# Temperature Dependence of the Dynamic Parameters of Contact Thermometers

^{*}

## Abstract

**:**

_{x}and time constants τ) depend on temperature. Several effects are superimposed. Constructional and material properties of the thermometer and the installation location affect the dynamic behavior as well as the type and material properties of the object to be measured. Thermal conductivity λ, specific heat capacity c, and density ρ depend on temperature. This temperature dependence can be mutually compensated for (see Section 3). At the same time, the dynamic behavior is also influenced by the temperature-dependent parameters of the medium. When the thermometers are installed in air, for example, the heat transfer coefficient α decreases with increasing temperature, owing to the temperature-dependent material data of the air, at constant speed v. At the same time, heat radiation effects are so strong that the heat transfer improves despite the decreasing convective heat transfer coefficient. In this paper, a number of examples are used to establish a model for the temperature dependence of the dynamic parameters for various thermometer designs. Both numerically and experimentally determined results for the determination of the dynamic characteristic values are included in the consideration.

## 1. Introduction

_{x}, time constants τ, or cut-off frequencies f

_{G}) are used. They can be described as both changing the medium temperature of the process and by generating a step response when the temperature sensor changes from one medium with the temperature T

_{1}to another medium with the temperature T

_{2}(T

_{1}≠ T

_{2}). In previous standards for contact thermometers for the determination of the dynamic behavior, the recording of step responses by ΔT ≈ 20–40 K in water or air was prescribed. However, conclusions cannot always be drawn from the obtained characteristic values about the dynamic behavior under other conditions (e.g., when using thermocouples in the hot steam range or in the exhaust gas systems of vehicles (temperatures up to 1100 °C)). The characteristic values of sensors at such high temperatures were not determined by using the equipment of the Institute for Process Measurement and Sensor Technology at the TU Ilmenau. Therefore, numerical calculations were carried out. At the beginning, a simple wire-wound measuring resistor was considered for these numerical calculations since only the temperature dependence of the material data of Al

_{2}O

_{3}needs to be taken into account, and analytical results can be used for the comparison of the numerically calculated ones [6]. Only theoretically determined results were described in [6], so in the present paper analytical, numerical, and experimentally determined results are presented.

- V—volume of the sensing element;
- ρ—density of the sensor;
- c—specific heat capacity of the sensor;
- α—heat transfer coefficient by convection;
- A
_{M}—sensor surface; - l—length of the sensor;
- λ—thermal conductivity of the sensor;
- r
_{a}, r_{i}—outer and inner radius of the sensor.

## 2. Test Equipment

_{S}(0) = 200 °C using a heat tube. At the beginning of the step, the tube drops down, driven by gravity, and the thermometer is cooled by forced convection in ambient air with different velocities between 1 m∙s

^{−1}and 10 m∙s

^{−1}(Figure 2).

_{x}were calculated by a normalized step response [8]:

- T
_{S}(t_{x})—temperature by time t_{x}; - T
_{S}(0)—temperature at the beginning of the step (t = 0 s); - T
_{M}—temperature of the medium (in this case: air).

- v
_{L}—air velocity; - v
_{M}—measured velocity; - Δv
_{S}—uncertainty of the velocity-measuring sensor; - Δv
_{MS}—difference between the velocity measurement and the velocity at measuring point; - Δv
_{SP}—influence of an inhomogeneous velocity profile.

- Δ(t
_{x})—uncertainty of the respective time-percent value; - Δ(h(t
_{x}))—uncertainty in determining the normalized temperature; - S(h(t
_{x}))—increase of the respective time-percent value; - Δ(t
_{A})—uncertainty of the sampling time; - Δ(t
_{MG})—uncertainty of the measuring device (HP 34410A); - Δ(t
_{MSU})—uncertainty of the measuring switch (PREMA 2024); - Δ(t
_{Fall})—uncertainty by falling of the heat tube.

## 3. Comparison of Analytical, Numerical, and Experimental Results for an Existing Sensor Element

_{2}O

_{3}.

