Write a closed-form expression for the work output of a solar-powered heat engine if the energy delivery of the solar collector at high temperature is given approximately by the expression

\[ q_{u}=(\tau \alpha)_{e f f} I_{c}-\frac{\sigma \varepsilon \bar{T}_{f}^{4}}{C R} \]

(convection and conduction losses are neglected) where \(\bar{T}_{f}\) is the average fluid temperature and \(C R\) is the \(C R\). Two heat engines are to be evaluated:

1. Carnot cycle

\[ \text { Cycle efficiency } \eta_{c}=1-\frac{T_{\infty}}{T_{f}} \]

2. Brayton cycle

\[ \text { Cycle efficiency } \eta_{B}=1-\frac{\mathrm{C}_{B} T_{\infty}}{T_{f}} C_{B} \geq 1 \]

where

\[ \begin{aligned} & C_{B}=\left(r_{p}\right)^{(k-1) / k} \\ & r_{p} \text { is the compressor pressure ratio } \\ & k \text { is the specific heat ratio of the working fluid } \end{aligned} \]

Write an equation with \(\bar{T}_{f}\) as the independent variable that, when solved, will specify the value of \(\bar{T}_{f}\) to be used for maximum work output as a function of \(C R\), surface emittance and \((\tau \alpha)_{\text {eff }}\) product, insolation level, and \(C_{B}\). Optional: solve the equation derived previously for \(C R=100, \varepsilon=0.5\), and \((\tau \alpha)_{\text {eff }}=0.70\), at an insolation level of \(1 \mathrm{~kW} / \mathrm{m}^{2}\). What is the efficiency of a solar-powered Carnot cycle and a Brayton cycle for which \(C_{B}=2\) ?