GUILLERMO F. UMBRICHT
Universidad Austral
FCE, Departamento de Matem´
atica
Paraguay 1950, Rosario
and CONICET
Godoy Cruz 2290, CABA,
ARGENTINA
0000-0002-8724-0909
DIANA RUBIO
Univ. Nacional de San Mart´
ın
Centro de Matem´
atica Aplicada
ITECA (CONICET-UNSAM)
25 de Mayo y Francia, San Mart´
ın
ARGENTINA
0000-0001-7180-1401
DOMINGO A. TARZIA
Universidad Austral
FCE, Departamento de Matem´
atica
Paraguay 1950, Rosario
and CONICET
Godoy Cruz 2290, CABA,
ARGENTINA
0000-0002-2813-0419
Abstract: The objective of this work is the determination of the materials that make up a three-layer body, based
on the simultaneous estimation of the thermal conductivity of the material of each layer. The body is exposed
to a one-dimensional stationary, non-invasive, heat transfer process. It is assumed that the union of each pair of
consecutive materials does not present thermal resistance. The parameters to be determined are estimated using
three temperature measurements, one at each interface and another at the right edge of the body. The estimation
is calculated analytically and a bound is given for the estimation error. In addition, an elasticity analysis is carried
out to analyze the local dependence of each estimated parameter with respect to the data. A numerical example is
included to illustrate and discuss the method proposed here.
Key–Words: Heat equation, solid-solid interface, inverse problems.
Received: May 15, 2022. Revised: October 16, 2022. Accepted: November 25, 2022. Published: December 31, 2022.
1 Introduction
The determination of the thermal conductivity in heat
transfer processes has several applications, for in-
stance, in optimal control design of thermal processes.
The thermal conductivity is a fundamental property
that has a determining influence on the temperature
distribution and heat flux density during thermal heat-
ing or cooling processes.
The estimation of thermal conductivity in heat
transfer processes has been widely addressed, during
the last decades. It was mainly studied using numer-
ical techniques of inverse problems, see for example
[3],[7],[14],[16]. In [4] the estimation was carried out
under particular conditions using the conjugate gradi-
ent method. In [7] a finite difference method was used
while in [18] an inverse linear model is proposed to
estimate the temperature dependence of thermal con-
ductivity. On the other hand, in [19] different iter-
ative methods are used. Particular problems of es-
timation of the thermal conductivity coefficient that
take into account multidimensional, inhomogeneous
and/or composite materials in [2], [5], [6], [8], [9],
[16], [17]. Other interesting estimation strategies ap-
plied to phase change materials can be seen in [10]-
[13].
This work deals with the simultaneous determina-
tion of the thermal conductivity coefficients, namely
κA,κB,κC[W/m◦C]of three materials, A,Band
C, that compose a three-layer body. The simultane-
ous estimation of the parameters is performed based
on three noisy temperature data; one at each interface
and one at the right edge of the body.
2 Mathematical Framework
The problem to be analyzed can be considered as a
stationary, one-dimensional transport process of ther-
mal energy. For this reason, the multilayer material is
modeled as a bar built with three consecutive sections
of homogeneous and isotropic materials, A,Band C,
so that the thermal diffusivity coefficients α2
A,α2
B,α2
C
[m2/s]are assumed to be constant. The left part of the
body (material A) has a length l1[m]; the middle sec-
tion (material B) has a length l2−l1[m]and the last
Determination of Thermal Conductivities in Multilayer
Materials
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Volume 17, 2022
section (material C) has a length L−l2[m].
On the other hand, it is assumed that the union
of the materials is perfectly assembled, i.e. no cracks
or roughness is present, so there is no thermal resis-
tance at the interfaces. Hence, continuity conditions
for temperature and heat flow are considered at the
solid-solid interfaces.
It is also assumed that the temperature at the left
edge of the body is kept constant, at temperature F
[◦C]and the right edge remains free, in contact with
the fluid, giving rise to the phenomenon of convection.
