Gerstenmaier York Christian, Wachutka Gerhard
Siemens AG, Corporate Technology, D-81739 Munich, Germany.
Phys Rev E Stat Nonlin Soft Matter Phys. 2012 Nov;86(5 Pt 2):056703. doi: 10.1103/PhysRevE.86.056703. Epub 2012 Nov 5.
A novel generalized Lagrange multiplier method for functional optimization with inclusion of subsidiary conditions is presented and applied to the optimization of material distributions in thermoelectric converters. Multistaged devices are considered within the same formalism by inclusion of position-dependent electric current in the legs leading to a modified thermoelectric equation. Previous analytical solutions for maximized efficiencies for generators and coolers obtained by Sherman [J. Appl. Phys. 31, 1 (1960)], Snyder [Phys. Rev. B 86, 045202 (2012)], and Seifert et al. [Phys. Status Solidi A 207, 760 (2010)] by a method of local optimization of reduced efficiencies are recovered by independent proof. The outstanding maximization problems for generated electric power and cooling power can be solved swiftly numerically by solution of a differential equation-system obtained within the new formalism. As far as suitable materials are available, the inhomogeneous TE converters can have increased performance by use of purely temperature-dependent material properties in the thermoelectric legs or by use of purely spatial variation of material properties or by a combination of both. It turns out that the optimization domain is larger for the second kind of device which can, thus, outperform the first kind of device.
提出了一种用于含附加条件的泛函优化的新型广义拉格朗日乘数法，并将其应用于热电转换器中材料分布的优化。通过在支腿中引入位置相关的电流，在相同形式体系内考虑多级器件，从而得到修正的热电方程。通过独立证明，恢复了先前由谢尔曼[《应用物理杂志》31, 1 (1960)]、斯奈德[《物理评论B》86, 045202 (2012)]以及塞弗特等人[《固体物理状态A》207, 760 (2010)]通过局部优化约化效率方法获得的发电机和冷却器最大效率的解析解。通过求解新形式体系内得到的微分方程组，可以快速数值求解发电功率和制冷功率的最大化问题。只要有合适的材料，非均匀热电转换器可以通过在热电支腿中使用仅与温度相关的材料特性、或使用仅材料特性的空间变化、或两者结合来提高性能。结果表明，第二种器件的优化域更大，因此其性能可能优于第一种器件。
