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The Developed Conservation Element and Solution Element Method in Two-Dimensional Spherical Coordinate and Its Application to the Analysis of Non-Fourier Heat Conduction

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Abstract

In solving the system of hyperbolic equations, highly accurate numerical methods which are easy to apply and have a shorter run time without causing numerical oscillations are more popular. In the present study, the space–time conservation element and solution element (CESE) method has been developed into two-dimensional spherical coordinates using polar elements. Then, the developed CESE method has been applied to investigate the propagation of a non-Fourier thermal waves in biological tissue. To evaluate the performance of the CESE method, the numerical results are compared to the existing semi-analytical results, and it is observed that the results are in good agreement. The experimental test is then conducted to measure the transient temperature behavior of the spherical Intralipid phantom irradiated by a near-infrared pulsed laser. A comparison of experimental and numerical results demonstrates the applicability of the dual phase lag model in the prediction of non-Fourier heat conduction in biological tissue. In addition, the contours of heat flux and temperature during and after laser irradiation are presented, and the propagation of thermal waves in the tissue is examined and discussed. The results indicate that the effects of the two-dimensional thermal wave appear after stopping the laser irradiation. Finally, the study of the effect of tissue type on wave progression in the tumor, muscle, and fat demonstrates that in the fat tissue, having the lowest thermal diffusivity, the thermal progression is less than the tumor and muscle. However, the temperature distribution in the fat tissue is greater than in the other tissues.

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Abbreviations

\(c\) :

Specific heat of tissue

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

Specific heat of blood

\({\varvec{F}}\) :

Flux vector in radial direction

\({\varvec{G}}\) :

Flux vector in angular direction

\({\varvec{H}}\) :

Space–time flux vector

\(k\) :

Thermal conductivity

\({\varvec{q}}\) :

Heat flux vector

\({\varvec{q}}_{{\mathbf{r}}}\) :

Heat flux in radial direction

\({\varvec{q}}_{{\varvec{\theta}}}\) :

Heat flux in angular direction

\(q_{{\text{L}}}\) :

Laser intensity

\(Q_{{\text{m}}}\) :

Metabolic heat generation

\(Q_{{\text{r}}}\) :

Dimensionless heat flux in radial direction

\(Q_{\theta }\) :

Dimensionless heat flux in angular direction

\(r\) :

Coordinate variable in radial direction

\({\varvec{r}}\) :

2-D position vector

\(R\) :

Diffuse reflectance

\({\varvec{S}}\) :

Source vector

\(t\) :

Time

\(T\) :

Temperature of tissue

\(T_{{\text{b}}}\) :

Blood temperature

\({\varvec{U}}\) :

Primary variable vector

\(\zeta\) :

Dimensionless radius

\(\eta\) :

Dimensionless time

\(\theta\) :

Coordinate variable in angular direction

\(\theta^{*}\) :

Dimensionless angle

\(\rho\) :

Density of tissue

\(\rho_{{\text{b}}}\) :

Density of blood

\(\tau_{q}\) :

Phase lag time of heat flux

\(\tau_{{\text{T}}}\) :

Phase lag time of temperature gradient

\({\Phi }\) :

Dimensionless temperature

\(\omega_{{\text{b}}}\) :

Blood perfusion rate

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Acknowledgements

Author M.A.A. was supported by Grant No. 98029460 of Iranian National Science Foundation (INSF).

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Authors and Affiliations

  1. Department of Mechanical Engineering, Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, Iran

    Amir Ghasemi Touran Poshti & Mohammad Bagher Ayani

  2. Optical Bio-Imaging Lab, Laser and Plasma Research Institute, Shahid Beheshti University, Daneshju Boulevard, Tehran, Iran

    Mohammad Ali Ansari

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  1. Amir Ghasemi Touran Poshti
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Correspondence to Mohammad Bagher Ayani.

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Poshti, A.G.T., Ansari, M.A. & Ayani, M.B. The Developed Conservation Element and Solution Element Method in Two-Dimensional Spherical Coordinate and Its Application to the Analysis of Non-Fourier Heat Conduction. Arab J Sci Eng 48, 12371–12387 (2023). https://doi.org/10.1007/s13369-023-07797-7

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