The current study presents an exact analytical solution for unsteady conductive heat transfer in multilayer spherical fiber-reinforced composite laminates. The orthotropic heat conduction equation in spherical coordinate is introduced. The most generalized linear boundary conditions consisting of the conduction, convection, and radiation heat transfer is considered both inside and outside of spherical laminate. The fibers' angle and composite material in each lamina can be changed. Laplace transformation is employed to change the domain of the solutions from time into the frequency. In the frequency domain, the separation of variable method is used and the set of equations related to the coefficients of Fourier–Legendre series is solved. Meromorphic function technique is utilized to determine the complex inverse Laplace transformation. Two functional cases are presented to investigate the capability of current solution for solving the industrial unsteady problems in different arrangements of multilayer spherical laminates.

References

References
1.
Ding
,
Y.
,
Chiu
,
W. K.
,
Liu
,
X. L.
, and
Whittingham
,
B.
,
2001
, “
Modeling of Thermal Response of Oil-Heated Tools Due to Different Flow Rates for the Manufacture of Composite Structures
,”
Compos. Struct.
,
54
(
4
), pp.
477
488
.10.1016/S0263-8223(01)00120-9
2.
Shojaei
,
A.
,
Ghaffarian
,
S. R.
, and
Karimian
,
S. M. H.
,
2004
, “
Three-Dimensional Process Cycle Simulation of Composite Parts Manufactured by Resin Transfer Molding
,”
Compos. Struct.
,
65
(
3–4
), pp.
381
390
.10.1016/j.compstruct.2003.12.001
3.
Joshi
,
S. C.
,
Liu
,
X. L.
, and
Lam
,
Y. C.
,
1999
, “
A Numerical Approach to the Modeling of Polymer Curing in Fiber-Reinforced Composites
,”
Compos. Sci. Technol.
,
59
(
7
), pp.
1003
1013
.10.1016/S0266-3538(98)00138-9
4.
Blanchard
,
P. J.
, and
Rudd
,
C. D.
,
1996
, “
Cycle Time Reduction in Resin Transfer Moulding by Phased Catalyst Injection
,”
Compos. Sci. Technol.
,
56
(
2
), pp.
123
133
.10.1016/0266-3538(95)00127-1
5.
Ju
,
D. Y.
,
2000
, “
Simulation of Thermo-Mechanical Behavior and Interfacial Stress of Metal Matrix Composite Under Thermal Shock Process
,”
Compos. Struct.
,
48
(
1
), pp.
113
118
.10.1016/S0263-8223(99)00082-3
6.
Agbossou
,
A.
, and
Pastor
,
J.
,
1997
, “
Thermal Stresses and Thermal Expansion Coefficients of n-Layered Fiber-Reinforced Composites
,”
Compos. Sci. Technol.
,
57
(
2
), pp.
249
260
.10.1016/S0266-3538(96)00137-6
7.
Dvorak
,
G. J.
,
Chen
,
T.
, and
Teply
,
J.
,
1992
, “
Thermomechanical Stress Fields in High Temperature Fibrous Composites. I: Unidirectional Laminates
,”
Compos. Sci. Technol.
,
43
(
4
), pp.
347
358
.10.1016/0266-3538(92)90058-B
8.
Ootao
,
Y.
, and
Tanigawa
,
Y.
,
2002
, “
Transient Thermal Stresses of Angle-Ply Laminated Cylindrical Panel Due to Nonuniform Heat Supply in the Circumferential Direction
,”
Compos. Struct.
,
55
(
1
), pp.
95
103
.10.1016/S0263-8223(01)00132-5
9.
Ageorges
,
C.
,
Ye
,
L.
, and
Hou
,
M.
,
2000
, “
Experimental Investigation of the Resistance Welding of Thermoplastic-Matrix Composites. Part II: Optimum Processing Window and Mechanical Performance
,”
Compos. Sci. Technol.
,
60
(
8
), pp.
1191
1202
.10.1016/S0266-3538(00)00025-7
10.
Dülk
,
I.
, and
Kovácsházy
,
T.
,
2013
, “
Steady-State Heat Conduction in Multilayer Bodies: An Analytical Solution and Simplification of the Eigenvalue Problem
,”
Int. J. Heat Mass Transfer
,
67
, pp.
784
797
.10.1016/j.ijheatmasstransfer.2013.08.070
11.
Ozisik
,
M. N.
,
1993
,
Heat Conduction
,
2nd ed.
,
Wiley
,
New York
.
12.
Carslaw
,
H. S.
, and
Jaeger
,
J. C.
,
1959
,
Conduction of Heat in Solids
,
2nd ed.
,
Clarendon Press
,
Oxford, UK
.
13.
