Analytical Solutions to Transient Conduction Heat Transfer Problems With Discrete Heat Generating Sources in Cylindrical Coordinates

2012 ◽  
Author(s):  
Eduardo Perez ◽  
Edwar Romero ◽  
Omar Meza
Keyword(s):  
Heat Transfer ◽  
Heat Conduction ◽  
Analytical Solutions ◽  
Separation Of Variables ◽  
Thermal Control ◽  
Conduction Heat Transfer ◽  
Aerospace Applications ◽  
Transient Conduction ◽  
The Times ◽  
Transfer Problems

Heat transfer mechanisms are virtually present anywhere where energy is involved; heat conduction is one of those mechanisms. Pure conduction heat transfer can be found mainly in aerospace applications especially when aircrafts are in shadows occulted from sunlight and temperatures drop to extremely low values. In these cases resistance heaters are activated in order to preserve temperatures inside the aircraft within values where circuitry does not stop working. Resistance heaters should be embedded such that appropriate thermal control is achieved. In addition conduction heat transfer is of special interest in fundamental theory for science and engineering. Typically this kind of problems are solved by means of numerical analysis or specialized software which most of the times are more time consuming and more expensive in terms of computational resources. In the other hand, traditional analytical methods such as separation of variables can be used to solve simple heat conduction problems, but do not work for problems where discrete heat generation sources are present. The purpose of this work is to present an analytical method to solve transient conduction heat transfer problems in geometries with embedded discrete heat generating sources. The Green’s Functions were used to formulate the mathematical model to find analytical solutions to transient conduction heat transfer problems for circular plates with embedded discrete heat generating sources. Two cases of embedded heat generating sources are discussed: a thin circular wire in the form of an arc, and a circular region. Results found in the present work were taken for long values of time and compared to those reported by Venkataraman et al who found results for similar problems under steady state conditions. Results were in excellent agreement. Final equations found with the technique presented here are in the form of summations series similar to those found using the separation of variables method.

Transient Two-Dimensional Heat Conduction Problem With Partial Heating Near Corners

2016 ◽  
Author(s):  
Robert L. McMasters ◽  
Filippo de Monte ◽  
James V. Beck ◽  
Donald E. Amos
Keyword(s):  
Heat Conduction ◽  
Numerical Integration ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Thermal Protection ◽  
Heat Conduction Problem ◽  
Separation Of Variables ◽  
Two Dimensional ◽  
Rectangular Region ◽  
Long Distance

This paper provides a solution for two-dimensional heating over a rectangular region on a homogeneous plate. It has application to verification of numerical conduction codes as well as direct application for heating and cooling of electronic equipment. Additionally, it can be applied as a direct solution for the inverse heat conduction problem, most notably used in thermal protection systems for re-entry vehicles. The solutions used in this work are generated using Green’s functions. Two approaches are used which provide solutions for either semi-infinite plates or finite plates with isothermal conditions which are located a long distance from the heating. The methods are both efficient numerically and have extreme accuracy, which can be used to provide additional solution verification. The solutions have components that are shown to have physical significance. The extremely precise nature of analytical solutions allows them to be used as prime standards for their respective transient conduction cases. This extreme precision also allows an accurate calculation of heat flux by finite differences between two points of very close proximity which would not be possible with numerical solutions. This is particularly useful near heated surfaces and near corners. Similarly, sensitivity coefficients for parameter estimation problems can be calculated with extreme precision using this same technique. Another contribution of these solutions is the insight that they can bring. Important dimensionless groups are identified and their influence can be more readily seen than with numerical results. For linear problems, basic heating elements on plates, for example, can be solved to aid in understanding more complex cases. Furthermore these basic solutions can be superimposed both in time and space to obtain solutions for numerous other problems. This paper provides an analytical two-dimensional, transient solution for heating over a rectangular region on a homogeneous square plate. Several methods are available for the solution of such problems. One of the most common is the separation of variables (SOV) method. In the standard implementation of the SOV method, convergence can be slow and accuracy lacking. Another method of generating a solution to this problem makes use of time-partitioning which can produce accurate results. However, numerical integration may be required in these cases, which, in some ways, negates the advantages offered by the analytical solutions. The method given herein requires no numerical integration; it also exhibits exponential series convergence and can provide excellent accuracy. The procedure involves the derivation of previously-unknown simpler forms for the summations, in some cases by virtue of the use of algebraic components. Also, a mathematical identity given in this paper can be used for a variety of related problems.

