6. Modeling approaches for thermoelectric cooling

Both system cooling power output and cooling COP should be considered for enhancing TE cooling system performance. There are three methods that can possibly lead to this enhancement. First, TE module design and optimization, such as number of thermocouples [63–66], thermoelement length [67–70] and thermoelement length to cross-sectional area ratio [71–73]. Second, cooling system thermal design and optimization [74], which includes investigation of heat sinks' geometry [75–77], identification of the heat transfer area and heat transfer coefficients of both hot and cold side heat sinks [78–80], more effective heat sinks (i.e. heat sink integrated with thermosyphon and phase change material) [16, 81, 82], thermal and electrical contact resistances and interface layer analysis [83–85]. Third, the TE cooling system working conditions (i.e. electric current input [86–88]), heat sink coolant and coolant's mass flow rate [10, 89].

In order to achieve this, a variety of system optimization methods have been adopted. The simplified energy equilibrium model for TE cooler can satisfy many different TE cooling applications including electronic devices cooling and air conditioning [90–94]. If the TE modules are employed with time-varying temperature distribution and cooling power output, either 1D or 3D transient modeling is needed to better capture the system performance. To capture the module performance, modeling temperature change in all thermoelements is very complicated. Therefore, energy equilibrium model can be applied to simplify the numerical analysis process, especially for those systems which include heat sinks in hot and cold sides.

Positive Thomson coefficient improves TE cooling performance by 5–7% [95], while negative Thomson coefficient reduces cooling performance [96]. However, for commercially available TE coolers, Thomson effect is often small and negligible. Dimensionless analysis is a powerful tool to evaluate the performance of TE cooling system. New dimensionless parameters, such as dimensionless entropy generation number [78], dimensionless thermal conductance ratio and dimensionless convection ratio [64] have been defined.

Both COP and cooling capacity are dependent on the length of thermoelement, and this dependence becomes highly significant with the decrease in the length of thermoelement [97]. As a result, a long thermoelement is preferred to obtain a large COP, while a short thermoelement would be preferable to achieve maximum heat pumping capacity. Therefore, it is obvious that the design of the optimum module will be a tradeoff between the requirements for the COP and the heat pumping capacity. Most commercially available TE modules have thermoelement length range from 1.0 to 2.5 mm. Cooling power density also increases with decreasing the ratio of thermoelement length to the cross-sectional area.

Typical TE modules have a size range from 4 � <sup>4</sup> � 3 mm<sup>3</sup> to about 50 � <sup>50</sup> � 50 mm<sup>3</sup> . The development of micro-TE devices to further reduce the dimensions, that is compatible with standard microelectronic fabrication technology [98], has the potential to improve the microelectronic systems performance, achieve considerable reductions in size and improve the TE devices performance, which opens up new commercial applications.

Electrical and thermal contact resistances, especially thermal contact resistance at the thermoelement interface layer, are critical to achieve a further improvement in both TE cooling capacity and COP [84]. An enhanced formula for the COP of a Peltier module which takes into consideration both the electrical and the thermal contact resistances can be written as [34]:

$$\text{COP}\_{opt} = \frac{l}{l + 2rl\_c} \left( \frac{T\_c}{T\_h - T\_c} \frac{\beta - T\_h/T\_c}{1 + \beta} - \frac{rl\_c}{l} \right) \tag{12}$$

where

introduced. The results of an experiment performed on a liquid like behavior of copper ions around a crystalline sublattice of Se in Cu2xSe showed a very low lattice thermal conductivity

The efficiency of TE devices can be further enhanced through nanostructural engineering [44] using two primary approaches: bulk materials containing nano-scale constitutes and nanoscale materials themselves. By the introduction of nanostructures, ZT was pushed to about 1.7

Many reviews have summed up progress on thermoelectric materials [49, 50], bulk thermoelectric materials [45] and low-dimensional thermoelectric materials [43, 51, 52]. Lowdimensional materials, including 2-D quantum wells, 1-D quantum wires and 0-D quantum dots, possess the quantum confinement effect of the electron charge carriers that would enhance the Seebeck coefficient and thus the power factor [53]. Furthermore, the introduced various interfaces will scatter phonons more effectively than electrons so that it reduces the

Two-dimensional Bi2Te3 quantum well improved ZT due to the enhancement of thermopower [54]. The ZT of Bi2Te3 quantum well structures are estimated to be much higher than its bulk material. The highest ZT observed was 2.4 using Bi2Te3–Sb2Te3 quantum well superlattices with a periodicity of 6 nm [55]. Similarly, the highest ZT value for its bulk material is only 1.1. Quantum-dot superlattices in the PbTe–PbSeTe system were developed under the quantum confinement may lead to an increased Seebeck coefficient and therefore higher ZT [56]. PbSe nanodots were embedded in a PbTe matrix and showed ZT of 1.6, which is much higher than their bulk materials of 0.34 [52]. Serial compound Ag1–<sup>x</sup>Pb18SbTe20 has a high ZT value of 2.2 at 800 K due to the special nanostructure that is still the most competitive TE material [57] and has ignited broad research interest [58–61]. These new technologies have pushed ZT to 2.4 [62] with predicted increase in the device conversion efficiency to a value between 15 and 20%.

