# One-Dimensional Systemic Modeling of Thermal Sensors Based on Miniature Bead-Type Thermistors

## Abstract

**:**

## 1. Introduction

- the steady-state methods, such as the guarded hot plate and the heated thermocouple methods.

- the self-heating mode [10], in which the sensitive element serves as both a heat source and a temperature sensor. The basic principle of this mode of operation is that the less heat conducting the surrounding materials to be characterized, the greater the self-heating of the sensor element. Thus, the average temperature ${T}_{c}\left(t\right)$ of the sensor and the electrical power ${P}_{e}\left(t\right)$ it dissipates are, in most situations, the basic informative signals considered in electro-thermal methods used for thermal characterization.

## 2. Materials and Circuits

#### 2.1. Glycerol

#### 2.2. NTC Bead-Type Thermistor

#### 2.2.1. Composition—Electrical Properties

- Radiative heat losses ${P}_{r}$ at the external surface of the NTC: These losses can be estimated using Stefan–Boltzmann’s law: ${P}_{r}=\sigma \epsilon {\mathrm{\Sigma}}_{e}({T}_{e}^{4}-{T}_{0}^{4})\approx 4\sigma \epsilon {\mathrm{\Sigma}}_{e}{T}_{0}^{3}\left({T}_{e}-{T}_{0}\right)$. Under the working conditions presented above, and considering a maximal emissivity $\epsilon =1$, we obtain ${P}_{r}\approx 0.1$ mW. Therefore, it is justified to neglect ${P}_{r}$ in front of ${P}_{e}$. Under the typical working conditions described in this work, it can be assumed that radiation losses have a negligible influence on the thermal characterization of materials using a miniature bead-type thermistor.
- Thermal losses ${P}_{L}$ at the electrical contacts between the thermistor and the control circuit: It is difficult to precisely estimate these heat losses because they are closely linked to the nature of the experimental system implemented: electrical insulation of the connecting wires (using a varnish or silicone) and length L of the connecting wires; use of metal or plastic mounting probe.Under the operating conditions described above, and for varnish insulated copper wires, with a diameter ${d}_{w}=0.18$ mm and a length of $L=1$ cm, immersed in pure glycerol at rest, a finite element modeling has led to heat losses typically in the order of 3 mW.Therefore, heat losses via the electrical connections are considerable and can in no way be neglected in the modeling of the thermistor.
- Convective losses ${P}_{c}$ when the material to characterize is a fluid: Convective losses always lead to an additional heat extraction from the sensor and are at the origin of an overestimation in the measurement of the conductivity of the fluid to be characterized. Therefore, it is essential to limit these convective losses if we want to correctly estimate the thermal conductivity of the fluid to be characterized. This requirement dictated the choice of glycerol as a test liquid for 1D systemic modeling of NTCs. The thermal power evacuated from the thermistor to the fluid can be evaluated in the presence of natural convection, from the Churchill correlation [33], which gives the following Nusselt number expression, valid for natural convection around a sphere:$$\mathrm{Nu}=2+\frac{0.589\phantom{\rule{0.166667em}{0ex}}{\mathrm{Ra}}_{{d}_{e}}^{1/4}}{{\left[1+{\left(0.469/\mathrm{Pr}\right)}^{9/16}\right]}^{4/9}},$$Using the values of Table 1 for $\Delta {T}_{e}=0.8\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$, we find in the case of glycerol that $\mathrm{Nu}\approx 2.5$, which remains close to the value obtained in the absence of natural convection, and ${P}_{c}=0.8\phantom{\rule{0.166667em}{0ex}}\mathrm{mW}$, which is negligible compared to ${P}_{e}$. It can be concluded that it is acceptable, for temperature variations $\Delta {T}_{e}\u2a7d1.0\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$ near room temperature, to neglect the contribution of natural convection around the NTC in the case of immersion in glycerol.In contrast, in the case of NTC immersion in water, considering a temperature difference $\Delta {T}_{e}=0.5\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$ (thermistor heating is less here than in the case of glycerol) near room temperature, we find that $\mathrm{Nu}\approx 3.7$, which is quite different from the values obtained in the absence of natural convection, and ${P}_{c}=3.8\phantom{\rule{0.166667em}{0ex}}\mathrm{mW}$, which can no longer be ignored.
- Poor thermal contacts: There are two main sources of poor thermal contact here: the thermal contacts between the active core of the NTC and its protective sheath and between the sheath and the surrounding medium to be characterized. If the latter is a fluid (which is the case in the present study), we can assume that the corresponding sheath/fluid thermal contact resistance is negligible. In contrast, the contact resistance ${R}_{c}$ between the sheath and the core is potentially significant, unknown and a priori different from one sensor to another and will be taken into account in the 1D systemic modeling proposed in this study.

