Thermocouples, their characteristic temperatures, and ...

Thermochimica Acta 603 (2015) 218–226

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Thermocouples, their characteristic temperatures, and simple approximation of the emf vs. T V.A. Drebushchak a,b, * a b

Institute of Geology and Mineralogy SB RAS, Pr. Ac. Koptyuga 3, Novosibirsk 630090, Russia Novosibirsk State University, Ul. Pirogova 2, Novosibirsk 630090, Russia

A R T I C L E I N F O

A B S T R A C T

Article history: Received 2 May 2014 Received in revised form 8 July 2014 Accepted 9 July 2014 Available online 11 July 2014

Calibration of the temperature and heat flux sensors for chip calorimetry is more effective if it is based on the optimal approximation of thermocouple's emf against temperature. Contact theory of thermoelectricity provides us with the formula for the approximation, but implies the characteristic temperature of a thermocouple constant QV . This assumption turned out to be incorrect in common practical use, what did makes the main problem for the approximation of the emf. At the same time, the theory explains that the rate of increase for the emf with temperature is not greater than T2. Three-point tests with the reference tables of type E and S thermocouples and many-point test with experimental data did not reveal evident preference of one procedure of approximation over the others. In considering all circumstances of the approximation, the 2nd-order polynomial is concluded to be the best way to approximate the emf of a thermocouple for practical use. Approximation of the characteristic temperature of type E thermocouple with square polynomial 277.01 0.63453T + 0.00044352T2 results in the accuracy of 0.5 K over the temperature range from 0 to 1000 C. ã 2014 Elsevier B.V. All rights reserved.

Keywords: Approximation Calibration Electromotive force Thermocouple Thermoelectricity

1. Introduction Chip calorimetry is the high-tech that combines nanotechnology with microchip in order to measure thermal properties of nanomaterials at a heating/cooling rate that is inaccessible to traditional DSCs [1–5]. In conventional calorimeters, thermal sensors (both for temperature and heat flow) are the thermocouples of conventional size that can be purchased from manufacturers, together with their certificates. Unconventional thermocouples are especially manufactured for chip calorimetry because their size and geometry makes it impossible to adopt the conventional thermocouples. Moreover, even the high-grade miniaturization of the pair of materials used in conventional thermocouples can lead to a significant modification of its voltage–temperature relationship from that approved by the standard surveys [6–9]. Calibration of such thermocouples is the part of proper operation of chip calorimeters. When using reference tables for the calibration of conventional thermocouples, it is suffice to compare the measured values with those listed in the reference table and then calculate the corrections. When calibrating nonstandard thermocouple, it is

* Corresponding author at: Institute of Geology and Mineralogy SB RAS, Pr. Ac. Koptyuga 3, Novosibirsk 630090, Russia. Tel.: +7 383 3332406; fax: +7 383 3332792. E-mail addresses: [email protected], [email protected] (V.A. Drebushchak). http://dx.doi.org/10.1016/j.tca.2014.07.008 0040-6031/ ã 2014 Elsevier B.V. All rights reserved.

necessary to generate a draft of the reference table from scratch. It is not a trivial task because the theory of thermoelectricity does not provide us with such an algorithm. Conventional polynomials of temperature are not suitable because a low-order polynomial is considered insufficiently accurate for a wide temperature range, and a high-order polynomial needs too much experimental points for the coefficients calculation, again without the warranty of accurate approximation [10]. For example, the reference tables of letter-designated thermocouples are divided into two or three temperature ranges fitted to the polynomials of high order (up to 13) with awkward polynomial coefficients (11 digits each). Additional problem is the boundary conditions between the temperature ranges, where the two values and their first and second derivatives with respect to temperature must be continuous. The generation of such a polynomial is very difficult and timeconsuming task. Theory of thermoelectricity in its gradient version is in no way involved in the generation of the reference tables for thermocouples. The whole work is usually done completely on empirical knowledge and the reference tables are improved after the trial and error method. In discussing thermoelectricity, one should separate metals (electronic conductivity) and semiconductors (ionic conductivity). The emf generated at the contact of two semiconductors is to a great extent the result of chemical reaction, solid-state

