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Fast Identification Method of Total Normal SurfaceAbsorptances Using Inverse TechniquesMohammed Boussaid a , Younès Boulaadjoul a , Mohamed Hedibel a & Tahar Loulou ba Université de Boumerdès , Boumerdès , Algeriab Université de Bretagne Sud, LIMATB , Lorient , FranceAccepted author version posted online: 11 Dec 2013.Published online: 06 Feb 2014.

To cite this article: Mohammed Boussaid , Younès Boulaadjoul , Mohamed Hedibel & Tahar Loulou (2014) Fast IdentificationMethod of Total Normal Surface Absorptances Using Inverse Techniques, Heat Transfer Engineering, 35:13, 1201-1208, DOI:10.1080/01457632.2013.870372

To link to this article: http://dx.doi.org/10.1080/01457632.2013.870372

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Heat Transfer Engineering, 35(13):1201–1208, 2014Copyright C©© Taylor and Francis Group, LLCISSN: 0145-7632 print / 1521-0537 onlineDOI: 10.1080/01457632.2013.870372

Fast Identification Method of TotalNormal Surface Absorptances UsingInverse Techniques

MOHAMMED BOUSSAID,1 YOUNES BOULAADJOUL,1

MOHAMED HEDIBEL,1 and TAHAR LOULOU2

1Universite de Boumerdes, Boumerdes, Algeria2Universite de Bretagne Sud, LIMATB, Lorient, France

This paper proposes a simple and fast method to identify the normal absorptance of various surfaces submitted to a radiationsource, using inverse techniques. The method consists of imposing during a lap of a few seconds a radiative flux on thefront face of a sample whose absorptance is to be identified. The time-dependent temperature on the rear face is measured,and the procedure of inversion is implemented to give a time function of absorbed flux. Only one time–temperature functionis measured using a current type K thermocouple. The normal absorptance of the front face is obtained by comparingthe time heat flux function of the source and the identified absorbed heat flux function. This method can be quickly andefficiently adopted for many practical applications without the need to use optical devices, which give accurate measurementbut at substantial cost. The inverse technique using a conjugate gradient method of minimization with adjoint problem isimplemented to estimate the absorbed heat flux. In order to achieve good values of radiative absorptances, reliable knowledgeof thermal diffusivities and adequately manufactured samples are required.

INTRODUCTION

For a good number of surfaces there is no need to accuratelymeasure their radiative properties, like building material sur-faces, packing products surfaces, and other paintings. In someindustrial processes, the color of the surfaces to be heated ordried up can change depending on the product. It becomes nec-essary to quickly identify the absorptances of those surfacesin relation to the radiant source in order to carry out relevantadjustments in exposure times and source intensity in particu-lar. Accurate measurement of thermal radiative properties likeabsorptances and emittances requires the use of expensive andcritical measuring devices. On the other hand, surface absorp-tances often change appreciably with time under the effect ofmoisture, aerosols, or oxidation, and by some manufacturingprocesses like cooking, heating, sterilization, and drying. Theradiant sources are fixed in space and in intensity; it is thereforenot necessary to determine the radiative properties in general

Address correspondence to Dr. Mohammed Boussaid, Faculte des Sciences,Universite de Boumerdes, 35000, Boumerdes, Algeria. E-mail: b asma89@yahoo.com

but rather from a practical point of view in relation to the partic-ular applications. Consequently, the measurement of radiativeabsorptance of surfaces with respect to radiant source does notrequire measuring reflectivities or emittances.

Various measuring techniques of thermal radiative absorp-tances of surfaces exist without using reflectances or emittances.An increasing number of authors use inverse techniques to pre-dict values of absorptances for samples under different radiativesources. Absorptance estimations are often obtained by imple-menting an inverse heat conduction problem (IHCP). Sometimesα is deduced as a parameter using the Levenberg–Marquardtmethod. Here, we identify first the absorbed and incident fluxfunctions using the conjugate gradient method. The absorbedand incident energies are then deduced. Absorptance is com-puted as the ratio of these two quantities in accordance withits definition. Chirov [1] uses an inverse technique to iden-tify the emittance and absorptance of different paints spreadin thin layers over a thin metallic plate that is assumed to beisothermic. An energy balance taking into account the radia-tive energies emitted and absorbed by the exposed surface andthe energy lost by the rear surface makes it possible to de-scribe the internal energy variation of the sample when main-tained in a vacuum. Measurement of the temperature variation

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and the emitted radiative energy permits identification ofthe absorptance and the emittance parameters using inversemethods.