^{−3}. The inverse thermal diffusivity a (${a}^{-1}=\frac{c\cdot \rho}{\lambda}$) increased with rising temperature for the material used.

^{−2}·K

^{−1}and ambient air temperature T = 20 °C were set as the boundary condition at the right line of the axial-symmetrical model and all other surface lines were insulated (Figure 4).

^{−8}and 0.05 s.

_{50}: blue line, t

_{63}: red line, and t

_{90}: green line) confirm the assumption that the time-percent values also increased with increasing temperature, but in this case the increase was very slight.

_{63}. To compare this value with the value of the time constant τ, the time constant was calculated analytically (see Equation (1), Section 1) according to [8]. The black points in Figure 5 show the analytically calculated results of the time constant τ. Up to a starting temperature of 400 °C there was good congruence with the values of t

_{63}.

## 4. Investigations with Typical Industrial Thermometers

_{S}(0) = 40–200 °C. The time-percent values were determined as the mean values of five step responses per temperature. Afterwards, these results were compared with FEA calculations. Only the sensor without a measuring insert was modeled for these calculations, as described in Section 3. The temperature-dependent material parameters of Al

_{2}O

_{3}were the same. The convective heat transfer coefficient was larger than the value in Section 2 due to the measuring insert. It changed with temperature in a range of α = 84.13–85.57 W∙m

^{−2}·K

^{−1}. The radiation between the thermometer and the surrounding area was considered. The step response was simulated for t = 400 s, with automatically selected time steps between 10

^{−6}s and 0.05 s.

_{50}and t

_{63}. But, in this simulation, the measuring insert, the influence of a differently temperature-controlled environment, and the place of installation were not considered. Therefore, there is a difference between the results of FEA-calculation (green line) and measurement (green dashed line) for the time-percent value t

_{90}(Figure 8).

^{−1}was different for the three materials used—it increased with rising temperature for MgO, and it decreased with rising temperature for the two metals (Figure 10).

## 5. Conclusions

- Temperature-dependent material properties of medium and thermometer;
- The thermometer design and installation conditions;
- Heat transfer conditions;
- Surrounding area.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