The problem described above can be modeled
with the following system:
u′′(x)=0,0< x < l1,
u′′(x)=0, l1< x < l2,
u′′(x)=0, l2< x < L,
u(x) = F, x = 0,
u(x−) = u(x+), x =l1,
u(x−) = u(x+), x =l2,
κAu′(x−) = κBu′(x+), x =l1,
κBu′(x−) = κCu′(x+), x =l2,
κCu′(x) = −h(u(x)−Ta), x =L,
(1)
where u[◦C]represents the stationary temperature, h
[W/(m2◦C)] the coefficient of heat transfer by con-
vection, Ta[◦C]the temperature outside the body and
u(l−
i) = lim
x→l−
i
u(x),
u(l+
i) = lim
x→l+
i
u(x).(2)
The analytical solution of the problem described
by equations (1)-(2) is given in the following result:
Theorem 1. Given κA, κB, κC, Ta, F, L, l1, l2, h ∈
IR+such that F > Ta,L > l2> l1and u(·)∈
C2((0, l1)∪(l1, l2)∪(l2, L)), the elliptic problem
(1)-(2) has a unique solution given by:
u(x) =
F+ζx,
F+ζd1l1+κA
κB
x,
F+ζd1l1+d2l2+κA
κC
x,
(3)
where the domains of each line are: 0≤x≤l1,
l1≤x≤l2,l2≤x≤L, respectively, and
d1= 1 −κA
κB
,(4)
d2=κA
κB
−κA
κC
,(5)
ζ=−F−Ta
L ζ0
,(6)
with
ζ0=κA
hL +d1
l1
L+d2
l2
L+κA
κC
.(7)
Proof. The theorem can be easily proved following
the idea developed in [17]. Observe that ζ0is strictly
positive, which assures the existence and uniqueness
of the solution.
3 Estimation of thermal conductivi-
ties
The solution to the forward problem, given by (3)-(7),
allows us to approach the estimation problem.
3.1 Determination of the parameters
In this subsection, an analytical expression for the
solution of the estimation problem is obtained from
three temperature measurements, one at each interface
(T1and T2) and another at the right edge of the body
(T3).
Theorem 2. Given κA,κB,κC,Ta,F,l1,l2,L,
h,T1,T2,T3∈IR+such that, 0< l1< l2< L and
Ta< T3< T2< T1< F. (8)
and the temperature function uthat satisfies u(·)∈
C2((0, l1)∪(l1, l2)∪(l2, L)), the solution to the
problem of determining the thermal conductivities κA,
κBand κCin the system (1) subjected to the over-
conditions
T1=u(x), x =l1,
T2=u(x), x =l2,
T3=u(x), x =L,
(9)
is
κA=h l1
T3−Ta
F−T1
,
κB=h(l2−l1)T3−Ta
T1−T2
,
κC=h(L−l2)T3−Ta
T2−T3
.
(10)
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Proof. Consider the system (1). Theorem 1 provides
an explicit analytical relationship between the temper-
ature function uand the physical parameters of the
model, given by the expressions (3)-(7). Applying the
conditions given in (9), it follows that
T1=F+ζ l1,(11)
T2=F+ζd1l1+κA
κB
l2,(12)
T3=F+ζd1l1+d2l2+κA
κC
L.(13)
From expressions (11)-(13) it results
ϑ1=κA
κB
=T2−T1
T1−F
l1
l2−l1
,(14)
ϑ2=κA
κC
=T3−T2
T1−F
l1
L−l2
.(15)
Replacing the expressions (14) and (15) in (6)-(7), it
follows
ζ=−F−Ta
κA
h+l1(1 −ϑ1) + l2(ϑ1−ϑ2) + ϑ2L,
(16)
Finally, (10) is derived by substituting (16) in equa-
tions (11)-(13).
3.2 Error estimate
An analytical expression is obtained for a bound of the
estimation error of the thermal conductivities κA,κB
and κC, when using three noisy temperature data Tϵ
1,
Tϵ
2and Tϵ
3, assuming
|T1−Tϵ
1| ≤ ϵ(F−Ta),
|T2−Tϵ
2| ≤ ϵ(F−Ta),
|T3−Tϵ
3| ≤ ϵ(F−Ta),
(17)
where ϵ > 0(small enough) denotes the noise level.
