3. Thermoelectric refrigeration in nonequilibrium thermodynamics framework

Theory of thermoelectric cooling is analysed according to out-of-equilibrium thermodynamics. Under isotropic conditions, when an electrical current density flows through the semiconductor material with a temperature gradient and steady-state condition, the heat transport and charge transport relations, consistent with the Onsager theory [17], are

$$j\_{\rm el} = \sigma E - \sigma \alpha \nabla T \tag{8}$$

and

The differential Seebeck coefficient, under open-circuit conditions, is defined as the ratio of the

<sup>α</sup>ab <sup>¼</sup> <sup>V</sup>

Electrons move through the n-type element towards the positive pole, attraction effect, while the negative pole of the voltage source repels them. Likewise, in the p-type semiconductor, the holes move to the negative potential of the voltage source, while positive potential acts as repel of the holes and they move in the contrary direction to the flow of electrons. As a result, in ptype semiconductors, α is positive and in n-type semiconductors, α is negative [15]. Peltier coefficient is equal to the rate of heating or cooling, Q, ratio at each junction to the electric

Peltier coefficient is regarded as positive if the junction at which the current enters is heated and the junction at which it leaves is cooled. When there is both an electric current and a

where x is a spatial coordinate and T the temperature. Thomson coefficient, known as the effect of liberate or absorb heat due to an electric current flux through a semiconductor material in

<sup>τ</sup><sup>a</sup> � <sup>τ</sup><sup>b</sup> <sup>¼</sup> <sup>T</sup> <sup>d</sup>αab

When Seebeck coefficient is considered independent of temperature, Thomson coefficient will

Seebeck effect is a combination of the Peltier and Thomson effects [16]. The relationship between temperature, Peltier, and Seebeck coefficient is given by the next Thomson relation

<sup>τ</sup> <sup>¼</sup> <sup>T</sup> <sup>d</sup><sup>α</sup>

To develop an irreversible thermodynamics theory, Thomson's theory of thermoelectricity plays a remarkable role, because this theorem is the first attempt to develop such theory.

dT

dQ dx <sup>¼</sup> <sup>τ</sup><sup>I</sup>

temperature gradient, the gradient of heat flux in the system is given by

which exist a temperature gradient, is given by the Kelvin relation as follows

These last effects have a relation to the Thomson coefficient, τ, given by

<sup>Δ</sup><sup>T</sup> (2)

Q ¼ πabI (3)

dx (4)

dT (5)

πab ¼ αabT (6)

dT (7)

voltage, V, to the temperature gradient,ΔT

292 Bringing Thermoelectricity into Reality

current, I. The rate of heat exchange at the junction is

not be taken into account in calculations, τ is zero.

2.1. Thomson relations

$$\mathbf{j}\_q = \alpha T \mathbf{j}\_{el} - \kappa \nabla T \tag{9}$$

where, α is the Seebeck coefficient, T is the temperature, κ is the thermal conductivity, E is the electric field, j el is the electric current density, j <sup>q</sup> is the heat flux and σ is the electric conductivity. Equation (9) is the essential equation for thermoelectric phenomena. The governing equations are

$$
\nabla \cdot \mathbf{j}\_{el} = \mathbf{0} \quad \text{and} \quad \nabla \cdot \mathbf{j}\_q = \mathbf{j}\_{el} \cdot E \tag{10}
$$

For one-dimensional model, from Equations (8) and (9), we get for the heat flux

$$\overrightarrow{\nabla} \cdot \left( \kappa \,\overrightarrow{\nabla} \, T \right) + \mathfrak{j}^2 \rho - T \frac{d\alpha}{dT} \, \overrightarrow{J} \cdot \overrightarrow{\nabla} \, T = 0 \tag{11}$$

where r is the electrical resistivity ð Þ r ¼ 1=σ and J is the electric current density. In Equation (11), the first term describes the thermal conduction due to the temperature gradient. According to Fourier's law, the second term is the joule heating and the third term is the Thomson heat, both depending on the electric current density [18]. Now, from Equation (11), the equation that governs the system for one-dimensional steady state is given by:

$$
\kappa(T)\frac{\partial^2 T}{\partial \mathbf{x}^2} + \frac{d\kappa}{dT} \left(\frac{\partial T}{\partial \mathbf{x}}\right)^2 - jT\frac{d\alpha}{dT}\frac{\partial T}{\partial \mathbf{x}} = -\frac{j^2}{\sigma(T)}\tag{12}
$$

#### 3.1. Cooling power

Thermoelectric coolers make use of the Peltier effect which origin resides in the transport of heat by an electric current. For this analysis, we assume that thermal conductivity, electrical resistivity, and Seebeck coefficient are all independent of temperature, that is, CPM model [19], and the metal that connects the p-type with the n-type leg has a low α value, therefore it is considered as zero. We assume that there is zero thermal resistance between the ends of the branches and the heat source and sink. Thus, only electrical resistance is considered for the thermocouple legs, thereby, the thermocouple legs are the only paths to transfer heat between the source and sink, conduction via the ambient, convection, and radiation are ignored. These considerations have been addressed in previous studies showing that the COP does not depend on the semiconductors length when the electrical and thermal contact resistances are not considered in calculations [20]. To determine the coefficient of performance (COP), which is defined as the ratio of the heat extracted from the source to the expenditure of electrical energy, a thermocouple model shown in Figure 1 is used. Thus, for the p-type and n-type legs, the heat transported from the source to the sink is

$$Q\_p = \alpha\_p IT - K\_p A\_p \frac{dT}{dx}; \qquad Q\_n = -\alpha\_n IT - K\_n A\_n \frac{dT}{dx} \tag{13}$$

where A is the cross-sectional area, K is the thermal conductivity, and dT=dx is the temperature gradient. Heat is removed from the source at the rate

$$Q\_c = \left(Q\_p + Q\_n\right)|\_{x=0} \tag{14}$$

W ¼ α<sup>p</sup> � α<sup>n</sup>

then, the coefficient of performance in a TEC system is defined as [21]

4.1. One-dimensional formulation of a physical two-stage TEC

positive to the negative terminal [22, 23].

4.2. TEC electrically connected in series

T1 1ð Þ ;<sup>2</sup> ¼ Tð Þ <sup>1</sup>;<sup>m</sup> ∓ A1 1ð Þ ;<sup>2</sup> x þ

T2 1ð Þ ;<sup>2</sup> ¼ Tð Þ <sup>m</sup>;<sup>2</sup> ∓ A2 1ð Þ ;<sup>2</sup> x þ

, Kij <sup>¼</sup> <sup>κ</sup>ijSij

, Aij <sup>¼</sup> RijI τijLij

where <sup>ω</sup>ij <sup>¼</sup> <sup>τ</sup>ijI

KijLij

4. Thomson effect impact on performance of a two-stage TEC

<sup>I</sup>Δ<sup>T</sup> <sup>þ</sup> <sup>I</sup>

Thermoelectric Cooling: The Thomson Effect in Hybrid Two-Stage Thermoelectric Cooler Systems with Different…

COP <sup>¼</sup> Qc

To determine analytical expressions of cooling power and coefficient of performance in a twostage TE system, we establish one-dimensional representation model, as shown in Figure 2. When a voltage is applied across the device, as a result, an electric current, I, flows from the

T<sup>1</sup> ¼ Tc, Qc1, and Qh<sup>1</sup> are, respectively, cold junction side temperature, amount of heat that can be absorb and amount of heat rejected from stage 1 to 2 of TEC. T<sup>2</sup> ¼ Th, Qh<sup>2</sup> and Qc<sup>2</sup> are, respectively, hot junction temperature, amount of heat rejected to the heat source and amount of heat absorbed from stage 1. It should be noted that Qm is the heat flow from stage 1 to stage 2, that is, Qm ¼ Qh<sup>1</sup> ¼ Qc2, and Tm is the average temperature in the system. For calculations, we use TDPM model [24] in order to show Thomson effect's role in the system. Arranging

Considering model from Figure 2, we get temperature distributions for p-type and n-type semiconductor legs in each stage. T<sup>11</sup> and T<sup>12</sup> are, respectively, the temperatures at the cold side junction for p-type and n-type legs in stage 1. T<sup>21</sup> and T<sup>22</sup> are, respectively, the temperatures at the hot side junction for p-type and n-type legs in stage 2 [25]. Solving with next boundary conditions: T11ð Þ¼ 0 T12ð Þ¼ 0 T<sup>1</sup> and T11ð Þ¼ L<sup>11</sup> T12ð Þ¼ L<sup>12</sup> Tm, we have for the first stage

> ΔT � A1 1ð Þ ;<sup>2</sup> L1 1ð Þ ;<sup>2</sup> <sup>1</sup> � <sup>e</sup><sup>∓</sup> <sup>ω</sup>1 1ð Þ ;<sup>2</sup> <sup>L</sup>1 1ð Þ ;<sup>2</sup>

and for the second stage, with T21ð Þ¼ L<sup>11</sup> T22ð Þ¼ L<sup>12</sup> Tm and T21ð Þ¼ L<sup>21</sup> T22ð Þ¼ L<sup>22</sup> Th

Lij , Rij <sup>¼</sup> Lij

ΔT � A2 1ð Þ ;<sup>2</sup> L2 1ð Þ ;<sup>2</sup> <sup>1</sup> � <sup>e</sup><sup>∓</sup> <sup>ω</sup>2 1ð Þ ;<sup>2</sup> <sup>L</sup>2 1ð Þ ;<sup>2</sup>

σijSij

1 and 2 describe cold temperature and hot temperature in the junctions. According to the theory of non-equilibrium thermodynamics, for the TEMC, we have for the first stage [26],

pairs of elements in this way allows the heat to be pumped in the same direction.

