Exact Analytical Solution for Unsteady Heat Conduction in Fiber-Reinforced Spherical Composites Under the General Boundary Conditions

2015 ◽  
Vol 137 (10) ◽  
Author(s):  
A. Amiri Delouei ◽  
M. Norouzi
Keyword(s):  
Heat Transfer ◽  
Heat Conduction ◽  
Analytical Solution ◽  
Boundary Conditions ◽  
Laplace Transformation ◽  
Exact Analytical Solution ◽  
Function Technique ◽  
Conductive Heat ◽  
Fiber Reinforced ◽  
Legendre Series

The current study presents an exact analytical solution for unsteady conductive heat transfer in multilayer spherical fiber-reinforced composite laminates. The orthotropic heat conduction equation in spherical coordinate is introduced. The most generalized linear boundary conditions consisting of the conduction, convection, and radiation heat transfer is considered both inside and outside of spherical laminate. The fibers' angle and composite material in each lamina can be changed. Laplace transformation is employed to change the domain of the solutions from time into the frequency. In the frequency domain, the separation of variable method is used and the set of equations related to the coefficients of Fourier–Legendre series is solved. Meromorphic function technique is utilized to determine the complex inverse Laplace transformation. Two functional cases are presented to investigate the capability of current solution for solving the industrial unsteady problems in different arrangements of multilayer spherical laminates.

Transient Heat Transfer between a Plate and a Fluid whose Temperature Varies Periodically with Time

1980 ◽  
Vol 102 (1) ◽  
pp. 126-131 ◽  
Author(s):  
J. Sucec
Keyword(s):  
Heat Transfer ◽  
Analytical Solution ◽  
Forced Convection ◽  
General Problem ◽  
Laplace Transformation ◽  
Exact Analytical Solution ◽  
Transient Heat Transfer ◽  
Convection Problem ◽  
Complex Temperature

Using the method of complex temperature in conjunction with the Laplace transformation, an exact analytical solution is found for the transient, conjugate, forced convection problem consisting of a plate, whose base is insulated, interacting with a fluid, moving in a steady slug fashion, whose temperature, at points far from the plate, varies sinusoidally with time. Simple quasi-steady results are derived for comparison. Also presented is a method for determining the qualitative conditions under which one might expect a quasi-steady analysis to be valid in a general problem.

Analytical Solution of the Problem of Conjugate Heat Transfer between a Gasdynamic Boundary Layer and Anisotropic Strip

2020 ◽  
pp. 44-59
Author(s):  
V.F. Formalev ◽  
S.A. Kolesnik ◽  
B.A. Garibyan
Keyword(s):  
Heat Transfer ◽  
Thermal Conductivity ◽  
Boundary Layer ◽  
Heat Conduction ◽  
Analytical Solution ◽  
Boundary Conditions ◽  
Conjugate Heat Transfer ◽  
Heat Fluxes ◽  
The Body ◽  
Conductivity Tensor

The paper focuses on the problem of conjugate heat transfer between the thermal-gas-dynamic boundary layer and the anisotropic strip in conditions of aerodynamic heating of aircraft. Under the assumption of an incompressible flow which takes place in the shock layer behind the direct part of the shock wave, we found a new analytical solution for the components of the velocity vector, temperature distribution, and heat fluxes in the boundary layer. The obtained heat fluxes at the interface between the gas and the body are included as boundary conditions in the problem of anisotropic heat conduction in the body. The study introduces an analytical solution to the second initial-boundary value problem of heat conduction in an anisotropic strip with arbitrary boundary conditions at the interfaces, with heat fluxes which are obtained by solving the problem of a thermal boundary layer used at the interface. An analytical solution to the conjugate problem of heat transfer between a boundary layer and an anisotropic body can be effectively used to control, e.g. to reduce, heat fluxes from the gas to the body if the strip material chosen is such that the longitudinal component of the thermal conductivity tensor is many times larger than the transverse component of the thermal conductivity tensor. Such adjustment is possible due to an increase in body temperature in the longitudinal direction, and, consequently, a decrease in the heat flow from the gas to the body, as well as due to a favorable change in the physical characteristics of the gas. Results of numerical experiments are obtained and analyzed

scholarly journals Modeling methodology of locally non-equilibrium heat conductivity processes

Vestnik IGEU ◽  
2020 ◽  
pp. 65-71
Author(s):  
A.V. Eremin
Keyword(s):  
Mathematical Modeling ◽  
Heat Transfer ◽  
Heat Conduction ◽  
Analytical Solution ◽  
Heat Equation ◽  
Exact Analytical Solution ◽  
Separation Of Variables ◽  
Infinite Plate ◽  
Transfer Processes ◽  
Non Equilibrium