#### 2.2.2. Heat Transfer through NTC and Surrounding Medium

#### 2.2.3. Mathematical Modeling

- (f) is an inert (non-biological) material at rest, with no heat source term. In this case, its temperature ${T}_{f}(r,t)$ also obeys a diffuse heat transfer equation, with spherical symmetry:$$\frac{1}{{r}^{2}}\frac{\partial}{\partial r}\left[{r}^{2}\frac{\partial}{\partial r}{T}_{f}(r,t)\right]=\frac{1}{{\alpha}_{f}}\frac{\partial}{\partial t}{T}_{f}(r,t)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{a}_{s}<r<{a}_{f},$$
- (f) is a biological material, which obeys a differential equation of the Penne type (bioheat transfer equation), for example [37,38]:$$\frac{1}{{r}^{2}}\frac{\partial}{\partial r}\left[{r}^{2}\frac{\partial}{\partial r}{T}_{f}(r,t)\right]+\frac{{\dot{q}}_{f}}{{k}_{f}}+\frac{\omega {\rho}_{b}{c}_{b}}{{k}_{f}}\left({T}_{0}-{T}_{f}\right)=\frac{1}{{\alpha}_{f}}\frac{\partial}{\partial t}{T}_{f}(r,t)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{a}_{s}<r<{a}_{f},$$The initial condition to be considered for the partial differential Equation (8) or (9) is ${T}_{f}(r,0)={T}_{0}$. The boundary condition to consider at $r={a}_{f}$ depends on the thermal characterization apparatus that is used. We suppose here that the medium to characterize is in perfect thermal contact at $r={a}_{f}$ with a thermostat at the temperature ${T}_{0}$:$${T}_{f}({a}_{f},t)={T}_{0}.$$While both situations (8) and (9) can be studied in the same way with the systemic modeling presented in this work, it is the (8) case that will be presented here in detail, both from a numerical and an experimental point of view.Finally, a perfect thermal contact is assumed between the thermistor sheath and the medium to be characterized (fluid at $r={a}_{s}$), the following continuity relationships must then be satisfied at all times:$${T}_{s}({a}_{s},t)={T}_{f}({a}_{s},t)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{and}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{k}_{s}{\left(\frac{\partial {T}_{s}}{\partial r}\right)}_{r={a}_{s}}={k}_{f}{\left(\frac{\partial {T}_{f}}{\partial r}\right)}_{r={a}_{s}}.$$

#### 2.3. Electronic Circuit

- Since the temperature variations due to self-heating are generally of small amplitude (a few Kelvins at most), it is necessary to use an analog-to-digital converter (ADC) with a resolution of at least 14 bits. Moreover, as thermal phenomena are generally relatively slow (with time constants typically in the order of a few tenths of a second or more), a sampling period of around 10 ms or more is, therefore, sufficient in most cases. A digital-to-analog converter (DAC) with a resolution of 12 bits or better can be used to provide the excitation signal ${v}_{e}\left(t\right)$ for transient methods. If the sensing element is to be excited in current rather than voltage, a voltage-to-current converter (of the Howland source type) can be used instead of the non-inverting amplifier shown in Figure 4.
- The inputs ADC0 and ADC1 must have a sufficiently large input impedance ${Z}_{e}$ (typically, ${Z}_{e}>{10}^{5}\phantom{\rule{0.166667em}{0ex}}\Omega $. In the case of USB-2537: ${Z}_{e}={10}^{7}\phantom{\rule{0.166667em}{0ex}}\Omega $) to not load the circuit. If necessary, a high impedance buffer (voltage follower) can be used to isolate the control circuit from the influence of the ADC.
- The operational amplifiers (OA) used in circuit Figure 4 must operate in the linear regime. If the self-heating of the sensor requires an electric current with an intensity $i\left(t\right)>10$ mA, then, the use of an operational amplifier capable of delivering high currents should be considered. This could be the case, for example, with low-resistance platinum wires, whose value is close to $1\phantom{\rule{0.166667em}{0ex}}\Omega $. In this case, a typical average electrical power ${P}_{e}=10$ mW requires about 100 mA electrical excitation current.To make the set-up as versatile as possible, a power OA of type L272 (delivering currents up to 1 A without significant harmonic distortion) was systematically used.
- The working (or baseline) temperature ${T}_{0}$ must be precisely regulated, usually by means of a temperature controlled bath. The variations $\Delta {T}_{0}$ of the working temperature must be negligible in front of the maximal temporal variations of the sensor core temperature $\delta {T}_{c}\left(t\right)$, due to self-heating. An accuracy of $\Delta {T}_{0}\approx 0.01\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$ is sufficient in most cases.