V.A. Drebushchak / Thermochimica Acta 603 (2015) 218–226

electrochemistry. The obvious difference between thermoelectric and electrochemical materials and devices is in their long-run stability. Freshly prepared metal thermocouple generates unstable signal, which becomes stable after annealing and remains stable for many and many years. Freshly prepared semiconductor thermoelectric material possesses high emf signal, which decreases with time, especially if the material is in use. In paying attention not to misuse thermoelectricity and electrochemistry, the semiconductor thermoelectrics are mentioned first of all because they possess unusually high value of the “Seebeck coefficient”. For example, the upper limit of the Seebeck coefficient in the contact theory is 0.08617 mV K1. This is the asymptotic limit for T ! 1 and will be discussed in Section 2.2. But “a Seebeck coefficient of 1.6 mV C1 was obtained” for K2SO4, twenty times higher than that attainable in true thermoelectricity [12]. In fact, this is the result of an increase in the reaction rate with temperature. And this process is common not only to the semiconductors. Cautionary example of electrochemical thermoelectricity is the pair of metals, copper and zinc. From the very start of thermoelectric investigations, the Cu–Zn pair was investigated together with other pairs of metals. But today we will not find any reference on the copper–zinc thermocouple. These metals react readily with one another producing brass. The reaction can be seen with the naked eye. It was observed by Ernest Kirkendall and did allow him to discover the effect named after him [13]. The emf generated on the contact of the alloys in Cu–Zn system is considered the manifestation of chemical reactions, namely the difference in the Gibbs energy between their reagents and products [14]. It is conventional to use the emf of semiconductors and metals for thermodynamic evaluation without any correction to thermoelectric contribution and the same emf for thermoelectric properties without any correction to the difference in their Gibbs energies. In this work, two mechanisms of the emf generation (thermoelectric and electrochemical) are distinguished, and we discuss only thermoelectricity without reactions. The objection of this report is to clarify which algorithm can improve the generation of the reference table for unconventional thermocouples. The discussion is based on the formula for the thermoelectric voltage derived from the contact theory of thermoelectricity [11]. 2. Theoretical background 2.1. Gradient theory Gradient theory of thermoelectricity is wide spread and is described in almost every book on the theme. It considers the thermoelectric voltage as the result of the difference in the absolute thermoelectric power (TEP), which is particular to each conductor. There are many contributions to the TEP, each with its own temperature relationship. Any attempt to recognize more or less universal function of the thermocouple voltage against temperature is meaningless. 2.2. Contact theory The contact theory of thermoelectricity is based on the Volta effect, when two different metals, say A and B, neutral from the very start, become charged with equal in magnitude but opposite in sign electricity after touching each other. This is because the conduction electrons in different metals possess different Fermi levels. When metals A and B are brought in contact with one another, their conduction electrons move freely to and fro through the contact boundary and tend to equilibrate their energies in A and B. The net increase in the number of electrons in one metal is equal to the net decrease in the other one,

219

generating simultaneously different electrostatic energy in these two metals. The equilibrium is attained when the difference in the Gibbs energy between two electronic subsystems in metals A and B becomes equal to zero:

DG ¼ 0 ¼ DH T DS þ DUDq:

(1)

Here, DH is the difference in the enthalpy of electrons in metal A (HA) and B (HB), similarly, DS is the difference in their entropy (SA and SB), DU is the voltage step between the metals A and B, and Dq is the net electric charge under equilibrium. To be successful, the evaluation of the equilibrium between two metals needs many simplifications. One of them is the low-temperature limit, because this allows us to use linear in temperature effects in electronic properties. First of all, this is valid for the “active” electrons that participate in the thermal agitation. Only electrons with the energy close to the Fermi level are of our interest. Their number is about kBT/eF, where kB is the Boltzmann constant, T is the temperature, and eF is the Fermi energy. According to the model of free electrons, only n¼

NA kB T

(2)

eF

in a mole (NA) of electrons in a metal possess the thermal energy of 3/2kBT each. Thus, the total thermal energy of the whole electronic subsystem is E¼

3kB T NA kB T ; 2 eF

(3)

and its heat capacity is 2

Cv ¼

dE 3NA kB ¼ T: dT eF

(4)

Eq. (4) was derived from the theory of free electrons in a metal. Experimental calorimetric data prove the linear increase of electronic heat capacity with temperature, but the expression for this relationship is arranged in some different way: C e ¼ g T;

(5)

where g is the electronic specific heat coefficient. Eq. (5) will be used in the evaluations of the equilibrium described by Eq. (1). Three types of energy contributions are there in Eq. (1), namely thermal (enthalpy and entropy), statistical (configurational entropy), and electrical (charge and voltage). We will discuss them one by one. Enthalpy of the electronic subsystem in a metal changes with temperature and can be derived from the electronic heat capacity (Ce) of metals: ZT HðTÞ ¼

C e dT:

(6)

