Modelling and analysis of thermoelectric modules for self-powered WSNs node with MPPT

The TEG module is composed of an array of thermocouples connected in series, and each thermocouple consists of a P-type and an N-type semiconductor material, shown as figure 2.

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When there is a temperature difference (${\rm{\Delta }}T$) across the TEG, a voltage proportional to ${\rm{\Delta }}T$ will be produced on the TEG output terminal. The equivalent circuit of a TEG with loads, namely the corresponding energy conversion circuits and the WSN or IoT node powered by the TEG, is shown as figure 3.

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${V}_{OC},$ ${R}_{in},$ ${V}_{out},$ ${I}_{out},$ and ${R}_{load}$ are the open circuit voltage, internal resistance, output voltage, output current, and the equivalent load resistance of the TEG, respectively. ${V}_{oc}$ and ${R}_{in}$ are both affected by the temperatures at the TEG hot side (${T}_{H}$) and the temperature at the TEG cold side (${T}_{C}$).

The physical effects during the working procedure TEG module mainly include the Seebeck effect, Peltier effect, Thomson effect, Kelvin effect, Joule effect, and Fourier effect.

2.2.1. Seebeck effects

The Seebeck effect is the basis of a TEG module. Shown as figure 4, as a closed circuit is composed of two metals or semiconductors of different materials, if there is a temperature difference between the two connections of the various materials, an electromotive force will be created in the closed circuit.

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Let the Seebeck coefficient of the P-type and N-type semiconductor be ${\alpha }_{a}$ and ${\alpha }_{b},$ the output voltage of a single thermocouple will be:

$$\begin{eqnarray}&&{\rm{\Delta }}U=\left({\alpha }_{a}-{\alpha }_{b}\right)* \left({T}_{H}-{T}_{C}\right)={\alpha }_{ab}* {\rm{\Delta }}T\end{eqnarray} \tag{ 3 }$$

where ${\alpha }_{ab}$ is the absolute Seebeck coefficient between the two semiconductors, and its unit is V/K. Generally, the Seebeck coefficient S of a TEG module is given by:

$$\begin{eqnarray}&&S=N* {\alpha }_{ab}\end{eqnarray} \tag{ 4 }$$

where N is the number of semiconductor thermocouples in the TEG module, and ${\alpha }_{ab}$ is the absolute Seebeck coefficient between the two semiconductors composing one thermocouple.

2.2.2. Peltier effect

Shown as figure 5, the Peltier effect describes when there is an electric current in a closed circuit composed of two different materials, the endothermic or exothermic phenomenon will occur at the connections of the two materials. Peltier effect is usually used for cooling or heating aims.

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The heat exchange value in Peltier effect is given by:

$$\begin{eqnarray}&&{\rm{\Delta }}Q=Q1-Q2=\pi * I\end{eqnarray} \tag{ 5 }$$

where π is the Peltier coefficient and I is the current in the loop. The unit of π is W/A.

2.2.3. Thomson effect

The Thomson effect describes the phenomenon that occurs when a current passes a conductor with a temperature gradient, the conductor will exchange heat with the surroundings through its surface. The amount of heat release or absorption is affected by the current direction and the conductor.

The amount of exchanged heat by the Thomson effect is given by:

$$\begin{eqnarray}&&{Q}_{T}=\beta I\displaystyle \frac{dT}{dx}\end{eqnarray} \tag{ 6 }$$

where ${Q}_{T}$ is Thomson heat with a unit of W, β is the Thomson coefficient with a unit of V/K, I is the current passing through the conductor with a unit of A, and $x$ is the longitudinal position in the galvanic couple with a unit of m, so $T\left(x\right)$ is a distribution function of the temperature in the galvanic couple.

2.2.4. Kelvin effect

The Kelvin effect summarizes the relationship between the Seebeck effect, Peltier effect, and Thomson effect. The relationship between the coefficients of these three effects is given by:

$$\begin{eqnarray}&&{\alpha }_{ab}=\displaystyle \frac{{\pi }_{ab}}{T}\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&\displaystyle \frac{d{\alpha }_{ab}}{dT}=\displaystyle \frac{{\beta }_{a}-{\beta }_{b}}{T}\end{eqnarray} \tag{ 8 }$$

where ${\alpha }_{ab}$ is the absolute Seebeck coefficient between the two materials (namely material a and b), ${\pi }_{ab}$ is the Peltier coefficient between the two materials, T is the temperature, and ${\beta }_{a}$ and ${\beta }_{b}$ are the Thomson coefficients of the two materials.

