scholarly journals An analytical solution to the problem of time-fractional heat conduction in a composite sphere

2017 ◽  
Vol 65 (2) ◽  
pp. 179-186 ◽  
Author(s):  
S. Kukla ◽  
U. Siedlecka
Keyword(s):  
Heat Conduction ◽  
Analytical Solution ◽  
Heat Source ◽  
Thermal Contact ◽  
Time Derivative ◽  
Solid Sphere ◽  
Composite Sphere ◽  
Problem Of Time ◽  
Time Space ◽  
Fractional Heat Conduction

Abstract An analytical solution to the problem of time-fractional heat conduction in a sphere consisting of an inner solid sphere and concentric spherical layers is presented. In the heat conduction equation, the Caputo time-derivative of fractional order and the Robin boundary condition at the outer surface of the sphere are assumed. The spherical layers are characterized by different material properties and perfect thermal contact is assumed between the layers. The analytical solution to the problem of heat conduction in the sphere for time-dependent surrounding temperature and for time-space-dependent volumetric heat source is derived. Numerical examples are presented to show the effect of the harmonically varying intensity of the heat source and the harmonically varying surrounding temperature on the temperature in the sphere for different orders of the Caputo time-derivative.

scholarly journals Heat conduction in a composite sphere - the effect of fractional derivative order on temperature distribution

2018 ◽  
Vol 157 ◽  
pp. 08008
Author(s):  
Urszula Siedlecka ◽  
Stanisław Kukla
Keyword(s):  
Boundary Condition ◽  
Heat Conduction ◽  
Temperature Distribution ◽  
Laplace Transform ◽  
Time Derivative ◽  
Spherical Layer ◽  
Solid Sphere ◽  
Liouville Derivative ◽  
The Laplace Transform ◽  
Fractional Heat Conduction

The aim of the contribution is an analysis of time-fractional heat conduction in a sphere with an inner heat source. The object of the consideration is a solid sphere with a spherical layer. The heat conduction in the solid sphere and spherical layer is governed by fractional heat conduction equation with a Caputo time-derivative. Mathematical (classical) or physical formulations of the Robin boundary condition and the perfect contact of the solid sphere and spherical layer is assumed. The boundary condition and the heat flux continuity condition at the interface are expressed by the Riemann-Liouville derivative. An exact solution of the problem under mathematical conditions is determined. A solution of the problem under physical boundary and continuity conditions using the Laplace transform method has been obtained. The inverse of the Laplace transform by using the Talbot method are numerically determined. Numerical results show the effect of the order of the Caputo and the Riemann-Liouville derivatives on the temperature distribution in the sphere.

scholarly journals Analytical Solution of the Hyperbolic Heat Conduction Equation for Moving Semi-Infinite Medium under the Effect of Time-Dependent Laser Heat Source

2009 ◽  
Vol 2009 ◽  
pp. 1-18 ◽  
Author(s):  
R. T. Al-Khairy ◽  
Z. M. AL-Ofey
Keyword(s):  
Heat Conduction ◽  
Analytical Solution ◽  
Heat Source ◽  
Heat Conduction Equation ◽  
Closed Form Solution ◽  
Infinite Medium ◽  
Form Solution ◽  
Time Dependent ◽  
Hyperbolic Heat Conduction ◽  
Laser Heat Source

This paper presents an analytical solution of the hyperbolic heat conduction equation for moving semi-infinite medium under the effect of time dependent laser heat source. Laser heating is modeled as an internal heat source, whose capacity is given by while the semi-infinite body has insulated boundary. The solution is obtained by Laplace transforms method, and the discussion of solutions for different time characteristics of heat sources capacity (constant, instantaneous, and exponential) is presented. The effect of absorption coefficients on the temperature profiles is examined in detail. It is found that the closed form solution derived from the present study reduces to the previously obtained analytical solution when the medium velocity is set to zero in the closed form solution.

