(Solved): SECTION B B1. (a) The maximum thermal efficiency of a thermoelectric generator with both geometry a ...

SECTION B B1. (a) The maximum thermal efficiency of a thermoelectric generator with both geometry and load resistance optimized can be expressed as $${?}_{T\max }=\frac{\left(m_{\text{opt }}?1\right)\left(?{}^2/T_H\right)}{m_{opt}+{}^{T}C/T_H}$$ where $$m_{opt}={\left(1+Z^{?}{T}_{av}\right)}^{1/2}$$ is the optimum value of the ratio of the load to internal (ie $$p$$ plus $$n$$ legs) resistances. $$Z^{?}$$ is the maximum value of the figure of merit $$Z$$. Write down an equation for the figure of merit, $$Z$$, in terms of the electrical resistance $$\mathrm{R}$$ and thermal conductance, $$\mathrm{K}$$, of the generator, giving equations for $$\mathrm{R}$$ and $$\mathrm{K}$$ in terms of the material properties and geometry of the $$n$$ and $$p$$ legs of the generator. Clearly define each of the parameters in the equations. [10 marks] (b) Define the figure of merit for a single material, clearly defining all the parameters and discuss how this figure of merit can be maximized, showing that the best type of materials are semiconductors. [15 marks] (c) A current of 3MA flows in a tokamak plasma which has $$\mathrm{n}=2\times 10^{20}{\mathrm{m}}^{?3}$$. Estimate the temperature of the plasma. [8 marks]