Abstract
The non-Fourier heat transfer in a half-space is analyzed under sudden heating or cooling on a local surface. The non-Fourier heat transfer effect is described by the time-fractional dual-phase-lag (DPL) model, where the fractional derivative without singular kernel is used. An axisymmetric mixed initial-boundary value problem is solved by the use of the Hankel and Laplace transforms. Two typical cases of sudden temperature rising on a circular zone of the surface or an instantaneous surface heat source are analyzed. For sudden temperature rises, the heat flux and temperature gradient exhibit an inverse square-root singularity near the boundary of the heating zone and their dynamic intensity factors are computed numerically in the time domain. For the instantaneous surface point heat source, an exact solution of the transient temperature at any position in the Laplace domain is obtained. The effects of the fractional order and relaxation time on the temperature distribution and heat flux response are elucidated. The singular behavior of the transient thermal response and the non-Fourier effect of heat transfer are shown.
Issue Section:
Research Papers
References
1.
McNelly
,
T. F.
,
Rogers
,
S. J.
,
Channin
,
D. J.
,
Rollefson
,
R. J.
,
Goubau
,
W. M.
,
Schmidt
,
G. E.
,
Krumhansl
,
J. A.
, and
Pohl
,
R. O.
, 1970
, “
Heatpulses in NaF: Onset of Second Sound
,” Phys. Rev. Lett.
,
24
(3
), pp. 100
–102
.10.1103/PhysRevLett.24.100
2.
Mitra
,
K.
,
Kumar
,
S.
,
Vedevarz
,
A.
, and
Moallemi
,
M. K.
, 1995
, “
Experimental Evidence of Hyperbolic Heat Conduction in Processed Meat
,” ASME J. Heat Mass Transfer-Trans. ASME
,
117
(3
), pp. 568
–573
.10.1115/1.2822615
3.
Liu
,
J.
,
Chen
,
X.
, and
Xu
,
L. X.
, 1999
, “
New Thermal Wave Aspects on Burn Evaluation of Skin Subjected to Instantaneous Heating
,” IEEE Trans. Biomed. Eng.
,
46
(4
), pp. 420
–428
.10.1109/10.752939
4.
Chandrasekharaiah
,
D. S.
, 1998
, “
Hyperbolic Thermoelasticity: A Review of Recent Literature
,” Appl. Mech. Rev.
,
51
(12
), pp. 705
–729
.10.1115/1.3098984
5.
Tzou
,
D. Y.
, 1995
, “
The Generalized Lagging Response in Small-Scale and High-Rate Heating
,” Int. J. Heat Mass Transfer
,
38
(17
), pp. 3231
–3240
.10.1016/0017-9310(95)00052-B
6.
Choudhuri
,
S. K. R.
, 2007
, “
On a Thermoelastic Three-Phase-Lag Model
,” J. Therm. Stresses
,
30
(3
), pp. 231
–238
.10.1080/01495730601130919
7.
Dong
,
Y.
,
Cao
,
B.-Y.
, and
Guo
,
Z.-Y.
, 2013
, “
Temperature in Nonequilibrium States and non-Fourier Heat Conduction
,” Phys. Rev. E
,
87
(3
), p. 032150
.10.1103/PhysRevE.87.032150
8.
Alvarez
,
F. X.
,
Jou
,
D.
, and
Sellitto
,
A.
, 2011
, “
Phonon Boundary Effects and Thermal Conductivity of Rough Concentric Nanowires
,” ASME J. Heat Mass Transfer-Trans. ASME
,
133
(2
), p. 022402
.10.1115/1.4002439
9.
Kovacs
,
R.
, and
Van
,
P.
, 2018
, “
Second Sound and Ballistic Heat Conduction: NaF Experiments Revisited
,” Int. J. Heat Mass Transfer
,
117
, pp. 682
–690
.10.1016/j.ijheatmasstransfer.2017.10.041
10.
Kozlov
,
V. P.
, and
Mandrik
,
P. A.
, 2000
, “
Nonstationary Temperature Fields in an Isotropic Half-Space Under Mixed Boundary Conditions Characteristic of Technologies of Laser Therapy in Medicine
,” J. Eng. Phys. Thermophys.
,
73
(3
), pp. 625
–633
.10.1007/BF02681807
11.
Yevtushenko
,
A.
