International Communications in Heat and Mass Transfer 38 (2011) 298–303

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International Communications in Heat and Mass Transfer

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Estimation of thermal contact resistance and temperature distributions in thepad/disc tribosystem☆

Yu-Ching Yang ⁎, Haw-Long Lee, Wen-Lih Chen, Jose Leon SalazarClean Energy Center, Department of Mechanical Engineering, Kun Shan University, Yung-Kang City, Tainan 710-03, Taiwan, Republic of China

☆ Communicated by W.J. Minkowycz.⁎ Corresponding author.

E-mail address: ycyang@mail.ksu.edu.tw (Y.-C. Yang

0735-1933/$ – see front matter © 2010 Elsevier Ltd. Aldoi:10.1016/j.icheatmasstransfer.2010.11.005

a b s t r a c t

a r t i c l e i n f o

Available online 3 December 2010

Keywords:Inverse problemBrakingThermal contact resistanceConjugate gradient method

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle isapplied to estimate the unknown time-dependent thermal contact resistance for the tribosystem consisting ofa semi-infinite foundation (disc) and a plane-parallel strip (pad) sliding over its surface, from the knowledgeof temperature measurements taken within the foundation. Subsequently, the temperature distributions inthe medium can be determined as well. The temperature data obtained from the direct problem are used tosimulate the temperature measurements, and the effect of the measurement errors upon the precision of theestimated results is also considered. Results show that an excellent estimation on the time-dependentthermal contact resistance can be obtained for the test case considered in this study. The current methodologycan be applied to the prediction of thermal contact resistance in engineering problems involving sliding-contact elements.

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© 2010 Elsevier Ltd. All rights reserved.

1. Introduction

The ceramic–metal frictional materials are now extensively used inbrake systems for their high thermal stability andwear resistance [1]. Inthe process of braking, the cermet patch is pressed to its counterbody(disc, brake drum, rim of the wheel, etc.). As a result of friction on thecontact surface, the kinetic energy is transformed into heat. Theelements of the brakes are heated and, hence, the operation conditionsfor the friction patches become less favorable: the friction coefficientdecreases and their wear intensifies, which may lead to emergencysituations. Thus, the correct calculation of temperature is one of themost important issues in the design of brakes [2]. In the past, there havebeenmany investigations focusing on thedeterminationof temperaturein brake systems [3–5]. However, to the best of the authors' knowledge,there are few researches describing the realistic situation in the brakesystems and considering the temperature drop at the interface betweenthe pad and the disc. The accurate determination of temperaturedistributions in the brake systems requires considering the imperfectconditions of thermal contact at the interface.

Over the past decades, inverse analysis has become a valuablealternative when the direct measurement of data is difficult or themeasuring process is very expensive, for example, the detection of

contact resistance, the determination of heat transfer coefficients, theestimation of unknown thermophysical properties of new materials,the prediction of damage in the structure fields, the detection offouling-layer profiles on the inner wall of a piping system, theoptimization of geometry, the prediction of crevice and pitting infurnace wall, the determination of heat flux at the outer surface of avehicle re-entry, and so on.

Although there have beenmany investigations considering the effectof contact resistance in different applications [6–9], nevertheless, thedetermination of thermal contact resistance at the interface has neverbeen an easy task. The main objective of the present study is to developan inverse analysis to estimate the thermal contact resistance for thetribosystem inRef. [5],which consists of a sliding strip (thepad) over thesurface of a semi-infinite foundation (the disc). An analysis of this kindposes significant implications on the study of the problems associatedwith sliding contact such as in a brake system. In this study, we presentthe conjugate gradient method [10–13] and the discrepancy principle[14] to estimate the time-varying thermal contact resistance by usingthe simulated temperature measurements. Subsequently, the distribu-tions of temperature in the strip and foundation can be determined aswell. The conjugate gradient method with an adjoint equation, alsocalled Alifanov's iterative regularization method, belongs to a class ofiterative regularization techniques, whichmeans that the regularizationprocedure is performed during the iterative processes, thus thedetermination of optimal regularization conditions is not needed. Noprior information is used in the functional form of the thermal contactresistance variation with time. On the other hand, the discrepancyprinciple is used to terminate the iteration process in the conjugategradient method.

d ks , αs

kf , α f

Vy

x

z

P

P

Fig. 1. Geometry and coordinate system.