2.2.5. Joule effect

The Joule effect depicts that the conductor generates heat when the current passes through the conductor due to the resistance of the conductor. The heat produced by Joule effect is given by:

$$\begin{eqnarray}&&{Q}_{J}={I}^{2}R\end{eqnarray} \tag{ 9 }$$

${Q}_{J}$ is Joule heat, the unit is W; I is the current through the conductor, the unit is A; R is the conductor resistance, the unit is Ω.

2.2.6. Fourier effect

The Fourier effect is a simple thermal effect, which describes the relationship between the heat flowing through a conductor and the temperature at both ends of the conductor. Its expression is given as follows:

$$\begin{eqnarray}&&{Q}_{K}=\displaystyle \frac{\lambda a}{L}{\rm{\Delta }}T\end{eqnarray} \tag{ 10 }$$

where ${Q}_{K}$ is the heat flowing through the conductor, the unit is W; λ is the thermal conductivity of the conductor, the unit is W/(m * K); L is the length of the conductor, the unit is m; a is the cross-sectional area of the conductor, the unit is m².

In addition to the above-mentioned effects, environmental factors will affect the characteristic of TEG modules. Therefore, it is difficult to derive the TEG output only from theoretical calculation. Recently, some researchers have obtained several approximate TEG models by ignoring some effects on the TEG module according to its working condition.

By ignoring the heat loss on the vertical direction of TEG current, paper [7] gives the internal temperature distribution function $T\left(x\right)$ of TEG and the relationship between absolute Seebeck coefficient α and temperature, $\alpha \left(T\right).$

$$\begin{eqnarray}&&T\left(x\right)={c}_{1}{e}^{{k}_{0}x}+{c}_{2}+\displaystyle \frac{{c}_{0}}{{k}_{0}}x\end{eqnarray} \tag{ 11 }$$

where

$$\begin{eqnarray}&&\begin{array}{l}{c}_{1}=\displaystyle \frac{{c}_{0}L-{k}_{0}\left({T}_{{\rm{H}}}-{T}_{C}\right)}{{k}_{0}\left(1-{e}^{{k}_{0}L}\right)}\\ {c}_{2}=\displaystyle \frac{{c}_{0}L-{k}_{0}\left({T}_{{\rm{H}}}-{T}_{C}{e}^{{k}_{0}L}\right)}{{k}_{0}\left({e}^{{k}_{0}L}-1\right)}\end{array}\end{eqnarray} \tag{ 12 }$$

In equation (12), ${c}_{0}$ and ${k}_{0}$ are constants, L is the length of the semiconductor material, ${T}_{H}$ and ${T}_{C}$ are the temperatures at TEG hot side and cold side, respectively.

$$\begin{eqnarray}&&\alpha \left(T\right)={\alpha }_{ref}+{\mu }_{E}\,\mathrm{ln}\left(\displaystyle \frac{T}{{T}_{H}}\right)\end{eqnarray} \tag{ 13 }$$

where ${\alpha }_{ref}$ is the reference absolute Seebeck coefficient and ${\mu }_{E}$ is a constant.

Then, the Seebeck coefficient of the whole semiconductor can be obtained:

$$\begin{eqnarray}&&S=N* \displaystyle \frac{1}{L}\displaystyle {\int }_{0}^{L}\alpha \left(T\left(x\right)\right)dx=N* {\alpha }_{ref}+\displaystyle \frac{N* {\mu }_{E}}{L}\displaystyle {\int }_{0}^{L}\mathrm{ln}\left(\displaystyle \frac{T\left(x\right)}{{T}_{H}}\right)dx\end{eqnarray} \tag{ 14 }$$

By ignoring the Thomson effect and letting the temperatures on both sides of the TEG to be stable, paper [8] defines the Seebeck coefficient S as below:

$$\begin{eqnarray}&&S=\displaystyle \frac{{V}_{\max }}{{T}_{H}}\end{eqnarray} \tag{ 15 }$$

where ${V}_{\max }$ is the maximum output voltage. For a given TEG module, ${V}_{\max }$ is constants. Therefore, the paper [8] considers that the Seebeck coefficient S of a TEG are functions that are only affected by the temperature on hot sideof TEG, namely,

$$\begin{eqnarray}&&{\rm{S}}\left({T}_{H}\right)=\displaystyle \frac{\alpha }{TH}\end{eqnarray} \tag{ 16 }$$

where a is constants.