2

R (20)

http://dx.doi.org/10.5772/intechopen.75440

295

<sup>W</sup> (21)

<sup>1</sup> � <sup>e</sup> <sup>∓</sup> <sup>ω</sup>1 1ð Þ ;<sup>2</sup> <sup>x</sup> ð Þ, <sup>0</sup> <sup>≤</sup> <sup>x</sup> <sup>≤</sup> <sup>L</sup>1 1ð Þ ;<sup>2</sup> (22)

<sup>1</sup> � <sup>e</sup><sup>∓</sup> <sup>ω</sup>2 1ð Þ ;<sup>2</sup> <sup>x</sup> ð Þ, L1 1ð Þ ;<sup>2</sup> <sup>≤</sup> <sup>x</sup> <sup>≤</sup> <sup>L</sup>2 1ð Þ ;<sup>2</sup> (23)

for i ¼ 1, 2 and j ¼ 1, 2 when i ≥ j. The subscripts

The rate of generation of heat per unit length from the Joule effect is I 2 r=A. This heat generation implies that there is a non-constant thermal gradient

$$-\kappa\_p A\_p \frac{d^2 T}{d\mathbf{x}^2} = \frac{I^2 \rho\_p}{A\_p}; \qquad -\kappa\_n A\_n \frac{d^2 T}{d\mathbf{x}^2} = \frac{I^2 \rho\_n}{A\_n} \tag{15}$$

Using next boundary conditions: T ¼ T<sup>1</sup> at x ¼ 0 and T ¼ T<sup>2</sup> at x ¼ L, we get

$$\kappa\_{n,p} A\_{n,p} \frac{dT}{d\mathbf{x}} = -\frac{\mathbf{I}^2 \rho\_{n,p} \left(\mathbf{x} - \mathbf{L}\_{n,p}/2\right)}{A\_{n,p}} + \frac{\kappa\_{n,p} A\_{n,p} (T\_2 - T\_1)}{L\_{n,p}} \tag{16}$$

where the subscripts n and p are for the n-type and p-type elements, respectively. From Equation (10), we find for the cooling power at the cold side x ¼ 0

$$Q\_c = (\alpha\_p - \alpha\_n)IT\_1 - K\Delta T - \frac{1}{2}I^2R\tag{17}$$

where ΔT ¼ T<sup>2</sup> � T1. The thermal conductance of the two legs in parallel is

$$K = \frac{\kappa\_p A\_p}{L\_p} + \frac{\kappa\_n A\_n}{L\_n} \tag{18}$$

and the electrical resistance of the two legs in series is

$$R = \frac{L\_p \rho\_p}{A\_p} + \frac{L\_n \rho\_n}{A\_n} \tag{19}$$

#### 3.2. Coefficient of performance

The total power consumption in the TEC system is

Thermoelectric Cooling: The Thomson Effect in Hybrid Two-Stage Thermoelectric Cooler Systems with Different… http://dx.doi.org/10.5772/intechopen.75440 295

$$\mathcal{W} = \left(\alpha\_{\mathcal{V}} - \alpha\_{\mathfrak{n}}\right) I \Delta T + I^2 R \tag{20}$$

then, the coefficient of performance in a TEC system is defined as [21]

$$\text{COP} = \frac{\text{Q}\_c}{\text{W}} \tag{21}$$

#### 4. Thomson effect impact on performance of a two-stage TEC

#### 4.1. One-dimensional formulation of a physical two-stage TEC

To determine analytical expressions of cooling power and coefficient of performance in a twostage TE system, we establish one-dimensional representation model, as shown in Figure 2. When a voltage is applied across the device, as a result, an electric current, I, flows from the positive to the negative terminal [22, 23].

T<sup>1</sup> ¼ Tc, Qc1, and Qh<sup>1</sup> are, respectively, cold junction side temperature, amount of heat that can be absorb and amount of heat rejected from stage 1 to 2 of TEC. T<sup>2</sup> ¼ Th, Qh<sup>2</sup> and Qc<sup>2</sup> are, respectively, hot junction temperature, amount of heat rejected to the heat source and amount of heat absorbed from stage 1. It should be noted that Qm is the heat flow from stage 1 to stage 2, that is, Qm ¼ Qh<sup>1</sup> ¼ Qc2, and Tm is the average temperature in the system. For calculations, we use TDPM model [24] in order to show Thomson effect's role in the system. Arranging pairs of elements in this way allows the heat to be pumped in the same direction.

#### 4.2. TEC electrically connected in series

considerations have been addressed in previous studies showing that the COP does not depend on the semiconductors length when the electrical and thermal contact resistances are not considered in calculations [20]. To determine the coefficient of performance (COP), which is defined as the ratio of the heat extracted from the source to the expenditure of electrical energy, a thermocouple model shown in Figure 1 is used. Thus, for the p-type and n-type legs,

where A is the cross-sectional area, K is the thermal conductivity, and dT=dx is the temperature

; � κnAn

<sup>r</sup>n,p <sup>x</sup> � Ln,p=<sup>2</sup> An, <sup>p</sup>

where the subscripts n and p are for the n-type and p-type elements, respectively. From

IT<sup>1</sup> � <sup>K</sup>Δ<sup>T</sup> � <sup>1</sup>

d2 T dx<sup>2</sup> <sup>¼</sup> <sup>I</sup>

þ

2 I 2

2 rn An

κn,pAn, <sup>p</sup>ð Þ T<sup>2</sup> � T<sup>1</sup> Ln, <sup>p</sup>

Qc ¼ Qp þ Qn 

> 2 rp Ap

Using next boundary conditions: T ¼ T<sup>1</sup> at x ¼ 0 and T ¼ T<sup>2</sup> at x ¼ L, we get

2

Qc ¼ α<sup>p</sup> � α<sup>n</sup>

where ΔT ¼ T<sup>2</sup> � T1. The thermal conductance of the two legs in parallel is

<sup>K</sup> <sup>¼</sup> <sup>κ</sup>pAp Lp þ κnAn Ln

<sup>R</sup> <sup>¼</sup> Lpr<sup>p</sup> Ap þ Lnr<sup>n</sup> An

dx ; Qn ¼ �αnIT � KnAn

dT

j<sup>x</sup>¼<sup>0</sup> (14)

2

dx (13)

r=A. This heat genera-

R (17)

(15)

(16)

(18)

(19)

dT

the heat transported from the source to the sink is

294 Bringing Thermoelectricity into Reality

gradient. Heat is removed from the source at the rate

tion implies that there is a non-constant thermal gradient

�κpAp

dT dx ¼ � <sup>I</sup>

Equation (10), we find for the cooling power at the cold side x ¼ 0

κn, pAn,p

and the electrical resistance of the two legs in series is

The total power consumption in the TEC system is

3.2. Coefficient of performance

Qp ¼ αpIT � KpAp

The rate of generation of heat per unit length from the Joule effect is I

d2 T dx<sup>2</sup> <sup>¼</sup> <sup>I</sup>

> Considering model from Figure 2, we get temperature distributions for p-type and n-type semiconductor legs in each stage. T<sup>11</sup> and T<sup>12</sup> are, respectively, the temperatures at the cold side junction for p-type and n-type legs in stage 1. T<sup>21</sup> and T<sup>22</sup> are, respectively, the temperatures at the hot side junction for p-type and n-type legs in stage 2 [25]. Solving with next boundary conditions: T11ð Þ¼ 0 T12ð Þ¼ 0 T<sup>1</sup> and T11ð Þ¼ L<sup>11</sup> T12ð Þ¼ L<sup>12</sup> Tm, we have for the first stage

$$T\_{1(1,2)} = T\_{(1,m)} \mp A\_{1(1,2)} \mathbf{x} + \frac{\Delta T \pm A\_{1(1,2)} L\_{1(1,2)}}{1 - e^{\mp \omega\_{1(1,2)} L\_{1(1,2)}}} (1 - e^{\mp \omega\_{1(1,2)} x}), 0 \le \mathbf{x} \le L\_{1(1,2)} \tag{22}$$

and for the second stage, with T21ð Þ¼ L<sup>11</sup> T22ð Þ¼ L<sup>12</sup> Tm and T21ð Þ¼ L<sup>21</sup> T22ð Þ¼ L<sup>22</sup> Th

$$T\_{2(1,2)} = T\_{(m,2)} \mp A\_{2(1,2)} \mathbf{x} + \frac{\Delta T \pm A\_{2(1,2)} L\_{2(1,2)}}{1 - e^{\mp a\_{2(1,2)} L\_{2(1,2)}}} (1 - e^{\mp a\_{2(1,2)} \mathbf{x}}), L\_{1(1,2)} \le \mathbf{x} \le L\_{2(1,2)}\tag{23}$$

where <sup>ω</sup>ij <sup>¼</sup> <sup>τ</sup>ijI KijLij , Aij <sup>¼</sup> RijI τijLij , Kij <sup>¼</sup> <sup>κ</sup>ijSij Lij , Rij <sup>¼</sup> Lij σijSij for i ¼ 1, 2 and j ¼ 1, 2 when i ≥ j. The subscripts 1 and 2 describe cold temperature and hot temperature in the junctions. According to the theory of non-equilibrium thermodynamics, for the TEMC, we have for the first stage [26],

Figure 2. Two-stage thermoelectric cooler (TEC), electrically connected in series.

$$Q\_{c1} = \alpha\_1^c T\_c I - K\_1^\*(T\_m - T\_c) - \left(R\_1^\* + R\_1\right) I^2 \tag{24}$$

Tm <sup>¼</sup> <sup>I</sup>

<sup>j</sup><sup>2</sup> , K<sup>∗</sup>

where K<sup>∗</sup>

son effect.

<sup>1610</sup> <sup>T</sup> � <sup>2</sup>:<sup>3</sup> <sup>T</sup><sup>2</sup>

R∗ <sup>j</sup><sup>2</sup> ¼ Rj<sup>2</sup> <sup>j</sup> <sup>¼</sup> <sup>K</sup><sup>∗</sup>

1 <sup>ω</sup>j2Lj<sup>2</sup> � <sup>1</sup> e <sup>ω</sup>j2Lj<sup>2</sup> �<sup>1</sup>

<sup>j</sup><sup>1</sup> <sup>þ</sup> <sup>K</sup><sup>∗</sup>

Table 1. Properties of thermoelectric (TE) elements.