## 3. 1D Systemic Modeling

#### 3.1. General Approach

#### 3.2. NTC Bead-Type 1D Systemic Modeling

#### 3.2.1. Ideal NTC Bead-Type 1D Systemic Modeling

#### 3.2.2. Realistic NTC Bead-Type 1D Systemic Modeling

## 4. Results

#### 4.1. Presentation

- the model does not require the knowledge of the active core thermal conductivity ${k}_{c}$ because the core temperature ${T}_{c}$ is assumed to be uniform (${k}_{c}\gg {k}_{s}$);
- the values of ${k}_{s}$ and ${\rho}_{s}{c}_{s}$ shown in Table 3 are compatible with usual epoxy values; and
- the value of the shell external radius ${a}_{s}$ was determined using a caliper. The NTC being a prolate spheroid, an average value was considered.

#### 4.2. Glycerol Thermal Characterization

#### 4.2.1. Immersion Length Tests

#### 4.2.2. Thermal Power Balance and Characteristics Times

- when $0\u2a7dt<{t}_{c1}$, time domain (I): only the thermal current density ${i}_{C}^{\mathrm{th}}$ is significant, and the thermal power supplied by the electrical control circuit mainly serves here to raise the temperature ${T}_{c}$ of the NTC active core. This time domain cannot be used to characterize the fluid surrounding the NTC because the evolution of ${T}_{c}\left(t\right)$ between 0 and ${t}_{c1}$ is mainly influenced here by the physical properties of the active core and not by the physical properties of the surrounding fluid to be characterized, that is not yet probed by the thermal waves emitted by the NTC. In addition, we can see in Figure 15a that the curves giving the experimental temporal evolution of ${T}_{c}$ coincide with $t\in [0,{t}_{c1})$ and this, whatever the value of ${L}_{i}$ and, thus, of ${R}_{L}$. This observation confirms that the evolution of ${T}_{c}\left(t\right)$ over the interval $[0,{t}_{c1})$ mainly reflects the physical properties of the thermistor active core only.
- When ${t}_{c1}<t<{t}_{c2}$, time domain (II): in the case of Figure 16, the three thermal current densities ${i}_{C}^{\mathrm{th}}$, ${i}_{L}^{\mathrm{th}}$, and ${i}_{S+F+{T}_{0}}^{\mathrm{th}}$ have comparable values; therefore, the thermal characterization of the surrounding fluid using the time evolution of the core temperature will be influenced by both the properties of the core, of the insulating shell and by the thermal losses via the connecting wires. Given these various influences, the usual transient methods of thermal characterization (which do not use systemic modeling) based on bead-type NTCs should avoid the use of domain (II) data.
- When $t>{t}_{c2}$, time domain (III): in this case, the thermal current density ${i}_{C}^{\mathrm{th}}$ has become negligible in front of ${i}_{L}^{\mathrm{th}}$ and ${i}_{S+F+{T}_{0}}^{\mathrm{th}}$, and it can be supposed that the properties of the core are then without any significant influence on the temporal evolution of its temperature ${T}_{c}$. Consequently, it is this time domain that should be used preferably for thermal characterization of materials by using bead-type NTCs self-heating methods. However, since thermal losses and the insulating sheath still have an influence on the temporal evolution of ${T}_{c}$ in the (III) domain, thermal characterization methods that do not use systemic modeling must imperatively resort to a prior calibration of the measuremen device by using several reference fluids, such as glycerol, ethanol, water–glycerol mixtures, and gelled water (using agar-agar, for example).Note that, in this work, the characteristic time ${t}_{c2}$ has been calculated using the following relation: ${i}_{C}^{\mathrm{th}}\left({t}_{c2}\right)\approx 3\xb7{10}^{-2}\times {i}_{C}^{\mathrm{th},\mathrm{max}}$, where ${i}_{C}^{\mathrm{th},\mathrm{max}}={i}_{C}^{\mathrm{th}}\left({0}_{+}\right)$.