Chen and Wu [2] have identified by inverse methods ra-diative absorptances of various metallic surfaces submitted tolaser sources during manufacturing processes. A numerical hy-brid finite difference–Laplace transform is used to obtain goodestimations of absorptances compared to other published exper-imental results.

de Monte et al. [3] have proposed a measurement methodof absorptance of metallic surfaces submitted to a short timeof concentrated solar heating. The goal is to improve the sur-faces’ hardness. Temperatures, measured by thermocouples, ofa stainless-steel sample are used to identify the absorbed fluxfunction using IHCP. The mathematical model yields a two-dimensions heat equation (r, z) with variable thermal propertiesbecause of the importance of temperature variation due to astrong heating effect caused by concentrated solar radiation.

Stenekes et al. [4] stressed the need for industry to have aquick and cheap means for the estimation of radiative absorp-tances under different laser sources. To achieve this objective,several paintings sensitive to laser radiations are used. The melt-ing points of these paintings in conjunction with an analyticalheat transfer model will cost-effectively and quickly give an es-timation of absorptances of different surfaces submitted to lasersources operating in cutting metal machines. The technique thatis developed in this work allows obtaining practical values ofradiative absorptances quickly and efficiently when accuracy isnot a prime concern.

The conjugate gradient method with adjoint problem hasbeen applied successfully by Mohammadiun et al. [5] in orderto estimate the time-dependent heat flux using the temperaturedistribution at a point. They show that obtained results presenta good accuracy and stability even if the input data includenoises up to 6%. Notice that we have used in our work the sameprocedure to identify the time-dependent flux absorbed in thefront face of the sample by means of a measured time-dependenttemperature in the rear face.

The procedure used here requires a known radiation source,a current measurement temperature device like a thermocoupleor an infrared (IR) detector, and a temperature data acquisitionsystem. The method consists in shedding light during a few sec-onds on the front face of the sample. Thermogram temperatureacquisition from the rear face starts before the heating processbegins. The conjugated gradient method with adjoint problemis used to identify the absorbed time heat flux function. Theabsorbed energy by the surface can then be calculated. Further,another experiment is implemented to identify incident timeflux function and incident energy in case these are not known.The absorptance is then evaluated as the ratio of the absorbedenergy over the incident energy. The absorptance thus obtainedis a total normal absorptance (in a small solid angle) at ambienttemperature with respect to radiant source at a temperature ofabout 3500 K.

DIRECT MODEL

A one-dimensional sample with thickness e in the x directionis considered. A radiant flux qi (t) (W/m2) is applied normally(in small solid angle) during a time τ in the front of the samplewhere total normal absorptance αn (Ti , TS) is to be found. Tem-perature variation with respect to x and t can be obtained bysolving the system of Eqs. (1) to (4). Because of small temper-ature increase (no more than 4◦C) during the brief experiment(no more than 60 s), the convection–radiation coefficient on thefront face is assumed to be small and constant. The rear face isinsulated. Mathematically speaking, the one-dimensional heatconduction problem is given by the following set of equations:

∂T (x, t)

∂t= a

∂T (x, t)

∂x20 < x < e t > 0 (1)

−k∂T (x, t)

∂x=

⎧⎨⎩

qa (t) + h [T (x, t) − Ta] 0 < t < τ

h [T (x, t) − Ta] τ < t < t f

@ x = 0 (2)

∂T (x, t)

∂x= 0 0 < t < t f @ x = e (3)

T (x, t) = T0 0 ≤ x ≤ e @ t = 0 (4)

where qa (t) is the heat flux absorbed by the sample, τ is theheating period, and the duration of the experience is greaterthan τ, named t f . Ta represents the ambient temperature.

The direct problem as represented by the system of equations(1) to (4) is solved numerically by a finite-difference methodusing a pure implicit scheme. The thermal diffusivity of thesample is assumed to be precisely known.

Figures 1a and 1b represent measured and calculated temper-atures by the direct problem when the estimated absorbed fluxfunction corresponds to the flux function actually applied on thefront face. The figures show good agreement between exper-imental and theoretical curves. Temperature residuals, that is,the difference between calculated and measured temperatures,are practically equal to the measured noise.

