New analysis for determination of hemispherical total emissivity by feedback-controlled pulse-heating technique

Transcription

1 REVIEW OF SCIENTIFIC INSTRUMENTS 76, New analysis for determination of hemispherical total emissivity by feedback-controlled pulse-heating technique Hiromichi Watanabe Thermophysical Properties Section, Metrology Institute of Japan, National Institute of Advanced Industrial Science and Technology (AIST), AIST Tsukuba Central 3, 1-1-1, Umezono, Tsukuba, Ibaraki , Japan Tsuyoshi Matsumoto International Metrology Cooperation Office, Metrology Management Center, National Institute of Advanced Industrial Science and Technology (AIST), AIST Tsukuba Central 3, 1-1-1, Umezono, Tsukuba, Ibaraki , Japan Received 14 December 2004; accepted 24 January 2005; published online 23 March 2005 A new analysis for the determination of hemispherical total emissivity has been developed on electrically conductive materials under a brief steady-state established by a feedback-controlled pulse-heating technique. The new analysis is based on an equation describing a more realistic situation of the thermal balance of the sample during the apparently static period. The dependency of sample temperature on time and space during the apparently static period is necessary in carrying out the new analysis, and is semiempirically estimated using experimental results for the electrical resistivity of the sample as a function of time. The new analysis was applied to the determination of the emissivity of molybdenum at temperatures beyond 1500 K. Contrary to the general expectation, the emissivity derived by the new analysis, which takes conductive heat loss into account, is systematically larger than that derived by the previous analysis, which neglects the heat loss. The deviation between the new and previous analyses is explained in terms of five factors. The dominant factor is associated with the decrease in the average temperature of the sample during the apparent temperature plateau and cannot be neglected even at the beginning of the apparently static period. The emissivity determined on molybdenum by the new analysis is in good agreement with the literature data obtained by two different methods American Institute of Physics. DOI: / I. INTRODUCTION The feedback-controlled pulse-heating technique has been developed to utilize the advantages of both transient and static measurements of thermophysical properties of electrically conductive materials at high temperatures. 1 4 The modified pulse-technique heats a conductive sample up to a desired high temperature beyond 1300 K in less than 1sby the passage of a large current through the sample, and then maintains the sample temperature at the preset value for a brief period typically 500 ms by a fast feedback control of the current. The resulting short exposure to high temperature reduces the damage to the sample and apparatus, and the hemispherical total emissivity can be calculated from a simple energy-balance equation of the steady-state conditions. In addition, averaging the emissivity data collected during the static period can reduce the random experimental error. Because of these advantages, the feedback-controlled pulse-heating technique has become an accepted method for the accurate measurement of hemispherical total emissivity of metallic materials at high temperatures. However, there is room for improvement of the analysis for the emissivity determination on the sample under such a brief steady state. The analysis 1 previously developed for the determination of hemispherical total emissivity of a sample under a brief steady state realized by using the feedback-controlled pulseheating technique is based on the energy balance of the sample during the steady-state period with the neglect of conductive heat loss 0=VI A SB T 4 T e 4, where V is the voltage drop across the length of the effective sample, I is the current passing through the sample, is the hemispherical total emissivity, A is the surface area of the effective sample, SB is the Stefan Boltzmann constant, T is the temperature of the sample, and T e is the ambient temperature 296 K. The effective sample means the space between a pair of the probes used for measuring V. Although the application of Eq. 1 to the emissivity measurement has been well accepted, we have recently noticed two drawbacks in the application. The first drawback is associated with the possible non-uniformity in the temperature within the sample. To begin with, it should be noted that T of Eq. 1 corresponds to not the temperature of a small partial portion but that of the whole zone of the effective sample. In the previous analysis, 1 however, the temperature of a small circular area at the center of the effective sample surface is inserted into Eq. 1 to calculate the emissivity, /2005/76 4 /043904/6/$ , American Institute of Physics

