The Laplace transform (LT) is a widely used methodology for analytical solutions of dual phase lag (DPL) heat conduction problems with consistent DPL boundary conditions (BCs). However, the inversion of LT requires a series summation with large number of terms for reasonably converged solution, thereby, increasing computational cost. In this work, an alternative approach is proposed for this inversion which is valid only for time-periodic BCs. In this approach, an approximate convolution integral is used to get an analytical closed-form solution for sinusoidal BCs (which is obviously free of numerical inversion or series summation). The ease of implementation and simplicity of the proposed alternative LT approach is demonstrated through illustrative examples for different kind of sinusoidal BCs. It is noted that the solution has very small error only during the very short initial transient and is (almost) exact for longer time. Moreover, it is seen from the illustrative examples that for high frequency periodic BCs the Fourier and DPL model give quite different results; however, for low frequency BCs the results are almost identical. For nonsinusoidal periodic function as BCs, Fourier series expansion of the function in time can be obtained and then present approach can be used for each term of the series. An illustrative example with a triangular periodic wave as one of the BC is solved and the error with different number of terms in the expansion is shown. It is observed that quite accurate solutions can be obtained with a fewer number of terms.

References

References
1.
Liu
,
W.
, and
Asheghi
,
M.
,
2004
, “
Thermal Modeling of Self-Heating in Strained-Silicon MOSFETs
,”
Ninth Intersociety Conference on Thermal and Thermomechanical Phenomena in Electronic Systems
, Las Vegas, NV, June 1–4, pp.
605
609
.
2.
Barron
,
R. F.
,
1999
,
Cryogenic Heat Transfer
,
Taylor & Francis
,
Washington, DC
.
3.
Piekarska
,
W.
, and
Kubiak
,
M.
,
2013
, “
Modeling of Thermal Phenomena in Single Laser Beam and Laser-Arc Hybrid Welding Processes Using Projection Method
,”
Appl. Math. Model.
,
37
(
4
), pp.
2051
2062
.
4.
Ghazanfarian
,
J.
,
Shomali
,
Z.
, and
Abbassi
,
A.
,
2015
, “
Macro- to Nanoscale Heat and Mass Transfer: The Lagging Behavior
,”
Int. J. Thermophys.
,
36
(
7
), pp.
1416
1467
.
5.
Zhou
,
J.
,
Chen
,
J. K.
, and
Zhang
,
Y.
,
2011
, “
Nonclassical Heat Transfer Models for Laser-Induced Thermal Damage in Biological Tissues
,”
ASME
Paper No. IMECE2011-62018
.
6.
Lee
,
H.-L.
,
Lai
,
T.-H.
,
Chen
,
W.-L.
, and
Yang
,
Y.-C.
,
2013
, “
An Inverse Hyperbolic Heat Conduction Problem in Estimating Surface Heat Flux of a Living Skin Tissue
,”
Appl. Math. Model.
,
37
(
5
), pp.
2630
2643
.
7.
Huang
,
C.-H.
, and
Huang
,
C.-Y.
,
2007
, “
An Inverse Problem in Estimating Simultaneously the Effective Thermal Conductivity and Volumetric Heat Capacity of Biological Tissue
,”
Appl. Math. Model.
,
31
(
9
), pp.
1785
1797
.
8.
Dehghan
,
M.
, and
Sabouri
,
M.
,
2012
, “
A Spectral Element Method for Solving the Pennes Bioheat Transfer Equation by Using Triangular and Quadrilateral Elements
,”
Appl. Math. Model.
,
36
(
12
), pp.
6031
6049
.
9.
Brorson
,
S. D.
,
Fujimoto
,
J. G.
, and
Ippen
,
E. P.
,
1987
, “
Femtosecond Electronic Heat-Transport Dynamics in Thin Gold Films
,”
Phys. Rev. Lett.
,
59
(
17
), pp.
1962
1965
.
10.
Qiu
,
T. Q.
,
Juhasz
,
T.
,
Suarez
,
C.
,
Bron
,
W. E.
, and
Tien
,
C. L.
,
1994
, “
Femtosecond Laser Heating of Multi-Layer Metals—II: Experiments
,”
Int. J. Heat Mass Transfer
,
37
(
17
), pp.
2799
2808
.
11.
Kaminski
,
W.
,
1990
, “
Hyperbolic Heat Conduction Equation for Materials With a Nonhomogeneous Inner Structure
,”
ASME J. Heat Transfer
,
112
(
3
), pp.
555
560
.
12.
Mitra
,
K.
,
Kumar
,
S.
,
Vedevarz
,
A.
, and
Moallemi
,
M. K.
,
1995
, “
Experimental Evidence of Hyperbolic Heat Conduction in Processed Meat
,”
ASME J. Heat Transfer
,
117
(
3
), pp.
568
573
.
13.