Figure 3. COP and Qc in function of the ratio <sup>τ</sup><sup>r</sup> <sup>¼</sup> <sup>τ</sup><sup>1</sup>

<sup>2</sup> R<sup>∗</sup> <sup>1</sup> � <sup>R</sup><sup>∗</sup>

<sup>j</sup><sup>1</sup> <sup>¼</sup> <sup>τ</sup>j1<sup>I</sup> 1�e

<sup>2</sup> � R<sup>2</sup> � Tc <sup>K</sup><sup>∗</sup>

�ωj1Lj<sup>1</sup> , K<sup>∗</sup>

<sup>2</sup> � τ<sup>1</sup> � <sup>K</sup><sup>∗</sup>

Thermoelectric Cooling: The Thomson Effect in Hybrid Two-Stage Thermoelectric Cooler Systems with Different…

between both stages according to average temperature Tm, which also depends on the Thom-

Two different materials were used for calculations, thermoelectric properties are shown in

With <sup>α</sup><sup>1</sup> <sup>¼</sup> <sup>2</sup> � <sup>10</sup>�<sup>4</sup> <sup>þ</sup> <sup>2</sup> � <sup>10</sup>�<sup>2</sup> ð Þ <sup>1</sup>=Tm � <sup>1</sup>=<sup>T</sup> for material Bi2Te<sup>3</sup> and <sup>α</sup><sup>2</sup> ¼ �½ <sup>62675</sup> <sup>þ</sup>

Figure 3 shows the COP and the cooling power Qc, in function of τr, at different electric current values Bi2Te<sup>3</sup> and ð Þ Bi<sup>0</sup>:<sup>5</sup>Sb<sup>0</sup>:<sup>5</sup> Te<sup>3</sup> [28]. It is clear that COP behaviour is influenced directly by the Thomson effect ratio of both stages. COP values increase when there is an increase in the ratio τ<sup>r</sup> for lower values of the electric current I. We must notice that for lower values of τ<sup>r</sup> < 1, COP values are very closely one with another, with a maximum difference of 17% as compared an

<sup>j</sup><sup>2</sup> <sup>¼</sup> <sup>τ</sup>j2<sup>I</sup> e

for <sup>j</sup> <sup>¼</sup> <sup>1</sup>, 2. Once again we must notice the relationship that exists

I α<sup>m</sup> <sup>1</sup> � <sup>α</sup><sup>m</sup>

4.2.1. Influence of Thomson effect on performance (COP) and cooling power (Qc)

Table 1, where only Seebeck coefficient is consider that depends on temperature.

� � <sup>10</sup>�<sup>6</sup> and for material ð Þ Bi<sup>0</sup>:<sup>5</sup>Sb<sup>0</sup>:<sup>5</sup> Te<sup>3</sup> [25].

τ2

, for different electric current values I.

<sup>1</sup> <sup>þ</sup> <sup>τ</sup>1<sup>I</sup> � <sup>K</sup><sup>∗</sup>

<sup>1</sup> <sup>þ</sup> <sup>K</sup><sup>∗</sup> 2

<sup>ω</sup>j2Lj<sup>2</sup> �<sup>1</sup> , <sup>R</sup><sup>∗</sup>

<sup>2</sup>Th

<sup>j</sup><sup>1</sup> ¼ Rj<sup>1</sup>

(29)

1 1�e

http://dx.doi.org/10.5772/intechopen.75440

�ωj1Lj<sup>1</sup> � <sup>1</sup> ωj1Lj<sup>1</sup> and 297

$$Q\_{h1} = \alpha\_1^m T\_m I - K\_1^\*(T\_m - T\_c) - \pi\_1(T\_m - T\_c) - R\_1^\* I^2 \tag{25}$$

$$Q\_{c2} = \alpha\_2^m T\_m I - K\_2^\*(T\_h - T\_m) - \left(R\_2^\* + R\_2\right)I^2 \tag{26}$$

$$Q\_{h2} = \alpha\_2^h T\_h I - K\_2^\*(T\_h - T\_m) - \tau\_2(T\_h - T\_m) - R\_2^\* I^2 \tag{27}$$

with α<sup>k</sup> <sup>1</sup> <sup>¼</sup> <sup>α</sup><sup>k</sup> <sup>12</sup> � <sup>α</sup><sup>k</sup> <sup>11</sup>, for <sup>k</sup> <sup>¼</sup> c, m and <sup>α</sup><sup>l</sup> <sup>2</sup> <sup>¼</sup> <sup>α</sup><sup>l</sup> <sup>22</sup> � <sup>α</sup><sup>l</sup> <sup>21</sup>, for l ¼ m, h; Rj ¼ Rj<sup>1</sup> þ Rj2; τ<sup>j</sup> ¼ τj<sup>2</sup> � τj<sup>1</sup> and R<sup>∗</sup> <sup>j</sup> <sup>¼</sup> <sup>R</sup><sup>∗</sup> <sup>j</sup><sup>1</sup> <sup>þ</sup> <sup>R</sup><sup>∗</sup> <sup>j</sup><sup>2</sup> � Rj<sup>1</sup> þ Rj<sup>2</sup> � � h i for <sup>j</sup> <sup>¼</sup> <sup>1</sup>, 2. A general solution for the heat fluxes in two-stage system is found in [27] where Thomson effect is studied. The system's coefficient of performance, COP, is determined by Qc and Qh as follows

$$\text{COP} = \frac{a\_2^h T\_h I - K\_2^\*(T\_h - T\_m) - \tau\_2 (T\_h - T\_m) - R\_2^\* I^2}{K\_1^\*(T\_m - T\_c) + (T\_h - T\_m) \left(-\tau\_2 I - K\_2^\*\right) + \left(a\_2^h T\_h - a\_1^c T\_c\right) I + \left(R\_1^\* + R\_1 - R\_2^\*\right) I^2} \tag{28}$$

Performance depends on Thomson coefficients values of both the first stage and the second stage. In our results, we show the role of the ratio values of the Thomson coefficients, τ<sup>r</sup> ¼ τ1=τ<sup>2</sup> between stages, on performance. Now solving for Tm, knowing that Qm ¼ Qh<sup>1</sup> ¼ Qc2, from Equations (25) and (26)

Thermoelectric Cooling: The Thomson Effect in Hybrid Two-Stage Thermoelectric Cooler Systems with Different… http://dx.doi.org/10.5772/intechopen.75440 297

$$T\_m = \frac{I^2 \left(R\_1^\* - R\_2^\* - R\_2\right) - T\_c \left(K\_1^\* + \tau\_1 I\right) - K\_2^\* T\_h}{I \left(\alpha\_1^m - \alpha\_2^m - \tau\_1\right) - \left(K\_1^\* + K\_2^\*\right)}\tag{29}$$

where K<sup>∗</sup> <sup>j</sup> <sup>¼</sup> <sup>K</sup><sup>∗</sup> <sup>j</sup><sup>1</sup> <sup>þ</sup> <sup>K</sup><sup>∗</sup> <sup>j</sup><sup>2</sup> , K<sup>∗</sup> <sup>j</sup><sup>1</sup> <sup>¼</sup> <sup>τ</sup>j1<sup>I</sup> 1�e �ωj1Lj<sup>1</sup> , K<sup>∗</sup> <sup>j</sup><sup>2</sup> <sup>¼</sup> <sup>τ</sup>j2<sup>I</sup> e <sup>ω</sup>j2Lj<sup>2</sup> �<sup>1</sup> , <sup>R</sup><sup>∗</sup> <sup>j</sup><sup>1</sup> ¼ Rj<sup>1</sup> 1 1�e �ωj1Lj<sup>1</sup> � <sup>1</sup> ωj1Lj<sup>1</sup> and R∗ <sup>j</sup><sup>2</sup> ¼ Rj<sup>2</sup> 1 <sup>ω</sup>j2Lj<sup>2</sup> � <sup>1</sup> e <sup>ω</sup>j2Lj<sup>2</sup> �<sup>1</sup> for <sup>j</sup> <sup>¼</sup> <sup>1</sup>, 2. Once again we must notice the relationship that exists between both stages according to average temperature Tm, which also depends on the Thomson effect.

#### 4.2.1. Influence of Thomson effect on performance (COP) and cooling power (Qc)

Two different materials were used for calculations, thermoelectric properties are shown in Table 1, where only Seebeck coefficient is consider that depends on temperature.

With <sup>α</sup><sup>1</sup> <sup>¼</sup> <sup>2</sup> � <sup>10</sup>�<sup>4</sup> <sup>þ</sup> <sup>2</sup> � <sup>10</sup>�<sup>2</sup> ð Þ <sup>1</sup>=Tm � <sup>1</sup>=<sup>T</sup> for material Bi2Te<sup>3</sup> and <sup>α</sup><sup>2</sup> ¼ �½ <sup>62675</sup> <sup>þ</sup> <sup>1610</sup> <sup>T</sup> � <sup>2</sup>:<sup>3</sup> <sup>T</sup><sup>2</sup> � � <sup>10</sup>�<sup>6</sup> and for material ð Þ Bi<sup>0</sup>:<sup>5</sup>Sb<sup>0</sup>:<sup>5</sup> Te<sup>3</sup> [25].

Figure 3 shows the COP and the cooling power Qc, in function of τr, at different electric current values Bi2Te<sup>3</sup> and ð Þ Bi<sup>0</sup>:<sup>5</sup>Sb<sup>0</sup>:<sup>5</sup> Te<sup>3</sup> [28]. It is clear that COP behaviour is influenced directly by the Thomson effect ratio of both stages. COP values increase when there is an increase in the ratio τ<sup>r</sup> for lower values of the electric current I. We must notice that for lower values of τ<sup>r</sup> < 1, COP values are very closely one with another, with a maximum difference of 17% as compared an


Table 1. Properties of thermoelectric (TE) elements.

Qc<sup>1</sup> <sup>¼</sup> <sup>α</sup><sup>c</sup>

Figure 2. Two-stage thermoelectric cooler (TEC), electrically connected in series.

