Analytical Solution of Heat Conduction for Hollow Cylinders with Time-Dependent Boundary Condition and Time-Dependent Heat Transfer Coefficient

An analytical solution for the heat transfer in hollow cylinders with time-dependent boundary condition and time-dependent heat transfer coefficient at different surfaces is developed for the first time. The methodology is an extension of the shifting function method. By dividing the Biot function into a constant plus a function and introducing two specially chosen shifting functions, the system is transformed into a partial differential equation with homogenous boundary conditions only. The transformed system is thus solved by series expansion theorem. Limiting cases of the solution are studied and numerical results are compared with those in the literature. The convergence rate of the present solution is fast and the analytical solution is simple and accurate. Also, the influence of physical parameters on the temperature distribution of a hollow cylinder along the radial direction is investigated

Direct and inverse heat conduction in rock materials using the finite element method

A finite element method is presented for solution of direct and inverse heat conduction problems in rock materials. Finite element programs for direct and inverse heat conduction problems were developed and demonstrated. Flow charts are given for the finite element programs. The direct finite element heat conduction program changes material properties with temperature to approximate the temperature dependent properties of rock materials. An example problem was solved by the finite element method and also using Kirchoff\u27s transformation. The finite element solution compared quite well with the Kirchoff transformation solution. The inverse method utilizes the direct finite element heat conduction program to iterate for the inverse solution. Data to test the inverse finite element program was generated with the direct finite element program of this research. The inverse program solved these generated problems very accurately. Internal temperature and fracture time measurements are presented on a series of hollow Dresser Basalt cylinders heated by a Kanthal wire heater. The internal temperature measurements were used in conjunction with the inverse method to obtain surface temperatures. The surface temperatures were used for thermal stress calculations to obtain time to fracture for an infinite cylinder --Abstract, page ii

Heat flux evaluation in high temperature ring-on-ring contacts

A comprehensive methodology to investigate heat flux in a ring-on-ring tribometer is presented. Thermal fluxes under high contact pressures and temperature differences were evaluated through an experimental campaign and by a numerical procedure of inverse analysis applied to surface temperature measurements. An approximation of a two-dimensional time-dependent analytical solution for the temperature distribution was first developed and subsequently adapted to mimic the specific testing configuration characteristics; the problem was finally simplified to enable further inverse analysis. Experiments were performed using an innovative high temperature ring-on-ring tribometer. The evaluated contact heat transfer rates were reported as a function of normal load and temperature difference between the discs under steady-state conditions; the results reported here show that, in the present test configuration, the temperature difference has stronger influence than the applied load in terms of heat transfer induced by contact

Simultaneous estimation of heat flux and heat transfer coefficient in irregular geometries made of functionally graded materials

A numerical inverse analysis based on explicit sensitivity coefficients is developed for the simultaneous estimation of heat flux and heat transfer coefficient imposed on different parts of boundary of a general irregular heat conducting body made of functionally graded materials with spatially varying thermal conductivity. It is assumed that the thermal conductivity varies exponentially with position in the body. The body considered in this study is an eccentric hollow cylinder. The heat flux is applied on the cylinder inner surface and the heat is dissipated to the surroundings through the outer surface. The numerical method used in this study consists of three steps: 1) to apply a boundary-fitted grid generation (elliptic) method to generate grid over eccentric hollow cylinder (an irregular shape) and then solve for the steady-state heat conduction equation with variable thermal conductivity to compute the temperature values in the cylinder, 2) to propose a new explicit sensitivity analysis scheme used in inverse analysis, and 3) to apply a gradient-based optimization method (in this study, conjugate gradient method) to minimize the mismatch between the computed temperature on the outer surface of the cylinder and simulated measured temperature distribution. The inverse analysis presented here is not involved with an adjoint equation and all the sensitivity coefficients can be computed in only one direct solution, without the need for the solution of the adjoint equation. The accuracy, efficiency, and robustness of the developed inverse analysis are demonstrated through presenting a test case with different initial guesses

Exact thermoelastic analysis of a thick cylindrical functionally graded material shell under unsteady heating using first order shear deformation theory

In this article, a new analytical formulation is presented for axisymmetric thick-walled FGM cylinder with power-law variation in mechanical and thermal properties under transient heating using first order shear deformation theory. Equilibrium equations are derived by virtual work principles and energy method. The unsteady heat conduction equation is solved using the method of separation of variables, generalized Bessel functions and an Eigen-function method. Validation of the analytical solutions is conducted with a finite element method (FEM). The effects of time on stress and displacement distribution are studied in detail. The numerical values used in this study are selected based on earlier studies. The influence of effect of transient heat transfer on heterogeneous thick-walled cylinder elasticity is clearly demonstrated. In particular the significant influence of time and heterogenous constant on radial displacement, hoop stress and temperature distributions is computed. The study is relevant to rocket chamber thermo-mechanics, propulsion duct thermophysical design, industrial thermal storage systems etc

Transient radiation and conduction in a slotted slab and a hollow cylinder

Transient radiation and conduction in slotted slab and hollow cylinde

Response-coefficient method for heat-conduction transients with time-dependent inputs

A theoretical overview of the response coefficient method for heat conduction transients with time-dependent input forcing functions is presented with a number of illustrative applications. The method may be the most convenient and economical if the same problem is to be solved many times with different input-time histories or if the solution time is relatively long. The method is applicable to a wide variety of problems, including irregular geometries, position-dependent boundary conditions, position-dependent physical properties, and nonperiodic irregular input histories. Nonuniform internal energy generation rates within the structure can also be handled by the method. The area of interest is long-time solutions, in which initial condition is unimportant, and not the early transient period. The method can be applied to one dimensional problems in cartesian, cylindrical, and spherical coordinates as well as to two dimensional problems in cartesian and cylindrical coordinates

Analysis of Transient Heat conduction in Different Geometries

Present work deals with the analytical solution of unsteady state one-dimensional heat conduction problems. An improved lumped parameter model has been adopted to predict the variation of temperature field in a long slab and cylinder. Polynomial approximation method is used to solve the transient conduction equations for both the slab and tube geometry. A variety of models including boundary heat flux for both slabs and tube and, heat generation in both slab and tube has been analyzed. Furthermore, for both slab and cylindrical geometry, a number of guess temperature profiles have been assumed to obtain a generalized solution. Based on the analysis, a modified Biot number has been proposed that predicts the temperature variation irrespective the geometry of the problem. In all the cases, a closed form solution is obtained between temperature, Biot number, heat source parameter and time. The result of the present analysis has been compared with earlier numerical and analytical results. A good agreement has been obtained between the present prediction and the available results