a fast non-intrusive method for estimating spatial thermal contact conductance by means of the...

7/23/2019 A fast non-intrusive method for estimating spatial thermal contact conductance by means of the reciprocity functi

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A fast non-intrusive method for estimating spatial thermal contact conductanceby means of the reciprocity functional approach and the methodof fundamental solutions

Marcelo J. Colao a,, Carlos J.S. Alves b

a Federal University of Rio de Janeiro, Department of Mechanical Engineering, PEM-COPPE-UFRJ, Cx. Postal 68503, Rio de Janeiro, RJ 21941-972, Brazilb Technical University of Lisbon, Instituto Superior Tcnico Department of Mathematics Av. Rovisco Pais, 1, 1049-001 Lisbon, Portugal

a r t i c l e i n f o

Article history:

Received 30 April 2012

Received in revised form 15 January 2013

Accepted 15 January 2013

Available online 13 February 2013

Keywords:

Inverse problems

Reciprocity function

Contact resistance

a b s t r a c t

Thermal contact conductance is very important in many heat transfer applications, such as electronic

packaging, nuclear reactors, aerospace and biomedicine, among others. The determination of the thermal

contact resistance/conductance is a very difficult task. The objective of this paper is to present a method-

ology to estimate the spatial variation of this parameter without intrusive measurements. The method-

ology presented is formulated in terms of a reciprocity functional approach together with the method of

fundamental solutions. The solution is composed of two steps. In the first step, two auxiliary problems,

which do not depend on the thermal conductance variation, are solved. With the results of this pre-pro-

cessing, different thermal conductances can be recovered by simply performing an integral. Thus, the

methodology is extremely fast and can be used to detect flaws in different materials in a short time.

2013 Elsevier Ltd. All rights reserved.

1. Introduction

Thermal contact conductance is very important in many heat

transfer applications, such as electronic packaging [20], nuclear

reactors [14], aerospace and biomedicine [17], among others. It

has been recognized for several years [13] that the increasing

power density of some electronic equipment requires cooling de-

vices able to remove great amounts of heat. In fact, there is an

interest in producing microchannel heat sinks with heat removal

capacities of more than 1 kW/cm2 [11]. An important factor in

obtaining such heat removal is to have a low thermal contact resis-

tance between the electronics and the cooling devices. In nuclear

reactors, resistance, which occurs in the gap between the nuclear

fuel and the metallic canning, has become a limiting factor in

exploiting reactor efficacy[14].When two materials are in contact, only fractions of them are

really touching each other. Thus, there is a discontinuity in the

temperature across the contact interface. Thermal contact resistanceis defined as the ratio of the temperature drop to the heat flow

across the interface

Rc DT=q 1

Thus, lower values ofRcindicate that the difference in the temper-

ature across the interface is low, which demonstrates a good

contact. Thermal contact conductance, in the context of this paper,

is defined as the inverse of the resistance (h= 1/Rc). Notice that both

Rcandh can vary spatially along the interface.

Some studies [15,20,10,19] used the definition given by Eq.

(1) to calculate the global thermal contact resistance of several

materials. These studies, in general, used an experimental appa-

ratus to measure the discontinuity in temperature and the heat

flux applied. Because the discontinuity at the exact location of

the interface is difficult to measure, they took temperature mea-

surements at several locations and extrapolated the value

of the temperature at the interface. Wolff and Schneider [6]used

the guarded hot plate method to determine temperature discon-

tinuities across interfaces. The disadvantage of such methods is

that they only predict global values of the thermal contact

resistance/conductance. In addition, they require complicatedexperimental apparatus and/or some intrusive temperature

measurements.

Milosevic et al. [14] used a non-intrusive method, the laser flash

method, together with the Gauss method to estimate a constant

value of the thermal contact resistance between two solids. In their

paper, they were able to estimate this parameter when the sample

materials were good heat conductors or when the thickness of the

layer was relatively small. In addition, the accuracy of the estimate

increased with higher values of the contact resistance. Thus, voids

with very small values of the thermal contact resistance could not

be very well captured. Milosevic also presented other results[22]

using the laser flash method.

