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High temperature resistivity determination of high-melting point metals and alloys
Vacuum 85 (2010) 498e501
Contents lists available at ScienceDirect
Vacuum journal homepage: www.elsevier.com/locate/vacuum
High temperature resistivity determination of high-melting point metals and alloys kos, Stanis1aw Ha1as*, Maciej Czarnacki Tomasz Pien Institute of Physics, Maria-Curie-Skłodowska University, pl. M. Curie -Skłodowskiej 1, 20-031 Lublin, Poland
a b s t r a c t Keywords: Resistivity Metals Alloys W ReW ReMo
A method of determination of temperature-dependent electrical resistivity for metals and alloys of highmelting points at high temperatures is presented. It is based on the computer simulation of wire heating by direct current and basic electrical measurements. The authors present new results for 25Re75W and 47.5Re52.5Mo alloys along with the results of test measurements for tungsten. Electrical resistivity for tungsten obtained with presented method is in good agreement with well known data for the metal. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Resistivity of metals and their alloys is an important parameter which determines their usability in different applications. Temperature dependence of resistivity of pure metals is well known and available from numerous publications. Such information is also accessible for a handful of alloys. Companies specializing in alloys production are able to provide alloys with different content of ingredients, although many of their physical properties are not available. Therefore such investigations are highly desirable. There are many methods of resistivity determination of metals. At room temperature bridge or four-point measurements are used. Usually, the investigated metal sample is placed in vacuum and heated with furnace or electric current. In both cases precise measurement of the metal temperature is necessary. The temperature is often determined with a pyrometer, which may be the main source of errors. Moreover, these methods require relatively long wire samples to eliminate cold end effects. We describe a method of obtaining the temperature dependence of metal resistivity based on simple electrical measurements. The method does not require measurement of sample temperature, temperature is determined from the electrical measurements instead. Obtained experimental data are analysed with a computer program which simulates the experiment and provides the temperature distribution along an electrically heated filament. The program takes temperature dependence of resistivity as a variable with two free parameters and compares experimental volteampere results with calculated ones. The agreement between
* Corresponding author. E-mail address: [email protected] (S. Ha1as). 0042-207X/$ e see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.vacuum.2010.01.020
experimental and calculated values is attained for particular values of the free parameters.
2. Experimental In our experiment, we use a piece of short and straight filament made of wire or ribbon being the sample of metal/alloy under consideration. The filament is connected to thick nickel end-pieces by spot-welding. In addition, two PtePtRh thermocouples are spotwelded at the sample ends, to provide the temperatures of ends (T1, T2). The sample is heated by direct current I. The Pt wires of the thermocouples are simultaneously used for the voltage measurement between the filament ends. The sample and the thermocouples were mounted on electrical feed-through and placed in a high vacuum chamber being evacuated by an ion pump. A schematic diagram of electric connections is shown in Fig. 1. We performed measurements for tungsten, ReMo and ReW wires. The wires were 29 mm, 37 mm, 26 mm length and 0.125 mm, 0.127 mm, 0.125 mm thick, respectively. The experiment provided temperatures T1, T2 and voltage drop U for different heating currents I.
3. Simulation The problem of temperature distribution along electrically heated filament has been often analysed since the beginning of the 20th century and related to investigations of thermoemission [1e5]. The most common method is based on a numerical solution of a differential equation determined by physical phenomena which occur during filament annealing. Various approximations were applied to overcome limitations in solving numerically the
kos et al. / Vacuum 85 (2010) 498e501 T. Pien
499
The main source of inhomogeneous temperature distribution along the filament is heat conductivity towards end-pieces. The filament ends are cooler than its central area. Heat loss or gain in the n-th element due to conduction towards two neighbouring elements, n 1 and n þ 1, per time unit is
T Tn Tn Tnþ1 ln ; Pncond: ¼ S ln n1 l l
where ln is thermal conductivity coefficient and it is correlated with resistivity by the WiedemanneFranz law
Fig. 1. Schematic diagram of the electric circuit of the apparatus.