Paper [9] proposes a new model of TEG and compares it with the results in [10]. The following functions are given in [9] by summarizing [10].

$$\begin{eqnarray}&&{\rm{S}}={\alpha }_{0}+{\alpha }_{1}T+{\alpha }_{2}{T}^{2}\,\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&T={T}_{AV}=\displaystyle \frac{{T}_{H}+{T}_{C}}{2}\end{eqnarray} \tag{ 18 }$$

where ${\alpha }_{0},{\alpha }_{1},{\rm{and}}\,{\alpha }_{2}$ are constant.

It can be seen that Seebeck coefficient of a TEG is considered as quadratic functions to TEG's temperature and average temperature ${T}_{AV}$ is used to replace the temperature distribution function T (x) of the TEG.

Using the new model, paper [9] gives the following conclusions:

$$\begin{eqnarray}&&S=\displaystyle \frac{{V}_{\max }}{{T}_{H}}\end{eqnarray} \tag{ 19 }$$

It is obtained by experimental testing on the TEG module proposed in [9].

It is clear that equation (19) given in paper [9] is the same to equation (15) presented in paper [8], and both can be written in the form of equation (16). In summary, although the authors in papers [7–10] try to generalize the relationship between the Seebeck coefficient of the TEG with its temperature distribution, their conclusions are not totally identical to each other.

To validate the proposed method, the experimental setup shown as figure 6 is used to conduct a series of experiments. The experimental setup consists of a heating furnace, a heat transfer box, two TEG modules with corresponding heat sinks, two thermocouples, two temperature measurement modules, and a laptop. The heat transfer box is a rectangular parallelepiped structure composed of stainless-steel plates to imitate the hot wall at industrial plants. Two TEG modules, TGM287–1.0–1.3 from European Thermodynamic [15], are pasted to the hot wall. However, only the copper heatsink on the top of the TEG modules can be seen in figure 5. The two TEG modules are independent of each other and can be used either alone or together in series or parallel connection. This research only uses the right TEG module. Two K-type thermocouples and two measurement devices (USB-TC01 from National Instruments Corporation), and a laptop are used to measure the hot and cold side temperatures of the TEG.

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The first experiment is performed to determine the relationship between TEG open circuit voltage ${V}_{OC}$ and the temperatures at the TEG hot and cold side, namely ${T}_{H}$ and ${T}_{C}.$ The setting temperature of the heating furnace is 200°C. The hot wall temperature will increase from room temperature to its peak value and then decrease to room temperature after switching off the furnace. In this procedure, ${V}_{oc},$ ${T}_{H},$ and ${T}_{C}$ are measured by the multimeter and USB-TC01 and recorded by the laptop.

The original data of ${V}_{OC},$ ${T}_{H},$ and ${T}_{C}$ are shown as figure 7. The horizontal x-axis represents the sample time in this experiment, while the vertical axes are the temperatures at the TEG hot and cold side, and TEG open circuit voltage.

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In the heating procedure (form start point to point C), ${T}_{H},$ ${T}_{C},$ and ${\rm{\Delta }}T$ gradually increases and reaches their maximum value at point C, while ${V}_{oc}$ peaks at point A. The positions of maximal ${V}_{oc}$ and ${\rm{\Delta }}T$ do not coincide because ${V}_{oc}$ is not only influenced by ${\rm{\Delta }}T,$ but also by ${T}_{C}.$ From point A to point B, ${V}_{oc}$ decreases because the growth rate of ${\rm{\Delta }}T$ is less than that of ${T}_{c}.$ From point B to point C, ${V}_{oc}$ maintains stable because the growth rate of ${\rm{\Delta }}T$ is balanced with the growth rate of ${T}_{c}.$

The TEG module begins its natural cooling stage from point C, where the heating furnace is turned off. Both ${V}_{oc}$ and ${\rm{\Delta }}T$ gradually decreased in the cooling phase. There is a clear change in the temperature curve slope in the cooling stage because of the different sample intervals used in the data acquisition process.