Theorem 3. The inverse problem of simultaneous de-
termination of the thermal conductivities κA,κBand
κCfrom (1),(8) and (9) is considered. Let cκA,cκB
and cκCbe the approximated solutions that depend on
noisy temperature measurements Tϵ
1at x=l1,Tϵ
2at
x=l2, and Tϵ
3at x=L, that satisfy the condition
(17).
There exist dimensionless constants
M1, M2, M3∈(0,1) such that
M1≤F−T1
F−Ta
, M2≤T1−T2
F−Ta
, M3≤T2−T3
F−Ta
,
(18)
that satisfy
|κA−cκA| ≤ 2h l1
M1(M1−ϵ)ϵ, (19)
|κB−cκB| ≤ 3h(l2−l1)
M2(M2−2ϵ)ϵ, (20)
and
|κC−cκC| ≤ 3h(L−l2)
M3(M3−2ϵ)ϵ, (21)
for
0< ϵ < min M1,M2
2,M3
2.(22)
Proof. By using the noisy temperature data Tϵ
1,Tϵ
2
and Tϵ
3in (10) it results
cκA=h l1
Tϵ
3−Ta
F−Tϵ
1
,
cκB=h(l2−l1)Tϵ
3−Ta
Tϵ
1−Tϵ
2
,
cκC=h(L−l2)Tϵ
3−Ta
Tϵ
2−Tϵ
3
.
(23)
From (10) and (23), the estimation errors are obtained
as follows
|κA−cκA|=h l1
T3−Ta
F−T1
−Tϵ
3−Ta
F−Tϵ
1.(24)
|κB−cκB|=h(l2−l1)
T3−Ta
T1−T2
−Tϵ
3−Ta
Tϵ
1−Tϵ
2,
(25)
|κC−cκC|=h(L−l2)
T3−Ta
T2−T3
−Tϵ
3−Ta
Tϵ
2−Tϵ
3.
(26)
Adding and subtracting Tϵ
1Tϵ
3in the numerator and
rewriting the expression in terms of the errors in the
data, (24) can be expressed by
|κA−cκA|
h l1
=
(F−Tϵ
1)(Tϵ
3−T3)+(T1−Tϵ
1)(Tϵ
3−Ta)
(F−T1)(F−Tϵ
1)
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and by the triangular inequality it follows that
|κA−cκA|
h l1
≤
|Tϵ
3−T3| |F−Tϵ
1|+|T1−Tϵ
1| |Tϵ
3−Ta|
(F−T1)|F−Tϵ
1|,
and assumptions (17) lead to
|κA−cκA|
h l1
≤F−Ta
F−T11 + |Tϵ
3−Ta|
|F−Tϵ
1|ϵ. (27)
Note that
Tϵ
1≤T1+ϵ(F−Ta) =⇒F−Tϵ
1≥F−T1−ϵ(F−Ta),
and F−T1−ϵ(F−Ta)>0for ϵsufficiently small.
Hence, from expresions (8), (17) and (27) it follows
that
|κA−cκA|
h l1
≤F−Ta
F−T11 + |Tϵ
3−T3|+|T3−Ta|
F−T1−ϵ(F−Ta)ϵ,
≤F−Ta
F−T11 + (1 + ϵ) (F−Ta)
F−T1−ϵ(F−Ta)ϵ,
≤F−Ta
F−T1F−T1+F−Ta
F−T1−ϵ(F−Ta)ϵ,
≤2F−Ta
F−T1F−Ta
F−T1−ϵ(F−Ta)ϵ.
Since 0< F −T1< F −Tathen 0<F−T1
F−Ta
<1and
so, there exists a constant M1∈(0,1) that satisfies
M1≤F−T1
F−Ta
,
yielding (19) for ϵ < M1. Analogously, there ex-
ist constants M2, M3∈(0,1) satisfying (18). Fi-
nally, from (25)-(26) we obtain (20) and (21) for
ϵ < 1
2min{M2, M3}.
