Estimating Relaxation Time and Fractionality Order Parameters in Fractional Non-Fourier Heat Conduction Using Conjugate Gradient Inverse Approach in Single and Three-Layer Skin Tissues
Abstract
:1. Introduction
2. Conjugate Gradient Inverse Method
3. Governing Equations and Discretization
3.1. Test Case 1
3.1.1. Direct Problem
3.1.2. Inverse Problem
3.2. Test Case 2
3.2.1. Direct Problem
3.2.2. Inverse Problem
4. Results and Discussion
4.1. Test Case 1
4.2. Test Case 2
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Nomenclature
Latin symbols | |
c | tissue heat capacity, Jkg−1 K−1 |
cb | blood heat capacity Jkg−1 K−1 |
D | coefficient of thermal diffusion WJ−1 m−3 |
time derivative order | |
f(t) | continuous function |
h | skin thickness (mm) |
sensitivity coefficient of heat respect to order of fractionality | |
sensitivity coefficient of heat respect to time lag | |
Jα | sensitivity coefficient respect to order of fractionality |
Jτ | sensitivity coefficient respect to time lag |
k | tissue thermal conductivity, Wm−1 K−1 |
m | number of iterations |
P | unknown parameters in inverse problem |
qgen | generated heat in tissue, Wm−3 |
qmet | metabolic heating source, Wm−3 |
Qin | laser intensity, Wcm−2 |
Rd | diffusion reflection |
t | time, s |
tf | time duration from the onset to the end, s |
tτ | time period of laser radiation on the skin, s |
u(t) | unit step function |
wb | blood perfusion rate, m3m−3 tissue |
wj | average of weighted arithmetic |
wr | weight function |
Greek symbols | |
α | order of fractional derivative |
α0 | initial guess for order of fractional derivative |
ε | error tolerance |
θ | temperature of tissue, °C |
θ0 | initial temperature, °C |
θb | blood temperature, °C |
θc | calculated temperature, °C |
θDPL | measured temperature, °C |
ρ | density of tissue, kgm−3 |
ρb | blood density, kgm−3 |
Γ | gamma function |
τ | time lag |
τT | temperature gradient time lag |
τq | heat flux time lag |
τ0 | initial guess for time lag |
Abbreviations | |
ACGM | Adjoint Conjugate Gradient Method |
CGIM | Conjugate Gradient inverse Method |
DF | Dermic-Fat interface |
DPL | Dual Phase Lag |
ED | Epidermis-Dermic interface |
FSPL | Fractional Single-Phase Lag |
SPL | Single Phase Lag |
Appendix A
Appendix B
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CGIM | DPL | |||
---|---|---|---|---|
α | τ | τT | τq | |
Sample A | 0.985707860045904 | 9.88794652835751 | 0.005 | 10 |
Sample B | 0.982593197834287 | 9.82287275801926 | 0.05 | 10 |
Sample C | 0.958513945252591 | 9.87599109128811 | 0.1 | 10 |
Sample D | 0.933915800219504 | 0.57923029381419 | 0.05 | 1 |
Sample E | 0.982593197834287 | 9.82287275801926 | 0.05 | 10 |
Sample F | 0.989762788979733 | 14.8313003950098 | 0.05 | 15 |
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Goudarzi, P.; Abidi, A.; Mehryan, S.A.M.; Ghalambaz, M.; Sheremet, M.A. Estimating Relaxation Time and Fractionality Order Parameters in Fractional Non-Fourier Heat Conduction Using Conjugate Gradient Inverse Approach in Single and Three-Layer Skin Tissues. Processes 2021, 9, 1877. https://doi.org/10.3390/pr9111877
Goudarzi P, Abidi A, Mehryan SAM, Ghalambaz M, Sheremet MA. Estimating Relaxation Time and Fractionality Order Parameters in Fractional Non-Fourier Heat Conduction Using Conjugate Gradient Inverse Approach in Single and Three-Layer Skin Tissues. Processes. 2021; 9(11):1877. https://doi.org/10.3390/pr9111877
Chicago/Turabian StyleGoudarzi, Piran, Awatef Abidi, Seyed Abdollah Mansouri Mehryan, Mohammad Ghalambaz, and Mikhail A. Sheremet. 2021. "Estimating Relaxation Time and Fractionality Order Parameters in Fractional Non-Fourier Heat Conduction Using Conjugate Gradient Inverse Approach in Single and Three-Layer Skin Tissues" Processes 9, no. 11: 1877. https://doi.org/10.3390/pr9111877