- Minkina, W.A. About the temperature sensor unit step response non-linearity during air temperature measurement. In Proceedings of the TEMPMEKO ’99, 7th International Symposium on Temperature and Thermal Measurements in Industry and Science, Delft, The Netherlands, 2–3 June 1999; Dubbeldam, J.F., de Groot, M.J., Eds.; Edauw Johannissen bv: Delft, The Netherlands, 1999; pp. 453–458. [Google Scholar]
- Michalski, L.; Eckersdorf, K. Dynamics of the Low-inertia Temperature Sensors in the Conditions of the Radiant Heat Exchange. In Proceedings of the 2nd Symposium on Temperature Measurement in Industry and Science, Suhl, Germany, 16–18 October 1984; pp. 305–314. [Google Scholar]
- Kerlin, T.W.; Shepard, R.L.; Hashemian, H.M.; Petersen, K.M. Response of Installed Temperature Sensors, Temperature, Its Measurement and Control in Science and Industry; Schooley, J.F., Ed.; AIP: New York, NY, USA, 1982; Part 2; Volume 5, pp. 1357–1366. [Google Scholar]
- Augustin, S.; Fröhlich, T.; Mammen, H.; Irrgang, K.; Meiselbach, U. Determination of the dynamic behaviour of high-speed temperature sensors. Meas. Sci. Technol.
**2012**, 23, 7. [Google Scholar] [CrossRef] - Augustin, S.; Fröhlich, T.; Ament, C.; Güther, T.; Irrgang, K.; Lippmann, L. Dynamic properties of contact thermometers for high temperatures. Measurement
**2013**, 51, 387–392. [Google Scholar] [CrossRef] - Fröhlich, T.; Augustin, S.; Ament, C. Temperature-Dependent Dynamic Behaviour of Process Temperature Sensors. Int. J. Thermophys.
**2015**, 36, 2115–2123. [Google Scholar] [CrossRef] - Lieneweg, F. Übergangsfunktion (Anzeigeverzögerung) von Thermometern—Aufnahmetechnik, Meßergebnisse, Auswertungen; Mitteilung aus dem Wernerwerk für Meßtechnik der Siemens & Halske AG: Karlsruhe, Germany, 1964; pp. R46–R53. [Google Scholar]
- Bernhard (Hrsg), F. Handbuch der Technischen Temperaturmessung; 2. Auflage; Springer: Berlin, Germany, 2014. [Google Scholar]
- Augustin, S.; Fröhlich, T.; Heydrich, M. Bestimmung der Messunsicherheit dynamischer Kennwerte von Berührungsthermometern in strömender Luft. In Technisches Messen; De Gruyter Oldenbourg: Oldenbourg, Germany, 2017; Volume 84. [Google Scholar]
- Landolt, H.; Madelung, O. Numerical Data and Functional Relationships, Science and Technology. In Group 4, Thermodynamic Properties of Inorganic Materials; Springer: Berlin, Germany, 2001; Volume 19. [Google Scholar]
- Dörre, E.; Hübner, H. Alumina: Processing, Properties and Applications; Springer: Berlin, Germany, 1984. [Google Scholar]
- Special Metals. Alloys Literature. Available online: http://www.specialmetals.com/tech-center/alloys.html (accessed on 3 April 2019).
- Touloukian, Y.S.; Kirby, R.K.; Taylor, E.R.; Lee, T.Y.R. Thermophysical Properties of Matter—The TPRC Data Series, Volume 13: Thermal Expansion—Nonmetallic Solids; Plenum Press: New York, NY, USA, 1977. [Google Scholar]
- Lippmann, L.; Meiselbach, U.; Irrgang, K.; Augustin, S.; Fröhlich, T. Konzeption und Installation einer Versuchsanlage zur Prüfung und Untersuchung von Temperaturfühlern in Heißgasumgebung. In Proceedings of the TEMPERATUR 2013, Berlin, Germany, 5–6 June 2013; PTB Berlin: Berlin, Germany, 2013; pp. 53–58. [Google Scholar]

**Figure 1.**Electrical analogy model of the first-order time delay element of a sensor (cylinder). where:

- R
_{α}—thermal resistance caused by convection; - R
_{1}—internal thermal resistance of the sensor caused by conduction; - C
_{1}—heat capacity of the sensor; - T
_{M}—medium temperature; - T
_{O}—temperature of the sensor surface; - T
_{S}—sensor temperature; - T
_{U}—ambient temperature.

- R
_{α}—thermal resistance caused by convection; - R
_{1}—internal thermal resistance of the sensor caused by conduction; - C
_{1}—heat capacity of the sensor; - T
_{M}—medium temperature; - T
_{O}—temperature of the sensor surface; - T
_{S}—sensor temperature; - T
_{U}—ambient temperature.

**Figure 2.**Schematic set-up for the experimental determination of the dynamic behavior of thermometers.

**Figure 4.**Finite element analysis (FEA) model, boundary condition, and an example of typical temperature gradient field.

**Figure 8.**Comparison of the results of FEA calculation and measurement (Me) for the industrial resistance thermometer shown in Figure 7.

**Figure 11.**Experimental results (air, steps from various temperature to ambient air, v = 3 m∙s

^{−1}).

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Augustin, S.; Fröhlich, T. Temperature Dependence of the Dynamic Parameters of Contact Thermometers. *Sensors* **2019**, *19*, 2299.
https://doi.org/10.3390/s19102299

**AMA Style**

Augustin S, Fröhlich T. Temperature Dependence of the Dynamic Parameters of Contact Thermometers. *Sensors*. 2019; 19(10):2299.
https://doi.org/10.3390/s19102299

**Chicago/Turabian Style**

Augustin, Silke, and Thomas Fröhlich. 2019. "Temperature Dependence of the Dynamic Parameters of Contact Thermometers" *Sensors* 19, no. 10: 2299.
https://doi.org/10.3390/s19102299