Quick, contactless thermal analysis of blends.

The measurements of thermal conductivity, heat capacity and thermal diffusivity play an important role in the rubber industry, mainly in tire construction, because these values and their changes directly influence the instantaneous value of viscosity, loss factor tg[delta], and because of that, the adhesion of the tire to a road surface.

General theoretical aspects of heat transport in solid materials have been described (ref. 1).

A very effective contactless method for the study of thermal properties of materials is infrared thermography, whose possibilities have been widely analyzed (ref. 2).

The temperature profile of a tire was tested by a scanning thermovision camera (ref. 3).

An interesting flash method was successfully tested for the thermal properties of metal samples. Obtained results have been presented (ref. 4). Nevertheless, this method is not absolutely contactless. From the theoretical point of view, presented theoretical results have been applied.

Contact methods of thermo parameters measurement need in general relatively complicated electronic equipment. One has been described (refs. 5 and 6). The present work uses this apparatus as a reference for determination of thermal conductivity [lambda] (W/m.K) and diffusivity [alpha] ([m.sup.2]/s).

In this work, a fully automatic and fully contactless method of heat capacity, thermal diffusivity and thermal conductivity determination based on one measurement is presented.

The method is proper for measurement of thermal properties of materials with small thermal conductivity, such as rubber blends, where it was applied.

Theory

The theory begins with a Fourier equation for non-stationary heat transport which has a form:

[partial derivative]T/[partial derivative]t = [alpha][[nabla].sup.2]T (1)

where T is the absolute temperature, and [alpha]([m.sup.2][s.sup.-1]) is thermal diffusivity. In the stationary case, the Fourier equation has a form of:

[[nabla].sup.2]T - 0 (2)

For thermal conductivity [lambda] (W/m.K), we can put down the well known equation:

[alpha] = [lambda]/[rho]c (3)

where [rho](kgm-3) is the sample density and c(J [kg.sup.-1][K.sup.-1]) is the specific heat capacity (ref. 2).

On the basis of this fundamental knowledge, the function describing the temperature distribution at an arbitrary time in a thermally insulated solid was introduced (ref. 7). This function has the form of:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

Two relations for determination of thermal diffusivity ct through the sample of thickness L were derived from this equation in the form (ref. 4):

[alpha] = 1.38 [L.sup.2]/[[pi].sup.2][t.sub.1/2] (5)

or

[alpha] = 0.48 [L.sup.2]/[[pi].sup.2][t.sub.x] (6)

where the sense of temperatures determined by both equations 5 and 6 is clear from figure 1. L is the sample thickness. According to equation 5 and corresponding time [t.sub.1/2], it is necessary to assume that in the real experiment we measure the effective value of thermal diffusivity [alpha] and also the effective time [t.sup.ef] corresponding to the maximum temperature. We assume that the reason for this statement is because when the heat pulse travels through the sample, the sample is heated and the amplitude of phonons is rising. Their mean free path is decreasing, which results in diffusivity [alpha] decreasing and finally in the rise of [t.sup.ef]. [t.sub.M], derived in the basic form (ref. 4), did not consider this physical fact. It has been shown that the relation between both temperatures can be found in the form (ref. 4):

[t.sup.ef] = [1.6t.sub.M] (7)

[FIGURE 1 OMITTED]

Finally, we postulate the following basic assumption in the frame of the above cited fundamental works:

* Enough short heat pulse, shorter than the time necessary for heat pulse transport through the sample.

* Enough thin sample allows for solving the heat transport problem as one-dimensional.

Experimental results and discussion

At the beginning of the discussion, it is necessary to underscore the fact that the CTA in its present form was developed for the measurement of the thermal parameters of rubber and other materials with a relatively small value of thermal conductivity (contactless 1). Presented measurements of metals are only supplementary in order to determine the absorbed heat Q.

In the first part of the present work (solution for relatively big samples), we have used a polystyrene calorimeter, where the test sample (Cu, Al or rubber blend) of rectangular shape with dimensions approximately equal to (0.09 x 0.11 x 0.0014)m was placed. The thickness of the rubber blend sample is given later. A schematic of the apparatus is shown in figure 2. The sample was illuminated through a halogen lamp (electrical power 1,500 w) switched on by a computer. We used a Raytek Thermalert MID 02 pyrosensor placed at the rear side near the surface of the measured sample in order to sense the temperature. The whole measuring process was controlled and evaluated by special software which automatically switches on the lamp, measures the time-temperature dependence of pyrosensor re-sponse and determines the temperature difference [DELTA]T from measured data. Every value was measured ten times in order to obtain the repetition ability of the apparatus. Then the full set of values was transferred to Matlab software. After application of the proper regression procedure of measured time-temperature dependence obtained from the pyrosensor response by Matlab, we obtained the following values: [t.sub.M] (according to equations 5 and 7) absorbed heat Q, specific heat capacity c, thermal diffusivity [alpha] and thermal conductivity [lambda].

