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Performance of a thermocouple subjected to a variable current
International Journal of Thermal Sciences 134 (2018) 440–452
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International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts
Performance of a thermocouple subjected to a variable current ∗
T
Youness Bouaanani, Philippe Baucour , Eric Gavignet, François Lanzetta FEMTO-ST Institute, CNRS, Univ. Bourgogne Franche-Comte, ENERGIE Department, 2 av. Jean Moulin, 90000, Belfort, France
A R T I C LE I N FO
A B S T R A C T
Keywords: Microthermocouple Thermoelectrical effects Transient model Transient measurement IR measurements Configurable transition
The aim of this study is to understand the various thermoelectrical phenomena by creating a 1D model that dynamically reacts like a thermocouple through which a current is passed. A new style of modelling is used in this study, which allows the characteristics transition between the two different alloys to be programmable. The main objective is to determine the best parameters that characterize the junction in terms of Seebeck coefficient and heat transfer coefficient in order to obtain a reliable model of the thermocouple. Experiments are performed on an E-type thermocouple of 80 μm in diameter. This thermocouple is subjected to three different types of variable currents. The 1D finite difference model results are compared with the experimental data acquired using an infrared camera. The development of an accurate dynamic model leads to a model exploration of the thermocouple response.
1. Introduction A thermocouple is a sensor that is widely used in several domains. The principal use of a thermocouple is temperature measurement owing to the thermoelectric effects and more specificly the Seebeck effect. A potential difference is created when there is a temperature difference at the junction of two different materials [1]. Based on this thermoelectric effect, several sensors are used to measure pressures [2], fluid flows [3,4], and heat flux [5–7] by using the same principle (see Figs. 12 and 13). However, there is another thermoelectrical effect that is less used in metrology applications: the Peltier effect. If an electric current flows across a junction of two dissimilar metals, the heat may either be absorbed (Peltier cooling) or given out (Peltier heating) in the junction volume, depending on the direction of the current. Then in the case of Peltier cooling, the junction absorbs heat from the surroundings and its temperature drops. Loffe [8] established the modern theory of thermoelectric conversion in 1949 and investigated the thermoelectric properties of semiconductors. Several years later, Goldsmid and Douglas demonstrated thermoelectric cooling below 0°C at room temperature environment [9]. Since then, the scientific community started to investigate the use of the Peltier effect as a cooling device [10,11]. However, its efficiency as a cooling device is low for commonly used materials [12]. Therefore, Peltier modules comprise structures with multiple junctions in order to increase the number of exchange surfaces and thus improve the impact of the current on the temperature drop [13]. Doped materials are also used in thermocouples to enhance their
∗
efficiency, and a large number of studies have been conducted to investigate the figures of merit [14]. Furthermore, some studies are focused on applications in which the Peltier effect is increasingly used for sensor applications such as [15] where the Peltier effect is used for anemometry [16], where it is used for the determination of fluid properties, and the work of [17–19] that concerns hygrometry measurement. Following the study conducted by S. Amrane [20], the aim of this new research is to understand the behaviour of an E-type thermocouple subjected to a variable current using a transient numerical model and specific experiments. A E-type of thermocouple (constantan/chromel) has been selected based on the electromotive force generated by it [1]. Therefore the choice of the materials (constatan and chromel) is driven by the selection of the highest Seebeck coefficient available. The drop in temperature caused by the Peltier effect is the parameter that is used for all sensor applications such as determining humidity parameters or water potential [21]; however, other thermal characteristics such as thermal conductivity [22] are also used. The aforementioned drop in temperature depends at first on the Seebeck coefficient value but also on several parameters such as the wire diameter and current injection as well as thermal effects such as conduction, convection, and radiation. The purpose of this study is to create a generalized thermoelectrical model of a thermocouple such that the position, length, wire diameter, material characteristics, and type of the junction are configurable. The results obtained with the model are then compared with a previous model prepared by Ref. [20] and also with the acquired experimental
Corresponding author. E-mail address: [email protected] (P. Baucour).
https://doi.org/10.1016/j.ijthermalsci.2018.07.029 Received 19 July 2017; Received in revised form 19 July 2018; Accepted 24 July 2018 1290-0729/ © 2018 Elsevier Masson SAS. All rights reserved.
International Journal of Thermal Sciences 134 (2018) 440–452
Y. Bouaanani et al.
Nomenclature
I J k L Nu Pr Ri , Rj Ra S T
Greek symbols
ε ρ ρelec σ σB τ
Radiative emissivity [-] Density [kg · m−3] Electrical resistivity [Ω · m−1] Seebeck coefficient [V · K−1] Stefan Boltzmann coefficient [W · m−2 · K−1] Thomson coefficient [V · K−1]
Roman symbols
Subscripts
a b c d Gr h
∞ c i j rad ref
Junction length [m] Temperature coefficient for electrical resistivity [K−1] Heat capacity [J · kg−1 · K−1] Diameter [m] Grashof number [−] Convective heat transfer coefficient [W · m−2 · K−1]
Current [A] Current density [A. m−2] Thermal conductivity [W · m−1 · K−1] Length [m] Nusselt number [−] Prandtl number [−] Electrical resistance [Ω] Rayleigh number [−] Section [m2] Temperature [K]
Ambient [−] Contact [−] Wire number or discretization index [−] Junction [−] Radiation [−] Reference [−]
that it is exposed to. Fig. 1 illustrates the experimental set-up, and the various devices used are as follows:
data. The experimental set-up involves a constantan wire aligned with a chromel wire and arc welded [23,24]. The junction created with a linear weld (see section 2.1) will result in a cylindrical junction whereas a traditional arc weld will result in a spherical junction. The thermocouple is then subjected to a variable current produced by a programmable power supply and the temperature drop realized is then evaluated using an infrared (IR) camera.
1. An IR camera FLIR SC7000 for the contactless temperature measurements. The camera is combined with a lens L0120 which has an aperture of F/2, a focal of 10.45 ± 0.50 mm and a spectral band of 3.5–5 ± 0.25 μm. 2. A temperature input module NI 9211 connected to a compactDAQ modular data acquisition system for performing temperature measurements using a microthermocouple. This device is used only for calibration purposes and is thus not shown in Fig. 1. 3. A PXI-4110 programmable power supply and PXI-1033 controller for generating the various voltage profiles. 4. The two software ALTAIR and LabVIEW Desktop. 5. E-type thermocouple (wires by Omega).
2. Experimental set-up In Ref. [20], a second thermocouple is glued on top of the junction that induces experimental issues. In contrast, the experimental set-up in the present study is designed to obtain temperature measurements while being as non-intrusive as possible. The device chosen to obtain the various data is an IR camera that facilitates data acquisition while remaining external to the system. For the model validation, an E-type thermocouple of 80 μm welded linearly is chosen (i.e. the two wires are welded end to end as shown on Fig. 2). To suspend the thermocouple horizontally, a metallic frame that is painted in black to reduce the radiance at the maximum is used. To insulate the system as well as possible, the latter is enclosed in black panels to reduce the radiation
This section comprises three parts: the first part consists of the description of the thermocouple welding and the device used for its manufacturing, the second part concerns the thermocouple characterization according to the IR camera numerical levels, and the third part
Fig. 1. Experimental set-up with the two different desktops and the experimental material. The IR camera and the thermocouple are shielded via black panels to prevent any parasite IR radiation. 441
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Fig. 2. Special device used for linear thermocouple welding. The electric supply is the TL-Welder from the Omega company.
While using a thin thermocouple, the camera lens was switched with the one that enables the visualization of pixels that approaches 7 μm. The data acquisition using an FLIR camera implies the use of the ALTAIR software for controlling the various camera parameters as well as the acquisition parameters. The default parameters are used for the thermocouple characterization, which implies an acquisition frequency of 50 Hz that corresponds to 50 frames per second with a response time of 2.2 m s. Working with a small thermocouple for the purpose of the dynamic study implies that the IR camera is programmed to acquire numerical levels instead of temperatures. This quantity represents the signal registered by the camera and this value is converted into temperature or luminance by using a conversion curve. The first step of the experiment involves determining the characterization curve between the temperature values obtained for the E-type thermocouple and the measurements obtained by the IR camera. To perform this study, the thermocouple is connected to the NI CompactDAQ provided with an NI 9211 module. This module is specially developed for obtaining temperature measurements with thermocouples and also integrates a cold junction compensation. This configuration allows a resolution of 24 bits for a sampling frequency of 14 Hz. Coupled with this system, the IR camera was used to acquire the various numerical levels at the three different thermocouple regions, the chromel side, constantan side, and junction side. The recorded numerical levels depend on the temperature measurements made using the CompactDAQ. To increase the reliability of the various values obtained, an arithmetic average of the temperature and the numerical levels was calculated. The thermocouple (a) and the three different zones (b) selected are presented in Fig. 3. On iterating this process, three characterization curves that represent the three different zones were obtained. The valid temperature range for those curves is between 18°C and 21°C. Fig. 4a and b shows the three different zones: as expected, the numerical level range varies for the various parts. These curves will be useful for the rest of the study and in particular for retrieving the temperature of the thermocouple
of the experimental set-up section presents the programming of the various current signals used. 2.1. Thermocouple welding The difficulty in this experimental set-up lies in the fabrication of a linear thermocouple. The E-type thermocouple is chosen because it presents the highest electromotive force (60 μV/ °C at 20°C). A 80 μm in diameter sensor is selected so that the sensor is situated in the mediumsize category. However, this constraint made the fabrication of the linear thermocouple more complicated. The use of a special device (shown on Fig. 2) was necessary to facilitate the linear arc welding. Fig. 2 shows the device used for the linear arc welding; it comprises two parts. One platform is fixed while the other is movable, and each platform possesses a chuck fixation that secure the wire in a horizontal position and insure wires alignment. The extremities of the other wire are then connected to the electric supply (TL-Welder from Omega), and the movable platform approaches the fixed one as the wire extremities connect. Then by switching on the electrical supply, an electrical arc causes the two wires to connect thus forming a linear junction. According to the TL-Welder specifications the energy output does not exceed 80 J. Afterwards, using image processing, the junction dimensions are measured: a diameter of 80 μm (equal to the wire diameter) and a length of 40 μm. 2.2. Thermocouple/IR camera characterization The studied E-type thermocouple in combination with the National Instrument NI 9211 module is used at various ambient temperatures in order to characterize the IR camera FLIR SC7000 in combination with the thermocouple. During these preliminary experiments, the thermocouple acts as a temperature reference for the IR camera and a current is — obviously — not passed through it.