Qc<sup>2</sup> <sup>¼</sup> <sup>α</sup><sup>m</sup>

Qh<sup>1</sup> <sup>¼</sup> <sup>α</sup><sup>m</sup>

Qh<sup>2</sup> <sup>¼</sup> <sup>α</sup><sup>h</sup>

<sup>j</sup><sup>2</sup> � Rj<sup>1</sup> þ Rj<sup>2</sup> � � h i

Qm ¼ Qh<sup>1</sup> ¼ Qc2, from Equations (25) and (26)

<sup>11</sup>, for <sup>k</sup> <sup>¼</sup> c, m and <sup>α</sup><sup>l</sup>

performance, COP, is determined by Qc and Qh as follows

<sup>1</sup>ð Þþ Tm � Tc ð Þ� Th � Tm <sup>τ</sup>2<sup>I</sup> � <sup>K</sup><sup>∗</sup>

with α<sup>k</sup>

and R<sup>∗</sup>

<sup>1</sup> <sup>¼</sup> <sup>α</sup><sup>k</sup>

<sup>j</sup> <sup>¼</sup> <sup>R</sup><sup>∗</sup>

<sup>12</sup> � <sup>α</sup><sup>k</sup>

296 Bringing Thermoelectricity into Reality

<sup>j</sup><sup>1</sup> <sup>þ</sup> <sup>R</sup><sup>∗</sup>

COP <sup>¼</sup> <sup>α</sup><sup>h</sup>

K∗

<sup>1</sup>TcI � <sup>K</sup><sup>∗</sup>

<sup>2</sup> TmI � <sup>K</sup><sup>∗</sup>

<sup>2</sup> <sup>¼</sup> <sup>α</sup><sup>l</sup>

<sup>22</sup> � <sup>α</sup><sup>l</sup>

two-stage system is found in [27] where Thomson effect is studied. The system's coefficient of

2 � � <sup>þ</sup> <sup>α</sup><sup>h</sup>

Performance depends on Thomson coefficients values of both the first stage and the second stage. In our results, we show the role of the ratio values of the Thomson coefficients, τ<sup>r</sup> ¼ τ1=τ<sup>2</sup> between stages, on performance. Now solving for Tm, knowing that

<sup>1</sup> TmI � <sup>K</sup><sup>∗</sup>

<sup>2</sup>ThI � <sup>K</sup><sup>∗</sup>

<sup>2</sup>ThI � <sup>K</sup><sup>∗</sup>

<sup>1</sup>ð Þ� Tm � Tc <sup>R</sup><sup>∗</sup>

<sup>2</sup>ð Þ� Th � Tm <sup>R</sup><sup>∗</sup>

<sup>1</sup>ð Þ� Tm � Tc <sup>τ</sup>1ð Þ� Tm � Tc <sup>R</sup><sup>∗</sup>

<sup>2</sup>ð Þ� Th � Tm <sup>τ</sup>2ð Þ� Th � Tm <sup>R</sup><sup>∗</sup>

<sup>2</sup>ð Þ� Th � Tm <sup>τ</sup>2ð Þ� Th � Tm <sup>R</sup><sup>∗</sup>

<sup>2</sup>Th � <sup>α</sup><sup>c</sup>

<sup>1</sup>Tc � �<sup>I</sup> <sup>þ</sup> <sup>R</sup><sup>∗</sup>

<sup>1</sup> þ R<sup>1</sup> � �I

<sup>2</sup> þ R<sup>2</sup> � �I

for j ¼ 1, 2. A general solution for the heat fluxes in

2I 2

1I

2I

<sup>21</sup>, for l ¼ m, h; Rj ¼ Rj<sup>1</sup> þ Rj2; τ<sup>j</sup> ¼ τj<sup>2</sup> � τj<sup>1</sup>

<sup>1</sup> <sup>þ</sup> <sup>R</sup><sup>1</sup> � <sup>R</sup><sup>∗</sup>

� �I

2

<sup>2</sup> (28)

<sup>2</sup> (24)

<sup>2</sup> (26)

<sup>2</sup> (25)

<sup>2</sup> (27)

Figure 3. COP and Qc in function of the ratio <sup>τ</sup><sup>r</sup> <sup>¼</sup> <sup>τ</sup><sup>1</sup> τ2 , for different electric current values I.

electric current value of 1 A with an electric current of 4 A, when τ<sup>r</sup> ¼ 1. Moreover, it is observed that from values of τ<sup>r</sup> > 2 COP values increase for all the different electric current values.

Similar behaviour, to what happens with the performance COP, happens for the cooling power Qc, where maximum values are obtained for higher values of τr, as shown in Figure 3. In this case, Qc value for an electric current value of 1 A is 11 % higher compared with electric current values of 4 A, when τ<sup>r</sup> ¼ 1.

#### 4.3. TECs electrically connected in parallel

Now, we analyse the case in which different electric currents flow in each stage of the system (Figure 4). The ratio of electric currents between each stage is given by

$$I\_r = \frac{I\_1}{I\_2} \tag{30}$$

Qm<sup>2</sup> <sup>¼</sup> <sup>α</sup><sup>m</sup>

Qh <sup>¼</sup> <sup>α</sup><sup>h</sup>

(32) and (33), we solve for the average temperature, Tm

The system's coefficient of performance, COP, is given by

Tm <sup>¼</sup> <sup>R</sup><sup>∗</sup> 1I1

COP <sup>¼</sup> <sup>α</sup><sup>h</sup>

<sup>2</sup>ThI<sup>2</sup> � <sup>K</sup><sup>∗</sup>

Figure 5. COP in function of the ratio Ir <sup>¼</sup> <sup>I</sup><sup>1</sup>

I2

, for different τrvalues.

αh

<sup>2</sup> TmI<sup>2</sup> � <sup>K</sup><sup>∗</sup>

<sup>2</sup> � <sup>τ</sup>1TcI<sup>1</sup> � <sup>K</sup><sup>∗</sup>

I<sup>1</sup> α<sup>m</sup> <sup>1</sup> � τ<sup>1</sup> � <sup>K</sup><sup>∗</sup>

<sup>2</sup>ThI<sup>2</sup> � <sup>K</sup><sup>∗</sup>

<sup>2</sup>ð Þ� Th � Tm <sup>τ</sup>2ð Þ� Th � Tm <sup>R</sup><sup>∗</sup>

� <sup>α</sup><sup>c</sup>

where I<sup>1</sup> is the electric current flow in stage 1 and I<sup>2</sup> is the electric current flow in stage 2.

According to the continuity of the heat flow between both stages, Qm<sup>1</sup> ¼ Qm2, from Equations

<sup>1</sup>Tc � <sup>K</sup><sup>∗</sup>

<sup>1</sup> <sup>þ</sup> <sup>K</sup><sup>∗</sup> 2 � <sup>α</sup><sup>m</sup>

<sup>2</sup>ð Þ� Th � Tm <sup>τ</sup>2ð Þ� Th � Tm <sup>R</sup><sup>∗</sup>

2I 2 2

In the previous section, it is shown that COP increases for higher values of Thomson coefficient ratio between both stages. The behaviour of the COP for the case where two different electric currents flow in the system, shown in Figure 5, is now analysed. Three different values of

<sup>2</sup>Th � <sup>R</sup><sup>∗</sup>

<sup>2</sup> þ R<sup>2</sup> I<sup>2</sup>

<sup>2</sup> I<sup>2</sup>

<sup>1</sup>TcI<sup>1</sup> � <sup>K</sup><sup>∗</sup>

<sup>2</sup>ThI<sup>2</sup> � <sup>K</sup><sup>∗</sup>

<sup>2</sup>ð Þ� Th � Tm <sup>R</sup><sup>∗</sup>

Thermoelectric Cooling: The Thomson Effect in Hybrid Two-Stage Thermoelectric Cooler Systems with Different…

<sup>2</sup>ð Þ� Th � Tm <sup>τ</sup>2ð Þ� Th � Tm <sup>R</sup><sup>∗</sup>

<sup>2</sup> þ R<sup>2</sup> I<sup>2</sup>

2I2

2

2I 2 2

<sup>1</sup>ð Þ� Tm � Tc <sup>R</sup><sup>∗</sup>

<sup>2</sup> (33)

http://dx.doi.org/10.5772/intechopen.75440

<sup>2</sup> (34)

(35)

299

2 1 (36)

<sup>1</sup> þ R<sup>1</sup> I

$$Q\_c = \alpha\_1^c T\_c I\_1 - K\_1^\*(T\_m - T\_c) - \left(R\_1^\* + R\_1\right) I\_1^{'2} \tag{31}$$

$$Q\_{m1} = \alpha\_1^m T\_m I\_1 - K\_1^\*(T\_m - T\_c) - \tau\_1 (T\_m - T\_c) - R\_1^\* I\_1^{'2} \tag{32}$$

Figure 4. Two-stage thermoelectric cooler (TEC), electrically connected in parallel. Now, in the same way as in the previous section, we solve for the heat fluxes in the system.

Thermoelectric Cooling: The Thomson Effect in Hybrid Two-Stage Thermoelectric Cooler Systems with Different… http://dx.doi.org/10.5772/intechopen.75440 299

$$Q\_{m2} = \alpha\_2^m T\_m I\_2 - K\_2^\*(T\_h - T\_m) - \left(R\_2^\* + R\_2\right) I\_2^{'2} \tag{33}$$

$$Q\_{\rm li} = \alpha\_2^h T\_h I\_2 - K\_2^\*(T\_h - T\_m) - \tau\_2 (T\_h - T\_m) - R\_2^\* I\_2^{\;2} \tag{34}$$

where I<sup>1</sup> is the electric current flow in stage 1 and I<sup>2</sup> is the electric current flow in stage 2.

According to the continuity of the heat flow between both stages, Qm<sup>1</sup> ¼ Qm2, from Equations (32) and (33), we solve for the average temperature, Tm

$$T\_m = \frac{R\_1^\* I\_1^{-2} - \tau\_1 T\_c I\_1 - K\_1^\* T\_c - K\_2^\* T\_h - \left(R\_2^\* + R\_2\right) I\_2^{-2}}{I\_1 \left(\alpha\_1^m - \tau\_1\right) - \left(K\_1^\* + K\_2^\*\right) - \alpha\_2^m I\_2} \tag{35}$$

The system's coefficient of performance, COP, is given by

electric current value of 1 A with an electric current of 4 A, when τ<sup>r</sup> ¼ 1. Moreover, it is observed that from values of τ<sup>r</sup> > 2 COP values increase for all the different electric current values.

