1. Introduction

The thermoelectric effect represents direct conversion of the temperature difference into voltage and vice versa and refers to phenomena with which the current flows through the thermoelements or legs of a thermoelectric module. The thermoelectric effect is formed due to free motion of the charge carriers (free electrons e considered as negative charge carriers and holes h+ considered as positive charge carriers) in metals and semiconductors while carrying energy and electric charge. In this case, the electric effects are accompanied by thermal effects and vice versa [1]. The thermoelectric effects are Peltier effect, Thomson effect and Seebeck effect. The Peltier effect is the phenomenon that converts current to temperature and occurs when an

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

electric current flows through a thermoelectric device. The Peltier effect is a reversible phenomenon, because the Peltier heat depends directly on the direction of the carrier flow or electrical current [2].

There is interdependence between the sense of the electric current and the temperature difference at the hot and cold ends of a thermoelectric device. In other words, if the current flow is changed, the temperature at the hot and cold ends is changed as well.

The heat flow rate is given by

$$
\dot{Q}\_{\text{Pellier}} = \pi\_{\text{AB}} \cdot I \cdot T = (\pi\_{\text{B}} - \pi\_{\text{A}}) \cdot I \cdot T \tag{1}
$$

With reference to Figure 1, the sign convention of the Thomson coefficient is positive for heat

Furthermore, the Joule heat Q\_ Joule, in W, is irreversible and takes place in a conductor with electrical current flow, regardless of the direction of the current. Its expression is given by

Joule ¼ R∙I

If a current density J exists through a homogeneous conductor, the heat production per unit

The Seebeck effect converts temperature to current and occurs like the Peltier effect, but the direction of the electric current is reversed. The Seebeck effect appears when a temperature gradient along a conductor provides a voltage increment. In other words, the Seebeck voltage drives the hole/electron flow due to a temperature difference which does exist in the conductor between the high- and low-temperature regions. The Seebeck voltage appearing at the circuit

<sup>¼</sup> <sup>r</sup><sup>∙</sup> <sup>I</sup> S � �<sup>2</sup>

� μ∙ I

ΔV ¼ αAB∙ΔT ¼ ð Þ α<sup>A</sup> � α<sup>B</sup> ∙ð Þ T<sup>H</sup> � T<sup>C</sup> (5)

� μ∙J∙∇T |fflfflffl{zfflfflffl} Q\_ Thomson

<sup>2</sup> (3)

<sup>S</sup> <sup>∙</sup>∇<sup>T</sup> (4)

Thermoelectric Refrigeration Principles http://dx.doi.org/10.5772/intechopen.75439 223

<sup>S</sup> is the current density in A�m�<sup>2</sup>

.

Q\_

absorbed (conductor A) and negative for heat dissipated (conductor B).

Figure 1. Schematic of Peltier effect and Thomson effect in a thermocouple.

where R is the conductor resistance in Ω.

volume or volumetric heat generation is

and S is the cross-sectional area in m<sup>2</sup>

junctions is

q\_

<sup>v</sup> ¼ r∙J

where <sup>r</sup> is the electrical resistivity of the material in <sup>Ω</sup>�m, <sup>J</sup> <sup>¼</sup> <sup>I</sup>

2 |{z} Joule heating

.

where <sup>α</sup>A, <sup>α</sup><sup>B</sup> are the Seebeck coefficients for the conductors A and B, in V�K�<sup>1</sup>

where Q\_ Peltier is the absorbed or dissipated heat flow rate, in W; I is the electric current that flows through the junctions, in A; πAB, π<sup>A</sup> and π<sup>B</sup> are the Peltier coefficients of the thermocouple and conductors A and B, in W�A�<sup>1</sup> ; and T is the absolute temperature, in K.

The Peltier coefficient π is defined as the amount of heat developed or absorbed at a junction of a thermocouple when a current of one ampere passes through this junction for one second. The Peltier coefficient π is positive for heat absorbed and negative for heat dissipated. The Peltier coefficient determines a cooling effect when the current flows from the N-type semiconductor material to a P-type semiconductor material and a heating effect when the current flows from the P-type semiconductor material to an N-type semiconductor material.

