**2. Influence of heat exchangers on thermoelectric devices**

A thermoelectric pair can be used to generate electric power, since Seebeck effect indicates that if the junctions of two thermoelectric legs type "p" and "n" are exposed to different temperatures, an electric current is induced. On the other hand, if an external electric source supplies power to the thermoelectric pair, Peltier effect states that one junction absorbs heat whereas the other one generates heat, so that the thermoelectric pair performs like a thermal machine that receives electric work, removes heat from a cold reservoir and emits heat to a hot reservoir. There are in the market different types of Peltier modules, composed of several thermoelectric pairs connected electrically in series and thermally in parallel. Figure 1 shows an average thermoelectric module working as refrigerator. In order to improve the heat transfer both in the hot and the cold side, a heat exchanger must be installed at either side of the Peltier module to increase the heat transfer area.

Fig. 1. Sketch of a Peltier module working as refrigerator.

The Peltier module is a small device that emits –or absorbs- large amounts of heat, so that the heat density or heat flux is significantly high. The face of a Peltier module is so small that increasing the heat transfer surface area of the heat exchanger (finned dissipator and cold plate in Figure 2) is virtually useless, since the effectiveness of the heat exchanger

Heat Exchangers for Thermoelectric Devices 293

The model solves the one-dimensional thermal conduction equation in transitory state, being *ρ* density, *cp* specific heat under constant pressure, *t* time, and *q* heat flux generated.

> *p* 2 *T T ck q <sup>t</sup> <sup>x</sup>*

1 1

*R R*

*i i ii i i i i i i ii*

1

*C V cp i ii* 

The generated (or absorbed) heat flux in node "i" is represented by the last member on the right of the equation (5), and its value is given by the Peltier, Joule and Thomson

Regarding thermoelectric generation, one can obtain the voltage and the electric power generated by a thermoelectric module with the following expressions, where *m* stands for the ratio of the electrical load resistance to the electrical resistance of the module, and *N*

> <sup>2</sup> <sup>1</sup> *hh cc h c <sup>m</sup> V N T T TT*

 

(8)

Regarding thermoelectric refrigeration, the thermal resistance of the insulating chamber of a thermoelectric refrigerator is provided by the following expressions, where *S* and *e* represents the surface area and thickness of the chamber, whereas *U* stands for the global heat transfer coefficient. The convection coefficients inside and outside the chamber (*hin, hout*)

,

*i i*

*R*

*C T TT T T T <sup>Q</sup>*

The thermal resistance between nodes "i" and "i+1", as well as the thermal capacity of node "i" are expressed by the following equations, being *Li,i+1* , *Ai* and *Vi* respectively the distance, cross section and volume between two connected nodes. Thermal resistances and capacities of commercial thermoelectric modules can be found in the scientific literature (Astrain et al., 2005, 2010; Min & Rowe, 2006). Thermal contact resistances between the components of the thermoelectric system can be found in the literature (Ritzer & Lau, 1994). The thermal resistances of the heat exchangers are parameters of study; therefore they are model

 

represents the number of thermoelectric pairs of the module.

*m*

applied to a node "i".

inputs.

effects.

2

1 1 , ,

1

,

*k A* 

*i i*

*i i*

*L*

(5)

(6)

*P VI* (9)

(7)

(4)

Firstly, the system must be discretized, that is, reduced into a set of representative nodes, as can be seen in Figure 3 for a thermoelectric generation system. Then, after using the first and second derivatives in finite differences (Özisik, 1994), the one-dimensional thermal conduction equation in transitory state is transformed into the finite differences form,

decreases as more heat transfer surface area is added. This makes difficult to attain proper heat transfer. Therefore, efficiencies of thermoelectric modules and thermoelectric devices in general, designed for either generation or refrigeration purposes, depend to a great extent on the thermal resistances of the heat exchangers installed at either side of the modules.

Fig. 2. Thermoelectric device.