0017-9310/$ - see front matter 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.01.026

Corresponding author.

E-mail addresses:[email protected](M.J. Colao),[email protected](C.J.S.

Alves).

International Journal of Heat and Mass Transfer 60 (2013) 653663

Contents lists available atSciVerse ScienceDirect

International Journal of Heat and Mass Transfer

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i j h m t
http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.01.026mailto:[email protected]:[email protected]://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.01.026http://www.sciencedirect.com/science/journal/00179310http://www.elsevier.com/locate/ijhmthttp://www.elsevier.com/locate/ijhmthttp://www.sciencedirect.com/science/journal/00179310http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.01.026mailto:[email protected]:[email protected]://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.01.026


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Fieberg and Kneer[2]solved an inverse heat conduction prob-

lem to estimate the heat flux at the interface between two solids

and used temperature measurements at the interface to estimate

the thermal contact resistance. In their work, the measurements

were taken by an infrared camera pointed at the location of the

interface. Thus, they needed access to the location of the interface.

In addition, a time dependent global contact resistance with con-

stant spatial distribution was estimated because no interior evalu-ation of the interface was performed.

Yang [21]also used an inverse heat conduction problem to esti-

mate a time dependent contact resistance in single-coated optical

fibers. Although good results were obtained, intrusive measure-

ments were required. No spatial variation was considered.

Gill et al.[9]solved an inverse heat conduction problem to esti-

mate the spatial distribution of the thermal contact resistance. The

authors mentioned that several models (as cited above) consider

the resistance constant, although it actually varies spatially. The re-

sults obtained by the authors were very sensitive to measurement

errors and required the use of a regularization technique. In addi-

tion, the temperatures were measured very close to the interface,

making the method very intrusive. However, the main contribution

was to estimate the spatial variation of the thermal contact resis-

tance instead of using a constant value.

According to the discussion above, the determination of the

thermal contact resistance/conductance is a very difficult task.

The objective of this paper is to present a methodology to estimate

the spatial variation of this parameter without intrusive measure-

ments. The methodology presented is formulated in terms of a rec-

iprocity functional approach [16] together with the method of

fundamental solutions [18] to solve two auxiliary problems. The

solution is composed of two steps. In the first step, auxiliary prob-

lems, which do not depend on the thermal conductance variation,

are solved. With the results of pre-processing, different thermal

conductances can be recovered by simply performing an integral.

Thus, the methodology is extremely fast and can be used to detect

flaws in different species using a short computational time.

2. Mathematical formulation

Let us consider a generic domain X, divided in three parts

X = X1 [C [X2, where X1is the first domain, with a thermal con-

ductivityK1, X2is the second domain, with a thermal conductivity

K2, and C is the contact surface between them. The boundary ofX1is @X1= C0 [C1 [ C, where the surface C0 is subjected to a pre-

scribed heat flux and its temperature is measured. C1is the lateral

surface ofX1, and C is the contact surface between X1and X2. On

the other hand, the boundary ofX2is @X2= C00 [C2 [ C, where

C00 is the lower surface, C2 is the lateral surface ofX2 and C is

the contact surface. Fig. 1shows the geometry for a two-dimen-

sional case.

The lateral surfaces C1 [C2are assumed to be thermally insu-lated while the lower surface C00 is subjected to a prescribed

temperature. The measurement surfaceC0 is assumed to have a pre-

scribedheatflux q imposedon it. ThecontactsurfaceC isassumedtohavea Robin boundarycondition, i.e.,K1@T1/@n= h(T1 T2), where

nis the normal derivative outward the boundary,K1is the thermal

conductivity of region 1, T1 and T2 are the temperatures at theinterfaceof domains one and two, respectively, and h is the thermalcontact conductance, which varies from zero (for an insulated

boundary) to infinity (for perfect contact). Typical values of thermalcontact conductances are given in Table 1.