The Thomson effect is given by generation or absorption of a heat quantity in a homogeneous conductor by which an electric current flows and where there is a temperature gradient. The heat absorbed or released depends on the electric current direction and the conductor material. The Thomson effect is a reversible thermoelectric phenomenon and is observed when the charge carriers change energy levels.

The convention for the Thomson effect is:


The Thomson heat flow rate Q\_ Thomson is proportional to the thermal gradient as well as to the intensity of the electric current which flows through the conductor:

$$
\dot{Q}\_{\text{Thomass}} = -\mu\_{\text{AB}} \cdot I \cdot \nabla T = \left(\mu\_{\text{B}} - \mu\_{\text{A}}\right) \cdot I \cdot \nabla T \tag{2}
$$

where <sup>μ</sup>AB is the Thomson coefficient in V�K�<sup>1</sup> , I is the electric current flowing through the circuit supplied by a voltage and <sup>∇</sup><sup>T</sup> <sup>¼</sup> <sup>d</sup><sup>T</sup> <sup>d</sup><sup>x</sup> is the temperature gradient along the conductor.

Figure 1. Schematic of Peltier effect and Thomson effect in a thermocouple.

With reference to Figure 1, the sign convention of the Thomson coefficient is positive for heat absorbed (conductor A) and negative for heat dissipated (conductor B).

Furthermore, the Joule heat Q\_ Joule, in W, is irreversible and takes place in a conductor with electrical current flow, regardless of the direction of the current. Its expression is given by

$$
\dot{Q}\_{\text{Joule}} = R \cdot I^2 \tag{3}
$$

where R is the conductor resistance in Ω.

electric current flows through a thermoelectric device. The Peltier effect is a reversible phenomenon, because the Peltier heat depends directly on the direction of the carrier flow or

There is interdependence between the sense of the electric current and the temperature difference at the hot and cold ends of a thermoelectric device. In other words, if the current flow is

where Q\_ Peltier is the absorbed or dissipated heat flow rate, in W; I is the electric current that flows through the junctions, in A; πAB, π<sup>A</sup> and π<sup>B</sup> are the Peltier coefficients of the thermocou-

The Peltier coefficient π is defined as the amount of heat developed or absorbed at a junction of a thermocouple when a current of one ampere passes through this junction for one second. The Peltier coefficient π is positive for heat absorbed and negative for heat dissipated. The Peltier coefficient determines a cooling effect when the current flows from the N-type semiconductor material to a P-type semiconductor material and a heating effect when the current flows from

The Thomson effect is given by generation or absorption of a heat quantity in a homogeneous conductor by which an electric current flows and where there is a temperature gradient. The heat absorbed or released depends on the electric current direction and the conductor material. The Thomson effect is a reversible thermoelectric phenomenon and is observed when the charge

• positive Thomson effect, when the hot end has a high voltage and the cold end has a low voltage; the heat is generated when the current flows from the hotter junction to the colder junction, while the heat is absorbed when the current flows from the colder end to the

• negative Thomson effect, when the hot end has a low voltage and the cold end has a high voltage; the heat is generated when the current flows from the colder junction to the hotter junction, while the heat is absorbed when the current flows from the hotter end to the colder end; some metals have negative Thomson coefficients (e.g. Co, Bi, Fe, and Hg) [3].

Thomson ¼ �μAB∙I∙∇T ¼ μ<sup>B</sup> � μ<sup>A</sup>

Peltier ¼ πAB∙I∙T ¼ ð Þ π<sup>B</sup> � π<sup>A</sup> ∙I∙T (1)

Thomson is proportional to the thermal gradient as well as to the

∙I∙∇T (2)

, I is the electric current flowing through the

<sup>d</sup><sup>x</sup> is the temperature gradient along the conductor.

; and T is the absolute temperature, in K.

changed, the temperature at the hot and cold ends is changed as well.

the P-type semiconductor material to an N-type semiconductor material.

intensity of the electric current which flows through the conductor:

Q\_

where <sup>μ</sup>AB is the Thomson coefficient in V�K�<sup>1</sup>

circuit supplied by a voltage and <sup>∇</sup><sup>T</sup> <sup>¼</sup> <sup>d</sup><sup>T</sup>

Q\_

electrical current [2].

222 Bringing Thermoelectricity into Reality

The heat flow rate is given by

carriers change energy levels.

The Thomson heat flow rate Q\_

hotter end.

