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

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Iranian Journal of Science and Technology, Transactions of Mechanical Engineering Aims and scope Submit manuscript

Abstract

This paper presents a pure analytical solution of one-dimensional hyperbolic heat conduction equation in a homogeneous finite medium under series of time pulsed heat source which is exponentially distributed and acts symmetrically on both sides. The solution is obtained without any numerical procedures, using the Eigenvalue function. The problem is solved under two types of step and exponential time pulse series functions, which are used in simulation of laser interaction of tissues, and the closed-form solutions are introduced. The ability of the solution to estimate the effect of pulse duration and intensity is investigated. The results can be applied as a verification branch for other numerical solutions such as pulse laser interaction phenomenon.

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Abbreviations

\(c_{\text{p}}\) :

Specific heat at constant pressure

\(H\) :

Heaviside function

\(I\left( t \right)\) :

Laser incident intensity

\(I_{r}\) :

Arbitrary reference laser intensity

\(L\) :

Dimensionless length of the slab

\(l\) :

Length of the slab

\(M\) :

Number of complete pulses

\(m\) :

Coefficient controlling Off period

\(N\) :

Counter of pulses

\(n\) :

Fourier series counter

\(Q\) :

Heat source function

\(q\) :

Heat flux vector

\(R\) :

Surface reflectance

T :

Temperature

\(T_{0} ,T_{m}\) :

Reference temperatures

T p :

Dimensionless step duration

t :

Time

w :

Speed of heat propagation

x :

Cartesian coordinate

X :

Dimensionless Cartesian coordinate

α :

Thermal diffusivity

ρ :

Density

\(\lambda\) :

Thermal conductivity

\(\tau_{q}\) :

Thermal relaxation time

\(\theta\) :

Dimensionless temperature

\(\delta\) :

Dirac function

\(\mu\) :

Penetration coefficient

\(\beta\) :

Dimensionless penetration coefficient

\(\tau\) :

Dimensionless time

\(\psi\) :

Dimensionless heat source function

\(\psi_{0}\) :

Constant coefficient

\(\eta \left( \tau \right)\) :

Dimensionless time-dependent laser function

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Funding

Funding was provided by Iran University of Science and Technology.

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Authors and Affiliations

  1. School of Railway Engineering, Rolling Stock, Iran University of Science and Technology (IUST), Tehran, 16846-13114, Iran

    Mohammad Reza Talaee, Ali Kabiri & Reza Khodarahmi

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  1. Mohammad Reza Talaee
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  2. Ali Kabiri
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  3. Reza Khodarahmi
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Correspondence to Mohammad Reza Talaee.

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Talaee, M.R., Kabiri, A. & Khodarahmi, R. Analytical Solution of Hyperbolic Heat Conduction Equation in a Finite Medium Under Pulsatile Heat Source. Iran J Sci Technol Trans Mech Eng 42, 269–277 (2018). https://doi.org/10.1007/s40997-017-0096-y

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  • DOI: https://doi.org/10.1007/s40997-017-0096-y

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