equation. Nowadays, the equation can be solved numerically in its original form, taking into account all phenomena. The electrically heated filament is treated in the simulation as a set of N short pieces of wire (elements), which have constant temperature and other parameters at a particular instant. Each element of the filament is heated by the electric current and, simultaneously, it is cooled by thermal radiation and undergoes heat exchange to its neighbours. At the border of two elements with different temperatures the Thomson effect appears. The temperature at the border between the filament ends and nickel end-pieces is also influenced by the Peltier effect. The temperature increment DT of n-th element during the time interval Dt depends on power Pn generated or dissipated in the element as follows
DTn ¼
Pn Dt ; mc
(7)
(1)
where m is mass of the element, c is the heat capacity of the filament material, Pn is the sum of partial powers related to all physical phenomena occuring in the filament. Taking into account current heating, radiation, thermal conduction and thermoelectric Thomson and Peltier effects, we obtain the following expression for the total power
ln ¼
Lz Tn
rn
;
(8)
where Lz is Lorenz number. Empirical Lz values for many metals are given in Ref. [6]. Temperature gradients along the filament and different metal junctions are sources of thermoelectric effects which additionally modify the temperature distribution and cause its asymmetry. The Thomson effect in respect to temperature gradients generates or dissipates power
PnTh: ¼ sT I
Tn Tn1 l; l
(9)
where sT is Thomson coefficient of the heated metal. The second inherent thermoelectric phenomenon, the Peltier effect, dissipates or generates power proportional to the electrical current I at a junction of two different metals
PnPe: ¼ PI;
(10)
where ro is resistivity at room temperature, a and b are empirical constants. The power dissipated due to thermal radiation in the n-th element is calculated according to the StefaneBoltzmann formula:
were P is the Peltier coefficient characteristic for these two metals. Hence PnPe: in equation (10) is different from zero only for n ¼ 1 and n ¼ N where different metals adjoin each other. Measurements are also perturbed by another thermoelectric effect: the Seebeck effect, which influences voltage determination between the filament ends. This is because the filament ends are welded to nickel holders and these junctions always have different temperatures indicated by thermocouples, so an additional voltage is generated. It is usually a very small effect, but it may be measured during a short pause in current supply to the filament. The simulation procedure allows one to calculate recurrently temperatures of all the elements according to equation (1) and, simultaneously, to update the temperature-dependent parameters. The calculations are performed until a steady temperature distribution is obtained. The observables which are required for the simulation are heating current I, the end-temperatures, T1 and T2, size and geometry of the filament and material properties: c(T), r (T), 31000, Lz, s(T), P(T), SAB(T). The simulation also provides time evolution of the temperature distribution. From the temperature distribution in the steady-state conditions the voltage on the filament may by calculated with the formula:
Pnrad: ¼ 3n s Tn4 Ta4 D;
VS ¼
Pn ¼ Pnel: Pnrad: Pncond: PnTh: PnPe: :
(2)
Power generated due to flow of current I along the n-th element is
l Pnel: ¼ I 2 rn ; S
(3)
where l is the element length, S is the element cross-section and rn its resistivity. In our simulation we assumed the following empirical formula for resistivity as a function of temperature:
rn ðTn Þ ¼ r0 1 þ aðTn 300 KÞ þ bðTn 300 KÞ2 ;
(4)
(5)
where Ta is ambient temperature, s is the StefaneBoltzmann constant and D is radiation area. 3n is the total emissivity of the surface of the filament material. This coefficient is temperature dependent and, for high-melting metals, we assumed the following formula
3n ¼ 31000 Tn =1000K;
(6)
where 31000 is an empirical constant which equals 0.12 for tungsten. The same value was assumed for ReMo and ReW alloys.
