A CONTRIBUTION TO THE STUDY OF THERMOELECTRICAL COALING

Academic Open Internet Journal

ISSN 1311-4360

www.acadjournal.com

Volume 20, 2007

A CONTRIBUTION TO THE STUDY OF THERMOELECTRICAL COOLING

Beguide Bonoma¹, Marc Tchetchoua¹, Sylvain Tchatchueng¹

¹ Applied physics laboratory of the National Higher Teacher’s Training College, BP 47 Yaounde, University of Yaounde I.

Abstract

In this work the expression of the performance coefficient of Peltier’s thermo electrical refrigeration is established. The examination of its optimisation is conducted according to its different parameters.

Keywords: Thermo electrical refrigeration; performance coefficient; Peltier effect.

1. Introduction

Thermoelectric cooling is based on two effects: the Seebeck and the Peltier. The former has its application in thermo batteries, while the latter has its own in refrigerators and heat pumps. An understanding of both effects is based on thermodynamics and the physics of solid.

The Peltier effect has a very wide range of application, amongst which can be cited heat pumps and thermoelectric coolers. The latter offers an invaluable advantage, in comparison with conventional refrigerators. First of all, they do not use gas which is a source of air pollution, and secondly they work with low intensity energy.

2. Presentation of models [1], [2], [3]

2.1. Seebeck effect

Let us consider the circuit presented in figure 1.

Figure 1: Physical model of Seebeck effect.

The circuit is made up of two semi-conductors, one of the type P, and the other of the type N, connected in series by a resistance R. one connection assures the contact of the two semi-conductors. A flow of the current I is observed in the circuit when the points of the conductors are subjected to a gradient of temperature. This is the Seebeck effect.

The heat , necessary to create the gradient of temperature in part is converted into electric energy with an output:

(1)

rd is the output of the conversion of thermal energy.

The thermoelectric power of the circuit is defined by:

(2)

T_h is the temperature of the connection, while a_pn is the thermoelectric power.

2.2. The Peltier effect

By replacing the resistance R in figure 1 with a source of continuous tension that supplies the current I (figure 2), what is noticed is a difference in temperature DT₀ between the points of the semi-conductors P and N which were initially at the same temperature: this is the Peltier effect.

Figure 2 : The physical model of the Peltier effect.

The temperature difference (DT₀ = T_h - T_c) at the terminals of the semi-conductors P and N induce the flow of an amount of heat in the circuit per unit of time defined by:

= - P I (3)

P represents the coefficient of Peltier linked to thermoelectric power by the relation:

P = a_pn T (4)

In the following, we will be considering the Peltier effect, which is the model of thermoelectric cooling.

3. Computation of the performance coefficient of Peltier’s ideal module.

3.1. Coefficient of performance [4], [5].

By examining the thermal power obtained from a cold source and the electric power P supplied by a source of continuous tension, the coefficient of performance (COP) of the thermocouple can be defined by:

(5)

j is the coefficient of performance.

The flow of the electric current I in the circuit produces three types of heat which are:

- The Peltier heat

(6)

- The Joule heat

(7)

r_n and r_p (W.m) are respectively the electric resistivities of the semi-conductors N and P, S_n and S_p their sections while ℓ is the length of the semi-conductors.

- The conduction heat.

(8)

This expression ensues from the law of Fourier. K_n and K_p are respectively the electric conductivities of the semi-conductors of the type N and P.

The thermal power is the sum of these different values from which:

(9)

The electric power by definition is:

P =U I (10)

where U is the continuous tension. This tension is divided into two parts. One is linked to the law of Ohm:

(11)

and the other to heat conversion into electric energy:

(12)

The expression P is deduced by:

(13)

From the relations (9) and (13) comes the coefficient of performance j:

(14)

Where R₀ is the expression:

(15)

The coefficient j is determined by the current I and the dimensions of the semi-conductors. For it to be optimised, the conditions to be fulfilled to get j at its maximum are:

(15a)

(l5b)

(15c)

Equation 15a is of a second degree in I and takes for a solution, the expression:

(16)

with

(17)

Z_pn is the merit factor defined by:

(18)

R and l are respectively the electric resistance and thermic conductance of the semi-conductors.

T_M is the arithmetic average of the temperatures T_c and T_h ; that is to say:

(19)

Considering the equations (16), (17), (18) and (19), the ideal maximum coefficient of performance of Peltier is expressed by:

(20)

3.2. Influence of electric contact resistance.

The equation (20) is got by disregarding the effects of contact between the semi-conductors. To study the influence of the electric resistance of contact, the resistance has to be taken account of in the total resistance thus:

R = R₀ + R_c (21)

with

(22)

r_c is the electric resistivity of contact, R_c the contact resistance per unit of length and the factor 4 is owning to the presence of four points of contact. It stands out for r_n and r_p that:

(23)

In putting down

(24)

The new merit factor Z' is connected to Z_pn by the relation:

(25)

Thus the coefficient of performance h₁ when the influences of contact resistance are considered by

(26)

3.3. Influence of thermic contact resistance

To consider this factor, two ceramic slabs which can conduct heat are attached to the points of the semi-conductors (figure 3).