Remark 4. Expressions (19)-(21) indicate that
if ϵ−→ 0then the estimation errors sat-
isfy |κA−cκA| −→ 0,|κB−cκB| −→ 0and
|κC−cκC| −→ 0.
Remark 5. Note that the condition (22) means that
precise measurements are required in order to obtain
the above results.
4 Elasticity analysis
The local relationships between the estimated param-
eters (cκA,cκB,cκC)and the data (Tϵ
1, T ϵ
2, T ϵ
3)used for
the estimation, are studied. For this purpose, the elas-
ticity function is used, which provides the percentage
estimation error when an error of 1% is made in the
measurement of the data.
Since three parameters are estimated from three
different data, nine elasticity functions arise for this
problem, they are the elasticity of each estimation cκA,
cκB,cκCwith respect to each datum Tϵ
1,Tϵ
2,Tϵ
3, given
by
ETϵ
j
bκi(Tϵ
1, T ϵ
2, T ϵ
3) = Tϵ
j
bκi(Tϵ
1, T ϵ
2, T ϵ
3)
∂bκi
∂T ϵ
j
(Tϵ
1, T ϵ
2, T ϵ
3),
(28)
where i=A, B, C and j= 1,2,3.
The system (23) lead to the following analytical
expressions for the elasticity functions defined in (28)
ETϵ
1
cκA(Tϵ
1, T ϵ
2, T ϵ
3) = Tϵ
1
F−Tϵ
1
,
ETϵ
2
cκA(Tϵ
1, T ϵ
2, T ϵ
3) = 0,
ETϵ
3
cκA(Tϵ
1, T ϵ
2, T ϵ
3) = Tϵ
3
Tϵ
3−Ta
,
(29)
ETϵ
1
cκB(Tϵ
1, T ϵ
2, T ϵ
3) = Tϵ
1
Tϵ
2−Tϵ
1
,
ETϵ
2
cκB(Tϵ
1, T ϵ
2, T ϵ
3) = Tϵ
1
Tϵ
1−Tϵ
2
,
ETϵ
3
cκB(Tϵ
1, T ϵ
2, T ϵ
3) = Tϵ
3
Tϵ
3−Ta
(30)
and
ETϵ
1
cκC(Tϵ
1, T ϵ
2, T ϵ
3) = 0,
ETϵ
2
cκC(Tϵ
1, T ϵ
2, T ϵ
3) = Tϵ
2
Tϵ
3−Tϵ
2
,
ETϵ
3
cκC(Tϵ
1, T ϵ
2, T ϵ
3) = Tϵ
3
Tϵ
3−Ta
Tϵ
2−Ta
Tϵ
2−Tϵ
3
.
(31)
Remark 6. Note that ETϵ
2
cκA(Tϵ
1, T ϵ
2, T ϵ
3) = 0
because the estimated value cκAis indepen-
dent of Tϵ
2. Analogously, ETϵ
1
cκC(Tϵ
1, T ϵ
2, T ϵ
3) = 0
since cκCis independent of Tϵ
1. Furthermore,
ETϵ
3
cκA(Tϵ
1, T ϵ
2, T ϵ
3) = ETϵ
3
cκB(Tϵ
1, T ϵ
2, T ϵ
3), while
ETϵ
3
cκC(Tϵ
1, T ϵ
2, T ϵ
3) = ETϵ
3
cκA(Tϵ
1, T ϵ
2, T ϵ
3)Tϵ
2−Ta
Tϵ
2−Tϵ
3
.
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5 Numerical example
Example 1. A Nickel-Lead-Iron material is assumed
where L= 10 m;F= 100 ◦Cand T a = 25 ◦C.
The convective fluid is assumed to be air and convec-
tive heat transfer coefficient (h)are determined as ex-
plained in [15]. It is also assumed that l1= 4 mand
l2= 7 m.