Fig. 3. Experimental set-up (a) and zone selection in the image software ALTAIR (b). 442
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Fig. 4. Characterization curve for the IR camera, i.e. temperature versus numerical levels, for the junction i.e. the central zone in Fig. 3. The confidence and prediction intervals are represented by shaded areas.
The numerical result of equation (1) gives an uncertainty of U = 0.51 K. To conclude this analysis, it's important to specify that the measurements are achieved on very small zones but they are not obtained simultaneously because it's necessary to modify the camera settings to focus the lens of the camera on each zone. The measurements are investigated on different spatial zones but we have no way of extracting a real temperature field or even a temperature profile with this experimental solution.
when a current is passed through it (see section 5). Several parameters should be considered to determine the measurements uncertainty. Two main types of errors contribute to uncertainty, the reading ones and those related to measurement devices [25]. The most important factors for our experimental work are the following. The thermocouple itself. This sensor is developed, characterized and calibrated in the laboratory. The process uses a calibration dry block calibrator (Gemini 550 LRI) associated with a secondary standard platinum resistance thermometer (AOIP PHP 601 with a temperature resolution of 0.001 K and an accuracy of 0.005 K) [26]. The calibration process provides an experimental uncertainty of the temperature value of 0.2 K. The data acquisition system (compactDAQ and the module NI9211). This device is connected to a thermocouple and is used to obtained the temperature acquisition. The error is given by National Instruments in the datasheet and is equal to 0.07 K. The FLIR SC7000 camera. This camera enables the temperature acquisition with a resolution of 0.02 K at the optimal utilization conditions. The regression curves. These curves are used to retrieve the thermocouple temperature from the numerical level. According to Fig. 4 the quality of the regression is evaluated by the interval of confidence and r 2 value. Using linear regression implies a wider interval i.e. the prediction interval [27]. The uncertainly is calculated to 0.128 K (see Fig. 4b). The external vibrations. The experimental set-up is submitted to external vibrations which increases the noise measurement on the numeric levels determined by the IR camera. In this experimental work, an arithmetic average of the temperature and the numeric levels is calculated for three different zones (junction, constantan, chromel). Each zone consists of 9 pixels (3 × 3) and a spatial average is calculated on these pixels with the ALTAIR software. By assuming a vertical or horizontal offset of 3 pixels we could estimated the uncertainty at 3 × 0.02= 0.06 K.
2.3. Variable voltage programming This section expands on the realization of the various voltage profiles used to excite the thermocouple junction. To inject a specific profile, the devices used are a PXI-1033 controller chassis and a PXI4110 module. The PXI-4110 module is a 16 bits programmable DC power supply with a range of 0–6 V and a resolution of 120 μV. It will be used to generate various voltage profiles and four excitation types are chosen. 1 2 3 4
Sinusoidal Square Constant Sawtooth
Each voltage profile is calculated using a mathematical model provided by LabVIEW that provides the point-to-point profile description at each iteration. The use of a formula node enables the generation of the signal at the maximum speed that the PXI allows (1 kHz) [28]. The obtained voltage is then generated using the PXI module with the corresponding current to excite the thermocouple junction. Each parameter, such as the excitation type, voltage amplitude, frequency, offset, and phase difference, is configurable using a graphical interface. The voltage profile is instantly generated by the PXI and presented on the graphical interface, thus allowing the user to observe it. The PXI is coupled with the IR camera that is used to determine the difference in the numerical levels when the current – according to the selected voltage profile – is passed through the thermocouple.
The expanded experimental uncertainty (U) is given by combining the individuals uncertainties with the following equation:
3. Governing equations and modelling
U=
⎛ 2 × ⎜∑ ui2, m + ⎜ i ⏟ coverage factor measurements ⎝ ⏟
⎞ ui2, d⎟ ⎟ devices ⏟⎠
1 2
3.1. Introductory work (1)
This study is an extension of the work of S. Amrane [17], in which a 443
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modelisation approach that can be used to tackle the various identified drawbacks.
three-part 1D model of a thermocouple excited by a current was designed. This model comprised three coupled equations: one equation for each wire (see eq. (2)) and one equation for the junction (eq. (3)).
3.2. A new approach
∂ 2T hP = ki 2i − i i (Ti − T∞ ) ⋯ ∂ x i S
∂T ρi ci i ∂ t Heat accumulation
Conduction
⋯−
The aim of this new study is to improve all those aspects for the purpose of generalization and to obtain a flexible and continuous model. We will discuss the thermoelectric phenomena in more detail (the Joule, Peltier, and Thomson effects). The Joule effect can be expressed as a function of the cross section, material length, current intensity, and electrical resistivity, which varies linearly with the temperature with respect to the references Tref ref and ρelec . It can be written as follows:
Convection
σB εi Pi 4 4 (Ti − Trad ) + ρelec, i J 2 i S
Joule effect
Radiation
(2)
In eq. (2), the i index represents the various parts of the thermocouple (wire 1 and wire 2). At the junction (j index), the continuity between these two equations is done via eq. (3). ∂Tj
ρj cj Vj⋅ ∂t Heat accumulation
=
∂T
−k1 S1⋅ ∂x1 x =− a Heat conduction from wire 1
+
∂T
k2 S2⋅ ∂x2 =+ x a
Radiation
Joule effect =
Peltier effect
Heat conduction from wire 2 4 σB ε j Sj⋅(T j4 − Trad ) +
⋯ − hj Sj⋅(Tj − T∞) − Convection
− σ12 I ⋅Tj ⋯
ref ρelec L
S
[1 + b (T − Tref )]⋅I 2
[W]
(4)
The Thomson effect is usually neglected — as is the case in the previous model (eqs. (2) and (3)) — but the study of Titov on semiconductors [29] shows that it should be taken into account. This effect, in contrast to the other two thermoelectrical effects (Seebeck and Peltier), does not require the presence of a junction. The Thomson effect is observed when a temperature gradient is present or when a current is passed through the material. This implies that a heat absorption or rejection occurs if a current is passed through the conductor material, or an electromotive force is produced if the conductor material has a temperature gradient across it.
Rj⋅I 2
⏟
Joule effect
(3) The phenomena described by eq. (2) and eq. (3) are considered to be unidimensional in the x direction. Considering a 2D/3D model will assume that a significant thermal gradient could exist inside the wires (in the radial direction if we work in cylindrical coordinates). This assumption could be easily rejected by calculating the Biot Number Bi ≈ 5.4⋅10−4 ≪ 1. The high thermal conductivity (k) of the wires and the junction, combined with the small size of the device (80 μm) will induced an homogeneous temperature in the wire section even for large values of the heat transfer coefficient (h). As can be seen, besides the Joule effect, no thermoelectrical effect is taken into account for the wires (except the Joule one), and the Peltier effect is only localized at the junction. The radiation heat exchange occurs between the surface at Ti and the surrounding at Trad . It should be noted that - in the model - T∞ (fluid temperature) differs from Trad . However during the experiments (see below) the test bench is IR-shielded so Trad = T∞. Fig. 5 shows the locations of the various effects that occur in the thermocouple. The problem encountered with this model is detailed in the subsection 3.2 To resume the main drawbacks are:
Thomson effect = −τ ∇TJ
[W. m−3]
(5)
Where J is the current density and τ is the Thomson coefficient, which is related to the Seebeck coefficient σ.
τ = −T
dσ dT
(6)
Intuitively, if the Seebeck coefficient evolution is more or less independent of the temperature or it changes slightly then the Thomson effect can be neglected. The Thomson and Peltier effects are linked and their individual terms can be combined to form one term that expresses the two effects as a function of the position (i.e. x in our study) [30,31]. If eq. (5) is integrated between the junction borders then the obtained result will be the Peltier effect expression used in eq. (3). Thus, using one equation allows us to take into consideration the two effects, and the position is the only factor that influences the type of thermoelectric effect that occurs.
1. Three different equations for one system 2. No Thomson effect in the wires 3. The model could not be used for more complex cases (such as 2 junctions or 2 wires with different diameters) 4. The junction is considered as homogeneous (a single point at the junction)
Peltier Thomson = −T
In order to enhance the previous study, we propose a new
∂σ J ∂x
[W. m−3]
(7)
With eq. (7), the two thermoelectric effects will be taken into
Fig. 5. Effects taken into account in various zones of the thermocouple for the previous model (adapted from Ref. [20]). 444
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account in the model, and it assumes that the value of the Seebeck coefficient along the thermocouple is known (see 3.2.2).
conditions for a current injection of 14 mA and an ambient temperature of 19.45°C.
3.2.1. Governing equation The lack of continuity at the junction induces to introduce the thermoelectrical effects everywhere (i.e. wires and junction) and meanwhile to consider the evolution of the different parameters along the x-axis. Such approach leads to a single equation model (eq. (8)).
3.2.4. Heat transfer coefficient Besides the transition at the junction, the other problem is the determination of the natural convection. Modelling natural convection is difficult because the correlations are dependent on the temperature difference between the surface of the object and the ambient temperature. However, the aim of this study is to determine this temperature difference, and hence, to find the value of the heat transfer coefficient, an assumption of the temperature drop at the beginning of the model has to be made. The difficulty faced in this case is that the correlations for small devices like microthermocouples do not fit well, as in Ref. [35], the difference between the experimental measurement and the correlation is approximately 24% for a K-type thermocouple of 50 μm. In the present work, the phenomena could be considered as an inverse plume effect (i.e. cold wire in a hot fluid), while the reverse has been studied much more (i.e. hot wire in a colder fluid). Usually, the heat transfer coefficient is determined from a finite element study on the flow [36] or experimental measurement [37,38]. The lack of data caused the authors to use the standard Churchill and Chu correlation [39] for free convection cited by Ref. [40].