Similar behaviour, to what happens with the performance COP, happens for the cooling power Qc, where maximum values are obtained for higher values of τr, as shown in Figure 3. In this case, Qc value for an electric current value of 1 A is 11 % higher compared with electric current

Now, we analyse the case in which different electric currents flow in each stage of the system

Ir <sup>¼</sup> <sup>I</sup><sup>1</sup> I2

Figure 4. Two-stage thermoelectric cooler (TEC), electrically connected in parallel. Now, in the same way as in the

<sup>1</sup>ð Þ� Tm � Tc <sup>R</sup><sup>∗</sup>

<sup>1</sup>ð Þ� Tm � Tc <sup>τ</sup>1ð Þ� Tm � Tc <sup>R</sup><sup>∗</sup>

<sup>1</sup> þ R<sup>1</sup> I<sup>1</sup>

1I1

(30)

<sup>2</sup> (31)

<sup>2</sup> (32)

(Figure 4). The ratio of electric currents between each stage is given by

<sup>1</sup> TmI<sup>1</sup> � <sup>K</sup><sup>∗</sup>

<sup>1</sup>TcI<sup>1</sup> � <sup>K</sup><sup>∗</sup>

Qc <sup>¼</sup> <sup>α</sup><sup>c</sup>

Qm<sup>1</sup> <sup>¼</sup> <sup>α</sup><sup>m</sup>

previous section, we solve for the heat fluxes in the system.

values of 4 A, when τ<sup>r</sup> ¼ 1.

298 Bringing Thermoelectricity into Reality

4.3. TECs electrically connected in parallel

$$\text{COP} = \frac{\alpha\_2^h T\_h I\_2 - K\_2^\*(T\_h - T\_m) - \tau\_2(T\_h - T\_m) - R\_2^\* I\_2^2}{\left(\alpha\_2^h T\_h I\_2 - K\_2^\*(T\_h - T\_m) - \tau\_2(T\_h - T\_m) - R\_2^\* I\_2^2\right) - \alpha\_1^c T\_c I\_1 - K\_1^\*(T\_m - T\_c) - \left(R\_1^\* + R\_1\right)I\_1^2} \tag{36}$$

In the previous section, it is shown that COP increases for higher values of Thomson coefficient ratio between both stages. The behaviour of the COP for the case where two different electric currents flow in the system, shown in Figure 5, is now analysed. Three different values of

Figure 5. COP in function of the ratio Ir <sup>¼</sup> <sup>I</sup><sup>1</sup> I2 , for different τrvalues.


<sup>γ</sup> <sup>¼</sup> <sup>I</sup> 2 R Aκ <sup>Δ</sup><sup>T</sup> L

5.1. Cooling power: the ideal equation and Thomson effect (τ)

Q\_ IE

<sup>c</sup> <sup>¼</sup> <sup>α</sup>T1<sup>I</sup> � <sup>1</sup>

Q\_ β

Figure 6. Schematic diagram of a thermocouple.

The resulting equation considering the Thomson effect is given by:

2 I 2 <sup>R</sup> � Ak

5.2. Geometric parameter between stages: area-length ratio (W = w1/w2)

Equation (14), which is called the ideal equation (IE) for cooling power

<sup>c</sup> <sup>¼</sup> <sup>α</sup>T1<sup>I</sup> � <sup>1</sup>

and <sup>ϕ</sup> <sup>¼</sup> <sup>Δ</sup><sup>T</sup>

Thermoelectric Cooling: The Thomson Effect in Hybrid Two-Stage Thermoelectric Cooler Systems with Different…

If we consider Seebeck coefficient independent of temperature, Thomson coefficient is negligible (β ¼ 0), we can obtain the exact result for the cooling power at the cold junction from

> 2 I 2 <sup>R</sup> � Ak

<sup>L</sup> ð Þþ <sup>T</sup><sup>2</sup> � <sup>T</sup><sup>1</sup> <sup>β</sup>

Figure 6 shows a simple thermocouple with length, L and cross-sectional area, A. Previous studies proved that an improvement on performance of TECs can be achieved by optimizing

T2

Ak

<sup>L</sup> ð Þ <sup>T</sup><sup>2</sup> � <sup>T</sup><sup>1</sup> (41)

http://dx.doi.org/10.5772/intechopen.75440

<sup>L</sup> ð Þ <sup>T</sup><sup>2</sup> � <sup>T</sup><sup>1</sup> (42)

(40)

301

Table 2. Maximum values of COP.

Thomson coefficients, τr, are considered. Table 2 shows maximum values, from Figure 5, for COP in function of the electric current ratio between both stages, Ir. Maximum COP value is obtained for higher values of the ratio τr, that is, a higher value of the electric current I<sup>2</sup> > I<sup>1</sup> is desirable to be able to achieve better COP.

#### 5. Dimensionless equations of a two-stage thermoelectric micro-cooler

Once it has been investigated the role of the Thomson heat on TEC performance, now a procedure to improve the performance of the micro-cooler based on optimum geometric parameters, cross-sectional area (A) and length (L), of the semiconductor elements is proposed. To optimal design of a TEMC, theoretical basis on optimal geometric parameters (of the p-type and n-type semiconductor legs) is required. Next analysis of a TEMC includes these optimization parameters. The configuration of a two-stage TE system considered in this work is shown in Figure 2. Each stage is made of different thermoelectric semiconductor materials. In order to make Equation (12) dimensionless using the boundary conditions Tð Þ¼ 0 T<sup>1</sup> and T Lð Þ¼ T2, in accordance with Figure 2, we define the dimensionless temperature, θ, and the ξ parameter as,

$$\theta = \frac{T - T\_1}{T\_2 - T\_1} \text{ and } \xi = \frac{\chi}{L} \tag{37}$$

Dimensionless differential equation corresponding to Equation (12) is given by:

$$\frac{d^2\theta}{d\xi^2} - \beta((\theta - 1)\phi + 1)\frac{d\theta}{d\xi} + \gamma = 0\tag{38}$$

where

$$\beta = \frac{I T\_2 \frac{d\alpha}{dT} \Delta T}{A \kappa \frac{\Delta T}{L}} \tag{39}$$

that is, β is the relation between Thomson heat with thermal conduction. From Equation (38), if β ¼ 0, we get the ideal equation (IE) when Thomson effect not considered. Dimensionless parameter, γ, is the relation between Joule heating to the thermal conduction, and the parameter ϕ, which is the ratio of temperature difference to the high junction temperature, defined as: Thermoelectric Cooling: The Thomson Effect in Hybrid Two-Stage Thermoelectric Cooler Systems with Different… http://dx.doi.org/10.5772/intechopen.75440 301

$$
\gamma = \frac{I^2 R}{A \kappa \frac{\Delta T}{L}} \quad \text{and} \quad \phi = \frac{\Delta T}{T\_2} \tag{40}
$$

#### 5.1. Cooling power: the ideal equation and Thomson effect (τ)

If we consider Seebeck coefficient independent of temperature, Thomson coefficient is negligible (β ¼ 0), we can obtain the exact result for the cooling power at the cold junction from Equation (14), which is called the ideal equation (IE) for cooling power

$$
\dot{Q}\_c^{IE} = \overline{a}T\_1 I - \frac{1}{2}I^2 R - \frac{Ak}{L}(T\_2 - T\_1) \tag{41}
$$

The resulting equation considering the Thomson effect is given by:

Thomson coefficients, τr, are considered. Table 2 shows maximum values, from Figure 5, for COP in function of the electric current ratio between both stages, Ir. Maximum COP value is obtained for higher values of the ratio τr, that is, a higher value of the electric current I<sup>2</sup> > I<sup>1</sup> is

5. Dimensionless equations of a two-stage thermoelectric micro-cooler

<sup>θ</sup> <sup>¼</sup> <sup>T</sup> � <sup>T</sup><sup>1</sup> T<sup>2</sup> � T<sup>1</sup>

Dimensionless differential equation corresponding to Equation (12) is given by:

<sup>d</sup>ξ<sup>2</sup> � β θð Þ � <sup>1</sup> <sup>ϕ</sup> <sup>þ</sup> <sup>1</sup> <sup>d</sup><sup>θ</sup>

<sup>β</sup> <sup>¼</sup> IT<sup>2</sup>

dα dT ΔT Aκ <sup>Δ</sup><sup>T</sup> L

that is, β is the relation between Thomson heat with thermal conduction. From Equation (38), if β ¼ 0, we get the ideal equation (IE) when Thomson effect not considered. Dimensionless parameter, γ, is the relation between Joule heating to the thermal conduction, and the parameter ϕ, which is the ratio of temperature difference to the high junction temperature, defined as:

d2 θ

where

and <sup>ξ</sup> <sup>¼</sup> <sup>x</sup>

<sup>L</sup> (37)

<sup>d</sup><sup>ξ</sup> <sup>þ</sup> <sup>γ</sup> <sup>¼</sup> <sup>0</sup> (38)

(39)

Once it has been investigated the role of the Thomson heat on TEC performance, now a procedure to improve the performance of the micro-cooler based on optimum geometric parameters, cross-sectional area (A) and length (L), of the semiconductor elements is proposed. To optimal design of a TEMC, theoretical basis on optimal geometric parameters (of the p-type and n-type semiconductor legs) is required. Next analysis of a TEMC includes these optimization parameters. The configuration of a two-stage TE system considered in this work is shown in Figure 2. Each stage is made of different thermoelectric semiconductor materials. In order to make Equation (12) dimensionless using the boundary conditions Tð Þ¼ 0 T<sup>1</sup> and T Lð Þ¼ T2, in accordance with Figure 2, we define the dimensionless temperature, θ, and the ξ parameter as,

desirable to be able to achieve better COP.

