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Materials Letters 62 (20

Measurement of thermal diffusivity of solids using infrared thermography

J.M. Laskar, S. Bagavathiappan, M. Sardar, T. Jayakumar, John Philip ⁎, Baldev Raj

Metallurgy and Materials Group, Indira Gandhi Centre for Atomic Research, Kalpakkam 603102, T.N., India

Received 5 December 2007; accepted 15 January 2008Available online 26 January 2008

Abstract

We report measurement of thermal diffusivity of solid samples by using a continuous heat source and infrared thermal imaging. In thistechnique, a continuous heat source is used for heating the front surface of solid specimen and a thermal camera for detecting the time dependenttemperature variations at the rear surface. The advantage of this technique is that it does not require an expensive thermal camera with highacquisition rate or transient heat sources like laser or flash lamp. The time dependent heat equation is solved analytically for the givenexperimental boundary conditions. The incorporation of heat loss correction in the solution of heat equation provides the values of thermaldiffusivity for aluminum, copper and brass, in good agreement with the literature values.© 2008 Elsevier B.V. All rights reserved.

Keywords: Thermal diffusivity; Infrared thermography; Thermal properties

1. Introduction

Understanding of thermal properties of materials is importantfor many practical applications [1–4]. Thermal diffusivity is animportant physical property that influences thermoelectric andheat transfer efficiencies, carrier properties in semiconductors,aging etc. Different experimental techniques such as photo-thermal [5], laser interferometer [6], laser flash [7] etc. have beendeveloped for thermal property measurements. Thermal diffu-sivity is extracted by correlating the change in temperature withthat obtained by solving the differential heat equation [8]. Veryrecently, the thermal properties of lotus-type, quasi-isotropicporous metals and porous metal spheres have been studied bynumerical and experimentally [9,10]. Quite recently, it has beendemonstrated that the Fourier transform infrared (FTIR) spectro-scopy can be used as an effective tool for characterization ofradiative thermal properties of thin polymer films for applicationwithin an intermediate temperature range [11].

In this work, we present a thermographic technique using acontinuous heat source for heating the front surface of solidspecimen and a thermal camera for detecting the time dependenttemperature variations at the rear surface. The advantage of this

⁎ Corresponding author. Fax: +91 44 27480356.E-mail address: philip@igcar.gov.in (J. Philip).

0167-577X/$ - see front matter © 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.matlet.2008.01.045

technique is that it does not require an expensive thermalcamera with high acquisition rate or transient heat sources likelaser or flash lamp. The thermal diffusivity has been estimatedprecisely by analyzing the time evolution of the rear surfacetemperature and incorporating heat loss corrections in thesolution of the heat equation.

2. Theory

For a sample of rectangular cross section having length l, theproblem of heat diffusion without considering any heat losswith constant heat flux in the x=0 plane can be presented by thefollowing equations

1aATAt

¼ A2TAx2

for0bxb1 ð1Þ

Q: ¼ �kA

ATAx

þ cqAATAt

dx when x ¼ 0; tN0; ð2Þ

kAAT

Ax¼ cqA

AT

Atdx when x ¼ l; tN0; ð3Þ

T ¼ T0 for 0bxb1; t ¼ 0; ð4Þwhere Q ˙ is the rate of absorption of heat energy on the samplesurface at x=0, k is the coefficient of thermal conductivity, c is

Fig. 1. The schematic of the experimental set up for measuring thermal diffusivity of solid samples.

2741J.M. Laskar et al. / Materials Letters 62 (2008) 2740–2742

the specific heat, A is the area of the rectangular cross section,ρ is the density of the material, α=k /ρc is the thermal dif-fusivity. After separating the variables, integrating and makinguse of the boundary conditions [Eqs. (2–4)], we finally arrive atthe solution

T x; tð Þ ¼:QkA

ffiffiffiffiffiffiffipat

perf

x

2ffiffiffiffiat

p� �

ð5Þ

where T(x,t) is the rise in temperature above the ambient at adistance x and time t. If we consider the heat loss from thesample surface during heating, the solution of the heat equation[Eq. (5)] at x= l can be modified in the following form

T l; tð Þ ¼:QkA

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip a� l2hð Þt

perf

l

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia� l2hð Þtp

!ð6Þ

where h is the heat loss coefficient and l2h is the heat losscorrection to thermal diffusivity. Since the thermal conductivityis not usually known and the rate at which heat energy isabsorbed is difficult to measure, it is necessary to scale thetemperature rise. The experimental temperature rise curve isscaled by dividing the measured temperature values by atemperature at a chosen time (te). Similarly, the theoretical

Fig. 2. The time dependent temperature rise curve for aluminum. The unfilledcircles show the experimental data. The solid line represents the best fit byconsidering the heat loss and the dotted line the fit without considering heat loss.

temperature rise is also scaled. The scaled theoretical tempera-ture rise equation takes the following form

T l; tð ÞT l; teð Þ ¼

1ffiffiffiffite

perf l

2ffiffiffiffiffiffiffiffiffiffiffia�l2hð Þ

pte

� �2664

3775 ffiffi

tp

erfl

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia� l2hð Þp

t

!