ρcS
ρ ∂T ∂ ⎛ ∂T ⎞ ∂σ I 4 kS ) + elec I 2 − T = − hP (T − T∞) − σB εP (T 4 − Trad S ∂t ∂x ⎝ ∂x ⎠ ∂x S
(8)
In eq. (8), all the bold symbols (such as k , S , P , ρ , c …) may vary along the x-axis in order to represent the change in material (i.e. wire, junction, and wire). By applying this equation along the geometry created for the model, at the junction, the equation will take into account the Peltier effect because of the difference in the Seebeck coefficient. In our application the ambient temperature T∞ and the radiative ambiance Trad are considered to be equal. Therefore for the temperature range used in the present study the radiation exchange is negligible compare to convection (∼ 276 W. m−2 for convection vs. ∼ 9 W. m−2 for radiation. Heat flux calculated with the parameters from Table 2). 3.2.2. Transition at the junction This formulation of the heat equation (see eq (8)) is in accordance with the theoretical work of Apertet [32]. The distinguishing feature of this model is that all the characteristics of the thermocouple are expressed as a function of the wire length. To obtain the best continuity possible, the functions chosen to translate the evolution of the characteristics are smoothed splines created with five points: two points at the very tip of the thermocouple wire, two points at the end of the junction length, and one point at the centre of the junction. The problem with this scheme is the type of transition that is applied at the junction. In fact, in this study, a transitional homogeneous scheme was designed. This is why all the characteristics at the junction are averaged as a function of the volume ratio between the two species according to [33] i.e. assuming 50% constantan and 50% chromel in our case. Therefore the characteristics of the junction can be deduced from those of the wires as shown on Table 2. However, the Seebeck coefficient is the only characteristic that uses another law of averaging, i.e. the Nordheim–Gorter law [34].
σj =
σ1. ρelec,1 + σ2. ρelec,2 ρelec,1 + ρelec,2
2
⎧ ⎪ 1 1 ⎪ 0.387⋅Gr 6 ⋅Pr 6 Nu = 0.6 + 8 ⎨ 9 27 ⎡1 + 0.559 16 ⎤ ⎪ Pr ⎪ ⎢ ⎥ ⎣ ⎦ ⎩
( )
4. Model implementation This part deals with special procedures that have been used to implement the model from eq. (8). Attention is first focused on the determination of the heat transfer coefficient and then on the numerical scheme. Finally, the following two models are compared:
[V . K−1] (9)
• The previous model from Amrane (eq. (2) and eq. (3)) [20]. • The new model described by eq. 8
3.2.3. Boundary conditions With respect to the establishment of the boundary conditions, in this study, two types of boundary conditions could be considered:
Table 2 Material characteristics of the chromel and constantan [1]. The characteristics of the junction are dedicated from these (mean value) except for the Seebeck coefficient σ which is determined via the Nordheim–Gorter law.
1. The extremities are insulated:
Symbol
=0 x =±L
Units
(10)
2. The temperatures at the extremities are imposed:
T x =±L = T∞
(12)
Where Nu , Gr , and Pr are the commonly used dimensionless Nusselt, Grashof, and Prandtl numbers, respectively. Equation (12) is valid for a wide range of Rayleigh (Ra = Gr⋅Pr ) As the weld is linear, the junction is cylindrical (as shown on Fig. 5), and therefore, eq. (12) is used all along the thermocouple.
Actually, this law averages the Seebeck coefficient of the coefficient (σj ) with the electrical resistivity of each wire ( ρelec, i ) and not the volume. The averaged value of σj = −2.67 × 10−6 V. K−1 is used along the junction as can be observed in Fig. 6.
∂T ∂x
⎫ ⎪ ⎪ with Gr⋅Pr = Ra⩽1012 ⎬ ⎪ ⎪ ⎭
(11)
Ni 90%/Cr 10%
Ni 50%/Cu 45%
Junction
ρ
kg.m−3
8730
8920
8825
J.kg−1. K−1
448
394
421
k
W.m−1. K−1 m .m
19.26
21.19
20.22
5 × 10−2 7.060 × 10−7
5 × 10−2 4.890 × 10−7
– 5.975 × 10−7
K−1 – m
4.1 × 10−4
−1. × 10−5
2.381 × 10−4
0.45 80 × 10−6 22 × 10−6
0.87 80 × 10−6 −38 × 10−6
0.66 80 × 10−6 −2.67 × 10−6
ref ρelec
b ε d σ
445
Constantan
c L
It should be noted that these boundary conditions have no impact on the result near the junction when an infinite thermocouple is considered (length of 10 cm for a diameter of 80 μm). These boundary conditions modify the temperature drop when the thermocouple length is less than 3 cm. Table 1 shows the difference found in the temperature drop according to the thermocouple length and the boundary
Chromel
V.K−1
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Fig. 6. Seebeck transition at the junction using the Nodheim–Gorter law for the Seebeck coefficient average.
4.1. Heat transfer coefficient determination Table 1 Impact of the boundary conditions on the relative deviation of the temperature at the junction (i.e. coolest point). Current injection = 14 mA, and ambient temperature = 19.45°C. Thermocouple length
Relative deviation
[m] 0.01 0.02 0.03 more than 0.04
% 0.5 0.3 0.01 0
In free convection, the heat transfer coefficient can be determined via the correlation in eq. (12) only if an evaluation of the temperature difference between the wire and the surroundings is given and is used to calculate the Grashof number (Gr ). Amrane et al. [20] have developed an analytical equation (see eq. (13)) that could be used to predict, in a steady state, the temperature drop at the junction (ΔTj = Tj − T∞) for a thermocouple through which a current of density J is passed.
J2 ΔTj =
S Ph
ref ref (ρelec ,1 k1 + ρelec,2 k2 ) − σ12 JT∞ Ph S
( k1 +
k2 ) − σ12 J
(13)
Fig. 7. Heat transfer coefficient versus the temperature drop at the junction for various diameters. Each curve corresponds to a range in current between 10 mA and 60 mA.
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to speed up the resolution, the TriDiagonal Matrix Algorithm (TDMA) [44] is implemented in Cython [45]. A 30000-point grid with a 100point grid at the junction and 2000 iterations over time is calculated. Owing to the Cython implementation the calculation time reduces from 20 min (with standard Python) to 3 min depending on the chosen time step.
To solve the non-linear circular dependency between eq. (13) and eq. (12) (the Gr number needs the ΔTj and Nu number the h), an iteration procedure (fixed point algorithm) is used to determine the heat transfer coefficient as a function of the temperature drop [20]. Convergence is commonly achieved in less than 10 iterations. As can be observed in Fig. 7, the range of the temperature difference is between 0.1°C and 1°C, and on considering this large range, the convergence procedure gives a heat transfer coefficient variation from 133 to 144 W.m−2. K−1. Using these values in the numerical model for a current amplitude of 10 mA, the temperature drop at the junction varies from 1.16 to 1.13°C. Therefore, a change of 8.27% in the heat transfer coefficient induces a change of only 2.82% in the temperature drop. This example shows that the model is only slightly sensitive to the heat transfer coefficient.
4.3. Comparison of models The first step of this study is to partially validate the new model by comparing the results with the older model [20]. According to [1], the multiphysics characteristics of the chromel and constantan are presented in Table 2. The Fig. 8 shows the comparison between the results obtained with two models at the junction with the previous experimental data. The comparison was made by applying the same multiphysical characteristics to the two models and the geometrical parameters of the older model into the new one. As can be observed, the final value is the same, which demonstrates how the new model can be easily modified. At the junction, the previous model uses a volume implementation that cannot be changed, whereas the new model depends only on the diameter value that is imposed. Both models provide results faster than the experiments but we have neglected the hypothesis that the delay observed is an experimental issue. During these experiments, another thermocouple (much thinner than the one studied i.e. of 7 μm in diameter) is glued at the top of the junction. The glue may induce an additional thermal resistance that delays the temperature variation; the use of an IR Camera aids in tackling this problem (see section 5). The second part of this partial validation consists of the general temperature profile — in steady state — along the thermocouple. The temperature profile is obtained by three different models:
4.2. Numerical scheme and simulation software The governing differential equation (eq. (8)) subject to the boundary conditions outlined in eq. (10) in 11 are solved using a finite difference approach with a fully implicit scheme. To carry out the proper discretization, all the properties are evaluated at midpoints. An example of the discretized version of the conduction differential for three nodes (i.e. i − 1, i, and i + 1) is as follows:
k −S −Ti − 1 − (ki−Si− + ki+Si+) Ti + ki+Si+Ti + 1 ∂ ⎛ ∂T ⎞ kS ⇒ i i Δx 2 ∂x ⎝ ∂x ⎠
(14)
where ki± , Si± are mean values between 2 adjacent nodes:
ki± =
ki ± 1 + ki Si ± 1 + Si and resp. Si± = 2 2
(15)
This procedure has been applied for all the parameters that may change along the x-axis. The difference between this new thermocouple model and the previous one is the fact that all the characteristics change along the x-axis (see eqs. (14) and (15)), and the junction is now considered as a transition. Therefore, the model is now continuous over the entire thermocouple length. The simulation has been performed on a standard Linux PC with Python. Python is a high-level language for numerical computations and has the ability to manipulate matrices and plot data using the Numpy [41,42], Scipy [43], and Matplotlib libraries. In order
1. the numerical model from Amrane (see eqs. (2) and (3)) [20]. 2. an analytical model developed by Amrane [20]. 3. the new numerical model (see eq. (8)) As shown in Fig. 8, no difference is observed between the results of the previous and the new models. The difference between the analytical
Fig. 8. Comparison between the previous and new models with the previous experimental data from Amrane [20]. Current injection starts at time t = 0 (i.e. step current). 447
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results and the model is caused by the capacitance term at the junction, which is absent in the analytical equations. The only difference that can be observed between the two models occurs at the junction. In fact, the previous model calculates only one point at the junction whereas the new model calculates various points because of the transition at the junction; therefore, there is a difference in temperature levels at the junction as shown in the zoomed-in Fig. 9.
Table 3 Comparison between the model and the experimental data at steady state. Current
Model temperature
Experimental temperature
Sparse
[mA]
decrease (eq. 8) [ °C]
decrease [°C]
[%]
10 20 26
−0.96 −1.38 −1.22
−0.95 −1.39 −1.21
1.02 0.7 0.8
5. Experimental results and model validation The results obtained with the model match well the experimental data, even the dynamic reduction in the temperature matches the predicted values calculated using the model. An observation that can be made is that the previous experimental data were delayed because of the thermal inertia of the glue between the two thermocouples [20] as can be observed in Fig. 8. After the comparison at 10 mA, where the temperature decrease approaches an exponential value, a comparison at 20 mA and 26 mA is of great interest. On reaching a certain current amplitude, a phenomenon of over-cooling is observed, but after a certain time, the temperature increases (Fig. 10b and c). As the junction cools down owing to the Peltier effect, the Joule effect in the wires tends to heat them up. The thermal gradient between the junction and the wires promotes heat fluxes by conduction and the junction temperature thus rises. The final temperature value reached at the steady state is a result of a balance between the thermoelectric effects and heat transfer phenomena (convection and conduction). The model can thus predict the cooling that occurs at the junction and is validated by the experimental data. The third part of the study concerns the validation of the model when the thermocouple is subjected to a variable current.