, and
Matysiak
,
S. J.
, 2005
, “
On Approximate Solutions of Temperature and Thermal Stresses in an Elastic Semi-Space Due to Laser Heating
,” Numer. Heat Transfer, Part A
,
47
(9
), pp. 899
–915
.10.1080/10407780590926147
12.
Banerjee
,
A.
,
Ogale
,
A. A.
,
Das
,
C.
,
Mitra
,
K.
, and
Subramanian
,
C.
, 2005
, “
Temperature Distribution in Different Materials Due to Short Pulse Laser Irradiation
,” Heat Transfer Eng.
,
26
(8
), pp. 41
–49
.10.1080/01457630591003754
13.
Zhang
,
S.
, and
Liu
,
Z.
, 2008
, “
An Analytical Model for Transient Temperature Distributions in Coated Carbide Cutting Tools
,” Int. Commun. Heat Mass Transfer
,
35
(10
), pp. 1311
–1315
.10.1016/j.icheatmasstransfer.2008.08.001
14.
Chandran
,
K. R.
, 2015
, “
Transient Joule Heating of Graphene, Nanowires and Filaments: Analytical Model for Current-Induced Temperature Evolution Including Substrate and End Effects
,” Int. J. Heat Mass Transfer
,
88
, pp. 14
–19
.10.1016/j.ijheatmasstransfer.2015.04.014
15.
Araya
,
G.
, and
Gutierrez
,
G.
, 2006
, “
Analytical Solution for a Transient, Three-Dimensional Temperature Distribution Due to a Moving Laser Beam
,” Int. J. Heat Mass Transfer
,
49
(21–22
), pp. 4124
–4131
.10.1016/j.ijheatmasstransfer.2006.03.026
16.
Laraqi
,
N.
, 2010
, “
Thermal Impedance and Transient Temperature Due to a Spot of Heat on a Half-Space
,” Int. J. Therm. Sci.
,
49
(3
), pp. 529
–533
.10.1016/j.ijthermalsci.2009.10.004
17.
Winczek
,
J.
, 2010
, “
Analytical Solution to Transient Temperature Field in a Half-Infinite Body Caused by Moving Volumetric Heat Source
,” Int. J. Heat Mass Transfer
,
53
(25–26
), pp. 5774
–5781
.10.1016/j.ijheatmasstransfer.2010.07.065
18.
Salimi
,
S.
,
Bahemmat
,
P.
, and
Haghpanahi
,
M.
, 2014
, “
A 3d Transient Analytical Solution to the Temperature Field During Dissimilar Welding Processes
,” Int. J. Mech. Sci.
,
79
, pp. 66
–74
.10.1016/j.ijmecsci.2013.11.015
19.
Kordkheili
,
H. M.
,
Amiri
,
G. G.
, and
Hosseini
,
M.
, 2017
, “
Axisymmetric Analysis of a Thermoelastic Isotropic Half-Space Under Buried Sources in Displacement and Temperature Potentials
,” J. Therm. Stresses
,
40
(2
), pp. 237
–254
.10.1080/01495739.2016.1234342
20.
Flint
,
T.
,
Francis
,
J.
,
Smith
,
M.
, and
Vasileiou
,
A.
, 2018
, “
Semi-Analytical Solutions for the Transient Temperature Fields Induced by a Moving Heat Source in an Orthogonal Domain
,” Int. J. Therm. Sci.
,
123
, pp. 140
–150
.10.1016/j.ijthermalsci.2017.09.012
21.
Laraqi
,
N.
,
Ramassamy
,
H.
,
Rajaoarisoa
,
O.
,
Wauthier
,
T.
, and
Bauzin
,
J.-G.
, 2019
, “
New Exact Analytical Solutions for the Transient Surface Temperature of Solids Subjected to a Non-Uniform Axisymmetric Circular Heat Source
,” Int. J. Therm. Sci.
,
145
, p. 106034
.10.1016/j.ijthermalsci.2019.106034
22.
Emanuel
,
M.
,
Bhouri
,
M.
,
Ackermann
,
S. L.
,
Groulx
,
D.
, and
Maassen
,
J.
, 2018
, “
Temperature Fields Generated by a Circular Heat Source (CHS) in an Infinite Medium: Analytical Derivation and Comparison to Finite Element Modeling
,” Int. J. Heat Mass Transfer
,
126
, pp. 1265
–1274
.10.1016/j.ijheatmasstransfer.2018.05.102
23.