0

Using Eq. (5) for the calculation of enthalpies of metals A and B, we have for the difference between the metals A and B in Eq. (1) 1 2

DH ¼ ðg A g B ÞT 2 :

(7)

Thermal entropy can also be derived from Ce: ZT SðTÞ ¼

Ce dT: T

(8)

0

Again, using Eq. (5) we have for Eq. (1) T DStherm ¼ ðg A g B ÞT 2 :

(9)

220


The configurational entropy depends on the number of microstates in the system. It can be calculated after general formula

As the Seebeck coefficient tends to zero for zero temperature, we have

Sconf ¼ kB ln N!;

gB gA

(10)

4ea

where N is the number of electrons. Here, we again simplify the model by accepting N equal for both metals, A and B. In equilibrium, the number of electrons in each metal in contact differs by a small fraction of active electrons (Eq. (2)): Dn n. As n is proportional to temperature, its small fraction is also proportional to temperature:

Dn ¼ aT:

lnðN aTÞ! N lnðN aTÞ:

(12)

For Eq. (1), the difference in the configurational entropy between two metals is N þ aT 2aT T DSconf ¼ kB NT ln kB NT ln 1 þ : (13) N aT N According to Eq. (11), the excess electric charge in equilibrium is +eaT for one metal and eaT for the other one, where e is the elementary charge. The difference in the electric charge between metals A and B is

Dq ¼ 2eaT;

(14)

and the electric contribution to Eq. (1) is

DUDq ¼ 2eaT DU:

(15)

Now we collect all the components of Eq. (1) (Eqs. (7), (9), (13), and (15)) and have 1 2aT ðg A g B ÞT 2 ðg A g B ÞT 2 kB NTln 1 þ þ 2eaT DU 2 N ¼0 (16) From this equation, the voltage is a function of temperature: g gA kB 2aT (17) T N ln 1 þ DU ¼ B N 4ea 2ea For the sake of convenience, the sign of the voltage in this equation is opposite to that in Eq. (16). In experiments, we usually measure the magnitude of the voltage, paying the attention to its sign only in special cases. Moreover, it is accepted in electrical engineering that an electric current flows from “plus” to “minus”, quite opposite to the physical nature of electric current, where negative electrons move from “minus” to “plus”. In the right-hand side of Eq. (17), the last term contains combinations N/2a and 2a/N. The equation becomes simpler when we denote

QV ¼

N 2a

with the dimension of temperature: g gA kB T T QV ln 1 þ DU ¼ B : 4ea e QV

kB e

and finally the voltage at the contact of two metals is k T DUðTÞ ¼ B T QV ln 1 þ : e QV

(21)

(22)

The Seebeck coefficient is

(11)

The number of conduction electrons in each of two metals changes after the equilibration in different ways, increasing in one metal (N + aT) and decreasing in the other one (N aT). Their configurational entropy also changes differently. Usually, the Stirling's formula is used for the calculations of entropy expressed in Eq. (10), but we will use much simplified approximation

¼

sðTÞ ¼

kB T : e QV þ T

(23)

2.3. Universal function of emf vs. T and characteristic temperature QV The emf generated with a thermocouple is the difference between the contact voltages on two junctions kept at two different temperatures, reference T0 and measured T. Reference tables for the letter-designated thermocouples are reported for the case of T0 = 273.15 K (t = 0 C). All functions DU(t) cross point t = 0 C and DU = 0 mV and cannot be fitted to a universal function. Theoretical equation for the emf voltage of letter designated thermocouple can be derived from their reference tables if attributing T0 = 0 K (t = 273.15 C). The only parameter that depends on the particular thermocouple in the expression for the emf (22) is its characteristic temperature (QV ). The rest values are two constants (kB and e) and one variable (T). Physical meaning of QV is somehow similar to the characteristic temperature in the heat capacity models (Einstein, Debye, Tarasov). If plotted against normalized temperature (T=QE or T=QD or T=Q1 ), the heat capacities of various substances (each with its own characteristic temperature) fall at the same line CP(T=Q). The analogy with the emf is incomplete because both axis, X and Y, must be normalized for thermocouples. Eq. (22) can be written as DUðTÞ kB T T ¼ ln 1 þ : (24) e QV QV QV By scaling both the emf and temperature, we obtain the universal plot of DU=QV vs. T=QV for all thermocouples. Both values range in very wide limits, from 3 to 3000 K for temperature and from 0.001 to 86 mV for the emf. Logarithm scales are more suitable for the graphical presentation of such values. The resulting plot is shown in Fig. 1. The algorithm of the scaling was described in [15]. In continuing the analogy with the heat capacity models, we can analyze the general functional relation between the emf and temperature. For very low temperatures, the Einstein model

[(Fig._1)TD$IG]

(18)

(19)

This is the voltage at the contact between metals A and B, and it depends on temperature. Its derivative with respect to temperature is the Seebeck coefficient: s¼

dDU g B g A kB QV ¼ : dT 4ea e QV þ T

(20) Fig. 1. Universal function of emf vs. T for letter-designated thermocouples.


