# A Fibonacci Wavelet Method for Solving Dual-Phase-Lag Heat Transfer Model in Multi-Layer Skin Tissue during Hyperthermia Treatment

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation of the Model

## 3. Fibonacci Wavelets and Operational Matrices of Integration

#### 3.1. Fibonacci Wavelets and Function Approximation

#### 3.2. Operational Matrices of Fibonacci Wavelets

## 4. Solution of the Dual-Phase Model

#### 4.1. Discretizing the Space Variable $\zeta $

#### 4.2. Implementation of the Fibonacci Wavelets

## 5. Numerical Results and Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Abbreviations

T | Temperature (${}^{\circ}$C) |

${T}_{b}$ | Arterial blood temperature (${}^{\circ}$C) |

${T}_{w}$ | Tissue wall temperature (${}^{\circ}$C) |

t | Time (s) |

r | Space coordinate (m) |

T | Temperature (${}^{\circ}$C) |

${T}_{b}$ | Arterial blood temperature (${}^{\circ}$C) |

${T}_{w}$ | Tissue wall temperature (${}^{\circ}$C) |

t | Time (s) |

r | Space coordinate (m) |

r${}^{*}$ | Tumor position |

${c}_{p}$ | Specific heat of tissue (J kg${}^{-1\circ}$C${}^{-1}$) |

L | Length of tissue (m) |

${w}_{b}$ | Mass flow rate of blood (kg m${}^{-3}$ s${}^{-1}$) |

${D}_{f}$ | Water diffusion in tissue (m${}^{2}$ s${}^{-1}$) |

${M}_{w}$ | Molar mass of water (g mol${}^{-1}$) |

${P}_{w}$ | Vapor pressure of water (Pa) |

${R}_{a}$ | Universal gas constant (J mol${}^{-1}$) |

RH | Relative humidity (%) |

$\rho $ | Tissue density (kg m${}^{-3}$) |

k | Thermal conductivity of tissue (Wm${}^{-1}$ ${}^{\circ}$C${}^{-1}$) |

${c}_{pb}$ | Specific heat of blood (J kg${}^{-1}$ ${}^{\circ}$C${}^{-1}$) |

${\tau}_{q}$ | Phase lag of heat flux (s) |

${\tau}_{t}$ | Phase lag of temperature gradient (s) |

${Q}_{m}$ | Heat generation due to metabolism in the skin tissue (Wm${}^{-3}$) |

${Q}_{d}$ | Heat loss due to water diffusion in the tissue (Wm${}^{-3}$) |

${Q}_{v}$ | Heat loss due to water vaporization in the tissue (Wm${}^{-3}$) |

${Q}_{r}$ | External heat source (Wm${}^{-3}$) |

S | Antenna constant (m${}^{-1}$) |

P | Antenna power (W) |

$\Delta $ m | Water vaporization rate of skin surface (g m${}^{-2}$ s${}^{-1}$) |

$\Delta $${H}_{vap}$ | Enthalpy of water vaporization (J kg${}^{-1}$) |

$\delta $ c | Average distance of momentum boundary layer (m) |

Dimensionless variables | |

$\zeta $ | Space coordinate |

$\eta $ | Fourier number or time |

${F}_{oq}$ | Phase-lag due to heat flux |

${F}_{ot}$ | Phase-lag due to temperature grad. |

$\theta $ | Local tissue temperature |

${\theta}_{b}$ | Arterial blood temperature |

${\theta}_{w}$ | Tissue wall temperature |

${P}_{f}$ | Blood perfusion coefficient |

${P}_{r}$ | External heat source coefficient |

${P}_{mo}$ | Metabolic heat source coefficient |

${K}_{i}$ | Kirchhoff number |

${B}_{i}$ | Biot number |

${\zeta}^{*}$ | Location of tumor |

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**Figure 5.**Temperature distributions for different values of ${F}_{oq}$ and ${F}_{ot}$= $1.741\times {10}^{-2}$, 0.05631, and $1.267\times {10}^{-2}$.

**Figure 6.**Temperature distributions for different values of ${F}_{ot}$ and ${F}_{oq}$= 69.638, 0.7883, and $1.773\times {10}^{-2}$.

**Table 1.**Some parameters of multilayer skin tissue used in this paper [30].

Parameters | Epidermis | Dermis | Subcutaneous |
---|---|---|---|

Thickness (m) | 0.00008 | 0.002 | 0.01 |

Blood perfusion rate (kg m${}^{-3}$s${}^{-1}$) | 0 | 0.00125 | 0.00125 |

Thermal conductivity (W m${}^{-1\phantom{\rule{0.166667em}{0ex}}\circ}$C${}^{-1}$) | 0.24 | 0.45 | 0.19 |

Specific heat (J kg${}^{-1\phantom{\rule{0.166667em}{0ex}}\circ}$C${}^{-1}$) | 3590 | 3330 | 2500 |

Water diffusivity (m${}^{2}$s${}^{-1}$) | 5 × 10${}^{-10}$ | 5 × 10${}^{-10}$ | 5 × 10${}^{-10}$ |

Density (kg m${}^{-3}$) | 1200 | 1200 | 1200 |

Water content (%) | 70 | 70 | 70 |

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**MDPI and ACS Style**

Srivastava, H.M.; Irfan, M.; Shah, F.A.
A Fibonacci Wavelet Method for Solving Dual-Phase-Lag Heat Transfer Model in Multi-Layer Skin Tissue during Hyperthermia Treatment. *Energies* **2021**, *14*, 2254.
https://doi.org/10.3390/en14082254

**AMA Style**

Srivastava HM, Irfan M, Shah FA.
A Fibonacci Wavelet Method for Solving Dual-Phase-Lag Heat Transfer Model in Multi-Layer Skin Tissue during Hyperthermia Treatment. *Energies*. 2021; 14(8):2254.
https://doi.org/10.3390/en14082254

**Chicago/Turabian Style**

Srivastava, Hari Mohan, Mohd. Irfan, and Firdous A. Shah.
2021. "A Fibonacci Wavelet Method for Solving Dual-Phase-Lag Heat Transfer Model in Multi-Layer Skin Tissue during Hyperthermia Treatment" *Energies* 14, no. 8: 2254.
https://doi.org/10.3390/en14082254