Thermodynamic Analysis of Thermoelectric Legs with Different Geometric Parameters

Abstract

The design of thermocouple legs is the central theme of this chapter, the methodology of reduced variables is shown, which allows the designer to obtain the dimensions of the geometric parameters, specifically the cross-sectional areas transversal An, Ap and length of the legs ln=lp. The main quantities used within this scheme calculation method are the reduced current density U, the thermoelectric potential Φ, and the thermal conductivity κ. Subsequently, the performance of the designed thermocouple is analyzed through the electrical power produced when the system is connected to an electrical resistance load. In this step, the condition of the dependence of thermoelectric properties on temperature is used. The results show specific values of the cross-sectional areas in which the maximum power value occurs. Although the main method has the advantage that it uses minimal computing and software requirements, a spreadsheet may be sufficient for the calculations. It is also important to mention that in addition to calculating the dimensional parameters, the technique allows the generation and evaluation of various designs for the same temperature range.

1. Introduction

The global energy demand is increasing every day [1] as a function of population growth. Most of the supply is achieved through fossil fuels (coal, natural gas, oil), [2]. These are distributed in different proportions over the planet, and it has an economic-environmental cost they imply for their exploitation. The projections made by various organizations indicate that energy dependence on fossil fuels will continue for many decades [2]. However, they also show encouraging results about better use of available energy resources (whether conventional or renewable) [3]. One form of this use is waste heat capture [1], which consists of implementing systems that capture residual heat from other generation systems and convert it into some form of useful energy. Thermoelectric generators have emerged as a response to this type of thermal use [1, 4].

A thermoelectric generator is a device capable of converting heat into electricity directly. Its basic structure [5, 6] is made up of a set of pairs of semiconductor materials (thermocouples), connected electrically in series and thermally in parallel. This entire arrangement is encapsulated between two rectangular plates of ceramic material. This device is linked on each side to a pair of heat exchangers, one of them being the hot side and the other the cold side. In this way, the module is under a temperature difference. When circulating a flow of heat Through the semiconductor materials (one type $N$ and the other type $P$), the module produces a direct electric current, giving rise to a phenomenon known as the Seebeck effect.

Thermoelectric phenomena were discovered in the early nineteenth century, first by Thomas J. Seebeck [6, 7]. Later, Jean C. A. Peltier discovered the reverse phenomenon of cooling. The first applications of these two phenomena occurred during and after the Second World War, mainly for military use [8] with $5\%$ efficiency for power generation. Advances continued towards domestic applications, developing some devices such as small radios that took advantage of the heat emitted by oil lamps. At the end of the 1960s, interest in thermoelectric technology declined, and many research programs in this field were dismantled. Despite this decline, Abram Fedorovich Ioffe and his research group at the institute in St. Petersburg carried out research on thermoelectrics in the USSR [9], the result of these efforts being the first thermoelectric generating and cooling devices. Around 1970, the reliability and simplicity of thermoelectric generators led them to consolidate among the technologies used for space exploration missions, such as the radioisotope thermoelectric generator used by NASA on the Apollo missions (1969–1972), Pioneer (1972–1973), Viking (1975), Voyager (1977), Galileo (1989), Cassini (1997), and Curiosity (2011) [8, 9]. Currently, the area of thermoelectricity involves both intense research and industry with a global impact because of the synergy between academia and industry. A variety of applications have been generated, including the automotive industry to harness the heat produced in automobile internal combustion engines, power supplies for electronic devices, self-powered wireless micro-platforms, and health monitoring systems [1].

Similarly, as various systems (mechanical, thermal, electrical, electronic) manufacture thermoelectric modules, it is necessary to elaborate the design of its structure, explicitly taking into account the influence of the physical phenomena that occur in the material network. For example, the design of thermocouple legs is carried out by taking advantage of the relationship between the transport mechanisms (of heat and electrical charge) and the geometric or dimensional parameters of the legs or thermoelectric pillars. Since the development of thermoelectric technology, thermocouples have been manufactured in a rectangular shape and with legs of the same cross-sectional area and length. However, with the development of new product concepts such as additive manufacturing, it has been possible to think of new geometric shapes for the legs.