With the development of laser technologies and the ability to carry out processing steps under extreme conditions (ul-trahigh temperatures, pressures and their gradients), the interest in studying the processes that occur under locally non-equilibrium conditions has grown significantly. The key directions for the description of locally non-equilibrium pro-cesses include thermodynamic, kinetic and phenomenological ones. The locally non-equilibrium transfer equations can also be derived from the Boltzmann equation by using the theory of random walks and molecular-kinetic methods. It should be noted that some options of locally non-equilibrium processes lead to conflicting results. This study aims to develop a method for mathematical modeling of locally nonequilibrium heat conduction processes in solids, which allows determining their temperature with high accuracy during fast and high-intensity heat transfer processes. As applied to heat transfer processes in solids, a generalized heat equation that takes into account the relaxation properties of materials is formulated. The exact analytical solution is obtained using the Fourier method of separation of variables. The methodology for mathematical modeling of locally non-equilibrium transfer processes based on modified conservation laws has been developed. The generalized differential heat equation which allows performing N-fold relaxation of the heat flow and temperature in the modified heat balance equation has been formulated. For the first time, an exact analytical solution to the unsteady heat conduction problem for an infinite plate was obtained taking into account many-fold relaxation. The analysis of the solution to the boundary value problem of locally nonequilibrium heat conduction enabled to conclude that it is impossible to instantly has establish a boundary condition of the first kind. It has been demonstrated that each of the following terms in the relaxed heat equation has an ever smaller effect on the heat transfer process. The obtained results can be used by the scientific and technical personnel of organizations and higher educational institutions in the study of fuel ignition processes, the development of laser processing of materials, the design of highly efficient heat transfer equipment and the description of fast-flowing heat transfer processes.

Microprocessor Silicon Die Temperature Prediction by Simplified Boundary Conditions

2015 ◽  
Author(s):  
Koji Nishi ◽  
Tomoyuki Hatakeyama ◽  
Shinji Nakagawa ◽  
Masaru Ishizuka
Keyword(s):  
Heat Transfer ◽  
Heat Conduction ◽  
Boundary Conditions ◽  
Thermal Resistance ◽  
Hot Spot ◽  
Three Dimensional ◽  
Thermal Design ◽  
Target Temperature ◽  
One Dimensional ◽  
Thermal Network

The thermal network method has a long history with thermal design of electronic equipment. In particular, a one-dimensional thermal network is useful to know the temperature and heat transfer rate along each heat transfer path. It also saves computation time and/or computation resources to obtain target temperature. However, unlike three-dimensional thermal simulation with fine pitch grids and a three-dimensional thermal network with sufficient numbers of nodes, a traditional one-dimensional thermal network cannot predict the temperature of a microprocessor silicon die hot spot with sufficient accuracy in a three-dimensional domain analysis. Therefore, this paper introduces a one-dimensional thermal network with average temperature nodes. Thermal resistance values need to be obtained to calculate target temperature in a thermal network. For this purpose, thermal resistance calculation methodology with simplified boundary conditions, which calculates thermal resistance values from an analytical solution, is also introduced in this paper. The effectiveness of the methodology is explored with a simple model of the microprocessor system. The calculated result by the methodology is compared to a three-dimensional heat conduction simulation result. It is found that the introduced technique matches the three-dimensional heat conduction simulation result well.

EVALUATION OF COMBINED DELAUNAY TRIANGULATION AND REMESHING FOR FINITE ELEMENT ANALYSIS OF CONDUCTIVE HEAT TRANSFER

2004 ◽  
Vol 27 (4) ◽  
pp. 319-339 ◽  
Author(s):  
Sutthisak Phongthanapanich ◽  
Pramote Dechaumphai
Keyword(s):  
Heat Transfer ◽  
Finite Element ◽  
Heat Conduction ◽  
Steady State ◽  
Delaunay Triangulation ◽  
Transient Heat Conduction ◽  
Conductive Heat ◽  
Element Analysis ◽  
Adaptive Remeshing ◽  
Solution Accuracy

A finite element method is combined with the Delaunay triangulation and an adaptive remeshing technique to solve for solutions of both steady-state and transient heat conduction problems. The Delaunay triangulation and the adaptive remeshing technique are explained in detail. The solution accuracy and the effectiveness of the combined procedure are evaluated by heat transfer problems that have exact solutions. These problems include steady-state heat conduction in a square plate subjected to a highly localized surface heating, and a transient heat conduction in a long plate subjected to a moving heat source. The examples demonstrate that the adaptive remeshing technique with the Delaunay triangulation significantly reduce the number of the finite elements required for the problems and, at the same time, increase the analysis solution accuracy as compared to the results produced using uniform finite element meshes.

scholarly journals An Analytical Solution for Transient Heat Conduction in a Composite Slab with Time-Dependent Heat Transfer Coefficient

2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
Ryoichi Chiba
Keyword(s):  
Heat Transfer ◽  
Heat Transfer Coefficient ◽  
Heat Conduction ◽  
Analytical Solution ◽  
Differential Equations ◽  
Transfer Coefficient ◽  
Transient Heat Conduction ◽  
External Boundary ◽  
Composite Slab ◽  
Composite Slabs

An analytical solution is derived for one-dimensional transient heat conduction in a composite slab consisting of n layers, whose heat transfer coefficient on an external boundary is an arbitrary function of time. The composite slab, which has thermal contact resistance at n-1 interfaces, as well as an arbitrary initial temperature distribution and internal heat generation, convectively exchanges heat at the external boundaries with two different time-varying surroundings. To obtain the analytical solution, the shifting function method is first used, which yields new partial differential equations under conventional types of external boundary conditions. The solution for the derived differential equations is then obtained by means of an orthogonal expansion technique. Numerical calculations are performed for two composite slabs, whose heat transfer coefficient on the heated surface is either an exponential or a trigonometric function of time. The numerical results demonstrate the effects of temporal variations in the heat transfer coefficient on the transient temperature field of composite slabs.

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