#### 4.2.3. Constant Voltage Excitation Signal Processing

#### 4.3. Liquids Thermal Conductivity Measurements

#### 4.3.1. Electro-Thermal Systemic Modeling (ESM) Method

- Determination of the model parameters values ${a}_{s}$, ${a}_{c}$, ${\rho}_{c}{c}_{c}$, ${\rho}_{s}{c}_{s}$, ${k}_{s}$, ${R}_{c}$, and ${R}_{L}$ from the thermal characterization of pure glycerol at rest, when ${L}_{i}=0.0$ mm (see Section 4.2).
- Voltage step excitation of the NTC (using the circuit of Figure 4) precisely immersed at ${L}_{i}=0.0$ mm in the liquid to characterize, at constant working temperature, ${T}_{0}$. The experimental time variations of the NTC core temperature ${T}_{c}^{\mathrm{mea}}\left(t\right)$ were extracted from the ${v}_{0}\left(t\right)$ and ${v}_{1}\left(t\right)$ voltages, recorded using a data acquisition board (see Section 2.3).
- Determination of the measured thermal conductivity value ${k}_{f}^{\mathrm{mea}}$ by minimization of $\delta {T}_{c}^{\mathrm{rms}}$, given by Equation (21), as a function of the thermal conductivity value ${k}_{f}$ used in the systemic model. The values of the fluid density ${\rho}_{f}$ and its specific heat ${c}_{f}$ are supposed known.

#### 4.3.2. Pure Water Liquid (100W0G)

#### 4.3.3. Glycerol–Water Mixtures 50W50G and 40W60G

#### Glycerol–Water Mixture 40W60G

#### Glycerol–Water Mixture 50W50G

#### 4.3.4. Synthesis

## 5. Concluding Remarks and Perspectives

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

HVAC | Heating, Ventilation, and Air Conditioning |

SPICE | Simulation Program with Integrated Circuit Emphasis |

THW | Transient Hot Wire |

THS | Transient Hot Strip |

TPS | Transient Plane Source |

NTC | Negative Temperature Coefficient |

HMW | Hot Metal Wire |

HMF | Hot Metal Film |

RTD | Resistance Temperature Detector |

ODE | Ordinary Differential Equation |

PDE | Partial Differential Equation |

ESM | Electro-thermal Systemic Modeling |

ADC | Analog to Digital Converter |

DAC | Digital to Analog Converter |

OA | Operational Amplifier |

BC | Boundary Condition |

CTHT | Constant Temperature Heating Technique |

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**Figure 1.**(

**a**) Precision miniature epoxy encapsulated bead-type 44004 NTC thermistor (2252@25) from Omega and (

**b**) its schematic composition. The length of the connecting copper wires is equal to $L=76$ mm in the present case.

**Figure 2.**Finite elements modeling of heat transfers through the multi-physics system: NTC + Fluid, with: (

**a**) a slab-shaped active-core; (

**b**) a spherical active core. In both cases, the same materials were considered for the sheath (epoxy, external radius ${a}_{s}=1$ mm), the active core (semiconductor material), and for the surrounding fluid (radius ${a}_{f}=5$ mm) to be characterized.

**Figure 3.**Ideal 1D effective model of a miniature bead-type NTC, immersed in a fluid at rest (with thermal conductivity ${k}_{f}$, density ${\rho}_{f}$, and specific heat ${c}_{f}$). The red curve represents effective variations of the $\delta T=T(r,t)-{T}_{0}$ temperature through the system, as a function of r, at a given time t.

