Modeling of Segmented Peltier Cooling with Discrete and Continuous Concentration Function
Authors: Walczak S. , Seifert Winfried , Müller Eckhard 
Published: March 2009

Commercialization of Peltier coolers has progressed during last years and special efforts have been undertaken to enhance the efficiency of thermoelectric (TE) devices. Along with the continued search for advanced TE materials, the concept of FGM offers a strategy of gradual improvement of device performance. In reality a functional gradient in a TE material means a related spatial variation of all TE properties – Seebeck coefficient, electrical, and thermal conductivity – whereas the most relevant effect is linked to the gradient of the Seebeck coefficient. Due to the spatial dependence of the Seebeck coefficient, Peltier heat is absorbed or released inside the TE element under current flow (distributed Peltier effect) which can be exploited to shape the internal temperature profile in a desired manner. Starting from the first principles of thermoelectricity, a differential equation governing the coupling of thermal and electrical transport is derived within the frame of a one-dimensional model. It is shown that this approach can be also used to model multi-segment Peltier cooling devices. Temperature profiles T(x) have been calculated for a segmented TE element within the framework of a constant parameters theory. The work presents an analytical model for performance evaluation of multiply-segmented Peltier elements. The problem is treated in a one-dimensional approach for a p-type stack containing N segments of different properties. Assuming constant TE material properties in each of the segments, the differential equation of TE transports has been solved to obtain the temperature profile T(x) in each segment. With the material properties values in each segment representing volume average values this model gives an excellent approximation also for continuously graded elements. The boundary conditions of the TE problem set-up, as conservation of heat at any intermediate junction between the segments, and fixed temperature at the cold and hot end of the element, lead to a linear equation system, which can be easily solved by means of standard methods. From the solution, all desired performance parameters can be deduced. Based on realistic material data exemplary calculations are presented for stacked and continuously graded elements. To demonstrate the developed numerical algorithm, gradients of the Seebeck coefficient are mainly considered. Calculations have been performed for N = 2, 5, 10, and continuous gradients. As target parameters, the C.O.P. and the cooling power have been calculated as functions of the electric current. As well, the minimum temperature of the cold side has been determined for various shape of the Seebeck gradient. It is shown that the TE FGM effect can be almost completely utilized already by a stack of two to five homogeneous segments. The results allow for giving an estimation on the order of magnitude of performance improvement of both discontinuously and continuously graded Peltier cooling devices. The model calculation was implemented with the software tool MATHEMATICA. The code provides an easy to handle convenient instrument for performance estimation of non-homogeneous Peltier pellets. Technological studies for controlled fabrication of those pellets are underway.