Nomenclature

Bi Biot's number (=dh/ks)Bis Biot's number (=dhs/ks)d thickness of the strip (m)f frictional coefficienth thermal contact conductance (Wm−2 K−1)hs convection heat transfer coefficient (Wm−2 K−1)J functionalJ′ gradient of functionalk thermal conductivity (Wm−1 K−1)P compressive pressure (N m−2)p direction of descentq intensity of the frictional heat flux (Wm−2)T temperature (K)T0 temperature scaling factor (K)t time coordinate (s)V sliding velocity (m s−1)x spatial coordinate (m)Y measured temperature (K)y spatial coordinate (m)z spatial coordinate (m)

Greek symbolsΔ small variation qualityα thermal diffusivity (m2 s−1)β step sizeγ conjugate coefficientη very small valueλ variable used in the adjoint problemσ standard deviationτ transformed time coordinateϖ random variable

Superscripts/subscriptsK iterative numberm measurement position* dimensionless quantity

299Y.-C. Yang et al. / International Communications in Heat and Mass Transfer 38 (2011) 298–303

2. Analysis

2.1. Direct problem

To illustrate the methodology for developing expressions for theuse in estimating the unknown time-dependent thermal contactresistance during frictional heating at the interface of a tribosystem,which consists of a sliding strip (the pad) over the surface of a semi-infinite foundation (the disc), we consider the following transientheat transfer problem of friction process as shown in Fig. 1. Here, theimperfect heat contact between the strip and the foundation isassumed. It is supposed that the constant compressive pressures P areapplied to the upper surface of the strip and to the infinity of thefoundation. The strip slides with uniform velocity V in the direction ofy-axis on the semi-infinite surface. Considering the thermal contactresistance between the strip and the foundation, then the dimen-sionless governing equations and the associated boundary and initialconditions for the system can be written as [5]:

∂2T�s z�; t�ð Þ∂z�2

=∂T�

s z�; t�ð Þ∂t� ; 0≤z�≤1; t� N 0; ð1Þ

∂2T�f z�; t�ð Þ∂z�2

=1αfs

∂T�f z�; t�ð Þ∂t� ;−∞bz�≤0; t� N 0; ð2Þ

∂T�s 1; t�ð Þ∂z� + Bis⋅T�

s 1; t�� �

= 0; z� = 1; t� N 0; ð3Þ

T�f z�; t�� �

→0; z�→−∞; t� N 0; ð4Þ

kfs∂T�

f 0; t�ð Þ∂z� −∂T�

s 0; t�ð Þ∂z� = 1; z� = 0; t� N 0; ð5Þ

kfs∂T�

f 0; t�ð Þ∂z� +

∂T�s 0; t�ð Þ∂z� = Bi t�

� �T�s 0; t�� �

−T�f 0; t�� �h i

; z� = 0; t� N 0;

ð6Þ

T�s z�;0� �

= 0; 0≤z�≤1; t� = 0; ð7Þ

T�f z�;0� �

= 0;−∞bz�≤0; t� = 0; ð8Þ

where the subscripts s and f refer to the regions of strip andfoundation, respectively.

The dimensionless variables used in the above formulation aredefined as follows:

z� = z = d; t� = αst = d2; T0 = qd= ks; T�

s = Ts = T0;

T�f = Tf = T0; αfs = αf =αs; kfs = kf = ks; Bi = hd= ks;

Bis = hsd= ks;

ð9Þ

where d is the thickness of the strip, q= fVP is the frictional heatgeneration at the interface of the strip and the foundation. k and α arethe thermal conductivity and thermal diffusivity, respectively. Thedirect problem considered here is concerned with the determinationof the medium temperature when the Biot numbers Bis and Bi(t*),thermophysical properties of the system, and initial and boundaryconditions are known. A hybrid numerical method of Laplacetransformation and finite difference used in our previous work canbe applied to solve the direct problem [13].

2.2. Inverse problem

For the inverse problem, the function of Biot number Bi(t*) isregarded as being unknown, while everything else in Eqs. (1)–(8) isknown. In addition, temperature readings taken at z=zm in thefoundation region are considered available. The objective of theinverse analysis is to predict the unknown time-dependent function

300 Y.-C. Yang et al. / International Communications in Heat and Mass Transfer 38 (2011) 298–303

of Biot number Bi(t*), merely from the knowledge of thesetemperature readings. Let the measured temperature at the mea-surement position z=zm and time t be denoted by Y(zm,t). Then thisinverse problem can be stated as follows: by utilizing the abovementioned measured temperature data Y(zm,t), the unknown Bi(t*) isto be estimated over the specified time domain.