For example, in [10] hollow prism geometries with square, triangular and circular bases are proposed, solid legs with circular, square, and triangular bases are also proposed, as well as legs of stacked layers with triangular, quadrangular, and circular shapes. Their results conclude that thermocouples with complex geometry (specifically stacked-layer legs) can achieve high electrical power values compared to legs of conventional shape. In [11] a study of optimization of the design of micro thermoelectric generators is shown, and it analyzes the role of structural geometry in the increase of the electrical power of the devices. One of its results shows that the cross-sectional area of the legs has a significant impact on the power and efficiency of the device.

The interest of the content of this chapter is to show that the design of thermoelectric legs is related to the needs of the application in which the thermoelectric generator will be used to harness heat. For example, in [12] it is shown that by varying the geometry of the leg of cascade-type thermoelectric modules, the performance of solar thermoelectric generators can be improved.

This work is focused on the design of thermocouple legs, presenting a first technique that is characterized by being of utility to the designer to carry out the calculations directly, preserving the physical information related to the phenomena of transport of electric charge and heat.

2. The thermoelectric generator as a heat engine

A heat engine is defined as a system that transforms heat into work in thermodynamics. It is a device that, operating cyclically, takes heat from a hot source, performs a certain amount of work (part of which is used to run the machine itself), and delivers waste heat to a cold source, usually the environment. The fluid known as the working substance must circulate through the structure of the machine and transports the heat from one source to another. In a broad sense, the term heat engine includes systems that produce work either through heat transfer or combustion, even though the devices do not operate in a thermodynamic cycle. According to what is mentioned in this last definition, it is possible then establish a thermoelectric generator as a heat engine, see Figure 1.

Figure 1.

Schematic to represent the thermoelectric generator as a heat engine.

Due to the way it works, a thermoelectric generator is between two heat sources, one at high temperature or hot side temperature $\left(T_H\right)$ and the other at low temperature or cold side temperature $\left(T_C\right)$. Thus the $\mathit{TEG}$ absorbs a quantity of heat, part of which is used to produce electrical work. It is important to clarify that unlike a traditional thermal engine, such as a steam turbine or an internal combustion engine, in the $\mathit{TEG}$, this role is played by the set of free electrons, which upon receiving the energy transferred by heat, manage to produce an electric current through the Seebeck effect. If some load $\left(R_{\mathit{\text{load}}}\right)$ is connected to the generator terminals, then a potential difference $\left(\mathrm{\Delta }V\right)$ will be produced, and finally, an electrical work.

2.1 Thermal efficiency $\left(\eta \right)$

As is known in thermodynamics, thermal efficiency $\left(\eta \right)$ measures the degree to which a thermal engine manages to take advantage of the heat that enters the hot side in the form of work. In terms of heat quantities, the thermal efficiency is formulated as follows,

$$\eta =\frac{Q_H-\mathit{QC}}{Q_H}=1-\frac{Q_C}{Q_H}$$E1

the theoretical limit of the thermal efficiency is imposed by the Carnot efficiency,

$${\eta }_{\mathit{max}}={\eta }_{\mathit{\text{Carnot}}}=1-\frac{T_C}{T_H}$$E2

the Carnot efficiency imposes the theoretical limit of the thermal efficiency,

$$\eta =\frac{\mathrm{\Delta }T}{T_H}\frac{\sqrt{1+Z\overline{T}}-1}{\sqrt{1+Z\overline{T}}+\frac{T_C}{T_H}}$$E3

where, $\mathrm{\Delta }T$ is the temperature difference, $\overline{T}$ is the average temperature at which the device is operating, $Z$ is the figure of merit and is given by,

$$Z=\frac{{\alpha }^2}{\mathit{\rho \kappa }}$$E4

it is observed that this quantity depends on the thermoelectric properties of the material, Seebeck coefficient $\left(\alpha \right)$, resistivity electrical $\rho$ and thermal conductivity $\kappa$.

Finally, it is worth mentioning that the thermal efficiency of any machine is affected by different factors. Specifically for the $\mathit{TEG}$, it is the irreversible mechanisms of heat transport that negatively affect the system’s performance. The following section deals with transport phenomena (reversible and irreversible) that occur in a thermoelectric generator.

3. Reversible and irreversible transport phenomena in thermocouples

In a thermoelectric material, two flows occur simultaneously, one of heat and the other of electric charge. There are three transport mechanisms for these flows in a thermoelectric generator: the Seebeck effect, the Joule effect, and the Fourier effect. Table 1 shows the classification of these effects, depending on whether they are reversible or irreversible.

