Scaling of thermoelectric inhomogeneities with temperature in platinum–rhodium alloyed thermocouples

Thermoelectric inhomogeneities of type S, type R and Pt-20%Rh/Pt thermocouples were determined in the temperature range between 200 °C and 1000 °C in temperature steps of 50 °C. Immersion profiles with each thermocouple at each of the 16 scanning temperatures were measured. From the measured temperature dependencies of the inhomogeneities, methods were derived how thermoelectric inhomogeneities measured at only one or two scanning temperatures are quantitatively transferred to other temperatures or temperature ranges. For this purpose, thermoelectric inhomogeneities were classified as irreversible and reversible inhomogeneities, as they must be treated differently. Irreversible thermoelectric inhomogeneities can be extrapolated linearly with temperature or electromotive force from only one immersion profile measurement at an arbitrary temperature to other temperatures in the temperature range investigated. Reversible inhomogeneities in Pt/Rh alloyed thermocouples must be taken as a kind of unavoidable background inhomogeneity (noise) whose amplitude essentially depends on the alloy composition. The distinction between reversible and irreversible inhomogeneities is made by measuring immersion profiles at two scanning temperatures: first at a temperature between 400 °C and 450 °C, where reversible inhomogeneities have a maximum value and at a temperature between 600 °C and 700 °C, where reversible inhomogeneities have a minimum in contrast to irreversible inhomogeneities.


Introduction
Thermoelectric inhomogeneity is the spatial variation ∂S of the Seebeck coefficient S along a thermoelement which Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. directly affects the measured Seebeck voltage (also called the electromotive force (emf)) [1]. Often, thermoelectric inhomogeneity is the dominant uncertainty component when calibrating a thermocouple [2]; determining this inhomogeneity reliably in quantitative terms is therefore of interest. Webster and White [3] published an overview of the methods, design and performance assessment of thermocouple inhomogeneity scanners used to determine thermoelectric inhomogeneities.
In general, thermoelectric inhomogeneity can be quantified by progressively passing the thermocouple through a narrow temperature gradient and monitoring the changes in the emf. During this process, the temperatures of the measuring and reference junctions of the thermocouple are held constant. In this way, every change in the emf is a measure of the thermoelectric inhomogeneity of the thermocouple. The scanning process generates immersion profiles for the thermocouple (emf or ∆emf versus position). In principle, inhomogeneity scans can be performed in liquid baths and heat pipes, which usually have a long zone of constant and homogeneous temperature, as well as in fixed-point cells during melting or freezing of the fixed-point material when the temperature is constant. However, because such scanning procedures are timeconsuming, they are often performed at only one temperature for routine calibrations. Therefore, the practical question arises as how the value of the thermoelectric inhomogeneity measured at one temperature is to be taken into account in the measurement uncertainty at other temperatures. Jahan and Ballico [4] first showed that, under certain conditions, a linear relationship between the thermoelectric inhomogeneity of type S and R thermocouples and the measured emf exists and the inhomogeneity ∆emf may be expressed as a percentage of the total emf measured. In another study [5], a linear scalability of irreversible thermoelectric inhomogeneities with temperature in type S and R thermocouples in the temperature range between 600 • C and 1000 • C was demonstrated. Thermoelectric inhomogeneities of type B and Land-Jewell thermocouples were investigated in [6] by using a high-resolution scanner operated between 600 • C and 900 • C. The results showed drift effects of between 0.2% and 0.6% for reversible oxidation effects, whereas drift caused by irreversible contamination effects was between 0.6% and 1.1%. It was concluded that the deviations in emfs caused by irreversible homogeneities in these thermocouples scale approximately linearly with temperature, which allows uncertainties assessed at one temperature to be extrapolated to other temperatures. Similar results were achieved with type B thermocouples in [7]. At high temperatures (between 600 • C and 960 • C), the relative inhomogeneities, ∆emf/emf, of the tested thermocouples were constant, which allowed a linear scaling with temperature. At low temperatures (between 180 • C and 400 • C), the relative inhomogeneities were significantly higher. Linear scaling from these low temperatures would result in overestimation of the uncertainty contributions caused by thermoelectric inhomogeneities at higher temperatures.