Antaki
,
P. J.
,
2005
, “
New Interpretation of Non-Fourier Heat Conduction in Processed Meat
,”
ASME J. Heat Transfer
,
127
(
2
), pp.
189
193
.
14.
Tzou
,
D. Y.
,
1995
, “
A Unified Field Approach for Heat Conduction From Macro- to Micro-Scales
,”
ASME J. Heat Transfer
,
117
(
1
), pp.
8
16
.
15.
Lewandowska
,
M.
, and
Malinowski
,
L.
,
2006
, “
An Analytical Solution of the Hyperbolic Heat Conduction Equation for the Case of a Finite Medium Symmetrically Heated on Both Sides
,”
Int. Commun. Heat Mass Transfer
,
33
(
1
), pp.
61
69
.
16.
Ghazanfarian
,
J.
, and
Abbassi
,
A.
,
2009
, “
Effect of Boundary Phonon Scattering on Dual-Phase-Lag Model to Simulate Micro- and Nano-Scale Heat Conduction
,”
Int. J. Heat Mass Transfer
,
52
(
15–16
), pp.
3706
3711
.
17.
Cattaneo
,
C.
, 1958, “
A Form of Heat Conduction Equation Which Eliminates the Paradox of Instantaneous Propagation
,” Comptes Rendus,
247
, pp. 431–433.
18.
Vernotte
,
P.
, 1958, “
Les paradoxes de la theorie continue de l'equa tion de la chaleur
,”
Comput. Rendus
,
246
, pp. 3154–3155.
19.
Vernotte
,
P.
, 1958, “
The True Heat Equation
,”
Comptes Rendus
,
247
, pp. 2103–2105.
20.
Vernotte
,
P.
, 1961, “
Some Possible Complications in the Phenomena of Thermal Conduction, Comptes
,”
Comptes Rendus
,
252
, pp. 2190–2091.
21.
Ho
,
J.-R.
,
Kuo
,
C.-P.
, and
Jiaung
,
W.-S.
,
2003
, “
Study of Heat Transfer in Multilayered Structure Within the Framework of Dual-Phase-Lag Heat Conduction Model Using Lattice Boltzmann Method
,”
Int. J. Heat Mass Transfer
,
46
(
1
), pp.
55
69
.
22.
Liu
,
K.-C.
, and
Chang
,
P.-C.
,
2007
, “
Analysis of Dual-Phase-Lag Heat Conduction in Cylindrical System With a Hybrid Method
,”
Appl. Math. Model.
,
31
(
2
), pp.
369
380
.
23.
Chou
,
Y.
, and
Yang
,
R.-J.
,
2009
, “
Two-Dimensional Dual-Phase-Lag Thermal Behavior in Single-/Multi-Layer Structures Using CESE Method
,”
Int. J. Heat Mass Transfer
,
52
(
1–2
), pp.
239
249
.
24.
Lee
,
Y.
,
Lin
,
P.
, and
Tsai
,
T.
,
2009
, “
Green's Function Solution of Dual-Phase-Lag Model
,”
ASME
Paper No. MNHMT2009-18425
.
25.
Alkhairy
,
R.
,
2012
, “
Green's Function Solution for the Dual-Phase-Lag Heat Equation
,”
Appl. Math.
,
3
(
10
), pp.
1170
1178
.
26.
Wang
,
L.
,
Xu
,
M.
, and
Zhou
,
X.
,
2001
, “
Well-Posedness and Solution Structure of Dual-Phase-Lagging Heat Conduction
,”
Int. J. Heat Mass Transfer
,
44
(
9
), pp.
1659
1669
.
27.
Wang
,
L.
, and
Xu
,
M.
,
2002
, “
Well-Posedness of Dual-Phase-Lagging Heat Conduction Equation: Higher Dimensions
,”
Int. J. Heat Mass Transfer
,
45
(
5
), pp.
1165
1171
.
28.
Kumar
,
S.
, and
Srivastava
,
A.
,
2017
, “
Finite Integral Transform-Based Analytical Solutions of Dual Phase Lag Bio-Heat Transfer Equation
,”
Appl. Math. Model.
,
52
, pp.
378
403
.
29.
Biswas
,
P.
, and
Singh
,
S.
,
2018
, “
Orthogonal Eigenfunction Expansion Method for One-Dimensional Dual-Phase Lag Heat Conduction Problem With Time-Dependent Boundary Conditions
,”
ASME J. Heat Transfer
,
140
(
3
), p.
034501
.
30.
Quaresma
,
J. N. N.
,
Macêdo
,
E. N.
,
da Fonseca
,
H. M.
,
Orlande
,
H. R. B.
, and
Cotta
,
R. M.
,
2010
, “
An Analysis of Heat Conduction Models for Nanofluids
,”
Heat Transfer Eng.
, 31(
14
), pp.
1125
1136
.