The statement of the interface heat transfer problem in the stea-

dy-state case for constant conductivities K1 and K2 can be formu-lated as the following direct problem:

r2T1 0 in X1 2:a

K1@T1@n

q at C0 2:b

@T1@n

0 at C1 2:c

K1@T1@n

hT1T2 at C 2:d

r2T2 0 in X2 2:e

@T2@n

0 at C2 2:f

T2 0 at C00 2:g

K2@T2@n

K1@T1@n

at C 2:h

The inverse problem consists of estimating the function h at the inac-

cessible contact surface C by using some extra temperature mea-

surements Y at the boundary C0. To estimate it, we will define

two auxiliary problems: the first one to determine the temperature

jump T1 T2at the interfaceC and the second one to determine the

heat flux K1@T1/@n at the same interface. According to Eq. (2.d),

the thermal contact conductance will be given as the ratio of these

two quantities. Note that if (T1 T2) is equal to zero, thenwe havea

perfect thermal contact between both surfaces and the definition of

thermal contact resistance does not make sense (h tends to infinity).

2.1. Obtaining T1 T2atC

Consider thefirst auxiliary problem for some harmonic test func-

tionsF1 2 C2(X1) andF2 2 C

2(X2):

r2F1 0 in X1 3:a

K1@F1@n

/ at C 3:b

@F1@n

0 at C1 3:c

F1 F2 at C 3:d

r2F2 0 in X2 3:e

@F2@n

0 at C2 3:f

F2 0 at C00 3:g

K2@F2@n

K1@F1@n

/ at C 3:h

Fig. 1. Geometry for a two-dimensional case.

Table 1

Typical values of thermal contact conductances [5].

Contacting faces Conductance[W/(m2C)]

Iron/ Aluminum 45,000

Cooper/ Cooper 10,00025,000

Aluminum/ Aluminum 2,20012,000

Stainless steel/ Stainless steel 2,0003,700

Stainless steel/ Stainless steel (evacuated gaps) 2001,100

Ceramic/ Ceramic 5003,000

654 M.J. Colao, C.J.S. Alves / International Journal of Heat and Mass Transfer 60 (2013) 653663
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There is no boundary condition for F1at C0. However, there are two

boundary conditions at C. ForF2, there are boundary conditions for

all boundaries ofX2. Thus, the problem forF1is a Cauchy problem.

Note that/ appearing in Eq.(3.b)is a generic basis function, which

will be defined later.

Let us write the following identity for the domain X1:

0 ZX1

F1r2T1 T1r

2F1dX1 4

Using Eqs.(2.a) and (3.a), both Laplacians are zero such that Eq.(4)

vanishes. By using Greens second identity, we can, however, obtainZX1

F1r2T1 T1r

2F1dX1

Z@X1

F1@T1@n

T1@F1@n

d@X1

0

ZC0 [C1[C

F1@T1@n

T1@F1@n

d@X1

5

and using the boundary conditions on C1, Eqs.(2.c) and (3.c),

0

ZC0 [C

F1@T1@n

T1@F1@n

d@X1 6

Using Eq.(2.b)and the fact the some measurements Yare availableat the boundary C0, such thatT1= Yat C0, we obtain

0

ZC0

F1q

K1

Y

@F1@n

dC0

ZC

F1@T1@n

T1@F1@n

dC 7

Let us now consider another identity, for the domain X2:

0

ZX2

F2r2T2 T2r

2F2dX2 8

where the Laplacians are taken from Eqs.(2.e) and (3.e)such that

Eq.(8) vanishes. By using Greens second identity, as well as Eqs.