The analytical (exact) data for this example are
T∗= (T1, T2, T3) = (87.71,64.01,52.65) [◦C], ob-
tained from (11)-(13) where the values of the coeffi-
cient of thermal conductivity are κA= 90 W/m◦C,
κB= 35 W/m◦Cand κC= 73 W/m◦C(see [1]).
Firstly, the estimation errors for different values
of Tϵ
1,Tϵ
2and Tϵ
3close to T1,T2and T3, are analyzed.
Let us define
ErrκA(Tϵ
1, T ϵ
2, T ϵ
3) = |κA−cκA|,
ErrκB(Tϵ
1, T ϵ
2, T ϵ
3) = |κB−cκB|,
ErrκC(Tϵ
1, T ϵ
2, T ϵ
3) = |κC−cκC|.
Table 1: Relative estimate errors Errκi
κi
,i=A, B, C
for Example 1.
Tϵ
1Tϵ
2Tϵ
3
ErrκA
κA
ErrκB
κB
ErrκC
κC
87.2 63.5 52.1 0.060 0.020 0.023
87.3 63.6 52.2 0.048 0.016 0.019
87.4 63.7 52.3 0.037 0.012 0.015
87.5 63.8 52.4 0.025 0.009 0.012
87.6 63.9 52.5 0.014 0.005 0.008
87.7 64.0 52.6 0.002 0.001 0.005
87.8 64.1 52.7 0.009 0.002 0.001
87.9 64.2 52.8 0.021 0.005 0.002
88.0 64.3 52.9 0.033 0.009 0.005
88.1 64.4 53.0 0.045 0.012 0.009
88.2 64.5 53.1 0.058 0.016 0.012
Table 1 shows the relative estimation errors for
some values Tϵ
1,Tϵ
2and Tϵ
3close to T1,T2and T3.
It can be seen that good relative estimation errors are
obtained for the conductivity values. Furthermore, it
is observed that the estimate worsens as the error in
data increases. In this range of temperature values, a
maximum error of 6% is obtained for the estimate of
κA, of 2% for the estimate of κBand of 2.3% for that
of κC.
It is interesting to determine the directions of
maximum increase in estimation errors. The theory
of calculus in several variables proves that the direc-
tion of maximum growth of a function at a point is
the direction of the gradient at that point. In this case,
they are the directions of ∇ErrκA(T∗),∇ErrκB(T∗)
and ∇ErrκC(T∗), respectively, shown in Figure 1.
86
88
90
92
94
96 62
64
66
68
70
50
55
60
65
T2
T1
T3
∇ Errκ
A
(T*)
∇ Errκ
B
(T*)
∇ Errκ
C
(T*)
Figure 1: Gradients of the estimation errors at (87.71,
64.01, 52.65) (Example 1).
Moreover, the maximum values of the derivatives
in these directions are
∥∇ErrκA(T∗)∥= 8.01,
∥∇ErrκB(T∗)∥= 2.44,
∥∇ErrκC(T∗)∥= 11.11.
Hence, taken into account the values for κA,κBand
κC, it follows that the maximum growth at T∗for the
relative errors of the estimates of κA,κB,κCare, re-
spectively, 0.089, 0.069 and 0.152, i.e., about 9%,7%
and 15%.
Finally, the local relationships between the
estimated parameters (cκA,cκB,cκC)and the data
(Tϵ
1, T ϵ
2, T ϵ
3)used for the estimation, are analyzed.
In Figures 2-4 the elasticity functions defined in
(29)-(31) for the Nickel-Lead-Iron material described
above, are plotted.
Figure 2 shows that the estimate error of κA
increases with Tϵ
1. It can also be observed that
ETϵ
1
cκA(T∗)≈7while ETϵ
1
cκB(T∗)≈3.7and ETϵ
1
cκC(T∗) =
0. In other words, an error of 1% in the measurement
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of T1leads to an error of about 7% in the estimate of
κAand 3.7 % in the estimate of κB, which means that
cκAis more sensitive to the error in Tϵ
1than the cκB. As
it was mentioned before κCdoes not depend on Tϵ
1.
