Solution of a two-dimensional equation of nonstationary heat conduction for orthotropic cylindrically bounded media

Analytical relationships are presented to determine two-dimensional (2-D) nonstationary thermal fields in a bounded orthotropic cylinder for various boundary conditions. The solution obtained generalizes a wide class of eighty boundary-value problems of nonstationary heat conduction in studies of heat exchange in the bodies being considered.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 56, No. 6, pp. 1014–1021, June, 1989.     

Green's function approach to unsteady thermal stresses in an infinite hollow cylinder of functionally graded material

Summary A Green's function approach based on the laminate theory is adopted for solving the two-dimensional unsteady temperature field (r, z) and the associated thermal stresses in an infinite hollow circular cylinder made of a functionally graded material (FGM) with radial-directionally dependent properties. The unsteady heat conduction equation is formulated as an eigenvalue problem by making use of the eigenfunction expansion theory and the laminate theory. The eigenvalues and the corresponding eigenfunctions obtained by solving an eigenvalue problem for each layer constitute the Green's function solution for analyzing the unsteady temperature. The associated thermoelastic field is analyzed by making use of the thermoclastic displacement potential function and Michell's function. Numerical results are carried out and shown in figures.     

The reproducing kernel particle method for two-dimensional unsteady heat conduction problems

The numerical solution to two-dimensional unsteady heat conduction problem is obtained using the reproducing kernel particle method (RKPM). A variational method is employed to furnish the discrete equations, and the essential boundary conditions are enforced by the penalty method. Convergence analysis and error estimation are discussed. Compared with the numerical methods based on mesh, the RKPM needs only the scattered nodes instead of meshing the domain of the problem. The effectiveness of the RKPM for two-dimensional unsteady heat conduction problems is examined by two numerical examples.     

Approximate solution of some nonlinear problems of heat conduction theory

In this paper we obtain approximate solutions for the problem of unsteady heat conduction in a heat-radiating plate, a solid cylinder, a hollow cylinder, and a sphere, in which internal heat sources are present. The solution involves the assumption of a parabolic temperature profile.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 17, No. 5, pp. 913–917, November, 1969.     

A Two-Dimensional Analytical Solution for the Transient Short-Hot-Wire Method

Unlike the conventional transient hot-wire method for measuring thermal conductivity, the transient short-hot-wire method uses only one short thermal-conductivity cell. Until now, this method has depended on numerical solutions of the two-dimensional unsteady heat conduction equation to account for end effects. In order to provide an alternative and to confirm the validity of the numerical solutions, a two-dimensional analytical solution for unsteady-state heat conduction is derived using Laplace and finite Fourier transforms. An isothermal boundary condition is assumed for the end of the cell, where the hot wire connects to the supporting leads. The radial temperature gradient in the wire is neglected. A high-resolution finite-volume numerical solution is found to be in excellent agreement with the present analytical solution.     

Ritz solutions for axisymmetric temperature and stress fields in finite elastic cylinders

Variational formulations are given for the stationary heat conduction and thermodelasticity problems of an orthotropic, non-homogeneous, finite length, cylinder. It is demonstrated that a single-element approach based upon Ritz approximations and the method of Lagrange multipliers offers and efficient means for computing the temperature and thermal stress distributions.     

Exchange of heat in an orthotropic bounded cylinder under combined boundary conditions of the first,second and third kinds

We present the results from a study of a two-dimensional nonsteady temperature field and the heat flows in an orthotropic bounded cylinder. A method is proposed for the determination of the thermophysical properties and their ratios, as well as for the determination of the coefficient of heat exchange at the side surface.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 57, No. 6, pp. 965–973, December, 1989.     

Precision methods of nondestructive control of thermophysical properties

We examine original solutions for two-dimensional problems of nonsteady heat conduction in the case of an orthotropic half-space with discontinuous mixed boundary conditions; these are used to develop new methods of controlling thermophysical properties in a nondestructive manner.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 57, No. 6, pp. 999–1005, December, 1989.     

Unsteady axisymmetric stagnation-point flow of a viscous fluid on a cylinder

The unsteady viscous flow in the vicinity of an axisymmetric stagnation point of an infinite circular cylinder is investigated when both the free stream velocity and the velocity of the cylinder vary arbitrarily with time. The cylinder moves either in the same direction as that of the free stream or in the opposite direction. The flow is initially (t=0) steady and then at t>0 it becomes unsteady. The semi-similar solution of the unsteady Navier–Stokes equations has been obtained numerically using an implicit finite-difference scheme. Also the self-similar solution of the Navier–Stokes equations is obtained when the velocity of the cylinder and the free stream velocity vary inversely as a linear function of time. For small Reynolds number, a closed form solution is obtained. When the Reynolds number tends to infinity, the Navier–Stokes equations reduce to those of the two-dimensional stagnation-point flow. The shear stresses corresponding to stationary and the moving cylinder increase with the Reynolds number. The shear stresses increase with time for the accelerating flow but decrease with increasing time for the decelerating flow. For the decelerating case flow reversal occurs in the velocity profiles after a certain instant of time.     

Heat transfer of an orthotropic bounded cylinder under boundary conditions of the first and third kinds

Results are presented for investigating the two-dimensional nonstationary temperature field in an orthotropic bounded cylinder. A method is proposed for determining the ratio of the heat conductivity (thermal diffusivity) coefficients and the absolute values of the thermal diffusivity along the axes of a cylindrical coordinate system.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 57, No. 5, pp. 830–837, November, 1989.     