Emanuel
,
M.
, and
Emanuel
,
A.
, 2019
, “
Temperature Fields Generated by a Circular Heat Source: Solution of a Composite Solid of Two Different Isotropic Semi-Infinite Media
,” ASME J. Heat Mass Transfer-Trans. ASME
,
141
(12
), p. 121401
.10.1115/1.4044829
24.
Zhang
,
L.
, and
Shang
,
X.
, 2015
, “
Analytical Solution to Non-Fourier Heat Conduction as a Laser Beam Irradiating on Local Surface of a Semi-Infinite Medium
,” Int. J. Heat Mass Transfer
,
85
, pp. 772
–780
.10.1016/j.ijheatmasstransfer.2015.02.024
25.
Antaki
,
P. J.
, 2005
, “
New Interpretation of Non-Fourier Heat Conduction in Processed Meat
,” ASME J. Heat Mass Transfer-Trans. ASME
,
127
(2
), pp. 189
–193
.10.1115/1.1844540
26.
Ramadan
,
K.
,
Tyfour
,
W. R.
, and
Al-Nimr
,
M. A.
, 2009
, “
On the Analysis of Short-Pulse Laser Heating of Metals Using the Dual Phase Lag Heat Conduction Model
,” ASME J. Heat Mass Transfer-Trans. ASME
,
131
(11
), p. 111301
.10.1115/1.3153580
27.
Akbarzadeh
,
A.
, and
Pasini
,
D.
, 2014
, “
Phase-Lag Heat Conduction in Multilayered Cellular Media With Imperfect Bonds
,” Int. J. Heat Mass Transfer
,
75
, pp. 656
–667
.10.1016/j.ijheatmasstransfer.2014.03.090
28.
Lee
,
H.-L.
,
Chen
,
W.-L.
,
Chang
,
W.-J.
,
Wei
,
E.-J.
, and
Yang
,
Y.-C.
, 2013
, “
Analysis of Dual-Phase-Lag Heat Conduction in Short-Pulse Laser Heating of Metals With a Hybrid Method
,” Appl. Therm. Eng
,
52
(2
), pp. 275
–283
.10.1016/j.applthermaleng.2012.12.019
29.
Li
,
S.-N.
, and
Cao
,
B.-Y.
, 2020
, “
Fractional-Order Heat Conduction Models From Generalized Boltzmann Transport Equation
,” Philos. Trans. R. Soc. A
,
378
(2172
), p. 20190280
.10.1098/rsta.2019.0280
30.
Sherief
,
H. H.
,
El-Sayed
,
A. M. A.
, and
Abd El-Latief
,
A. M.
, 2010
, “
Fractional Order Theory of Thermoelasticity
,” Int. J. Solids Struct.
,
47
(2
), pp. 269
–275
.10.1016/j.ijsolstr.2009.09.034
31.
Youssef
,
H. M.
, 2010
, “
Theory of Fractional Order Generalized Thermoelasticity
,” ASME J. Heat Mass Transfer-Trans. ASME
,
132
(6
), p. 061301
.10.1115/1.4000705
32.
Ezzat
,
M. A.
, 2011
, “
Magneto-Thermoelasticity With Thermoelectric Properties and Fractional Derivative Heat Transfer
,” Phys. B
,
406
(1
), pp. 30
–35
.10.1016/j.physb.2010.10.005
33.
Povstenko
,
Y.
, 2015
, Fractional Thermoelasticity
, Vol.
219
,
Springer
.34.
Povstenko
,
Y.
, and
Kyrylych
,
T.
, 2020
, “
Fractional Thermoelasticity Problem for an Infinite Solid With a Penny-Shaped Crack Under Prescribed Heat Flux Across Its Surfaces
,” Philos. Trans. R. Soc. A
,
378
(2172
), p. 20190289
.10.1098/rsta.2019.0289
35.
Zhang
,
X.-Y.
,
Peng
,
Y.
, and
Li
,
X.-F.
, 2018
, “
Time-Fractional Hygrothermoelastic Problem for a Sphere Subjected to Heat and Moisture Flux
,” ASME J. Heat Mass Transfer-Trans. ASME
,
140
(12
), p. 122002
.10.1115/1.4041419
36.
Zhang
,
X.-Y.
,
Peng
,
Y.
,
Xie
,
Y.-J.
