Question

Consider a simple ideal Brayton cycle with air as the working fluid. The pressure ratio of the cycle is \(6,\) and the minimum and maximum temperatures are 300 and \(1300 \mathrm{K}\) respectively. Now the pressure ratio is doubled without changing the minimum and maximum temperatures in the cycle. Determine the change in \((a)\) the net work output per unit mass and ( \(b\) ) the thermal efficiency of the cycle as a result of this modification. Assume variable specific heats for air.

Short answer

Answer: The change in net work output per unit mass and thermal efficiency for the cycle can be calculated using the following expressions: - Change in net work output per unit mass: \(\Delta W_{cycle} = c_p\left((T_2' - T_1 - T_4' + T_3) - (T_2 - T_1 - T_4 + T_3)\right)\) - Change in thermal efficiency: \(\Delta \eta_{th} = \eta_{th}' - \eta_{th} = 1- \frac{T_4'-T_3}{T_2'-T_1} - \left(1-\frac{T_4-T_3}{T_2-T_1}\right)\) To find the actual values for the change, one would need the specific heat ratios, \(\gamma_t\) and \(\gamma_c\), the isentropic efficiencies, \(\eta_t\) and \(\eta_c\), and the specific heat at constant pressure, \(c_p\).

Step-by-step solution

- Relevant equations and properties

Before starting, let's list out the relevant equations. For air with variable specific heats: 1. Temperature ratio: \(T_2=T_1\left(1+(\gamma_c-1) \frac{(P_2/P_1)^{(\gamma_c-1)/\gamma_c}-1}{\eta_c}\right)\) 2. Net work output (isentropic): \(W_{cycle} = W_c - W_t = c_p \left(T_2 - T_1 - T_4 + T_3\right)\) 3. Thermal efficiency: \(\eta_{th} = \frac{W_{cycle}}{Q_{in}} =1-\frac{T_4-T_3}{T_2-T_1}\) 4. Pressure ratio: \(\frac{P_2}{P_1}=\frac{P_3}{P_4}=r_p\) Here, \(\gamma_c\) is the specific heat ratio, \(\eta_c\) is the isentropic efficiency of the compressor, \(W_c\) and \(W_t\) are the work done by the compressor and turbine respectively, and \(c_p\) is the specific heat at constant pressure.

- Initial cycle calculation

For the initial cycle, we are given the pressure ratio \(r_p = 6\) and the minimum and maximum temperatures, \(T_1 = 300K\) and \(T_3 =1300K\). Let us now find the other temperatures and the net work output per unit mass and the thermal efficiency. Temperature at state 2 (\(T_2\)): $$T_2=T_1\left(1+(\gamma_c-1) \frac{(P_2/P_1)^{(\gamma_c-1)/\gamma_c}-1}{\eta_c}\right)=300\left(1+(\gamma_c-1) \frac{(6)^{(\gamma_c-1)/\gamma_c}-1}{\eta_c}\right)$$ Temperature at state 4 (\(T_4\)): $$T_4=T_3-\left(T_2-T_1\right)\frac{\eta_t}{\gamma_t-1}=1300- \left(T_2-300\right)\frac{\eta_t}{\gamma_t-1}$$ Net work output per unit mass (\(W_{cycle}\)): $$W_{cycle} = c_p\left(T_2 - T_1 - T_4 + T_3\right)$$ Thermal efficiency (\(\eta_{th}\)): $$\eta_{th} =1-\frac{T_4-T_3}{T_2-T_1}$$

- Modified cycle calculation

For the modified cycle, the pressure ratio is doubled, i.e., \(r_p = 12\). We can calculate the other temperatures and the net work output per unit mass and the thermal efficiency using the same equations as in the initial cycle. Temperature at state 2 (\(T_2'\)): $$T_2'=T_1\left(1+(\gamma_c-1) \frac{(P_2'/P_1)^{(\gamma_c-1)/\gamma_c}-1}{\eta_c}\right)=300\left(1+(\gamma_c-1) \frac{(12)^{(\gamma_c-1)/\gamma_c}-1}{\eta_c}\right)$$ Temperature at state 4 (\(T_4'\)): $$T_4'=T_3-\left(T_2'-T_1\right)\frac{\eta_t}{\gamma_t-1}=1300- \left(T_2'-300\right)\frac{\eta_t}{\gamma_t-1}$$ Net work output per unit mass (\(W_{cycle}'\)): $$W_{cycle'} = c_p\left(T_2' - T_1 - T_4' + T_3\right)$$ Thermal efficiency (\(\eta_{th}'\)): $$\eta_{th}' =1-\frac{T_4'-T_3}{T_2'-T_1}$$

- Determine the change in net work output per unit mass and thermal efficiency

Finally, let's find the change in net work output per unit mass and thermal efficiency as a result of doubling the pressure ratio. Change in net work output per unit mass (\(\Delta W_{cycle}\)): $$\Delta W_{cycle} = W_{cycle}' - W_{cycle} = c_p\left((T_2' - T_1 - T_4' + T_3) - (T_2 - T_1 - T_4 + T_3)\right)$$ Change in thermal efficiency (\(\Delta \eta_{th}\)): $$\Delta \eta_{th} = \eta_{th}' - \eta_{th} = 1- \frac{T_4'-T_3}{T_2'-T_1} - \left(1-\frac{T_4-T_3}{T_2-T_1}\right)$$ Thus, we have analytically determined the change in net work output per unit mass and thermal efficiency of the cycle when the pressure ratio is doubled. To find the actual values for the change, one would need the specific heat ratios, \(\gamma_t\), and \(\gamma_c\), the isentropic efficiencies, \(\eta_t\) and \(\eta_c\), and the specific heat at constant pressure, \(c_p\).