Method for determining the thermophysical characteristics of orthotropic bodies

The solution for the inverse coefficient problem of heat conduction for an orthotropic body is proposed.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 54, No. 4, pp. 596–600, April, 1988.     

An Accelerated Two-Dimensional Unsteady Heat Conduction Calculation Procedure for Thermal-Conductivity Measurement by the Transient Short-Hot-Wire Method

A fast and accurate procedure is proposed for solution of the two-dimensional unsteady heat conduction equation used in the transient short-hot-wire method for measuring thermal conductivity. Finite Fourier transforms are applied analytically in the wire-axis direction to produce a set of one-dimensional ordinary differential equations. After discretization by the finite-volume method in the radial direction, each one-dimensional algebraic equation is solved directly using the tri-diagonal matrix algorithm prior to application of the inverse Fourier transform. The numerical procedure is shown to be very accurate through comparison with an analytical solution, and it is found to be an order of magnitude faster than the usual numerical solution.     

Solution of the heat conduction problem for laminar orthotropic slabs in a spatial formulation

A solution is obtained for the stationary heat conduction problem in a spatial formulation for rectangular slabs with an arbitrary quantity of orthotropic layers.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 58, No. 5, pp. 859–862, May, 1990.     

Coupled heat conduction and deformation in a viscoelastic composite cylinder

We study the coupled problem of deformation due to mechanical and thermal loading of a composite cylinder made up of two layers of linear isotropic viscoelastic materials. The effect of a time-varying temperature field due to unsteady heat conduction on the short term and long term material response is examined in terms of the stress, displacement, and strain fields. The material properties of the two layers of the composite cylinder at any given location and time are assumed to depend on the temperature at that location at that given instant of time. Sequentially coupled analyses of heat conduction and deformation of the viscoelastic composite cylinder are carried out. Analytical solutions for the stress, strain and displacement fields of the viscoelastic composite cylinder are obtained from the corresponding solution of the linear elasticity problem by applying the Correspondence Principle. We examine the discontinuity in the hoop stress and the radial strain at the interface of the two layers caused by mismatches in material properties, during transient heat conduction. We find that the discontinuities change over time as the mismatch in the moduli of the two layers changes due to the material properties which are time-dependent. We also investigate the effect of the thermal field on the time-dependent field variables in the composite body.     

Effect of axial heat flux on the temperature field of a hollow cylinder

Solutions are given for two-dimensional problems of heat conduction in a hollow cylinder with quasi-steady heating (constant heating rate) for a boundary condition of the second kind at the inner surface of the cylinder, and boundary conditions of the third or first kind at the outer surface. The effect of axial heat flux on the temperature field of the hollow cylinder is estimated.     

Transient solution of temperature field in functionally graded hollow cylinder with finite length using multi layered approach

In this paper by using a multi-layered approach based on the theory of laminated composites, the solution of temperature in functionally graded circular hollow cylinders subjected to transient thermal boundary conditions are presented. The material properties are assumed to be temperature-independent and radially dependent. The cylinder has finite length and is subjected to axisymmetric thermal loads. It is assumed that the functionally graded circular hollow cylinder is composed of N fictitious layers and the properties of each layer are assumed to be homogeneous and isotropic. Employing Laplace transform techniques and series solving method for two-dimensional heat conduction equation in the cylinder, solutions for the variation of temperature with time as well as temperature distribution through the cylinder are obtained.     

Method of determining the thermophysical properties of orthotropic materials from the solution of a two-dimensional inverse heat-conduction problem

An unsteady two-dimensional inverse coefficient problem of heat conduction is formulated mathematically and solved.Translated from Inzhenefno-Fizicheskii Zhurnal, Vol. 56, No. 3, pp. 483–491, March, 1989.     

General boundary element method for non-linear heat transfer problems governed by hyperbolic heat conduction equation

The general Boundary Element Method (BEM) for strongly non-linear problems proposed by Liao (1995) is further applied to solve a two-dimensional unsteady non-linear heat transfer problem in the time domain, governed by the hyperbolic heat conduction equation (HHCE) with the temperature-dependent thermal conductivity coefficients which are different in the x and y directions. This paper confirms that the general BEM can be used to solve even those non-linear unsteady heat transfer problems whose governing equations do not contain any linear terms in spatial domain.     

Calculating the heating of a massive cylinder by the sweep method

Numerical solution of the heat conduction problem for a massive cylinder with allowance for heat transfer by radiation at the cylinder surface and for the furnace time lag is considered.     

Numerical Simulation of Heat Conduction to Liquids from a Thin Vertical Cylinder

The paper presents numerical simulations of heat conduction around a circular vertical cylinder immersed in liquids. A finite volume formulation is used, and the numerical analysis is performed in unsteady state with an explicit scheme. The numerical predictions are compared with experiments performed on liquids to find the temperature inside the cylinder, where a thermocouple is located, and at the wall of the insulated coaxial container, where the liquid is poured. The cylinder is immersed vertically. The numerical results are in good agreement with the temperature at the wall of the container. The experimental temperature measurement of the thermocouple located inside the probe is intermediate between the numerical temperatures on the axis and on the surface of the probe. The natural convection phenomenon is evidenced in the experiments, after a certain time from the beginning of heating, in some of the liquids used, except glycerol. Natural convection is not considered in the present numerical simulations, which solve only the heat conduction equation.