Effect of a Heat-Conducting Well Casing on Temperature Distribution in an Observation Well

1979 ◽  
Vol 101 (1) ◽  
pp. 20-27
Author(s):  
P. J. Closmann ◽  
E. R. Jones ◽  
E. A. Vogel
Keyword(s):  
Heat Conduction ◽  
Temperature Distribution ◽  
Heat Source ◽  
Constant Temperature ◽  
Thermal Contact ◽  
Heat Conducting ◽  
Formation Temperature ◽  
Maximum Difference ◽  
Observation Well ◽  
Perfect Thermal Contact

The effect of heat conduction on temperature along the wall of a well casing has been determined by solution of the equations of heat conduction. The casing was assumed to pass vertically through a planar heat source of constant temperature. The casing and formation were assumed to be in perfect thermal contact. Numerical results were obtained for two sizes of steel casing and one size of aluminum casing. At any given distance from the heat source, the casing temperature differs most at early times from the formation temperature computed in the absence of casing. This difference decreases rapidly with time. Furthermore, the maximum difference occurs at greater distances from the heat source as time increases. In general, after about three months of heating, errors in measured temperatures due to conduction along the casing wall are negligible.

Analytical Solution for Three-Dimensional Hyperbolic Heat Conduction Equation with Time-Dependent and Distributed Heat Source

2016 ◽  
Vol 33 (1) ◽  
pp. 65-75 ◽  
Author(s):  
M. R. Talaee ◽  
V. Sarafrazi
Keyword(s):  
Heat Conduction ◽  
Analytical Solution ◽  
Heat Source ◽  
Heat Conduction Equation ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Closed Form Solution ◽  
Time Dependent ◽  
Conduction Equation ◽  
Hyperbolic Heat Conduction

AbstractThis paper is devoted to the analytical solution of three-dimensional hyperbolic heat conduction equation in a finite solid medium with rectangular cross-section under time dependent and non-uniform internal heat source. The closed form solution of both Fourier and non-Fourier profiles are introduced with Eigen function expansion method. The solution is applied for simple simulation of absorption of a continues laser in biological tissue. The results show the depth of laser absorption in tissue and considerable difference between the Fourier and Non-Fourier temperature profiles. In addition the solution can be applied as a verification branch for other numerical solutions.

Analytical solution to thermal stresses of high speed PM rotor considering thermal load and heat convection

2020 ◽  
Vol 64 (1-4) ◽  
pp. 533-540
Author(s):  
Wenjie Cheng ◽  
Zhikai Deng ◽  
Guangdong Cao ◽  
Ling Xiao ◽  
Huimin Qi ◽  
...  
Keyword(s):  
Temperature Field ◽  
Heat Conduction ◽  
Analytical Solution ◽  
Stress Field ◽  
Heat Source ◽  
Heat Conduction Equation ◽  
High Speed ◽  
Heat Convection ◽  
Conduction Equation ◽  
Air Cooling

Aiming at the high speed permanent magnet (PM) rotor with the heat source, this work investigates the analytic solution to the transient temperature field and thermal stress field of the rotor, considering the influence of the forced air cooling of rotor surface on the stress field. Firstly, dimensionless formulation of the transient heat conduction equation including interior heat source is derived, where the axially non-uniform heat convection coefficient and the temperature of main flow region are equivalent to their mean values. Secondly, the Fourier integral transform method is used to solve the dimensionless heat conduction equation. Then, the obtained temperature field is loaded into the analytical solution of strength, in which three types of stress sources such as interference fit, centrifugal force and temperature gradient are included. Finally, examples are carried out to verify the analytical solutions and relative results are discussed.

Analytical solution to laser short-pulse heating of microsized metal wire: volumetric and surface heat source considerations

2012 ◽  
Vol 90 (9) ◽  
pp. 911-918 ◽  
Author(s):  
B.S. Yilbas ◽  
A.Y. Al-Dweik
Keyword(s):  
Analytical Solution ◽  
Heat Source ◽  
Pulse Heating ◽  
Short Pulse ◽  
Thermal Contact ◽  
Lower Surface ◽  
Surface Heat ◽  
Metal Wire ◽  
Fourier Cosine ◽  
Volumetric Heat Source

Analytical solution for laser short-pulse heating of a micro-sized metal wire is presented. In the analysis, volumetric and surface heat sources are incorporated for the same pulse intensity. The volumetric heat source resembles absorption by irradiated field according to Lambert’s Beer law while a surface heat source represents short pulse heating through high intensity thermal contact at the surface. The method of Lie point symmetries is combined with a Fourier cosine transformation to solve the temperature equation with appropriate boundary conditions. It is found that temperature profiles differ significantly for volumetric heat source and surface heat source considerations; in which case, volumetric heat source consideration results in considerably lower surface temperatures than that of the surface heat source consideration.

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