In another study, several platinum-rhodium alloyed thermocouples with different rhodium contents were investigated with regard to their thermoelectric homogeneity by using a water heat pipe scanner at a temperature of 100 • C [8]. The scanning temperature chosen (about 100 • C) was low enough to avoid additional changes in the Seebeck coefficients caused by thermal effects during the scanning process itself. Before this experiment, the Pt/Rh alloys were annealed at 1100 • C for 2 h and removed (quenched) directly from the annealing furnace. They were then subjected to a gradient annealing for up to 100 h in a special tube furnace with a linear temperature gradient (approx. 150 • C-950 • C). Figure 1 shows the compilation of scanning results of different Pt/Rh alloys in a comparative presentation based on scanning results presented and published in other ways (separately for individual Pt/Rh alloys) in [8]. The location-dependent (and therefore temperature-dependent) emf changes, ∆emf, of the respective Pt/Rh alloys against pure platinum after 100 hours' exposure in the gradient furnace are shown. The temperature on the y-axis corresponds to the annealing temperature in the gradient furnace.
Of note in figure 1 is the compactness of the results presented. The qualitative influence on the thermoelectric voltage to be measured when the thermocouple has been exposed to different temperatures is shown in only one diagram. Additionally, a quantitative estimation of the resulting temperature error at approx. 100 • C can be performed by integrating the ∆emf curves. At least an estimation of the inhomogeneityrelated measurement uncertainty is possible. The aim of the work presented here was to determine thermoelectric inhomogeneities at different scanning temperatures and to derive a rule or principle for scaling the thermoelectric inhomogeneities with temperature from the data obtained.
One type S thermocouple, one type R thermocouple and three Pt-20%Rh/Pt thermocouples were scanned for thermoelectric inhomogeneities at different temperatures in the temperature range between 200 • C and 1000 • C in 50 • C intervals. The experimental details are described in section 2. After the measurement results are presented in section 3 and are discussed in section 4, a summary is given in section 5.

Thermocouples
The five thermocouples investigated were constructed at the Physikalisch-Technische Bundesanstalt (PTB) by using thermoelements 2 m in length and with diameters of 0.5 mm. The thermoelements were cleaned by using pure alcohol followed by distilled water. The thermoelements were then electrically annealed for 4 h (the platinum thermoelements at about 1200 • C and the platinum/rhodium (Pt/Rh) alloyed wires at about 1300 • C). The annealed thermoelements were assembled in two-hole insulation tubes about 700 mm in length and outfitted with a matching protective tube closed on one side, all of which were made of pure alumina (Al 2 O 3 , 99.7%). Prior to assembly, the alumina tubes were baked at 1200 • C for about 5 h. The measuring junctions were prepared using an oxygen-hydrogen flame. All thermocouples were outfitted with a reference junction about 23 cm in length which can be used in a dewar vessel filled with an ice-water mixture to maintain a highly stable reference temperature of 0 • C.
The thermocouples were used routinely at PTB at different temperatures and thus had different homogeneity signatures. Two Pt-20%Rh/Pt thermocouples No. 01/91 and 02/91, were constructed in 1991. Thermocouple No. 01/91 was used more often than No. 02/91, especially for determining a provisional emf-temperature function up to temperatures of 1450 • C. Thermocouple No. 02/91 was used only at temperatures up to the freezing point of copper (1084.62 • C). A third Pt-20%Rh/Pt thermocouple (No. 22-02) was constructed in 2022. It was subjected to thermal heat treatments at different temperatures up to 1100 • C to investigate the thermoelectric stability via repeated measurements at the freezing point of copper. In addition, preliminary investigations of the thermoelectric homogeneity were carried out. The type S thermocouple S18-01 and the type R thermocouple R15-02 were used at PTB for different purposes up to 1100 • C (S18-01) and up to 1340 • C (R15-02), respectively. A summary of the data for the thermocouples investigated can be found in table 1.
The five thermocouples underwent a preparatory thermal heat treatment at 1100 • C for 45-70 h to achieve a uniform thermoelectric state before starting the homogeneity investigation. They were removed directly and quickly (within 2-3 min) from the annealing furnace to maintain the quenched state [8] and to minimize Rh oxidation of the alloyed thermoelements, which takes place in the temperature range between about 550 • C and 900 • C.