Effect	Formula	Classification
$\mathit{\text{Seebeck}}$	$\alpha T_{H}I$	$\mathit{\text{Reversible}}$
$\mathit{\text{Joule}}$	$\frac{1}{2}{\mathit{RI}}^2$	$\mathit{\text{Irreversible}}$
$\mathit{\text{Fourier}}$	$K\left(T_H-T_C\right)$	$\mathit{\text{Irreversible}}$

Table 1.

Mechanisms of heat and electric charge transport in a $\mathit{TEG}$.

For Table 1 the following variables are defined.

$\alpha$ $\equiv$ Seebeck coefficient.

$T_H$ $\equiv$ hot side temperature.

$I$ $\equiv$ electrical current.

$R$ $\equiv$ electrical resistance.

$K$ $\equiv$ thermal conductance.

$T_C$ $\equiv$ cold side temperature.

The basic model is shown to show how these transport phenomena occur in a $\mathit{TEG}$, which is formed by two legs, one type $n$ and another type $p$. This piece retains all the thermoelectric properties and maintains the dimensions of the original system, Figure 2.

Figure 2.

Diagram showing the basic model of a thermocouple and the phenomena of heat and electric charge transport. The terms that are observed are the following: ${\alpha }_{n/p}\left(T_h-T_c\right)$ is the Seebeck voltage, $T_h-T_c$ is the temperature difference, ${\alpha }_{T_h/T_c}$ is the Peltier heat flow, $\frac{1}{2}{\mathit{RI}}^2$ is the heat dissipated by joule effect, $K\left(T_h-T_c\right)$ is the Fourier effect conduction heat.

The figure shows the direction in which each of the three transport mechanisms occurs, the Seebeck effect heat and the Fourier effect heat are due to the difference in temperature between the hot and cold sources. In contrast, the Joule effect heat is produced by the electrical resistance of the thermoelectric material itself.

4. The thermoelectric potential

As for other systems and processes that involve energy, thermoelectricity has sought to develop calculation schemes that are solid and that have the greatest possibility of being able to work with the information of the variables that correspond to the physics of the system. A treatment that has arisen from thermodynamic principles and that has shown a great scope demonstrated by its application to the design of thermocouples is the method of reduced variables, where one of the principal quantities is the thermoelectric potential $\left(\mathrm{\Phi }\right)$ [13, 14]; this quantity is a state function given by the following equation,

$$\mathrm{\Phi }=\frac{1}{u}+\mathit{\alpha T}$$E5

where, $u$ is the reduced current density and $\alpha$ is the Seebeck coefficient.

A characteristic that shows the usefulness and importance of $\mathrm{\Phi }$ is that it manages to link the two flows that occur simultaneously in the thermoelectric material of the leg, this quality becomes evident when deriving expressions for the volumetric production of heat $\left({\nu }_q\right)$ and the electric field $\left(E\right)$; as a function of thermoelectric potential $\mathrm{\Phi }$ and current density $J$,

$${\nu }_q=\nabla \cdot \left(\mathrm{\Phi }J\right)=\nabla \mathrm{\Phi }\cdot J$$E6

$$E=\nabla \mathrm{\Phi }$$E7

The advantage of the thermoelectric potential $\left(\mathrm{\Phi }\right)$ is that it allows working with classical thermodynamics, formulated from average quantities, even when the phenomenon has a microscopic nature and that the formalism that is applied for its treatment is the linear thermodynamics of coupled irreversible processes out of balance. Being formulated $\left(\mathrm{\Phi }\right)$ as a state function, it is possible to analyze the system’s evolution without taking into account the process path.

The following section shows the fundamentals of the reduced variables scheme, within which the formulation of the thermoelectric potential is generated as a result.

5. Reduced variables

When a system is analyzed in thermodynamics, a first step is to identify the variables that describe its behavior; subsequently, from this set, those that turn out to be the independent variables and whose numerical values can be obtained from an experiment or direct measurements are determined. Then, by applying the laws of thermodynamics, mathematical relationships between the dependent and independent variables are obtained, managing to deduce an equation of state. Although this equation can have a useful form to derive the quantities that control the thermodynamics of the system, in some instances, it is desirable to obtain a more straightforward form that allows a better understanding of the physics of the system. To achieve this goal, a useful technique is the application of reduced variables, which are intensive properties and thanks to which the equation of state can have a simpler form. However, it is also an equation that maintains the information of the process and allows analytical calculations of properties or design parameters in a practical and direct way. Snyder and collaborators [13] have built on solid foundations the method of reduced variables and have proposed a series of equations that allow calculations for the design of thermocouples.