(2.f), (2.g), (3.f) and (3.g), we can obtain

0 ZC

F2@T2

@nT2

@F2

@n dC 9

AsK1andK2are constants, summing Eqs.(7) and (9), we obtain

ZC0

K1 F1q

K1

Y

@F1@n

dC0

ZC

K1 F1@T1@n

T1@F1@n

dC

ZC

K2 F2@T2@n

T2@F2@n

dC 0 10

orZC0

K1 F1q

K1

Y

@F1@n

dC0

ZC

K2F2@T2@n

K1F1@T1@n

dC

ZC

K2T2@F2@n

K1T1@F1@n

dC

11

Using Eqs. (2.h) and (3.d), the first term on the right hand side is

equal to zero, so we haveZC0

K1 F1q

K1

Y

@F1@n

dC0

ZC

K2T2@F2@n

K1T1@F1@n

dC 12

Using Eq.(3.h), we finally obtainZC0

K1 F1q

K1

Y

@F1@n

dC0

ZC

K1@F1@n

T1T2dC 13

Now we can define RF1 as the reciprocity functional, a notion

used by Andrieux and Abda[16], in terms of the test functions F1as

RF1 ZC0

F1q

K1

Y

@F1@n

dC0 14

For the calculation of the reciprocity functional, no information

regarding the boundary C is needed. In addition, once the function

F1is specified, only the conductivity K1, the imposed heat flux qand

the measured temperatures Y are needed for the calculation of

RF1. Using Eqs.(13) and (14), we obtain

RF1K1 T1T2;K1@F1@n

L2C

15

Now take F1,jsuch that K1@F1,j/@n= /jat C [Eq.(3.b)], where (/j) i s a

L2(C) orthonormal basis system. Then, taking the projection of Eq.

(15)over /j, the discontinuityT1 T2can be written as

T1T2C Xj

hT1T2;/jiL2 C/j Xj

RF1;jK1/j 16

To obtain the functions F1,j, we must solve a Cauchy problem with

double conditions K1@F1,j/@n= /j andF1,j= F2,j at the boundary C.

The problem has to be solved thus for several functions/j. The solu-

tion of the auxiliary problem is independentof the direct problem,

except by the geometry and the thermal conductivity K1. Thus, once

the auxiliary problem is solved and the functions F1,j are obtained,

different discontinuity configurations T1 T2 can be obtained by

simply evaluating a different integral in Eq. (14). Eq. (16) also allows

to identify situations of perfect thermal contact (T1= T2) where wecan avoid the division in the definition ofh.

Remark A. We can identify the situation where T1= T2 everywherein C, by considering the analytical solution found in Section2.4.

This case occurs when the measured temperatures (Y) in C0 areequal toq(a/K2+ b/K1), whereais the height of the domain X2and

bis the height of the domain X1

2.2. ObtainingK1@T1/@n atC

Consider now the second auxiliary problem, for some harmonictest functionsG1 2 C

2(X1):

r2G1 0 in X1 17:a

G1 / at C 17:b

@G1@n

0 at C1 17:c

@G1@n

0 at C 17:d

Following the same procedure to obtain Eq.(11), we haveZC0

K1 G1q

K1

Y

@G1@n

dC0

ZC

K1G1@T1@n

dC

ZC

K1T1@G1@n

dC 18

Using Eq.(17.d), we obtainZC0

K1 G1q

K1

Y

@G1@n

dC0

ZC

K1G1@T1@n

dC 19

Now we can define RG1 as the reciprocity functional in terms of

the test functionsG1as

RG1

ZC0

G1q

K1

Y

@G1@n

dC0 20

Using Eqs.(19) and (20), we obtain

RG1K1 G1;K1@T1@n

L2C

21

As before, taking G1,jsuch that G1,j= /jat C [Eq. (17.b)], where (/j) is

aL2

(C) orthonormal basis system, the term K1@T1/@ncan be writ-ten as

M.J. Colao, C.J.S. Alves / International Journal of Heat and Mass Transfer 60 (2013) 653663 655
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K1@T1@n

C

Xj

K1@T1@n

; /j

L2C

/j Xj

RG1;jK1/j 22

To find the functions G1,j, we must solve a Cauchy problem with double

conditions G1,j = /j and @G1,j/@n = 0 at the boundaryC. Once again, the

solution of the auxiliary problem isindependentof the direct problem,

except intermsof the geometry, which is the same. Thus, once the aux-

iliary problem is solved and the functionsG1,j are obtained, different

heat fluxesK1@T1/@n canbe obtained by simply evaluatinga different

integral in Eq.(20), whichis themainadvantageof thismethod because

the pre-processing involved in solving the auxiliary problems is per-

formed only once. Then, different thermal contact conductances can

be recovered by simply solving different integrals.