Paper [16] explores the characteristic of a TEG exposed to a transient heat source on the hot side and natural convection on the cold side. The ${V}_{oc}$ curve depicted in figure 6 is similar to the results in [16], namely ${V}_{oc}$ reaches its maximum value and then fells to a stable stage.

To explore the relationship between ${V}_{oc},$ ${T}_{H}$ and ${T}_{C},$ the above experimental data are fitted in MATLAB. The results are given as follows:

$$\begin{eqnarray}&&{V}_{oc}=3840-79.4* {T}_{c}+66.54* {T}_{h}=3840+66.54* {\rm{\Delta }}T-12.86* {T}_{c}\,\,\left({R}^{2}=0.9992\right)\end{eqnarray} \tag{ 20 }$$

where ${R}^{2}$ is the coefficient of determination, the closer ${R}^{2}$ is to 1 means the higher fitting accuracy.

The comparison of the fitting result by using equation (20) and the measured value of ${V}_{oc}$ is shown as figure 8. It can be seen that the fitted result is very accurate.

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According to figure 2, the TEG internal resistance ${R}_{in}$ can be calculated by:

$$\begin{eqnarray}&&{R}_{in}=\displaystyle \frac{{V}_{oc}-{V}_{out}}{{V}_{out}}{R}_{load}\end{eqnarray} \tag{ 21 }$$

where, V_oc can be calculated using equation (20) and V_out will be measured on-site.

From equation (20), the change of Seebeck coefficient S with the temperature difference ${\rm{\Delta }}T$ can be calculated by:

$$\begin{eqnarray}&&S=\displaystyle \frac{3840}{{\rm{\Delta }}T}+66.54-12.86* \displaystyle \frac{{T}_{c}}{{\rm{\Delta }}T}\end{eqnarray} \tag{ 22 }$$

It can be seen that the forms of equation (22) is different from equations (14), (16), and (17). Such an accurate and linear relationship described in equation (22) has not been reported by other authors.

According to equations (15) and (19) proposed by paper [8] and [9], $S* {T}_{H}$ is a constant. However, our research depicts the actual changing procedure of $S* {T}_{H}$ in figure 9. It can be seen that the value of ${S* T}_{{\rm{H}}}$ varies with T_H and T_C .

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Since ${c}_{0}$ and ${k}_{0}$ in equation (12) cannot be obtained, the method proposed in [10] is adopted and the temperature distribution function T(x) is replaced by ${T}_{AV},$ so that equation (14) can be written as:

$$\begin{eqnarray}&&S\left(T\right)=N* {\alpha }_{ref}+N* {\mu }_{E}* \,\mathrm{ln}\left(\displaystyle \frac{{T}_{AV}}{{T}_{H}}\right)\end{eqnarray} \tag{ 23 }$$

Using the experimental data shown as figure 6 to equation (23), we can obtain that $N* {\alpha }_{ref}$ = 50.71 and $N* {\mu }_{E}$ = 105.7.

Therefore, the model in [7] is approximately given by:

$$\begin{eqnarray}&&S\left(T\right)=50.71+105.7* \,\mathrm{ln}\left(\displaystyle \frac{{T}_{AV}}{{T}_{H}}\right)\,\left({R}^{2}=0.00305\right)\end{eqnarray} \tag{ 24 }$$

Using the experimental data shown as figure 6 to the equation (17) given in [10], we can obtain the value of ${\alpha }_{0},{\alpha }_{1},{\alpha }_{2}$ and rewritten equation (17) as:

$$\begin{eqnarray}&&S\left(T\right)=2876-16.55* T+0.0241* {T}^{2}\,\left({R}^{2}=0.2474\right)\end{eqnarray} \tag{ 25 }$$

The Seebeck coefficients calculated by equations (22), (24), and (25) are shown as figure 10. Comparing with the results obtained by using the methods in paper [7] and [10], the Seeback coefficient obtained by using the model proposed in this paper and equation (22) fits better with the experimental data.

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