Fundamental solutions for transient heat transfer by conduction and convection in an unbounded, halfspace, slab and layered media in the frequency domain
ABSTRACT Analytical Green's functions in the frequency domain are presented for the threedimensional diffusion equation in an unbounded, halfspace, slab and layered media. These proposed expressions take into account the conduction and convection phenomena, assuming that the system is subjected to spatially sinusoidal harmonic heat line sources and do not require any type of discretization of the space domain. The application of time and spatial Fourier transforms along the two horizontal directions allows the solution of the threedimensional time convectiondiffusion equation for a heat point source to be obtained as a summation of onedimensional responses. The problem is recast in the time domain by means of inverse Fourier transforms using complex frequencies in order to avoid aliasing phenomenon. Further, no restriction is placed on the source time dependence, since the static response is obtained by limiting the frequency to zero and the high frequency contribution to the response is small.The proposed functions have been verified against analytical time domain solutions, known for the case of an unbounded medium, and the Boundary Element Method solutions for the case of the halfspace, slab and layered media.

Conference Paper: New method for basic detection and characterization of flaws in composite slabs through finite difference thermal contrast (FDTC)
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ABSTRACT: In this paper, a technique based on a new model from finite differences discretization of Fourier heat propagation in 3D, is presented in order to be applied on a sequence of infrared images to enhance contrast and automatically detect and characterize flaws in composite slabs. The performance of this technique is evaluated using artificial thermal sequences from a simulated CFRP slab generated by ThermoCalc6L software. Results show that this technique offers a better contrast between defect and background than other common techniques like modified differentiated absolute contrast, and it runs faster than the classic 3D thermal filtering method.11th Quantitative Infrared Thermography conference, Naples, Italy; 06/2012  [Show abstract] [Hide abstract]
ABSTRACT: This article presents an experimental validation of a semianalytical solution for transient heat conduction in multilayer systems. The semianalytical solution is obtained using the heat conduction equation Green’s functions in the frequency domain. These solutions are obtained after time and spatial Fourier transforms are then applied in the two horizontal directions when the space domain has no kind of discretization. The problem is recast in the time domain by means of inverse Fourier transforms, using complex frequencies to avoid aliasing.The multilayer system used in the experimental validation is built by superimposing layers of different materials whose thermal properties were previously estimated experimentally. The external surfaces of the full system are then subjected to heat variation. The generated variation of the temperature field across the multilayer system was recorded using thermocouples. The full system was also simulated using the semianalytical solution. Comparison of the results showed that the semianalytical solution agrees with the experimental one.International Journal of Thermal Sciences 07/2012; 57:192–203. · 2.56 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: The Finite Difference Thermal Contrast (FDTC) is a new technique based on the approximation to the discretization of the Fourier heat propagation model in 3D, in order to be applied on a sequence of infrared images to enhance contrast for automatic detection and characterization of flaws in composite slabs. This contrast enhancement is performed by the calculus of relative error between predicted and real temperature over the heated surface only and for each pixel, in such a way that defective regions will exhibit greater errors than sound ones. Thermal sequences from a simulated Carbon Fiber Reinforced Plastic (CFRP) slab with airfilled defects, and from a real CFRP slab sample with Teflon squared defects, are used to evaluate and compare the enhancement obtained from FDTC, Normalized Contrast (NC) and Modified Differential Absolute Contrast (mDAC). In spite of the need of executing an additional background compensation in case of real slabs, results show that the proposed technique offers a better contrast between defects and background than the other techniques (about 33 % less residuary thermal nonuniformity with the adjusted version—FDTCa), mainly because of the more energy of the resulting thermal profiles. Also, as this technique does not estimate the temperature distribution along depth axis, but approximates temperature after a spatial step only, it can run faster than other thermal reconstruction methods like the classic 3D thermal filtering.Journal of Nondestructive Evaluation 03/2014; 33(1):62. · 1.02 Impact Factor
Page 1
Fundamental solutions for transient heat transfer by conduction
and convection in an unbounded, halfspace, slab and layered
media in the frequency domain
Nuno Simo ˜es, Anto ´nio Tadeu*
Department of Civil Engineering, University of Coimbra, Pinhal de Marrocos, 3030290 Coimbra, Portugal
Received 19 October 2004; revised 31 May 2005; accepted 5 June 2005
Available online 26 August 2005
Abstract
Analytical Green’s functions in the frequency domain are presented for the threedimensional diffusion equation in an unbounded, half
space, slab and layered media. These proposed expressions take into account the conduction and convection phenomena, assuming that the
system is subjected to spatially sinusoidal harmonic heat line sources and do not require any type of discretization of the space domain. The
application of time and spatial Fourier transforms along the two horizontal directions allows the solution of the threedimensional time
convectiondiffusion equation for a heat point source to be obtained as a summation of onedimensional responses. The problem is recast in
the time domain by means of inverse Fourier transforms using complex frequencies in order to avoid aliasing phenomenon. Further, no
restriction is placed on the source time dependence, since the static response is obtained by limiting the frequency to zero and the high
frequency contribution to the response is small.
The proposed functions have been verified against analytical time domain solutions, known for the case of an unbounded medium, and the
Boundary Element Method solutions for the case of the halfspace, slab and layered media.
q 2005 Elsevier Ltd. All rights reserved.
Keywords: Transient heat transfer; Conduction; Convection; 2.5D Green’s functions; Layered media
1. Introduction
The transient heat diffusion is a fundamental phenom
enon observed in several applications, such as building
physics and thermal engineering. One of the most important
reference works for transient heat transfer is by Carslaw and
Jaeger [1]. This book contains a set of analytical solutions
and Green’s functions for the diffusion equation, which
gives the response, i.e. temperature field or/and heat fluxes,
of the diffusion equation in the presence of a transient heat
process.
Most of the known techniques to solve transient
convectiondiffusion heat problems have been formulated
in the time domain or using Laplace transforms. In the ‘time
marching’ approach, the solution is assessed step by step at
consecutive time intervals after an initially specified state
has been assumed.Using the Laplace transform, a numerical
transform inversion is required to calculate the physical
variables in the real space, after the solution has been
obtained for a sequence of values of the transformed
parameter.
In the ‘time marching’ approach, the result at each time
step is computed directly in the time domain. Chang et al.
[2] and Shaw [3] used a timedependent fundamental
solution for studying transient heat processes. Later, Wrobel
and Brebbia [4] implemented a Boundary Element Method
BEM formulation for axisymmetric diffusion problems.
Dargush and Banerjee [5] proposed a BEM approach in the
time domain, where planar, threedimensional and axisym
metric analyses are all addressed with a timedomain
convolution. Lesnic et al. [6] studied the unsteady diffusion
equation in both one and two dimensions by a time
marching BEM model, taking into account the treatment
of singularities. Davies [7] used a timedomain analysis to
compute the heat flow across a multilayer wall, considering
surface films to model the radiant and convective exchange.
