, we present an alternative approach using the finite integral transform method to solve the asymmetric heat conduction problem in a multilayer annulus with time-dependent boundary conditions and/or heat sources. An eigenfunction expansion approach satisfying periodic boundary condition in the angular direction is used. After integral transformation and subsequent weighted summation over the radial layers, partial derivative with respect to r-variable is eliminated and, first order ordinary differential equations (ODEs) are formed for the transformed temperatures. Solutions of ODEs are then inverted to obtain the temperature distribution in each layer. Since the proposed solution requires the same eigenfunctions as those in the similar problem with time-independent sources and/or boundary conditions—a problem solved using the SOV method—it is also “free” from imaginary eigenvalues.     

Transient thermal stresses in a bonded composite hollow circular cylinder under symmetrical temperature distribution

A general treatise for the transient thermal stress problem of a composite circular cylinder made by bonding is described under a symmetrical temperature distribution. Analysis is developed by stress function method by the aid of Laplace transforms. Numerical examples are illustrated for the hollow composite cylinder with several radius-ratios made by the different materials due to transient heating at the internal surface by liquid.     

Thermal stress in a bimodulus thick cylinder

Certain materials are known to behave differently under tension and compression. This reports an investigation of a simple thermal stress problem based on the bimodulus elasticity in the framework of the classical-type uncoupled thermoelasticity. Because of an unavoidable complexity of three-dimensional analysis in bimodulus elasticity, as a fundamental illustration we consider herein an axisymmetric plane strain state of a thick orthotropic circular cylinder under axisymmetric heat conditions. A system of transcendental simultaneous algebraic equations with respect to eight unknowns are derived. Numerical analyses are carried out for a steady-state temperature field and for a quasi-stationary state with internal heat generation. Results of calculations show that the states of a cylinder are affected significantly by the difference between tensile and compressive moduli of elasticity.     

Transient thermal stresses in a composite hollow cylinder subjected to γ-ray heating

Transient temperature and thermal-stress distributions arising in an infinite long composite hollow circular cylinder subjected to internal heat generation decaying exponentially along the wall thickness resulting from γ-ray radiation are analyzed, under the thermal conditions of cooling by convection on the inner surface and insulation on the outer surface. Numerical calculations of the transient temperature and thermal-stress distributions are carried out for the case of a composite cylinder of stainless steel and carbon steel. The influence of the absorption coefficient, the Biot number and the splicing radius on the results are considered.     

Dynamic response of a clamped/free hollow circular cylinder under travelling torsional impact loads

Impact-induced vibrations in the casing of a gas centrifuge due to a sudden failure of the spinning rotor (crash) can cause structural disintegrity of the casing. In order to study the influence of the rotor failure behaviour and the impact load histories on the dynamic response of the casing, a simple crash model is proposed in this paper to analyse the transient torsional response due to tangential components of the impact loads. The casing is modeled as a linear-elastic hollow circular cylinder, clamped at the lower end and free at the upper end. The rotor is thought to break up in identical sections in a sequence determined by its fracture behaviour. Each section is assumed to cause an axi-symmetric load distribution at the inner surface of the casing. Therefore the problem is essentially reduced to the analysis of a clamped/free cylinder under travelling torsional impact loads. The problem is solved by representing the impact loads as local pulses acting over the length of the sections. A perturbation method is used to show that the general two-dimensional theory of axi-symmetric torsional wave propagation in circular cylinders, for the problem under consideration, may be approximated by the elementary one-dimensional theory. Solutions are obtained according to the usual modal expansion approach. Measurements of transient torsional responses are shown to be in good agreement with the calculated responses by choosing a suitable shape of the pulses. The effects of travelling velocity and pulse shape are investigated. Finally the transfer of kinetic energy in the rotor to vibrational energy of torsion in the casing is studied.     