Figure 3 : A practical thermocouple diagram.

The temperature gradients will appear in the two slabs, thus influencing the coefficient of performance and the power of the Peltier module:

(27)

Where DT_max is got from solving the equation h₀ = 0 of the relation (20).

What obtains is:

(28)

Using r to designate the ratio:

(29)

Where K stands for the thermic conductivity of the semi-conductors and K_c the thermic conductivity of the ceramic slabs.

h₂ can also be expressed as:

(30)

with ℓ_c as the thickness of the ceramic slabs and ℓ as the length of the semi-conductors.

The simultaneous effects of the electric and thermic contact resistance can be taken into account in one expression just by replacing h₀ in (30) by the expression of h₁ in (20). This gives:

(31)

This relation shows that the coefficient of performance h is conditioned by the length ℓ of the semi-conductors, the thickness ℓ_c of the ceramic slabs, the relation n of electric contact resistivities and the semi-conductors, and the ratio r of the thermic conductivity of semi-conductors to that of contact. The amelioration of the COP will therefore require a mastery of these four parameters.

As it concerns the power of the thermocouple related to the surface unit of the semi-conductors, it is shown that it is connected to the COP by the relation:

(32)

4. Results of the influences of some parameters on the coefficient of performance and the power of the thermocouple [5].

To arrive at this study, it is necessary to set at least two of the four parameters cited previously. From this, the variations of COP with the length for the different values of the parameter n are represented in figure 4. In setting ℓ_c = 0.7 mm and in giving the parameter r the value 0.2, it can be deduced from the curves of this figure that the COP is at its best when the contact electrical effects are minimised during the production of thermoelectric refrigeration.

Figure 4 : The influence of the length of thermo elements on the performance coefficient COP.

The effect of contact thermic resistance is also represented by setting the parameter n and the thickness ℓ_c of the ceramic slabs (figure 5). The curves show the variations of COP in accordance with the length of semi-conductors for different values of the ratio r. It is observed that the COP rises when the ratio r falls. Since thermic contact is unavoidable, the optimum COP for a given length of semi-conductor can be got by reducing the ratio r; this is by choosing a ceramic slab of very considerable thermic contact conductivity.

Figure 5 : The reaction of COP with thermic resistance and the length of thermo elements.

When the ratios n and r are determined, the variations of COP with the length ℓ of semi-conductors and the thickness ℓ_c of the ceramic slabs can be observed. The optimal COP is obtained here by using a very light ceramic slab (figure 6).

Figure 6 : The variation of COP with the thickness of the ceramic slab and the length of thermo elements.

The study of variations of power per surface unit according to length accords to certain values of ℓ tending toward zero, negative values. This has no physical meaning. What is considered here are the limits zero and infinity. The curves of the practical modules of figure 7 each present a maximum, as opposed to the curve of the ideal module which decreases constantly. This result confirms the theorem of Rolle which states the existence of a maximum power. The difference between the ideal module and the practical one is more highlighted when the length of the semi-conductors is not considerable.

Figure 7 : The variation of power with the length of thermoelements.

5. Conclusion

This study reveals that the coefficient of performance and the power of the thermocouple depend on the length ℓ of the semi-conductors. The practical module tends toward the ideal one when this length becomes too long. For economic reasons, the optimisation of COP shall not be based on this parameter. Elsewhere, the unavoidable effects of electric and thermic contacts being responsible for a reduction in COP, another way out lies in mastery of the n, r and ℓ_c parameters which control these contacts, as their reduction results in a considerable amelioration of the coefficient of performance.

References

[1] Zely Didier; Monchoux Françoise. Modelling of thermoelectric’s modules. Application to analysis of space scientific experiment. Thèse de doctorat, Université de Toulouse 3 (1992) 130 p.

[2] H. J. Goldsmid. Timeless in the development of thermoelectric cooling. Seventeenth International conference on thermoelectric’s proceedings, ICT 98 (1998) pp. 25-28.

[3] Goktun S. Design considerations for a thermoelectric refrigerator, energy conversion and management, (1995) vol. 36 n°12, pp.1197-1200.

[4] Fujishiro H.; Kusaka K.; Ikebe M.; Ogasawara H. Thermoelectric refrigerator, Proceedings of the international cryogenic engineering conference, (1994) vol. 34, pp.231-234.

[5] Chen Jinca; Yan Zijun. Optimal performance of thermoelectric refrigerator, AIP conference proceedings, (1994) vol. 316 n°1; pp. 199-202.

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