N X n¼1
Vn ¼
N X n¼1
I rn
4l
pd2
:
(11)
The simulation is executed with the values of heating currents used in the experiment. The calculated voltage drops VS are compared with experimental ones VE and the root mean square error (RMSE) is calculated
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 u M uX ViS ViE RMSE ¼ t ; M i¼1
(12)
500
kos et al. / Vacuum 85 (2010) 498e501 T. Pien
where M is a number of points in the volteampere characteristics. Multiple steps of simulation performed with different values of a and b provide a map of RMSE. The smallest value of RMSE identifies the best values of the determined factors a and b. The calculation of the error map is not a fast method of minimalization of the error function, but it shows the shape of RMSE function and avoids the error of being incidentally trapped in a local minimum. The error map may be additionally zoomed by further calculations to provide more precise results. 4. Influence of thermoelectric effects on simulation results In the experiment, temperatures of filament ends were different even for small currents. The sources of this asymmetry were the Peltier and Thomson effects. Usually it is difficult to gain precise thermoelectric coefficients for investigated metals or alloys over a wide range of temperature. Therefore the simulation is limited due to these thermoelectric effects. To avoid this problem we determined the influence of Peltier and Thomson effects on the simulation results. Temperature distributions for a ReW filament simulated with one of the thermoelectric effects are shown in Fig. 2. The thermoelectric coefficients, used in the simulations, were obtained after matching simulated and experimental temperatures of the filament ends. The temperature difference of the ends may be achieved by simulating either the Peltier or Thomson effect. However, the calculated temperature distributions in the two cases will be somewhat different. We noticed that the influence of the Peltier effect may be totally eliminated by assuming boundary conditions for the temperature in the simulation. By using the experimental temperatures of the sample ends, T1 and T2, as simulation parameters, the result is equivalent to simulating the Peltier effect. Unfortunately the Thomson effect is not directly correlated with any measured quantity, therefore it is difficult to separate it from the Peltier effect. However, the Thomson coefficients for investigated metals are small [7,8] and the effect is almost compensated
a
Fig. 3. Currentevoltage characteristics of investigated tungsten, ReMo and ReW wires.
along the whole filament, we have thus omitted, the Thomson effect in our simulation.
5. Results The obtained currentevoltage characteristics for tungsten, ReMo and ReW are plotted in Fig. 3. For three metals a non-linear dependence U(I) is visible, which corresponds to the variable resistivity of the investigated wires with temperature. The dependence does not correspond directly to formula (4) through Ohm’s law because of variable temperature along the wire. Therefore, the simulation of wire heating has to be performed. In the calculations we used room temperature values r0 ¼ 2.04 107 Um for ReMo alloy, r0 ¼ 2.85 107 Um for ReW [9] and r0 ¼ 5.65 108 Um for tungsten [10]. The number of wire elements N in the simulation was 100. The results of simulations performed for tungsten wire are plotted in Fig. 4. The error map for tungsten, expressed as log (RMSE), has a stretched minimum, which limits the accuracy of determination of a and b parameters. Large change of a and b may lead to small error increment. Two pairs of a and b are marked on the error map. The first one (the solid line cross-hair) corresponds to the best fit obtained with the simulation (a ¼ 4.427 103 K1, b ¼ 4.5 107 K2) and the second one (the dashed line cross-hair)
b
Fig. 2. (a) Temperature distributions along the filament for two currents simulated with one thermoelectric effect only: Peltier (solid line) or Thomson (doted line). (b) The differences of the distributions.
Fig. 4. The error map expressed as log(RMSE) for tungsten wire. The solid line crosshair points the best fit obtained with the simulation and the dashed line cross-hair points the known parameters for tungsten.
kos et al. / Vacuum 85 (2010) 498e501 T. Pien
501
Fig. 6. Resistivity for tungsten, ReMo and ReW obtained with the simulation (solid line on upper graph). Dashed line represents known values of r for tungsten. The relative error between calculated and known value of tungsten resistivity is plotted on the lower graph.