The convention for the Thomson effect is:

ple and conductors A and B, in W�A�<sup>1</sup>

If a current density J exists through a homogeneous conductor, the heat production per unit volume or volumetric heat generation is

$$\dot{q}\_{\text{v}} = \underbrace{\rho \cdot \mathbf{J}^2}\_{\text{Joule heating}} - \underbrace{\mu \cdot \mathbf{J} \cdot \nabla T}\_{\text{Jtuam}} = \rho \cdot \left(\frac{I}{S}\right)^2 - \mu \cdot \frac{I}{S} \cdot \nabla T \tag{4}$$

where <sup>r</sup> is the electrical resistivity of the material in <sup>Ω</sup>�m, <sup>J</sup> <sup>¼</sup> <sup>I</sup> <sup>S</sup> is the current density in A�m�<sup>2</sup> and S is the cross-sectional area in m<sup>2</sup> .

The Seebeck effect converts temperature to current and occurs like the Peltier effect, but the direction of the electric current is reversed. The Seebeck effect appears when a temperature gradient along a conductor provides a voltage increment. In other words, the Seebeck voltage drives the hole/electron flow due to a temperature difference which does exist in the conductor between the high- and low-temperature regions. The Seebeck voltage appearing at the circuit junctions is

$$
\Delta V = \alpha\_{\rm AB} \cdot \Delta T = (\alpha\_{\rm A} - \alpha\_{\rm B}) \cdot (T\_{\rm H} - T\_{\rm C}) \tag{5}
$$

where <sup>α</sup>A, <sup>α</sup><sup>B</sup> are the Seebeck coefficients for the conductors A and B, in V�K�<sup>1</sup> . The Seebeck coefficient or thermoelectric power is a very important parameter for the thermoelectric materials, determining the performance of Peltier elements. For a good thermoelectric material, the Seebeck coefficient has to be high in order to obtain the desired voltage more easily, the electrical conductivity has to be high, and the thermal conductivity has to be small to reduce the thermal losses in the junctions of the thermocouple [4].

The relationship of the Seebeck coefficient to the electric field E and the temperature gradient ∇T is

$$
\alpha\_{\rm AB} = \frac{E}{\nabla T} \tag{6}
$$

each other by junctions and are sandwiched between two ceramic substrates to form a thermoelectric module (TEM). These ceramic substrates act as insulator from electrical point of view but allow the thermoelectric couples to be thermally in parallel. The number of thermoelectric couples is influenced by the needed cooling capacity and the maximum electric current [5]. When a low voltage DC power source is applied to the free end of the TEM, the heat flow rate is transferred from one side to other side of the device through the N- and P-semiconductor legs and junctions. In this case, one side of the TEM is cooled, and the other side is heated [7]. In the cooling mode, the sense of the electrical current is from the N-type semiconductor to the P-type semiconductor (Figure 2). The Seebeck voltage is generated in the device when there is a temperature difference between

The direction of the current is then essential to establish the functionality of the device. If the direction of the electrical current is reversed, the compartment would be heated instead of

At the top of every junction, the temperature is the same (Tc), and at the bottom of every junction, the temperature is the same (Th). At the cold junction, the temperature T<sup>c</sup>

Through the cold junction, the electrons are transported from a low energy level inside the P-type semiconductor legs to a high energy level inside the N-type semiconductor

This heat is dissipated at the heat sink (a passive heat exchanger that cools a device by dissipating heat into the environment), and the free electrons flow to an inferior energy

<sup>c</sup> is absorbed from the compartment which must be cooled.

<sup>h</sup> is transferred according to the transport electrons.

Thermoelectric Refrigeration Principles http://dx.doi.org/10.5772/intechopen.75439 225

the junctions of the thermoelements [8].

being cooled.

decreases, and the heat Q\_

legs. At the hot junction, the heat Q\_

level in the P-type semiconductor.

Figure 2. Schematic of a thermoelectric module (TEM) operating in cooling mode.