Remark B. Note that this auxiliary problem is independently

solvable. The extension of the solution to C0 by some harmonic

functions (as for example when one uses the method of funda-

mental solutions that will be described in Subsection 2.5) insures

the compatibility of this solution in the whole domain.

2.3. Obtaining h atC

From the previous results, the value ofh can be obtained fromEqs.(2.d), (16) and (22) as

h

Xj

RG1;j/j

Xj

RF1;j/j23

whenever the denominator is non-zero. Since we first find the value

ofRF1 by using Eq. (16), if it happens that (T1 T2) is equal to

zero, then we have a perfect contact between both surfaces and

Eq.(23)is not necessary (htends to infinity).

2.4. Analytical 1D solution

If we consider a constant value of thermal contact conductance

h along the horizontal direction, an analytical solution can be

found. Considering the boundary C1thermally insulated, the tem-

peratures in the domains X1and X2 can be written as

T1y C1yC2 26:a

T2y C3yC4 26:b

where theCs are constants. If measured temperaturesYand an im-

posed heat fluxq are available at boundary C0, these constants can

be obtained and the value ofh can be simply obtained by the fol-

lowing equation

h q

Yqa=K2b=K1 27

where a is the height of the domain X2 and b is the height of the

domain X1. Note thatT1= T2 all over C if

Y q

K1b

K1K2

a

28

This condition allows us to identify a perfect transmission contact.

Note that ifK1= K2, the material is the same, and the condition Y=

(q/K1)(a+ b) is just the compatible condition between the Dirichlet

and Neumann data. In that case, we have perfect transmission and

the solution is the same T1(y) = T2(y) = (q y)/K1.

2.5. Solution methodology

The previous sections showed how to obtain the discontinuityT1 T2 by the first auxiliary problem and the term K1@T1/@n by

the second auxiliary problem. In both problems, the solution forthe domain X2is straightforward and can be obtained by any tra-

ditional method, such as finite difference, integral transform, etc.

However, the solution for the domainX1is more involved because

there is no boundary condition atC0. Instead, there are two bound-

ary conditions atC, and a special technique is required to solve this

Cauchy problem. One possible strategy is to use the method of fun-

damental solutions[4,12].The method of fundamental solutions (MFS) is a meshless tech-

nique, which is integration free, for the numerical solution of cer-

tain elliptic boundary value problems. It was first proposed by

Krupadze and Aleksidze[18], and it is applicable if the fundamen-

tal solution of the governing equation is known. The MFS has been

used to solve different kinds of problems such as biharmonic prob-

lems, radiation-type boundary conditions, diffusive-convective

problems, free boundary problems, potential problems and elasto-

static, acoustic problems[7], and steady-state heat conduction in

composite materials[8,1]. The MFS is also known as the desingu-

larized method, the charge simulation method or the superposition

method in the mathematical and engineering literature [3].

The main idea of the MFS consists of approximating the solution

of the problem by means of a linear combination of fundamental

solutions with respect to some source points that are placed on a

fictitious boundary outside or inside the domain. Good results

are usually obtained with equally distributed collocation points

on the boundary and a similar distribution of source points on

the artificial boundary.

The solutions of the auxiliary problems are then approximatedby

F1x XNi1

biMix xi for x2 X1 29:a

G1x XNi1

ciMix xi for x2 X1 29:b

where b

i andci are the unknown coefficients to be determined,Mi(x) is the fundamental solution of the elliptic partial differentialequation considered andN is the number of source points. For the

Laplace equations, such as the case considered in this paper, the

fundamental solutions in two and three dimensions are given as

Mix xi 1

2pln jx xij for 2D 30:a

Mix xi 1

4pjx xij for 3D 30:b

Eqs.(29) and (30)can be then used to solve the auxiliary problems

for the domain X1 with a proper choice of source points located

outside of the domain.