Fig. 5. The error map expressed as log(RMSE) for ReMo and ReW alloys. The solid line cross-hair indicates the best fit obtained with the simulation.
decreases at higher temperatures. We suppose that the main source of the disagreement is uncertainty of measurement of a wire dimensions. Moreover, the wire may have variable diameter which also generates additional error. However, the obtained result for tungsten is satisfactory, which gives us confidence in the high quality of the method proposed here. 6. Conclusion
corresponds to values a ¼ 4.568 103 K1 and b ¼ 4.1 107 K2, which were found by fitting the experimental data published in CRC Handbook [10]. The RMSE for the best fit is 5.14 105 V and for a and b values from the literature, we obtained RMSE equal to 2.48 104 V. The obtained error maps for the ReMo and ReW wires are shown in Fig. 5. The maps also have not very well localized mimimum. The lowest value of the error for ReMo is 4.82 104 V and it corresponds to a ¼ 1.47 103 K1, b ¼ 6 108 K2. The obtained values for ReW are a ¼ 8.67 104 K1, b ¼ 1.13 107 K2 at the RMSE ¼ 3.83 104 V. The resistivity of the investigated metals calculated using the a and b parameter obtained is presented in Fig. 6. As shown for tungsten, the agreement between known and calculated values is good; the relative difference between them is plotted on the lower graph. The maximum relative error is 1.83% at 950 K and it
The method discussed is based on simple electrical measurements and simple computer simulation. It provides results of acceptable accuracy and it is very easy to perform. It is not limited to use on equation (4), but it also enables application to other functions which may be appropriate for different metals and alloys. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
Bush V, Gould KE. Phys Rev 1927;29:337. Russell DC, Schofield FH. Philos Trans R Soc Lond A 1960;252:463e98. Halas S, Durakiewicz T. Vacuum 1998;49:331. Durakiewicz T, Halas S. J Vac Sci Technol A 1998;16:194. Jain SC, Krishnan KS. Proc Royal Soc London A 1954;222:167e80. Kumar GS, Prasad G, Pohl RO. J Mater Sci 1993;28:4261e72. Worthing AG. Phys Rev 1915;5:445e51. Lander JJ. Phys Rev 1948;74:479. Roberts S, Effgen M. IEEE 2006;25e27:315e6. Weast RC, editor. CRC handbook of chemistry and physics. 61st ed. CRC Press, Inc.; 1980e1981.
Contents lists available at ScienceDirect
Vacuum journal homepage: www.elsevier.com/locate/vacuum
High temperature resistivity determination of high-melting point metals and alloys kos, Stanis1aw Ha1as*, Maciej Czarnacki Tomasz Pien Institute of Physics, Maria-Curie-Skłodowska University, pl. M. Curie -Skłodowskiej 1, 20-031 Lublin, Poland
a b s t r a c t Keywords: Resistivity Metals Alloys W ReW ReMo
A method of determination of temperature-dependent electrical resistivity for metals and alloys of highmelting points at high temperatures is presented. It is based on the computer simulation of wire heating by direct current and basic electrical measurements. The authors present new results for 25Re75W and 47.5Re52.5Mo alloys along with the results of test measurements for tungsten. Electrical resistivity for tungsten obtained with presented method is in good agreement with well known data for the metal. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Resistivity of metals and their alloys is an important parameter which determines their usability in different applications. Temperature dependence of resistivity of pure metals is well known and available from numerous publications. Such information is also accessible for a handful of alloys. Companies specializing in alloys production are able to provide alloys with different content of ingredients, although many of their physical properties are not available. Therefore such investigations are highly desirable. There are many methods of resistivity determination of metals. At room temperature bridge or four-point measurements are used. Usually, the investigated metal sample is placed in vacuum and heated with furnace or electric current. In both cases precise measurement of the metal temperature is necessary. The temperature is often determined with a pyrometer, which may be the main source of errors. Moreover, these methods require relatively long wire samples to eliminate cold end effects. We describe a method of obtaining the temperature dependence of metal resistivity based on simple electrical measurements. The method does not require measurement of sample temperature, temperature is determined from the electrical measurements instead. Obtained experimental data are analysed with a computer program which simulates the experiment and provides the temperature distribution along an electrically heated filament. The program takes temperature dependence of resistivity as a variable with two free parameters and compares experimental volteampere results with calculated ones. The agreement between
* Corresponding author. E-mail address: [email protected] (S. Ha1as). 0042-207X/$ e see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.vacuum.2010.01.020
experimental and calculated values is attained for particular values of the free parameters.
