The relative cooling of performance of phi/phi(0) for a thermoelectric (TE) element was derived analytically by taking the linear and non-linear components in the temperature (T(z)) dependences of TE properties into the thermal rate equations (TRE) on the assumption that all of the TE properties are expressed by a quadratic function of T(z) at a position z along a TE element and the temperature profile along a TE element is linear or non-linear, where phi and phi(0) are the coefficients of performance (COP) derived from the new and conventional TRE, respectively. The linear and non-linear components in the T(z)-dependences of TE properties were estimated experimentally for Bi-Te alloys. When a TE element has a linear temperature profile, phi/phi(0) estimated using the linear and/or non-linear components in the T(z)-dependences of their TE properties increases with an increase of Delta T and reached great values of 1.3-1.6 at ZT = 1 under the condition of T = 300 K and Delta T = 80 k. As a result, it was found that the linear component in the electrical resistivity rho and the non-linear one in the Seebeck coefficient alpha have a significant effect on phi/phi(0). When phi/phi(0) was estimated for a non-linear temperature profile of a module fabricated using these Bi-Te alloys, however, it was slightly lower than that obtained for a linear temperature profile. The formulas obtained for phi and phi/phi(0) are applicable even for the practical TE coolers and refrigerators with a strong non-linearity in the temperature profile. (C) 2008 Elsevier Ltd. All rights reserved.