INVERSE MODEL

Incident and absorbed fluxes are identified as time functionsby implementing the conjugated gradient method where the gra-dient of the functional to be minimized is computed by solvingthe adjoint problem [5, 6]. It is not necessary to have somea priori information on the missing function in order to pro-ceed to its identification. Only needed in this case is reliableadditional information, which is represented by measured tem-peratures obtained on the rear face of the sample, thus greatlysimplifying the experimental setup. The inverse heat conduction

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Figure 1 Measured and calculated Thermograms and their residues for (a)brass sample of thickness 3.4 mm and (b) a stainless steel sample of thickness3.5 mm.

problem, considered here, consists in estimating the unknowntime-dependent function qa (t). This aims to minimize the an-alytical expression (5) under the constraints specified by theequations of the direct problem:

J (qa (t)) =∫ t f

0[T (e, t) − Y (t)]2 dt (5)

For this purpose, one must consider two additional problems:(1) the sensitivity problem, and (2) the adjoint problem. Theappendix at the end of this paper presents in more details theminimization procedure and the derivation of additional prob-lems needed in this solution of this inverse problem. Special-ized literature, like reference [6], presents this method clearlythrough several basic examples.

EXPERIMENTAL PROCEDURE

Samples are irradiated by a known source represented bya commercial halogen xenon lamp with tungsten filament attemperature of 3500 K whose front face surface area is equalto 45.57 mm2. The total power is 400 W at 36 V max. Thebulb glass is of fused silica type with transmittance equal tounity between 0.22 and 3 μm and equal to 0 outside this wave-length range. Flux density of the source varies between 5000to 21,000 W/m2, depending on the applied voltage. Flux fallsonto the sample inside a small solid angle equal to 2.03 × 10−3

srd. This solid angle is calculated using the surface area of thefilament (45.57 mm2) and the distance between the source andthe sample (15 cm). Prior to the experimental measurements,samples are cleaned and roughly polished with emery 1200 pol-ishing paper. A thermogram is measured from the rear face ofthe sample by a K-type thermocouple of 1/10 mm diameter withseparated contacts.

The inversion procedure allows us to identify the absorbedflux function qa(t). In case the incident flux function is notknown, the same procedure as the one of the absorbed flux func-tion can be applied. The sample is then covered by a thin layerof candle soot, which gives it a known absorptance, typically ofthe order of 0.95.

Various metallic samples such as mild cast, stainless steel,brass, and duralumin were tested. With the thermal diffusivitiesof pure copper and Armco being well known, they are takenhere as references. Figure 2 shows the variation of the esti-mated incident fluxes with respect to time for three samples,copper, Armco, and brass. The samples are irradiated for thesame period of time (7 s) in the same conditions. One can no-tice that the discrepancy between the three sets of measurement,for Armco (2 mm thick), copper (3 mm), and brass (3.4 mm),does not exceed 2%. The time step value used in thermogramsmeasurement is important for the inversion procedure. Varioustime steps were tested, in view of assessing their effects on the

Figure 2 Incident flux qi(t) received normally on the sample in W/m2 fromthe lamp.

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Figure 3 Effect of time step choice on identified flux function for (a) Armcosample and (b) copper sample.

identified flux function. Figures 3a and b show that for copperand Armco samples, measurement values taken appear close for0.03-, 0.04-, and 0.05-s time steps. Discrepancies do not exceed3%. For large time steps (0.1 s), discrepancies that appear onthe ascending and descending parts of the estimated functionbecome more important. For nonmetallic samples, time stepsgreater than 0.1 s may be required because of the small valuesof their thermal diffusivities.

For a given time step, the time duration of the irradiationof the sample front face is also chosen. Theoretically, any timeduration is possible, even impulses. Practically, however, a fewseconds are needed in order to obtain a good thermogram al-lowing then good identification of the absorbed flux function;time durations ranging from 7 to 20 s were used in this work.Durations longer than 20 s can weaken the assumptions of aconstant convection coefficient and a negligible emitted flux.

The effect of the space step in the finite-difference schemeon the flux functions is also assessed. Figure 4 shows that this

Figure 4 Grid dependence of the solutions for Armco sample.

effect is very small for a number of nodes ranging from 21 to201. A grid of 81 nodes is used, although for sizes of about 31,results are practically the same.

ERROR ANALYSIS

Overall estimation error is proposed by identifying the mostsignificant error sources. The main source errors come from ther-mal diffusivity, temperature measurement, convection–radiationheat flux lost by the front face, sample thickness, time measure-ment, numerical method used to solve direct problems, andinversion procedure.

The thermal diffusivity has a significant effect on time-dependent flux identification. Any measurement error of thethermal diffusivity will adversely affect flux identification andhence the accuracy of the absorptance. This effect is depicted inFigure 5, where the thermal diffusivity value of a stainless-steelsample is varied between –10% and +10%. For Armco and

Figure 5 Diffusivity variation effect on identified time flux functions.

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copper samples, thermal diffusivities values are known within3% accuracy; the uncertainty on the total normal absorptancesolely due to the effect of thermal diffusivity is of the order of 6%when both absorbed and incident flux functions are identifiedaccording to Eq. (6). For other material samples, the uncertaintyon the evaluation of absorptance varies between 8 and 12% [7].This is the reason why it is important for the samples diffusivityto be accurately measured or carefully chosen.

To evaluate convection coefficient, we use the following clas-sical correlation for horizontal plates: Nu = 0.54Ra0.25

D . Themaximal difference temperature between the front face duringirradiation experiments and ambient temperature is about 4◦C;then the convection coefficient is calculated to be equal h =4.5 W/m2-◦C. Heat convection flux lost by the front face is then18 W/m2. On the other hand, for metallic polished surfaces,emittances are less than 0.1. Heat flux lost by emission is about2 W/m2. Total heat flux lost by the front face is then 20 W/m2.Therefore, a linearized convection–radiation coefficient is about5 W/m2-◦C. Numerical experiments were carried out by varyingthis coefficient from 3 to 7 W/m2-◦C and show that the maxi-mum effect on identified heat flux was equal to 1%. Temperaturemeasurement error is estimated to 0.4◦C and induces 1.5% er-ror on heat flux. All experimented samples are manufacturedwith a tolerance of at least 0.01 mm. Numerical simulationshave shown that the effect on the absorptance is less than 0.5%.The uncertainties on the time measurement, numerical method,and inversion procedure are neglected. By using the addingprinciple of uncertainties, we can estimate the overall error foridentifying total normal absorptance varying between 8 and16%.

RESULTS

Total normal radiative absorptance relatively to the lamp isdefined by Eq. (6) (see [8]):

αn (Ts, Ti ) = Total normal absorbed energy

Total normal incident energy(6)

Both absorbed and incident energies are obtained using numer-ical integration with respect to time.

Figures 6a and 6b show incidents and absorbed fluxes fora Armco sample for two different values of the incident flux.Figures 6 to 10 show results for five samples namely shown inTable 1.

Table 1 Measured surface absorptances

Surface Present work Uncertainties References [7], [9], [10]

Copper 0.34 8.4% 0.35Armco 0.44 8% 0.445Stainless steel 18/8 0.33 16.2% 0.38Brass 0.33 15.8% 0.4Mild cast 0.5 16% 0.55

Figure 6 Armco sample (2 mm) surface exposed to (a) 21,000 W/m2 or (b)7000 W/m2.

Figure 7 Copper sample of thickness 3 mm.

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1206 M. BOUSSAID ET AL.

Figure 8 Stainless-steel sample of thickness 3.5 mm.

Figure 9 Mild cast sample of thickness 5.5 mm.

Figure 10 Brass sample of thickness 3.4 mm.

For two different incident fluxes and time irradiation, ab-sorptance αn (Ts, Ti ) was found to be 0.43 as in Figure 6a(21,000 W/m2) and 0.44 as in Figure 6b (7000 W/m2). These re-sults were obtained with several repeated experiments, as repre-sented in Figure 6a. For the copper sample surface, absorptancewas found to be 0.4.

Table 1 recapitulates results after identification of total nor-mal absorptance of five different surfaces. Information corre-sponding to the values of total normal absorptances relative tothe sun irradiation is given in Table 1; the listed values weretaken from references [7], [9], and [10].

CONCLUSIONS

Our goal was to develop a fast and cheap method for identi-fying total normal absorptances of various surfaces with respectto a given radiative source. An experimental procedure is pro-posed, consisting of measuring only one thermogram on the rearface of the sample in a way similar to the flash method [11]. Afront face is exposed to a radiative flux stemming from a xenonlamp source at a temperature of approximately 3500 K, for atime period of a few seconds. Absorptance is obtained as a ratioof absorbed energy to incident energy received during the sameexposure time. To obtain the absorbed flux, the inverse methodused to identify the flux function with respect to time is im-plemented. Metallic samples were tested with exposed surfacescleaned and roughly polished. However, the method can also beused successfully in estimating various absorptances of surfacescoated with different paintings or thin layers that can be usedwith well-known diffusivities of support material.

The accuracy of the method depends on the accuracy ofthe thermal diffusivity, the care taken in the geometrical real-ization of the sample, the temperature measurement, and theinversion procedure. The error in the absorptance identificationis estimated to be between 8 and 16%. It is possible to reducethis error by using more sensitive temperature detectors suchas semiconductor or infrared ones, more sophisticated tempera-ture data acquisition apparatus, and finally on the analytical levelthe use of a two-dimensional direct model with more realisticboundary conditions.

NOMENCLATURE

a thermal diffusivity, m2/se thickness of the sample, mh convection–radiation coefficient, W/m2-◦CJ functionalk thermal conductivity,W/m-◦Cq heat flux, W/m2

Nu mean Nusselt numberRaD Rayleigh number based on diametert time, s

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M. BOUSSAID ET AL. 1207

T temperature, ◦Cx: abscissa, mY measured temperature, ◦C

Greek Symbols

αn (Ts, Ti ) total normal absorptance of the surface at Ti rela-tively to the source at Ts

τ exposure duration, s

Subscripts

a absorbedD characteristic dimension of the samplei incidentn normals sourcef final0 initial

REFERENCES

[1] Chirov, A. A., A Method to Measure Absorptance andEmittance of Temperature-Controlling Coatings of Space-craft Using Radiation Flux of Variable Intensity, CosmicResearch, vol. 41, no. 6, 2003, pp. 584–592. Translatedfrom Kosmicheskie Issledovaniya, vol. 41, no. 6, 2003, pp.622–663.

[2] Chen, H. T., and Wu, X. Y., Estimation of Surface Absorp-tance in Laser Surface Heating Process With ExperimentalData, Journal of Physics D: Applied Physics, vol. 39, pp.1141–1148, 2006.

[3] de Monte, F., Ferriere, A., and Beck, J. V., Solar Absorp-tance of Metallic Layers Subject to a Short-Flash of Con-centrated Solar Energy. Theoretical-Experimental Calcu-lation, Journal of Physics: Conference Series 135, 012034,6th International Conference on Inverse Problems in En-gineering: Theory and Practice, 2008.

[4] Stenekes, J., Koshy, P., and Elbestawi, M. A., A SimpleMethod for the Estimation of Laser Absorptance UsingHeat-Sensitive Paints, Measurement Science and Technol-ogy, vol. 20, 117002, 2009.

[5] Mohammadiun, M., Esfahani, J. A., Mahian, O., and Mo-hammadiun, H., A Numerical Solution to Modify the In-verse Heat Conduction Problem With Noisy Data Duringthe Heating of a Solid Bar, European Journal of ScientificResearch, vol. 57, no. 3, pp. 391–400, 2011.

[6] Ozisik, M. N., and Orlande, H. R. B., Inverse Heat Transfer,Fundamental and applications, Taylor & Francis, NewYork, NY, 2000.

[7] Touloukian, Y. S., and Dewitt, D. T., Thermophysical Prop-erties of Matter, Thermal Radiative Properties–MetallicElements and Alloys, vol. 7, Plenum, New York, NY, 1970.

[8] Modest, M. F., Radiative Heat Transfer, 2nd ed., AcademicPress, San Diego, 2003.

[9] Gardon, G. D., Measurement of Ratio of Absorptance ofSunlight to Thermal Emittance, Review of Scientific Instru-ments, vol. 31, no. 11, pp. 1204–121., 1960.

[10] Nelson, K. E., and Bevans, J. T., Errors of the CalorimetricMethod of Total Emittance Measurement, Measurement ofThermal Radiation Properties of Solids, NASA SP-31, pp.55–65, 1963.

[11] Parker, W. J., Jenkins, R. J., Butler, C. P., and Abbott, G.L, Flash Method of Determining Thermal Diffusivity, HeatCapacity, and Thermal Conductivity, Journal of AppliedPhysics, vol. 32, pp. 1679–1684, 1961.

APPENDIX

The minimization of the functional J (qa (t))defined by

J (qa (t)) =∫ t f

0[T (e, t) − Y (t)]2 dt (7)

is performed through the iterative procedure of the conjugategradient method as follows:

qk+1a (t) = qk

a (t) + γk Dk (t) (8)

where γk is the search step size and the superscript k is theiteration number. The quantity Dk (t) represents the descentdirection, given by

Dk (t) = J ′ (qa (t))ka − βk Dk−1 (t) (9)

The gradient of the function J ′(qa (t)) is obtained by solving theadjoint problem. The parameter γk designates the conjugationcoefficient. In this work, the Fletcher–Reeves version of thisparameter is used. This procedure is developed in reference [6].In addition to the direct problem, it consists in deriving twoother problems: the direct problem in variations and the adjointproblem. The direct problem is given by

∂T (x, t)

∂t= a

∂T (x, t)

∂x20 < x < e t > 0 (10)

−k∂T (x, t)

∂t=

⎧⎨⎩

qa(t) + h [T (x, t) − Ta] 0 < t < τ

h [T (x, t) − Ta] τ < t < t f

@ x = 0 (11)

∂T (x, t)

∂x= 0 0 < t < t f x = e (12)

T (x, t) = T0 0 ≤ x ≤ e @ t = 0 (13)

Calculus of variations is used to derive the direct problem invariations expressed here as

∂�T (x, t)

∂t= a

∂�T (x, t)

∂x20 < x < e t > 0 (14)

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−k∂�T (x, t)

∂t=

⎧⎨⎩

�qa(t) + h�T (x, t) 0 < t < τ

h�T (x, t) τ < t < t f

@ x = 0 (15)

∂�T (x, t)

∂x= 0 0 < t < t f x = e (16)

�T (x, t) = 0 0 ≤ x ≤ e @ t = 0 (17)

The corresponding variation of the functional J (qa (t))is ex-pressed as

�J (qa (t)) =∫ t f

0[T (e, t) − Y (t)] �T (e, t) dt (18)

From the study of the conditions under which the augmentedLagrangian functional vanishes, one can deduce easily the fol-lowing adjoint problem:

−∂ P (x, t)

∂t= a

∂ P (x, t)

∂x2+ [T (x, t) − Y (t)] δ (x − e)

0 < x < e 0 < t < t f (19)

−k∂ P (x, t)

∂t= h P (x, t) 0 < t < t f x = 0 (20)

∂ P (x, t)

∂x= 0 0 < t < t f x = e (21)

P (x, t) = 0 0 ≤ x ≤ e @ t = t f (22)

Finally, the gradient of the functional J (qa (t))is obtained as

J ′ (qa (t)) = P (0, t) (23)

The different steps of the minimization algorithm are summa-rized as:

1. With a given initial guess of qa (t)solve the direct problemgiven in Eqs. (10)–(13).

2. Compute the functional J (qa (t)), Eq. (7).3. Check the stopping criterion with the discrepancy principle.4. Solve the adjoint problem given in Eqs. (19)–(22).5. Deduce the gradient of the functional J (qa (t))given in Eq.

(23).

6. Compute the temperature variations fields given in Eqs.(14)–(17).

7. Increment the unknown function, as displayed in Eqs. (8)and (9), go to step 1.

As the standard deviation of the measurements is known, thediscrepancy principle can be used as the stopping criterion. Thisrecommendation is given in reference [6].

Mohammed Boussaid is an associate professorof mechanical engineering at the University ofBoumerdes in Algeria. His research interests in-clude computational and experimental fluid dynam-ics, numerical heat and mass transfer, particularlynatural convection, radiative transfer, inverse tech-niques, and heat exchangers. He taught during manyyears heat transfer and applied numerical methods inBoumerdes University.

Younes Boulaadjoul is a master’s degree studentin mechanical engineering at the University ofBoumerdes in Algeria. His main work is in inversetechniques applied to conduction heat transfer.

Mohamed Hedibel is an associate professor ofphysics at the University of Boumerdes in Algeria.He specializes in thermodynamics and transport phe-nomena. His research fields are diffusion betweensolid materials, and nonequilibrium thermodynam-ics.

Tahar Loulou is professor of mechanical engineer-ing at the University of South Brittany in Lorient,France. His research is in the broad field of heat andmass transfer. He has published more than 40 archivalpapers in the field of heat transfer and the solution ofinverse problems in general. He is a member of theFrench Society of Heat Transfer (SFT) and the Amer-ican Society of Mechanical Engineers (ASME).

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