**4. How to avoid thermal bridges**

In the case of having four modules in an assembly with only four screws, the fact that pressure is not uniformly distributed becomes more noticeable. Only the four corners, where screws

If an additional screw is located in the center so that the bending of the dissipater is reduced and the pressure is increased in order to improve the contact, the improvements obtained are minimal (**Figure 7b**) due to the big distance between the additional screw and the modules.

Visual analysis of **Figures 6** and **7** states a pressure distribution which is far from uniformity. In this section, this fact is demonstrated with a statistical analysis based on the median. Since the pressure films used for the experiments are ranged from 0.6 to 2.5 MPa, those parts that work with a pressure out of scale will not be appropriately represented. Thus, the median is the most significant parameter to analyze: half of the pixels work under that pressure while

On the one hand, for the heat exchanger with two modules, in the case in which there are only four screws, most of the module works under a pressure of 0.6 MPa or less. This value improves if the bending is restricted by means of an additional pair of screws located between the modules. As shown in **Figure 8**, and accordingly to the previous states, the median of the

On the other hand, in the heat exchanger with four modules, the lack of macroscopic contact is evident with the median value of 0.6 MPa, i.e. most parts of the modules have a clamping pressure of less than 0.6 MPa. Nonetheless, although the visual analysis evinces a slight improvement when the fifth screw is introduced and a more significant torque exerted, this fact is not appreciable in the median value since the range of pressures of both configurations

Hence, despite not being considered a critical aspect, screw distribution and torque are aspects to definitely take into account. The assembly configuration determines the pressure

are actually located, are working under an appreciable pressure (**Figure 7a**).

Thus, it is recommended that each module has its own tightening.

pressure increases with the torque in a proportion that seems linear.

is much lower than 0.6 MPa, the lower limit of the scale of the film.

**Figure 7.** Pressure films for (a) four screws, 1 Nm (b) five screws, 2 Nm.

*3.2.2. Quantitative analysis*

132 Bringing Thermoelectricity into Reality

the other half works above it.

Last section has concluded that it is recommended to have an individual tightening of the modules in order to ensure a uniform pressure distribution that leads to a good thermal contact. In order to achieve it, screws are necessary although they represent a type of thermal bridge.

By definition, a thermal bridge is an area or component of an object which has higher thermal conductivity than the surrounding materials, creating a path of least resistance for heat transfer [45, 46]. In thermoelectric generators, there are two main sources for thermal bridges. On the one hand, the screws used to ensure a good contact and pressure distribution are normally metallic, and therefore, highly thermal conductive. In order to reduce the amount of heat lost through these screws, it is typical to use nylon washers. Nonetheless, even with this nylon rings, it is estimated that the thermal resistance of each screw is 52 K/W.

On the other hand, due to the small thickness of the thermoelectric modules (most commonly 3 mm), it can occur that part of the heat directly flows within the heat exchangers, instead of through the thermoelectric modules. Therefore, insulating materials are usually inserted between the heat exchangers.

In the present section, assuming there is a good thermal contact, three different alternatives of insulating materials that are normally used in order to avoid thermal bridges will be studied:


#### **4.1. Methodology**

For each of these insulating materials, the assembly depicted in **Figure 9** was mounted. A heating plate acts as a heat source, providing a power, *Q*̇ *source*, of 100, 150 or 200 W. This plate is in direct contact with two thermoelectric modules surrounded by the insulating material of study in each case. In the cold side of the thermoelectric modules, there is a fin dissipater assisted by a ventilator. In order to minimize direct thermal losses from the heating plate to the environment, a 50 mm layer of rock wool has been used.

**Figure 9** also shows the location of the type K thermocouples installed. Hence, the three involved heat fluxes can be determined: the heat flux that goes through the modules and therefore is responsible of the electric generation, *Q*̇ mod; the heat flux lost due to the thermal bridges, *Q*̇ *tb*; and the heat flux that is directly lost from the source to the ambient, *Q*̇ *amb*. This last heat flux has been calculated thanks to the insulation and the ambient temperatures in conjunction with a convection coefficient of *h* = 5 W/m2 K [49].

$$
\dot{Q}\_{amb} = h \cdot A \cdot \left( T\_{ins} - T\_{amb} \right) \tag{3}
$$

Secondly, the heat that goes through the modules is computed with the mentioned thermal