Engineering Analysis with Boundary Elements 29 (2005) 1130–1142
www.elsevier.com/locate/enganabound
09557997/$  see front matter q 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.enganabound.2005.06.002
*Corresponding author.Tel.: C351 239 797201;fax: C351 239 797190.
Email address: tadeu@dec.uc.pt (A. Tadeu).
Page 2
An alternative to the ‘time marching’ approach is to
remove the time dependent derivative, using instead a
transformed variable. The Laplace transform technique
has been extensively reported in the literature, to shift the
solution from the time domain to a transformed domain,
for solving diffusion problems. However, an inverse
transform is then required to reconstitute the solution in
the time domain, which is associated with a loss of
accuracy and may lead to the amplification of small
truncation errors. Several researchers have tried to
overcome this major drawback: algorithms for Laplace
inversion have been proposed by Stehfest [8] and Papoulis
[9]. Rizzo and Shippy [10] used a numerical approach that
incorporates a Laplace transform to create a time
independent boundary integration in a transform domain.
Other authors have proposed different solutions for
analyzing the diffusiontype problem by means of Laplace
transforms, such as those presented by Cheng et al. [11],
Zhu and Satravaha ([12,13]).
The search for Green’s functions has been extensively
researched, given their interest as benchmark solutions
and in the development of numerical methods, such as
the BEM (e.g. Ochiai [14]) and the Method of the
Fundamental Solutions (MFS) (e.g. Sˇarler [15]). Feng
[16] used a method based on modified Green’s functions
to compute the unsteady heat transfer of a homogeneous
or a composite solid body. HajiSheikh et al. [17] present
different types of Green’s functions that are solutions of
the heat conduction diffusion equation in multidimen
sional, multilayer bodies for different boundary con
ditions, calculating eigenvalues.
This work presents Green’s functions for calculating the
transient heat transfer wave field in the presence of an
unbounded, halfspace, slab and multilayer formations,
with the occurrence of conduction and convection phenom
ena. The problem is formulated in the frequency domain
using time Fourier transforms. The proposed technique
allows the use of any type of heat source, and deals with the
static response.
The problem of multilayer heat transfer has been
broadly studied. The O¨zisik’s book [18] includes a review
of onedimensional composite media, referring to orthog
onal expansions, Green’s functions and Laplace transform
techniques. Monte [19] analysed the transient heat
conduction of multilayer composite slabs, applying the
method of separation of variables to the heat conduction
partial differentialequation.
regression method was developed by Wang and Chen
[20] to compute the heat flow for a onedimensional
multilayer model.
The present work defines the transient heat transfer in a
multilayer system subject to a point, a linear or a plane
source in the presence of both convection and conduction.
This work extends previous work carried out by the
authors to define the response of layered solid media
subjected to a spatially sinusoidal harmonic heat
A frequencydomain
conduction line source, e.g. Tadeu et al. [21], where
only the conduction phenomenon was addressed. The
proposed fundamental solutions relate the heat field
variables (fluxes or temperatures) at some position in
the domain caused by a heat source placed elsewhere in
the media, in the presence of both conduction and
convection phenomena.
As in the previous work, the technique requires the
knowledge of the Green’s function for the unbounded
media, which are developed by first applying a time
Fourier transform to the time diffusion equation for a
heat point source and then a spatial Fourier transform to
the resulting Helmholtz equation, along the z direction, in
the frequency domain. So these functions are written first
as a superposition of cylindrical heat waves along one
horizontal direction (z) and then as a superposition of
heat plane sources.
The Green’s functions for a layered formation are
formulated as the sum of the heat source terms equal to
those in the fullspace and the surface terms required to
satisfy the boundary conditions at the interfaces, i.e.
continuity of temperatures and normal fluxes between
layers, and null normal fluxes or null temperatures at the
outer surface. The total heat field is achieved by adding the
heat source terms, equal to those in the unbounded space, to
the sets of surface terms arising within each layer and at
each interface.
The scope of this paper is to present first the three
dimensional formulation, explaining the mathematical
manipulation to obtain the Green’s functions for a heat
line source as a continuous superposition of heat plane
sources in the frequency domain. The procedure to retrieve
the time domain solutions is also given. Thismethodology is
verified by comparing the results obtained with the exact
time solutions for one, two and threedimensional point heat
sources placed in an unbounded medium.
This paper then goes on to describe the formulation of a
sinusoidal line heat load applied to a halfspace, a slab, a
slab over a halfspace medium and a layered formation. The
continuity of temperature and heat fluxes need to be
achieved between two neighbouring layers, while null
temperatures or null heat fluxes may be prescribed along an
external interface of the boundary. The full set of
expressions is corroborated by comparing its solutions
with those provided by the Boundary Element Method,
which requires the discretization of all layer interfaces.
2. Threedimensional problem formulation and Green’s
functions in an unbounded medium
The transient heat transfer by conduction and convection
in the domain with constant velocities along the x, y and z
directions is expressed by the equation
N. Simo ˜es, A. Tadeu / Engineering Analysis with Boundary Elements 29 (2005) 1130–11421131
Page 3
v2
vx2Cv2
K1
vy2Cv2
?
vT
vt;
vz2
v
vxCVy
??
T
K
Vx
v
vyCVz
v
vz
?
T
Z1
K
(1)
in which Vx, Vyand Vxare the velocity components in the
direction x, y and z, respectively, t is time, T(t,x,y,z) is
temperature, KZk/(rc) is the thermal diffusivity, k is the
thermal conductivity, r is the density and c is the specific
heat. The application of a Fourier transformation in the time
domain to the Eq. (1) gives the equation
v2
vx2Cv2
vy2Cv2
!2!
vz2
??
K1
K
Vx
v
vxCVy
v
vyCVz
v
vz
???
