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The compressor pressure ratio is a key parameter in the Brayton cycle. It represents the ratio of the pressure after compression to the pressure before compression. In mathematical terms, it is denoted as \text{PR} and defined as \text{PR}=\frac{P_2}{P_1}\. This ratio is crucial because it determines the temperature rise in the compressor according to the relation: \ T_2 = T_1 \times \text{PR}^{(\frac{k-1}{k})}\, where \(k=1.4\). The higher the pressure ratio, the higher the temperature rise, and consequently, the higher the efficiency of the cycle. Given the compressor inlet temperature (\(T_1=300\) K) and pressure ratio (\(\text{PR}=10\)), the compressor exit temperature is determined using the formula above, resulting in \(T_2=791.7\) K. This high exit temperature forms the basis for subsequent calculations in the Brayton cycle.

The turbine inlet temperature (TIT) is the temperature at which air enters the turbine. It is a critical parameter for the Brayton cycle as it directly affects the power output of the turbine and the overall efficiency of the cycle. For our exercise, the turbine inlet temperature is given as \(T_3=1400\, K\). To find the turbine exit temperature (\(T_4\)), we use the pressure ratio in reverse and the formula: \ T_4 = T_3 \times \frac{1}{\text{PR}^{(\frac{k-1}{k})}}\. This gives us an exit temperature of \(T_4\approx 530.6\, K\). This temperature is lower than the inlet temperature, showing that the turbine produces work by expanding high-temperature air to a lower temperature.

The thermal efficiency of the Brayton cycle indicates how well the cycle converts heat into work. It is calculated using the formula: \ \text{Efficiency} (\eta) = 1 - \frac{T_1}{T_3}\, where \(T_1\) is the compressor inlet temperature and \(T_3\) is the turbine inlet temperature. From the given values (\(T_1=300\, K\) and \(T_3=1400\, K\)), the efficiency is: \ \eta = 1 - \frac{300}{1400} = 1 - 0.214 \approx 0.786\ or \ 78.6\text{%}\. This relatively high efficiency highlights the effectiveness of the Brayton cycle under the given conditions.

The back work ratio (BWR) is the ratio of the work consumed by the compressor to the work produced by the turbine. It is a measure of how much of the turbine's work is used to drive the compressor. The BWR is given by: \ \text{BWR} = \frac{W_c}{W_t}\, where \(W_c\) is the compressor work and \(W_t\) is the turbine work. In the exercise, we calculate these as follows: \(W_c = C_p (T_2 - T_1)\ and\ W_t = C_p (T_3 - T_4)\). Given \(C_p=1.005\, kJ/kg \cdot K\), we find: \(W_c \approx 493.96\, kJ/kg\) and \(W_t \approx 873.7\, kJ/kg\). Thus, the BWR is \( \text{BWR} \approx 0.565\). This ratio tells us that about 56.5% of the turbine work is used to drive the compressor.

The net power developed (\text{P}_net) is the useful power output of the Brayton cycle after accounting for the power used by the compressor. It is calculated by the formula: \ \text{P}_{net} = \text{mass flow rate} \times (W_t - W_c)\. Given the mass flow rate of \(6\, kg/s\), \(W_t=873.7\, kJ/kg\), and \(W_c=493.96\, kJ/kg\), the net power developed is: \ \text{P}_{net} = 6 \times (873.7 - 493.96) \approx 2278.44\, kW\. This is the effective power that can be used for doing useful work, such as driving generators or propulsion systems.