Several computational models have been developed (Stockholm & Stockholm, 1992; Astrain et al., 2005, 2010; Crane & Bell, 2006; Crane et al., 2009) to study the whole thermoelectric system, including the heat exchangers. These models serve as study and design tools for both thermoelectric refrigeration and generation devices.

## **2.1 Development and validation of a computational model for thermoelectric systems**

The computational model described in this section is an example that serves to assess the influence of the thermal resistances of the heat exchangers on the performance of thermoelectric devices. This model (Astrain et al., 2010) solves the non-linear system composed of thermoelectric and heat transfer equations, using the implicit finitedifferences method. When used to simulate thermoelectric generation devices, the model requires the following inputs: geometric data, material properties, number and type of thermoelectric modules, thermal resistances of the heat exchangers, ambient temperature and energy introduced into the system. Then, the model outputs are: efficiency, voltage, electric current, electric power generated, temperatures and heat fluxes, all of them timedependent. On the other hand, when the model simulates thermoelectric refrigeration devices, the inputs must be: geometric data, material properties, number and type of thermoelectric modules, thermal resistances of the heat exchangers, ambient temperature and voltage supplied to the modules. Then, the model provides temperatures and heat fluxes, coefficient of performance (COP), electric current and electric power consumed, again all of them time-dependent.

decreases as more heat transfer surface area is added. This makes difficult to attain proper heat transfer. Therefore, efficiencies of thermoelectric modules and thermoelectric devices in general, designed for either generation or refrigeration purposes, depend to a great extent on the thermal resistances of the heat exchangers installed at either side of the modules.

Several computational models have been developed (Stockholm & Stockholm, 1992; Astrain et al., 2005, 2010; Crane & Bell, 2006; Crane et al., 2009) to study the whole thermoelectric system, including the heat exchangers. These models serve as study and design tools for

**2.1 Development and validation of a computational model for thermoelectric systems**  The computational model described in this section is an example that serves to assess the influence of the thermal resistances of the heat exchangers on the performance of thermoelectric devices. This model (Astrain et al., 2010) solves the non-linear system composed of thermoelectric and heat transfer equations, using the implicit finitedifferences method. When used to simulate thermoelectric generation devices, the model requires the following inputs: geometric data, material properties, number and type of thermoelectric modules, thermal resistances of the heat exchangers, ambient temperature and energy introduced into the system. Then, the model outputs are: efficiency, voltage, electric current, electric power generated, temperatures and heat fluxes, all of them timedependent. On the other hand, when the model simulates thermoelectric refrigeration devices, the inputs must be: geometric data, material properties, number and type of thermoelectric modules, thermal resistances of the heat exchangers, ambient temperature and voltage supplied to the modules. Then, the model provides temperatures and heat fluxes, coefficient of performance (COP), electric current and electric power consumed,

**Peltier Module**

**Insulation** 

Fig. 2. Thermoelectric device.

again all of them time-dependent.

**Cold Extender**

both thermoelectric refrigeration and generation devices.

**Fan**

**Finned dissipator** 

**Cold plate** 

The model solves the one-dimensional thermal conduction equation in transitory state, being *ρ* density, *cp* specific heat under constant pressure, *t* time, and *q* heat flux generated.

$$
\rho \, c \, c\_p \, \frac{\partial T}{\partial t} = k \left( \frac{\partial^2 T}{\partial \mathbf{x}^2} \right) + \dot{q} \tag{4}
$$

Firstly, the system must be discretized, that is, reduced into a set of representative nodes, as can be seen in Figure 3 for a thermoelectric generation system. Then, after using the first and second derivatives in finite differences (Özisik, 1994), the one-dimensional thermal conduction equation in transitory state is transformed into the finite differences form, applied to a node "i".

$$\frac{C\_i}{\delta \tau} (T\_i' - T\_i) = \frac{T\_{i-1} \stackrel{\prime}{\ } -T\_i \stackrel{\prime}{\ }}{R\_{i-1,i}} + \frac{T\_{i+1} \stackrel{\prime}{\ } -T\_i \stackrel{\prime}{\ }}{R\_{i,i+1}} + \dot{Q}\_i \tag{5}$$