C
ffiffiffiffiffiffiffiffi ffi
K
K iu
r
^Tðu;x;y;zÞ Z0;
(2)
where iZ
the Helmholtz equation by the insertion of a convective
term. For a heat point source, applied at (0,0,0) in an
unbounded medium, of the form p(u, x, y, z, t)Zd(x) d(y)
d(z)i(ut), where d(x), d(y) and d(z) are Diracdelta functions,
the fundamental solution of Eq. (2) can be expressed as
ffiffiffiffiffiffi
K 1
p
and u is the frequency. Eq. (2) differs from
^Tfðu;x;y;zÞ
ZeðVxxCVyyCVzzÞ=2K
2k
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
As the geometry of the problem remains constant along
the z direction, the full threedimensional problem can be
expressed as a summation of simpler twodimensional
solutions. This requires the application of a Fourier
transformation along that direction, writing this as a
summation of twodimensional solutions with different
spatial wavenumbers kz (Tadeu and Kausel [22]). The
application of a spatial Fourier transformation to
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
along the z direction, leads to this fundamental solution
x2Cy2Cz2
p
eK i
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
K ðV2
xCV2
yCV2
zÞ=4K2K iu=K
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x2Cy2Cz2
p
:
(3)
eK i
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x2Cy2Cz2
p
K ðV2
xCV2
yCV2
zÞ=4K2K iu=Kx2Cy2Cz2
p
;
(4)
~Tfðu;x;y;kzÞZKieðVxxCVyyCVzzÞ=2K
4k
s
!H0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4K2
KV2
xCV2
yCV2
z
Kiu
KKðkzÞ2
r0
0
@
1
A;
(5)
where H0() are Hankel functions of the second kind and
order 0, and r0Z
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x2Cy2
p
.
The full threedimensional solution is then achieved by
applying an inverse Fourier transform along the kxdomain
to the expression ðK i=2ÞH0>
K ðiu=KÞKðkzÞ2r0
can be expressed as a discrete summation if we assume
the existence of virtual sources, equally spaced at Lz, along
z, which enables the solution to be obtained by solving a
limited number of twodimensional problems,
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
K ðV2
xCV2
yCV2
zÞ=4K2
q
?
?
. This inverse Fourier transformation
^Tðu;x;y;zÞZ2p
L
eðVxxCVyyCVzzÞ=2K
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4K2
@
2kx2Cy2Cz2
p
KV2
X
Kiu
M
mZKM
H0
!
xCV2
yCV2
z
KKðkzmÞ2
s
r0
01
AeKikzmz;
(6)
with kzmbeing the axial wavenumber given by KzmZ(2p/Lx)m.
The distance Lz is chosen so as to prevent spatial
contamination from the virtual sources, i.e. it must be
sufficiently large (Bouchon and Aki [23]). This technique is
an adaptation and extension of other mathematical and
numerical formulations used to solve problems such as wave
propagation (Tadeu et al. [24] and Godinho et al. [25]).
Note that Eq. (5) with VzZ0 becomes the fundamental
solution of the differential equation obtained from Eq. (2)
after the application of a spatial Fourier transformation
along the z direction, namely
v2
vx2Cv2
vy2
??
K1
K
Vx
v
vxCVy
v
vy
?
!2!
??
C
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
K
K iu
KðkzÞ2
r
~Tðu;x;y;kzÞ Z0;
(7)
when VzZ0.
Eq. (5), which results when a spatially sinusoidal
harmonic heat line source is applied at the point (0,0)
along the z direction, subject to convection velocities
Vx, Vyand Vz, can be further manipulated and written as a
continuous superposition of heat plane phenomena,
~Tfðu;x;y;kzÞ ZK ieðVxxCVyyCVzzÞ=2K
4pk
!
ð
C N
K N
eK injyj
n
!
eK ikxðxK x0Þdk0
x
(8)
where
n Z
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2K
KV2
xCV2
yCV2
z
Kiu
KKðkzÞ2Kk2
x
s
and Im(n)%0, and the integration is performed with respect
to the horizontal wavenumber (kx) along the x direction.
Assuming the existence of an infinite number of virtual
sources, we can discretize these continuous integrals.
The integral in the above equation can be transformed into
N. Simo ˜es, A. Tadeu / Engineering Analysis with Boundary Elements 29 (2005) 1130–11421132
Page 4
a summation if an infinite number of such sources are
distributed along the x direction, spaced at equal intervals
Lx. The above equation can then be written as
~Tfðu;x;y;kzÞ ZK i eðVxxCVyyCVzzÞ=2K
4k
E0
X
EdZeK ikxnðxÞ,
nZC N
nZ K N
E
nn
??
Ed;
(9)
where
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
%0, and kxnZ(2p/Lx)n,which can in turn be approximated
by a finite sum of equations (N). Notice that kzZ0
corresponds to the twodimensional case.
E0ZKi/2kLx,EZeK innjyj,
nnZ
K ðV2
xCV2
yCV2
zÞ=4K2Kðiu=KÞKðkzÞ2Kk2
xn
q
and Im(nn)
2.1. Responses in the time domain
The heat in the spatialtemporal domain is calculated by
applying a numerical inverse fast Fourier transform in kz, kx
and in the frequency domain. The computations are
performed using complex frequencies with a small
imaginary part of the form ucZuih (with hZ0.7Du, and
Du being the frequency step) to prevent interference from
aliasing phenomena. In the time domain, this effect is
removed by rescaling the response with an exponential
window of the form eht. The time variation of the source can
be arbitrary. The time Fourier transformation of the source
heat field defines the frequency domain to be computed. The
response may need to be computed from 0.0 Hz up to very
high frequencies. However, as the heat responses decay very
rapidly with increasing frequency, we may limit the upper
frequency for which the solution is required. The frequency
0.0 Hz corresponds to the static response that can be
computed when the frequency is zero. The use of complex
frequencies allows the solution to be obtained because,
when ucZuih (for 0.0 Hz), the arguments of the Hankel
function of the equations are not zero.
The technique proposed in this paper uses Fourier
transformations, which can be written as discrete sum
mations over wavenumbers and frequencies. The math
ematical formulation entails the use of sources equally
spaced in the zaxis and xaxis by spatial separations
LzZ2p/Dkzand LxZ2p/Dkx, and also by temporal intervals
TZ2p/Du, with Dkzand Dkxbeing the wavenumber steps.
Note that the use of complex frequencies diminishes the
contribution from the periodic (fictitious) sources to the
response at the time window T.
2.2. Verification of the solution
The formulation described above was implemented and
used to compute the heat field in an unbounded medium. In
order to verify this formulation, the solution is compared
with the analytical response in the time domain.
The exact solution of the convective diffusion, expressed
by Eq. (1), in an unbounded medium subjected to a unit heat
sourceiswellknownanditallowsthecomputationoftheheat
field given by both conduction and convection phenomena in
thepresenceofthree,twooronedimensionalproblems.When
the heat source is applied at the point (0,0,0) at time tZt0, the
temperature at (x,y,z) is given by the expression
Tðt;x;y;zÞ ZeðK ðK tVxCxÞ2K ðK tVyCyÞ2K ðK tVzCzÞ2Þ=4Kt
rcð4pKtÞd=2
;
if tOt0;
(10)
where tZtt0,dZ3,dZ2anddZ1wheninthepresenceofa
three, two and onedimensional problems, respectively
(Carslaw and Jaeger [1] and Hagentoft [26]).