**Figure 9.** Schematic view of the studied configuration with specification of the position of the type K thermocouples

mod <sup>=</sup> *<sup>M</sup>* <sup>⋅</sup> (*Th* <sup>−</sup> *Tc*) \_\_\_\_\_\_\_\_\_\_\_\_ *Rco*,*<sup>h</sup>* <sup>+</sup> *<sup>R</sup>*mod <sup>+</sup> *Rco*,*<sup>c</sup>*

Finally, the heat that is lost in the thermal bridge can be computed as the difference between the heat that is supposed to flow through the modules, and the heat that actually passes

*<sup>h</sup>* <sup>−</sup> *<sup>Q</sup>*̇

Nonetheless, the percentage values of the heat that goes through the thermal bridges instead

*tb* <sup>=</sup> *<sup>Q</sup>*̇ \_\_\_*tb Q*̇ *h*

The repetitiveness of the experiments has been ensured by means of mounting and dismounting the ensemble three times for each configuration, which also reduces the uncertainty of the

mod (6)

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⋅ 100 (7)

*tb* <sup>=</sup> *<sup>Q</sup>*̇

(5)

resistances, the measured temperature difference and the number of modules *M*.

*Q*̇

installed and the heat fluxes considered.

*Q*̇

of the modules acquires more interest:

%*Q*̇

measurements and the experimental procedure.

through them.

In contrast, the calculation of the other two heat fluxes has required the use of a computational model. Temperatures *Tc* and *Th* are known, but thermal resistances depend on the thermal contacts between the different parts, *Rco,h* and *Rco,c*, as well as on the existing temperature distribution in each case. Thus, the computational model based on the finite difference method of **Figure 10** has been used. This model calculates all the possible thermal resistances of the modules, *R*mod, and their generating voltage depending on the temperature difference and the load resistance, with an error less than 10% [50]. By comparing the experimental temperature difference and the open circuit voltage determined by the model, both the thermal contact resistances and the thermoelectric modules resistance have been estimated.

Based on them, the different heat fluxes can be computed as follows. Firstly, the heat that is supposed to flow through the modules is the heat provided by the source minus the heat directly lost to the environment.

$$
\dot{Q}\_h = \dot{Q}\_{sava} - \dot{Q}\_{amb} \tag{4}
$$

• Mineral wool fiber cardboard manufactured by Nefalit. This material is made up of highly thermal insulating fibers bound together with fillers. These plates are suitable for temperatures up to 750 or 1000°C and are easy to handle in the assembly. They present a thermal

• Acrylic wool made by Flexiband: this insulating material is manufactured from pure refractory fibers which provide a low thermal conductivity (0.09 W/mK) and flexibility [48]. • Air: known as one of the best insulators, air presents a thermal conductivity of 0.024 W/mK. Nevertheless, if convection currents are created, heat transfer coefficients improve and the effective value of the thermal conductivity can be considerably increased.

For each of these insulating materials, the assembly depicted in **Figure 9** was mounted. A

is in direct contact with two thermoelectric modules surrounded by the insulating material of study in each case. In the cold side of the thermoelectric modules, there is a fin dissipater assisted by a ventilator. In order to minimize direct thermal losses from the heating plate to

**Figure 9** also shows the location of the type K thermocouples installed. Hence, the three involved heat fluxes can be determined: the heat flux that goes through the modules and

*tb*; and the heat flux that is directly lost from the source to the ambient, *Q*̇

heat flux has been calculated thanks to the insulation and the ambient temperatures in con-

In contrast, the calculation of the other two heat fluxes has required the use of a compu-

thermal contacts between the different parts, *Rco,h* and *Rco,c*, as well as on the existing temperature distribution in each case. Thus, the computational model based on the finite difference method of **Figure 10** has been used. This model calculates all the possible thermal resistances of the modules, *R*mod, and their generating voltage depending on the temperature difference and the load resistance, with an error less than 10% [50]. By comparing the experimental temperature difference and the open circuit voltage determined by the model, both the thermal

Based on them, the different heat fluxes can be computed as follows. Firstly, the heat that is supposed to flow through the modules is the heat provided by the source minus the heat

*source* <sup>−</sup> *<sup>Q</sup>*̇

contact resistances and the thermoelectric modules resistance have been estimated.