2 H. Watanabe and T. Matsumoto Rev. Sci. Instrum. 76, VI p = A SB T 4 p T 4 e, 2 where p is the hemispherical total emissivity derived by the previous analysis, and T p is the temperature measured on the sample center using a combination of a pyrometer and a high speed ellipsometer. Even if the sample is rapidly heated by a large current pulse, heat transfer by conduction from the center to the ends of the sample establishes a nonuniform temperature distribution within the effective sample. In fact, Righini et al. 5 have observed a change in the form of temperature profile of a pulse-heated sample from flat to convex. In such a case, the average temperature of the effective sample should be substituted for T in Eq. 1. The second but more critical drawback is as follows: the average temperature of the effective sample varies with time even though the temperature of the sample center is kept constant by using feedback control of current. We have recently observed that the electrical resistivity of a metal sample decreases with time during the brief steady-state period realized by the feedback-controlled pulse-heating technique. The decrease in electrical resistivity definitely indicates that the effective sample is in a nonsteady state condition. Thus, the term related to the heat capacity, in other words, the left-hand side of Eq. 1, should not be zero. It can be concluded that the neglect of the heat-capacity term and the substitution of T p for T of Eq. 1 could lead to significant error in the emissivity determined by the previous analysis. Accordingly, an analysis based on more practical thermal conditions has been required to improve the accuracy in the emissivity determination of the sample under such a brief steady-state. The aim of this work is to propose a more detailed analysis for the determination of hemispherical total emissivity with the feedback-controlled pulse-heating technique. Contrary to the previous analysis, the new analysis is based on an equation describing the energy transport under dynamic conditions. The new analysis was applied to the emissivity measurement on molybdenum. The effects of the conductive heat loss on the emissivity determination were investigated by comparing the emissivities derived by the new and previous analyses. The advantages of the new analysis are described in this article. In addition, the emissivity determined on molybdenum using the new analysis was compared with the literature data obtained by two different experimental methods. II. THEORY A. Basic equations of new analysis The basis of the new analysis is the long-thin-rod approximation 6 of the general equation describing the energy transport in a directly heated homogeneous sample under dynamic conditions as follows: c p T t = I2 S 2 SBp T 4 T 4 e + S x x T I T S x, 3 where c p is the specific heat capacity at constant pressure, t is time, is the density, is the electrical resistivity, S is the FIG. 1. a A possible variation of temperature profile along the axial direction of the half-sample with time during the apparent temperature plateau. b A possible curved-profile of temperature of the half-sample and the specific position x av at which the temperature is identical to the average temperature T av t of the effective sample. cross-sectional area, p is the perimeter, x is the position along the longitudinal axis of the sample, is the thermal conductivity, and is the Thomson coefficient. Several theoretical 4,7 and experimental 5 works have indicated that the temperature profile along the axial direction of a pulse-heated sample becomes a convex form having a maximum at the center of the effective sample. An expected variation of the profile of the half sample with time during the apparent temperature plateau is shown in Fig. 1 a. Since Eq. 3 is valid for any position within the effective sample, we focus on a specific position x av at which the temperature is identical to the average temperature T av of the effective sample, as illustrated in Fig. 1 b. In the new analysis, the transformed form of Eq. 3 at x av is employed to calculate the emissivity: av = S SB p T 4 av T 4 e + dt d T 2 x xav avi 2 S 2 c p T I S T x xav, t xav + 2 T x 2 xav where av and av are the hemispherical total emissivity and the electrical resistivity at x av, respectively, and the sub index xav means x av. The reason for choosing x av is because av can be measured as the average electrical resistivity of the effective sample. Equation 4 can be transformed into another form including p and T p : 4 av = T 4 4 p T av p 1+ T 4 av T 4 1 mc p e T + F VI t xav VI 2 T x 2 xav + F d VI dt T 2 L T 5 V x xav, x xav where m and F are the mass and volume of the effective sample, respectively. To simplify this expression, the following identifications are made:

3 Hemispherical total emissivity Rev. Sci. Instrum. 76, C1= T 4 4 p T av T 4 4 av T, C2= mc p 0 T, VI t xav C3= VI F 2 T x 2 xav, C4= F d VI dt T 2 x xav C5= L T, V x xav and C c = 1+C1 1+C2+C3+C4+C5. Comparisons of the five individual correction-factors C1, C2, C3, C4, and C5 and of the combined correction-factors C c s at different temperatures are useful for understanding how the conductive heat loss affects the emissivity determination. In addition, these five factors are dimensionless quantities. B. Model of temperature as functions of time and space The basic equation of the new analysis Eq. 4 or 5 involves the dependency of temperature on time and space, as well as some physical properties. In the new analysis the time and space function of temperature within the effective sample are semiempirically estimated. To begin with, we assume that the possible curved profile of the temperature along the axial direction of the sample during the apparently static period is expressed by a second-order polynomial function: T t,x = c t x 2 + T p, 6 where the origin of x is the center of the effective sample, c t is the coefficient of the second-order term at t, and T p is the constant temperature of the sample center during the static period. To determine c t, we can use the following relation: L/2 T t,x T av t dx, 7 0= 0 where L is the length of effective sample and T av t is the average temperature of the effective sample at t. Combining Eqs. 6 and 7 yields c t =12 T av t T p L 2. 8 Inserting Eq. 8 into Eq. 6 at x av, we can determine x av as a constant position: x av = Tav t T p c t = L/ From Eqs. 6 and 8, we can obtain T x,t if T av t is known. In most experiments with the pulse-heating technique, T av t cannot be directly measured, because of the difficulty in measuring the temperature distribution of the sample. In the new analysis, therefore, T av t is estimated from the simultaneously measured results of the average electrical resistivity of the effective sample as a function of time during the apparently static period. At high temperatures the electrical resistivity of metals would exhibit a linear dependency on temperature unless any phase transition happens. Therefore,, FIG. 2. Experimental results for the average electrical resistivity av for molybdenum as a function of the temperature T p of the sample center. the average electrical resistivity of the effective sample as a function of time can be expressed by a linear function of T av t : av t = at av t + b, 10 where av t is the average electrical resistivity measured at t during an apparently static period and a and b are coefficients of the linear function. Using Eq. 10, then T av t is given by T av t = av t av t 0 a + T av t 0, 11 where t 0 is the starting time of the apparent plateau. Unfortunately, a cannot be directly determined, because T av t is unknown. As an alternative, the average electrical resistivity as a function of T p is experimentally determined. Figure 2 shows that measured values of av on molybdenum exhibit a linear dependency on T p. In the new analysis, the measurable slope of av against T p was substituted for a in Eq. 11. On the other hand, T av t 0 is expressed by T av t 0 = T p + T 0, 12 where T 0 is the difference between T p and T av t at t 0.In the new analysis T 0 is roughly estimated using the measured variation rate of av t during the apparent temperature plateau. Figure 3 shows the results of av t of molybdenum as a function of time during the plateau of 1507 K. Inspection of Fig. 3 indicates that av t linearly decreases with time from the beginning to the end of the apparent plateau. Regardless of the value of T p, such a linear dependency has been observed. Therefore, av t can be expressed by the following linear function of time: av t = e Tp t + f Tp, 13 where e Tp and f Tp are the slope and the intercept of the time axis during the plateau of T p, respectively. Inserting Eqs. 12 and 13 at t 0 into Eq. 11 gives T av t = e Tp a t t 0 + T p + T Equation 14 indicates that T av linearly decreases with time at the rate of e Tp /a during the plateau of T p. In the present work t 0 is defined as the time at which T p starts being stable within 50 mk. Before attaining the static level, T p always

