Entropy Generation Minimization of a Thermoelectric Cooler

In the present work, we develop a theoretical analysis to minimize the entropy generation of a thermo-electric cooler (TEC), which can be used to cool a processor. We have applied an analysis of first and second law of the Thermodynamics to a TEC. We consider the entropy generation equation as the objective function and the first law as a restriction, in accordance with the variational calculus theory. The numerical predictions show that minimum entropy generation and the better performance coefficients (COPs) are achieved with the largest figure of merit, which represents a relationship between the Peltier effect (cooling) and the Joule effect (heating). INTRODUCTION Thermoelectric cooling allows changing directly electricity into refrigeration. In 1834 Peltier noticed, that when he applied electric current to a circuit, which was made with the junction of two distinct metals, one of the junctions was heated and the other was cooled. Thermoelectric cooling is one application of thermoelectricity; it is the set of phenomena related with processing electricity into heat and vice versa. When an electric current circulates through one or more semiconductors pairs type n and p, it produces a temperature gradient between the junctions. It is important to note here that a p-type semiconductor represents a type of semiconductor in which current passes through a solid via electron flow into positive 'holes' in a crystal. The holes are introduced using electron-deficient impurity atoms (doped crystal). Meanwhile an n-type semiconductor is obtained by carrying out a process of doping, that is, by adding an impurity of valence-five elements to a valence-four semiconductor in order to increase the number of free (in this case negative) charge carriers. Energy crosses easily from the low temperature reservoir to the high temperature reservoir through the semiconductor type n, meanwhile energy flows hardly through material p from high temperature to low temperature reservoir [1]. A typical device showing the above operation is represented in Fig. 1. Thermoelectric Studies Nowadays, it is very well-known that an increment in the capacity and reduction size of electronic equipment produces higher heat transfer rates than before [2, 3]. Phelan and others [4] made a review study about different and nonconventional refrigeration systems; concluding that only TECs can meet the requirements imposed by the electronic miniaturization and to be commercially available. Omer and others [5] proposed an experimental design, in which a phase change of a material was used in combination *Address correspondence to this author at the Departamento de Termofluidos, Facultad de Ingeniería, Universidad Nacional Autónoma de México, 04510, México D. F., MEXICO; Tel: 52-55-56228103; Fax: 52-5556228106; E-mail: [email protected] with one thermosyphon in order to that the system achieves a refrigeration capacity between 150 and 200 W. Cheng and Shih [6] optimized a set of TECs coupled in cascade using genetic algorithms. Gordon and others [7] coupled a thermoelectric refrigerator with an absorption system; they achieved higher CPOs than those attained by the systems individually. Yang and others [8] reviewed the performance of TECs employing electric currents varying with time and with values higher to the recommended. Vikhor and Anatychuk [9] used segmented semiconductors n and p elements, and they gotten a better performance of TEC, because they changed discreetly TEC properties. Cheng and Lin [10] optimized a TEC geometrically and took the volume used by TEC as a restriction, solving also the problem with genetic algorithms. They got the best cooling capacities when they reduced the characteristic length of semiconductors materials. Fig. (1). Outline of a thermoelectric cooler (TEC) device. In the present work, we have developed a theoretical analysis based on minimization of the entropy generation to ! " # !