In the verification procedure, a homogeneous unboun
ded medium, with thermal properties that allowed
kZ1.4 W mK18CK1, cZ880.0 J KgK18CK1and rZ
2300 Kg mK3, was excited at tZ277.8 h by a unit heat
source placed at xZ0.0 m, yZ0.0 m, zZ0.0 m. The
convection velocities applied in the x, y and z direction
were equal to 1!10K6m sK1.
Theresponseswerecalculatedalongalineof40receivers
placed from (xZK1.5,yZ0.35,zZ0.0) to (xZ1.5,yZ0.35,
zZ0.0), for a plane (dZ1), cylindrical (dZ2) and spherical
(dZ3) unit heat source.
The calculations were first performed in the frequency
range [0, 1024!10K7]Hz with an increment of DuZ10K7
Hz, which defines a time window of TZ2777.8 h. The
solution for the twodimensional case (cylindrical unit heat
source) was found with Eq. (5), while the results for a plane
unit heat source propagating along the y axis was obtained
ascribing kzZ0 and kxnZ0 to Eq. (9), multiplied by Lx.
Complex frequencies of the form ucZuKi0.7Du have
been used to avoid the aliasing phenomenon. The spatial
period has been set as LxZLzZ2
In Fig. 1, the solid line represents the exact time solution
given by Eq. (10) while the marks show the response
obtained using the proposed Green’s functions. There is
good agreement between these two solutions. Notice that, as
we have assigned a convection velocity in the x direction,
the temperature response along the line of receivers is not
symmetrical. In addition, lower temperatures were regis
tered at the threedimensional case, since the energy emitted
by the heat source is dissipated in the three directions.
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
k=ðrcDfÞ
p
.
3. Green’s functions in a halfspace
In this section, a semiinfinite medium bounded by a
surface with null heat fluxes or null temperatures is
considered. The required Green’s functions for a half
space can be expressed as the sum of the surface terms and
the source terms. The surface terms need to satisfy the
boundary condition of the surface (null heat fluxes or null
temperatures), while the source terms are equal to those
presented for the infinite unbounded medium. The surface
terms for a heat source located at (x0,y0)can be expressed by
N. Simo ˜es, A. Tadeu / Engineering Analysis with Boundary Elements 29 (2005) 1130–11421133
Page 5
~T1ðu;x;y;kzÞ
ZE0eððVxðxK x0ÞCVyðyK y0ÞÞ=2KÞX
where EaZeK inny. Anis the unknown coefficient to be com
puted, so that the heat field produced simultaneously by the
source and surface terms should produce~T1ðu;x;y;kzÞZ0
or ðv~T1ðu;x;y;kzÞ=vyÞZ0 at yZ0.
The computation of the unknown coefficient is obtained
for each value of n. These coefficients are given below for
the two cases of null heat fluxes and null temperatures at the
surface yZ0.
Null normal flux at yZ0,
nZC N
nZ K N
Ea
nn
An
??
Ed;
(11)
AnZeK inny0;
Null temperature at yZ0,
AnZK eK inny0:
(12)
Replacing these coefficients in Eq. (11), we may compute
the heat terms associated with the surface.
Null normal flux at yZ0,
~T1ðu;x;y;kzÞ ZE0eððVxðxK x0ÞCVyðyK y0ÞÞ=2KÞX
nZC N
nZ K N
Eaf
nn
??
Ed;
Null temperature at yZ0,
~T1ðu;x;y;kzÞ
ZE0eððVxðxK x0ÞCVyðyK y0ÞÞ=2KÞX
nZC N
nZ K N
Eat
nn
??
Ed
(13)
where EafZeK innðyCy0Þand EatZK eK innðyCy0Þ.
The final fundamental solutions for a halfspace are
given by adding both terms: the source and the surface
terms, which leads to
Null normal flux at yZ0,
~Tðu;x;y;kzÞ
ZE0eððVxðxK x0ÞCVyðyK y0ÞÞ=2KÞX
nZC N
nZ K N
ECEaf
nn
??
Ed;
(14)
Null temperature at yZ0,
~Tðu;x;y;kzÞ
ZE0eððVxðxK x0ÞCVyðyK y0ÞÞ=2KÞX
nZC N
nZ K N
EKEat
nn
??
Ed:
(15)
0
–1.5
0.2
0.4
0.6
0.8(a)
–1.0–0.50 0.51.01.5
350 h
450 h
550 h
650 h
350 h
450 h
550 h
650 h
350 h
450 h
550 h
650 h
X (m)
–1.5–1.0 –0.50 0.51.01.5
X (m)
–1.5–1.0 –0.50 0.51.01.5
X (m)
Temperature (ºC)
0
0.2
0.1
0.3
0.4
0.5(b)
Temperature (ºC)
0
0.2
0.1
0.3(c)
Temperature (ºC)
Fig. 1. Temperature along a line of 40 receivers, at times 350, 450, 550 and 650 h: (a) for a plane (dZ1) unit heat source; (b) for a cylindrical (dZ2) unit heat
source and (c) for a spherical (dZ3) unit heat source.
N. Simo ˜es, A. Tadeu / Engineering Analysis with Boundary Elements 29 (2005) 1130–11421134
Page 6
3.1. Verification of the solution
A BEM model was used to compute the temperature field
when a heat source is placed in a semiinfinite medium
bounded by a surface with null temperatures or null normal
heat fluxes. This technique entails high computational costs,
since it needs the discretization of the boundary. In order to
simulate the halfspace, the boundary is modelled through a
large number of elements distributed along as much of the
surface as necessary. The limited discretization of the
interfaces was achieved by introducing an imaginary part to
the frequencies, ucZuKih (with hZ0.7Du), which intro
duces damping.
The verification of the solution is illustrated for a
homogeneous halfspace medium with thermal material
properties that allow kZ1.4 W mK18CK1, cZ880.0 J KgK1
8CK1andrZ2300 Kg mK3.Theconvectionvelocity,inthey
direction,appliedtothehalfspacemediumis5!10K7m sK1.
It is assumed that the origin of convection coincides with the
source position. This structure is excited at (xZ0.0 m,yZ
1.0 m) by a line heat source with spatial sinusoidal variation
(kzZ0.4 rad mK1). Fig. 2 gives the results obtained at the
receiver(xZ0.2 m,yZ0.5 m)inthefrequencyrange[0,032!
10K7] Hz with a frequency increment of 1!10K7Hz. In this
plot, the solid lines represent the results provided by the
proposed solutions while the markers correspond to the
solution computed using the Boundary Element technique.
These results show that the responses are similar.
4. Green’s functions in a slab formation
For a slab structure with thickness h, the Green’s
functions can be achieved taking into account the boundary
conditions prescribed at each surface, i.e. null heat fluxes or
null temperatures. They can be expressed by adding the
surface and source terms, which are equal to those in the
fullspace.