**Figure 4.**Amplified divider bridge circuit used both to control the self-heating of the NTC bead-type thermistor and measure the time variations of its electrical resistance $R\left({T}_{c}\right)$. This circuit can be used in both transient and frequency modes. ADC0 and ADC1 are two inputs of an Analog to Digital Converter (ADC). The circuit can be excited either by using a Digital To Analog Converter (DAC) or by using an Arbitrary Waveforms Generator (AWG).

**Figure 5.**Spatial discretization of the spherical shell $(\Omega )=\left({a}_{1},{a}_{2}\right)$ into M slices of equal thickness ${\delta}_{r}=({a}_{2}-{a}_{1})/M$.

**Figure 6.**Electro-thermal modeling of a slice (m) of a material obeying the partial differential Equation (12).

**Figure 7.**(

**a**) Electro-thermal modeling of a shell subjected to Dirichlet’s boundary conditions, without heat volumetric source. (

**b**) Stationary temperature profile through the shell. (

**c**) Temperature variations as a function of time at different points throughout the shell.

**Figure 8.**Electro-thermal modeling of a shell subjected to Dirichlet boundary conditions, with a constant heat volumetric source. (

**a**) Steady state temperature profile through the shell. (

**b**) Temperature variations as a function of time at different points throughout the shell.

**Figure 9.**(

**a**) Electro-thermal modeling of a shell subjected to Dirichlet boundary condition ($\delta {T}_{1}=10\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$) at $r={a}_{1}$ and homogeneous Neumann condition at $r={a}_{2}$: $\Phi \left({a}_{2}\right)={\left({\partial}_{r}T\right)}_{r={a}_{2}}=0$. (

**b**) Steady state temperature profile through the shell. (

**c**) Temperature variations as a function of time at $r={a}_{2}$.

**Figure 10.**Electro-thermal modeling of bioheat transfer through a shell subjected to Dirichlet boundary conditions and constant heat volumetric source. (

**a**) Steady state temperature profile through the shell. (

**b**) Temperature variations as a function of time at $r={a}_{1}+{\delta}_{r}$.

**Figure 11.**One-dimensional electro-thermal systemic modeling of an ideal NTC active core with electrical resistance $R\left({T}_{c}\right)$ given by Equation (1) or (2). Recall that the effective radius ${a}_{c}$ is chosen so that the volume ${V}_{c}$ of the equivalent spherical core is equal to that of the slab.

**Figure 12.**One-dimensional electro-thermal systemic modeling of a bead-type 44004 NTC immersed into a liquid at rest.

**Figure 14.**Evolution of the active core temperature $\delta {T}_{c}\left(t\right)={T}_{c}\left(t\right)-{T}_{0}$ as a function of time t when the self-heated thermistor NTC1 is immersed with zero immersion length (${L}_{i}=0.0$ mm) in pure glycerol at rest, considering a working temperature ${T}_{0}=25.0{\phantom{\rule{0.166667em}{0ex}}}^{\circ}\mathrm{C}$ and constant excitation voltages (

**a**) ${v}_{0}=7.63\phantom{\rule{0.166667em}{0ex}}\mathrm{V}$ and (

**b**) ${v}_{0}=6.70\phantom{\rule{0.166667em}{0ex}}\mathrm{V}$. Measurements were conducted using the control circuit of Figure 4.

**Figure 15.**Influence of the immersion length ${L}_{i}$ on the self-heating of NTC3 immersed in glycerol at rest, at a working temperature ${T}_{0}=25.0{\phantom{\rule{0.166667em}{0ex}}}^{\circ}\mathrm{C}$. A constant excitation voltage ${v}_{0}=6.90$ V was used to heat the NTC. (

**a**) Active core temperature evolution as a function of time and length ${L}_{i}$. The solid black curves correspond to the results given by the 1D electro-thermal systemic modeling while the colored curves correspond to the experimental data. (

**b**) Evolution of the loss resistance ${R}_{L}$ as a function of the immersion length ${L}_{i}$.

**Figure 16.**Evolution of the different thermal current densities as a function of time $t\in [0,30]$ s, calculated at $r={a}_{c}$, in the case of NTC3 immersed in pure glycerol at rest, when: ${L}_{i}=5.0\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$, ${v}_{0}=6.90\phantom{\rule{0.166667em}{0ex}}\mathrm{V}$, and ${T}_{0}=25{\phantom{\rule{0.166667em}{0ex}}}^{\circ}\mathrm{C}$.