*<sup>h</sup>* <sup>=</sup> *<sup>Q</sup>*̇

*source*, of 100, 150 or 200 W. This plate

mod; the heat flux lost due to the thermal

*amb* (4)

*amb* = *h* ⋅ *A* ⋅ (*Tins* − *Tamb*) (3)

are known, but thermal resistances depend on the

*amb*. This last

conductivity of 0.15 W/mK [47].

134 Bringing Thermoelectricity into Reality

heating plate acts as a heat source, providing a power, *Q*̇

the environment, a 50 mm layer of rock wool has been used.

therefore is responsible of the electric generation, *Q*̇

*Q*̇

tational model. Temperatures *Tc*

directly lost to the environment.

*Q*̇

junction with a convection coefficient of *h* = 5 W/m2 K [49].

and *Th*

**4.1. Methodology**

bridges, *Q*̇

**Figure 9.** Schematic view of the studied configuration with specification of the position of the type K thermocouples installed and the heat fluxes considered.

Secondly, the heat that goes through the modules is computed with the mentioned thermal resistances, the measured temperature difference and the number of modules *M*.

$$\dot{Q}\_{\text{mod}} = \frac{M \cdot (T\_h - T\_c)}{R\_{\text{on}h} + R\_{\text{mod}} + R\_{\text{on}c}} \tag{5}$$

Finally, the heat that is lost in the thermal bridge can be computed as the difference between the heat that is supposed to flow through the modules, and the heat that actually passes through them.

$$
\dot{Q}\_{th} = \dot{Q}\_h - \dot{Q}\_{\text{mod}} \tag{6}
$$

Nonetheless, the percentage values of the heat that goes through the thermal bridges instead of the modules acquires more interest:

$$\% \dot{Q}\_{\text{o}} = \frac{\dot{Q}\_{\text{o}}}{\dot{Q}\_{\text{i}}} \cdot 100 \tag{7}$$

The repetitiveness of the experiments has been ensured by means of mounting and dismounting the ensemble three times for each configuration, which also reduces the uncertainty of the measurements and the experimental procedure.

**Figure 10.** Thermal-electric analogy of the computational model.

#### **4.2. Results and discussion**

In a first approach, **Figure 11** shows the results obtained for the configuration in which the insulating material has the same thickness as the module, i.e. 3 mm (**Figure 12**). As it can be observed, for all the studied materials, the percentage of heat lost through the insulation increases with the heat dissipated by the source. Nonetheless, this amount differs for each material. Thus, cardboard presents the most significant heat losses, with almost 30% lost due to thermal bridges. This is the expected result since it presents the highest conductivity of the studied materials. However, despite the air having the smallest thermal resistance, the thickness between both heat exchangers is enough to create convection currents that improve the heat transfer. As a result, thermal losses with either acrylic wool or air as insulators are similar. Furthermore, it is worth mentioning that in the case of having air, which is equivalent to not putting anything, the assembly is more complicated since modules can easily move, as they are not held by the insulating material.

Since thermal losses represent a considerable part of the heat that should go through the modules, a second configuration (**Figure 13**) with a higher distance between the heat exchangers increases was also studied. Increasing the thickness of the insulation causes an increment of the thermal resistance, and therefore, thermal losses should decrease. This increment of thickness

is compensated with highly conductive aluminum heat extenders in the case of the modules. Nonetheless, this leads to two additional contacts, with their associated thermal contact resistance, which can negatively influence the thermal transfer if contact is not appropriated. In this case, graphite sheets have again been used as TIM in order to ensure the microscopic contact. As it can be observed in **Figure 14**, the increment of distance between the heat exchangers supposes a reduction of approximately 5% in the percentage of heat that is lost due to thermal bridges since, as expected, thermal resistances of the insulation increase leading to less thermal losses. For this configuration, the tendency remains similar: heat losses increase with the power from the heat source and acrylic wool is still the best insulator and cardboard the worst. However, differences between acrylic wool and air are now more evident. Since the distance between the exchangers has increased, there is more space for the convection currents to flow, this improving the heat transfer and leading to more thermal losses.

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**Figure 12.** Exploded view of the studied generator without heat extenders.

**Figure 13.** Exploded view of the studied generator with heat extenders.

**Figure 11.** Percentage of heat loss through insulation for different heat powers in absence of heat extenders.

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**Figure 12.** Exploded view of the studied generator without heat extenders.