4 H. Watanabe and T. Matsumoto Rev. Sci. Instrum. 76, TABLE I. Data of e Tp, t 0, and t M obtained during the apparent temperature plateaus of 1507 K and 2230 K. T p K e Tp µ cm/ms t 0 ms t M ms FIG. 3. Variations of the average electrical resistivity av and the temperature T p of the sample center of a molybdenum sample during the apparent temperature plateau of 1507 K. overshoots its target and reaches a maximum value at a time t M. In the new analysis we assume that the temperature distribution of the sample is practically uniform at t M and that the variation rate of the average temperature from t M to t 0 is equal to that after t 0, i.e., e Tp /a. On the basis of these assumptions, T 0 is given by T 0 = e Tp a t 0 t M. Combining Eqs. 6, 8, 14, and 15 yields 15 T t,x = 12e Tp t t M al 2 x 2 + T p. 16 From Eq. 16, we can obtain the first and second derivatives of T with respect to x and first derivative of T with respect to t at x av as follows: T/ x xav =2c t x av =4 3eTp t t M al, 2 T/ x 2 xav =2c t =24e Tp t t M al 2, and T/ t xav = e Tp /a the spectral response of the signal in the wavelength range from 350 to 1250 nm and by measuring the output voltage of a black body cavity at the melting point of copper K. Normal spectral emissivity of the sample used for determining the true temperature was measured using a highspeed ellipsometer. The pyrometer signal and voltage drop across the effective sample were recorded every 100 µs. An appropriate voltage was simultaneously computed by a PID feedback control algorithm and transferred to the solid-state switch to maintain the radiance temperature at a constant value. On the other hand, the signal from the ellipsometer was collected every 500 µs. A strip-shaped sample of molybdenum was used in this work. The purity of the sample is mass %. The sample was 80.0 mm long, 5.10 mm wide, and mm thick. An optical flat whose roughness was less than 100 nm was fabricated on the sample surface. The distance L between the pair of voltage probes was 37.4 mm. Prior to the experiments, the sample was preheated close to the melting point to remove contaminations. All experiments were conducted in vacuum at about Pa. From preliminary measurements, a was determined to be cm/k on the same sample using the same pulse-heating system. Measurements of hemispherical total emissivity were conducted while T p was maintained at 1507 K and 2230 K, respectively. In each case, values of e Tp, t 0, and t M were deduced from the simultaneously measured results of T p and av as a function of time. The measured data of e Tp, t 0, and t M are listed in Table I. All emissivities were computed without correction for thermal expansion. C. Estimates of parameters There are three physical properties c p,, and required to bring the new analysis into practice. In the present work on molybdenum, our previously measured data 4 for c p were employed, was determined from the present results of av according to the Wiedemann Franz law, and was calculated from the literature value 8 of the Seebeck coefficient using the Kelvin relations. III. EXPERIMENTS The apparatus used in the present work is almost the same as that used in previous works. 1 4 The current to heat the sample was supplied from a condenser bank and regulated by a solid-state switch that consists of field-effectivetransistors. The feedback control of current, as well as data acquisition, was performed using a personal computer with an A/D analog-to-digital and a D/A digital-to-analog converter. A high-speed pyrometer was used to measure radiance temperatures. The pyrometer was calibrated by measuring IV. RESULTS AND DISCUSSION A. Comparison between emissivities determined by new and previous analyses Figures 4 a and 4 b show the experimental results obtained while T p was stabilized to 1507 K and 2230 K, respectively. The thick and thin solid lines indicate av and p calculated using Eq. 4 and Eq. 2, respectively, from the same experimental data. The thick and thin broken lines represent T av and T p as a function of time. The linear decrease in T av corresponds to that in av. The important findings in this work are as follows: 1 av is systematically larger than p and 2 the discrepancy between av and p is fairly large even at the beginning of the apparently static period. These results are in contrast to the general expectation: the effect of the conductive heat loss on the calorimetric measurement with pulse-heating techniques is negligibly small. To reduce the experimental error due to conductive heat loss, the data used for calculating p is limited to those obtained during the initial period of the brief steady state in the previous analy-