Metrological equipment
The thermoelectric homogeneity of the thermocouples at temperatures between 200 • C and 500 • C was investigated using a salt bath. At high temperatures (between 600 • C and 990 • C), a pressure-controlled sodium heat pipe (PC-HP) was used. The immersion profiles of the thermocouples were measured by withdrawing them stepwise from metal protection tubes (salt bath) and ceramic protection tubes (PC-HP) manually. The emfs were measured by using a Keithley 2182 nanovoltmeter. The temperature of the reference junctions at all measurements was 0 • C, realized via an ice/water mixture. The thermocouples were scanned at increasing temperatures starting with the lowest temperature (200 • C) in 50 • C steps. At 550 • C, measurements were not possible due to the limited temperature ranges of the salt bath (T ⩽ 500 • C) and the PC- The temperature of the salt bath was recorded via a standard platinum resistance thermometer (SPRT) (No. 4727) during the immersion profile measurements. The temporal stability of the scanning temperatures in the salt bath was high. The standard deviations of the mean scanning temperatures were ±2 mK at temperatures from 200 • C to 300 • C. These standard deviations increased continuously with rising temperatures up to about ±12 mK at 500 • C. All individual emf values measured during recording of the immersion profiles were corrected for this temporary instability of the salt bath temperature by an emf equivalent calculated from the individual temperature deviation from the calculated mean value of the temperature indication during the immersion profile measurements.
The spatial temperature homogeneity of the salt bath was within ±5 mK over a length of about 25 cm between the maximum immersion (40 cm) and an immersion depth in the salt bath of >15 cm. However, depending on the scanning temperature, at immersion depths of less than about 16-20 cm, environmental effects (parasitic heat flux along the thermocouples) resulted in slightly decreasing temperatures. Therefore, only emf values obtained from immersion depths between 40 cm  and 20 cm ensuring a constant measuring junction temperature were considered for the measurement evaluation. The temporal stability of the PC-HP was measured continuously with Au/Pt thermocouple No. 157-03. Corrections resulting from temporal fluctuations of the temperature were also made here for the emfs of the scanned thermocouples. The mean standard deviation of the scanning temperatures amounted to ±13 mK. The homogeneous temperature zone (±10 mK) extended from immersion depths between about 50 cm and 27 cm. Figure 2 shows examples of the spatial temperature profiles of the salt bath and of the PC-HP measured with the SPRT 4724 and the Au/Pt thermocouple 157-03, respectively. The maximum possible immersion depth of the scanned thermocouples in the salt bath was 40 cm. In order to ensure that the temperature gradients of both scanning devices were locally over approximately the same ranges of the thermocouples investigated, the installation depth in the PC-HP was chosen to be 47 cm. In this way, the comparability of the results of the homogeneity tests in the salt bath and in the PC-HP was ensured, as can be seen in figure 2.

Immersion profiles
3.1.1. Type S and type R thermocouples. The interpretation of the following immersion profile data is based on the spatial range of immersions between the maximum immersion depth at 40 cm and an immersion depth of 20 cm; in this range, no parasitic heat flow effects occurred which could have influenced the stability of the measuring junction temperature. The emf deviations (∆emf) from the initial emf at the maximum immersion of the thermocouple measured at different immersion depths during the scanning procedure can be positive or negative. The largest difference, ∆emf max , between the positive and negative deviations (∆emf(+) and ∆emf(−)) measured during the scanning procedure is a measure of the absolute thermoelectric inhomogeneity of a thermocouple at the particular scanning temperature. The ratio ∆emf max /emf(∆T), with the emf(∆T) as the difference of electromotive voltage at the corresponding scanning temperature T S and the emf at room temperature (∆T-temperature gradient), is a measure of the relative inhomogeneity of a thermocouple.
