Abstract
A practical unified method is applied to the hyperbolic heat conduction of insulated semi-infinite functionally graded body under the effect of a time-dependent laser heat source. It is assumed that the material properties of the body vary exponentially through the axial direction, except thermal relaxation parameter, which is taken to be constant. These conditions produce a linear partial differential equation that cannot be solved analytically with conventional methods except for some simple grading functions. Therefore, numerical solution becomes essential to solve the problem. First, the problem is transformed to the Laplace domain, and then, Chebyshev pseudospectral collocation method is employed, yielding the final results that are transformed to the time domain using the modified Durbin’s method. The results of the temperature distribution are discussed for different inhomogeneity parameters and different time characteristics of the heat source capacity. Homogeneous solutions that are available in the literature are used to verify the results and to emphasize the convergence of the numerical solutions.
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Abbreviations
- \(c_{\text{p}}\) :
-
Specific heat
- \(g\) :
-
Capacity of internal heat source
- \(I\) :
-
Laser incident intensity
- \(I_{\text{r}}\) :
-
Arbitrary reference laser intensity
- \(k\) :
-
Thermal conductivity
- \(R\) :
-
Surface reflectance
- \(q\) :
-
Heat flux
- \(s\) :
-
Laplace variable
- \(\psi\) :
-
Dimensionless capacity of internal heat source
- \(\psi_{ 0}\) :
-
Constant coefficients related to the dimensionless capacity of internal heat source
- \(t\) :
-
Time
- \(T\) :
-
Temperature
- \(T_{ 0} , T_{\text{m}}\) :
-
Arbitrary reference temperature
- \(c\) :
-
Speed of heat propagation \(\left( {c = \sqrt {\left\{ {\kappa_{i} /\tau } \right\}} } \right)\)
- \(x\) :
-
Cartesian coordinate
- \(\beta\) :
-
Dimensionless absorption coefficient
- \(\gamma\) :
-
Inhomogeneity parameter
- \(\kappa_{i}\) :
-
Thermal diffusivity \(\left( {\kappa_{i} = k_{i} /\rho c_{{{\text{p}}_{i} }} } \right)\)
- \(\eta\) :
-
Dimensionless axial coordinate
- \(\mu\) :
-
Absorption coefficient
- \(\rho\) :
-
Density
- \(\tau\) :
-
Thermal relaxation time
- \(\xi\) :
-
Dimensionless time
- \(\xi_{i}\) :
-
Dimensionless duration of laser pulse
- \(w\) :
-
Frequency of a periodic heat source
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Yarımpabuç, D. A Unified Approach to Hyperbolic Heat Conduction of the Semi-infinite Functionally Graded Body with a Time-Dependent Laser Heat Source. Iran J Sci Technol Trans Mech Eng 43, 729–737 (2019). https://doi.org/10.1007/s40997-019-00312-0
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DOI: https://doi.org/10.1007/s40997-019-00312-0