3. Results

In this paper, we considered a steady-state heat conduction

problem in a medium composed of two layers, as shown in

Fig. 1. Boundaries C1 and C2 are thermally insulated while Co is

subjected to a heat flux q. The boundary C00 is kept at a constanttemperature equal to 0oC. The domains X1andX2may have differ-

ent thermal conductivities. The contact between the two domains

is made by the boundary C, which is assumed to have a Robin

boundary condition, i.e., K1@T1/@n= h(T1 T2), where n is thenormal derivative outward of the boundary, K1is the thermal con-ductivity of region 1, T1 and T2 are the temperatures at the inter-face of domains one and two, respectively, and h is thermal

contact conductance, which varies from zero (for an insulated

boundary) to infinity (for a perfect contact). The inverse problemthus is to determine h by means of the reciprocity functional

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approach. To circumvent the lack of information on the boundary

C, temperature measurements, Y, at the boundary C0 are alsoavailable.

As discussed before, the reciprocity functional approach re-

quires the solution of two auxiliary problems: one for the determi-

nation of (T1 T2) and the other for the determination ofK1@T1/

@n, both at the interface C. Thus, the heat transfer coefficient can

be obtained simply by dividing one quantity by the other. For the

solution of the auxiliary problem related to the boundary X

2,which is a boundary value problem, we used a finite-difference ap-

proach, where a GaussSeidel approach with SOR was used to

solve the linear system. The problem for the boundary X1is more

complex because it is a Cauchy problem, and we used the method

of fundamental solutions (MFS), described in the previous section.

In this work, in order to alleviate the ill-conditioned character of

the inverse problem, we first filtered the measured temperature,

by using the method of fundamental solutions. In other words,

we used the measured temperature and the imposed heat flux to

solve a Cauchy problem for the domainX1 (the boundary condition

at C is still unknown, but we have two boundary conditions atC0).

Then we used the obtained solution of this procedure to calculate

(using again the MFS) new temperatures at the boundary C0 that

are compatible with the imposed heat flux. It is important to note

that the MFS will in this way regularize the measured tempera-

tures in order to guarantee this compatibility condition. Those

new temperatures were then used in place of the measured ones.

Besides, we used a Tikhonov regularization method to solve the

systemAx= b:

kI ATAx A

Tb 31

where the Tikhonov parameter kwas initially chose small (k= 108)

and it was continually updated to higher value (k= 2k) if the Dis-

crepancy principle fails (i.e. if the mean squared difference between

the calculated and the measured temperature is less than the vari-ance of the measured data), or if the inverse of the condition num-

ber ofATA is zero in floating point representation.

In the above procedure we used the IMSL subroutines DLFIRG

and DLFCRG for solving and estimating the condition number of

the matrices, respectively. This approach was able to produce sta-

ble solutions, as we will show later.

Six different thermal contact conductance profiles were used in

this paper.Table 2shows the definitions of the profiles, where L isthe length of the material and hmaxis the maximum value ofh. Note

that test case 5 corresponds to the one used in the analytical solu-

tion presented in Section2.4.

The reciprocity functional approach requires the solution of theauxiliary problems only once. Once the solution is obtained, the same

solution can be used to capture different heat transfer coefficient pro-files, even with different imposed heat fluxes and different measuredtemperatures.

The measured temperatures were obtained by solving the direct

problem with a known value ofh. The solution was obtained by thefinite-difference approach. Because different methodologies were

used for the solution of the direct and inverse problems, the so-

called inverse crime was avoided. A grid convergence analysis

was performed for a case where the heat transfer coefficient was

given by profile 1 in Table 2, where hmax was set to 10,000 W/(m2 C) in a geometry with a length equal to 0.04 m and a total

height equal to 0.02 m (0.01 m for domain 1 and 0.01 m for do-

main 2). It was assumed that the thermal conductivity was

54 W/(m C) for both domains, and a constant heat flux of

10 W/m2 was imposed. After some tests, a grid with 120 60

points was considered converged for this problem.