5.1. Introduction A series of experiments were performed with various parameters to facilitate the validation of the model. The programmable generator generated four current profiles. During the experimental acquisition, various frequencies from 0.2 Hz to 1 Hz with current amplitudes from 10 mA to 26 mA were generated. The aim had been to determine if the model can react, from a dynamic point of view, like a thermocouple through which a current is passed. 5.2. Steady state validation The results for the steady state validation are presented in Table 3. It can be seen that the results obtained using the model and the experimental results are similar and that the corresponding error is approximately 1%. In the case of the steady state, the model is accurate even for different profiles of current amplitude. 5.3. Step current validation The goal of this study is to create a model that can dynamically responds as an experimental thermocouple. Moreover, this will result in better comprehension and knowledge regarding the various phenomena that occur at the junction of two different materials. The comparison results for a step current of 10 mA, 20 mA, and 26 mA between the model and the experimental data is shown in Fig. 10.
6. Variable current model validation The Table 4 shows the various currents injected with their frequencies and amplitudes.
Fig. 9. Evolution of the temperature in steady state as a function of the length: comparison between the new and the previous models and the analytical values. A zoom located at the junction x = 0 is done to show the little differences between the two models.
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Fig. 10. Thermocouple junction temperature versus time for a step current of 10 (a), 20 (b) and 26 mA (c). Comparison between the model and the experimental data for three different currents.
The next figures (from Figs. 11 to 13) show the comparison between the experimental data and the thermocouple model on applying the current variation shown in Table 4.
6.2. Square current For the square current the sparse between the model and the data at 1 Hz is approximately 17%, which is close to that in the case of the sinus current (20%). For high frequencies, the model underestimates or overestimates the cooling realized. The reason for this can be explained by the cutting frequency, which is not well handed by the 1D model. However, the square current temperature evolution with a high current amplitude injection is better reproduced by the model. In the case of the square model, the dynamic is good but the errors in the values are approximately 5%.
6.1. Sinus current The model handles the variation in temperature well when the current is injected at a low frequency. The sparse is the most important at a high frequency, and thus, at 1 Hz, the relative sparse is approximately 20% against 1% at 0.2 Hz. This result could be compared to that obtained in the study of Rojo [46], which shows similar results and a frequency analysis in nanowires. Moreover, at a high current, the data demonstrates a phase in which the temperature increase is staged. The model shows the same phenomena, but in a much more smoothed manner, such that the sparse at the temperature levels is not high but the dynamic sparse is more important in this case.
6.3. Sawtooth current For the sawtooth current, the same conclusions can be obtained as that of the preceding current type. At a high frequency, the relative sparse is important, for 1 Hz the sparse is about 23% as against 3% at 449
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Fig. 11. Sinus current model validation. Comparison between the model and experimental data for a sinus current injection.
The other causes that can be taken into account include, first of all, the acquisition quality; the IR camera is really sensitive with respect to its surroundings. Secondly, the heat transfer coefficient could be the cause of this temperature difference; at high cooling and heating rates, hot and cold air layers may be created around the junction thus varying the heat transfer coefficient. In conclusion, owing to the fact that the model is simplified by only one equation and that it is easy to modulate it, the comparison of its results with the experimental data was quite simple to perform. Furthermore, the maximum sparse makes the model valid for the dynamic response for various types of current. Moreover, the sparse at 1 Hz can be reduced by performing some tests under vacuum, in which case, the convection can be ignored, and it can be determined if the various air layers around the junction are responsible for this temperature difference.
0.2 Hz. As can be observed, the model reacts pretty well depending on the signal injected. The sparse between the model and experimental data is caused by the different assumptions made for the characteristics and the transition chosen at the junction. The transition at the junction is based on an assumption of averaged homogeneous metals but the reality is different; after the electrical weld, the junction between the two metals is not perfectly homogeneous, and thus, the heterogeneity of the junction can play a role on the cooling factor and on the transient response of the thermocouple. However, the determination of the heterogeneity of the junction is complicated and involves probabilistic equations [47]. A study performed to investigate the influence of the difference in the characteristics found that it causes an error of only 0.3%, such that, the variation of the thermophysical characteristics can be neglected in the present study.
Fig. 12. Square current model validation. Comparison between the model and experimental data for a square current injection. 450
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Fig. 13. Sawtooth current model validation. Comparison between the model and the experimental data for a sawtooth current injection.
for exploration. The perspectives of this study are to extend the Peltier effect and to use it for sensor applications.
Table 4 Experimental tests. Type of current
Sinus Square Sawtooth
Frequency/Amplitude
Frequency/Amplitude
[Hz/mA]
[Hz/mA]
0.2/10 0.2/26 0.2/10
1/10 1/10 1/10
References [1] R.E. Bentley, Handbook of temperature measurement, The Theory and Practice of Thermoelectric Thermometry, vol. 3, Springer Science & Business Media, 1998. [2] E. Gavignet, F. Lanzetta, Dynamic operation of a micro-thermocouple sensor as a vacuum gauge, Vacuum 100 (2014) 18–21. [3] T. Marcelli, P. Santoni, E. Leoni, A. Simeoni, Viegas (Ed.), Gas Temperature and Velocity Measurements in the Plume of a Spreading Fire with Double Thermocouple Probe Technique, Forest Fire Research & Wildland Fire Safety, 2002, pp. 1–12. [4] M. Tagawa, Y. Ohta, Two-thermocouple probe for fluctuating temperature measurement in combustion–rational estimation of mean and fluctuating time constants, Combust. Flame 109 (4) (1997) 549–560. [5] J.P. Ploteau, P. Glouannec, H. Noel, Conception of thermoelectric flux meters for infrared radiation measurements in industrial furnaces, Appl. Therm. Eng. 27 (2) (2007) 674–681. [6] T.E. Diller, Advances in heat flux measurements, Adv. Heat Tran. 23 (1993) 279–368. [7] T.E. Diller, Heat flux, Measurement, Instrumentation, and Sensors Handbook, Second Edition: Spatial, Mechanical, Thermal, and Radiation Measurement, CRC Press, 201467–1. [8] A. Loffe, A. Gelbtuch, Semiconductor Thermoelements and Thermoelectric Cooling, (1957). [9] H.J. Goldsmid, R.W. Douglas, The use of semiconductors in thermoelectric refrigeration, Br. J. Appl. Phys. 5 (11) (1954) 386. [10] D. Astrain, J. Vián, J. Albizua, Computational model for refrigerators based on peltier effect application, Appl. Therm. Eng. 25 (17) (2005) 3149–3162. [11] A. Majumdar, Thermoelectric devices: helping chips to keep their cool, Nat. Nanotechnol. 4 (2009) 214–215. [12] S. Tassou, J. Lewis, Y. Ge, A. Hadawey, I. Chaer, A review of emerging technologies for food refrigeration applications, Appl. Therm. Eng. 30 (4) (2010) 263–276. [13] G.J. Snyder, J.-P. Fleurial, T. Caillat, R. Yang, G. Chen, Supercooling of peltier cooler using a current pulse, J. Appl. Phys. 92 (3) (2002) 1564–1569. [14] W. Zhu, Y. Deng, Y. Wang, A. Wang, Finite element analysis of miniature thermoelectric coolers with high cooling performance and short response time, Microelectron. J. 44 (9) (2013) 860–868. [15] S. Guths, D. Leclercq, E. Gaviot, A new sensor to measure fluid velocities: the peltier anemometer, Meas. Sci. Technol. 8 (5) (1997) 530. [16] D. Leclercq, P. Thery, Utilization of the peltier effect for measuring a fluid property. application for designing new sensors, Meas. Sci. Technol. 4 (2) (1993) 158. [17] S. Amrane, Developpement d’un capteur pour la mesure simultanée de température et d’humidité, phdthesis, Université de Franche-Comté - France, (2014). [18] R.W. Brown, D.L. Bartos, A Calibration Model for Screen-caged Peltier Thermocouple Psychrometers, US. Department of Agriculture, Forest Service, Intermountain Forest and Range Experimental Station. Pap. INT, 293 (1982), p. 155p. [19] J. Comstock, Correction of thermocouple psychrometer readings for the interaction of temperature and actual water potential, Crop Sci. 40 (3) (2000) 709–712. [20] S. Amrane, P. Baucour, E. Gavignet, F. Lanzetta, Experimental and theoretical study of microthermocouple used as cooling device, High. Temp. - High. Press. 43 (2014)
7. Conclusion and perspectives In this study, a transient thermoelectrical model that can be applied to thermocouples was designed. The general purpose of this model is to create a generalized model to develop new type of sensor based on the Peltier effect. To generalize the model, various thermal effects such as conduction, convection, and radiation as well as the thermoelectrical effects like the Joule effect, Thomson effect, and Peltier effect are taken into consideration. The specificity of this model is that only one governing equation is necessary to translate all the effects in the thermocouple, thus causing the model to be continuous. The main objective is to create a reliable model, from a transient point of view, that can provide a frequency analysis of the thermocouple response. Such analysis in a complex environment (such as an alternating flow) will allow to determine if some information regarding temperature, or velocity can be obtained. Another application of the model could be the simulation of a thermocouple with several junctions. As a matter of fact a sensor that induces a temperature gradient might be a reliable candidate for a heat flux sensor. Such device should be carefully designed via modelisation - to obtain the maximal temperature difference between the cold and hot junction. The model validation was carried out using an 80 μm E-type thermocouple through which current generated by a programmable tension generator was passed. The data was acquired by using an IR camera and then compared with the result obtained with the model. The validation is carried out for two cases: the first validation is carried out at steady state conditions, for which the model provides accurate results and the error is approximately 1%. The second model validation is carried out at transient conditions. At 1 Hz, the relative error is approximately 20%, whereas at 0.2 Hz, it is approximately 1%. In conclusion, this model is accurate at low frequencies and demonstrates great potential 451
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[34] T. Datta, A. Hermann, R. Deck, R. Somoano, Nordheim-gorter like behavior and one Carrier conduction in a quasi-one-dimensional system, Solid State Commun. 35 (11) (1980) 883–885. [35] L. Thiery, Microsondes Thermoelectriques, Techniques Ingénieur, 2006. [36] A. Prhashanna, R. Chhabra, Free convection in power-law fluids from a heated sphere, Chem. Eng. Sci. 65 (23) (2010) 6190–6205. [37] C.-S. Yih, Laminar free convection due to a line source of heat, Eos, Trans. Am. Geophys. Union 33 (5) (1952) 669–672. [38] S. Grafsrønningen, A. Jensen, B.A.P. Reif, Piv investigation of buoyant plume from natural convection heat transfer above a horizontal heated cylinder, Int. J. Heat Mass Tran. 54 (23–24) (2011) 4975–4987. [39] S.W. Churchill, H.H. Chu, Correlating equations for laminar and turbulent free convection from a horizontal cylinder, Int. J. Heat Mass Tran. 18 (9) (1975) 1049–1053. [40] T.L. Bergman, F.P. Incropera, A.S. Lavine, Fundamentals of Heat and Mass Transfer, John Wiley & Sons, 2011. [41] T.E. Oliphant, Guide to NumPy, second ed., CreateSpace Independent Publishing Platform, USA, 2015. [42] P.F. Dubois, K. Hinsen, J. Hugunin, et al., Numerical python, Comput. Phys. 10 (1996) 262–267. [43] E. Jones, T. Oliphant, P. Peterson, et al., Open Source Scientific Tools for Python, Scipy, (2001). [44] A. Quarteroni, R. Sacco, F. Saleri, Numerical Mathematics 37 Springer Science & Business Media, 2010. [45] S. Behnel, R. Bradshaw, C. Citro, L. Dalcin, D.S. Seljebotn, K. Smith, Cython: the best of both worlds, Comput. Sci. Eng. 13 (2) (2011) 31–39. [46] M.M. Rojo, J.J. Romero, D. Ramos, D.-A. Borca-Tasciuc, T. Borca-Tasciuc, M.M. Gonzalez, Modeling of transient thermoelectric transport in harman method for films and nanowires, Int. J. Therm. Sci. 89 (2015) 193–202. [47] F. Rosi, B. Abeles, R. Jensen, Materials for thermoelectric refrigeration, J. Phys. Chem. Solid. 10 (2) (1959) 191–200.