2. Experimental In our experiment, we use a piece of short and straight filament made of wire or ribbon being the sample of metal/alloy under consideration. The filament is connected to thick nickel end-pieces by spot-welding. In addition, two PtePtRh thermocouples are spotwelded at the sample ends, to provide the temperatures of ends (T1, T2). The sample is heated by direct current I. The Pt wires of the thermocouples are simultaneously used for the voltage measurement between the filament ends. The sample and the thermocouples were mounted on electrical feed-through and placed in a high vacuum chamber being evacuated by an ion pump. A schematic diagram of electric connections is shown in Fig. 1. We performed measurements for tungsten, ReMo and ReW wires. The wires were 29 mm, 37 mm, 26 mm length and 0.125 mm, 0.127 mm, 0.125 mm thick, respectively. The experiment provided temperatures T1, T2 and voltage drop U for different heating currents I.
3. Simulation The problem of temperature distribution along electrically heated filament has been often analysed since the beginning of the 20th century and related to investigations of thermoemission [1e5]. The most common method is based on a numerical solution of a differential equation determined by physical phenomena which occur during filament annealing. Various approximations were applied to overcome limitations in solving numerically the
kos et al. / Vacuum 85 (2010) 498e501 T. Pien
499
The main source of inhomogeneous temperature distribution along the filament is heat conductivity towards end-pieces. The filament ends are cooler than its central area. Heat loss or gain in the n-th element due to conduction towards two neighbouring elements, n 1 and n þ 1, per time unit is
T Tn Tn Tnþ1 ln ; Pncond: ¼ S ln n1 l l
where ln is thermal conductivity coefficient and it is correlated with resistivity by the WiedemanneFranz law
Fig. 1. Schematic diagram of the electric circuit of the apparatus.
equation. Nowadays, the equation can be solved numerically in its original form, taking into account all phenomena. The electrically heated filament is treated in the simulation as a set of N short pieces of wire (elements), which have constant temperature and other parameters at a particular instant. Each element of the filament is heated by the electric current and, simultaneously, it is cooled by thermal radiation and undergoes heat exchange to its neighbours. At the border of two elements with different temperatures the Thomson effect appears. The temperature at the border between the filament ends and nickel end-pieces is also influenced by the Peltier effect. The temperature increment DT of n-th element during the time interval Dt depends on power Pn generated or dissipated in the element as follows
DTn ¼
Pn Dt ; mc
(7)
(1)
where m is mass of the element, c is the heat capacity of the filament material, Pn is the sum of partial powers related to all physical phenomena occuring in the filament. Taking into account current heating, radiation, thermal conduction and thermoelectric Thomson and Peltier effects, we obtain the following expression for the total power
ln ¼
Lz Tn
rn
;
(8)
where Lz is Lorenz number. Empirical Lz values for many metals are given in Ref. [6]. Temperature gradients along the filament and different metal junctions are sources of thermoelectric effects which additionally modify the temperature distribution and cause its asymmetry. The Thomson effect in respect to temperature gradients generates or dissipates power
PnTh: ¼ sT I
Tn Tn1 l; l
(9)
where sT is Thomson coefficient of the heated metal. The second inherent thermoelectric phenomenon, the Peltier effect, dissipates or generates power proportional to the electrical current I at a junction of two different metals
PnPe: ¼ PI;
(10)
where ro is resistivity at room temperature, a and b are empirical constants. The power dissipated due to thermal radiation in the n-th element is calculated according to the StefaneBoltzmann formula:
were P is the Peltier coefficient characteristic for these two metals. Hence PnPe: in equation (10) is different from zero only for n ¼ 1 and n ¼ N where different metals adjoin each other. Measurements are also perturbed by another thermoelectric effect: the Seebeck effect, which influences voltage determination between the filament ends. This is because the filament ends are welded to nickel holders and these junctions always have different temperatures indicated by thermocouples, so an additional voltage is generated. It is usually a very small effect, but it may be measured during a short pause in current supply to the filament. The simulation procedure allows one to calculate recurrently temperatures of all the elements according to equation (1) and, simultaneously, to update the temperature-dependent parameters. The calculations are performed until a steady temperature distribution is obtained. The observables which are required for the simulation are heating current I, the end-temperatures, T1 and T2, size and geometry of the filament and material properties: c(T), r (T), 31000, Lz, s(T), P(T), SAB(T). The simulation also provides time evolution of the temperature distribution. From the temperature distribution in the steady-state conditions the voltage on the filament may by calculated with the formula:
Pnrad: ¼ 3n s Tn4 Ta4 D;
VS ¼
Pn ¼ Pnel: Pnrad: Pncond: PnTh: PnPe: :
(2)
Power generated due to flow of current I along the n-th element is
l Pnel: ¼ I 2 rn ; S
(3)
where l is the element length, S is the element cross-section and rn its resistivity. In our simulation we assumed the following empirical formula for resistivity as a function of temperature:
rn ðTn Þ ¼ r0 1 þ aðTn 300 KÞ þ bðTn 300 KÞ2 ;
(4)
(5)
where Ta is ambient temperature, s is the StefaneBoltzmann constant and D is radiation area. 3n is the total emissivity of the surface of the filament material. This coefficient is temperature dependent and, for high-melting metals, we assumed the following formula
3n ¼ 31000 Tn =1000K;
(6)
where 31000 is an empirical constant which equals 0.12 for tungsten. The same value was assumed for ReMo and ReW alloys.
N X n¼1
Vn ¼
N X n¼1
I rn
4l
pd2
:
(11)
The simulation is executed with the values of heating currents used in the experiment. The calculated voltage drops VS are compared with experimental ones VE and the root mean square error (RMSE) is calculated
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 u M uX ViS ViE RMSE ¼ t ; M i¼1
(12)
500
kos et al. / Vacuum 85 (2010) 498e501 T. Pien
where M is a number of points in the volteampere characteristics. Multiple steps of simulation performed with different values of a and b provide a map of RMSE. The smallest value of RMSE identifies the best values of the determined factors a and b. The calculation of the error map is not a fast method of minimalization of the error function, but it shows the shape of RMSE function and avoids the error of being incidentally trapped in a local minimum. The error map may be additionally zoomed by further calculations to provide more precise results. 4. Influence of thermoelectric effects on simulation results In the experiment, temperatures of filament ends were different even for small currents. The sources of this asymmetry were the Peltier and Thomson effects. Usually it is difficult to gain precise thermoelectric coefficients for investigated metals or alloys over a wide range of temperature. Therefore the simulation is limited due to these thermoelectric effects. To avoid this problem we determined the influence of Peltier and Thomson effects on the simulation results. Temperature distributions for a ReW filament simulated with one of the thermoelectric effects are shown in Fig. 2. The thermoelectric coefficients, used in the simulations, were obtained after matching simulated and experimental temperatures of the filament ends. The temperature difference of the ends may be achieved by simulating either the Peltier or Thomson effect. However, the calculated temperature distributions in the two cases will be somewhat different. We noticed that the influence of the Peltier effect may be totally eliminated by assuming boundary conditions for the temperature in the simulation. By using the experimental temperatures of the sample ends, T1 and T2, as simulation parameters, the result is equivalent to simulating the Peltier effect. Unfortunately the Thomson effect is not directly correlated with any measured quantity, therefore it is difficult to separate it from the Peltier effect. However, the Thomson coefficients for investigated metals are small [7,8] and the effect is almost compensated
a
Fig. 3. Currentevoltage characteristics of investigated tungsten, ReMo and ReW wires.
along the whole filament, we have thus omitted, the Thomson effect in our simulation.