The thermal resistance between nodes "i" and "i+1", as well as the thermal capacity of node "i" are expressed by the following equations, being *Li,i+1* , *Ai* and *Vi* respectively the distance, cross section and volume between two connected nodes. Thermal resistances and capacities of commercial thermoelectric modules can be found in the scientific literature (Astrain et al., 2005, 2010; Min & Rowe, 2006). Thermal contact resistances between the components of the thermoelectric system can be found in the literature (Ritzer & Lau, 1994). The thermal resistances of the heat exchangers are parameters of study; therefore they are model inputs.

$$R\_{i,i+1} = \frac{L\_{i,i+1}}{k\_i A\_i} \tag{6}$$

$$\mathbf{C}\_{i} = V\_{i}\rho\_{i}\mathbf{c}p\tag{7}$$

The generated (or absorbed) heat flux in node "i" is represented by the last member on the right of the equation (5), and its value is given by the Peltier, Joule and Thomson effects.

Regarding thermoelectric generation, one can obtain the voltage and the electric power generated by a thermoelectric module with the following expressions, where *m* stands for the ratio of the electrical load resistance to the electrical resistance of the module, and *N* represents the number of thermoelectric pairs of the module.

$$
\Delta V = \left(\frac{m}{1+m}\right) 2N \left( \left( \alpha\_h T\_h - \alpha\_c T\_c \right) - \tau \left( T\_h - T\_c \right) \right) \tag{8}
$$

$$P = \Delta VI\tag{9}$$

Regarding thermoelectric refrigeration, the thermal resistance of the insulating chamber of a thermoelectric refrigerator is provided by the following expressions, where *S* and *e* represents the surface area and thickness of the chamber, whereas *U* stands for the global heat transfer coefficient. The convection coefficients inside and outside the chamber (*hin, hout*)

Heat Exchangers for Thermoelectric Devices 295

simulations were complemented with experimental tests conducted with a prototype thermoelectric refrigerator supplied with 12 V, and different configurations of heat exchangers. A controllable fan was installed over the heat exchanger outside the refrigerator (consisting of a finned dissipator), which allows the control of the air flowing through the fins of this heat exchanger and, therefore, its thermal resistance. Inside the refrigerator, the studied configurations of heat exchanger were: flat plate (no fins) without fan, finned

Table 1 shows experimental and simulated temperatures inside the refrigerator (*Tint*) and at either side of the Peltier module (*Thp* and *Tcp*), as well as the temperature difference between the inside and the ambient (*Tamb-Tint*). The model predicts temperatures accurately, and the maximum error in temperature differences reaches 1.2 K. What is more, the study highlights the significant influence of the thermal resistances of the heat exchangers on the performance of thermoelectric refrigerators; particularly, the temperature difference between the inside of the refrigerator and the ambient increases by more than 40% when

**Configuration Test characteristics Temperature (K) Model Prototype** 

Table 1. Experimental and simulated results for 12 V of supplied voltage to the Peltier

**2.3 Influence of heat exchangers on the performance of thermoelectric generators**  A methodology similar to that used for thermoelectric refrigerators was applied to study the influence of the thermal resistances of the heat exchangers at either side of the thermoelectric modules on the electric power generated by a thermoelectric generator. Figure 4 shows the electric power as a function of both thermal resistances, where one can observe the significant increase in the electric power that occurs when decreasing the thermal resistances of both heat exchangers. As an example, if both thermal resistances improved from 0.5 to 0.4 K/W, the electric power generated would increase by 20 %. On average, a decrease by 10 % in both thermal resistances entails an increase in the electric power by around 8 %. This serves to illustrate the importance of the design of heat