The solution of the present inverse problem is to be obtained insuch a way that the following functional is minimized:

J Bi t�� �� �

= ∫ t�ft� = 0 T�

f z�m;t�� �−Y� z�m;t

�� �h i2dt�; ð10Þ

where Y*(zm* ,t*)=Y(zm,t)/T0, and Tf*(zm* ,t*) is the estimated (or

computed) temperature at the measurement location z=zm* . In thisstudy, Tf*(zm* ,t*) are determined from the solution of the direct problemgiven previously by using an estimated BiK(t*) for the exact Bi(t*), hereBiK(t*) denotes the estimated quantities at the Kth iteration. tf* is thefinal time of the measurement. In addition, in order to developexpressions for the determination of the unknown Bi(t*), a ‘sensitivityproblem’ and an ‘adjoint problem’ are constructed as described below.

2.3. Sensitivity problem and search step size

The sensitivity problem is obtained from the original directproblem defined by Eqs. (1)–(8) in the following manner: it isassumed that when Bi(t*) undergoes a variationΔBi t�ð Þ, Ts*(z*,t*) and Tf

*

(z*,t*)are perturbed by T�s + ΔT�

s and T�f + ΔT�

f , respectively. Thenreplacing in the direct problem Bi(t*) by Bi t�ð Þ + ΔBi t�ð Þ, Ts

* byT�s + ΔT�

s , and Tf* by T�

f + ΔT�f , subtracting from the resulting

expressions the direct problem, and neglecting the second-orderterms, the following sensitivity problem for the sensitivity functionΔT�

s and ΔT�f can be obtained:

∂2ΔT�s

∂z�2=

∂ΔT�s

∂t� ; 0≤z�≤1; t� N 0; ð11Þ

∂2ΔT�f

∂z�2=

1αfs

∂ΔT�f

∂t� ;−∞bz�≤0; t� N 0; ð12Þ

∂ΔT�s

∂z� + Bis⋅ΔT�s = 0; z� = 1; t� N 0; ð13Þ

ΔT�f →0; z�→−∞; t� N 0; ð14Þ

kfs∂ΔT�

f

∂z� −∂ΔT�s

∂z� = 0; z� = 0; t� N 0; ð15Þ

kfs∂ΔT�

f

∂z� +∂ΔT�

s

∂z� = Bi⋅ ΔT�s −ΔT�

f

h i+ ΔBi⋅ T�

s−T�f

h i; z� = 0; t� N 0;

ð16Þ

ΔT�s = 0; 0≤z�≤1; t� = 0; ð17Þ

ΔT�f = 0;−∞bz�≤0; t� = 0: ð18Þ

The sensitivity problem of Eqs. (11)–(18) can be solved by thesame method as the direct problem of Eqs. (1)–(8).

2.4. Adjoint problem and gradient equation

To formulate the adjoint problem, Eqs. (1) and (2) are multipliedby the Lagrange multipliers (or adjoint functions) λs

* and λf*,

respectively, and the resulting expressions are integrated over the

time and correspondent space domains. Then the results are added tothe right hand side of Eq. (10) to yield the following expression for thefunctional J[Bi(t*)]:

J Bi t�ð Þ½ � = ∫ t�ft� = 0∫

0z� = −∞ T�

f z�;t�ð Þ−Y� z�;t�ð Þh i2⋅δðz�−z�mÞdz�dt�

+ ∫ t�ft� = 0∫

1z� = 0λ

�s z�;t�ð Þ⋅ ∂2T�

s

∂z�2−∂T�

s

∂t�

" #dz�dt�

+ ∫ t�ft� = 0∫

0z� = −∞λ

�f z�;t�ð Þ⋅ ∂2T�

f

∂z�2− 1

αfs

∂T�f

∂t�

" #dz�dt�:

ð19Þ

The variation ΔJ is derived after Bi(t*) is perturbed by ΔBi t�ð Þ,Ts* and Tf

* are perturbed by T�s + ΔT�

s and T�f + ΔT�

f , respectively, inEq. (19). Subtracting from the resulting expression the originalEq. (19) and neglecting the second-order terms, we thus find:

ΔJ Bi t�ð Þ½ � = ∫ t�ft� = 0∫

0z� = −∞2 T�

f z�;t�ð Þ−Y� z�;t�ð Þh i

⋅ΔT�f ⋅δ z�−z�mð Þdz�dt�

+ ∫ t�ft� = 0∫

1z� = 0λ

�s z�;t�ð Þ⋅ ∂2ΔT�

s

∂z�2−∂ΔT�

s

∂t�

" #dz�dt�

+ ∫ t�ft� = 0∫

0z� = −∞λ

�f z�;t�ð Þ⋅ ∂2ΔT�

f

∂z�2− 1

αfs

∂ΔT�f

∂t�

" #dz�dt�;

ð20Þ

where δ(⋅) is the Dirac function. We can integrate the second and thethird double integral terms in Eq. (20) by parts, utilizing the initial andboundary conditions of the sensitivity problem. The vanishing of theintegrands containing ΔT�

s and ΔT�f leads to the following adjoint

problem for the determination of λs* and λf

*:

∂2λ�s

∂z�2+

∂λ�s

∂t� = 0; 0≤z�≤1; t� N 0; ð21Þ

∂2λ�f

∂z�2+

1αfs

∂λ�f

∂t� + 2 T�f z�;t�� �

−Y� z�m;t�� �h i

⋅δ z�−z�m� �

= 0;−∞bz�≤0; t� N 0;

ð22Þ

∂λ�s

∂z� + Bis⋅λ�s = 0; z� = 1; t� N 0; ð23Þ

λ�f→0; z�→−∞; t� N 0; ð24Þ

∂λ�f

∂z� −∂λ�

s

∂z� = 0; z� = 0; t� N 0; ð25Þ

∂λ�f

∂z� +∂λ�

s

∂z� = Bi⋅ λ�s−

λ�f

kfs

" #; z� = 0; t� N 0; ð26Þ

λ�s = 0; 0≤z�≤1; t� = t�f ; ð27Þ

λ�f = 0;−∞bz�≤0; t� = t�f : ð28Þ

The adjoint problem is different from the standard initial valueproblem in that the final time condition at time t*= tf* is specifiedinstead of the customary initial condition at time t*=0. However, thisproblem can be transformed to an initial value problem by thetransformation of the time variable as τ*= tf*− t*. Then the adjointproblem can be solved by the same method as the direct problem.

Finally the following integral term is left:

ΔJ = ∫ t�ft� = 0

14

T�s 0; t�� �

−T�f 0; t�� �h i

λ�f 0;t�� �

= kfs−λ�s 0;t�� �h i

ΔBi t�� �

dt�:

ð29Þ

301Y.-C. Yang et al. / International Communications in Heat and Mass Transfer 38 (2011) 298–303

From the definition used in Ref. [13], we have:

ΔJ = ∫ t�ft� = 0 J′ t�

� �ΔBi t�

� �dt�; ð30Þ

where J′(t*) is the gradient of the functional J(Bi). A comparison ofEqs. (29) and (30) leads to the following form:

J′ t�� �

=14

T�s 0; t�� �

−T�f 0; t�� �h i

λ�f 0;t�� �

= kfs−λ�s 0;t�� �h i

: ð31Þ

2.5. Conjugate gradient method for minimization

The following iteration process based on the conjugate gradientmethod is now used for the estimation of Bi(t*) by minimizing theabove functional J[Bi(t*)]:

BiK + 1 t�� �

= BiK t�� �

−βKpKðt�Þ; K = 0; 1; 2; :::; ð32Þ

whereβK is the search step size in going from iteration K to iteration K+1, and pK(t*) is the direction of descent (i.e., search direction) given by:

pK t�� �

= J′K t�� �

+ γKpK−1ðt�Þ; ð33Þ

which is the conjugation of the gradient direction J′K(t*) at iteration Kand the direction of descent pK−1(t*)at iteration K−1. The conjugatecoefficient γK is determined from:

γK =∫ t�f

t� = 0 J′K t�ð Þh i2

dt�

∫ t�ft� = 0 J′K−1 t�ð Þ

h i2dt�

; with γ0 = 0: ð34Þ

The convergence of the above iterative procedure in minimizingthe functional J is proved in Ref. [14]. To perform the iterationsaccording to Eq. (32), we need to compute the step size βK and thegradient of the functional J′K(t*).