Both system cooling power output and cooling COP should be considered for enhancing TE cooling system performance. There are three methods that can possibly lead to this enhancement. First, TE module design and optimization, such as number of thermocouples [63–66], thermoelement length [67–70] and thermoelement length to cross-sectional area ratio [71–73]. Second, cooling system thermal design and optimization [74], which includes investigation of heat sinks' geometry [75–77], identification of the heat transfer area and heat transfer coefficients of both hot and cold side heat sinks [78–80], more effective heat sinks (i.e. heat sink integrated with thermosyphon and phase change material) [16, 81, 82], thermal and electrical contact resistances and interface layer analysis [83–85]. Third, the TE cooling system working conditions (i.e. electric current input [86–88]), heat sink coolant and coolant's mass flow rate [10, 89].

In order to achieve this, a variety of system optimization methods have been adopted. The simplified energy equilibrium model for TE cooler can satisfy many different TE cooling

which increased the value of ZT in this simple semiconductor [47].

thermal conductivity more than the electrical conductivity [18].

6. Modeling approaches for thermoelectric cooling

[48] with power conversion efficiency of 11–15%.

254 Bringing Thermoelectricity into Reality

<sup>β</sup> = 1 <sup>þ</sup> lZTM nþl <sup>1</sup>=<sup>2</sup> , n = <sup>2</sup>Rc <sup>R</sup> , r <sup>¼</sup> <sup>k</sup> kc , l is the thermoelement length, lc is the thickness of the contact layer, k is the thermal conductivity of the thermoelements, kc is the thermal conductivity of the contact layers and T<sup>M</sup> = ð Þ ThþTc <sup>2</sup> .

In addition, an accurate fabrication technique is needed to provide high-quality and highperformance TE modules. The requirements include: precise measurements of the internal resistance for each module at ambient temperature; determination of the module supply leads resistance; consideration of optimum values for voltage and current of each module; verification of thermal efficiency of each module and calculations of temperature difference, maximum cooling capacity according to the measurement results, figure of merit and values of internal resistance [99].

transferring the heat between the two surfaces. When the TE hot side heats the block, the liquid coolant absorbs the heat as it flows through all the channels, which will be dissipated through a radiator. The same system can be applied at the cold side for the transfer of the cool due to

Recently, heat transfer in mini channels within heat exchangers is drawing substantial attention trying to improve their performance. The proper selection of channel dimensions and nonuniform distribution of the channels can improve the cooling power [107]. Therefore, thermal and hydrodynamic characteristics of channels need to be examined and developed. A TE system using liquid cooling for electronic application using micro-channel heat sink was proposed and its experimental analysis performance was investigated [108]. The effect of channel width,

high thermal resistance between the cold side of the TE and the space being cooled.

coolant flow rate and heat sink material on the heat transfer rate was also examined [76].

experimental results confirmed the superiority of this cooling technique [111, 112].

the coolant and carried away by the circulation.

toward the channel exits. These can be expressed as:

here h is the convective heat transfer coefficient:

the surface area will be:

Although micro-channel heat exchangers are able to dissipate higher heat flux densities, the slow flow rate creates a large increase in the temperature alongside the direction of the coolant flow in both channel material and the coolant. Surface roughness also participates in the heat transfer characteristics and the drop of pressure of coolant flow in a channel. Many studies clearly reported that the roughness has an effect on the flow of the coolant and heat transfer characteristics, in addition to the laminar and turbulent transition [109, 110]. Micro channel heat exchangers with different designs and coolants were manufactured and tested and the

Heat removal through parallel channels involves a complex combination of convection, conduction and coolant flow. In a rectangular channel plate with width W, height H and length L, taking the advantage of the symmetry of the channels, a unit cell containing only one channel with the surrounding metal is chosen. The results obtained can easily be applied to the whole plate. Heat transport in the unit cell is a conjugate problem that mixes heat conduction in the metal and convective heat transfer to the coolant. The dissipated heat in the surrounding regions conducts to the channel side walls, which is then absorbed, through convection, by

These parameters can be summarized by stating them as thermal resistances. Conductive resistance, Rcond, is determined by thermal characteristics of aluminum that conducts the dissipated heat in the region surrounding the sidewalls of the channel. Convection resistance, Rconv, is a result of the convection from sidewalls of the channel to the coolant. Heat resistance, Rheat, is a result of heating up of coolant in the downstream direction as the flow is pushed

Rconv <sup>¼</sup> <sup>1</sup>

where A is the channel surface area. Assuming that heat is transmitted from all the sidewalls,

h A (13)

Thermoelectric Cooling

257

http://dx.doi.org/10.5772/intechopen.75791

A ¼ 2 L Wð Þ þ H (14)

Heat sink performance at the hot side is more important than heat sink at the cold side because the heat flux density at hot side is higher. Allocation of the heat transfer area or heat transfer coefficients between hot and cold sides is particularly important. For given hot and cold side fluid temperatures, there exists an optimum cooling capacity which leads to maximum COP [64, 80, 92].

The COP of TE devices could be improved by minimizing the difference in temperature between their hot and cold faces [100]. The hot side of the TE cooler exhibits very high power densities that demands sophisticated cooling infrastructure with high pumping power.