Three scenarios can be considered: the prescription of
null heat fluxes at the top and bottom interfaces (Case I); or
null temperatures in both surface boundaries (Case II); or
even the consideration of different conditions at each
surface (Case III). At the top and bottom interfaces, surface
terms can be generated and expressed in a form similar to
that of the source term.
Top surface medium
~T1ðu;x;y;kzÞ ZE0eððVxðxK x0ÞCVyðyK y0ÞÞ=2KÞX
nZC N
nZ K N
Ea
nn
At
n
??
Ed;
Bottom surface medium
~T2ðu;x;y;kzÞ
ZE0eððVxðxK x0ÞCVyðyK y0ÞÞ=2KÞX
where EbZeK innjyK hj. At
coefficients to be determined by imposing the appropriate
boundary conditions, so that the field originated simul
taneously by the source and the surface terms guarantees
null heat fluxes or null temperatures at yZ0 and at yZh.
This problem formulation leads to a system of two
equations in the two unknown constants for each value
of n.
nZC N
nZ K N
Eb
nn
Ab
n
??
Ed;
(16)
n and Ab
n are as yet unknown
Case I. null heat fluxes at the top and bottom surfaces.
Vy
2KKinn
0
@
Vy
2KKinn
1
A
Vy
2KCinn
0
@
1
AeK innh
1
A
0
@
1
AeK innh
Vy
2KCinn
Vy
2KCinn
0
@
2
6666664
3
7777775
Atn
Ab
n
"#
Z
K
0
@
1
AeK inny0
1
K
Vy
2KKinn
0
@AeK innjhK y0j
2
6666664
3
7777775
:
(17)
–0.030
–0.015
0
0.015
0.030
0.045
0.060(a) (b)
0 1x10–6
2x10–6
3x10–6
real part
imaginary part
Frequency (Hz)
01x10–6
2x10–6
3x10–6
Frequency (Hz)
Amplitude (°C)
–0.03
–0.02
–0.01
0
0.01
0.02
Amplitude (°C)
real part
imaginary part
Fig. 2. Real and imaginary parts of the response for a halfspace formation (KzZ0.4 rad mK1): (a) Null normal flux at yZ0; and (b) Null temperature at yZ0.
N. Simo ˜es, A. Tadeu / Engineering Analysis with Boundary Elements 29 (2005) 1130–11421135
Page 7
Case II. null temperatures at the top and bottom surfaces.
1
K eK innh
eK innh
1
"#
Atn
Ab
n
"#
Z
K eK inny0
K eK innjhK y0j
"#
(18)
Case III. null heat fluxes at the top surface and null
temperatures at the bottom surface.
Vy
2KKinn
0
@
1
A
Vy
2KCinn
0
@
1
AeK innh
eK innh
1
2
664
3
775
At
n
Ab
n
"#
Z
K
Vy
2KCinn
0
@
1
AeK inny0
K eK innjhK y0j
2
664
3
775:
(19)
Once this system of equations has been solved, the
amplitude of the surface terms has been fully defined, and
the heat in the slab can thus be found. The final expressions
for the Green’s functions are then derived from the sum of
the source terms and the surface terms originated in the two
slab surfaces, which leads to the following expressions,
~Tðu;x;y;kzÞ ZK i
4keððVxðxK x0ÞCVyðyK y0ÞÞ=2KÞH0ðKtr0Þ
nZC N
CE0eððVxðxK x0ÞCVyðyK y0ÞÞ=2KÞX
where
nZ K N
Ea
nn
At
nCEb
nn
Ab
n
??
Ed;
(20)
KtZ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2K
KV2
xCV2
yCV2
z
K iu
K
KðkzÞ2
s
:
4.1. Verification of the solution
The formulation described above related to the slab
formation was used to compute the responses at a receiver
placedinaslab3.0 mthick,subjectedtoaspatiallyharmonic
varying line load in the z direction. The results for the three
different cases of boundary conditions were then compared
with those achieved by using a Boundary Element Method.
In this verification procedure, the medium properties and
convectionvelocityremainthesameasthoseassumedforthe
halfspace (kZ1.4 W mK18CK1, cZ880.0 J KgK18CK1
and rZ2300 Kg mK3). The slab is heated by a harmonic
point heat source applied at (xZ0.0 m, yZ1.0 m).
Theresponseisperformedinthefrequencyrange[0,32!
10K7]HzwithafrequencyincrementofDuZ10K7Hz.The
imaginarypartofthefrequencyhasbeensettohZ0.7Du.To
validate the results, the response is computed for a single
value of kz(kzZ0.4 rad mK1). Fig. 3 shows the real and
imaginary parts of the responses at the receiver (xZ0.2 m,
yZ0.5 m). The solid lines represent the discrete analytical
responses, while the marked points correspond to the
Boundary Element Method. The results confirm that the
solutions to the three cases are in very close agreement.
5. Green’s functions in a layered formation
The solutions for more complex structures, such as a
layer over a halfspace, a layer bounded by two semiinfinite
media and a multilayer can be established imposing the
required boundaryconditions at the interfaces and at the free
surface.
–0.050
–0.025
0
0.025
0.050 (a)
(b)
(c)
Amplitude (°C)
Amplitude (°C)
Amplitude (°C)
–0.02
–0.01
0
0.01
0.02
–0.050
–0.025
0
0.025
0.050
0 1x10–6
2x10–6
3x10–6
Frequency (Hz)
0 1x10–6
2x10–6
3x10–6
Frequency (Hz)
0 1x10–6
2x10–6
3x10–6
Frequency (Hz)
real part
imaginary part
real part
imaginary part
real part
imaginary part
Fig. 3. Real and imaginary parts of the responses for a slab formation, when
a heat source is applied at the point (xZ0.0 m, yZ0.0 m): (a) Case I (null
heat fluxes at the top and bottom surfaces); (b) Case II (null temperatures at
the top and bottom surfaces); and (c) Case III (null temperatures at the top
surface and null heat fluxes at the bottom surface).
N. Simo ˜es, A. Tadeu / Engineering Analysis with Boundary Elements 29 (2005) 1130–11421136
Page 8
5.1. Layer over a halfspace
Assuming the presence of a layer, h1thick, over a half
space, we may prescribe null temperature or null heat fluxes
at the free surface (top), while at the interface we need to
satisfy the continuity of temperature and normal heat fluxes.
The solution is again expressed as the sum of the source
terms (the incident field) equal to those in the fullspace and
the surface terms. At the interfaces 1 and 2, surface terms
are generated, which can be expressed in a form analogous
to that of the source term.
Layer interface 1
~T11ðu;x;y;kzÞ ZE01eðVy1ðyK y0Þ=2K1ÞX
nZC N
nZ K N
E11
nn1
At
n1
??