[(Fig._2)TD$IG]

predicted the exponential increase in the heat capacity C V expðQ=TÞ, the Debye model predicted the cubic function C V T 3 , and the Tarasov model predicted linear increase C V T. At very high temperatures, the heat capacity is constant (CV = 3R) for all the models. Exponential function was not found in any solid, and the Einstein model is used today only for high-temperature approximation as a contribution, additional to the Debye model. The Debye model was found to predict exactly the cubic increase in the experimental data for most solids, and function C V ¼ aT 3 þ g T is used for the approximation of low-temperature heat capacity. The Tarasov model was found to be valid for polymers, and their heat capacity is usually fitted to a straight line. Which functional relation for the emf of a thermocouple is predicted by the contact theory? In neglecting the factors (because they do not affect the functional relation), we can rewrite Eq. (24) as y ¼ x lnð1 þ xÞ:

[(Fig._3)TD$IG]

(25)

Fig. 2. The characteristic temperature for type S and type R thermocouples.

For the low-temperature limit, x 1 and we can expand the logarithm 1 lnð1 þ xÞ ¼ x x2 þ x3 . . . 2

(26)

and use it in Eq. (25): 1 y ¼ x x þ x2 x3 þ . . . or y x2 : 2

(27)

Thus, for the extremely low temperatures, the emf is proportional to the square of temperature: DU T2. For the high-temperature limit, the logarithm in the right-hand part of Eq. (25) is negligible as compared with x, and the emf turns out to be proportional to temperature: DU T. Thus, the quadratic polynomial is to be enough to fit the emf against temperature. The characteristic temperature QV is the single parameter, which defines the emf as a function of temperature for every thermocouple. Again, like it was for the theories of heat capacity of solids, the experimental data on the emf do not fit theoretical predictions exactly. There are discrepancies between the reference tables and the emf values generated from the single value of QV for every thermocouple. It is interesting to analyze the variations in QV values with temperature. The value of QV for a thermocouple at any temperature can be calculated according to equation k 1 1 (28) QV ðTÞ ¼ T B e sðTÞ derived from Eq. (23). In routine calculations with tabulated values, it is common to use the difference quotients instead of true derivatives. In our case, the estimate of the characteristic temperature over the temperature interval from T1 to T2 is k T2 T1 1 ; (29) QV ðT Þ ¼ T B e UðT 2 Þ UðT 1 Þ where T is the arithmetic mean between T1 and T2. The values of QV ðTÞ for the letter-designated thermocouples are shown in Figs. 2–7. They were calculated from the reference tables (http:// srdata.nist.gov/its90/download/download.htmlhttp://srdata.nist. gov/its90/download/download.html) with using 3-point moving average. Every temperature in the table was used as T, with T1 = T 1 and T2 = T + 1 K ( C). The emf values in the tables are rounded to 1 mV. This results in the steps on the QV ðTÞ line, especially at high temperatures. The scattering of the points at extremely low temperatures is also explained with the rounding, but the deflections upward (types N, R, and S) and downward (types E and T) are probably because of not quite correct extrapolation of the reference tables to the lowest temperature limit of the tabulation.

Fig. 3. The characteristic temperature for type K thermocouple.

[(Fig._4)TD$IG]

[(Fig._5)TD$IG]

221

Fig. 4. The characteristic temperature for type N thermocouple.

Fig. 5. The characteristic temperature for type J thermocouple.

222

[(Fig._6)TD$IG]


2.4. Characteristic temperature QV as a tool for the approximation

Fig. 6. The characteristic temperature for type E thermocouple.