**Figure 13.** Exploded view of the studied generator with heat extenders.

**Figure 11.** Percentage of heat loss through insulation for different heat powers in absence of heat extenders.

In a first approach, **Figure 11** shows the results obtained for the configuration in which the insulating material has the same thickness as the module, i.e. 3 mm (**Figure 12**). As it can be observed, for all the studied materials, the percentage of heat lost through the insulation increases with the heat dissipated by the source. Nonetheless, this amount differs for each material. Thus, cardboard presents the most significant heat losses, with almost 30% lost due to thermal bridges. This is the expected result since it presents the highest conductivity of the studied materials. However, despite the air having the smallest thermal resistance, the thickness between both heat exchangers is enough to create convection currents that improve the heat transfer. As a result, thermal losses with either acrylic wool or air as insulators are similar. Furthermore, it is worth mentioning that in the case of having air, which is equivalent to not putting anything, the assembly is more complicated since modules can easily move, as

Since thermal losses represent a considerable part of the heat that should go through the modules, a second configuration (**Figure 13**) with a higher distance between the heat exchangers increases was also studied. Increasing the thickness of the insulation causes an increment of the thermal resistance, and therefore, thermal losses should decrease. This increment of thickness

**4.2. Results and discussion**

136 Bringing Thermoelectricity into Reality

they are not held by the insulating material.

**Figure 10.** Thermal-electric analogy of the computational model.

is compensated with highly conductive aluminum heat extenders in the case of the modules. Nonetheless, this leads to two additional contacts, with their associated thermal contact resistance, which can negatively influence the thermal transfer if contact is not appropriated. In this case, graphite sheets have again been used as TIM in order to ensure the microscopic contact.

As it can be observed in **Figure 14**, the increment of distance between the heat exchangers supposes a reduction of approximately 5% in the percentage of heat that is lost due to thermal bridges since, as expected, thermal resistances of the insulation increase leading to less thermal losses. For this configuration, the tendency remains similar: heat losses increase with the power from the heat source and acrylic wool is still the best insulator and cardboard the worst. However, differences between acrylic wool and air are now more evident. Since the distance between the exchangers has increased, there is more space for the convection currents to flow, this improving the heat transfer and leading to more thermal losses.

increase the distance between them with the aid of a conductive heat extender and insulate with acrylic wool. Nonetheless, if this material is not added, air has demonstrated to also be

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The authors are indebted to the Ministry of Economy, Industry and Competitiveness-Government of Spain and FEDER Funds for economic support of this work, included in the DPI2014-53158-R Research Project, as well as to the FPU Program of the Spanish Ministry of

K

/W

a good insulator, cheaper but which leads to a less precise assembly.

**Acknowledgements**

**Nomenclature**

*Q*̇

*Q*̇

*Q*̇

*Q*̇

*Q*̇

*Rco*,*<sup>h</sup>*

Education, Culture and Sport (FPU16/05203).

*α* Seebeck coefficient, V/K

*ρ* electrical resistivity, Ωm

*A* contact area, m2

*λ* thermal conductivity, W/mK

*h* free convection coefficient, W/m<sup>2</sup>

*M* number of thermoelectric modules *n* relative to type-n semiconductor *p* relative to type-p semiconductor *Q*̇ heat flux through the interface, W

*amb* heat flux losses to ambient through insulation, W

; *Rco*,*<sup>c</sup>* contact thermal resistance of hot and cold side, K/W

*<sup>h</sup>* heat through thermoelectric generator, W

mod heat through thermoelectric modules, W

*source* heat provided by heating plate, W

*tb* heat flux due to thermal bridges, W *<sup>R</sup><sup>a</sup>* arithmetical mean roughness (μm) *Rz* ten-point mean roughness (μm) *Rco* thermal contact resistance, Km2

*<sup>R</sup>*mod thermal resistance of the modules, K/W

*Tc* module cold face temperature, K

*<sup>T</sup>amb* ambient temperature, K

**Figure 14.** Percentage of heat loss through insulation for different heat powers in the case of introducing heat extenders.

In summary, losses due to thermal bridges definitely need to be taken into account: around one-fourth of the power provided from the source goes through the insulation instead of the thermoelectric modules. As a consequence, the generation is reduced.

In order to decrease the thermal losses, it has been demonstrated that it is better to increase the thickness of the insulation, despite adding two additional contacts. Among the materials studied, the best one is acrylic wool. Nonetheless, air (equivalent to not adding any insulation) should be considered, since the cost is reduced and there is not such a significant difference among them. However, it presents the disadvantage of a more complicated assembly.