[FIGURE 2 OMITTED]

We started the experimental analysis of results from the measurement of heat absorbed in the sample. We calculated it from the calorimetric equation Q = mc[DELTA]T for the Cu sample, with the table value of [c.sub.cu] = 383 J/kg.K. The surface of the Cu sample was covered on both sides of the sample by mat black spray. The heat absorbed by the Cu sample was approximately equal to [Q.sub.cu] = (61.81 [+ or -] 0.02) J. We determined [DELTA]T to be the temperature difference of ambient temperature and maximum surface temperature measured by a pyrosensor on the rare sample surface. The validity of such an experimental procedure was tested on the Al sample of the same dimensions, also covered by the same mat black spray. In this case, we calculated the specific heat capacity of the A1 sample according to a calorimetric equation at known absorbed heat [Q.sub.Cu], determined on the basis of a previous experiment. The mean measured value of [c.sub.A1] was (883.67 [+ or -] 0.04) J/kg.K. The table value was 896J/kg.K, corresponding to the difference of both values of [c.sub.A1] approximately equal to 1.4%. Therefore, the described measurement of Q gives the relevant values.

Later, the rubber blend sample of rectangular shape, with a thickness of 0.002 m was measured the same way. The geometry of the experiment and the value of the active surface were the same in all experiments. For the evaluation of [c.sub.rubb], we used the same absorbed heat determined for the Cu sample [Q.sub.Cu]. The surface of the rubber was also covered by black spray on both sides of the sample, as in the previous cases. It is also necessary to use the black spray film on the surface of other, non-black materials. Using the method described above, we obtained the mean value of [C.sub.rubb] = (1,514.80 [+ or -] 30.31) J/kg.K.

Because we didn't have the reference table value for this blend, we analyzed the reference measurement with a Perkin Elmer, type Diamond DSC (differential scanning calorimeter). Mean value at 38[degrees]C was [c.sub.rubb] = 1,678 [+ or -] 0.043) [Jkg.sup.-1][K.sup.-1], corresponding to a 10% deviation of the values obtained by both methods. It is necessary to underscore that the other sample of the same composition has been used for DSC measurement and for CTA. The influence of local sample inhomogeneity is possible, because the mass necessary for DSC measurement is approximately on the level of 10 milligrams.

In the next step, we will judge the results of [alpha] and [lambda] measurements on the rubber blend sample obtained by CTA and an independent contact method used in other work (refs. 5 and 6) (referred to as contact). A brief description of the apparatus used in both works is as follows. Other details may be found in the cited works (refs. 5 and 6).

The configuration of the experiment is shown in figure 3. The nickel disk (Ni disk) serves as the heat source and, at the same time, as a thermometer. Two identical samples in the cylindrical shape cause symmetrical division of the heat flow into aluminum blocks (A1 blocks), which supply the isothermal border conditions of an experiment.

[FIGURE 3 OMITTED]

Now we can compare results obtained by both methods. The mean value of both [alpha] and [lambda] obtained from CTA were [alpha] = (1.81 [+ or -] 0.03). [10.sup.-7] [m.sup.2]s and [lambda] = (0.344 [+ or -] 0.004) W/mK. Sample density was [rho] = (0.99973 [+ or -] 0.00006). [10.sup.3] kg/[m.sup.3]. The rubber thickness was 2.2 [10.sup.-3] m, and other dimensions were the same as for metal samples. The results obtained by the apparatus described in other works (refs. 5 and 6) for both values are [alpha] = (1.85 [+ or -] 0.02). [10.sup.-7][m.sup.2]s and [lambda] = (0.31105 [+ or -] 0.0004) W/mK. Specific thermal heat capacity calculated from these measurements according to equation 3 is equal to c = 1,691.8 [Jkg.sup.-1][K.sup.-1], representing a deviation from the CTA value equal to approximately 12%. The sample specific density calculated from equation 3 equals [rho] = 993.66 [kgm.sup.-3], said to be is in excellent agreement with the value reported above. The difference between diffusivities [alpha] obtained from both experiments is approximately equal to 2%. The difference between both values of thermal conductivity Z is on the level of 10%. From the presented results obtained for c, [lambda] and [alpha], it is clearly seen that the shown method gives reproducible results with repeatability. The obtained results coincide well with other independent methods for measurement of c, [alpha] and [lambda].