For a complete understanding of the deduction of this method, it is suggested to review, for the scope of this chapter, it is sufficient to show and indicate the usefulness of the following equations. See Table 2.

Equation	Name	Interpretation
$u=\frac{J}{\kappa \nabla T}$	reduced current density	converts to total current density in an intensive variable
${\eta }_r=\frac{u\left(\alpha -\mathit{u\rho \kappa }\right)}{\mathit{u\alpha }+\frac{1}{T}}$	reduced efficiency	a totally intensive quantity in terms of the reduced current density and of the temperature
${\eta }_{r-\mathit{max}}=\frac{\sqrt{1+\mathit{zT}}-1}{\sqrt{1+\mathit{zT}}+1}$	reduced maximum efficiency	the highest value of the reduced efficiency
$s=\frac{\sqrt{1+\mathit{zT}}-1}{\mathit{\alpha T}}$	compatibility factor	the reduced current value that maximized to reduced efficiency

Table 2.

Main equations of the reduced variables method.

As can be seen in the Table 2, the new formulation of the efficiency ${\eta }_r$ is found in terms of intensive quantities. This characteristic eliminates a considerable part of the interdependence between variables, and then the scheme becomes a practical tool for calculating thermocouple design. Specifically, this chapter shows its usefulness for designing conventional and segmented legs. The following section shows the application of this calculation scheme to design a conventional leg.

6. Geometric parameters of thermocouples

Figure 3 shows the basic structure of a thermocouple. In the diagram shown, it can be seen that the model is made up of a pair of legs of semiconductor material. Each of these legs has doping (one is $n$-type and the other $p$-type) and are connected (to each other) electrically in series employing a metal bridge.

Figure 3.

Basic structure of a thermocouple.

Regarding system design, it is essential to say that this process can be as simple or complex as required for the application it seeks to serve. Table 3 classifies the aspects that are taken into account to carry out the task of thermocouple design.

Design aspect	Parameter type	Parameters
Composition	Material	Material of the metal bridge, material of the legs Ceramic plate material, amount of material to use
Size	Geometric parameters	Leg length, leg cross-sectional area, Ceramic plate thickness, metal bridge thickness
Architecture	shape	shape of the cross section of the leg, shape of the $\mathit{TEG}$, separation between the legs of the thermocouples

Table 3.

Classification of aspects and the corresponding parameters, for the design of thermocouples.

This chapter focuses on the design aspect of the size of the legs of the thermocouple. The geometric parameters of interest that will be calculated are the length ($l$) and cross-sectional area ($A$) of the legs and the thickness of the metal bridge. It is worth mentioning that this is the first scope and that the study can be extended and more complete when considering other parameters such as the thickness of the ceramic plate. In the following section, the design of a conventional thermocouple is carried out, taking into account the geometric parameters ($l$, $A$, $l_{\mathit{\text{metal}}}$).

7. Modeling of conventional thermocouples

In Figure 3 structure of a conventional thermocouple is shown. In this section, the design of a system is made to operate in the range of ($498K-623K$). For this purpose, the following parameters are calculated.

$l$ is leg length.

$A_n$ is cross-sectional area of $n$-type leg.

$A_p$ is cross-sectional area of $p$-type leg.

$l_{\mathit{\text{metal}}}$ is thickness of the metal bridge that electrically connects the two legs.

The materials selected for this first design are:

$p-\mathit{\text{type}}$: ${\mathit{Zn}}_4{\mathit{Sb}}_3$.

$n-\mathit{\text{type}}$: ${\mathit{\text{CoSb}}}_3$.

For the calculations, the equations of the Table 2 are applied. Employing the definition of reduced current density, equation of the first row, the following integrals are obtained,

$$l\left(T\right)J_n={\int }_{T_c}^{T_h}{u}_n{\kappa }_n\mathit{dT}$$E8

$$l\left(T\right)J_p={\int }_{T_c}^{T_h}{u}_p{\kappa }_p\mathit{dT}$$E9

for the calculation of the integrals (8 and 9) the data of Table 4 are necessary.