Most attractive advantage of the characteristic temperature is its ability to reveal the irregular behavior of the thermocouple sensitivity in “anomalous” temperature ranges. There are no thermocouples with a single constant value of QV , but we can recognize wide temperature ranges where the values of QV change nearly linear with temperature and those where QV changes abruptly. The linear ranges are exemplified with 200–400 K for type E, 450–730 K and 1100–1300 K for type J, 200–670 K for type T, etc. Irregularities are from 750 to 870 K and 1030–1100 K for type J, from 225 to 410 K for types S and R, from 320 to 500 K for type K, etc. It is difficult and probably impossible in principle to develop the general rules for the approximation of the emf over such irregular regions. But, we can realize how to change the formulas for the emf and the Seebeck coefficient for the temperature ranges where the characteristic temperature behaves regularly. If the QV changes linearly with temperature, we have

QV ¼ a þ bT: The characteristic temperatures for types S, R, K, and N behave in a similar way. Slightly decreasing from the very beginning, the line reaches its minimum near 300 K and then increases reaching its value at the upper temperature limit of the reference tables. The plot for type J (Fig. 5) looks like those for S, R, K, and N, but with the depression forming the second minimum near 1060 K. This probably is the manifestation of the Curie point of iron (1043 K). It is necessary to remind here that type J thermocouple is made of iron and constantan. The characteristic temperature of type E thermocouple (Fig. 6) increases slightly from the very zero reaching its local maximum near 35 K and then decreases with a minimum near 700 K. Then, it increases again, reaching its maximum at the upper temperature limit of its reference table. The picture for type T thermocouple is most simple (Fig. 7). Its characteristic temperature increases up to a maximum near 130 K and then decreases in a straight line. The QV values characterize generally a particular thermocouple, and allow one to compare sensitivity of various thermocouples with one another. The higher the characteristic temperature of a thermocouple, the less its sensitivity. The characteristic temperature of type E thermocouple is the lowest among all those shown in Figs. 2–7. It is the most sensitive thermocouple, especially for its lowest QV values near 700 K. Contrary, type S thermocouple has the highest QV values, and its sensitive is the least. Type S thermocouple was the main sensor for high-temperature measurements for a long time, but type R was found to possess some higher sensitivity and its characteristic temperature is some less than that for type S (Fig. 2).

[(Fig._7)TD$IG]

(30)

We can use it in Eq. (23) for the Seebeck coefficient: sðTÞ ¼

kB T kB 1 T ¼ : b þ 1 a=ðb þ 1Þ þ T e a þ bT þ T e

(31)

To obtain the emf as a function of temperature with the derivative described in Eq. (31), we have to integrate it: kB 1 e bþ1

ZT

T dT a=ðb þ 1Þ þ T T0 kB 1 a a þ Tðb þ 1Þ T T0 ln : ¼ bþ1 a þ T 0 ðb þ 1Þ e bþ1

DUðTÞ DUðT 0 Þ ¼

(32) This equation should be tested against the experimental data on the emf of thermocouples. If valid, it can be used for the approximation of empirical reference tables or the generation of draft reference tables for new thermocouples. 3. Examples of approximation Most attractive is the approximation that allows us to derive the reference table of a thermocouple with maximal accuracy after minimal number of measurements. On the one hand, the accuracy of approximation after experimental measurements depends on the accuracy of the measurements themselves. It is unreasonable to expect small approximation error if the reference points of the approximation were measured with large errors. On the other hand, we investigate the accuracy of the approximation, not of the experimental procedure. So, let us consider the measurements as accurate as possible in order to diminish the factor of experimental accuracy and investigate only the accuracy of the approximation itself. This can be simulated with using the approved reference tables, i.e., we consider the experiment that provides us with the accurate emf values not distinguished from the values reported in the reference table. 3.1. Narrow temperature interval

Fig. 7. The characteristic temperature for type T thermocouple.

Minimal number of points allowing one to detect non-linear relationship is three. So let us have three values of the emf measured with type E thermocouple at three temperatures, namely 25, 50, and 75 C (298.15, 323.15, and 348.15 K), denoted T0, T1, and T2, respectively. It is conventional to use the degree Celsius (or even Fahrenheit) in the operations with the emf of thermocouples, but we need temperature in kelvins because the


contact theory operates only with thermodynamic scale of temperature. To be more convenient for the specialists acquainted with the thermoelectric reference tables, all the explanations in this Section will be explained simultaneously both in K (T) and C (t). If we use the first point (25 C) as the reference temperature for the thermocouple, we have DU(T0) = 0 mV, DU(T1) = 1.533 mV, and DU(T2) = 3.161 mV. The mean temperature for the interval from T0 to T1 is 310.65 K (T 1 ). The characteristic temperature for T 1 is calculated according Eq. (29) by using T0, T1, DU(T0), and DU(T1): QV ðT 1 Þ = 120.289 K. Similar calculations are carried out for the interval from T1 to T2, yielding QV ðT 2 Þ = 114.044 K. With two values, QV ðT 1 Þ and QV ðT 2 Þ, we can calculate the coefficients of Eq. (30) for the drift in the characteristic temperature QV ðTÞ ¼ a þ bT: b¼