Ed:
(21)
Layer interface 2
~T12ðu;x;y;kzÞ ZE01eðVy1ðyK y0Þ=2K1ÞX
nZC N
nZ K N
E12
nn1
Ab
n1
??
Ed:
(22)
Halfspace (interface 2)
~T21ðu;x;y;kzÞ ZE02eðVy2ðyK y0Þ=2K2ÞX
nZC N
nZ K N
E21
nn2
At
n2
??
Ed:
(23)
where E0jZðK i=2kjLxÞ, E11ZeK inn1yE12ZeK inn1jyK h1j, E21Z
eK inn2jyK h1j
nnjZ
K ðVyj=2KjÞ2Kðiu=KjÞKk2
Im(nnj)%0 and h1is the layer thickness (jZ1 stands for
the layer (medium 1) while jZ2 indicates the halfspace
(medium 2)). Meanwhile, KjZkj/rjcj) is the thermal
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
zKk2
xn
q
with
diffusivity in the medium j (kj, rjand cjare the thermal
conductivity, the density and the specific heat of the
material in the medium j, respectively).
The coefficients At
unknown. They are defined in order to ensure the
appropriate boundary conditions: the field produced
simultaneously by the source and surface terms allows
the continuity of heat fluxes and temperatures at yZh1,
and null heat fluxes (Case I) or null temperatures (Case
II) at yZ0.
Imposing the three stated boundary conditions for each
value of n, a system of three equations in the three unknown
coefficients is defined.
n1, Ab
n1
and At
n2
are as yet
Case I. null heat fluxes at yZ0.
where
b1ZKVy1
2K1
Cinn1
?
?
?
?
eK inn1y0;
b2ZKVy1
2K1Kinn1
eK inn1jh1K y0j;
b3ZK eK inn1jh1K y0j
when the source is in the layer ðy0!h1Þ;
while
b1Z0;
b2Z
Vy2
2K2
Cinn2
??nn1
nn2
eðVy2ðhK y0Þ=2K2Þ
eðVy1ðhK y0Þ=2K1ÞeK inn2jh1K y0j;
b3Zk1nn1
k2nn2
eðVy2ðhK y0Þ=2K2Þ
eðVy1ðhK y0Þ=2K1ÞeK inn2jh1K y0j;
when the source is in the halfspace (y0Oh1).
Case II. null temperatures at yZ0.
Vy1
2K1
Kinn1
0
@
Vy1
2K1
1
A
Vy1
2K1
Cinn1
0
@
1
AeK inn1h1
Cinn1
0
Kinn1
0
@
1
AeK inn1h1
Vy1
2K1
0
@
1
A
Knn1
nn2
Vy2
2K2
Kinn2
0
@
Kk1nn1
k2nn2
1
AeðVy2ðh1K y0Þ=2K2Þ
eðVy2ðh1K y0Þ=2K2Þ
eðVy1ðh1K y0Þ=2K1Þ
eðVy1ðh1K y0Þ=2K1Þ
eK inn1h1
1
2
666666666664
3
777777777775
At
n1
Ab
n1
At
n2
2
664
3
775Z
b1
b2
b3
2
64
3
75;
(24)
1eK inn1h1
0
Vy1
2K1Kinn1
0
@
1
AeK inn1h1
Vy1
2K1
Cinn1
0
@
1
A Knn1
nn2
Vy2
2K2Kinn2
0
@
Kk1nn1
k2nn2
1
AeðVy2ðh1K y0Þ=2K2Þ
eðVy2ðh1K y0Þ=2K2Þ
eðVy1ðh1K y0Þ=2K1Þ
eðVy1ðh1K y0Þ=2K1Þ
eK inn1h1
1
2
6666666664
3
7777777775
At
n1
Ab
n1
At
n2
2
664
3
775Z
b1
b2
b3
2
64
3
75:
(25)
N. Simo ˜es, A. Tadeu / Engineering Analysis with Boundary Elements 29 (2005) 1130–11421137
Page 9
where
b1ZK eK inn1y0;
b2ZKVy1
2K1
Kinn1
??
eK inn1jh1K y0j;
b3ZK eK inn1jh1K y0j;
when the source is in the layer ðy0!h1Þ;
while
b1Z0;
b2Z
Vy2
2K2
Cinn2
??nn1
nn2
eðVy2ðhK y0Þ=2K2Þ
eðVy1ðhK y0Þ=2K1ÞeK inn2jh1K y0j;
b3Zk1nn1
k2nn2
eðVy2ðhK y0Þ=2K2Þ
eðVy1ðhK y0Þ=2K1ÞeK inn2jh1K y0j;
when the source is in the half–space ðy0Oh1Þ:
The heat field produced within the two media results
from the contribution of both the surface terms generated at
the various interfaces and the source term.
y0!h1(source in the medium 1)
~Tðu;x;y;kzÞ ZK i
4k1
eðVy1ðyK y0Þ=2K1ÞH0ðKt1r0Þ
nZC N
E11
nn1
~Tðu;x;y;kzÞ ZE02eðVy2ðyK y0Þ=2K2Þ
E21
nn2
CE01eðVy1ðyK y0Þ=2K1ÞX
if y!h1;
nZC N
?
nZ K N
At
n1CE12
nn1
Ab
n1
??
Ed;
!
X
nZ K N
At
n2
?
Ed;
if y!h1:
(26)
y0Oh1(source in the medium 2)
~Tðu;x;y;kzÞZE01eðVy1ðyKy0Þ=2K1ÞX
if y!h1;~Tðu;x;y;kzÞZKi
nZCN
nZKN
E11
nn1
At
n1CE12
nn1
Ab
n1
??
Ed;
4k2
?
eðVy2ðyKy0Þ=2K2ÞH0ðKt2r0Þ
CE02eðVy2ðyKy0Þ=2K2ÞX
nZCN
nZK N
E21
nn2
At
n2
?
Ed;if yOh1:
(27)
with
KtjZ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2Kj
K
Vyj
??2
Kiu
KjKðkzÞ2
s
ðjZ1;2Þ:
5.2. Layer bounded by two semiinfinite media
For the case of the layer placed between two semi
infinite media, the solution needs additionally to account
for the continuity of temperature and heat fluxes at the
interface 1, since the heat propagation also occurs
through the top semiinfinite space (medium 0), which
can be expressed by
~T02ðu;x;y;kzÞ
ZE00eðVy0ðyK y0Þ=2K0ÞX
nZC N
nZ K N
E01
nn0
Ab
n0
??
Ed;
(28)
where
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
The surface terms produced in the interfaces 1 and 2, in
the media 1 and 2 (bottom semiinfinite medium) are
expressed in Eqs. (21)–(23).
The coefficients Ab
respecting the continuity of heat fluxes and temperatures at
yZh1 and yZ0. These surface conditions lead to the
following system of four equations when the heat source is
placed within the layer.