$\mathbf{T}\left(K\right)$	${\mathbf{u}}_p{k}_p\mathit{dT}\left(A/\mathit{cm}\right)$	${\mathbf{u}}_n{k}_n\mathit{dT}$ (A/cm)
$T_0=498$	$u_p{k}_p\left(T_0\right)=0\cdot 4999$	$u_n{k}_n\left(T_0\right)=-2\cdot 4164$
$T_1=523$	$u_p{k}_p\left(T_1\right)=0\cdot 5107$	$u_n{k}_n\left(T_1\right)=-2\cdot 3984$
$T_2=548$	$u_p{k}_p\left(T_2\right)=0\cdot 5257$	$u_n{k}_n\left(T_2\right)=-2\cdot 3855$
$T_3=573$	$u_p{k}_p\left(T_3\right)=0\cdot 5430$	$u_n{k}_n\left(T_3\right)=-2\cdot 3783$
$T_4=598$	$u_p{k}_p\left(T_4\right)=0\cdot 5607$	$u_n{k}_n\left(T_4\right)=-2\cdot 3774$
$T_5=623$	$u_p{k}_p\left(T_5\right)=0\cdot 5768$	$u_n{k}_n\left(T_5\right)=-2\cdot 3832$

Table 4.

Numerical data of the product $\mathit{uk}$ for the p-type and n-type materials, in the temperature range $498-623K$.

Source [14].

Combining the data in Table 4 with the fourth-order Newton-Cotes numerical method to calculate the integrals (8 and 9),

$$l\left(T\right){\int }_{T_c}^{T_h}{u}_p{\kappa }_p\mathit{dT}=52.7163\mathit{mA}/\mathit{cm}$$E10

$$l\left(T\right){\int }_{T_c}^{T_h}{u}_n{\kappa }_n\mathit{dT}=-238.93\mathit{mA}/\mathit{cm}$$E11

then the quotient is calculated.

$$\frac{-J_n}{J_p}=4.53237$$E12

Next, it is to know the values of the total heat flux ($W$) and the thermoelectric potential ($\mathrm{\Phi }$) at the hot spot ($T_h$) of the thermocouple for each of the two legs ($n$) and ($p$). It is important to mention that the value of (W) is an input parameter that is obtained depending on the heat flow that is required for the application or final use of the $\mathit{TEG}$; the value of ($\mathrm{\Phi }$) is calculated with Eq. (5). For the particular case of the thermocouple designed, the values used for the design are given by the following Table 5.

Parameter	Numerical value
$W$	20 (W/cm^2)
${\mathrm{\Phi }}_p$	$0.39643$ (V)
${\mathrm{\Phi }}_n$	$-0.50663$ (V)

Table 5.

Numerical values of thermoelectric potentials and heat flux, used for calculation of current density $J_p$.

The following calculation is that of the current densities $\left(J_p\right)$ and $\left(J_n\right)$, applying the following eq. (13), $\left(J_p\right)$ is calculated.

$$J_p=W\frac{1+\frac{A_n}{A_p}}{{\mathrm{\Phi }}_p-{\mathrm{\Phi }}_n}$$E13

to calculate $\left(J_n\right)$ the definition of the quotient (14) is applied

$$\frac{-J_n}{J_p}$$E14

then the following values are obtained from the Table 6.

Current density	Numerical value
$J_p$	27.0333 (mA/cm^2)
$J_n$	122.525 (mA/cm^2)

Table 6.

Calculated values of the current densities, $J_p$ and $J_n$.

7.1 Calculation of geometric parameters

With the results obtained in the previous section, it is now possible to determine the numerical values of the geometric parameters. First, the length ($l$) of the thermocouples is calculated.

$$l=\frac{{\int }_{T_c}^{T_h}{u}_p{\kappa }_p\mathit{dT}}{J_p}$$E15

combining the result of (10) and ($J_p$) from Table 6.

$$l=1.95\mathit{mm}$$E16

The following parameter to calculate is the cross-sectional area, $A_t$ is defined as the cross-sectional area,

$$A_t=A_p+A_n$$

For a first calculation, a value $A_t=1{\mathit{mm}}^2$ is proposed later to calculate $A_p$ and $A_n$, the following system of linear equations is constructed,

$$A_p=4.53A_n$$E17

$$A_p+A_n=1$$E18

When solving the system, the following values are obtained,

$$A_p=0.82{\mathit{mm}}^2$$E19

$$A_n=0.18{\mathit{mm}}^2$$E20

7.2 Thermoelectric properties of the thermocouple

This section is continued with the design of the thermocouple. The next step is to carry out part of the composition design (first row of Table 3) for the scope of this chapter, which is to show a first approach to the design of thermocouples, an of the most used techniques and with a certain degree of reliability, which is the calculation of the average. The formula for calculating the average of a thermoelectric property $\left(x\right)$ in a temperature range $\left(T_c-T_h\right)$ is defined as,