The thermocouples were thermally stabilized at the respective scanning temperatures for about 2 h before starting the immersion profile measurements. The measured immersion profiles of the type S thermocouple S-18-01 in the salt bath at different temperatures in steps of 50 • C are shown in figure 3. The thermoelectric inhomogeneity at temperatures between 200 • C and 300 • C is in the order of only a few tenths of a µV. With increasing temperatures, the maximum of thermoelectric inhomogeneity (∆emf max = +2.7 µV) occurs at 450 • C. At 500 • C, the maximum value of ∆emf max decreases and amounts to approx. 2 µV, similar to the value at 400 • C. Figure 4 shows the immersion profiles of the same S-18-01 thermocouple measured in the pressure-controlled heat pipe. Here, the algebraic signs of the deviations (∆emf) change    A similar dependence of the thermoelectric inhomogeneity on the scanning temperature was found with the type R thermocouple (R-15-02). Figure 5 shows the immersion profiles in 100 • C steps for the R-15-02 thermocouple measured in the salt bath and in the PC-HP. At low temperatures (200 • C to 300 • C) only small inhomogeneities in the order of a few tenths of a µV occurred, which became larger with increasing temperatures and reached their maximum at 400 • C to 500 • C with ∆emf values of up to +2.3 µV. With further rising scanning temperatures, the inhomogeneities became smaller, their algebraic signs changed and the maximum negative ∆emf max values occurred at 800 • C. At high temperatures (990 • C), the inhomogeneities were recovered within a range of only a few tenths of a µV.       section between 20 cm and 25 cm shows better thermoelectric homogeneity than the section between 25 cm and 35 cm. The homogeneity between 20 cm and 25 cm is in the same order of magnitude as that found for the other two Pt-20%Rh/Pt thermocouples. It is interesting to consider not only the absolute values of the maximum deviations ∆emf max but also the direction of the deviations (positive or negative). For this purpose, the measured deviations ∆emf(+) and ∆emf(−) for the R-15-02 thermocouple at arbitrary immersion depths were plotted against  the scaling temperature. Figure 10 shows the results at immersion depths of 24 cm and 30 cm. For better visual orientation, 3rd order polynomials have been added. Very similar results were obtained with the S-18-01 thermocouple.
The reversible character of the thermoelectric inhomogeneity of the R-15-02 thermocouple is clearly visible in figure 10.
Increasing inhomogeneity values were found up to temperatures between 400 • C and 500 • C, which are associated with ordering effects in the crystal lattice (e.g. vacancy healings). At temperatures above 500 • C, the amount of inhomogeneity again becomes smaller, the algebraic sign changes at about 650 • C and its maximum negative value is reached at approx. 800-850 • C. These negative emf deviations from the initial voltage are associated with the Rh oxidation of the alloyed wire. This process reduces the thermoelectrically active rhodium component, causing the thermoelectric voltage to decrease. The highest oxidation rate is reached at approximately 800-850 • C. At even higher temperatures of approximately 950-1000 • C, the rhodium oxide dissociates and the initial situation of thermoelectric inhomogeneity (as at the quenched state after annealing at 1100 • C) is restored.  order polynomials have been added. Very similar results were obtained with thermocouple No. 02/91. As pointed out before, the degree of thermoelectric inhomogeneity for the Pt-20%Rh/Pt thermocouples is lower than for the type R and type S thermocouples by a factor of 2.

Pt
Pt-20%Rh thermocouple No. 01/91, which was previously used up to 1450 • C, showed significant increasing absolute ∆emf max values with scanning temperatures over the entire temperature range as shown in figure 13. A straight-line approximation curve shows the linear character of the increase of the inhomogeneities with respect to temperature and the

Discussion
When discussing the results presented here regarding the scalability of thermoelectric inhomogeneities in Pt/Rh-alloyed thermocouples measured at only one or two temperatures to other temperatures or temperature ranges, a distinction must be made between irreversible and reversible inhomogeneities. Irreversible inhomogeneities are permanent, static, and mostly locally limited changes of the Seebeck coefficient in a thermoelement. Their presence does not depend on the temperature, but the extent of their influence on the measured emf varies with the temperature. They can be caused by cold work, mechanical deformations or impurities that permanently change the crystalline structure and composition of the metals and therefore the Seebeck coefficient. Reversible inhomogeneities in Pt/Rh alloyed thermocouples are unavoidable and temperature-and time-dependent. They are dynamic changes of the Seebeck coefficient occurring over the entire length of the thermoelements. They include changes in the crystalline lattice structure caused by mild cold work, changes in the composition of the Pt-Rh alloys and by temperature-dependent ordering processes. Ordered lattice states of an alloy often have a slightly higher Seebeck coefficient than disordered states of the same alloy composition. In contrast to changes in the Seebeck coefficient caused by irreversible inhomogeneities, changes in the Seebeck coefficient caused by reversible inhomogeneities can be eliminated by annealing, which places the whole length of the thermocouple into the same (homogeneous) metallurgical and oxidative state.