E0jZðK i=2kjLxÞ,E00ZeK inn0y, and
nn0Z
K ðVy0=2K0Þ2Kðiu=K0ÞKk2
zKk2
xn
q
.
n0, At
n1, Ab
n1and At
n2are defined by
where
c3jZe
Vyj
P
j
lZ1
hlK y0
??
=2Kj
??
nnj
;
c4jZe
Vyj
P
jK 1
lZ1
hlK y0
??
=2Kj
??
nnj
;
Vy0
2K0
Cinn0
0
@
1
Ac30
c41
K
Vy1
2K1
? inn1
0
@
1
A
K
Vy1
2K1
Cinn1
0
@
1
AeK inn1h1
0
k1c30
k0c41
K 1
K eK inn1h1
0
0
Vy1
2K1
? inn1
0
@
1
AeK inn1h1
Vy1
2K1
Cinn1
0
@
1
A
K
Vy2
2K2
? inn2
2
4
3
5c42
c31
0eK inn1h1
1
Kk1c42
k2c31
2
66666666666666664
3
77777777777777775
Ab
n0
At
n1
Ab
n1
At
n2
2
666664
3
777775
Z
b1
b2
b3
b4
2
66664
3
77775
(29)
N. Simo ˜es, A. Tadeu / Engineering Analysis with Boundary Elements 29 (2005) 1130–11421138
Page 10
b1Z
Vy1
2K1
Cinn1
??
eK inn1y0;
b2ZeK inn1y0;
b3ZKVy1
2K1
Kinn1
??
eK inn1jh1K y0j;
b4ZK eK inn1jh1K y0j;
when the source is in the layer ð0!y0!h1Þ;
while
b1ZKVy0
2K0Kinn0
??c30
c41
eK inn0jy0j;
b2ZKk1c30
k0c41
eK inn0jy0j;
b3Z0;
b4Z0
when the source is in the half–space ðy0!h1Þ:
and
b1Z0;
b2Z0;
b3Z
Vy2
2K2
Cinn2
??c42
c31
eK inn2jh1K y0j;
b4Zk1c42
k2c31
eK inn2jh1K y0j;
when the source is in the half–space ðy0Oh1Þ:
The temperatures for the three media are then computed
by adding the contribution of the source terms to those
associated with the surface terms originated at the various
interfaces. This procedure produces the following
expressions for the temperatures in the three media.
~Tðu;x;y;kzÞZE00eðVy0ðyKy0Þ=2K0ÞX
~Tðu;x;y;kzÞZKi
4k1
nZCN
E11
nn1
nZCN
nZK N
E01
nn0
Ab
n0
??
Ed;ify!0;
eðVy1ðyKy0Þ=2K1ÞH0ðKt1r0ÞCE01eðVy1ðyKy0Þ=2K1Þ
!
X
nZKN
At
n1CE12
nn1
Ab
n1
??
Ed;if0!y!h1;
~Tðu;x;y;kzÞZE02eðVy2ðyKy0Þ=2K2ÞX
nZCN
nZK N
E21
nn2
At
n2
??
Ed;ifyOh1:
(30)
The derivation presented assumed that the spatially
sinusoidal harmonic heat source is placed within the layer.
However, the equations can be easily manipulated to
accommodate another position of the source.
5.3. Multilayer
The Green’s functions for a multilayer are established
using the required boundary conditions at all interfaces.
Consider a system built from a set of m plane layers of
infinite extent bounded by two flat, semiinfinite media, as
shown in Fig. 4. The top semiinfinite medium is called
medium 0, while the bottom semiinfinite medium is
assumed to be the medium mC1. The thermal material
properties and thickness of the various layers may differ.
Different vertical convection velocities can be ascribed at
each layer. The convection is computed assuming that the
origin of convection coincides with the conduction source.
The system of equations is achieved considering that the
multilayer is excited by a spatially sinusoidal heat source
located in the first layer (medium 1). The heat field at some
position in the domain is computed, taking into account both
the surface heat terms generated at each interface and the
contribution of the heat source term.
For the layer j, the heat surface terms on the upper and
lower interfaces can be expressed as
~Tj1ðu;x;y;kzÞ ZE0jeðVyjðyK y0Þ=2KjÞX
nZC N
nZ K N
Ej1
nnj
At
nj
??
Ed;
~Tj2ðu;x;y;kzÞ ZE0jeVyjðyK y0Þ=2KjX
nZC N
nZ K N
Ej2
nnj
Ab
nj
??
Ed;
(31)
where E0jZK i=2kjLx, Ej1Ze
nnjZ
K ðVy=2KjÞ2CðK iu=KjÞKkJ2
and hlis the thickness of the layer l. The heat surface terms
produced at interfaces 1 and mC1, which govern the heat
that propagates through the top and bottom semiinfinite
media, are ,respectively, expressed by
K innjyKP
jK 1
lZ1
hl
????????, Ej2Ze
xn
; with Im(nxj)%0
K innjyKP
j
lZ1
hl
????????,
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
zKk2
q
~T02ðu;x;y;kzÞ ZE00eðVy0ðyK y0Þ=2K0ÞX
~TðmC1Þ2ðu;x;y;kzÞ ZE0ðmC1ÞeðVyðmC1ÞðyK y0Þ=2KðmC1ÞÞX
!
nnðmC1Þ
nZC N
nZ K N
E01
nn0
Ab
n0
??
Ed;
nZC N
nZ K N
EðmC1Þ2
At
nðmC1Þ
??
Ed:
(32)
Medium 1
Y
X
h1
Medium m+1
Medium 0
Interface 1
Interface m+1
Medium 2
Medium m
Interface 2
Interface m
hm
Fig. 4. Geometry of the problem for a multilayer bounded by two semi
infinite media.
N. Simo ˜es, A. Tadeu / Engineering Analysis with Boundary Elements 29 (2005) 1130–11421139
Page 11
A system of 2(mC1) equations is derived, ensuring the
continuity of temperatures and heat fluxes along the mC1
interfaces between layers. Each equation takes into account
the contribution of the surface terms and the involvement of
the incident field. All the terms are organized according to
the form FaZb
where
c1jZ
Vyj
2Kj
Cinnj
??
?
;
c2jZ
Vyj
2Kj
? innj
??
;
c3jZe
Vyj
P
j
lZ1
hlK y0
?
=2Kj
nnj
;
c4jZe
Vyj
P
jK 1
lZ1
hlK y0
??
=2Kj
nnj
and
c5jZeK innjhj:
The resolution of the system gives the amplitude of the
surface terms in each interface. The temperature field for
each layer formation is obtained by adding these surface
terms to the contribution of the incident field, leading to the
following equations.