13–37. [21] T. Chow, J. De Vries, Dynamic measurement of soil and leaf water potential with a double loop peltier type thermocouple psychrometer, Soil Sci. Soc. Am. J. 37 (2) (1973) 181–188. [22] T. Kobari, J. Okajima, A. Komiya, S. Maruyama, Development of guarded hot plate apparatus utilizing peltier module for precise thermal conductivity measurement of insulation materials, Int. J. Heat Mass Tran. 91 (2015) 1157–1166. [23] G. Hostache, P. Prenel, R. Porcar, Couples thermoélectriques à définition spatiotemporelle fine. réalisation. réponse impulsionnelle de microjonctions cylindriques, Rev. Gen. Therm. 25 (299) (1986) 539–543. [24] F. Lanzetta, E. Gavignet, Temperature Measurements: Thermoelectricity and Microthermocouples, CRC Press, 2011, p. 95. [25] L. Kirkup, R.B. Frenkel, An Introduction to Uncertainty in Measurement: Using the GUM (Guide to the Expression of Uncertainty in Measurement), first ed., Cambridge University Press, 2006. [26] M. Doghmane, F. Lanzetta, E. Gavignet, Dynamic characterization of a transient surface temperature sensor, Procedia Engineering 120 (2015) 1245–1248. [27] J.E. Gentle, W. Härdle, Y. Mori, Handbook of Computational Statistics, Springer, 2004. [28] N. Instruments, Labview: Data Acquisition Basics Manual, (1996). [29] O. Titov, J. Velazquez-Perez, Y. Gurevich, Mechanisms of the thermal electromotive force, heating and cooling in semiconductor structures, Int. J. Therm. Sci. 92 (2015) 44–49. [30] C. Goupil, W. Seifert, K. Zabrocki, E. Müller, G.J. Snyder, Thermodynamics of thermoelectric phenomena and applications, Entropy 13 (8) (2011) 1481–1517. [31] C. Goupil, Thermodynamics of the thermoelectric potential, J. Appl. Phys. 106 (10) (2009) 104907–104910. [32] Y. Apertet, C. Goupil, On the fundamental aspect of the first kelvin's relation in thermoelectricity, Int. J. Therm. Sci. 104 (2016) 225–227. [33] J. Wang, J.K. Carson, M.F. North, D.J. Cleland, A new approach to modelling the effective thermal conductivity of heterogeneous materials, Int. J. Heat Mass Tran. 49 (17) (2006) 3075–3083.
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Contents lists available at ScienceDirect
International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts
Performance of a thermocouple subjected to a variable current ∗
T
Youness Bouaanani, Philippe Baucour , Eric Gavignet, François Lanzetta FEMTO-ST Institute, CNRS, Univ. Bourgogne Franche-Comte, ENERGIE Department, 2 av. Jean Moulin, 90000, Belfort, France
A R T I C LE I N FO
A B S T R A C T
Keywords: Microthermocouple Thermoelectrical effects Transient model Transient measurement IR measurements Configurable transition
The aim of this study is to understand the various thermoelectrical phenomena by creating a 1D model that dynamically reacts like a thermocouple through which a current is passed. A new style of modelling is used in this study, which allows the characteristics transition between the two different alloys to be programmable. The main objective is to determine the best parameters that characterize the junction in terms of Seebeck coefficient and heat transfer coefficient in order to obtain a reliable model of the thermocouple. Experiments are performed on an E-type thermocouple of 80 μm in diameter. This thermocouple is subjected to three different types of variable currents. The 1D finite difference model results are compared with the experimental data acquired using an infrared camera. The development of an accurate dynamic model leads to a model exploration of the thermocouple response.
1. Introduction A thermocouple is a sensor that is widely used in several domains. The principal use of a thermocouple is temperature measurement owing to the thermoelectric effects and more specificly the Seebeck effect. A potential difference is created when there is a temperature difference at the junction of two different materials [1]. Based on this thermoelectric effect, several sensors are used to measure pressures [2], fluid flows [3,4], and heat flux [5–7] by using the same principle (see Figs. 12 and 13). However, there is another thermoelectrical effect that is less used in metrology applications: the Peltier effect. If an electric current flows across a junction of two dissimilar metals, the heat may either be absorbed (Peltier cooling) or given out (Peltier heating) in the junction volume, depending on the direction of the current. Then in the case of Peltier cooling, the junction absorbs heat from the surroundings and its temperature drops. Loffe [8] established the modern theory of thermoelectric conversion in 1949 and investigated the thermoelectric properties of semiconductors. Several years later, Goldsmid and Douglas demonstrated thermoelectric cooling below 0°C at room temperature environment [9]. Since then, the scientific community started to investigate the use of the Peltier effect as a cooling device [10,11]. However, its efficiency as a cooling device is low for commonly used materials [12]. Therefore, Peltier modules comprise structures with multiple junctions in order to increase the number of exchange surfaces and thus improve the impact of the current on the temperature drop [13]. Doped materials are also used in thermocouples to enhance their
∗
efficiency, and a large number of studies have been conducted to investigate the figures of merit [14]. Furthermore, some studies are focused on applications in which the Peltier effect is increasingly used for sensor applications such as [15] where the Peltier effect is used for anemometry [16], where it is used for the determination of fluid properties, and the work of [17–19] that concerns hygrometry measurement. Following the study conducted by S. Amrane [20], the aim of this new research is to understand the behaviour of an E-type thermocouple subjected to a variable current using a transient numerical model and specific experiments. A E-type of thermocouple (constantan/chromel) has been selected based on the electromotive force generated by it [1]. Therefore the choice of the materials (constatan and chromel) is driven by the selection of the highest Seebeck coefficient available. The drop in temperature caused by the Peltier effect is the parameter that is used for all sensor applications such as determining humidity parameters or water potential [21]; however, other thermal characteristics such as thermal conductivity [22] are also used. The aforementioned drop in temperature depends at first on the Seebeck coefficient value but also on several parameters such as the wire diameter and current injection as well as thermal effects such as conduction, convection, and radiation. The purpose of this study is to create a generalized thermoelectrical model of a thermocouple such that the position, length, wire diameter, material characteristics, and type of the junction are configurable. The results obtained with the model are then compared with a previous model prepared by Ref. [20] and also with the acquired experimental
Corresponding author. E-mail address: [email protected] (P. Baucour).
https://doi.org/10.1016/j.ijthermalsci.2018.07.029 Received 19 July 2017; Received in revised form 19 July 2018; Accepted 24 July 2018 1290-0729/ © 2018 Elsevier Masson SAS. All rights reserved.
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Nomenclature
I J k L Nu Pr Ri , Rj Ra S T
Greek symbols
ε ρ ρelec σ σB τ
Radiative emissivity [-] Density [kg · m−3] Electrical resistivity [Ω · m−1] Seebeck coefficient [V · K−1] Stefan Boltzmann coefficient [W · m−2 · K−1] Thomson coefficient [V · K−1]
Roman symbols
Subscripts
a b c d Gr h
∞ c i j rad ref
Junction length [m] Temperature coefficient for electrical resistivity [K−1] Heat capacity [J · kg−1 · K−1] Diameter [m] Grashof number [−] Convective heat transfer coefficient [W · m−2 · K−1]
Current [A] Current density [A. m−2] Thermal conductivity [W · m−1 · K−1] Length [m] Nusselt number [−] Prandtl number [−] Electrical resistance [Ω] Rayleigh number [−] Section [m2] Temperature [K]
Ambient [−] Contact [−] Wire number or discretization index [−] Junction [−] Radiation [−] Reference [−]
that it is exposed to. Fig. 1 illustrates the experimental set-up, and the various devices used are as follows:
data. The experimental set-up involves a constantan wire aligned with a chromel wire and arc welded [23,24]. The junction created with a linear weld (see section 2.1) will result in a cylindrical junction whereas a traditional arc weld will result in a spherical junction. The thermocouple is then subjected to a variable current produced by a programmable power supply and the temperature drop realized is then evaluated using an infrared (IR) camera.
1. An IR camera FLIR SC7000 for the contactless temperature measurements. The camera is combined with a lens L0120 which has an aperture of F/2, a focal of 10.45 ± 0.50 mm and a spectral band of 3.5–5 ± 0.25 μm. 2. A temperature input module NI 9211 connected to a compactDAQ modular data acquisition system for performing temperature measurements using a microthermocouple. This device is used only for calibration purposes and is thus not shown in Fig. 1. 3. A PXI-4110 programmable power supply and PXI-1033 controller for generating the various voltage profiles. 4. The two software ALTAIR and LabVIEW Desktop. 5. E-type thermocouple (wires by Omega).