$$\overline{X}=\frac{{\int }_{T_c}^{T_h}X\mathit{dT}}{T_h-T_c}$$E21

Eq. (21) is applied to calculate the averaged properties: Average Seebeck coefficient $\left(\overline{\alpha }\right)$, thermal conductivity average $\left(\overline{\kappa }\right)$, average electrical resistivity $\left(\rho \right)$. To complete this task, it is necessary to know the measurement data of the properties mentioned above. The following segment shows the calculation of the averaged amounts.

7.2.1 Average Seebeck coefficient

Measurement data for this quantity is shown in the Tables 7–12 for the materials ${\mathit{Zn}}_4{\mathit{Sb}}_3$, ${\mathit{\text{CoSb}}}_3$;

Temperature (K)	Seebeck coefficient $\mu$V/K $\left({{Zn}}_4{{Sb}}_3\right)$
498	168
523	173
548	178
573	182
598	187

Table 7.

Numerical data of Seebeck coefficient of p-type material.

From reference [14].

Temperature (K)	Seebeck coefficient $\mu$ V/K $\left({{\text{CoSb}}}_3\right)$
498	−168
523	−174
548	−179
573	−184
598	−188

Table 8.

Numerical data of Seebeck coefficient of n-type material.

From reference [14].

Temperature (K)	Electrical resistivity $\mathrm{\Omega }$cm $\left({{Zn}}_4{{Sb}}_3\right)$
498	2.691
523	2.760
548	2.825
573	2.889
598	2.949

Table 9.

Numerical data of electrical resistivity of p-type material.

From reference [14].

The results of the averages are as follows.

${\overline{\alpha }}_p$ = $177.56\mathit{\mu V}/K$.

${\overline{\alpha }}_n$ = $-178.84\mathit{\mu V}/K$.

7.2.2 Average electrical resistivity

Measurement data for this quantity is shown in the Tables 9 and 10 for the materials ${\mathit{Zn}}_4{\mathit{Sb}}_3$, ${\mathit{\text{CoSb}}}_3$;

Temperature (K)	Electrical resistivity $\mathrm{\Omega }$cm $\left({{\text{CoSb}}}_3\right)$
498	0.845
523	0.867
548	0.888
573	0.907
598	0.924

Table 10.

Numerical data of electrical resistivity of n-type material.

From reference [14].

the results of the averages are as follows.

${\overline{\rho }}_p$ = $2.82\mathrm{\Omega }\mathit{cm}$.

${\overline{\rho }}_n$ = $0.89\mathrm{\Omega }\mathit{cm}$.

7.2.3 Average thermal conductivity

Measurement data for this quantity is shown in the Tables for the materials ${\mathit{Zn}}_4{\mathit{Sb}}_3$, ${\mathit{\text{CoSb}}}_3$; (Table 11).

Temperature (K)	Thermal conductivity (m W/cm K) $\left({{Zn}}_4{{Sb}}_3\right)$
498	5.91
523	5.93
548	6.00
573	6.10
598	6.22

Table 11.

Numerical data of thermal conductivity of p-type material.

From reference [14] (Table 12).

Temperature (K)	Thermal conductivity (m W/cm K) $\left({{\text{CoSb}}}_3\right)$
498	39.925
523	39.10
548	38.37
573	37.75
598	37.25

Table 12.

Numerical data of thermal conductivity of n-type material.

From reference [14].

the results of the averages are as follows.

${\overline{\kappa }}_p$ = $6.02mW/\mathit{cm}K$.

${\overline{\kappa }}_n$ = $38.44mW/\mathit{cm}K$.

8. The metallic bridge and its properties

The connection between two legs, one type ($n$) and the other type ($p$), to form a thermocouple is achieved by making a bridge-type connection between a pair of ends (one of each leg) through a segment of metallic material, as shown in Figure 4.

Figure 4.

Connection between the legs of the thermocouple.

