Question

For fixed maximum and minimum temperatures, what is the effect of the pressure ratio on \((a)\) the thermal efficiency and ( \(b\) ) the net work output of a simple ideal Brayton cycle?

Short answer

Answer: In a simple ideal Brayton cycle, increasing the pressure ratio generally leads to increased thermal efficiency. The net work output, on the other hand, has an optimal value at a certain pressure ratio. The specific effects will depend on other input parameters, such as the specific heat ratio and the maximum and minimum temperatures.

Step-by-step solution

Define the given variables

Let's define the fixed maximum and minimum temperatures as \(T_{max}\) and \(T_{min}\), and the pressure ratio as \(rp\).

Calculate the temperature after isentropic compression

We know that for an isentropic process, \(T_2/T_1 = (P_2/P_1)^{(\gamma-1)/\gamma}\), where \(\gamma\) is the specific heat ratio, and \(T_1\) and \(P_1\) are the initial temperature and pressure. Since the pressure ratio \(rp = P_2/P_1\), we get \(T_2 = T_1 * (rp)^{(\gamma-1)/\gamma}\).

Calculate the temperature after isentropic expansion

Similarly, we can find the temperature after isentropic expansion using the relation \(T_4 = T_3*(1/rp)^{(\gamma-1)/\gamma}\), where \(T_3\) is the maximum temperature (\(T_{max}\)).

Calculate the work done during compression and expansion

The work done during the compression process, \(W_{comp}\), can be calculated using \(W_{comp} = C_p * (T_2 - T_1)\), where \(C_p\) is the specific heat at constant pressure. The work done during the expansion process, \(W_{exp}\), can be calculated using \(W_{exp} = C_p * (T_3 - T_4)\).

Calculate the net work output

The net work output, \(W_{net}\), is the difference between the work done during expansion and compression: \(W_{net} = W_{exp} - W_{comp}\).

Calculate the heat added during the cycle

The heat added during the constant-pressure heat addition process, \(Q_{in}\), can be calculated using \(Q_{in} = C_p * (T_3 - T_2)\).

Calculate the thermal efficiency

The thermal efficiency, \(\eta\), can be calculated using the formula \(\eta = \frac{W_{net}}{Q_{in}}\).

Examine the effect of pressure ratio

Now that we have the expressions for net work output and thermal efficiency in terms of pressure ratio, we can analyze how these parameters change as the pressure ratio changes. In general, the thermal efficiency increases with increasing pressure ratio, while the net work output has an optimal value at a certain pressure ratio. However, the specific effects will depend on the other input parameters, such as the specific heat ratio and the maximum and minimum temperatures. In conclusion, increasing the pressure ratio in a simple ideal Brayton cycle generally leads to increased thermal efficiency and may lead to an optimal net work output at a certain pressure ratio.