The sign of the Seebeck coefficient depends on the hole and electron flow:


There is interdependence between the Peltier coefficient and the Seebeck coefficient, as well as between the Seebeck coefficient and the Thomson coefficient, given by the following relationships [5, 6]:

$$
\pi\_{\rm AB} = \alpha\_{\rm AB} \cdot T \tag{7}
$$

$$
\mu\_{\rm AB} = T \cdot \frac{\mathbf{d} \alpha\_{\rm AB}}{\mathbf{d}T} \tag{8}
$$

A thermoelectric cooler (TEC) is a semiconductor composed of an electronic component which transforms electrical energy into a temperature gradient. The TEC consists of one or more thermoelectric couples. A thermoelectric couple is a couple having one P-type thermoelectric leg (an excess of holes h<sup>+</sup> , positive Seebeck coefficient αP, electrical resistivity r<sup>P</sup> and thermal conductivity kN) and one N-type thermoelectric leg (an excess of free electrons e�, negative Seebeck coefficient αN, electrical resistivity r<sup>N</sup> and thermal conductivity k<sup>N</sup> linked to each other by an electrical conductor (a conductive metallic strip) forming a junction. The thermoelectric couples are connected in such a way that when the current flows through the device, both the P-type holes and the N-type electrons move towards the same side of the device.

The two legs are made of two different thermoelectric materials. A thermoelectric material is defined as an alloy of materials that generates thermoelectric properties (thermal conductivity, electric conductivity and Seebeck coefficient). The quality as a semiconductor material to be cooled strictly depends on the transport properties of the material (Seebeck voltage, electrical resistivity and thermal conductivity) as well as the operational temperature field between the cold and hot ends [5]. Considering that the input voltage of a single thermoelectric couple is reduced, many thermoelectric couples are connected to each other by junctions and are sandwiched between two ceramic substrates to form a thermoelectric module (TEM). These ceramic substrates act as insulator from electrical point of view but allow the thermoelectric couples to be thermally in parallel. The number of thermoelectric couples is influenced by the needed cooling capacity and the maximum electric current [5]. When a low voltage DC power source is applied to the free end of the TEM, the heat flow rate is transferred from one side to other side of the device through the N- and P-semiconductor legs and junctions. In this case, one side of the TEM is cooled, and the other side is heated [7]. In the cooling mode, the sense of the electrical current is from the N-type semiconductor to the P-type semiconductor (Figure 2). The Seebeck voltage is generated in the device when there is a temperature difference between the junctions of the thermoelements [8].

The Seebeck coefficient or thermoelectric power is a very important parameter for the thermoelectric materials, determining the performance of Peltier elements. For a good thermoelectric material, the Seebeck coefficient has to be high in order to obtain the desired voltage more easily, the electrical conductivity has to be high, and the thermal conductivity has to be small to

The relationship of the Seebeck coefficient to the electric field E and the temperature gradient

<sup>α</sup>AB <sup>¼</sup> <sup>E</sup>

• A negative Seebeck coefficient is obtained in semiconductors negatively doped (e.g. N-type

• A positive Seebeck coefficient is obtained in semiconductors positively doped (e.g. P-type

There is interdependence between the Peltier coefficient and the Seebeck coefficient, as well as between the Seebeck coefficient and the Thomson coefficient, given by the following relation-

μAB ¼ T∙

A thermoelectric cooler (TEC) is a semiconductor composed of an electronic component which transforms electrical energy into a temperature gradient. The TEC consists of one or more thermoelectric couples. A thermoelectric couple is a couple having one P-type ther-

and thermal conductivity kN) and one N-type thermoelectric leg (an excess of free electrons e�, negative Seebeck coefficient αN, electrical resistivity r<sup>N</sup> and thermal conductivity k<sup>N</sup> linked to each other by an electrical conductor (a conductive metallic strip) forming a junction. The thermoelectric couples are connected in such a way that when the current flows through the device, both the P-type holes and the N-type electrons move towards

The two legs are made of two different thermoelectric materials. A thermoelectric material is defined as an alloy of materials that generates thermoelectric properties (thermal conductivity, electric conductivity and Seebeck coefficient). The quality as a semiconductor material to be cooled strictly depends on the transport properties of the material (Seebeck voltage, electrical resistivity and thermal conductivity) as well as the operational temperature field between the cold and hot ends [5]. Considering that the input voltage of a single thermoelectric couple is reduced, many thermoelectric couples are connected to

dαAB

<sup>∇</sup><sup>T</sup> (6)

πAB ¼ αAB∙T (7)

, positive Seebeck coefficient αP, electrical resistivity r<sup>P</sup>

<sup>d</sup><sup>T</sup> (8)

reduce the thermal losses in the junctions of the thermocouple [4].