In the next part of our study (contactless 2), we have used compact measuring, a fully automatic system presented in figure 4. The tested samples were Cd (as the reference) and a rubber blend, all of cylindrical shape with dimensions [empty set] = 12 mm and thickness approximately 2 mm. The sample was illuminated through a halogen lamp (electrical power 200 w) switched on by a computer. We used the Raytek Thermalert MID 02 pyrosensor placed at the rear side near the surface of the measured sample in order to sense the temperature. The whole measuring process, as in the previous case, was controlled and evaluated by special software which automatically switches the lamp on, measures the time-temperature dependence of the pyrosensor response and determines the temperature difference [DELTA]T from measured data. Every value was measured ten times in order to obtain repeatability of the apparatus.

[FIGURE 4 OMITTED]

The Cd sample was covered on both sides of the sample by mat black spray. The measured value of [c.sub.Cd] was (225.33 [+ or -] 1.6) J/kg.K, and the corresponding table value is equal to 231 J/kg.K. The difference between both values is approximately 2.5%. So we can conclude that the described measurement of Q gives the relevant values.

Tested rubber blends of a diameter of 12 mm (covered by mat black spray on both sides) and the same composition as in the above described experiments offer very close values of measured thermal parameters [alpha] = (1.91 [+ or -] 0.02). [10.sup.-7] [m.sup.2]s and [lambda] = (0.37 [+ or -] 0.006) W/mK. A summary of all the obtained results appears in tables 1 and 2.

Conclusions

Presented CTA (in both versions) is fully contactless, fully automatic and it is proper for the testing of materials with relatively small thermal conductivity. From one measurement it is possible to determine specific heat capacity, thermal conductivity and thermal diffusivity with relatively high precision and very good reproducibility. Absorbed heat Q and density are input parameters.

References

(1.) L. Kubicar, "Rychla metoda merania zakladnych termofyzikalnych parametrov," Veda, Bratislava (1988).

(2.) X. Maldague, Theory and Practice of Infrared Technology for Non-Destructive Testing, Wiley, New York (2001).

(3.) M. Skulec, S. Rosina, Z. Jonsta, I. Kopal and P. Kostial, "Temperature distribution measurement in personal tires," Materials Engineering, 1 (2004) 167, ISSN 1335-0803.

(4.) W.J. Parker, R.J. Jenkins, C.P. Butler and G.L. Abbott, "Flash method of determining thermal diffusivity, heat capacity and thermal conductivity," Journal of Applied Physics, 32 (1961) 1,679.

(5.) E. Karawacki, B.M. Suleiman, I. Ul-Hag and B. Nhi, "An extension to the dynamic plate source technique for measuring thermal conductivity, thermal diffusivity and specific heat of dielectric solids," Rev. Sci. Instrum., 63 (1992) 4,390.

(6.) S. Malinaric and P. Kostial, "Measuring of thermo physical properties of HDPE," Materials' Engineering, 1 (2004) 151, ISSN 1335-0803.

(7.) H.S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids, 2nd Edition, Clarendon Press, Oxford, U.K. (1959).

P. Kostial, J. Hutyra, I. Kopal, M. Mokrysova, Z. Svecova, I. Ruziak and J. Kucerova, University of Alexander Dubcek

Table 1

Material       Cp          Cp          Cp
            [J/kg. K]   [J/kg. K]   [J/kg. K]
            Contact-    Contact-     Contact
             less 1      less 2

Cu             --          --          --
Al           883.67        --          --
Cd             --        225.33        --
Rubber        1,515       1,614       1,692

Material       Cp          Cp
            [J/kg. K]   [J/kg. K]
               DSC        Table
                          value

Cu             --          383
Al             --          896
Cd             --          231
Rubber        1,678        --

Table 2

Material     [lambda]     [lambda]     [lambda]

             [W/m.K]      [W/m.K]      [W/m.K]
             Contact      Contact      Contact
              less 1       less 2

Rubber        0.344         0.37       0.31105

Material     [alpha]      [alpha]      [alpha]

            [m.sup.2s]   [m.sup.2s]   [m.sup.2s]
             Contact-     Contact-     Contact
              less 1       less 2

Rubber       1.81E-7      1.91E-7      1.85E-7