**Figure 17.**Signal processing of the experimental thermal signals obtained by constant voltage excitation in the case of NTC3 immersed in pure glycerol at rest, when ${L}_{i}=5.0\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$, ${v}_{0}=6.90\phantom{\rule{0.166667em}{0ex}}\mathrm{V}$, and ${T}_{0}=25{\phantom{\rule{0.166667em}{0ex}}}^{\circ}\mathrm{C}$: (

**a**) Kharalkar et al. signal processing; (

**b**) CTHT-type signal processing. Fitting curves (straight lines) were calculated using data belonging only to time domain (III).

**Figure 18.**Experimental and 1D ESM computed thermal signals in the case of pure water at rest. These signals were obtained using a voltage step excitation ${v}_{0}=8.46\phantom{\rule{0.166667em}{0ex}}\mathrm{V}$ at a working temperature ${T}_{0}=24.0{\phantom{\rule{0.166667em}{0ex}}}^{\circ}\mathrm{C}$ and immersion length ${L}_{i}=0.0\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$: (

**a**) measurements and electro-thermal systemic modeling results; (

**b**) computed thermal current densities when the fluid thermal conductivity ${k}_{f}$ is set to the value ${k}_{f}^{\mathrm{mea}}=0.586$ in the ESM model. Thermal conductivities ${k}_{f}$ are given in $\mathrm{W}\xb7{\mathrm{K}}^{{}^{-1}}\xb7{\mathrm{m}}^{-1}$.

**Figure 19.**Principle of the determination of pure water thermal conductivity ${k}_{f}^{\mathrm{mea}}$ by the present 1D electro-thermal systemic modeling (ESM) approach. The values of pure water density ${\rho}_{f}$ and specific heat ${c}_{f}$ at ${T}_{0}=24.0{\phantom{\rule{0.166667em}{0ex}}}^{\circ}\mathrm{C}$ used in the systemic modeling are given in Table 1.

**Figure 20.**Experimental and 1D ESM computed thermal signals in the case of water–glycerol mixture 40W60G at rest. These signals were obtained using a voltage step excitation ${v}_{0}=8.39\phantom{\rule{0.166667em}{0ex}}\mathrm{V}$ at a working temperature ${T}_{0}=24.0{\phantom{\rule{0.166667em}{0ex}}}^{\circ}\mathrm{C}$ and immersion length ${L}_{i}=0.0\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$: (

**a**) measurements and ESM results; (

**b**) computed thermal current densities when the fluid thermal conductivity ${k}_{f}$ is set to the value ${k}_{f}^{\mathrm{mea}}=0.384$ in the ESM model. Thermal conductivities ${k}_{f}$ are given in $\mathrm{W}\xb7{\mathrm{K}}^{{}^{-1}}\xb7{\mathrm{m}}^{-1}$.

**Figure 21.**Experimental and 1D ESM computed thermal signals in the case of water–glycerol mixture 50W50G at rest. These signals were obtained using a voltage step excitation ${v}_{0}=8.39\phantom{\rule{0.166667em}{0ex}}\mathrm{V}$ at a working temperature ${T}_{0}=22.0{\phantom{\rule{0.166667em}{0ex}}}^{\circ}\mathrm{C}$ and immersion length ${L}_{i}=0.0\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$: (

**a**) measurements and ESM results; (

**b**) computed thermal current densities when the fluid thermal conductivity ${k}_{f}$ is set to the value ${k}_{f}^{\mathrm{mea}}=414$ in the ESM model. Thermal conductivities ${k}_{f}$ are given in $\mathrm{W}\xb7{\mathrm{K}}^{{}^{-1}}\xb7{\mathrm{m}}^{-1}$.

**Table 1.**Typical physical properties values water (at 20 ${}^{\circ}\mathrm{C}$ and 24 ${}^{\circ}\mathrm{C}$, from reference [35]) at atmospheric pressure. The units of $\alpha $ and $\beta $ are given in the text.