QV ðT 2 Þ QV ðT 1 Þ

T2 T1 ¼ 197:897K:

¼ 0:24982; a ¼ QV ðT 1 Þ bT 1 (33)

Coefficients a and b were used for the calculation of the emf according to Eq. (32). In order to examine the accuracy of the approximation, the difference between the tabulated and calculated values is shown in Fig. 8. All the point are within the limits of 0.001 mV. As the round-off error of the values reported in the reference table is 0.0005 mV, the value of 0.001 mV is exactly explained with this reason. Thus, this result assured us in the validity of the approximation based on the linear change in the characteristic temperature. For comparison, the difference between the data in the reference table and quadratic polynomial

223

The characteristic temperatures QV ðT 1 Þ = 6232.631 K and QV ðT 2 Þ = 7643.726 K were calculated according Eq. (29). The characteristic temperature changes with temperature QV ðTÞ = 2360.82 + 4.70365 T. The results of the approximation are shown in Fig. 9, together with the quadratic polynomial fitted to the three reference points. The difference between the reference tables for type S thermocouple and the approximation algorithm based on the characteristic temperature approaches +15 mV, which is equivalent to 1.5 C. Such a difference is rather significant, exceeding the limits of 1 C recommended by the NIST [16]. Surprisingly, the quadratic polynomial with the coefficients derived from three reference points (400, 700, and 1000 C) turned out to fit the reference table within the limits from 2 to +2.5 mV, which is equivalent to 0.25 C. It is very high accuracy of the approximation over such a wide temperature range, especially for such a simple mathematical tool. 3.3. Calibration against many experimental points

The next example of the approximation with the characteristic temperature is for type S thermocouple. Reference temperatures for the approximation are T0 = 400, T1 = 700, and T2 = 1000 C (673.15, 973.15, and 1273.15 K). If we use again the first point (400 C) as the reference temperature for the thermocouple, we have DU(T0) = 0 mV, DU(T1) = 3.016 mV, and DU(T2) = 6.328 mV.

Now, let us consider the case of several experimental points measured with errors. The example of these data was borrowed from literature. In the “Series in Measurement Science and Technology” published by Institute of Physics, monograph “Evaluating the Measurement Uncertainty: Fundamentals and Practical Guidance” by I. Lira contains section “Calibration of a Thermocouple” [17]. The section describes the calibration of type S thermocouple against eleven reference points in a temperature range from 0 to 180 C with a step of 18 C (ti + 1 = ti + 18 C; i = 0, 1, 2, . . . , 10). The reference junction is taken at 0 C, and DU(t0) = 0 mV. All the input data for the approximation of these experimental values are listed Table 1. It is noted in the monograph that the function for the emf approximation is not standardized, but a high-order polynomial of temperature is the standard function de facto, and the eleven reference points were approximated with a polynomial of the 4th order: DU(t) = 5.8631 103t + 8.574 106t2 1.44 108t3 + 2.9 1011t4 mV. There were no arguments in favor of exactly 4th-order polynomial as compared with 3rd- or 5th-order polynomial. It was recently explained that the high-order polynomials are unsuitable for the approximation of the emf of thermocouples because the logarithm function belongs to the type of functions that generate Runge's phenomenon [18]. Thus, one can expect that the polynomial of the 4th order will turn out not to be the best one for the approximation of this case of eleven reference points. We start our analysis with the calculation of the characteristic temperature. Eleven points yield ten values of QV ðT i Þ, with T i = 273.15 + (ti + ti + 1)/2. They are shown in Fig. 10, together with the

[(Fig._8)TD$IG]

[(Fig._9)TD$IG]

Fig. 8. The difference between two procedures of approximation and reference table for type E thermocouple: squares – Eq. (32), circles – quadratic polynomial.

Fig. 9. The difference between two procedures of approximation and reference table for type S thermocouple: squares – Eq. (32), circles – quadratic polynomial.

