The computation of the analytical solution of the steady temperature distribution in multilayered media can become numerically unstable if there are different longitudinal (i.e., the directions parallel to the layers) boundary conditions for each layer. In this study, we develop a method to resolve these computational difficulties by approximating the temperatures at the junctions step-by-step and solving for the thermal field separately in only the single layers. First, we solve a two-layer medium problem and then show that multilayered media can be represented as a hierarchy of two-layered media; thus, the developed method is generalized to an arbitrary number of layers. To improve the computational efficiency and speed, we use varying weighting coefficients during the iterations, and we present a method to decompose the multilayered media into two-layered media. The developed method involves the steady-state solution of the diffusion equation, which is illustrated for 2D slabs using separation of variables (SOV). A numerical example of four layers is also included, and the results are compared to a numerical solution.

References

References
1.
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
, pp.
1455
1469
.10.1016/S0017-9310(02)00417-9
2.
Shupikov
,
A. N.
,
Smetankina
,
N. V.
, and
Svet
,
Y. V.
,
2007
, “
Nonstationary Heat Conduction in Complex-Shape Laminated Plates
,”
ASME J. Heat Transfer
,
129
(
3
), pp.
335
341
.10.1115/1.2427073
3.
Haji-Sheikh
,
A.
,
Beck
,
J. V.
, and
Agonafer
,
D.
,
2003
, “
Steady-State Heat Conduction in Multi-Layer Bodies
,”
Int. J. Heat Mass Transfer
,
46
, pp.
2363
2379
.10.1016/S0017-9310(02)00542-2
4.
Desai
,
A.
,
Geer
,
J.
, and
Sammakia
,
B.
,
2007
, “
Heat Conduction in Multilayered Rectangular Domains
,”
ASME J. Electron. Packag.
,
129
(
4
), pp.
440
451
.10.1115/1.2804094
5.
Desai
,
A.
,
Geer
,
J.
, and
Sammakia
,
B.
,
2006
, “
Models of Steady Heat Conduction in Multiple Cylindrical Domains
,”
ASME J. Electron. Packag.
,
128
(
1
), pp.
10
17
.10.1115/1.2159003
6.
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. Transfer
,
53
, pp.
2133
2142
.10.1016/j.ijheatmasstransfer.2009.12.035
7.
Jain
,
P. K.
,
Singh
,
S.
, and
Rizwan-uddin
,
2009
, “
Analytical Solution to Transient Asymmetric Heat Conduction in a Multilayer Annulus
,”
ASME J. Heat Transfer
,
131
(
1
), p. 011304.10.1115/1.2977553
8.
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
, pp.
261
273
.10.1016/j.ijthermalsci.2007.01.031
9.
Kayhani
,
M. H.
,
Shariati
,
M.
,
Nourozi
,
M.
, and
Demneh
,
M. K.
,
2009
, “
Exact Solution of Conductive Heat Transfer in Cylindrical Composite Laminate
,”
Heat Mass Transfer
,
46
, pp.
83
94
.10.1007/s00231-009-0546-1
10.
Mikhailov
,
M. D.
, and
Özisik
,
M. N.
,
1986
, “
Transient Conduction in a Three-Dimensional Composite Slab
,”
Int. J. Heat Mass Transfer
,
29
, pp.
340
342
.10.1016/0017-9310(86)90242-5
11.
Salt
,
H.
,
1983
, “
Transient Heat Conduction in a Two-Dimensional Composite Slab-I. Theoretical Development of Temperatures Modes
,”
Int. J. Heat Mass Transfer
,
26
, pp.
1611
1616
.10.1016/S0017-9310(83)80080-5
12.
van der Tempel
,
L.
,
2002
, “
Transient Heat Conduction in a Heat Generating Layer Between Two Semi-Infinite Media
,”
ASME J. Heat Transfer
,
124
(
2
), pp.
299
309
.10.1115/1.1447930
13.
Lu
,
X.
,
Tervola
,
P.
, and
Viljanen
,
M.
,
2006
, “
Transient Analytical Solution to Heat Conduction in Multi-Dimensional Composite Cylinder Slab
,”
Int. J. Heat Mass Transfer
,
49
, pp.