$\mathbf{k}\phantom{\rule{0.166667em}{0ex}}(\mathbf{W}\xb7{\mathbf{m}}^{-1}\xb7{\mathbf{K}}^{-1})$ | $\mathit{\rho}\phantom{\rule{0.166667em}{0ex}}(\mathbf{kg}/{\mathbf{m}}^{3})$ | $\mathbf{c}\phantom{\rule{0.166667em}{0ex}}(\mathbf{kJ}\xb7{\mathbf{kg}}^{-1}\xb7{\mathbf{K}}^{-1})$ | $\mathit{\eta}\phantom{\rule{0.166667em}{0ex}}(\mathbf{Pa}\xb7\mathbf{s})$ | $\mathit{\alpha}$$/{10}^{-8}$ | $\mathit{\beta}$$/{10}^{-4}$ | |
---|---|---|---|---|---|---|

Water (20 ${}^{\circ}\mathrm{C}$) | $0.598$ | $998.2$ | $4.19$ | $1.00\times {10}^{-3}$ | $14.3$ | $2.1$ |

Water (24 ${}^{\circ}\mathrm{C}$) | $0.605$ | $997.3$ | $4.18$ | $9.11\times {10}^{-4}$ | $14.5$ | $2.5$ |

Thermal | Electro-Thermal |
---|---|

$\frac{k}{{\delta}_{r}}\left({T}_{m-1}^{n}-{T}_{m}^{n}\right)$ | ${i}_{m-\frac{1}{2}}^{\mathrm{th}}$ |

$\frac{k}{{\delta}_{r}}\left({T}_{m}^{n}-{T}_{m+1}^{n}\right)$ | ${i}_{m+\frac{1}{2}}^{\mathrm{th}}$ |

$\frac{k}{{r}_{m}}\left({T}_{m-1}^{n}-{T}_{m+1}^{n}\right)$ | ${i}_{m}^{\mathrm{th}}$ |

${\dot{q}}_{m}^{n}\delta r$ | ${i}_{{\dot{q}}_{m}}^{\mathrm{th}}$ |

${h}_{\omega}{\delta}_{r}\left({T}_{m}^{n}-{T}_{0}\right)$ | ${i}_{{h}_{\omega}m}^{\mathrm{th}}$ |

$\rho c{\delta}_{r}\frac{{T}_{m}^{n+1}-{T}_{m}^{n}}{\Delta {t}_{n}}$ | ${i}_{Cm}^{\mathrm{th}}$ |

**Table 3.**Optimal parameters values deduced from glycerol thermal characterization tests. The units of the two volumetric heat capacities ${\rho}_{c}{c}_{c}$ and ${\rho}_{s}{c}_{s}$ are given in $\mathrm{J}\xb7{\mathrm{K}}^{-1}\xb7{\mathrm{m}}^{-3}$.

${\mathbf{R}}_{\mathit{c}}\phantom{\rule{0.166667em}{0ex}}({\mathbf{m}}^{2}\xb7\mathbf{K}/\mathbf{W})$ | ${\mathit{\rho}}_{\mathbf{s}}{\mathbf{c}}_{\mathbf{s}}$ | ${\mathit{\rho}}_{\mathit{c}}{\mathbf{c}}_{\mathbf{c}}$ | ${\mathbf{a}}_{\mathbf{c}}\phantom{\rule{0.166667em}{0ex}}\left(\mathbf{mm}\right)$ | ${\mathbf{a}}_{\mathbf{s}}\phantom{\rule{0.166667em}{0ex}}\left(\mathbf{mm}\right)$ | ${\mathbf{k}}_{\mathbf{s}}\phantom{\rule{0.166667em}{0ex}}(\mathbf{W}\xb7{\mathbf{K}}^{-1}\xb7{\mathbf{m}}^{-1})$ |
---|---|---|---|---|---|

$3.0\times {10}^{-4}$ | $6.72\times {10}^{6}$ | $3.56\times {10}^{6}$ | $1.04$ | $1.17$ | $0.95$ |

**Table 4.**NTC3 self-heating 1D modeling results, when immersed at ${L}_{i}$ in glycerol at rest, at ${T}_{0}=25.0{\phantom{\rule{0.166667em}{0ex}}}^{\circ}\mathrm{C}$ and ${v}_{0}=6.90$ V. The thermal power current densities values (in $\mathrm{W}\xb7{\mathrm{m}}^{-2}$) were computed at ${t}_{f}=30\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$ and $r={a}_{c}$. The values of $\delta {T}_{c}^{\mathrm{rms}}$ were calculated from the results shown in Figure 15a.