Table 2. Maximum values of COP.

300 Bringing Thermoelectricity into Reality

$$
\dot{Q}\_c^\delta = \overline{a}T\_1 I - \frac{1}{2}I^2 R - \frac{Ak}{L}(T\_2 - T\_1) + \beta \frac{Ak}{L}(T\_2 - T\_1) \tag{42}
$$

#### 5.2. Geometric parameter between stages: area-length ratio (W = w1/w2)

Figure 6 shows a simple thermocouple with length, L and cross-sectional area, A. Previous studies proved that an improvement on performance of TECs can be achieved by optimizing

Figure 6. Schematic diagram of a thermocouple.

geometric size of the semiconductor legs [29, 30]. A geometric parameter, ω, is defined as the area-length ratio of the legs in the thermocouple in each stage of the TEMC

$$
\omega\_1 = \frac{A\_1}{L\_1} \quad \text{and} \quad \omega\_2 = \frac{A\_2}{L\_2} \tag{43}
$$

for the first and second stage, respectively.

We define the geometric parameter, W, which allows us to determine the optimal geometric parameters of the stages, which is expressed as,

$$W = \frac{\omega\_1}{\omega\_2} \tag{44}$$

were provided by the manufacturer [21], and material M2, ð Þ Bi<sup>0</sup>:<sup>5</sup>Sb<sup>0</sup>:<sup>5</sup> <sup>2</sup>Te<sup>3</sup> [17], Tavg ¼ ð Þ <sup>T</sup><sup>1</sup> <sup>þ</sup> <sup>T</sup><sup>2</sup> <sup>=</sup>2, where <sup>α</sup> <sup>¼</sup> <sup>α</sup> Tavg and Seebeck coefficients are dependent on temperature while the electrical resistivity and the thermal conductivity are constant. The sign of n-type elements coefficient is negative while the sign of p-type element coefficients is positive for the Seebeck

Thermoelectric Cooling: The Thomson Effect in Hybrid Two-Stage Thermoelectric Cooler Systems with Different…

http://dx.doi.org/10.5772/intechopen.75440

303

<sup>α</sup><sup>1</sup> <sup>¼</sup> <sup>½</sup>0:<sup>2068</sup> <sup>T</sup> <sup>þ</sup> <sup>138</sup>:78� � <sup>10</sup>�<sup>6</sup> and <sup>α</sup><sup>2</sup> ¼ �<sup>62675</sup> <sup>þ</sup> <sup>1610</sup> <sup>T</sup> � <sup>2</sup>:<sup>3</sup> <sup>T</sup><sup>2</sup> � <sup>10</sup>�<sup>6</sup> (51)

In this section, we analyse a single-stage system to compare with two-stage system to show the differences between both systems. Thereby, we calculate the two important parameters: COP and Qc versus electric current; and COP and Qc versus geometric parameter (w), for both materials. CPM models are compared with TDPM model, for this purpose, in all figures are shown results obtained considering Thomson effect (solid lines) and results using the ideal equation (dashed lines). Figure 7 shows the COP1, Qc,<sup>1</sup> and COP2, Qc, <sup>2</sup> for materials, M<sup>1</sup> and M<sup>2</sup> respectively, as a function of the electric current. The maximum values of COP and Qc are reached when the Thomson effect is considered, better cooling power is obtained with lower values of β. Results show that material M<sup>1</sup> achieves higher values of cooling power Qc and COP than material M2. The COP for material M<sup>1</sup> is 15.1% more than for material M<sup>2</sup> and Qc for

Now, according to optimal electric current values, determined in the previous section, we show the effect of the semiconductor geometric parameters on the COP(w) and Qc(w) of the system. Figure 8 shows that, for COP and Qc, material M<sup>1</sup> has better results in both cases than material M2. The COP of material M<sup>1</sup> is 21.18% higher than that for material M<sup>2</sup> and the Qc

Now, we analyse a hybrid two-stage TEMC, that is, a system with a different thermoelectric material in each stage. Homogeneous system can also be analysed, this can be achieved by placing the same materials in both stages, as is shown in [27]. We focus on analysing two-stage hybrid systems, where two temperature gradients are generated and, therefore we must

coefficients values. Then, for materials 1 and 2, we have next equations

5.4. Special case: single-stage TEMC performance analysis

Table 3. Properties of thermoelectric (TE) elements used in the TEMC device.

material M<sup>1</sup> is 40.12% more than for materialM2.

5.5. Hybrid two-stage TEMC system

value in material M<sup>1</sup> is 14.85% higher than for materialM2.

In terms of the geometric parameters, ω<sup>1</sup> and ω2, we get:

$$R = R\_p + R\_n = \frac{L\_p}{\sigma\_p A\_p} + \frac{L\_n}{\sigma\_n A\_n} = \frac{1}{\sigma\_p \omega\_1} + \frac{1}{\sigma\_n \omega\_2} \tag{45}$$

$$K = K\_p + K\_n = \frac{A\_p k\_p}{A\_p} + \frac{A\_n k\_n}{A\_n} = \omega\_1 k\_p + \omega\_2 k\_n \tag{46}$$

We have for the cooling power, in terms of the geometric parameters,ω1andω<sup>2</sup>

$$\dot{Q}\_c^{\text{IE}} = a \left( T\_{w\text{g}} \right) T\_1 I - \frac{1}{2} I^2 \left( \frac{1}{\sigma\_p \omega\_1} + \frac{1}{\sigma\_n \omega\_2} \right) - \left( \omega\_1 k\_p + \omega\_2 k\_n \right) (T\_2 - T\_1) \tag{47}$$

For ideal equation, Q\_ IE <sup>c</sup> , and Thomson effect, <sup>Q</sup>\_ <sup>β</sup> <sup>c</sup> , we have

$$\dot{Q}\_c^\beta = a \left( T\_{\text{avg}} \right) T\_1 I - \frac{1}{2} I^2 \left( \frac{1}{\sigma\_p \omega\_1} + \frac{1}{\sigma\_n \omega\_2} \right) - \left( \omega\_1 k\_p + \omega\_2 k\_n \right) (T\_2 - T\_1) + \beta \left( \omega\_1 k\_p + \omega\_2 k\_n \right) (T\_2 - T\_1) \tag{48}$$

Finally, we introduce the ratio, M, of the number of thermocouples in the first stage, n1, to the number of thermocouples in second stage,n<sup>2</sup>

$$M = \frac{n\_1}{n\_2} \tag{49}$$

The total number of thermocouples, N, for both stages is given by,

$$N = n\_1 + n\_2 \tag{50}$$

#### 5.3. Material properties considerations: CPM and TDPM models

The two different semiconductor materials and their properties are given in Table 3: Material M1, which is obtained from commercial module of laird CP10 � 127 � 05 and its properties


Table 3. Properties of thermoelectric (TE) elements used in the TEMC device.

geometric size of the semiconductor legs [29, 30]. A geometric parameter, ω, is defined as the

We define the geometric parameter, W, which allows us to determine the optimal geometric

<sup>W</sup> <sup>¼</sup> <sup>ω</sup><sup>1</sup> ω2

σpAp þ Ln σnAn

> Ap þ Ankn An

� ω1kp þ ω2kn

Finally, we introduce the ratio, M, of the number of thermocouples in the first stage, n1, to the

<sup>M</sup> <sup>¼</sup> <sup>n</sup><sup>1</sup> n2

The two different semiconductor materials and their properties are given in Table 3: Material M1, which is obtained from commercial module of laird CP10 � 127 � 05 and its properties

<sup>c</sup> , we have

and <sup>ω</sup><sup>2</sup> <sup>¼</sup> <sup>A</sup><sup>2</sup>

L2

<sup>¼</sup> <sup>1</sup> σpω<sup>1</sup> þ 1 σnω<sup>2</sup>

� ω1kp þ ω2kn

ð Þþ <sup>T</sup><sup>2</sup> � <sup>T</sup><sup>1</sup> β ω1kp <sup>þ</sup> <sup>ω</sup>2kn

N ¼ n<sup>1</sup> þ n<sup>2</sup> (50)

¼ ω1kp þ ω2kn (46)

ð Þ <sup>T</sup><sup>2</sup> � <sup>T</sup><sup>1</sup> (47)

ð Þ <sup>T</sup><sup>2</sup> � <sup>T</sup><sup>1</sup>

(43)

(44)

(45)

(48)

(49)

area-length ratio of the legs in the thermocouple in each stage of the TEMC

<sup>ω</sup><sup>1</sup> <sup>¼</sup> <sup>A</sup><sup>1</sup> L1

for the first and second stage, respectively.

302 Bringing Thermoelectricity into Reality

Q\_ IE

<sup>T</sup>1<sup>I</sup> � <sup>1</sup>

For ideal equation, Q\_ IE

<sup>c</sup> ¼ α Tavg

Q\_ β

<sup>c</sup> ¼ α Tavg

2 I <sup>2</sup> 1 σpω<sup>1</sup> þ 1 σnω<sup>2</sup>

number of thermocouples in second stage,n<sup>2</sup>

parameters of the stages, which is expressed as,

In terms of the geometric parameters, ω<sup>1</sup> and ω2, we get:

<sup>T</sup>1<sup>I</sup> � <sup>1</sup>

<sup>R</sup> <sup>¼</sup> Rp <sup>þ</sup> Rn <sup>¼</sup> Lp

<sup>K</sup> <sup>¼</sup> Kp <sup>þ</sup> Kn <sup>¼</sup> Apkp

2 I <sup>2</sup> 1 σpω<sup>1</sup> þ 1 σnω<sup>2</sup>

<sup>c</sup> , and Thomson effect, <sup>Q</sup>\_ <sup>β</sup>

The total number of thermocouples, N, for both stages is given by,

5.3. Material properties considerations: CPM and TDPM models

We have for the cooling power, in terms of the geometric parameters,ω1andω<sup>2</sup>

were provided by the manufacturer [21], and material M2, ð Þ Bi<sup>0</sup>:<sup>5</sup>Sb<sup>0</sup>:<sup>5</sup> <sup>2</sup>Te<sup>3</sup> [17], Tavg ¼ ð Þ <sup>T</sup><sup>1</sup> <sup>þ</sup> <sup>T</sup><sup>2</sup> <sup>=</sup>2, where <sup>α</sup> <sup>¼</sup> <sup>α</sup> Tavg and Seebeck coefficients are dependent on temperature while the electrical resistivity and the thermal conductivity are constant. The sign of n-type elements coefficient is negative while the sign of p-type element coefficients is positive for the Seebeck coefficients values. Then, for materials 1 and 2, we have next equations

$$a\_1 = \begin{bmatrix} 0.2068 \ T + 138.78 \end{bmatrix} \times 10^{-6} \quad \text{and} \quad a\_2 = \begin{bmatrix} -62675 + 1610 \ T - 2.3 \ T^2 \end{bmatrix} \times 10^{-6} \tag{51}$$

#### 5.4. Special case: single-stage TEMC performance analysis

In this section, we analyse a single-stage system to compare with two-stage system to show the differences between both systems. Thereby, we calculate the two important parameters: COP and Qc versus electric current; and COP and Qc versus geometric parameter (w), for both materials. CPM models are compared with TDPM model, for this purpose, in all figures are shown results obtained considering Thomson effect (solid lines) and results using the ideal equation (dashed lines). Figure 7 shows the COP1, Qc,<sup>1</sup> and COP2, Qc, <sup>2</sup> for materials, M<sup>1</sup> and M<sup>2</sup> respectively, as a function of the electric current. The maximum values of COP and Qc are reached when the Thomson effect is considered, better cooling power is obtained with lower values of β. Results show that material M<sup>1</sup> achieves higher values of cooling power Qc and COP than material M2. The COP for material M<sup>1</sup> is 15.1% more than for material M<sup>2</sup> and Qc for material M<sup>1</sup> is 40.12% more than for materialM2.

Now, according to optimal electric current values, determined in the previous section, we show the effect of the semiconductor geometric parameters on the COP(w) and Qc(w) of the system. Figure 8 shows that, for COP and Qc, material M<sup>1</sup> has better results in both cases than material M2. The COP of material M<sup>1</sup> is 21.18% higher than that for material M<sup>2</sup> and the Qc value in material M<sup>1</sup> is 14.85% higher than for materialM2.

#### 5.5. Hybrid two-stage TEMC system

Now, we analyse a hybrid two-stage TEMC, that is, a system with a different thermoelectric material in each stage. Homogeneous system can also be analysed, this can be achieved by placing the same materials in both stages, as is shown in [27]. We focus on analysing two-stage hybrid systems, where two temperature gradients are generated and, therefore we must

5.5.1. Average system temperature, Tm

and for the second stage,

the temperature between stages, Tm,

5.5.2. Dimensionless temperature distribution

cross-sectional area of Ac <sup>¼</sup> <sup>4</sup>:<sup>9</sup> � <sup>10</sup>�<sup>9</sup>

Tm <sup>¼</sup> � <sup>1</sup> 2 I 2

A two-stage TEMC consists of n<sup>1</sup> and n<sup>2</sup> thermocouples in the first and second stages, respectively. The heat flux at the cold side, Qc1, and the heat flux at the hot side, Qh1, in the first stage, and for the second stage, Qc<sup>2</sup> and Qh2, respectively. Thus, heat flux equations in the first stage are [31],

Thermoelectric Cooling: The Thomson Effect in Hybrid Two-Stage Thermoelectric Cooler Systems with Different…

For a hybrid system (different materials in each stage), from equations (53) and (54), we obtain

1 <sup>2</sup> τ<sup>2</sup> � α<sup>2</sup>

For the hybrid two-stage TEMC system, the best configuration of semiconductor thermoelectric materials and its optimal geometric parameters is found in this section. For calculations we use a

of thermocouples of n<sup>1</sup> ¼ n<sup>2</sup> ¼ 100 in the first and second stages, respectively. Figure 9 shows the dimensionless spatial temperature distributions, for cases (a) and (b) mentioned earlier. An important factor to analyse in the graphic is the maximum values of the temperature distribution in each stage. When the value of the derivative is to be dθ=dξ > 0, the semiconductor material is able to absorb a certain amount of heat, that is, Thomson heat acts by absorbing heat. For the case when the value of the derivative is to be dθ=dξ < 0, a release of heat occurs in the semiconductor, that is, Thomson heat acts by liberating heat. From Figure 9, maximum temperature distribution values in stage 1, θ ¼ 1:06, is near to the junction with stage 2, which is desirable because in the first stage, the system must absorb higher amount of heat to later be released in stage 2. Thereby, dimensionless temperature distribution, θ, as a function of the length, ξ, shows that a lower temperature distribution is required in the first stage and that higher values of temperature distribution are required in the second stage; this is achieved by choosing the optimal arrangement of materials between the two stages. According to this last statement, case

Figure 10 shows COP and Qc for the TEMC system for cases (a) and (b) described previously. Case (a) reaches best cooling power and coefficient of performance values. Notice that the

<sup>2</sup> <sup>þ</sup> <sup>τ</sup>1I Tð Þ <sup>m</sup> � Tc<sup>1</sup>

http://dx.doi.org/10.5772/intechopen.75440

<sup>2</sup> � <sup>τ</sup>1I Tð Þ <sup>m</sup> � Tc<sup>1</sup>

<sup>2</sup> <sup>þ</sup> <sup>τ</sup>2I Tð Þ <sup>h</sup><sup>2</sup> � Tm

<sup>2</sup> � <sup>τ</sup>2I Tð Þ <sup>h</sup><sup>2</sup> � Tm

<sup>m</sup><sup>2</sup> and element length of <sup>L</sup> <sup>¼</sup> <sup>15</sup>μm, with a total number

(56)

305

(52)

(53)

(54)

(55)

<sup>2</sup> Iðτ2n1Th<sup>2</sup> � τ1n1Tc1Þ � K2Th2n<sup>1</sup> � K1Tc1n1Þ

� <sup>K</sup>1n<sup>1</sup> � <sup>K</sup>2n<sup>2</sup>

Qc<sup>1</sup> ¼ n<sup>1</sup> α1ITc<sup>1</sup> � K1ð Þ� Tm � Tc<sup>1</sup> 1=2R1I

Qh<sup>1</sup> ¼ n<sup>1</sup> α1ITm � K1ð Þþ Tm � Tc<sup>1</sup> 1=2R1I

Qc<sup>2</sup> ¼ n<sup>2</sup> α2ITm � K2ð Þ� Th<sup>2</sup> � Tm 1=2R2I

Qh<sup>2</sup> ¼ n<sup>2</sup> α2ITh<sup>2</sup> � K2ð Þþ Th<sup>2</sup> � Tm 1=2R2I

<sup>2</sup> τ<sup>1</sup> <sup>þ</sup> In<sup>2</sup>

<sup>ð</sup>R1n<sup>1</sup> <sup>þ</sup> <sup>R</sup>2n2Þ þ <sup>1</sup>

(a) is the best configuration of materials to improve the TEMC.

5.5.3. Analysis and coefficient of performance and cooling power (Qc)

In<sup>1</sup> <sup>α</sup><sup>1</sup> � <sup>1</sup>

Figure 7. Single-stage coefficient of performance, COP Ið Þ, and cooling power, Qc ð ÞI , for both materials M<sup>1</sup> and M2. Solid lines calculated with Thomson effect and dashed lines considering ideal equation.

Figure 8. Single-stage COP(w) and Qcð Þ ω for both materials, using optimal electric currents I cop opt . Solid lines calculated with Thomson effect and dashed lines considering ideal equation.

analyse which material works better in each stage. Thus, we determine the optimum thermoelectric material arrangement for the best performance of the TEMC system. For this purpose, two configurations of materials in the hybrid two-stage TEMC model are considered: (a) materials M<sup>1</sup> and M<sup>2</sup> are used in the first and the second stage, respectively; and the inverse system (b) materials M<sup>2</sup> and M<sup>1</sup> are used in the first and the second stage, respectively.

#### 5.5.1. Average system temperature, Tm

A two-stage TEMC consists of n<sup>1</sup> and n<sup>2</sup> thermocouples in the first and second stages, respectively. The heat flux at the cold side, Qc1, and the heat flux at the hot side, Qh1, in the first stage, and for the second stage, Qc<sup>2</sup> and Qh2, respectively. Thus, heat flux equations in the first stage are [31],

$$Q\_{c1} = n\_1 \left[ \alpha\_1 T\_{c1} - K\_1 (T\_m - T\_{c1}) - 1/2 \mathcal{R}\_1 I^2 + \tau\_1 I (T\_m - T\_{c1}) \right] \tag{52}$$

$$Q\_{h1} = n\_1 \left[ \alpha\_1 T\_m - K\_1 (T\_m - T\_{c1}) + 1/2 \mathcal{R}\_1 I^2 - \tau\_1 I (T\_m - T\_{c1}) \right] \tag{53}$$

and for the second stage,

analyse which material works better in each stage. Thus, we determine the optimum thermoelectric material arrangement for the best performance of the TEMC system. For this purpose, two configurations of materials in the hybrid two-stage TEMC model are considered: (a) materials M<sup>1</sup> and M<sup>2</sup> are used in the first and the second stage, respectively; and the inverse system (b) materials M<sup>2</sup> and M<sup>1</sup> are used in the first and the second stage, respectively.

cop

opt . Solid lines calculated

Figure 8. Single-stage COP(w) and Qcð Þ ω for both materials, using optimal electric currents I

with Thomson effect and dashed lines considering ideal equation.

Figure 7. Single-stage coefficient of performance, COP Ið Þ, and cooling power, Qc ð ÞI , for both materials M<sup>1</sup> and M2. Solid

lines calculated with Thomson effect and dashed lines considering ideal equation.

304 Bringing Thermoelectricity into Reality

$$Q\_{c2} = n\_2 \left[ \alpha\_2 T\_m - K\_2 (T\_{h2} - T\_m) - 1/2 R\_2 I^2 + \tau\_2 I (T\_{h2} - T\_m) \right] \tag{54}$$

$$Q\_{h2} = n\_2 \left[ \alpha\_2 I T\_{h2} - K\_2 (T\_{h2} - T\_m) + 1/2 R\_2 I^2 - \tau\_2 I (T\_{h2} - T\_m) \right] \tag{55}$$

For a hybrid system (different materials in each stage), from equations (53) and (54), we obtain the temperature between stages, Tm,

$$T\_m = \frac{-\frac{1}{2}I^2(R\_1n\_1 + R\_2n\_2) + \frac{1}{2}I(\tau\_2n\_1T\_{h2} - \tau\_1n\_1T\_{c1}) - K\_2T\_{h2}n\_1 - K\_1T\_{c1}n\_1}{I n\_1(\alpha\_1 - \frac{1}{2}\tau\_1) + I n\_2(\frac{1}{2}\tau\_2 - \alpha\_2) - K\_1n\_1 - K\_2n\_2} \tag{56}$$

#### 5.5.2. Dimensionless temperature distribution

For the hybrid two-stage TEMC system, the best configuration of semiconductor thermoelectric materials and its optimal geometric parameters is found in this section. For calculations we use a cross-sectional area of Ac <sup>¼</sup> <sup>4</sup>:<sup>9</sup> � <sup>10</sup>�<sup>9</sup> <sup>m</sup><sup>2</sup> and element length of <sup>L</sup> <sup>¼</sup> <sup>15</sup>μm, with a total number of thermocouples of n<sup>1</sup> ¼ n<sup>2</sup> ¼ 100 in the first and second stages, respectively. Figure 9 shows the dimensionless spatial temperature distributions, for cases (a) and (b) mentioned earlier. An important factor to analyse in the graphic is the maximum values of the temperature distribution in each stage. When the value of the derivative is to be dθ=dξ > 0, the semiconductor material is able to absorb a certain amount of heat, that is, Thomson heat acts by absorbing heat. For the case when the value of the derivative is to be dθ=dξ < 0, a release of heat occurs in the semiconductor, that is, Thomson heat acts by liberating heat. From Figure 9, maximum temperature distribution values in stage 1, θ ¼ 1:06, is near to the junction with stage 2, which is desirable because in the first stage, the system must absorb higher amount of heat to later be released in stage 2. Thereby, dimensionless temperature distribution, θ, as a function of the length, ξ, shows that a lower temperature distribution is required in the first stage and that higher values of temperature distribution are required in the second stage; this is achieved by choosing the optimal arrangement of materials between the two stages. According to this last statement, case (a) is the best configuration of materials to improve the TEMC.

#### 5.5.3. Analysis and coefficient of performance and cooling power (Qc)

Figure 10 shows COP and Qc for the TEMC system for cases (a) and (b) described previously. Case (a) reaches best cooling power and coefficient of performance values. Notice that the

COP<sup>β</sup>

maxð Þ<sup>a</sup> is 19.05% better than COP<sup>β</sup>

5.5.4. Optimization analysis according to the geometric parameter W

maxð Þb . It is clear from the graphic that for the same

http://dx.doi.org/10.5772/intechopen.75440

307

current values, cooling power values for the case (a) are always over those of the case (b).

Thermoelectric Cooling: The Thomson Effect in Hybrid Two-Stage Thermoelectric Cooler Systems with Different…

In this section, we analyse the physical sizes, length and the cross-sectional area of the thermocouples, when the two stages are related each other. We present an optimization procedure of a two-stage TEMC system, on COP and Qc, by introducing a geometric parameter, W ¼ ω1=ω2. The effect of the parameter W on COP and Qc is analysed when (1) ω<sup>1</sup> ¼ ω<sup>2</sup> and (2) when ω<sup>1</sup> 6¼ ω2. Figure 11 (a) shows best optimal material configuration for COP(w) and Qc (w), which turns out to be case (a) where material M<sup>1</sup> is placed in the first stage and material M<sup>2</sup> in the second stage. Results proved that, higher area-length ratio values do not improve Qc, on the contrary, the cooling power improves for lower values of w. COP and Qc increases by 19 and 10.5%, respectively, from case (a) to case (b). The most relevant case, geometric parameters <sup>ω</sup><sup>1</sup> 6¼ <sup>ω</sup>2, is analysed. In this case, we set <sup>ω</sup><sup>2</sup> <sup>¼</sup> <sup>3</sup>:<sup>26</sup> � <sup>10</sup>�<sup>4</sup> <sup>m</sup> to be a constant value. Figure 11 (b) shows COP Wð Þ and Qcð Þ ω where it is noted that COP increases by 8.9% and Qc increases 6.27% in case (a) compared with case (b). From this last result, it is important to note that although the performance of TEC systems is affected by combination of different materials, it is also affected by the material configuration and the system geometry as well. These results offer a novel alternative in the improvement of thermoelectric systems, when they are used as coolers. Results shown in this chapter are based only on theory of thermoelectricity to optimize a TEMC system, according to geometric parameters. However, parameters as length and crosssectional area of the semiconductor elements are based on studies which validated similar results with experimental data [32, 33]. In micro-refrigeration, an important problem is the fact of handle heat flux in a small space and it has been proved that thermal interface resistance has beneficial or detrimental effects on cooling performance [34]. For calculation, contact resistances between stages are not considered, since it is known that thermal resistances exist in the interfaces, which are large when the cross-sectional areas are very dissimilar in the stages and negligible for similar cross-sectional areas [35, 36]. Present work can be useful as theoretical

Figure 11. Hybrid two-stage TEMC. (a) COPð Þ ω and Qc ð Þ ω when ω<sup>1</sup> ¼ ω2. (b) COP Wð Þ and Qc ð Þ ω when ω<sup>1</sup> 6¼ ω2. Solid

lines calculated with Thomson effect and dashed lines considering ideal equation.

Figure 9. Hybrid two-stage TEMC. Dimensionless temperature distribution, θ ξð Þ. Case (a): material M<sup>1</sup> is placed in stage 1 (black line) and material M<sup>2</sup> in stage 2 (blue line). Case (b): material M<sup>2</sup> is placed in stage 1 (blue line) and material M<sup>1</sup> in stage 2 (black line). Solid lines calculated with Thomson effect and dashed lines considering ideal equation.

Figure 10. Hybrid two-stage TEMC. COP(I) and Qc ð ÞI for cases (a) and (b) with w = w1 = w2. Solid lines calculated with Thomson effect and dashed lines considering ideal equation.

COP<sup>β</sup> maxð Þ<sup>a</sup> is 19.05% better than COP<sup>β</sup> maxð Þb . It is clear from the graphic that for the same current values, cooling power values for the case (a) are always over those of the case (b).

#### 5.5.4. Optimization analysis according to the geometric parameter W

In this section, we analyse the physical sizes, length and the cross-sectional area of the thermocouples, when the two stages are related each other. We present an optimization procedure of a two-stage TEMC system, on COP and Qc, by introducing a geometric parameter, W ¼ ω1=ω2. The effect of the parameter W on COP and Qc is analysed when (1) ω<sup>1</sup> ¼ ω<sup>2</sup> and (2) when ω<sup>1</sup> 6¼ ω2. Figure 11 (a) shows best optimal material configuration for COP(w) and Qc (w), which turns out to be case (a) where material M<sup>1</sup> is placed in the first stage and material M<sup>2</sup> in the second stage. Results proved that, higher area-length ratio values do not improve Qc, on the contrary, the cooling power improves for lower values of w. COP and Qc increases by 19 and 10.5%, respectively, from case (a) to case (b). The most relevant case, geometric parameters <sup>ω</sup><sup>1</sup> 6¼ <sup>ω</sup>2, is analysed. In this case, we set <sup>ω</sup><sup>2</sup> <sup>¼</sup> <sup>3</sup>:<sup>26</sup> � <sup>10</sup>�<sup>4</sup> <sup>m</sup> to be a constant value. Figure 11 (b) shows COP Wð Þ and Qcð Þ ω where it is noted that COP increases by 8.9% and Qc increases 6.27% in case (a) compared with case (b). From this last result, it is important to note that although the performance of TEC systems is affected by combination of different materials, it is also affected by the material configuration and the system geometry as well. These results offer a novel alternative in the improvement of thermoelectric systems, when they are used as coolers. Results shown in this chapter are based only on theory of thermoelectricity to optimize a TEMC system, according to geometric parameters. However, parameters as length and crosssectional area of the semiconductor elements are based on studies which validated similar results with experimental data [32, 33]. In micro-refrigeration, an important problem is the fact of handle heat flux in a small space and it has been proved that thermal interface resistance has beneficial or detrimental effects on cooling performance [34]. For calculation, contact resistances between stages are not considered, since it is known that thermal resistances exist in the interfaces, which are large when the cross-sectional areas are very dissimilar in the stages and negligible for similar cross-sectional areas [35, 36]. Present work can be useful as theoretical

Figure 11. Hybrid two-stage TEMC. (a) COPð Þ ω and Qc ð Þ ω when ω<sup>1</sup> ¼ ω2. (b) COP Wð Þ and Qc ð Þ ω when ω<sup>1</sup> 6¼ ω2. Solid lines calculated with Thomson effect and dashed lines considering ideal equation.

Figure 10. Hybrid two-stage TEMC. COP(I) and Qc ð ÞI for cases (a) and (b) with w = w1 = w2. Solid lines calculated with

Figure 9. Hybrid two-stage TEMC. Dimensionless temperature distribution, θ ξð Þ. Case (a): material M<sup>1</sup> is placed in stage 1 (black line) and material M<sup>2</sup> in stage 2 (blue line). Case (b): material M<sup>2</sup> is placed in stage 1 (blue line) and material M<sup>1</sup> in

stage 2 (black line). Solid lines calculated with Thomson effect and dashed lines considering ideal equation.

Thomson effect and dashed lines considering ideal equation.

306 Bringing Thermoelectricity into Reality

basis for future research in the experimental area, development and design of thermoelectric multistage coolers.

References

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