DUðTÞ ¼ a þ bT þ cT 2

(34)

are also shown in the same figure. The coefficients of the polynomial were calculated from the reference points T0, T1, and T2 according to the conventional way of solution for the system of three equations with three variables. Again, the errors of this approximation are within the limits of 0.001 mV. One can conclude that the quadratic polynomial approximation is as good as that based on the characteristic temperature. 3.2. Wide temperature interval

224


Table 1 Comparison among three procedures of the approximation of 11 experimental points for type S thermocouple. Input data

4th-order polynomial

2nd-order polynomial

Eq. (32)

Ref. table

t ( C)

DU(t) (mV)

DU(t) (mV)

DU(t) (mV)

DU(t) (mV)

DU(t) (mV)

0 18 36 54 72 90 108 126 144 162 180

0 0.1081 0.2221 0.3387 0.4629 0.5872 0.7206 0.8517 0.9936 1.1308 1.2803

0 0.1082 0.2216 0.3396 0.4620 0.5885 0.7190 0.8534 0.9915 1.1336 1.2796

0 0.1093 0.2228 0.3404 0.4622 0.5881 0.7182 0.8524 0.9909 1.1334 1.2801

0 0.1083 0.2214 0.3391 0.4613 0.5877 0.7184 0.8530 0.9915 1.1338 1.2797

0 0.101 0.210 0.325 0.446 0.573 0.705 0.841 0.982 1.126 1.273

characteristic temperature derived from the reference table for type S thermocouple. The QV ðT i Þ values are scattered in a way similar to that for the characteristic temperature of the reference table, though with some higher drift. The values are fitted to a linear regression

QV ðTÞ ¼ 2855 þ 3:203T:

(35)

For comparison, the values of the reference table are fitted to QV ðTÞ = 3452 + 1.685 T. Regression (35) was used for the calculation of the emf after Eq. (32). The results of the approximation are listed in Table 1. And finally, the values calculated after the 2nd-order polynomial approximating the same experimental points are also listed in Table 1. The reference data of type S thermocouple for the input temperatures are listed in the last column of Table 1. Thus, we have three different approximations. The first one is the 4th-order polynomial. It does not have any physical sense. It is just routine and conventional mathematical tool for the approximation of any array of data, no matter what they refer to. Contrary, the contact theory considers the 2nd-order polynomial as the highest polynomial, which is necessary for the approximation of the emf. Simultaneously, the 2nd-order polynomial is also routine and conventional mathematical tool for the approximation of any array of data, no matter what they refer to. The difference between the 4th and 2nd-order polynomials is also in the quality and quantity of the labor necessary to derive the coefficients of the polynomials. The less the number of coefficients, the less the efforts to derive the coefficients. The case of Eq. (32) is completely different. This approximation is valid and applicable only to the data on the emf of thermocouples. It is the part of the contact

[(Fig._10)TD$IG]

Fig. 10. The characteristic temperature for type S thermocouple calculated after the reference table (small circles) and the experimental points for the approximation (diamonds). The lines show the drift in the QV ðTÞ for the reference table (thin line) and experimental points (thick line).

theory of thermoelectricity. It contains the fundamental constants and specific properties of a particular thermocouple. Eq. (32) is a part of a theory that unites different phenomena. For instance, the characteristic temperature of a thermocouple depends on the difference in the electronic specific heat coefficient of the metals forming the couple. This equation surely will be useful for the development of the theory of thermoelectricity. As for the application in the approximation of the emf values, the importance of Eq. (32) is not so evident. In comparing the results of three different approximations, we see that they differ insignificantly. All the calculated values are within the limits of 0.8 mV as compared with each other. Fig. 11 shows the difference between every approximation procedure and the reference table for type S thermocouple. It is difficult to choose a single preferable procedure of the approximation. All they should be considered nearly equal in their approximation ability. Here, we have to involve other factors characterizing these procedures. If comparing two polynomials, of 2nd and 4th order, we surely prefer the former just because it is simpler. If comparing the 2nd-order polynomial with Eq. (32), we have to prefer the former again. The polynomial did approximate the wide temperature range of type S thermocouple (400–1000 C) better than Eq. (32). Besides, Eq. (32) is more complex for the calculations than the 2nd-order polynomial. Let us remember that before the calculations after Eq. (32), we have to calculate the characteristic temperatures for several temperature intervals, then to fit the values to a linear regression of QV = a + bT, and finally to calculate the emf values after rather complex Eq. (32). Thus, we prefer the 2nd-order polynomial without doubts.

[(Fig._1)TD$IG]

Fig. 11. The difference between three procedures of the approximation of experimental points and reference table for type S thermocouple: squares – Eq. (32), circles – quadratic polynomial, diamonds – the 4th-order polynomial.