To verify the stability of the solution, experimental errors were

introduced in the measured temperatures according to Eq. (32)

Table 2

Heat transfer coefficient profiles.

Profile h[W/(m2C)]

1 hmax for x < L/4 andx > 3L/4; 0 for L/4


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Y TC0 er 32

wheree is a random variable with a Gaussian distribution and r isthe standard deviation of the measurements. In order to generate

the Gaussian random number, with zero mean and unit variance,we used the Box-Muller transform

effiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 lnuq

cos2pv 30

whereu and vare two uniformly distributed random numbers (be-

tween 0 and 1) generated by the Fortran intrinsic functionrandom_number ().

Fig. 2. EstimatedK1(T1 T2) for test cases 16a.

Fig. 3. EstimatedK1@T1/@n for test cases 16a.



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As discussed before, the solution of the auxiliary problems in-

volves their solution for a set of orthonormal functions. In this

study, we chose a combination of sine and cosine-wave functions.

As we will see, this choice does not provide the best estimate fordiscontinuous functions, mainly due to the Gibbs phenomenon.

The number of test functions being used were choose such

as the reciprocity functions given by Eqs. (15) and (21)reached a

stable value. Thus, the two auxiliary problems were solved

several times for different values of /j, as showed in Eqs. (3.b)and (17.b).

Fig. 4. Estimatedh for test cases 16a.

Fig. 5. Estimatedh for test cases 16b (r = 0.1 % ofjYmaxj).

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Different test cases were chosen to test different values ofhmax,

q(the imposed heat flux), the size of the material, and the levels ofuncertainties.Table 3shows the test cases analyzed, andTable 4

presents the maximum values of the measured temperatures (Y)

for each test case.

Initially, we will analyze the results of (T1 T2) for cases 16a.

Fig. 2shows these values using the heat transfer coefficients for

the six profiles presented in Table 1, where hmax was set to1,000 W/(m2 C) in a geometry with a length equal to 0.04 m and

a total height equal to 0.02 m (0.01 m for domain 1 and 0.01 m

Fig. 6. Estimated h for test cases 16c (r = 0.5 % ofjYmaxj).

Fig. 7. Estimatedh for test cases 16d (smaller domain).



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for domain 2). In addition, the thermal conductivity was assumed

to be 54 W/(m C) for both domains, and a constant heat flux equal

to10 W/m2 was imposed.Fig. 2shows that the general behavior

of (T1 T2) is well captured. In addition, even for functions where his discontinuous, the variation of (T1 T2) is continuous.

Fig. 3shows the estimated values for K1@T1/@n corresponding

to the test cases analyzed in Fig. 2. Here, the estimate ofK1@T1/@n

is worse than the one for (T1 T2). It can be observed however that

the oscillations are reasonably well captured, although the integra-

tion process smoothes the discontinuities.

Finally,Fig. 4shows the estimated value of the thermal contact

conductance, which is basically a computation of the previous val-

ues ofK1@T1/@nand (T1 T2). The general behavior of the function

is well captured. Although the discontinuity is not predicted by this

Fig. 8. Estimatedh for test cases 16e (larger hmax).

Fig. 9. Estimatedh for test cases 16f (larger hmax andr = 0.1 % ofjYmaxj).



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technique, the location of the maximum and minimum values ofhis predicted well in all cases. This problem is mainly due to the

choice of the orthogonal functions (Fourier series). Thus, disconti-

nuity functions could be better estimated by a proper choice of the

orthogonal functions in Eqs. (3.b) and (17.b). For test function

number 5, the analytical solution obtained by Eq. (27) is alsoshowed, where it can be seen that it perfectly match the exact va-

lue ofh.Next, we will analyze the influence of the measurement errors

on the estimated values of the thermal conductance. Figs. 5 and

6 show the results for a standard deviation r equal to 0.1%and 0.5% of jYmaxj, respectively, corresponding to test cases 16band 16c. The real function is reasonably well captured for a

standard deviation of 0.1% of jYmaxj, although the estimate is not

as good for 0.5% ofjYmaxj. However, even for this value ofr, resultscould be used to identify the regions of maximum and minimum

values of the thermal conductance.