The functional J[BiK+1(t*)] for iteration K+1 is obtained byrewriting Eq. (10) as:

J BiK + 1 t�� �h i

= ∫ t�ft� = 0 T�

f BiK−βKpK� �

−Y� z�m;t�� �h i2

dt�; ð35Þ

where we replace BiK+1 by the expression given by Eq. (32). Iftemperature Tf

*(BiK−βKpK) is linearized by a Taylor expansion,Eq. (35) takes the form:

J BiK + 1 t�� �h i

= ∫ t�ft� = 0 T�

f BiK� �

−βKΔT�f pK� �

−Y� z�m;t�� �h i2

dt�; ð36Þ

where Tf*(BiK) is the solution of the direct problem at z*=zm

* by usingestimated BiK(t*) for exact Bi(t*) at time t*. The sensitivity functionΔT�

f pK� �

are taken as the solution of Eqs. (11)–(18) at the measuredposition z*=zm

* by letting ΔBi = pK [10]. The search step size βK isdetermined by minimizing the functional given by Eq. (36) withrespect to βK. The following expression can be obtained:

βK =∫ t�f

t� = 0ΔT�f pK� �

T�f BiK� �

−Y�h idt�

∫ t�ft� = 0 ΔT�

f pK� �h i2

dt�: ð37Þ

2.6. Stopping criterion

If the problem contains no measurement errors, the traditionalcheck condition specified as:

J BiK + 1� �

bη; ð38Þ

where η is a small specified number, can be used as the stoppingcriterion. However, the observed temperature data contains mea-surement errors; as a result, the inverse solution will tend to approachthe perturbed input data, and the solution will exhibit oscillatorybehavior as the number of iteration is increased [14]. Computationalexperience has shown that it is advisable to use the discrepancyprinciple for terminating the iteration process in the conjugategradient method. Assuming Tf

*(zm* ,t*)−Y*(zm* ,t*)≅σ, the stoppingcriteria η by the discrepancy principle can be obtained from Eq. (10) as:

η = σ2t�f ; ð39Þ

where σ is the standard deviation of the measurement error. Then thestopping criterion is given by Eq. (38) with η determined from Eq. (39).

3. Results and discussion

The objective of this article is to validate the present approachwhen used in estimating the unknown time-dependent thermalcontact resistance at the interface of a strip (the pad) and a semi-infinite foundation (the disc) during a sliding contact accurately withno prior information on the functional form of the unknownquantities, a procedure called function estimation. In the presentstudy, we consider the simulated exact value of Bi(t*) over the timeperiod t*=0 to 1 as:

Bi t�� �

= 1−e−10t�� �

: ð40Þ

Thematerial of the foundation is assumed being cast iron CHhMKh(kf=51 Wm−1 K−1 and αf=1.4×10−5 m2 s−1), while the materialof the strip is ceramic–metal FMK-11 of thickness d=5 mm(ks=34.3Wm−1 K−1 andαs−1.52×10−5 m2 s−1). TheBiot's numberat the strip surface is taken as Bis=1.0 in this study [5]. A singlethermocouple is assumed to be located at the interface (zm* =0). In termsof the timedomain, the total dimensionlessmeasurement time is chosenas tf*=1.0 and measurement time step is taken to be 0.02. Besides, thesame computational procedure as in Ref. [7] is used in the numericalcalculations.

In the analysis, we do not have a real experimental set up tomeasurethe temperature Y*(zm* ,t*) in Eq. (10). Instead, we assume a real thermalcontact resistance, Bi(t*) of Eq. (40), and substitute the exact Bi(t*) intothe direct problem of Eqs. (1)–(8) to calculate the temperatures at thelocation where the thermocouple is placed. The results are taken asthe computed temperature Yexact

* (zm* ,t*). Nevertheless, in reality, thetemperaturemeasurements always contain somedegree of error,whosemagnitude depends upon the particular measuring method employed.In order to consider the situation ofmeasurement errors, a randomerrornoise is added to the above computed temperature Yexact* (zm* ,t*) to obtainthe measured temperature Y*(zm* ,t*). Hence, the measured temperatureY*(zm* ,t*) is expressed as

Y� z�m;t�� �

= Y�exact z�m;t

�� �+ ϖσ; ð41Þ

where ϖ is a random variable within −2.576 to 2.576 for a 99%confidence bounds, and σ is the standard deviation of the measure-ment. The measured temperature Y*(zm* ,t*) generated in such way isthe so-called simulated measurement temperature.