5 Hemispherical total emissivity Rev. Sci. Instrum. 76, FIG. 4. Variations of hemispherical total emissivities av and p derived by the new and previous analyses on a molybdenum sample during the apparent temperature plateau of a 1507 K and b 2230 K. sis. However, the present results indicate that the scheme for error reduction used in the previous analysis is not very effective. The positive deviation of av from p can be explained in detail by comparing the magnitudes of individual correctionfactors C1, C2, C3, C4, and C5 included in Eq. 5. Figures 5 a and 5 b show the variations of five correction-factors with time during the apparent plateau of 1507 K and 2230 K, respectively. It should be firstly noted that C2 exhibits positive values much larger than the others during the initial part of the plateau. It is clear that C2 is the dominant factor for the positive deviation of av from p. In addition, C1 also contributes to the positive deviation, especially during the final part of the plateau. In contrast, C3 linearly decreases with time and has negative values. C4 and C5 are negligibly small during the whole period of the plateau. Figure 6 shows a comparison between the combined correction-factor C c s obtained during the plateau of 1507 K and 2230 K. Inspection of Fig. 6 indicates that C c decreases with increasing temperature. Accordingly, it can be stated that the new analysis is indispensable for the emissivity determination of a sample at lower temperatures T p 2000 K. Among the five correction-factors, C1 and C2 are categorized into those associated with the variation of average temperature of the effective sample with time, and the others FIG. 5. Variations of individual correction-factors C1, C2, C3, C4, and C5 during the apparent temperature plateau of a 1507 K and b 2230 K. into those associated with the temperature gradient within the effective sample. In the previous analysis, the possible variation of T av has received much less attention than the possible temperature gradient. However, the contribution of the first-category factors is much larger than that of the second-category factors, and C2 cannot be neglected even at the beginning of the static period. Therefore, it must be emphasized that accuracy in the parameters involved in C2 T/ t xav and c p is essential for increasing the accuracy in the emissivity determined by the new analysis. An advantage FIG. 6. Comparison between the values of combined correction-factors C c s obtained during the apparent temperature plateau of 1507 K and 2230 K.

6 H. Watanabe and T. Matsumoto Rev. Sci. Instrum. 76, FIG. 7. Comparison with the literature data of hemispherical total emissivity of molybdenum reported since of the new analysis is the important parameter T/ t xav capable of being determined from the experimental data on a and e Tp. In addition, c p is also experimentally determined using the same apparatus used in the emissivity measurement. Other important parameters T av,, and 2 T/ x 2 xav included into C1 and C3 may be less accurate, because their determinations involve some assumptions. However, inspection of Fig. 5 indicates that their effects can be neglected by limiting the data used in the calculation to those obtained during the initial part of the static period. B. Comparison with previously reported data Figure 7 shows a comparison with the literature data of hemispherical total emissivity of molybdenum reported since For technical reasons, the literature data reported over 30 years ago were not included in the comparison. In Fig. 7, the closed circles and triangles show the average av and p of the data collected in the period from t 0 to t ms, respectively. The open triangles 9 and diamonds 10 represent the literature data determined by the direct measurements of the emissivity spectrum over a broad wavelength range. The broken line shows the literature data 11 obtained by using the traditional pulse-heating technique. In the traditional method, the emissivity is determined from the heating and cooling rates of the sample. 7,11 The error bars of the broken line represent the magnitude of uncertainty 5% of the data. It should be noted that p is in poor agreement with the emissivity determined by the traditional pulse technique. The discrepancy may not be explained, even by taking into account the possible difference in the sample surface conditions and experimental errors. On the other hand, it can be stated that av is in relatively good agreement with the traditional pulse-technique data. In conclusion, values of av exist between the literature data measured using the two different methods. ACKNOWLEDGMENT A part of this study was financially supported by the Budget for Nuclear Research of the Ministry of Education, Culture, Sports, Science and Technology, based on screening and counseling by the Atomic Energy Commission. 1 T. Matsumoto and A. Cezairliyan, Int. J. Thermophys. 18, T. Matsumoto, A. Cezairliyan, and D. Basak, Int. J. Thermophys. 20, T. Matsumoto and A. Ono, High Temp. - High Press. 32, T. Matsumoto and A. Ono, Meas. Sci. Technol. 12, F. Righini, G. C. Bussolino, A. Rosso, and R. B. Roberts, Int. J. Thermophys. 11, R. E. Taylor, High Temp. - High Press. 4, A. Cezairliyan, J. Res. Natl. Bur. Stand., Sect. C 75C, N. Cusack and P. Kendall, Proc. Phys. Soc. 72, A. M. Rakov and B. A. Khrustalev, Issledovaniya po Teplomassoobmenu 53, J.-P. Hiernaut, R. Beukers, M. Hoch, T. Matsui, and R. W. Ohse, High Temp. - High Press. 18, A. Cezairliyan, Int. J. Thermophys. 4,