[(Fig._12)TD$IG]

225

3.4. Quadratic polynomial for the characteristic temperature Theory predicts that a quadratic polynomial (Eq. (34)) is enough to fit well the emf of a thermocouple (Eq. (22)) if the characteristic temperature is constant. Unfortunately, it is not constant. But fortunately, if the characteristic temperature of a thermocouple can be approximated with a quadratic polynomial, the emf can be described rather accurately with a function derived completely from this quadratic polynomial. The reference table for thermocouple E consists of two parts, one from 270 to 0 C (3.15–273.15 K) and the other from 0 to 1000 C (273.15–1273.15 K), approximated with polynomials of 13th and 10th orders, respectively. The characteristic temperature of type E thermocouple changes regularly over the wide temperature interval and can be rather accurately approximated with a quadratic polynomial

QV ðTÞ ¼ a þ bT þ cT ; 2

(36)

where a = 277.01, b = 0.63453, and c = 0.00044352 over the temperature interval from 273.15 to 1273.15 K. The sensitivity of the thermocouple is kB T kB T ¼ e QV ðTÞ þ T e a þ ðb þ 1ÞT þ cT 2 kB 1 T 2 ¼ ; c T þ pT þ q e

sðTÞ ¼

(37)

where p = (b + 1)/c and q = a/c. This expression for the sensitivity can be used for the evaluation of the emf according to Z Z k 1 T DUðTÞ ¼ sðTÞdT ¼ B dT: (38) c T 2 þ pT þ q e The integration yields k 1 1 DUðTÞ ¼ B lnðT 2 þ pT þ qÞ e c 2 !# p T þ ðp=2Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi p arctg þ C; 2ðq ðp2 =4ÞÞ q ðp2 =4Þ

(39)

where C is the constant of integration. Its value depends on the accepted reference temperature. In following the NIST reference tables, we accept " kB 1 1 lnð273:152 þ p273:15 þ qÞ C¼ e c 2 p 273:15 þ ðp=2Þ arctg pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðq ðp2 =4ÞÞ q ðp2 =4Þ

!# :

(40)

The difference between the emf of type E thermocouple calculated after Eq. (39) and NIST reference table is shown in Fig. 12 as divided by the sensitivity s(T). This yields the difference in temperature units. We can see that the simple quadratic polynomial (36) with three coefficients five digits each allows us to generate equation approximating the reference table within the limits 0.5 K over the temperature range of 1000 K. For the same temperature interval, similar error (discrepancy) of 0.5 K was between the current (ITS-90) and previous (IPTS-68) temperature scales [19]. In comparing the 10th order NIST polynomial with Eq. (39), we may conclude that the latter is more convenient for practical use than the former, providing practically the same accuracy of approximation.

Fig. 12. The difference between 10th-order polynomial by NIST and Eq. (39) for type E thermocouple.

4. Conclusions Contact theory of thermoelectricity provides us with the formula that expresses the voltage at the contact of two metals against temperature. In theory, the single particular parameter that defines completely the emf of a thermocouple is its characteristic temperature QV . In reality, the characteristic temperature turns out to be not constant, but changing with temperature, i.e., QV ðTÞ. This is the reason why it is impossible to suggest a simple and simultaneously accurate expression for the approximation of the emf vs. T. The functions QV ðTÞ of letter-designated thermocouples behave differently in different temperature ranges, being constant, decreasing or increasing in rather wide temperature intervals, and changing its values in unpredictable way in a narrow temperature intervals. The equation was derived from the theory that allows one to approximate the emf with sufficient practical accuracy over the temperature intervals where QV ðTÞ changes linearly with temperature. The equation was tested over three-point approximation for thermocouples of types E (narrow temperature interval) and S (wide interval). The 2nd-order polynomial approximation was used for comparison. The equation was shown to be valid and suitable for the approximation, but not having any advantages over the 2nd-order polynomial. The approximation of many points with experimental errors was tested for the new equation and two polynomials, of 2nd and 4th order. All three ways were found to approximate the experimental points with almost identical accuracy. In considering all the circumstances of the approximation for every procedure, we have to conclude that the best way to approximate the emf of a thermocouple is the 2nd-order polynomial. In case when the characteristic temperature can be accurately approximated with a quadratic polynomial, the reference table can be generated only from the three coefficients of the polynomial. This is shown for type E thermocouple. The error of this approximation for the reference table is within the limits of 0.5 K. The same procedure can be applied to any new thermocouple for chip calorimetry without approved reference table. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.tca.2014.07.008. References [1] H. Huth, A. Minakov, C. Schick, High sensitive differential AC-chip calorimeter for nanogram samples, Netsu Sokutei 32 (2005) 70–76. http:// netsu.org/j+/Jour_J/pdf/32/32-2-70.

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