From the figure, it is possible to understand that the metallic material is responsible for transporting charge carriers from one terminal to another when the thermocouple is under heat transport. So the designer must consider the effect of this part on the structure of a thermoelectric system. Table 13 shows the numerical values of the thermoelectric properties of the metallic bridge,

Component	Electrical resistivity ($\mathrm{\Omega }$cm)	Thermal conductivity (W/cm K)
Metal	$1\cdot 7\ast {10}^{-6}$	$3\cdot 98$

Table 13.

Thermoelectric properties of the metal bridge.

Data obtained from [15, 16].

9. Final design

The previous sections have shown the first thermocouple design procedure in the most detailed way possible. Figure 5 shows a sketch of the design generated so far,

Figure 5.

Thermocouple structure designed. The dimensioning of the length of the legs, the cross-sectional area of each one and the thickness of the metal bridge are shown.

The system obtained is characterized by having legs of the same length but of different cross-sectional areas. It is important to note that the cross-sectional areas remain constant throughout the length of each of the legs. The legs are joined (one end of each) by the metal bridge, which has an area that is adjusted to each of the cross-sectional areas ($A_p$, $A_n$). Table 14 shows a file of the design characteristics,

Characteristic	Symbol	Numerical value
Average Seebeck coefficient of p-type material	${\overline{\alpha }}_p$	177.56 $\mu$ V/K
Average Seebeck coefficient of the n-type material	${\overline{\alpha }}_n$	- 178.84 $\mu$ V/K
Average electrical resistivity of p-type material	${\overline{\rho }}_p$	2.82 $\mathrm{\Omega }$ cm
Average electrical resistivity of n-type material	${\overline{\rho }}_n$	0.89 $\mathrm{\Omega }$ cm
Average thermal conductivity of p-type material	${\overline{\kappa }}_p$	6.02 m W/cm K
Average thermal conductivity of n-type material	${\overline{\kappa }}_n$	38.44 m W/cm K
Cross-sectional area of p-type leg	$A_p$	0.82 mm^2
Cross-sectional area of the n-type leg	$A_n$	0.18 mm^2
Leg length	$l$	1.95 mm

Table 14.

Characteristics of the designed thermocouple.

The designed system is already characterized. The next step is to analyze its performance. In the next section, it is performed this analysis considers certain conditions regarding the geometric parameters.

10. Analysis of the performance of the designed thermocouple

The performance of the designed thermocouple is evaluated using the power $\left(P\right)$, which has the following equation,

$$P=\mathit{VI}$$E22

where,

$P$: electrical power generated by the thermocouple, $\left(W\right)$,

$V$: voltage produced by the thermocouple, $\left(\mathit{\text{Volts}}\right)$,

$I$: current produced by the thermocouple; $\left(\mathit{\text{Amperes}}\right)$.

The corresponding formulation for the quantities ($V$), ($I$) is as follows:

$$V=\frac{\left({\alpha }_p+{\alpha }_n\right)R_{\mathit{\text{load}}}}{R_n+R_p+R_{\mathit{\text{metal}}}+R_{\mathit{\text{load}}}}\left(T_H-T_C\right)$$E23

$$I=\frac{\left({\alpha }_p+{\alpha }_n\right)}{R_n+R_p+R_{\mathit{\text{metal}}}+R_{\mathit{\text{load}}}}\left(T_H-T_C\right)$$E24

in eqs. (23 and 24) the following electrical resistances are found, (Table 15).

Symbol	Electrical resistance type
$R_p$	Electrical resistance of the p-type leg
$R_n$	Electrical resistance of the n-type leg
$R_{\mathit{\text{metal}}}$	Electrical resistance of the metal bridge
$R_{\mathit{\text{load}}}$	System load resistance connected to the thermocouple

Table 15.

Types of electrical resistance.

Eqs. (23 and 24) are combined with eq. (22) to formulate the electrical power of the thermocouple. In the following subsections show specific cases with imposed conditions for the parameters.

10.1 Electrical power with variation of total cross-sectional area and electrical load resistance

In this first case, it is intended to observe what happens when the thermocouple is connected to a system that is fed with the current produced, as shown in Figure 6.

Figure 6.

Thermocouple connected to a resistance electrical load.

In the design calculations, carried out in the previous sections, a total area value was established $\left(A_{\mathit{\text{total}}}=1{\mathit{mm}}^2\right)$, and determining the corresponding values of ($A_p$) and ($A_n$). Graph 7 shows how the power generated by the thermocouple varies with respect to the ranges $\left(0.1\le A_n\le 0.18\right)$ and $\left(0.1\le A_p\le 0.82\right)$ for a fixed value of resistance load $R_{\mathit{\text{load}}}$.