5. Results The obtained currentevoltage characteristics for tungsten, ReMo and ReW are plotted in Fig. 3. For three metals a non-linear dependence U(I) is visible, which corresponds to the variable resistivity of the investigated wires with temperature. The dependence does not correspond directly to formula (4) through Ohm’s law because of variable temperature along the wire. Therefore, the simulation of wire heating has to be performed. In the calculations we used room temperature values r0 ¼ 2.04 107 Um for ReMo alloy, r0 ¼ 2.85 107 Um for ReW [9] and r0 ¼ 5.65 108 Um for tungsten [10]. The number of wire elements N in the simulation was 100. The results of simulations performed for tungsten wire are plotted in Fig. 4. The error map for tungsten, expressed as log (RMSE), has a stretched minimum, which limits the accuracy of determination of a and b parameters. Large change of a and b may lead to small error increment. Two pairs of a and b are marked on the error map. The first one (the solid line cross-hair) corresponds to the best fit obtained with the simulation (a ¼ 4.427 103 K1, b ¼ 4.5 107 K2) and the second one (the dashed line cross-hair)
b
Fig. 2. (a) Temperature distributions along the filament for two currents simulated with one thermoelectric effect only: Peltier (solid line) or Thomson (doted line). (b) The differences of the distributions.
Fig. 4. The error map expressed as log(RMSE) for tungsten wire. The solid line crosshair points the best fit obtained with the simulation and the dashed line cross-hair points the known parameters for tungsten.
kos et al. / Vacuum 85 (2010) 498e501 T. Pien
501
Fig. 6. Resistivity for tungsten, ReMo and ReW obtained with the simulation (solid line on upper graph). Dashed line represents known values of r for tungsten. The relative error between calculated and known value of tungsten resistivity is plotted on the lower graph.
Fig. 5. The error map expressed as log(RMSE) for ReMo and ReW alloys. The solid line cross-hair indicates the best fit obtained with the simulation.
decreases at higher temperatures. We suppose that the main source of the disagreement is uncertainty of measurement of a wire dimensions. Moreover, the wire may have variable diameter which also generates additional error. However, the obtained result for tungsten is satisfactory, which gives us confidence in the high quality of the method proposed here. 6. Conclusion
corresponds to values a ¼ 4.568 103 K1 and b ¼ 4.1 107 K2, which were found by fitting the experimental data published in CRC Handbook [10]. The RMSE for the best fit is 5.14 105 V and for a and b values from the literature, we obtained RMSE equal to 2.48 104 V. The obtained error maps for the ReMo and ReW wires are shown in Fig. 5. The maps also have not very well localized mimimum. The lowest value of the error for ReMo is 4.82 104 V and it corresponds to a ¼ 1.47 103 K1, b ¼ 6 108 K2. The obtained values for ReW are a ¼ 8.67 104 K1, b ¼ 1.13 107 K2 at the RMSE ¼ 3.83 104 V. The resistivity of the investigated metals calculated using the a and b parameter obtained is presented in Fig. 6. As shown for tungsten, the agreement between known and calculated values is good; the relative difference between them is plotted on the lower graph. The maximum relative error is 1.83% at 950 K and it
The method discussed is based on simple electrical measurements and simple computer simulation. It provides results of acceptable accuracy and it is very easy to perform. It is not limited to use on equation (4), but it also enables application to other functions which may be appropriate for different metals and alloys. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
Bush V, Gould KE. Phys Rev 1927;29:337. Russell DC, Schofield FH. Philos Trans R Soc Lond A 1960;252:463e98. Halas S, Durakiewicz T. Vacuum 1998;49:331. Durakiewicz T, Halas S. J Vac Sci Technol A 1998;16:194. Jain SC, Krishnan KS. Proc Royal Soc London A 1954;222:167e80. Kumar GS, Prasad G, Pohl RO. J Mater Sci 1993;28:4261e72. Worthing AG. Phys Rev 1915;5:445e51. Lander JJ. Phys Rev 1948;74:479. Roberts S, Effgen M. IEEE 2006;25e27:315e6. Weast RC, editor. CRC handbook of chemistry and physics. 61st ed. CRC Press, Inc.; 1980e1981.