*Tint* 278.9 277.7 *Thp* 311.2 308.4 *Tcp* 263.3 261.6 *(Tamb-Tint) 17.8 19* 

*Tint* 280.6 279.8 *Thp* 316.2 312.1 *Tcp* 265.2 264 *(Tamb-Tint) 16.1 16.9* 

*Tint* 278.5 277.3 *Thp* 312.6 309.4 *Tcp* 265.3 264.2 *(Tamb-Tint) 18.1 19.3* 

*Tint* 273.7 272.7 *Thp* 311.4 308.8 *Tcp* 266.6 266 *(Tamb-Tint) 22.9 23.9* 

dissipator without fan, finned dissipator with a fan supplied with 3 V (0.5 W).

configuration 4 is used instead of configuration 2.

supplied with 12 V.

supplied with 4 V.

supplied with 12 V.

supplied with 12 V.

supplied with 3 V.

exchangers in thermoelectric generation applications.

Inside: Flat plate, no fan.

Inside: Flat plate, no fan.

Outside: Finned dissipator, fan

Outside: Finned dissipator, fan

Outside: Finned dissipator, fan

Inside: Finned dissipator, no fan.

Outside: Finned dissipator, fan

Inside: Finned dissipator, fan

module, ambient temperature 296.6 K and 60% of relative humidity.

1

2

3

4

are calculated with equation (12), which is an experimental expression of the convection coefficient in a plane plate that considers laminar flow and dismisses viscosity dissipation (Parmelee & Huebscher, 1947).

$$R\_{ins} = \frac{1}{LIS} \tag{10}$$

$$\text{LII} = \frac{1}{\text{l/h}\_{in} + \text{e}/\text{k} + \text{l/h}\_{out}} \tag{11}$$

$$\begin{aligned} \text{Nu}\_{\text{L}} &= 0.664 \cdot \text{Pr}^{1/3} \cdot \text{Re}\_{\text{L}}^{1/2} \\ 0.6 \le \text{Pr} \le 50, \text{Re} &< \text{Re}\_{\text{x}, \text{c}} \approx 5 \cdot 10^5 \end{aligned} \tag{12}$$

Fig. 3. Sketch and discretization of a thermoelectric generator.

The experimental validation proved that this computational model predicts the experimental values of the output parameters with errors always lower than 10%. Therefore, this model is an appropriate tool to study and present the significant influence that the heat exchangers have on the performance of both thermoelectric refrigerators and generators.

#### **2.2 Influence of heat exchangers on the performance of thermoelectric refrigerators**

Different values of thermal resistances of the heat exchangers at either side of the thermoelectric module are introduced in the computational model in order to study their influence on the performance of a thermoelectric refrigerator. Moreover, resulting model

are calculated with equation (12), which is an experimental expression of the convection coefficient in a plane plate that considers laminar flow and dismisses viscosity dissipation

> 1 1 1 *in out*

L L

0.6 Pr 50,Re Re 5 10

Nu 0.664 Pr Re

1/3 1/2

x,c

*U*

<sup>1</sup> *Rins US* (10)

*h ek h* (11)

5

(12)

(Parmelee & Huebscher, 1947).

C1 R1,2

C2 R2,3

C3 R3,4

T1 T2 T3 T4 T5

C4 R4,5

Heat exchanger

> C5 R5,6

Fig. 3. Sketch and discretization of a thermoelectric generator.

C6 R6,7

The experimental validation proved that this computational model predicts the experimental values of the output parameters with errors always lower than 10%. Therefore, this model is an appropriate tool to study and present the significant influence that the heat exchangers have on the performance of both thermoelectric refrigerators and generators.