Lu
,
X.
,
Tervola
,
P.
, and
Viljanen
,
M.
,
2006
, “
Transient Analytical Solution to Heat Conduction in Composite Circular Cylinder
,”
Int. J. Heat Mass Transfer
,
49
(
1–2
), pp.
341
348
.10.1016/j.ijheatmasstransfer.2005.06.019
14.
Desgrosseilliers
,
L.
,
Groulx
,
D.
, and
White
,
M. A.
,
2013
, “
Heat Conduction in Laminate Multilayer Bodies With Applied Finite Heat Source
,”
Int. J. Therm. Sci.
,
72
, pp.
47
59
.10.1016/j.ijthermalsci.2013.05.001
15.
Sharma
,
A. K.
,
Agarwal
,
V.
,
Das
,
A. K.
,
Ghosh
,
S.
, and
Das
,
P. K.
,
2013
, “
Conduction in Composite Slabs: Reliability of 1D and 2D Calculations
,”
Energy Build.
,
65
, pp.
242
247
.10.1016/j.enbuild.2013.05.044
16.
Salt
,
H.
,
1983
, “
Transient Conduction in a Two-Dimensional Composite Slab-I. Theoretical Development of Temperature Modes
,”
Int. J. Heat Mass Transfer
,
26
(
11
), pp.
1611
1616
.10.1016/S0017-9310(83)80080-5
17.
Salt
,
H.
,
1983
, “
Transient Conduction in a Two-Dimensional Composite Slab-II. Physical Interpretation of Temperature Modes
,”
Int. J. Heat Mass Transfer
,
26
(
11
), pp.
1617
1623
.10.1016/S0017-9310(83)80081-7
18.
de Monte
,
F.
,
2003
, “
Unsteady Heat Conduction in Two-Dimensional Two Slab-Shaped Regions. Exact Closed-Form Solution and Results
,”
Int. J. Heat Mass Transfer
,
46
(
8
), pp.
1455
1469
.10.1016/S0017-9310(02)00417-9
19.
Kayhani
,
M. H.
,
Norouzi
,
M.
, and
Amiri Delouei
,
A.
,
2012
, “
A General Analytical Solution for Heat Conduction in Cylindrical Multilayer Composite Laminates
,”
Int. J. Therm. Sci.
,
52
, pp.
73
82
.10.1016/j.ijthermalsci.2011.09.002
20.
Kayhani
,
M. H.
,
Norouzi
,
M.
, and
Amiri Delouei
,
A.
,
2010
, “
An Exact Solution of Axi-Symmetric Conductive Heat Transfer in Cylindrical Composite Laminate Under the General Boundary Condition
,”
Word Acad. Sci. Eng. Technol.
,
69
, pp.
55
62
.
21.
Kayhani
,
M. H.
,
Norouzi
,
M.
, and
Amiri Delouei
,
A.
,
2010
, “
On Heat Conduction Problem in Multi–Layer Composite Pin Fins
,”
3th International Conference of Advanced Composite Materials Engineering
,
Brasov, Romani
, pp.
89
94
.
22.
Amiri Delouei
,
A.
,
Kayhani
,
M. H.
, and
Norouzi
,
M.
,
2012
, “
Exact Analytical Solution of Unsteady Axi-Symmetric Conductive Heat Transfer in Orthotropic Cylindrical Composite Laminates
,”
Int. J. Heat Mass Transfer
,
55
(
15–16
), pp.
4427
4436
.10.1016/j.ijheatmasstransfer.2012.04.012
23.
Tarn
,
J. Q.
,
2001
, “
Exact Solution for Functionally Graded Anisotropic Cylinders Subjected to Thermal and Mechanical Loads
,”
Int. J. Solids Struct.
,
38
(
46–47
), pp.
8189
8206
.10.1016/S0020-7683(01)00182-2
24.
Tarn
,
J. Q.
, and
Wang
,
Y. M.
,
2004
, “
End Effects of Heat Conduction in Circular Cylinders of Functionally Graded Materials and Laminated Composites
,”
Int. J. Heat Mass Transfer
,
47
(
26
), pp.
5741
5747
.10.1016/j.ijheatmasstransfer.2004.08.003
25.
Wang
,
H. M.
, and
Liu
,
C. B.
,
2013
, “
Analytical Solution of Two-Dimensional Transient Heat Conduction in Fiber-Reinforced Cylindrical Composites
,”
Int. J. Therm. Sci.
,
69
, pp.
43
52
.10.1016/j.ijthermalsci.2013.02.001
26.
Kayhani
,
M. H.
,
Shariati
,
M.
,
Norouzi
,
M.
, and
Karimi Demneh
,
M.