2. Experimental set-up In Ref. [20], a second thermocouple is glued on top of the junction that induces experimental issues. In contrast, the experimental set-up in the present study is designed to obtain temperature measurements while being as non-intrusive as possible. The device chosen to obtain the various data is an IR camera that facilitates data acquisition while remaining external to the system. For the model validation, an E-type thermocouple of 80 μm welded linearly is chosen (i.e. the two wires are welded end to end as shown on Fig. 2). To suspend the thermocouple horizontally, a metallic frame that is painted in black to reduce the radiance at the maximum is used. To insulate the system as well as possible, the latter is enclosed in black panels to reduce the radiation
This section comprises three parts: the first part consists of the description of the thermocouple welding and the device used for its manufacturing, the second part concerns the thermocouple characterization according to the IR camera numerical levels, and the third part
Fig. 1. Experimental set-up with the two different desktops and the experimental material. The IR camera and the thermocouple are shielded via black panels to prevent any parasite IR radiation. 441
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Fig. 2. Special device used for linear thermocouple welding. The electric supply is the TL-Welder from the Omega company.
While using a thin thermocouple, the camera lens was switched with the one that enables the visualization of pixels that approaches 7 μm. The data acquisition using an FLIR camera implies the use of the ALTAIR software for controlling the various camera parameters as well as the acquisition parameters. The default parameters are used for the thermocouple characterization, which implies an acquisition frequency of 50 Hz that corresponds to 50 frames per second with a response time of 2.2 m s. Working with a small thermocouple for the purpose of the dynamic study implies that the IR camera is programmed to acquire numerical levels instead of temperatures. This quantity represents the signal registered by the camera and this value is converted into temperature or luminance by using a conversion curve. The first step of the experiment involves determining the characterization curve between the temperature values obtained for the E-type thermocouple and the measurements obtained by the IR camera. To perform this study, the thermocouple is connected to the NI CompactDAQ provided with an NI 9211 module. This module is specially developed for obtaining temperature measurements with thermocouples and also integrates a cold junction compensation. This configuration allows a resolution of 24 bits for a sampling frequency of 14 Hz. Coupled with this system, the IR camera was used to acquire the various numerical levels at the three different thermocouple regions, the chromel side, constantan side, and junction side. The recorded numerical levels depend on the temperature measurements made using the CompactDAQ. To increase the reliability of the various values obtained, an arithmetic average of the temperature and the numerical levels was calculated. The thermocouple (a) and the three different zones (b) selected are presented in Fig. 3. On iterating this process, three characterization curves that represent the three different zones were obtained. The valid temperature range for those curves is between 18°C and 21°C. Fig. 4a and b shows the three different zones: as expected, the numerical level range varies for the various parts. These curves will be useful for the rest of the study and in particular for retrieving the temperature of the thermocouple
of the experimental set-up section presents the programming of the various current signals used. 2.1. Thermocouple welding The difficulty in this experimental set-up lies in the fabrication of a linear thermocouple. The E-type thermocouple is chosen because it presents the highest electromotive force (60 μV/ °C at 20°C). A 80 μm in diameter sensor is selected so that the sensor is situated in the mediumsize category. However, this constraint made the fabrication of the linear thermocouple more complicated. The use of a special device (shown on Fig. 2) was necessary to facilitate the linear arc welding. Fig. 2 shows the device used for the linear arc welding; it comprises two parts. One platform is fixed while the other is movable, and each platform possesses a chuck fixation that secure the wire in a horizontal position and insure wires alignment. The extremities of the other wire are then connected to the electric supply (TL-Welder from Omega), and the movable platform approaches the fixed one as the wire extremities connect. Then by switching on the electrical supply, an electrical arc causes the two wires to connect thus forming a linear junction. According to the TL-Welder specifications the energy output does not exceed 80 J. Afterwards, using image processing, the junction dimensions are measured: a diameter of 80 μm (equal to the wire diameter) and a length of 40 μm. 2.2. Thermocouple/IR camera characterization The studied E-type thermocouple in combination with the National Instrument NI 9211 module is used at various ambient temperatures in order to characterize the IR camera FLIR SC7000 in combination with the thermocouple. During these preliminary experiments, the thermocouple acts as a temperature reference for the IR camera and a current is — obviously — not passed through it.
Fig. 3. Experimental set-up (a) and zone selection in the image software ALTAIR (b). 442
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Fig. 4. Characterization curve for the IR camera, i.e. temperature versus numerical levels, for the junction i.e. the central zone in Fig. 3. The confidence and prediction intervals are represented by shaded areas.
The numerical result of equation (1) gives an uncertainty of U = 0.51 K. To conclude this analysis, it's important to specify that the measurements are achieved on very small zones but they are not obtained simultaneously because it's necessary to modify the camera settings to focus the lens of the camera on each zone. The measurements are investigated on different spatial zones but we have no way of extracting a real temperature field or even a temperature profile with this experimental solution.
when a current is passed through it (see section 5). Several parameters should be considered to determine the measurements uncertainty. Two main types of errors contribute to uncertainty, the reading ones and those related to measurement devices [25]. The most important factors for our experimental work are the following. The thermocouple itself. This sensor is developed, characterized and calibrated in the laboratory. The process uses a calibration dry block calibrator (Gemini 550 LRI) associated with a secondary standard platinum resistance thermometer (AOIP PHP 601 with a temperature resolution of 0.001 K and an accuracy of 0.005 K) [26]. The calibration process provides an experimental uncertainty of the temperature value of 0.2 K. The data acquisition system (compactDAQ and the module NI9211). This device is connected to a thermocouple and is used to obtained the temperature acquisition. The error is given by National Instruments in the datasheet and is equal to 0.07 K. The FLIR SC7000 camera. This camera enables the temperature acquisition with a resolution of 0.02 K at the optimal utilization conditions. The regression curves. These curves are used to retrieve the thermocouple temperature from the numerical level. According to Fig. 4 the quality of the regression is evaluated by the interval of confidence and r 2 value. Using linear regression implies a wider interval i.e. the prediction interval [27]. The uncertainly is calculated to 0.128 K (see Fig. 4b). The external vibrations. The experimental set-up is submitted to external vibrations which increases the noise measurement on the numeric levels determined by the IR camera. In this experimental work, an arithmetic average of the temperature and the numeric levels is calculated for three different zones (junction, constantan, chromel). Each zone consists of 9 pixels (3 × 3) and a spatial average is calculated on these pixels with the ALTAIR software. By assuming a vertical or horizontal offset of 3 pixels we could estimated the uncertainty at 3 × 0.02= 0.06 K.
2.3. Variable voltage programming This section expands on the realization of the various voltage profiles used to excite the thermocouple junction. To inject a specific profile, the devices used are a PXI-1033 controller chassis and a PXI4110 module. The PXI-4110 module is a 16 bits programmable DC power supply with a range of 0–6 V and a resolution of 120 μV. It will be used to generate various voltage profiles and four excitation types are chosen. 1 2 3 4
Sinusoidal Square Constant Sawtooth
Each voltage profile is calculated using a mathematical model provided by LabVIEW that provides the point-to-point profile description at each iteration. The use of a formula node enables the generation of the signal at the maximum speed that the PXI allows (1 kHz) [28]. The obtained voltage is then generated using the PXI module with the corresponding current to excite the thermocouple junction. Each parameter, such as the excitation type, voltage amplitude, frequency, offset, and phase difference, is configurable using a graphical interface. The voltage profile is instantly generated by the PXI and presented on the graphical interface, thus allowing the user to observe it. The PXI is coupled with the IR camera that is used to determine the difference in the numerical levels when the current – according to the selected voltage profile – is passed through the thermocouple.
The expanded experimental uncertainty (U) is given by combining the individuals uncertainties with the following equation:
3. Governing equations and modelling
U=
⎛ 2 × ⎜∑ ui2, m + ⎜ i ⏟ coverage factor measurements ⎝ ⏟
⎞ ui2, d⎟ ⎟ devices ⏟⎠
1 2
3.1. Introductory work (1)
This study is an extension of the work of S. Amrane [17], in which a 443
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modelisation approach that can be used to tackle the various identified drawbacks.
three-part 1D model of a thermocouple excited by a current was designed. This model comprised three coupled equations: one equation for each wire (see eq. (2)) and one equation for the junction (eq. (3)).
3.2. A new approach
∂ 2T hP = ki 2i − i i (Ti − T∞ ) ⋯ ∂ x i S
∂T ρi ci i ∂ t Heat accumulation
Conduction
⋯−
The aim of this new study is to improve all those aspects for the purpose of generalization and to obtain a flexible and continuous model. We will discuss the thermoelectric phenomena in more detail (the Joule, Peltier, and Thomson effects). The Joule effect can be expressed as a function of the cross section, material length, current intensity, and electrical resistivity, which varies linearly with the temperature with respect to the references Tref ref and ρelec . It can be written as follows:
Convection
σB εi Pi 4 4 (Ti − Trad ) + ρelec, i J 2 i S
Joule effect
Radiation
(2)
In eq. (2), the i index represents the various parts of the thermocouple (wire 1 and wire 2). At the junction (j index), the continuity between these two equations is done via eq. (3). ∂Tj
ρj cj Vj⋅ ∂t Heat accumulation
=
∂T
−k1 S1⋅ ∂x1 x =− a Heat conduction from wire 1
+
∂T
k2 S2⋅ ∂x2 =+ x a
Radiation
Joule effect =
Peltier effect
Heat conduction from wire 2 4 σB ε j Sj⋅(T j4 − Trad ) +
⋯ − hj Sj⋅(Tj − T∞) − Convection
− σ12 I ⋅Tj ⋯
ref ρelec L
S
[1 + b (T − Tref )]⋅I 2
[W]
(4)
The Thomson effect is usually neglected — as is the case in the previous model (eqs. (2) and (3)) — but the study of Titov on semiconductors [29] shows that it should be taken into account. This effect, in contrast to the other two thermoelectrical effects (Seebeck and Peltier), does not require the presence of a junction. The Thomson effect is observed when a temperature gradient is present or when a current is passed through the material. This implies that a heat absorption or rejection occurs if a current is passed through the conductor material, or an electromotive force is produced if the conductor material has a temperature gradient across it.