For the next result, we used a domain with a length of 0.004 m

and a total height of 0.002 m (0.001 m for domain 1 and 0.001 m

for domain 2), corresponding to cases 16d, to verify the sensitivityof the method with respect to the distance between the measure-

ments and the unknown boundary condition interface. All other

physical parameters remained constant, including the number of

grid points. For this case, we considered measurements without er-

rors, and the results are presented inFig. 7, corresponding to test

cases 16d. These figure shows little variance with the respect to

the previous figures presented in Fig. 4 for cases 16a, demonstrat-

ing that the estimate is not dependent on the thickness of the

material.

Fig. 8 shows the influence ofhmaxon the solution, correspondingto test cases 16e. This figure shows the results for the original do-

main (the same used to obtain the results up to Fig. 6) but for

hmax = 10,000 W/(m2C). These figures show that the estimate be-

comes worse as the discontinuity increases, which was expected.

In addition, larger values of the thermal conductance indicate an

interface with lower thermal resistance. Thus, in the limiting case

of an infiniteh, it is expected that the method will not be able to

identify it because it indicates a perfect continuous material.

Fig. 9show the results for test cases 16f, which are the same as

those presented in Fig. 8 but with measurement errors with a stan-

dard deviation of 0.1% of the maximum temperature. These resultsdemonstrate the stability of the method with respect to the errors

in the measurements.

As presented inTable 4, the maximum value of the measured

temperature in all test cases is extremely low because the applied

heat flux is not sufficient to heat the material above 1 C. The first

set of results then showed the capability of estimating the un-

known thermal conductance profile, even for extremely small vari-

ations of the temperature. However, it is not practical because no

instrument is able to measure such low values of temperatures

with small measurement errors. Thus, Fig. 10 shows the results

for test cases 16g, where the applied heat flux was increased to

100 kW/m2, resulting in a maximum value of the measured tem-

perature of approximately 200 C. For this case, a standard devia-

tion for the measurement errors of 0.1% of the maximummeasured temperature (0.2 C) was assumed. This value is repre-

sentative of the measurement errors found in modern infrared

cameras. The analysis of Fig. 10shows that the estimated value

ofh is very good, even in the presence of experimental errors. Thus,this method could be used in real experiments by simply increas-

ing the value of the applied heat flux such that the measurement

errors can fit within the maximum values required for the stability

of the method (0.1% ofjYmaxj in the present case).

4. Conclusions

In this study, we used an approach based on the reciprocity

functional together with the method of fundamental solutions to

estimate the spatial variation of the thermal contact conductancebetween two materials. The solution is composed of two steps. In

Fig. 10. Estimatedh for test cases 16g (larger heat flux and r = 0.1 % ofjYmaxj)

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the first step, the auxiliary problems, which do not depend on the

thermal conductance variation, are solved. With the results of pre-

processing, different thermal conductances can be recovered by

simply performing an integral. Thus, the methodology is extremely

fast and can be used to detect flaws in different samples using low

computational time. The results obtained were very good, even for

data with measurement errors. In order to improve the quality of

the results, especially for the discontinuous thermal conductances,different orthonormal functions shall be investigated. Also, the

authors are already working on the extension of the method pro-

posed to transient heat conduction in composite materials. Preli-

minary results are very good and shall be published soon.

Acknowledgements

The authors thank the Brazilian agencies, Conselho Nacional deDesenvolvimento Cientfico e Tecnolgico (CNPq), Coordenao de

Aperfeioamento de Pessoal de Nvel Superior(CAPES) andFundaoCarlos Chagas Filho de Amparo Pesquisa do Estado do Rio de Janeiro(FAPERJ), for fostering science and for financial support for this

work. This work was part of an international cooperation project

between Brazil and Portugal (CAPES/FCT 305).

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a fast non-intrusive method for estimating spatial thermal contact conductance by means of the...

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