, and
Li
,
X.-F.
, 2019
, “
Hygrothermoelastic Response of a Hollow Cylinder Based on a Coupled Time-Fractional Heat and Moisture Transfer Model
,” Z. Angew. Math. Phys.
,
70
(1
), p. 2
.10.1007/s00033-018-1047-1
37.
Hussein
,
E. M.
, 2019
, “
Effect of Fractional Parameter on Thermoelastic Half-Space Subjected to a Moving Heat Source
,” Int. J. Heat Mass Transfer
,
141
, pp. 855
–860
.10.1016/j.ijheatmasstransfer.2019.06.094
38.
Zhang
,
X.-Y.
, and
Li
,
X.-F.
, 2017
, “
Transient Thermal Stress Intensity Factors for a Circumferential Crack in a Hollow Cylinder Based on Generalized Fractional Heat Conduction
,” Int. J. Therm. Sci.
,
121
, pp. 336
–347
.10.1016/j.ijthermalsci.2017.07.015
39.
Zhang
,
X.-Y.
,
Chen
,
Z.-T.
, and
Li
,
X.-F.
, 2019
, “
Non-Fourier Fractional Heat Conduction in Two Bonded Dissimilar Materials With a Penny-Shaped Interface Crack
,” Int. J. Therm. Sci.
,
140
, pp. 319
–328
.10.1016/j.ijthermalsci.2019.03.001
40.
Ezzat
,
M. A.
,
El-Karamany
,
A. S.
, and
Ezzat
,
S. M.
, 2012
, “
Two-Temperature Theory in Magneto-Thermoelasticity With Fractional Order Dual-Phase-Lag Heat Transfer
,” Nucl. Eng. Des.
,
252
, pp. 267
–277
.10.1016/j.nucengdes.2012.06.012
41.
Xu
,
H.-Y.
, and
Jiang
,
X.-Y.
, 2015
, “
Time Fractional Dual-Phase-Lag Heat Conduction Equation
,” Chin. Phys. B
,
24
(3
), p. 034401
.10.1088/1674-1056/24/3/034401
42.
Yang
,
W.
, and
Chen
,
Z.
, 2020
, “
Investigation of Transient Thermal-Mechanical Behavior of a Cracked Viscoelastic Material Using Time-Fractional Dual-Phase-Lag Theory
,” Theor. Appl. Fract. Mech
,
106
, p. 102500
.10.1016/j.tafmec.2020.102500
43.
Hu
,
Y.
,
Zhang
,
X.-Y.
, and
Li
,
X.-F.
, 2022
, “
Thermoelastic Response of Skin Using Time-Fractional Dual-Phase-Lag Bioheat Heat Transfer Equation
,” J. Therm. Stresses
,
45
(7
), pp. 597
–615
.10.1080/01495739.2022.2078452
44.
Hu
,
Y.
,
Zhang
,
X.-Y.
, and
Li
,
X.-F.
, 2022
, “
Thermoelastic Analysis of Biological Tissue During Hyperthermia Treatment for Moving Laser Heating Using Fractional Dual-Phase-Lag Bioheat Conduction
,” Int. J. Therm. Sci.
,
182
, p. 107806
.10.1016/j.ijthermalsci.2022.107806
45.
Tzou
,
D. Y.
, 2015
, Macro- to Microscale Heat Transfer
,
Wiley
, West Sussex, UK
.46.
Caputo
,
M.
, and
Fabrizio
,
M.
, 2016
, “
A New Definition of Fractional Derivative Without Singular Kernel
,” Progr. Fract. Differ. Appl.
,
2
(1
), pp. 1
–11
.10.18576/pfda/020101
47.
Tzou
,
D. Y.
, 1990
, “
The Singular Behavior of the Temperature Gradient in the Vicinity of a Macrocrack Tip
,” Int. J. Heat Mass Transfer
,
33
(12
), pp. 2625
–2630
.10.1016/0017-9310(90)90198-4
48.
Carslaw
,
H. S.
, and
Jaeger
,
J. C.
, 1959
, Conduction of Heat in Solids
,
Oxford University Press
, London, UK
.49.
Antaki
,
P. J.
, 2000
, “
Effect of Dual-Phase-Lag Heat Conduction on Ignition of a Solid
,” J. Thermophys. Heat Transfer
,
14
(2
), pp. 276
–278
.10.2514/2.6520
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