1107
1114
.10.1016/j.ijheatmasstransfer.2005.08.033
14.
Nourozi
,
M.
,
Niya
,
S. M. R.
,
Kayhani
,
M. H.
,
Shariati
,
M.
,
Demneh
,
M. K.
, and
Naghavi
,
M. S.
,
2012
, “
Exact Solution of Unsteady Conductive Heat Transfer in Cylindrical Composite Laminates
,”
ASME J. Heat Transfer
,
134
(
10
), p. 101301.
15.
Jain
,
P. K.
,
Singh
,
S.
, and
Rizwan-uddin
,
2011
, “
Finite Integral Transform Method to Solve Asymmetric Heat Conduction in a Multilayer Annulus With Time-Dependent Boundary Conditions
,”
Nucl. Eng. Des.
,
241
, pp.
144
154
.10.1016/j.nucengdes.2010.10.010
16.
Kayhani
,
M. H.
,
Norouzi
,
M.
, and
Delouei
,
A. 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
17.
Abdul Azeez
,
M. F.
, and
Vakakis
,
A. F.
,
2000
, “
Axisymmetric Transient Solutions of the Heat Diffusion Problem in Layered Composite Media
,”
Int. J. Heat Mass Transfer
,
43
, pp.
3883
3895
.10.1016/S0017-9310(99)00386-5
18.
Leturcq
,
P. H.
,
Dorkel
,
J. M.
,
Ratolojanahary
,
F. E.
, and
Tounsi
,
S.
,
1993
, “
A Two-Port Network Formalism for 3D Heat Conduction Analysis in Multilayered Media
,”
Int. J. Heat Mass Transfer
,
36
, pp.
2317
2326
.10.1016/S0017-9310(05)80116-4
19.
Ma
,
C. C.
, and
Chang
,
S. W.
,
2004
, “
Analytical Exact Solutions of Heat Conduction Problems for Anisotropic Multi-Layered Media
,”
Int. J. Heat Mass Transfer
,
47
, pp.
1643
1655
.10.1016/j.ijheatmasstransfer.2003.10.022
20.
Haji-Sheikh
,
A.
,
Beck
,
J. V.
, and
Agonafer
,
D.
,
2002
, “
Temperature Solution in Multi-Dimensional Multi-Layer Bodies
,”
Int. J. Heat Mass Transfer
,
45
, pp.
1865
1877
.10.1016/S0017-9310(01)00279-4
21.
Huang
,
S. C.
, and
Chang
,
Y. P.
,
1980
, “
Heat Conduction in Unsteady, Periodic and Steady States in Laminated Composites
,”
ASME J. Heat Transfer
,
102
(
4
), pp.
742
748
.10.1115/1.3244383
22.
Chang
,
Y. P.
, and
Tsou
,
R. C. H.
,
1977
, “
Heat Conduction in an Anisotropic Medium Homogeneous in Cylindrical Regions-Steady State
,”
ASME J. Heat Transfer
,
99
(
1
), pp.
132
134
.10.1115/1.3450636
23.
Chang
,
Y. P.
, and
Tsou
,
R. C. H.
,
1977
, “
Heat Conduction in an Anisotropic Medium Homogeneous in Cylindrical Coordinates-Unsteady State
,”
ASME J. Heat Transfer
,
99
(
1
), pp.
41
46
.10.1115/1.3450652
24.
Lu
,
X.
,
Tervola
,
P.
, and
Viljanen
,
M.
,
2006
, “
Transient Analytical Solution to Heat Conduction in Composite Circular Cylinder
,”
Int. J. Heat Mass Transfer
,
49
, pp.
341
348
.10.1016/j.ijheatmasstransfer.2005.06.019
25.
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
, pp.
2873
2890
.10.1088/0305-4470/38/13/004
26.
de Monte
,
F.
,
2004
, “
Transverse Eigenproblem of Steady-State Heat Conduction for Multi-Dimensional Two-Layered Slabs With Automatic Computation of Eigenvalues
,”
Int. J. Heat Mass Transfer
,
47
(2), pp.
191
201
.10.1016/j.ijheatmasstransfer.2003.07.002
27.
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.
787
797
.10.1016/j.ijheatmasstransfer.2013.08.070
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