85 86 87 88 89 90
−1
0
1
2
3
4
5
6
7
8
9
10
Tǫ
1(◦C)
Elasticity
E
κ
A
T
1
ε
E
κ
B
T
1
ε
E
κ
C
T
1
ε
Figure 2: Elasticity of the conductivities with respect
to T1for a Nickel-Lead-Iron material (Example 1).
In Figure 3 it is observed that the estimate of κCis
more sensitive to the measured value Tϵ
2than the other
estimates since ETϵ
2
cκC(T∗)≈6(about 6% of estimate
error when there is a 1% in the measurement error),
while ETϵ
2
cκB(T∗)≈2.8(about 2.8% of error in the es-
timate when there is a 1% in the measurement error),
and ETϵ
2
cκA(T∗) = 0 (κAdoes not depend on Tϵ
2).
62 63 64 65 66 67
−1
0
1
2
3
4
5
6
7
8
9
10
Tǫ
2(◦C)
Elasticity
E
κ
A
T
2
ε
E
κ
B
T
2
ε
E
κ
C
T
2
ε
Figure 3: Elasticity of the conductivities with respect
to T2for a Nickel-Lead-Iron material (Example 1).
Figure 4 indicates that the estimate of κCis also
more sensitive to the measurement of T3than the other
estimates since ETϵ
3
cκC(T∗)≈6.5(about 6.5% of esti-
mate error when there is a 1% in the measurement er-
ror), while ETϵ
3
cκA(T∗) = ETϵ
3
cκB(T∗)≈2(about 2% of
error in the estimates when there is a 1% in the mea-
surement error).
50 51 52 53 54 55
−1
0
1
2
3
4
5
6
7
8
9
10
Tǫ
3(◦C)
Elasticity
E
κ
A
T
3
ε
E
κ
B
T
3
ε
E
κ
C
T
3
ε
Figure 4: Elasticity of the conductivities with respect
to T3for a Nickel-Lead-Iron material (Example 1).
The graphs show that the estimates of κAand κB
are more sensitive to measurement error in T1while
the estimate of κCis more sensitive to errors in the
measurements of T2and T3.
Denoting Ebκi= (ETϵ
1
bκi, ETϵ
2
bκi, ETϵ
3
bκi), i =A, B, C
from the numerical experiment for this example re-
sults in ∥EcκA(T∗)∥= 7.39,∥EcκB(T∗)∥= 4.96,
∥EcκC(T∗)∥= 8.63, so larger errors in the tempera-
ture measurements lead to larger errors in the estima-
tion of κAand κCthan in κB. These results agree with
the conclusion reached previously when discussing
the maximum growth values of the relative errors for
the estimation.
6 Conclusion
This article analyzes the simultaneous estimation of
the thermal conductivity coefficients for a stationary
heat transfer problem with two solid-solid interfaces.
A technique is proposed for the estimation of these
physical properties based on three noisy temperature
over-conditions, one at each interface and another at
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Guillermo F. Umbricht, Diana Rubio,
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193
the right end of the material. The necessary and suf-
ficient conditions for the existence and uniqueness of
the solution to the estimation problem are provided,
and analytical bounds for the determination errors are
derived.
The local influence of the data on the estimated
parameters is studied by means of elasticity analysis.
For a numerical example, the directions and the val-
ues of maximum growth of the relative errors are also
studied, noting that the results are consistent with the
elasticity analysis.
The results obtained suggest that the approach
presented here is useful to determine the three thermal
conductivities for each body material. However, it is
important to measure the temperatures as accurately
as possible, since the estimated values are sensitive to
measurement errors.
Acknowledgements: The research was supported by
the Universidad de San Mart´
ın, Universidad Austral
and, in the case of the first and second authors, by
SOARD/AFOSR (Grant FA9550-18-1-0523). The
third author acknowledges support from European
Union’s Horizon 2020 Research and Innovation Pro-
gramme under the Marie Sklodowska-Curie Grant
Agreement No. 823731 CONMECH and by the
Project PIP No. 0275 from CONICET-UA, Rosario,
Argentina.
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