**2.2 Influence of heat exchangers on the performance of thermoelectric refrigerators**  Different values of thermal resistances of the heat exchangers at either side of the thermoelectric module are introduced in the computational model in order to study their influence on the performance of a thermoelectric refrigerator. Moreover, resulting model

 R12,13

Peltier module

C13 R13,14

C14 R14,15

C15 R15,16

C16 R16,17

C17

T17

T16

T15

dissipator

T14

 Q14

Semiconductor Heat Ambient

Ceramic layer

T13

 Q13

T6

 Q6

 Q5

Ceramic layer

 Q2

Ambient Heat

source

simulations were complemented with experimental tests conducted with a prototype thermoelectric refrigerator supplied with 12 V, and different configurations of heat exchangers. A controllable fan was installed over the heat exchanger outside the refrigerator (consisting of a finned dissipator), which allows the control of the air flowing through the fins of this heat exchanger and, therefore, its thermal resistance. Inside the refrigerator, the studied configurations of heat exchanger were: flat plate (no fins) without fan, finned dissipator without fan, finned dissipator with a fan supplied with 3 V (0.5 W).

Table 1 shows experimental and simulated temperatures inside the refrigerator (*Tint*) and at either side of the Peltier module (*Thp* and *Tcp*), as well as the temperature difference between the inside and the ambient (*Tamb-Tint*). The model predicts temperatures accurately, and the maximum error in temperature differences reaches 1.2 K. What is more, the study highlights the significant influence of the thermal resistances of the heat exchangers on the performance of thermoelectric refrigerators; particularly, the temperature difference between the inside of the refrigerator and the ambient increases by more than 40% when configuration 4 is used instead of configuration 2.


Table 1. Experimental and simulated results for 12 V of supplied voltage to the Peltier module, ambient temperature 296.6 K and 60% of relative humidity.

#### **2.3 Influence of heat exchangers on the performance of thermoelectric generators**

A methodology similar to that used for thermoelectric refrigerators was applied to study the influence of the thermal resistances of the heat exchangers at either side of the thermoelectric modules on the electric power generated by a thermoelectric generator. Figure 4 shows the electric power as a function of both thermal resistances, where one can observe the significant increase in the electric power that occurs when decreasing the thermal resistances of both heat exchangers. As an example, if both thermal resistances improved from 0.5 to 0.4 K/W, the electric power generated would increase by 20 %. On average, a decrease by 10 % in both thermal resistances entails an increase in the electric power by around 8 %. This serves to illustrate the importance of the design of heat exchangers in thermoelectric generation applications.

Heat Exchangers for Thermoelectric Devices 297

 Increasing the height of the fins is also beneficial from both a thermal and hydrodynamic point of view. Specifically, if this parameter rises from 40 mm to 60 mm

in a dissipator with an axial fan over it, the thermal resistance reduces by 10.4 %.

Fig. 5. Temperature distribution in a finned dissipator, with and without thermosyphon.

exchanger installed at the hot side of the Peltier modules decreases by 13 %.

phase-change thermosyphons, described in the following section.

**3.2 Thermosyphon for the hot end of a Peltier module** 

Finally, this work presents a prototype thermoelectric refrigerator that served to experimentally prove that the COP improves by 10 % if the thermal resistance of the heat

In conclusion, this work makes evident the important role that holds the thermal resistance of the heat exchangers in the efficiency of a thermoelectric refrigeration device. Likewise, it also indicates that the thermal resistance of a finned dissipator is too high despite the optimization process. This fact indicates that it is absolutely necessary to design new types of heat exchangers in order to reduce even more the thermal resistance and increase the efficiency of thermoelectric devices. In this line of work, there have been developed the

A thermosyphon is a hermetically sealed container in the shape of a straight prism, enclosing a fluid. The Peltier module is attached to the bottom of the rear surface, so that the heat flux produced by the module is transmitted to the fluid, which begins to boil. Vapour produced in the process rises up to the top of the thermosyphon by natural convection. Likewise, the cold reservoir (usually the ambient) is connected to the front surface of the thermosyphon, where several fins are installed. Thus, when the vapour touches this cold

Fig. 4. Electric power generated (Pmax) versus thermal resistance of the hot side heat exchanger (Rdc) and thermal resistance of the cold side heat exchanger (Rdf).