Rj⋅I 2
⏟
Joule effect
(3) The phenomena described by eq. (2) and eq. (3) are considered to be unidimensional in the x direction. Considering a 2D/3D model will assume that a significant thermal gradient could exist inside the wires (in the radial direction if we work in cylindrical coordinates). This assumption could be easily rejected by calculating the Biot Number Bi ≈ 5.4⋅10−4 ≪ 1. The high thermal conductivity (k) of the wires and the junction, combined with the small size of the device (80 μm) will induced an homogeneous temperature in the wire section even for large values of the heat transfer coefficient (h). As can be seen, besides the Joule effect, no thermoelectrical effect is taken into account for the wires (except the Joule one), and the Peltier effect is only localized at the junction. The radiation heat exchange occurs between the surface at Ti and the surrounding at Trad . It should be noted that - in the model - T∞ (fluid temperature) differs from Trad . However during the experiments (see below) the test bench is IR-shielded so Trad = T∞. Fig. 5 shows the locations of the various effects that occur in the thermocouple. The problem encountered with this model is detailed in the subsection 3.2 To resume the main drawbacks are:
Thomson effect = −τ ∇TJ
[W. m−3]
(5)
Where J is the current density and τ is the Thomson coefficient, which is related to the Seebeck coefficient σ.
τ = −T
dσ dT
(6)
Intuitively, if the Seebeck coefficient evolution is more or less independent of the temperature or it changes slightly then the Thomson effect can be neglected. The Thomson and Peltier effects are linked and their individual terms can be combined to form one term that expresses the two effects as a function of the position (i.e. x in our study) [30,31]. If eq. (5) is integrated between the junction borders then the obtained result will be the Peltier effect expression used in eq. (3). Thus, using one equation allows us to take into consideration the two effects, and the position is the only factor that influences the type of thermoelectric effect that occurs.
1. Three different equations for one system 2. No Thomson effect in the wires 3. The model could not be used for more complex cases (such as 2 junctions or 2 wires with different diameters) 4. The junction is considered as homogeneous (a single point at the junction)
Peltier Thomson = −T
In order to enhance the previous study, we propose a new
∂σ J ∂x
[W. m−3]
(7)
With eq. (7), the two thermoelectric effects will be taken into
Fig. 5. Effects taken into account in various zones of the thermocouple for the previous model (adapted from Ref. [20]). 444
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account in the model, and it assumes that the value of the Seebeck coefficient along the thermocouple is known (see 3.2.2).
conditions for a current injection of 14 mA and an ambient temperature of 19.45°C.
3.2.1. Governing equation The lack of continuity at the junction induces to introduce the thermoelectrical effects everywhere (i.e. wires and junction) and meanwhile to consider the evolution of the different parameters along the x-axis. Such approach leads to a single equation model (eq. (8)).
3.2.4. Heat transfer coefficient Besides the transition at the junction, the other problem is the determination of the natural convection. Modelling natural convection is difficult because the correlations are dependent on the temperature difference between the surface of the object and the ambient temperature. However, the aim of this study is to determine this temperature difference, and hence, to find the value of the heat transfer coefficient, an assumption of the temperature drop at the beginning of the model has to be made. The difficulty faced in this case is that the correlations for small devices like microthermocouples do not fit well, as in Ref. [35], the difference between the experimental measurement and the correlation is approximately 24% for a K-type thermocouple of 50 μm. In the present work, the phenomena could be considered as an inverse plume effect (i.e. cold wire in a hot fluid), while the reverse has been studied much more (i.e. hot wire in a colder fluid). Usually, the heat transfer coefficient is determined from a finite element study on the flow [36] or experimental measurement [37,38]. The lack of data caused the authors to use the standard Churchill and Chu correlation [39] for free convection cited by Ref. [40].
ρcS
ρ ∂T ∂ ⎛ ∂T ⎞ ∂σ I 4 kS ) + elec I 2 − T = − hP (T − T∞) − σB εP (T 4 − Trad S ∂t ∂x ⎝ ∂x ⎠ ∂x S
(8)
In eq. (8), all the bold symbols (such as k , S , P , ρ , c …) may vary along the x-axis in order to represent the change in material (i.e. wire, junction, and wire). By applying this equation along the geometry created for the model, at the junction, the equation will take into account the Peltier effect because of the difference in the Seebeck coefficient. In our application the ambient temperature T∞ and the radiative ambiance Trad are considered to be equal. Therefore for the temperature range used in the present study the radiation exchange is negligible compare to convection (∼ 276 W. m−2 for convection vs. ∼ 9 W. m−2 for radiation. Heat flux calculated with the parameters from Table 2). 3.2.2. Transition at the junction This formulation of the heat equation (see eq (8)) is in accordance with the theoretical work of Apertet [32]. The distinguishing feature of this model is that all the characteristics of the thermocouple are expressed as a function of the wire length. To obtain the best continuity possible, the functions chosen to translate the evolution of the characteristics are smoothed splines created with five points: two points at the very tip of the thermocouple wire, two points at the end of the junction length, and one point at the centre of the junction. The problem with this scheme is the type of transition that is applied at the junction. In fact, in this study, a transitional homogeneous scheme was designed. This is why all the characteristics at the junction are averaged as a function of the volume ratio between the two species according to [33] i.e. assuming 50% constantan and 50% chromel in our case. Therefore the characteristics of the junction can be deduced from those of the wires as shown on Table 2. However, the Seebeck coefficient is the only characteristic that uses another law of averaging, i.e. the Nordheim–Gorter law [34].
σj =
σ1. ρelec,1 + σ2. ρelec,2 ρelec,1 + ρelec,2
2
⎧ ⎪ 1 1 ⎪ 0.387⋅Gr 6 ⋅Pr 6 Nu = 0.6 + 8 ⎨ 9 27 ⎡1 + 0.559 16 ⎤ ⎪ Pr ⎪ ⎢ ⎥ ⎣ ⎦ ⎩
( )
4. Model implementation This part deals with special procedures that have been used to implement the model from eq. (8). Attention is first focused on the determination of the heat transfer coefficient and then on the numerical scheme. Finally, the following two models are compared:
[V . K−1] (9)
• The previous model from Amrane (eq. (2) and eq. (3)) [20]. • The new model described by eq. 8
3.2.3. Boundary conditions With respect to the establishment of the boundary conditions, in this study, two types of boundary conditions could be considered:
Table 2 Material characteristics of the chromel and constantan [1]. The characteristics of the junction are dedicated from these (mean value) except for the Seebeck coefficient σ which is determined via the Nordheim–Gorter law.
1. The extremities are insulated:
Symbol
=0 x =±L
Units
(10)
2. The temperatures at the extremities are imposed:
T x =±L = T∞
(12)
Where Nu , Gr , and Pr are the commonly used dimensionless Nusselt, Grashof, and Prandtl numbers, respectively. Equation (12) is valid for a wide range of Rayleigh (Ra = Gr⋅Pr ) As the weld is linear, the junction is cylindrical (as shown on Fig. 5), and therefore, eq. (12) is used all along the thermocouple.
Actually, this law averages the Seebeck coefficient of the coefficient (σj ) with the electrical resistivity of each wire ( ρelec, i ) and not the volume. The averaged value of σj = −2.67 × 10−6 V. K−1 is used along the junction as can be observed in Fig. 6.
∂T ∂x
⎫ ⎪ ⎪ with Gr⋅Pr = Ra⩽1012 ⎬ ⎪ ⎪ ⎭
(11)
Ni 90%/Cr 10%
Ni 50%/Cu 45%
Junction
ρ
kg.m−3
8730
8920
8825
J.kg−1. K−1
448
394
421
k
W.m−1. K−1 m .m
19.26
21.19
20.22
5 × 10−2 7.060 × 10−7
5 × 10−2 4.890 × 10−7
– 5.975 × 10−7
K−1 – m
4.1 × 10−4
−1. × 10−5
2.381 × 10−4
0.45 80 × 10−6 22 × 10−6
0.87 80 × 10−6 −38 × 10−6
0.66 80 × 10−6 −2.67 × 10−6
ref ρelec
b ε d σ
445
Constantan
c L
It should be noted that these boundary conditions have no impact on the result near the junction when an infinite thermocouple is considered (length of 10 cm for a diameter of 80 μm). These boundary conditions modify the temperature drop when the thermocouple length is less than 3 cm. Table 1 shows the difference found in the temperature drop according to the thermocouple length and the boundary
Chromel
V.K−1
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Fig. 6. Seebeck transition at the junction using the Nodheim–Gorter law for the Seebeck coefficient average.
4.1. Heat transfer coefficient determination Table 1 Impact of the boundary conditions on the relative deviation of the temperature at the junction (i.e. coolest point). Current injection = 14 mA, and ambient temperature = 19.45°C. Thermocouple length
Relative deviation
[m] 0.01 0.02 0.03 more than 0.04
% 0.5 0.3 0.01 0
In free convection, the heat transfer coefficient can be determined via the correlation in eq. (12) only if an evaluation of the temperature difference between the wire and the surroundings is given and is used to calculate the Grashof number (Gr ). Amrane et al. [20] have developed an analytical equation (see eq. (13)) that could be used to predict, in a steady state, the temperature drop at the junction (ΔTj = Tj − T∞) for a thermocouple through which a current of density J is passed.
J2 ΔTj =
S Ph
ref ref (ρelec ,1 k1 + ρelec,2 k2 ) − σ12 JT∞ Ph S
( k1 +
k2 ) − σ12 J
(13)
Fig. 7. Heat transfer coefficient versus the temperature drop at the junction for various diameters. Each curve corresponds to a range in current between 10 mA and 60 mA.
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to speed up the resolution, the TriDiagonal Matrix Algorithm (TDMA) [44] is implemented in Cython [45]. A 30000-point grid with a 100point grid at the junction and 2000 iterations over time is calculated. Owing to the Cython implementation the calculation time reduces from 20 min (with standard Python) to 3 min depending on the chosen time step.
To solve the non-linear circular dependency between eq. (13) and eq. (12) (the Gr number needs the ΔTj and Nu number the h), an iteration procedure (fixed point algorithm) is used to determine the heat transfer coefficient as a function of the temperature drop [20]. Convergence is commonly achieved in less than 10 iterations. As can be observed in Fig. 7, the range of the temperature difference is between 0.1°C and 1°C, and on considering this large range, the convergence procedure gives a heat transfer coefficient variation from 133 to 144 W.m−2. K−1. Using these values in the numerical model for a current amplitude of 10 mA, the temperature drop at the junction varies from 1.16 to 1.13°C. Therefore, a change of 8.27% in the heat transfer coefficient induces a change of only 2.82% in the temperature drop. This example shows that the model is only slightly sensitive to the heat transfer coefficient.