Irreversible thermoelectric inhomogeneities
Pt-20%Rh/Pt thermocouple No 01/91 was the only thermocouple investigated which had significant irreversible thermoelectric inhomogeneities. The approach of a linear extrapolation of the ∆emf max values measured at one scanning temperature T S to arbitrary temperatures T was applied successfully. The extrapolation was carried out according to equation (1): with ∆emf T as the deviation resulting from the electromotive force at an arbitrary temperature T calculated using the thermoelectric inhomogeneity measured at the scanning temperature T S and emf T as the electromotive voltage at T. The ratio ∆emf max (T S )/emf(∆T) is the relative inhomogeneity and was calculated at every scaling temperature. The mean value of the relative inhomogeneity was 0.046% with a standard deviation of 0.009%. The results of the linear extrapolation to determine the thermoelectric inhomogeneity at different temperatures on the basis of the relative inhomogeneities ∆emf max (T S )/emf(∆T) calculated at selected scanning temperatures T S are shown in figure 15 for thermocouple No. 01/91. The ∆emf max values measured at the different scanning temperatures are also shown for comparison. The linear extrapolation of the measured maximum deviation ∆emf max at an arbitrary scanning temperature T S to other temperatures according to equation (1) delivers good results and clearly confirms that this approach can be used for irreversible inhomogeneities [4][5][6][7].
The largest deviations of the linear extrapolations from the measured values ∆emf max were obtained using the data at scanning temperatures T S of 200 • C and 250 • C. The difference between these two slopes ∆emf max (T S )/emf(∆T) is 0.000 21, which corresponds to a relative thermoelectric inhomogeneity of 0.021%. For thermocouple No. 01/91, this value can be considered as the uncertainty of the scaling procedure (rectangular distribution). This rather high uncertainty of almost half the mean relative thermoelectric inhomogeneity (0.046%) reaffirms that extreme care must be taken when determining thermoelectric inhomogeneities.
Some deviations from the linear extrapolation can be explained via the reversible inhomogeneities superimposed on the irreversible inhomogeneities, as can be seen in figure 16. These inhomogeneities are in the order of 1.5 µV at each scanning temperature and correspond in magnitude to the  reversible inhomogeneities detected in the other two Pt-20%Rh/Pt thermocouples (see figure 12).

Reversible thermoelectric inhomogeneities
The reversible thermoelectric inhomogeneities in Pt/Rh alloyed thermocouples are difficult to quantify due to their dynamic behaviour with temperature and time. However, because of their inevitability, they are always present and must be taken into account in some way as the measurement uncertainty caused by thermoelectric inhomogeneity.
The attempt at a linear extrapolation as performed in section 4.1 (irreversible inhomogeneity) on the basis of relative inhomogeneities leads to erroneous results as shown in figure 17. Specifically, extrapolation on the basis of the measured inhomogeneities at temperatures between 400 • C and 500 • C results in an overestimation of the thermoelectric inhomogeneities and the extrapolations on the basis of 300 • C and at temperatures above 900 • C result in an underestimation of thermoelectric inhomogeneities.
Considering the absolute values of ∆emf max as a measure of thermoelectric inhomogeneity in type R, S and Pt-20%Rh/Pt thermocouples, almost no systematic dependence on the temperature could be identified (see figures 9 and 11). Therefore, the simple mean of the ∆emf max values would be an effective way to estimate the thermoelectric inhomogeneity. Table 2 summarizes the mean ∆emf max values of the four thermocouples and their standard deviations. It should be noted that the absolute inhomogeneities and the standard deviations of the Pt-20%Rh/Pt thermocouples were lower by about a factor of 2 than those found for the type S and R thermocouples. In practice, a low scanning temperature is recommended to avoid additional changes to the Seebeck coefficients due to the thermal effects during the scanning process itself. From this point of view, the optimum scanning temperatures should not exceed about 300 • C. However, this approach leads to an underestimation of the reversible thermoelectric inhomogeneities of Pt/Rh alloyed thermocouples over the entire temperature range. Table 3 summarizes the ∆emf max values measured with all thermocouples at all scanning temperatures. The ∆emf max values taken from the immersion profiles at 200 • C-300 • C are low and not yet significantly influenced by reversible inhomogeneities. These values increase strongly at 350 • C and 400 • C and reach the previously calculated mean value of ∆emf max (table 2) at temperatures between 350 • C and 400 • C. Three of the four thermocouples without irreversible inhomogeneities showed maximum ∆emf max values at temperatures between 400 • C and 450 • C (shaded fields in table 3). Therefore, measurements of immersion profiles in this temperature range allow the maximum thermoelectric inhomogeneities to be estimated for the entire temperature range. Only for thermocouple No. 02/91 would this lead to a misjudgement.