Transient Conduction From Parallel Isothermal Cylinders

2012 ◽  
Vol 134 (12) ◽  
Author(s):  
Rajai S. Alassar
Keyword(s):  
Heat Transfer ◽  
Fourier Series ◽  
Heat Conduction ◽  
Energy Equation ◽  
Transient Heat Conduction ◽  
Infinite Medium ◽  
Basis Functions ◽  
Cylindrical Coordinates ◽  
Transient Conduction ◽  
Isothermal Cylinders

The transient heat conduction from two parallel isothermal cylinders is studied using the naturally fit bipolar cylindrical coordinates system. The energy equation is expanded in a Fourier series using appropriate basis functions to eliminate one of the physical coordinates. The resulting modes of the expansion are solved using a finite difference scheme. It is shown that, as is the case with a single isothermal cylinder in an infinite medium, steady states for two isothermal cylinders are not possible and heat transfer changes indefinitely with time.

A Meshless Finite Difference Method for Conjugate Heat Conduction Problems

2009 ◽  
Author(s):  
Chandrashekhar Varanasi ◽  
Jayathi Y. Murthy ◽  
Sanjay Mathur
Keyword(s):  
Heat Transfer ◽  
Heat Conduction ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Conjugate Heat Transfer ◽  
Sparse Matrix ◽  
Complex Geometries ◽  
Meshless Finite Difference Method ◽  
Transfer Problems

In recent years, there has been a great deal of interest in developing meshless methods for computational fluid dynamics (CFD) applications. In this paper, a meshless finite difference method is developed for solving conjugate heat transfer problems in complex geometries. Traditional finite difference methods (FDMs) have been restricted to an orthogonal or a body-fitted distribution of points. However, the Taylor series upon which the FDM is based is valid at any location in the neighborhood of the point about which the expansion is carried out. Exploiting this fact, and starting with an unstructured distribution of mesh points, derivatives are evaluated using a weighted least squares procedure. The system of equations that results from this discretization can be represented by a sparse matrix. This system is solved with an algebraic multigrid (AMG) solver. The implementation of Neumann, Dirichlet and mixed boundary conditions within this framework is described, as well as the handling of conjugate heat transfer. The method is verified through application to classical heat conduction problems with known analytical solutions. It is then applied to the solution of conjugate heat transfer problems in complex geometries, and the solutions so obtained are compared with more conventional unstructured finite volume methods. Metrics for accuracy are provided and future extensions are discussed.

scholarly journals A NOTE ON THE CONSTANT THERMAL CONDUCTIVITY HYPOTHESIS AND ITS CONSEQUENCE FOR CONDUCTION HEAT TRANSFER PROBLEMS

2014 ◽  
Vol 13 (2) ◽  
pp. 48
Author(s):  
R. M. S. Gama
Keyword(s):  
Heat Transfer ◽  
Thermal Conductivity ◽  
High Temperature ◽  
Heat Generation ◽  
Strong Dependence ◽  
Temperature Gradients ◽  
Conduction Heat Transfer ◽  
Kirchhoff Transformation ◽  
Constant Thermal Conductivity ◽  
Transfer Problems

This work discuss the usual constant conductivity assumption and its consequences when a given material presents a strong dependence between the temperature and the thermal conductivity. The discussion is carried out considering a sphere of silicon with a given heat generation concentrated in a vicinity of its centre, giving rise to high temperature gradients. This particular case is enough to show that the constant thermal conductivity hypothesis may give rise to very large errors and must be avoided. In order to surpass the mathematical complexity, the Kirchhoff transformation is used for constructing the solution of the problem. In addition, an equation correlating thermal conductivity and the temperature is proposed.

Effect of enrichment functions on GFEM solutions of time-dependent conduction heat transfer problems

2020 ◽  
Vol 85 ◽  
pp. 89-106 ◽  
Author(s):  
M. Iqbal ◽  
K. Alam ◽  
H. Gimperlein ◽  
O. Laghrouche ◽  
M.S. Mohamed
Keyword(s):  
Heat Transfer ◽  
Time Dependent ◽  
Conduction Heat Transfer ◽  
Transfer Problems

Analytical solutions of some heat transfer problems in mining practice

2014 ◽  
Vol 50 (1) ◽  
pp. 81-86 ◽  
Author(s):  
N. N. Smirnova ◽  
N. V. Nikolaeva ◽  
V. N. Brichkin ◽  
V. B. Kuskov
Keyword(s):  
Heat Transfer ◽  
Analytical Solutions ◽  
Transfer Problems
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