4.3. Comparison of models The first step of this study is to partially validate the new model by comparing the results with the older model [20]. According to [1], the multiphysics characteristics of the chromel and constantan are presented in Table 2. The Fig. 8 shows the comparison between the results obtained with two models at the junction with the previous experimental data. The comparison was made by applying the same multiphysical characteristics to the two models and the geometrical parameters of the older model into the new one. As can be observed, the final value is the same, which demonstrates how the new model can be easily modified. At the junction, the previous model uses a volume implementation that cannot be changed, whereas the new model depends only on the diameter value that is imposed. Both models provide results faster than the experiments but we have neglected the hypothesis that the delay observed is an experimental issue. During these experiments, another thermocouple (much thinner than the one studied i.e. of 7 μm in diameter) is glued at the top of the junction. The glue may induce an additional thermal resistance that delays the temperature variation; the use of an IR Camera aids in tackling this problem (see section 5). The second part of this partial validation consists of the general temperature profile — in steady state — along the thermocouple. The temperature profile is obtained by three different models:
4.2. Numerical scheme and simulation software The governing differential equation (eq. (8)) subject to the boundary conditions outlined in eq. (10) in 11 are solved using a finite difference approach with a fully implicit scheme. To carry out the proper discretization, all the properties are evaluated at midpoints. An example of the discretized version of the conduction differential for three nodes (i.e. i − 1, i, and i + 1) is as follows:
k −S −Ti − 1 − (ki−Si− + ki+Si+) Ti + ki+Si+Ti + 1 ∂ ⎛ ∂T ⎞ kS ⇒ i i Δx 2 ∂x ⎝ ∂x ⎠
(14)
where ki± , Si± are mean values between 2 adjacent nodes:
ki± =
ki ± 1 + ki Si ± 1 + Si and resp. Si± = 2 2
(15)
This procedure has been applied for all the parameters that may change along the x-axis. The difference between this new thermocouple model and the previous one is the fact that all the characteristics change along the x-axis (see eqs. (14) and (15)), and the junction is now considered as a transition. Therefore, the model is now continuous over the entire thermocouple length. The simulation has been performed on a standard Linux PC with Python. Python is a high-level language for numerical computations and has the ability to manipulate matrices and plot data using the Numpy [41,42], Scipy [43], and Matplotlib libraries. In order
1. the numerical model from Amrane (see eqs. (2) and (3)) [20]. 2. an analytical model developed by Amrane [20]. 3. the new numerical model (see eq. (8)) As shown in Fig. 8, no difference is observed between the results of the previous and the new models. The difference between the analytical
Fig. 8. Comparison between the previous and new models with the previous experimental data from Amrane [20]. Current injection starts at time t = 0 (i.e. step current). 447
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results and the model is caused by the capacitance term at the junction, which is absent in the analytical equations. The only difference that can be observed between the two models occurs at the junction. In fact, the previous model calculates only one point at the junction whereas the new model calculates various points because of the transition at the junction; therefore, there is a difference in temperature levels at the junction as shown in the zoomed-in Fig. 9.
Table 3 Comparison between the model and the experimental data at steady state. Current
Model temperature
Experimental temperature
Sparse
[mA]
decrease (eq. 8) [ °C]
decrease [°C]
[%]
10 20 26
−0.96 −1.38 −1.22
−0.95 −1.39 −1.21
1.02 0.7 0.8
5. Experimental results and model validation The results obtained with the model match well the experimental data, even the dynamic reduction in the temperature matches the predicted values calculated using the model. An observation that can be made is that the previous experimental data were delayed because of the thermal inertia of the glue between the two thermocouples [20] as can be observed in Fig. 8. After the comparison at 10 mA, where the temperature decrease approaches an exponential value, a comparison at 20 mA and 26 mA is of great interest. On reaching a certain current amplitude, a phenomenon of over-cooling is observed, but after a certain time, the temperature increases (Fig. 10b and c). As the junction cools down owing to the Peltier effect, the Joule effect in the wires tends to heat them up. The thermal gradient between the junction and the wires promotes heat fluxes by conduction and the junction temperature thus rises. The final temperature value reached at the steady state is a result of a balance between the thermoelectric effects and heat transfer phenomena (convection and conduction). The model can thus predict the cooling that occurs at the junction and is validated by the experimental data. The third part of the study concerns the validation of the model when the thermocouple is subjected to a variable current.
5.1. Introduction A series of experiments were performed with various parameters to facilitate the validation of the model. The programmable generator generated four current profiles. During the experimental acquisition, various frequencies from 0.2 Hz to 1 Hz with current amplitudes from 10 mA to 26 mA were generated. The aim had been to determine if the model can react, from a dynamic point of view, like a thermocouple through which a current is passed. 5.2. Steady state validation The results for the steady state validation are presented in Table 3. It can be seen that the results obtained using the model and the experimental results are similar and that the corresponding error is approximately 1%. In the case of the steady state, the model is accurate even for different profiles of current amplitude. 5.3. Step current validation The goal of this study is to create a model that can dynamically responds as an experimental thermocouple. Moreover, this will result in better comprehension and knowledge regarding the various phenomena that occur at the junction of two different materials. The comparison results for a step current of 10 mA, 20 mA, and 26 mA between the model and the experimental data is shown in Fig. 10.
6. Variable current model validation The Table 4 shows the various currents injected with their frequencies and amplitudes.
Fig. 9. Evolution of the temperature in steady state as a function of the length: comparison between the new and the previous models and the analytical values. A zoom located at the junction x = 0 is done to show the little differences between the two models.
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Fig. 10. Thermocouple junction temperature versus time for a step current of 10 (a), 20 (b) and 26 mA (c). Comparison between the model and the experimental data for three different currents.
The next figures (from Figs. 11 to 13) show the comparison between the experimental data and the thermocouple model on applying the current variation shown in Table 4.
6.2. Square current For the square current the sparse between the model and the data at 1 Hz is approximately 17%, which is close to that in the case of the sinus current (20%). For high frequencies, the model underestimates or overestimates the cooling realized. The reason for this can be explained by the cutting frequency, which is not well handed by the 1D model. However, the square current temperature evolution with a high current amplitude injection is better reproduced by the model. In the case of the square model, the dynamic is good but the errors in the values are approximately 5%.
6.1. Sinus current The model handles the variation in temperature well when the current is injected at a low frequency. The sparse is the most important at a high frequency, and thus, at 1 Hz, the relative sparse is approximately 20% against 1% at 0.2 Hz. This result could be compared to that obtained in the study of Rojo [46], which shows similar results and a frequency analysis in nanowires. Moreover, at a high current, the data demonstrates a phase in which the temperature increase is staged. The model shows the same phenomena, but in a much more smoothed manner, such that the sparse at the temperature levels is not high but the dynamic sparse is more important in this case.
6.3. Sawtooth current For the sawtooth current, the same conclusions can be obtained as that of the preceding current type. At a high frequency, the relative sparse is important, for 1 Hz the sparse is about 23% as against 3% at 449
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Fig. 11. Sinus current model validation. Comparison between the model and experimental data for a sinus current injection.
The other causes that can be taken into account include, first of all, the acquisition quality; the IR camera is really sensitive with respect to its surroundings. Secondly, the heat transfer coefficient could be the cause of this temperature difference; at high cooling and heating rates, hot and cold air layers may be created around the junction thus varying the heat transfer coefficient. In conclusion, owing to the fact that the model is simplified by only one equation and that it is easy to modulate it, the comparison of its results with the experimental data was quite simple to perform. Furthermore, the maximum sparse makes the model valid for the dynamic response for various types of current. Moreover, the sparse at 1 Hz can be reduced by performing some tests under vacuum, in which case, the convection can be ignored, and it can be determined if the various air layers around the junction are responsible for this temperature difference.
0.2 Hz. As can be observed, the model reacts pretty well depending on the signal injected. The sparse between the model and experimental data is caused by the different assumptions made for the characteristics and the transition chosen at the junction. The transition at the junction is based on an assumption of averaged homogeneous metals but the reality is different; after the electrical weld, the junction between the two metals is not perfectly homogeneous, and thus, the heterogeneity of the junction can play a role on the cooling factor and on the transient response of the thermocouple. However, the determination of the heterogeneity of the junction is complicated and involves probabilistic equations [47]. A study performed to investigate the influence of the difference in the characteristics found that it causes an error of only 0.3%, such that, the variation of the thermophysical characteristics can be neglected in the present study.
Fig. 12. Square current model validation. Comparison between the model and experimental data for a square current injection. 450
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Fig. 13. Sawtooth current model validation. Comparison between the model and the experimental data for a sawtooth current injection.
for exploration. The perspectives of this study are to extend the Peltier effect and to use it for sensor applications.
Table 4 Experimental tests. Type of current
Sinus Square Sawtooth
Frequency/Amplitude
Frequency/Amplitude
[Hz/mA]
[Hz/mA]
0.2/10 0.2/26 0.2/10
1/10 1/10 1/10
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7. Conclusion and perspectives In this study, a transient thermoelectrical model that can be applied to thermocouples was designed. The general purpose of this model is to create a generalized model to develop new type of sensor based on the Peltier effect. To generalize the model, various thermal effects such as conduction, convection, and radiation as well as the thermoelectrical effects like the Joule effect, Thomson effect, and Peltier effect are taken into consideration. The specificity of this model is that only one governing equation is necessary to translate all the effects in the thermocouple, thus causing the model to be continuous. The main objective is to create a reliable model, from a transient point of view, that can provide a frequency analysis of the thermocouple response. Such analysis in a complex environment (such as an alternating flow) will allow to determine if some information regarding temperature, or velocity can be obtained. Another application of the model could be the simulation of a thermocouple with several junctions. As a matter of fact a sensor that induces a temperature gradient might be a reliable candidate for a heat flux sensor. Such device should be carefully designed via modelisation - to obtain the maximal temperature difference between the cold and hot junction. The model validation was carried out using an 80 μm E-type thermocouple through which current generated by a programmable tension generator was passed. The data was acquired by using an IR camera and then compared with the result obtained with the model. The validation is carried out for two cases: the first validation is carried out at steady state conditions, for which the model provides accurate results and the error is approximately 1%. The second model validation is carried out at transient conditions. At 1 Hz, the relative error is approximately 20%, whereas at 0.2 Hz, it is approximately 1%. In conclusion, this model is accurate at low frequencies and demonstrates great potential 451
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