ð7Þ

The diffusivity value is obtained from the best fit on the timedependent temperature data.

3. Experimental

Fig. 1 shows the schematic of the experimental set up formeasuring thermal diffusivity of solid samples. A hot air gunwith controllable heat output is used as a heating source, whichis operated at 450 °C for 6 min for each sample. A lead slit ofwidth 5 mm and height 35 mm is used as a slit for the heat flux.The slit is placed 5 mm away from the gun for getting good heatoutput, without melting of the slit. The distance between theheat source and the sample which is 0.5 cm, heating rate and theduration of heating are kept same throughout the measurements.The specimens with dimensions of 150 mm×40 mm×8 mm

Fig. 3. The time dependent temperature rise curve for copper. The unfilledcircles show the experimental data. The solid line represents the best fit byconsidering the heat loss and the dotted line the fit without considering heat loss.

Table 1The thermal diffusivity values obtained from the experiment in aluminum,copper and brass and the literature values [12,13]

Sample Thermal diffusivity,α(measured)×10−4 m2/s

Thermal diffusivity,α(literature)×10−4 m2/s

Aluminum 0.9990 0.9786Copper 1.230 1.1623Brass 0.3820 0.3752

2742 J.M. Laskar et al. / Materials Letters 62 (2008) 2740–2742

(L×B×H) were prepared from copper, aluminum and stainlesssteel and polished with 400 grit emery paper for uniform surfacefinish. Thermal images are captured using an Agema Thermo-vision −550 system, capable of measuring object temperaturefrom 253 K to 1473 K, with a measurement accuracy of ±2 K.This is a compact light weight system with a built-in 20° lensand focal plane array based detector having spectral range of3.6–5 µm with a thermal sensitivity less than 0.1 K at 300 K.The camera is placed at a distance of 0.55 m from the sample forbest spatial and thermal resolution. The thermal images areanalysed using Irwin software. Proper care has been taken toavoid sudden air drifts and temperature changes. Thetemperature is measured at 300 K with a relative humidity ofabout 60%. Since only the relative differences in thetemperatures are mapped, emissivity is not a parameter ofmajor concern.

4. Results and discussion

To measure the thermal diffusivity of a material using the technique,as proposed in the theory, it is necessary to calibrate the heat losscoefficient (h) first. For this purpose, an aluminum sample is usedwhose thermal diffusivity value (αAl=0.9786×10

−4 m2/s) is known.The best fit on scaled temperature rise curve [Eq. (7)] gives value ofheat loss coefficient (h=28.78×10−4 s−1). This calibrated value of h isused in the scaled theoretical Eq. (7) to find out the value of the thermaldiffusivity of other materials. The use of same value of h for theanalysis of experimental data of all the samples is justified as thesurface area in contact with the surrounding and all other experimentalconditions such as heating rate, duration of heating, nature ofsurrounding medium (air) are the same.

Figs. 2 and 3 show the temperature rise curve on the scaledexperimental data for Al and Cu samples respectively. The solid lineshows the best fit obtained on the experimental data by incorporating theheat loss coefficient(h=28.78×10−4 s−1). The dotted line shows the curvefit without considering any heat loss (h=0). The three thermographs atdifferent time intervals are also shown in the inset of Figs. 2 and 3. Similarresults are obtained for brass sample. It is clear from the Figs. 2 and 3 thatthe dotted curve (without considering heat loss) shows faster temperaturerise than the experimental one during the initial rise time. However, aftersaturation (above 100 s) the dotted line falls below the experimental curve.

The reason for the slow rise, in the case of the experimental curve, isattributed to the heat loss from the samples to the surroundings during thecourse of the experiment. The loss of the heat is mostly by convection asthe faces of the sample are in contact with air. In addition to this, the samplealso losses some heat by conduction as it is placed on an insulatingplatform. So, the rise of the scaled experimental curve is slower than thedotted curve that represents the theoretical curve without taking intoaccount of the heat loss.

When the heat loss correction is incorporated in the scaled analyticalsolution of the heat equation [Eq. (7)], the theoretical curve fits quite wellwith the experimental data and the obtained values of diffusivity are ingood agreement with the literature values. The same treatment is alsoapplied to brass to extract its thermal diffusivity value. The thermaldiffusivity values obtained from our experimental data and the literaturevalues are shown in Table 1.

5. Conclusions

In conclusion, we show a new and simple approach formeasuring thermal diffusivity of solids using a continuous heatsource and infrared thermography. We have also proposed aphenomenological approach for incorporating the heat losscorrection to the thermal diffusivity that can provide accuratevalues of thermal diffusivity from the experimental thermal profile.The model is validated for three solid samples of aluminum,copper and brass.

Acknowledgements

We thankDr. P.R.VasudevaRao for support and encouragements.

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