Analytical Solution of Hyperbolic Heat Conduction Equation in a Finite Medium Under Pulsatile Heat Source

2017 ◽  
Vol 42 (3) ◽  
pp. 269-277 ◽  
Author(s):  
Mohammad Reza Talaee ◽  
Ali Kabiri ◽  
Reza Khodarahmi
Keyword(s):  
Heat Conduction ◽  
Analytical Solution ◽  
Heat Source ◽  
Heat Conduction Equation ◽  
Conduction Equation ◽  
Hyperbolic Heat Conduction ◽  
Finite Medium

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 of Hyperbolic Heat Conduction Equation for a Finite Slab With Arbitrary Boundaries, Initial Condition, and Stationary Heat Source

2008 ◽  
Author(s):  
Hossein Shokouhmand ◽  
Seyed Reza Mahmoudi ◽  
Kaveh Habibi
Keyword(s):  
Heat Conduction ◽  
Analytical Solution ◽  
Heat Source ◽  
Heat Conduction Equation ◽  
Conduction Equation ◽  
Transient Temperature ◽  
Hyperbolic Heat Conduction ◽  
High Heat Flux ◽  
Adequate Model ◽  
Finite Slab

This paper presents an analytical solution of the hyperbolic heat conduction equation for a finite slab that sides are subjected to arbitrary heat source, boundary, and initial conditions. In the mathematical model used in this study, the heating on both sides treated as an apparent heat source while sides of the slab assumed to be insulated. Distribution of the apparent heat source for a problem with arbitrary heating on two boundaries is solved. The solution obtained by separation of variable method using appropriate Fourier series. Being a Sturm-Liouville problem in x-direction, suitable orthogonal functions can be allocated to hyperbolic heat conduction equation depending on the type of boundary conditions. Despite ease of proposed method, very few works has been done to solve hyperbolic heat conduction problems using this method by authors. The main feature of the method is straightforward formulation. In the analysis of heat conduction involving extremely short times, the parabolic heat conduction equation breaks down. By increasing the applications of the fast heat sources such as laser pulse for annealing of semiconductors and high heat flux applications, the need for adequate model of heat conduction has arisen. The hyperbolic heat conduction equation eliminates the paradox of an infinite speed of propagation of thermal disturbances which contradicts with Einstein’s theory of relativity. Moreover, it describes the highly transient temperature distribution in a finite medium more accurately.

Analytical Solution of the Hyperbolic Heat Conduction Equation for a Moving Finite Medium Under the Effect of Time Dependent Laser Heat Source

2012 ◽  
Vol 134 (12) ◽  
Author(s):  
R. T. Al-khairy
Keyword(s):  
Heat Conduction ◽  
Analytical Solution ◽  
Heat Source ◽  
Heat Conduction Equation ◽  
Time Dependent ◽  
Hyperbolic Heat Conduction ◽  
Laser Heat Source ◽  
Reference Curves ◽  
Finite Medium ◽  
Phase Differences

This paper presents an analytical solution of the hyperbolic heat conduction equation for a moving finite medium under the effect of a time-dependent laser heat source. Laser heating is modeled as an internal heat source, whose capacity is given by g(x,t) = I(t) (1 – R)μe−μx while the finite body has an insulated boundary. The solution is obtained by the Laplace transforms method, and the discussion of solutions for two time characteristics of heat source capacities (instantaneous and exponential) is presented. The effect of the dimensionless medium velocity on the temperature profiles is examined in detail. It is found that there exists clear phase shifts in connection with the dimensionless velocity U in the spatial temperature distributions: the temperature curves with negative U values lag behind the reference curves with zero U, while the ones with positive U values precedes the reference curves. It is also found that the phase differences are the sole products of U, with increasing U predicting larger phase differences.

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.

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.

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