Top semiinfinite medium (medium 0)
~Tðu;x;y;kzÞ ZE00eðVy0ðyK y0Þ=2K0ÞX
if y!0;
nZC N
nZ K N
E01
nn0
Ab
n0
??
Ed;
Layer 1 (source position)
~Tðu;x;y;kzÞZKi
nZCN
4k1
eðVy1ðyKy0Þ=2K1ÞH0ðK1r0ÞCE01eðVy1ðyKy0Þ=2K1Þ
!
X
nZK N
E11
nn1
At
n1CE12
nn1
Ab
n1
??
Ed;if 0!y!h1;
Layer j (js1)
~Tðu;x;y;kzÞZE0jeðVyjðyKy0Þ=2KjÞX
if
X
nZCN
nZKN
Ej1
nnj
At
njCEj2
nnj
Ab
nj
??
Ed;
jK1
lZ1
hl!y!
X
j
lZ1
hl;
Bottom semiinfinite medium (medium mC1)
~TðmC1Þ2ðu;x;y;kzÞ
ZE0ðmC1ÞeðVyðmC1ÞðyKy0Þ=2KmC1ÞX
nZCN
nZK N
EðmC1Þ2
nnðmC1Þ
At
nðmC1Þ
??
Ed:
(34)
Notice that when the position of the heat source
is changed, the matrix F remains the same, while
the independent terms of?b are different. However, as the
equations can be easily manipulated to consider another
position for the source, they are not included here.
5.4. Verification of the solution
Next, the results are found for the three scenarios. First, a
flat layer, 3.0 m thick, is assumed to be bounded by one half
space. Null heat fluxes or null temperatures are prescribed at
the top surface. Then, a flat layer, also 3.0 m thick, bounded
by two semiinfinite media, is used to evaluate the accuracy
of the proposed formulation. The convection velocities
applied to the three media were 5!10K7, 8!10K7and 1!
10K6m sK1for the top medium, intermediate layer and
bottom medium, respectively. The thermal material proper
tiesusedintheintermediatelayerwerekZ1.4 W mK18CK1,
cZ880.0 J KgK18CK1and rZ2300.0 Kg mK3, while at
the top and bottom media were kZ63.9 W mK18CK1, cZ
434.0 J KgK18CK1and rZ7832.0 Kg mK3.
c10c30
K c21c41
K c11c41c51
.
000
c30
k0
Kc41
k1
Kc41c51
k1
.
000
0c21c31c51
c11c31
.
000
0
c31c51
k1
.
c31
k1
.
.
000
.....
000
.
K c2mc4m
K c1mc4mc5m
0
000
.
Kc4m
km
Kc41c5m
km
0
000
.
c2mc3mc5m
c1mc3m
K c2ðmC1Þc4ðmC1Þ
000
.
c3mc5m
km
c3m
km
Kc4ðmC1Þ
kðmC1Þ
2
66666666666666666666666666666664
3
77777777777777777777777777777775
Ab
n0
At
n1
Ab
.
n1
Atnm
Ab
nm
At
nðmC1Þ
2
66666666666664
3
77777777777775
Z
c11c41eK inn1y0
c41
k1
eK inn1y0
K c21c31eK inn1jh1K y0j
Kc31
k1
eK inn1jh1K y0j
.
0
0
0
0
2
666666666666666666666666666664
3
777777777777777777777777777775
(33)
N. Simo ˜es, A. Tadeu / Engineering Analysis with Boundary Elements 29 (2005) 1130–11421140
Page 12
The calculations have been performed in the frequency
domain from 0 to 32!10K7Hz, with a frequency increment
of DuZ10K7Hz and considering a single value of kzequal
to 0.4 rad mK1. The amplitude of the response for two
receivers placed in two different media was computed for a
heat point source applied at (xZ0.0 m, yZ1.0 m). The
real and imaginary parts of the response at the receiver 1
(xZ0.2 m, yZ3.5 m) and the receiver 2 (xZ0.2 m, yZ
3.5 m) are displayed in Figs. 5 and 6, with the imaginary
part of the frequency set to hZ0.7Du. The solid lines
represent the analytical responses, while the marked points
correspond to the BEM solution.
As can be seen, these two solutions seem to be in very
close agreement, and equally good results were obtained
Receiver 1 Receiver 2
–0.050
–0.025
0
0.025
0.050(a)
(b)
Amplitude (°C)
Amplitude (°C)
Amplitude (°C)
Amplitude (°C)
–0.002
–0.001
0
0.001
0.002
0.003
–0.02
–0.01
0
0.01
0.02
–0.002
–0.001
0
0.001
0.002
01x10–6
2x10–6
3x10–6
Frequency (Hz)
01x10–6
2x10–6
3x10–6
Frequency (Hz)
0 1x10–6
2x10–6
3x10–6
Frequency (Hz)
01x10–6
2x10–6
3x10–6
Frequency (Hz)
real part
imaginary part
real part
imaginary part
real part
imaginary part
real part
imaginary part
Fig. 5. Real and imaginary parts of the responses for a layer over a halfspace: (a) Case Inull heat fluxes at yZ0, responses at receivers 1 and 2; and (b) Case II
null temperatures at yZ0, responses at receivers 1 and 2.
–0.010
–0.005
0
0.005
0.010 (a)(b)
Amplitude (°C)
Amplitude (°C)
–0.002
–0.001
0
0.001
0.002
01x10–6
2x10–6
3x10–6
Frequency (Hz)
01x10–6
2x10–6
3x10–6
Frequency (Hz)
real part
imaginary part
real part
imaginary part
Fig. 6. Real and imaginary parts of the responses for a layer bounded by two semiinfinite media: (a) Receiver 1; and (b) Receiver 2.
N. Simo ˜es, A. Tadeu / Engineering Analysis with Boundary Elements 29 (2005) 1130–11421141
Page 13
from tests in which heat sources and receivers were situated
at different points.
6. Conclusions
This paper has presented the 2.5 Green’s functions for
computing the transient heat transfer by conduction and
convection in an unbounded medium, halfspace, slab and
layered media. In this approach the calculations are first
performed in the frequency domain. The results for a
layered formation are obtained adding the heat source term
and the surface terms, required to satisfy the interface
boundary conditions (temperature and heat fluxes con
tinuity). Notice that null convection velocities can be
prescribed for the different layers, allowing solid layers to
be modelled.
The unbounded medium formulation was corroborated
by comparing its time responses and the exact time
solutions. In turn, the analytical solutions used in the half
space, slab systems and layered media formulation were
verified using a BEM algorithm. Very good agreement was
found between the solutions.
The 2.5 Green’s functions for a layered medium can be
useful for solving problems such as the heat performance of
a multilayer construction element. Using these funda
mental solutions together with the BEM or MFS algorithms
can be helpful in resolving engineering problems, such as
the case of layered formations with buried inclusions.
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