,
2009
, “
Exact Solution of Conductive Heat Transfer in Cylindrical Composite Laminate
,”
Heat Mass Transfer
,
46
(
1
), pp.
83
94
.10.1007/s00231-009-0546-1
27.
Norouzi
,
M.
,
Rezaei Niya
,
S. M.
,
Kayhani
,
M. H.
,
Karimi Demneh
,
M.
, and
Naghavi
,
M. S.
,
2012
, “
Exact Solution of Unsteady Conductive Heat Transfer in Cylindrical Composite Laminates
,”
ASME J. Heat Transfer
,
134
(
10
), p.
101301
.10.1115/1.4006009
28.
Singh
,
S.
,
Jain
,
P. K.
, and
Rizwan-uddin
,
2008
, “
Analytical Solution to Transient Heat Conduction in Polar Coordinates With Multiple Layers in Radial Direction
,”
Int. J. Therm. Sci.
,
47
(
3
), pp.
261
273
.10.1016/j.ijthermalsci.2007.01.031
29.
Jain
,
P. K.
,
Singh
,
S.
, and
Rizwan-uddin
,
2010
, “
An Exact Analytical Solution for Two-Dimensional, Unsteady, Multilayer Heat Conduction in Spherical Coordinates
,”
Int. J. Heat Mass Transfer
,
53
(
9–10
), pp.
2133
2142
.10.1016/j.ijheatmasstransfer.2009.12.035
30.
Norouzi
,
M.
,
Amiri Delouei
,
A.
, and
Seilsepour
,
M.
,
2013
, “
A General Exact Solution for Heat Conduction in Multilayer Spherical Composite Laminates
,”
Compos. Struct.
,
106
, pp.
288
295
.10.1016/j.compstruct.2013.06.005
31.
Fung
,
Y. C.
,
1965
,
Foundation of Solid Mechanics
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
32.
Powers
,
J. M.
,
2004
, “
On the Necessity of Positive Semi-Definite Conductivity and Onsager Reciprocity in Modeling Heat Conduction in Anisotropic Media
,”
ASME J. Heat Transfer
,
126
(
5
), pp.
670
675
.10.1115/1.1798913
33.
Herakovich
,
C. T.
,
1998
,
Mechanics of Fibrous Composites
,
Wiley
,
New York
.
34.
Halpin
,
J. C.
,
1992
,
Primer on Composite Materials Analysis
,
CRC Press
,
Boca Raton, FL
.
35.
Bruggeman
,
D. A. G.
,
1935
, “
Calculation of Various Physical Constants in Heterogeneous Substances. I. Dielectric Constants and Conductivity of Composites From Isotropic Substances (German)
,”
Ann. Phys.
,
416
(
8
), pp.
665
679
.10.1002/andp.19354160802
36.
Maxwell
,
J. C.
,
1954
,
A Treatise on Electricity and Magnetism
,
3rd ed.
,
Dover Publications
,
New York
.
37.
Hashin
,
Z.
, and
Shtrikman
,
S.
,
1962
, “
A Variational Approach to the Theory of the Effective Magnetic Permeability of Multiphase Materials
,”
J. Appl. Phys.
,
33
(
10
), pp.
1514
1517
.10.1063/1.1728579
38.
Torquato
,
S.
,
1984
, “
Bulk Properties of Two-Phase Disordered Media. I. Cluster Expansion for the Effective Dielectric Constant of Dispersions of Penetrable Spheres
,”
J. Chem. Phys.
,
81
(
11
), pp.
5079
5088
.10.1063/1.447497
39.
Myint-U
,
T.
, and
Debnath
,
L.
,
2007
,
Linear Partial Differential Equations for Scientists and Engineers
,
Birkhauser
,
Boston, MA
.
40.
Arpaci
,
V. S.
,
1996
,
Conduction Heat Transfer
,
Addison-Wesley
,
Boston, MA
.
41.
Bronshtein
,
I. N.
,
Semendyayev
,
K. A.
,
Musiol
,
G.
, and
Muehlig
,
H.
,
1998
,
Hand Book of Mathematics
,
5th ed.
,
Springer
,
Berlin
.
42.
Siegel
,
R.
, and
Howell
,
J. R.
,
1992
,
Thermal Radiation Heat Transfer
,
3rd ed.
,
Taylor & Francis
,
Washington, DC
.
43.
Bahadur
,
R.
, and
Bar-Cohen
,
A.
,
2007
, “
Orthotropic Thermal Conductivity Effect on Cylindrical Pin Fin Heat Transfer
,”
Int. J. Heat Mass Transfer
,
50
(
5–6
), pp.
1155
1162
.10.1016/j.ijheatmasstransfer.2006.04.025
You do not currently have access to this content.