31.
Liu
,
K.-C.
, and
Cheng
,
P.-J.
,
2006
, “
Numerical Analysis for Dual-Phase-Lag Heat Conduction in Layered Films
,”
Numer. Heat Transfer Part A Appl.
,
49
(
6
), pp.
589
606
.
32.
Kumar
,
S.
, and
Srivastava
,
A.
,
2015
, “
Thermal Analysis of Laser-Irradiated Tissue Phantoms Using Dual Phase Lag Model Coupled With Transient Radiative Transfer Equation
,”
Int. J. Heat Mass Transfer
,
90
, pp.
466
479
.
33.
Ramadan
,
K.
,
2008
, “
Treatment of the Interfacial Temperature Jump Condition With Non-Fourier Heat Conduction Effects
,”
Int. Commun. Heat Mass Transfer
,
35
(
9
), pp.
1177
1182
.
34.
Askarizadeh
,
H.
, and
Ahmadikia
,
H.
,
2014
, “
Analytical Analysis of the Dual-Phase-Lag Heat Transfer Equation in a Finite Slab With Periodic Surface Heat Flux
,”
Int. J. Eng.
,
27
(
6
), pp.
971
978
.http://www.ije.ir/abstract/%7BVolume:27-Transactions:C-Number:6%7D/=1709
35.
Biswas
,
P.
, and
Singh
,
S.
,
2016
, “
Analytical Solution of 1-D Multiple Layer Dual Phase Lag Heat Conduction Problem With Generalized Time Dependent Boundary Condition
,”
12th International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics
, Costa Del Sol, Spain, July 11–14, pp.
1557
1561
.https://repository.up.ac.za/bitstream/handle/2263/62017/Biswas_Analytical_2016.pdf?sequence=1&isAllowed=y
36.
Ramadan
,
K.
,
2009
, “
Semi-Analytical Solutions for the Dual Phase Lag Heat Conduction in Multilayered Media
,”
Int. J. Therm. Sci.
,
48
(
1
), pp.
14
25
.
37.
Asan
,
H.
, and
Sancaktar
,
Y. S.
,
1998
, “
Effects of Wall's Thermophysical Properties on Time Lag and Decrement Factor
,”
Energy Build.
,
28
(
2
), pp.
159
166
.
38.
Asan
,
H.
,
2006
, “
Numerical Computation of Time Lags and Decrement Factors for Different Building Materials
,”
Build. Environ.
,
41
(
5
), pp.
615
620
.
39.
Glass
,
D. E.
,
Özişik
,
M. N.
, and
Vick
,
B.
,
1987
, “
Non-Fourier Effects on Transient Temperature Resulting From Periodic On-Off Heat Flux
,”
Int. J. Heat Mass Transfer
,
30
(
8
), pp.
1623
1631
.
40.
Barletta
,
A.
, and
Zanchini
,
E.
,
1996
, “
Hyperbolic Heat Conduction and Thermal Resonances in a Cylindrical Solid Carrying a Steady-Periodic Electric Field
,”
Int. J. Heat Mass Transfer
,
39
(
6
), pp.
1307
1315
.
41.
Liu
,
J.
, and
Xu
,
L. X.
,
1999
, “
Estimation of Blood Perfusion Using Phase Shift in Temperature Response to Sinusoidal Heating at the Skin Surface
,”
IEEE Trans. Biomed. Eng.
,
46
(
9
), pp.
1037
1043
.
42.
Yuen
,
W. W.
, and
Lee
,
S. C.
,
1989
, “
Non-Fourier Heat Conduction in a Semi-Infinite Solid Subjected to Oscillatory Surface Thermal Disturbances
,”
ASME J. Heat Transfer
,
111
(
1
), pp.
178
181
.
43.
Roetzel
,
W.
,
Putra
,
N.
, and
Das
,
S. K.
,
2003
, “
Experiment and Analysis for Non-Fourier Conduction in Materials With Non-Homogeneous Inner Structure
,”
Int. J. Therm. Sci.
,
42
(
6
), pp.
541
552
.
44.
Ordóñez-Miranda
,
J.
, and
Alvarado-Gil
,
J. J.
,
2009
, “
Thermal Wave Oscillations and Thermal Relaxation Time Determination in a Hyperbolic Heat Transport Model
,”
Int. J. Therm. Sci.
,
48
(
11
), pp.
2053
2062
.
45.
Ziaei Poor
,
H.
,
Moosavi
,
H.
, and
Moradi
,
A.
,
2016
, “
Analysis of the Dual Phase Lag Bio-Heat Transfer Equation With Constant and Time-Dependent Heat Flux Conditions on Skin Surface
,”
Therm. Sci.
,
20
(
5
), pp.
1457
1472
.
46.
Tang
,
D. W.
, and
Araki
,
N.
,
1996
, “
Non-Fourier Heat Conduction in a Finite Medium Under Periodic Surface Thermal Disturbance
,”
Int. J. Heat Mass Transfer
,
39
(
8
), pp.
1585
1590
.
47.
Tang
,
D. W.
, and
Araki
,
N.
,
1996
, “
Non-Fourier Heat Conduction in a Finite Medium Under Periodic Surface Thermal Disturbance—II: Another Form of Solution
,”
Int. J. Heat Mass Transfer
,
39
(
15
), pp.
3305
3308
.
48.
Antaki
,
P. J.
,
1997
, “
Analysis of Hyperbolic Heat Conduction in a Semi-Infinite Slab With Surface Convection
,”
Int. J. Heat Mass Transfer
,
40
(
13
), pp.
3247
3250
.
49.
Mikhailov
,
M. D.
, and
Cotta
,
R. M.
,
1997
, “
Steady-Periodic Hyperbolic Heat Conduction in a Finite Slab
,”
Int. Commun. Heat Mass Transfer
,
24
(
5
), pp.
725
731
.
50.
Abdel-Hamid
,
B.
,
1999
, “
Modelling Non-Fourier Heat Conduction With Periodic Thermal Oscillation Using the Finite Integral Transform
,”
Appl. Math. Model.
,
23
(
12
), pp.
899
914
.
51.
Cossali
,
G. E.
,
2004
, “
Periodic Conduction in Materials With Non-Fourier Behaviour
,”
Int. J. Therm. Sci.
,
43
(
4
), pp.
347
357
.
52.
Cossali
,
G. E.
,
2009
, “
Periodic Heat Conduction in a Solid Homogeneous Finite Cylinder
,”
Int. J. Therm. Sci.
,
48
(
4
), pp.
722
732
.
53.
Li
,
J.
,
Cheng
,
P.
,
Peterson
,
G. P.
, and
Xu
,
J. Z.
,
2005
, “
Rapid Transient Heat Conduction in Multilayer Materials With Pulsed Heating Boundary
,”
Numer. Heat Transf. Part A Appl.
,
47
(
7
), pp.
633
652
.
54.
Moosaie
,
A.
,
2007
, “
Non-Fourier Heat Conduction in a Finite Medium Subjected to Arbitrary Periodic Surface Disturbance
,”
Int. Commun. Heat Mass Transfer
,
34
(
8
), pp.
996
1002
.
55.
Moosaie
,
A.
,
2008
, “
Non-Fourier Heat Conduction in a Finite Medium Subjected to Arbitrary Non-Periodic Surface Disturbance
,”
Int. Commun. Heat Mass Transfer
,
35
(
3
), pp.
376
383
.
56.
Shirmohammadi
,
R.
, and
Moosaie
,
A.
,
2009
, “
Non-Fourier Heat Conduction in a Hollow Sphere With Periodic Surface Heat Flux
,”
Int. Commun. Heat Mass Transfer
,
36
(
8
), pp.
827
833
.
57.
Shen
,
B.
, and
Zhang
,
P.
,
2008
, “
Notable Physical Anomalies Manifested in Non-Fourier Heat Conduction Under the Dual-Phase-Lag Model
,”
Int. J. Heat Mass Transfer
,
51
(
7–8
), pp.
1713
1727
.
58.
Al-Nimr
,
M. A.
,
Naji
,
M.
, and
Abdallah
,
R. I.
,
2004
, “
Thermal Behavior of a Multi-Layered Thin Slab Carrying Periodic Signals Under the Effect of the Dual-Phase-Lag Heat Conduction Model
,”
Int. J. Thermophys.
,
25
(
3
), pp.
949
966
.
59.
Lu
,
X.
,
Tervola
,
P.
, and
Viljanen
,
M.
,
2005
, “
A New Analytical Method to Solve the Heat Equation for a Multi-Dimensional Composite Slab
,”
J. Phys. A. Math. Gen.
,
38
(
13
), pp.
2873
2890
.
60.
Lu
,
X.
,
Tervola
,
P.
, and
Viljanen
,
M.
,
2005
, “
An Efficient Analytical Solution to Transient Heat Conduction in a One-Dimensional Hollow Composite Cylinder
,”
J. Phys. A. Math. Gen.
,
38
(
47
), pp.
10145
10155
.
61.
Lu
,
X.
, and
Viljanen
,
M.
,
2006
, “
An Analytical Method to Solve Heat Conduction in Layered Spheres With Time-Dependent Boundary Conditions
,”
Phys. Lett. A
,
351
(
4–5
), pp.
274
282
.
62.
,
X.
,
Lu
,
T.
, and
Viljanen
,
M.
,
2006
, “
A New Analytical Method to Simulate Heat Transfer Process in Buildings
,”
Appl. Therm. Eng.
,
26
(
16
), pp.
1901
1909
.
You do not currently have access to this content.