The sign of the Seebeck coefficient depends on the hole and electron flow:

∇T is

semiconductors).

224 Bringing Thermoelectricity into Reality

semiconductors).

moelectric leg (an excess of holes h<sup>+</sup>

the same side of the device.

ships [5, 6]:

The direction of the current is then essential to establish the functionality of the device. If the direction of the electrical current is reversed, the compartment would be heated instead of being cooled.

At the top of every junction, the temperature is the same (Tc), and at the bottom of every junction, the temperature is the same (Th). At the cold junction, the temperature T<sup>c</sup> decreases, and the heat Q\_ <sup>c</sup> is absorbed from the compartment which must be cooled. Through the cold junction, the electrons are transported from a low energy level inside the P-type semiconductor legs to a high energy level inside the N-type semiconductor legs. At the hot junction, the heat Q\_ <sup>h</sup> is transferred according to the transport electrons. This heat is dissipated at the heat sink (a passive heat exchanger that cools a device by dissipating heat into the environment), and the free electrons flow to an inferior energy level in the P-type semiconductor.

Figure 2. Schematic of a thermoelectric module (TEM) operating in cooling mode.

2. Theoretical assessment of a thermoelectric cooler

Consider a non-uniformly heated thermoelectric material having isotropic properties (the

!

þ α∙ ∇ ! T |fflffl{zfflffl} Seebeck voltage

> þ α∙T∙ J ! |fflfflffl{zfflfflffl} Peltier heat

> > ! ¼ r∙J

∂T

.

<sup>2</sup> <sup>þ</sup> <sup>J</sup> ! ∙α∙ ∇ !

.

! <sup>þ</sup> <sup>q</sup>\_ vol <sup>¼</sup> <sup>r</sup>∙cp<sup>∙</sup>

where r is the electrical resistivity in Ω�m, c<sup>p</sup> is the specific heat capacity at constant pressure in

þ α∙ ∇ ! T � �<sup>∙</sup> <sup>J</sup>

Based on Thomson's relationship and Osanger's relationship, the heat flow rate vector is

and is expressed as

!

!

!

Thermoelectric Refrigeration Principles http://dx.doi.org/10.5772/intechopen.75439

¼ 0 (9)

<sup>∂</sup><sup>z</sup> (10)

.

T and is expressed as

<sup>∂</sup><sup>t</sup> (13)

T (15)

T (14)

. This electric field depends on the temper-

is the gradient vector, a differential operator with

[2, 14–

227

(11)

(12)

same transport properties in all directions) crossed by a constant current density J

∇ ! ∙ J !

!

∇ ! ¼ i ! ∂ ∂x þ j ! ∂ ∂y þ k ! ∂

2.1. The general heat diffusion equation

respect to the three orthogonal directions:

is the current density vector and ∇

A temperature gradient generates an electric field E

where <sup>k</sup> is the thermal conductivity in W�(m�K)�<sup>1</sup>

The volumetric heat generation is also given by

q\_ vol ¼ E ! ∙ J ! ¼ r∙ J !

The general heat diffusion equation for transient state [14] is

T and on the current density J

E !

> q !¼ �k<sup>∙</sup> <sup>∇</sup> ! T |fflfflfflffl{zfflfflfflffl} Fourier0 s law

� ∇ ! ∙ q

vol is the volumetric heat generation, in W�m�<sup>3</sup>

q !¼ <sup>α</sup>∙T<sup>∙</sup> <sup>J</sup> ! �k∙ ∇ !

Substituting Eq. (14) and Eq. (15) into Eq. (13), with successive elaborations, yields

¼ r∙ J ! |ffl{zffl} Ohm's law

! also depends on the temperature gradient ∇

with the electrical resistivity <sup>r</sup> in <sup>Ω</sup>�m and the Seebeck coefficient <sup>α</sup> in V�K�<sup>1</sup>

16]. The continuity equation is

!

where J !

ature gradient ∇

The heat flow rate q

<sup>J</sup>�(kg�K)�<sup>1</sup> and <sup>q</sup>\_

written as [16]

Figure 3. Schematic of a TEM used for the refrigeration unit.

The main components of a refrigeration unit (Figure 3) are [7, 9, 10]:

