2.2.7 The electrical power generated at the load PL

The electrical power generated at the load PL can be found using:

$$P\_L = I^2 R\_L \text{ measured in watts} \tag{13}$$

If we substitute (Eq. (11)) into (Eq. (13)), we obtain:

$$P\_L = \left\{ \frac{(a\_p - a\_n)(T\_h - T\_c)}{R\_p + R\_n + R\_L} \right\}^2 R\_L \text{ measured in watts} \tag{14}$$

Alternatively, the electrical power generated at the load can be found by:

$$P\_L = V\_L \times I\_L \text{ measured in watts} \tag{15}$$

In electrical and electronic engineering, the maximum power transfer between the generator and the load normally occurs when the load resistance is equal to the generator resistance, and if we consider the thermoelectric couple as an ideal model, with no heat lost through thermal radiation or conduction, and ignoring any effect of contact resistances, the maximum efficiency of a thermoelectric generator will not exceed 50%. It should be noted that if the load resistance is increased away from the value that gives rise to maximum power transfer, the power output of the thermoelectric generator will be reduced [8].

#### 2.2.8 Calculating the thermoelectric couple efficiency

The efficiency η of a thermoelectric couple can be found by:

$$\eta = \frac{\text{Energy supplied to the load}}{\text{Heat energy absorbed at the hot junction}} \tag{16}$$

The electrical power supplied to the load is PL, and we now need to find the heat energy absorbed at the hot junction. A proportion of the heat that is absorbed at the Thermoelectric Energy Harvesting DOI: http://dx.doi.org/10.5772/intechopen.85670

<sup>I</sup> <sup>¼</sup> <sup>V</sup> RT

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moelectric couple and the load. It is possible to find RT using:

<sup>I</sup> <sup>¼</sup> <sup>α</sup><sup>p</sup> � <sup>α</sup><sup>n</sup>

2.2.7 The electrical power generated at the load PL

PL <sup>¼</sup> <sup>α</sup><sup>p</sup> � <sup>α</sup><sup>n</sup>

thermoelectric generator will be reduced [8].

18

2.2.8 Calculating the thermoelectric couple efficiency

The efficiency η of a thermoelectric couple can be found by:

<sup>η</sup> <sup>¼</sup> Energy supplied to the load

The electrical power supplied to the load is PL, and we now need to find the heat energy absorbed at the hot junction. A proportion of the heat that is absorbed at the

load VL can be found by:

Using (Eq. (1)) and (Eq. (10)), the current I can be found by:

The electrical power generated at the load PL can be found using:

PL ¼ I 2

If we substitute (Eq. (11)) into (Eq. (13)), we obtain:

 ð Þ Th � Tc Rp þ Rn þ RL <sup>2</sup>

Alternatively, the electrical power generated at the load can be found by:

In electrical and electronic engineering, the maximum power transfer between the generator and the load normally occurs when the load resistance is equal to the generator resistance, and if we consider the thermoelectric couple as an ideal model, with no heat lost through thermal radiation or conduction, and ignoring any effect of contact resistances, the maximum efficiency of a thermoelectric generator will not exceed 50%. It should be noted that if the load resistance is increased away from the value that gives rise to maximum power transfer, the power output of the

 ð Þ Th � Tc Rp þ Rn þ RL

2.2.6 The load current IL and the voltage generated at the load VL

where V is the closed-circuit voltage, and RT is the total resistance of the ther-

As the load resistor RL is connected across the output terminals of the couple, the load current IL is the same as the circuit current I, and the voltage generated at the

measured in amperes (9)

measured in amperes (11)

RT ¼ Rp þ Rn þ RL measured in Ohms (10)

VL ¼ IL � RL measured in volts (12)

PL ¼ VL � IL measured in watts (15)

Heat energy absorbed at the hot junction (16)

RL measured in watts (13)

RL measured in watts (14)

hot junction from the heat source Th is used to balance the Peltier cooling effect in the thermoelectric couple, and an opposing flow of heat occurs due to the thermal conduction of the thermoelectric legs, and by Joule heating within the device. The cooling effect at the source Qsource can be found by:

$$Q\_{source} = (a\_p - a\_n)I\_L T\_h \text{ measured in watts} \tag{17}$$

The opposing heat by conduction in the thermoelectric legs (or pellets) can be found by:

$$Q\_{\rm leg^\circ} = (T\_h - T\_c) \left(K\_p + K\_n\right) \text{ measured in watts} \tag{18}$$

where Kp and Kn are the thermal conductance of the p-type and n-type thermoelectric legs, respectively. The cooling effect is opposed by Joule heating Qj within the thermoelectric legs, found by:

$$Q\_j = \left(\frac{I^2 \left(R\_p + R\_n\right)}{2}\right) \text{ measured in watts} \tag{19}$$

According to [8], it can be shown that half of the Joule heating passes to the sink Tc and half to the source Th, with each half equal to (Eq. (19)). The expression for the heat energy absorbed at the hot junction, which is the same as the cooling power at the hot side of the thermoelectric couple Qh can now be found by:

$$Q\_h = \left(a\_p - a\_n\right)IT\_h - \left(T\_c - T\_h\right)\left(K\_p + K\_n\right) - \frac{I^2\left(R\_p + R\_n\right)}{2}\text{ measured in watts} \tag{20}$$

Hence, the efficiency of the thermoelectric couple η can now be found by (Eq. (16)), and written as:

$$\eta = \frac{P\_L}{Q\_h} \tag{21}$$

Substituting Qh from (Eq. (20)) and PL from (Eq. (14)) into (Eq. (21)), we obtain:

$$\eta = \frac{\left\{\frac{\left(a\_p - a\_n\right)(T\_h - T\_c)}{R\_p + R\_n + R\_L}\right\}^2 R\_L}{\left(a\_p - a\_n\right)IT\_h - \left(T\_c - T\_h\right)\left(K\_p + K\_n\right) - \frac{I^2\left(R\_p + R\_n\right)}{2}}\tag{22}$$

In thermoelectricity, efficiency is expressed as the dimensionless figure-of-merit Z, or more commonly, expressed as a function of the temperature over which the device is operated ZT. The thermoelectric figure-of-merit Z describes the thermoelectric efficiency of a thermoelectric couple for a given pair of p-type and n-type thermoelectric materials, and is normally shown as:

$$Z = \frac{a^2}{\lambda \rho} \tag{23}$$

where α is the Seebeck coefficient of the thermoelectric couple, λ is the thermal conductivity, and ρ is the couple's electrical resistivity. In the context of this

discussion, the Seebeck coefficient of the couple is equal to (αp�αn), the thermal conductivity λ is (Kp + Kn), and the electrical resistivity ρ is equal to (Rp + Rn), therefore (Eq. (23)) can be rewritten as:

$$Z = \frac{\left(a\_p - a\_n\right)^2}{\left(K\_p + K\_n\right)\left(R\_p + R\_n\right)}\tag{24}$$

schematic diagram of a thermoelectric module, operating as a thermoelectric power

According to [11] the thermoelectric module open-circuit voltage Voc can be

where α<sup>m</sup> is the thermoelectric module's average Seebeck coefficient in volts per kelvin and may be found quoted in manufacturer's literature. If a load resistance RL is now connected across the output terminals of the thermoelectric module, the

where V is the closed-circuit voltage and Rm is the thermoelectric module's average resistance in Ohms. As the Seebeck coefficient and module resistance are temperature dependent, their values should be calculated at the average or mean

where the current at the load IL is equal to the circuit current I. The electrical

<sup>η</sup> <sup>¼</sup> PL Qh

The cooling power at the hot side of the thermoelectric module Qh can be found by:

I 2 ð Þ Rm

ð Þ α<sup>m</sup> ð Þ Tm RmþRL n o<sup>2</sup>

ð Þ <sup>α</sup><sup>m</sup> ITh � ð Þ Tc � Th ð Þ� Km <sup>I</sup>

ZT <sup>¼</sup> <sup>α</sup>m<sup>2</sup>σ<sup>m</sup> λm

RL

The efficiency η of the thermoelectric module can now be found by:

The thermoelectric figure-of-merit ZT can be found using:

Voc ¼ αmð Þ Th � Tc measured in volts (26)

measured in amperes (27)

<sup>2</sup> measured in kelvin (28)

<sup>L</sup> � RL measured in watts (30)

<sup>2</sup> measured in watts (33)

<sup>2</sup>ð Þ Rm 2

(32)

(34)

(35)

VL ¼ IL � RL measured in volts (29)

PL ¼ VL � IL measured in watts (31)

generator, is shown in Figure 4.

DOI: http://dx.doi.org/10.5772/intechopen.85670

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current I can be found by:

module temperature Tm given by:

or by using:

21

<sup>I</sup> <sup>¼</sup> <sup>V</sup> Rm þ RL

Tm <sup>¼</sup> Th <sup>þ</sup> Tc

power generated by the module at the load can then be found by:

The efficiency η of the thermoelectric module can be found by:

PL ¼ I 2

Qh ¼ ð Þ α<sup>m</sup> ITh � ð Þ Tc � Th ð Þ� Km

η ¼

The voltage at the load VL can be found by:

obtained by:

The thermoelectric figure-of-merit Z is commonly used to optimize the performance of materials used in the manufacture of thermoelectric couples. However, when stating the thermoelectric efficiency, it is more common to express it as a function of the temperature over which the device is operated, referred to as ZT, which can be found by using (Eq. (23)) at a specific temperature, and is normally written as:

$$Z\_T = \frac{a^2 \sigma}{\lambda} \tag{25}$$

where α is the Seebeck coefficient, σ is the electrical conductivity, and λ is the thermal conductivity of the couple at a specific operating temperature.

#### 2.3 Thermoelectric module power generation

If we now consider a thermoelectric module that contains several thermoelectric couples connected electrically in series, and thermally in parallel, as shown in Figure 2, a small amount of electrical power, typically in the milliwatt range (mW), can be generated from a thermoelectric module if a temperature difference is maintained between two sides of the module. Normally, one side of the module is attached to a heat source and is referred to as the 'hot'side or TH. The other side of the module is typically attached to a heat sink and is called the 'cold'side or TC. The heat sink is used to create a temperature difference between the hot and cold sides of the module. If a resistive load RL is connected across the module's output terminals, electrical power will be generated at the load when a temperature difference exists between the hot and cold sides of the module due to the Seebeck effect. A

Figure 4. A thermoelectric module configured for thermoelectric power generation.

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discussion, the Seebeck coefficient of the couple is equal to (αp�αn), the thermal conductivity λ is (Kp + Kn), and the electrical resistivity ρ is equal to (Rp + Rn),

<sup>2</sup>

The thermoelectric figure-of-merit Z is commonly used to optimize the performance of materials used in the manufacture of thermoelectric couples. However, when stating the thermoelectric efficiency, it is more common to express it as a function of the temperature over which the device is operated, referred to as ZT, which can be found by using (Eq. (23)) at a specific temperature, and is normally

ZT <sup>¼</sup> <sup>α</sup><sup>2</sup><sup>σ</sup>

where α is the Seebeck coefficient, σ is the electrical conductivity, and λ is the

If we now consider a thermoelectric module that contains several thermoelectric

couples connected electrically in series, and thermally in parallel, as shown in Figure 2, a small amount of electrical power, typically in the milliwatt range (mW), can be generated from a thermoelectric module if a temperature difference is maintained between two sides of the module. Normally, one side of the module is attached to a heat source and is referred to as the 'hot'side or TH. The other side of the module is typically attached to a heat sink and is called the 'cold'side or TC. The heat sink is used to create a temperature difference between the hot and cold sides of the module. If a resistive load RL is connected across the module's output terminals, electrical power will be generated at the load when a temperature difference exists between the hot and cold sides of the module due to the Seebeck effect. A

thermal conductivity of the couple at a specific operating temperature.

2.3 Thermoelectric module power generation

A thermoelectric module configured for thermoelectric power generation.

(24)

<sup>λ</sup> (25)

<sup>Z</sup> <sup>¼</sup> <sup>α</sup><sup>p</sup> � <sup>α</sup><sup>n</sup>

Kp þ Kn Rp <sup>þ</sup> Rn

therefore (Eq. (23)) can be rewritten as:

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written as:

Figure 4.

20

schematic diagram of a thermoelectric module, operating as a thermoelectric power generator, is shown in Figure 4.

According to [11] the thermoelectric module open-circuit voltage Voc can be obtained by:

$$V\_{oc} = a\_m(T\_h - T\_c) \text{ measured in volts} \tag{26}$$

where α<sup>m</sup> is the thermoelectric module's average Seebeck coefficient in volts per kelvin and may be found quoted in manufacturer's literature. If a load resistance RL is now connected across the output terminals of the thermoelectric module, the current I can be found by:

$$I = \frac{V}{R\_m + R\_L} \text{ measured in amperes} \tag{27}$$

where V is the closed-circuit voltage and Rm is the thermoelectric module's average resistance in Ohms. As the Seebeck coefficient and module resistance are temperature dependent, their values should be calculated at the average or mean module temperature Tm given by:

$$T\_m = \frac{T\_h + T\_c}{2} \text{ measured in kelvin} \tag{28}$$

The voltage at the load VL can be found by:

$$V\_L = I\_L \times R\_L \text{ measured in volts} \tag{29}$$

where the current at the load IL is equal to the circuit current I. The electrical power generated by the module at the load can then be found by:

$$P\_L = I\_L^2 \times R\_L \text{ measured in watts} \tag{30}$$

or by using:

$$P\_L = V\_L \times I\_L \text{ measured in watts} \tag{31}$$

The efficiency η of the thermoelectric module can be found by:

$$\eta = \frac{P\_L}{Q\_h} \tag{32}$$

The cooling power at the hot side of the thermoelectric module Qh can be found by:

$$Q\_h = (a\_m)IT\_h - (T\_c - T\_h)(K\_m) - \frac{I^2(R\_m)}{2} \text{ measured in watts} \tag{33}$$

The efficiency η of the thermoelectric module can now be found by:

$$\eta = \frac{\left\{\frac{(a\_m)(T\_m)}{R\_m + R\_L}\right\}^2 R\_L}{(a\_m)IT\_h - (T\_c - T\_h)(K\_m) - \frac{I^2(R\_m)}{2}}\tag{34}$$

The thermoelectric figure-of-merit ZT can be found using:

$$Z\_T = \frac{\alpha\_m^{-2} \sigma\_m}{\lambda\_m} \tag{35}$$

where α<sup>m</sup> is the module's average Seebeck coefficient, σ<sup>m</sup> is the average electrical conductivity, and λ<sup>m</sup> is the module average thermal conductivity, at a specific operating temperature.

### 2.4 A numerical analysis of a thermoelectric power generation module

If we consider a thermoelectric module, [12] have published some general material properties data for a 127 thermoelectric couple module. For the purpose of this analysis, we will use a hot side temperature TH of 400 K, and a cold side temperature TC of 300 K. The Seebeck coefficient and module resistance is temperature dependent, hence, their values should be calculated at the average module temperature Tm given by:

$$T\_m = \frac{T\_h + T\_c}{2} \text{ measured in kelvin}$$

$$T\_m = \frac{400 + 300}{2} \tag{36}$$

$$T\_m = 350 \text{ Kelvin}$$

<sup>I</sup> <sup>¼</sup> <sup>V</sup> RM þ RL

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2.4.6 The electrical power generated at the load PL

2.4.7 The efficiency of the thermoelectric module η

PL ¼ I 2

(Eq. (30)) as:

by (Eq. (33)) as:

Eq. (32) as:

23

or by using (Eq. (31)) as:

2.4.5 The load current IL and the voltage generated at the load VL

<sup>¼</sup> <sup>5</sup>:<sup>544</sup>

I ¼ 0:596 A

IL ¼ 0:596 A

VL ¼ IL � RL ¼ 0:596 � 4:6491 VL ¼ 2:7710V

The electrical power generated by the module at the load can now be found by

PL ¼ 1:6514 W

PL ¼ VL � IL ¼ 2:7710 � 0:596 PL ¼ 1:6515W

The cooling power at the hot side of the thermoelectric couple Qh can be found

I 2 ð Þ Rm 2

2

ð Þ 4:6491 2

Qh ¼ ð Þ α<sup>m</sup> ITh � ð Þ Tc � Th ð Þ� Km

Qh <sup>¼</sup> ðð Þ� <sup>0</sup>:<sup>05544</sup> <sup>0</sup>:<sup>596</sup> � <sup>400</sup>Þ � ðð Þ <sup>300</sup> � <sup>400</sup> ð Þ <sup>0</sup>:<sup>4422</sup> Þ � <sup>0</sup>:596<sup>2</sup>

Qh <sup>¼</sup> <sup>13</sup>:<sup>2169</sup> � �ð <sup>100</sup> � ð Þ <sup>0</sup>:<sup>4422</sup> Þ � <sup>0</sup>:<sup>3552</sup> � ð Þ <sup>4</sup>:<sup>6491</sup>

Qh ¼ 13:2169 þ 44:22 � 0:8257 Qh ¼ 56:62W

Hence, the efficiency of the thermoelectric module η can now be found by

<sup>¼</sup> <sup>1</sup>:<sup>6515</sup> 56:62

<sup>η</sup> <sup>¼</sup> PL Qh

<sup>L</sup> � RL <sup>¼</sup> <sup>0</sup>:596<sup>2</sup> � <sup>4</sup>:<sup>6491</sup>

The load current IL is the same value as the circuit current I, hence:

The voltage generated at the load can be found by (Eq. (29)) as:

<sup>4</sup>:<sup>6491</sup> <sup>þ</sup> <sup>4</sup>:<sup>6491</sup> <sup>¼</sup> <sup>5</sup>:<sup>544</sup>

9:2982

According to [12], with an average module temperature of 350 K, the module thermoelectric parameters are; a Seebeck coefficient α<sup>m</sup> of 0.05544 V/K; a module resistance RM of 4.6491 Ohms; and a thermal conductivity KM of 0.4422 W/K.