Fig. 2 shows the estimated values of the unknown function Bi(t*),obtained with the initial guesses Bi0=0.0, temperature measurementtaken at zm* =0.0, and measurement error of deviation σ=0.0 and0.01, respectively. These results confirm that the estimated results arein very good agreement with those of the exact values. For atemperature of unity and 99% confidence, that standard deviation,σ=0.01, corresponds to measurement error of 2.58%. The results inFig. 2 also demonstrate that, for the cases considered in this study, anincrease in the measurement error does not cause obvious

t*

Bi

0 0.2 0.4 0.6 0.8 10

2

4

6

ExactEstimated σ = 0.00Estimated σ = 0.01

Fig. 2. Estimated Bi(t*) at 10th iteration with initial guesses Bi0(t*)=0.0, zm*=0.0, andσ=0.00 and 0.01, respectively.

t*

Bi

0 0.2 0.4 0.6 0.8 10

2

4

6

ExactEstimated

Fig. 3. Estimated Bi(t*) at 10th iteration with initial guesses Bi0(t*)=0.0, zm*=−0.1and σ=0.00.

z*

T*

-3 -2 -1 0 1 20

0.1

0.2

0.3

0.4

0.5

ExactEstimated

t*=

0.2

t*=

0.5

t*=

0.8

Fig. 4. Estimated temperature distributions at 10th iteration with initial guesses Bi0

(t*)=0.0, zm*=0.0, and σ=0.00.

302 Y.-C. Yang et al. / International Communications in Heat and Mass Transfer 38 (2011) 298–303

,

deterioration on the accuracy of the inverse solution. Meanwhile, inorder to investigate the influence of measurement location uponthe estimated results, Fig. 3 illustrates the estimated unknown functionBi(t*), with temperature measurement taken at zm* =−0.1. Here,the initial guesses Bi0=0.0 and measurement error σ=0.00, andsatisfactory results are still returned which has proved that differentmeasurement locations pose no influence on the accuracy of thepresentinverse method.

The estimated temperature distributions in the strip and semi-infinite foundation for t*=0.20, 0.50, and 0.80, respectively, aredemonstrated in Fig. 4. The results in Fig. 4 are obtained with theinitial guesses Bi0=0.0, temperature measurement taken at zm* =0.0,and measurement error of deviation σ=0.00. These results confirmthat the estimated temperature values are in very good agreementwith those of the exact values for the case considered in this study. Itcan be found in Fig. 4 that, overall, the temperature rises rapidly at theinterface as a consequence of the rapid rise of its internal energy byheat generation, but it drops sharply as the distance from the interfaceincreases.

In order to demonstrate the capability of the presented method-ology in obtaining an accurate estimation no matter how complex theunknown function is, we consider another case of Bi(t*) with thefollowing form:

Bi t�� �

= 5t�=0:6; for 0≤t�≤0:6;5; for 0:6≤t�≤1:

�ð42Þ

Fig. 5 shows the estimated results of Bi(t*), obtained with theinitial guesses Bi0=0.0, temperature measurement taken at zm* =0.0,and measurement error of deviation σ=0.00. It can be found in Fig. 5that an excellent estimation still can be obtained with this complexunknown function.

t*

Bi

0 0.2 0.4 0.6 0.8 10

2

4

6

ExactEstimated

Fig. 5. Estimated Bi(t*) at 10th iteration with initial guesses Bi0(t*)=0.0, zm*=0.0, andσ=0.00.

303Y.-C. Yang et al. / International Communications in Heat and Mass Transfer 38 (2011) 298–303

4. Conclusion

An inverse algorithm based on the conjugate gradient method andthe discrepancy principle was successfully applied to estimate theunknown time-dependent thermal contact resistance for the tribo-system consisting of a semi-infinite foundation and a plane-parallel

strip sliding over its surface, while knowing the temperature historyat some measurement locations. Subsequently, the temperaturedistributions in the system can be calculated. Numerical resultsconfirm that the method proposed herein can accurately estimate thetime-dependent thermal contact resistance and temperature distri-butions for the problem even involving the inevitable measurementerrors.

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