${\mathbf{L}}_{\mathbf{i}}$ $\left(\mathbf{mm}\right)$ | ${\mathbf{R}}_{\mathbf{L}}\phantom{\rule{0.166667em}{0ex}}({\mathbf{m}}^{2}\xb7\mathbf{K}/\mathbf{W})$ | $\mathit{\delta}{\mathbf{T}}_{\mathbf{c}}^{\mathbf{rms}}\phantom{\rule{0.166667em}{0ex}}\left(\mathbf{mK}\right)$ | ${\mathbf{i}}_{{\dot{\mathbf{q}}}_{\mathbf{e}}}^{\mathbf{th}}$ | ${\mathbf{i}}_{\mathbf{L}}^{\mathbf{th}}$ | ${\mathbf{i}}_{\mathbf{C}}^{\mathbf{th}}$ | ${\mathbf{i}}_{\mathbf{S}+\mathbf{F}+{\mathbf{T}}_{0}}^{\mathbf{th}}$ |
---|---|---|---|---|---|---|

$0.0$ | $80.0\times {10}^{-4}$ | $18.3$ | $567.2$ | $144.4$ | $7.7$ | $415.1$ |

$0.5$ | $26.5\times {10}^{-4}$ | $19.6$ | $568.5$ | $289.4$ | $2.5$ | $276.6$ |

$5.0$ | $18.0\times {10}^{-4}$ | $18.6$ | $569.9$ | $345.8$ | $1.5$ | $222.6$ |

$10.0$ | $15.8\times {10}^{-4}$ | $24.0$ | $571.7$ | $365.3$ | $1.2$ | $205.2$ |

**Table 5.**Determination of the thermal conductivities of several liquids (pure water and water–glycerol mixtures) by the present 1D ESM approach, with: $\overline{k}$ the mean value of measurements, $\delta k={\sigma}_{k}/\sqrt{N-1}$ the standard error of measurement, ${\sigma}_{k}$ the standard deviation of measurements, and ${k}_{\mathrm{ref}}$ the reference thermal conductivity value (all these quantities are given in $\mathrm{W}\xb7{\mathrm{K}}^{{}^{-1}}\xb7{\mathrm{m}}^{-1}$).

Liquid | ${\mathbf{T}}_{0}$${(}^{\circ}\mathbf{C})$ | $\overline{\mathbf{k}}$ | $\mathit{\delta}\mathbf{k}$ | $\mathit{\delta}\mathbf{k}/\overline{\mathbf{k}}$ (%) | ${\mathbf{k}}_{\mathbf{ref}}$ | $\left|{\mathbf{k}}_{\mathbf{ref}}/\overline{\mathbf{k}}-1\right|$ (%) |
---|---|---|---|---|---|---|

Water (100W0G) | $24.0$ | $0.607$ | $0.014$ | $2.3$ | $0.605$ | $0.3$ |

50W50G | $22.0$ | $0.4114$ | $0.0049$ | $1.2$ | $0.4189$ | $1.8$ |

40W60G | $24.0$ | $0.3905$ | $0.0027$ | $0.7$ | $0.3876$ | $0.7$ |

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Heyd, R.
One-Dimensional Systemic Modeling of Thermal Sensors Based on Miniature Bead-Type Thermistors. *Sensors* **2021**, *21*, 7866.
https://doi.org/10.3390/s21237866

**AMA Style**

Heyd R.
One-Dimensional Systemic Modeling of Thermal Sensors Based on Miniature Bead-Type Thermistors. *Sensors*. 2021; 21(23):7866.
https://doi.org/10.3390/s21237866

**Chicago/Turabian Style**

Heyd, Rodolphe.
2021. "One-Dimensional Systemic Modeling of Thermal Sensors Based on Miniature Bead-Type Thermistors" *Sensors* 21, no. 23: 7866.
https://doi.org/10.3390/s21237866