If only one temperature is used to determine the thermoelectric inhomogeneity of the Pt-Rh alloyed thermocouples in the entire temperature range between 200 • C and 1000 • C, this should be done by measuring an immersion profile at a temperature between 400 • C and 450 • C. The ∆emf max value obtained at this scanning temperature can be used as a good maximum estimation of the thermoelectric inhomogeneity over the whole temperature range.
At this point, it should be noted that a compensation of the reversible inhomogeneities occurs if the homogeneities are located in the temperature gradient. They provide a positive voltage contribution in the temperature range between approx. 200 • C and 650 • C and a negative contribution between approx. 700 • C and 950 • C (figures 10 and 12). In this respect, the ∆emf max value determined at 400 • C-450 • C can be used as the maximum uncertainty contribution for the thermoelectric inhomogeneity; the risk of underestimating this contribution is low.
It is not possible to distinguish irreversible thermoelectric inhomogeneities or reversible inhomogeneities of Pt-Rh alloyed thermocouples by measuring the inhomogeneity at only one temperature when the thermocouples are in the quenched state like in this work. Therefore, an additional measurement is needed to determine the thermoelectric inhomogeneity at a further scanning temperature. The temperature should be chosen within a temperature range at which the reversible inhomogeneities become very small (i.e. in the temperature range between 600 • C and 700 • C). In this way, a distinction can be made between reversible and irreversible inhomogeneities. If the ∆emf max values at a scanning temperature between 600 • C and 700 • C are higher than those at scanning temperatures between 400 • C and 450 • C, it is most likely that the dominant inhomogeneities are of irreversible character and can be scaled with the temperature based on relative inhomogeneities.

Summary
The uncertainty contribution caused by thermoelectric inhomogeneities is estimated, as is common practice, by measuring immersion profiles. Before starting such measurements, the thermocouple should be in a defined thermoelectric state: annealed at 1100 • C for ⩾2 h and removed from the annealing furnace within 2-3 min (quenched state). Furthermore, before starting the scanning procedures, the thermocouple should be stabilized at the respective scanning temperature for about 2 h.
It is necessary to distinguish between irreversible and reversible inhomogeneities to choose the right approach to quantifying the thermoelectric inhomogeneities in Pt/Rh alloyed thermocouples. Irreversible inhomogeneities can be scaled linearly according to equation (1) over the entire temperature range as shown with thermocouple No. 01/92 based on only one immersion profile measurement at a temperature that is in principle arbitrary (see figure 13). This confirms the common practice [4][5][6][7] of treating irreversible inhomogeneities. For reversible inhomogeneities, the ∆emf max value measured at a scanning temperature between 400 • C and 450 • C is sufficient to present a maximum estimate of the thermoelectric inhomogeneity in the whole temperature range.
The distinction between irreversible and reversible inhomogeneities is made by measuring a second immersion profile at a scanning temperature between 600 • C and 700 • C. This takes advantage of the fact that the reversible inhomogeneities show a minimum in this temperature range (change of algebraic sign, see figures 10 and 12). Therefore, they are usually smaller than at the first scanning temperature (400 • C to 450 • C), or at least in the same order of magnitude. Irreversible inhomogeneities, however, increase with temperature and are higher at the second scanning temperature (600 • C-700 • C) than at the first scanning temperature (400 • C-450 • C) (table 3, last row). In this way, a distinction between irreversible and reversible inhomogeneities is possible in most cases and the corresponding procedure for scaling the measured ∆emf max values can be chosen to determine the uncertainty contribution due to thermoelectric inhomogeneity for the entire temperature range between approx. 200 • C and 1000 • C.
The uncertainty of the inhomogeneity should be estimated as a rectangular contribution wherein the half width is the maximum ∆emf max value, since the immersion profiles are usually measured over only a relatively small length of the thermocouple [9].
In principle, the reversible inhomogeneities in Pt/Rh alloyed thermocouples can be taken as a kind of unavoidable background inhomogeneity (noise) whose amplitude essentially depends on the alloy composition. The type S and R thermocouples investigated here showed maximum reversible inhomogeneities within about ±2 µV (figure 10) which is consistent with the results in [10]. The Pt-20%Rh/Pt thermocouples showed maximum reversible inhomogeneities lower by a factor of 2 in the order of ±1 µV (figure 12) and therefore confirmed the findings in [8]. When using Pt/Rh alloyed thermocouples (Type S and R and Pt-20%Rh/Pt), an inevitable minimum inhomogeneity of this order of magnitude must be expected.