Group theory transformation for Soret and Dufour effects on free convective heat and mass transfer with thermophoresis and chemical reaction over a porous stretching surface in the presence of heat source/sink

The group theoretic method is applied for solving the problem of combined effect of thermal diffusion and diffusion thermo on free convective heat and mass transfer over a porous stretching surface in the presence of thermophoresis particle deposition with variable stream conditions. The application of one-parameter groups reduces the number of independent variables by one and consequently, the system of governing partial differential equations with the boundary conditions reduces to a system of ordinary differential equations with appropriate boundary conditions. The equations along with the boundary conditions are solved numerically by using Runge Kutta Gill integration scheme with shooting technique. Impact of Soret and Dufour effects in the presence of thermophoresis particle deposition with chemical reaction plays an important role on the flow field. The results thus obtained are presented graphically and discussed.     

Effects of slip on unsteady mixed convective flow and heat transfer past a porous stretching surface

This paper investigates the unsteady mixed convective boundary layer flow and heat transfer over a porous stretching vertical surface in presence of slip. Similarity solutions for the transformed governing equations are obtained and the reduced equations are then solved numerically. With increasing values of the unsteadiness parameter, fluid velocity and the temperature are found to decrease in both the presence and absence of slip at the boundary. Fluid velocity decreases due to increasing values of the velocity slip parameter resulting an increase in the temperature field. Skin-friction decreases with the velocity slip parameter whereas it increases with unsteadiness parameter. The rate of heat transfer decreases with the velocity slip parameter while increases with unsteadiness parameter. Same feature is also noticed for thermal slip parameter. Effects of increasing mixed convection parameter on the velocity boundary layer is to increase the velocity field and the temperature decreases in this case.     

Stress analysis for end effects in cylindrical rods by the lanczos-chebyshev method

The Lanczos-Chebyshev method is used to reduce the two-dimensional displacement equilibrium equations in cylindrical co-ordinates, to a set of coupled ordinary differential equations. The eigenvalues and eigenvectors defining the axial decay of self-equilibrating stress systems in a solid cylinder are obtained, and used to solve a particular problem of specified self-equilibrating end stresses applied to a semi-infinite cylinder, for which an accurate reference solution is known. The method is compared with a Galerkin and a finite element method of achieving the same result, and is shown to have definite advantages in simplicity, speed and accuracy, over the two approximation methods.     

Transient natural convection cooling of a vertical circular cylinder

A numerical solution is presented for transient natural convection cooling of a finite vertical circular cylinder standing on a semi-infinite horizontal base. The method used is the quasi-finite element approach introduced by Mansour et al. in the determination of the steady state temperature field in metal cutting. The method is able to solve heat transfer problems in a more economical way (memory and computing time requirements) than the classical finite element method and the finite difference techniques.     

Heat transfer analysis for unsteady MHD flow past a non-isothermal stretching surface

An analysis is made for the unsteady two-dimensional magneto-hydrodynamic flow of an incompressible viscous and electrically conducting fluid over a stretching surface having a variable and general form of surface temperature which removes the restrictions of the particular forms of prescribed surface temperature. Similarity solutions for the transformed governing equations are obtained. The transformed boundary layer equations are solved numerically for some values of the involved parameters, namely the unsteadiness parameter, magnetic parameter, the temperature exponent parameters. The features of the flow and heat transfer characteristics for different values of the governing parameters are analysed and discussed. It is found that the fluid velocity and temperature decrease for increasing unsteadiness parameter. Fluid velocity decreases with the increasing values of the Hartman number resulting an increase in the temperature field in steady as well in unsteady case. It is observed that the variation of the sheet temperature in respect of distance and time has analogous effects both on the free surface temperature and on the heat transfer rate (Nusselt number) at the sheet.     

Thermal stresses in a cylindrical shell containing a circular hole or a rigid inclusion

The thermal stress problem of a circular discontinuity in a cylindrical shell has been solved by continuum approach. Two types of discontinuities are considered: (i) a circular hole and (ii) a circular regid inclusion. The effect of a uniform temperature or a linearly varying temperature across the thickness has been studied. The problem is converted into an equivalent boundary value problem and boundary conditions are specified around the discontinuity. The results are presented in a graphical form for ready use.