It is observed in the Figure 7 that at the beginning of each of the ranges, the power surface grows rapidly and stabilizes when it reaches a maximum value. Table 16 shows the maximum power value $\left(P_{\mathit{max}}\right)$ and the values of $\left(A_n\right)$ and ($A_p$) that allow the thermocouple to reach this maximum value.

Figure 7.

Power surface produced by the thermocouple, as a function of cross-sectional area.

Parameter	Numerical value
$P_{\mathit{max}}$	0.00016
$A_p$	0.67
$A_{\mathit{\text{total}}}$	0.14

Table 16.

Maximum power value and corresponding cross-sectional area values.

It is important to note that the above results have been calculated for a specific value of load resistance $\left(R_{\mathit{\text{load}}}\right)$. Below are two graphs that show the effect of the variation of $\left(R_{\mathit{\text{load}}}\right)$ on the power.

The previous Figures 8 and 9 show that as the load resistance increases, the electrical power produced decreases. Figure 8 has been obtained for three different values of $R_{\mathit{\text{load}}}$ in the following order: $\left(5\mathrm{\Omega }\right)$ red color, $\left(10\mathrm{\Omega }\right)$ blue color, $\left(15\mathrm{\Omega }\right)$ orange color.

Figure 8.

Power surfaces for different values of electrical load resistance.

Figure 9.

Electrical power for fixed values of Ap and an, with variation of the electrical load resistance.

11. CPU time and compute resource characteristics

One type of information that is important to document when developing codes for numerical calculations is the characteristics of the computer equipment and the execution time of the codes. The design calculations were made using a personal computer, which has the characteristics shown in Table 17.

CPU	AMD Ryzen 33,200 U with Radeon Vega Mobile Gfx 2.60 GHz
Installed RAM	8.00 GB (5.95 GB usable)
Type of system	64-bit operating system, x64 processor

Table 17.

Characteristics of the computer equipment used for design calculations.

Figure 10 show the CPU time required for the execution of the calculation code of eqs. (10 and 11).

Figure 10.

CPU time required for the execution of the calculation code of eqs. (10 and 12).

12. Conclusions

This chapter has shown a methodology for the design of thermocouple legs. This methodology is based on the approach of reduced variables. The reduced current density $\left(u\right)$ is the main quantity combined with the thermoelectric potential $\left(\mathrm{\Phi }\right)$ and the thermal conductivity. Then it is possible to calculate the length of the legs (l). Subsequently, setting a fixed value for the total cross-sectional area (An + Ap) and again using the quantities from the reduced variables approach has been possible to calculate the cross-sectional areas corresponding to each of the legs $\left(A_n\right)$ and $\left(A_p\right)$. It is important to note that numerical values of the thermoelectric properties $\left(\alpha ,\rho ,\kappa \right)$ of each of the two materials $\left({\mathit{Zn}}_4{\mathit{Sb}}_3\right)$ y $\left({\mathit{\text{CoSb}}}_3\right)$ were also used to perform the calculations. In order to have support for the classification of thermocouple design tasks, a Table 3 has been prepared; with the help of this resource, the designer can label the type of design being carried out. The methodology is also characterized by considering the dependence of thermoelectric properties on temperature. This aspect allows generating a specific design for a temperature range according to the application required to be served. In the example of the designed thermocouple, notice that the design of the metal bridge was included. In a later work, it is intended to add the sizing of the ceramic plate. The analysis of the power of the thermocouple shows that there are specific values of $\left(A_n\right)$ and $\left(A_p\right)$ that allow it to reach its maximum value. That is, it is possible to determine optimal values of the cross-sectional area of the legs for a particular load resistance $\left(R_{\mathit{\text{load}}}\right)$. Graphs 8 and 9 lead to the conclusion that $\left(R_{\mathit{\text{load}}}\right)$ also determines the maximum power value that the designed system can reach. The Figure 11 shows the steps of the methodology that has been shown in this work for the design of thermocouples.

Figure 11.

Flowchart with the steps to design the thermocouple.

Acknowledgments

This research was funded by Instituto Politecnico Nacional, Mexico grant number 20220099 and CONACYT-Mexico grant number CVU 444915. The APC was funded by Instituto Politecnico Nacional, Mexico.

The authors acknowledge the editorial assistance in improving the manuscript.