Question

A gas-turbine power plant operates on the simple Brayton cycle between the pressure limits of 100 and 800 kPa. Air enters the compressor at \(30^{\circ} \mathrm{C}\) and leaves at \(330^{\circ} \mathrm{C}\) at a mass flow rate of \(200 \mathrm{kg} / \mathrm{s}\). The maximum cycle temperature is \(1400 \mathrm{K}\). During operation of the cycle, the net power output is measured experimentally to be 60 MW. Assume constant properties for air at \(300 \mathrm{K}\) with \(c_{\mathrm{v}}=0.718 \mathrm{kJ} / \mathrm{kg} \cdot \mathrm{K}, c_{p}=\) \(1.005 \mathrm{kJ} / \mathrm{kg} \cdot \mathrm{K}, R=0.287 \mathrm{kJ} / \mathrm{kg} \cdot \mathrm{K}, k=1.4\) (a) Sketch the \(T\) -s diagram for the cycle. (b) Determine the isentropic efficiency of the turbine for these operating conditions. (c) Determine the cycle thermal efficiency.

Short answer

**Question**: Sketch the T-s diagram for a given Brayton cycle and determine the isentropic efficiency of the turbine and the cycle thermal efficiency, given the following values: \(T_1 = 303.15 K\), \(T_2 = 603.15 K\), \(T_3 = 1400 K\), \(P_1 = 100 kPa\), \(P_2 = 800 kPa\), and \(m\dot{ = 200 kg/s}\). **Answer**: 1. Sketch the T-s diagram, showing vertical lines connecting Points 1 and 2 (compression process) and Points 3 and 4 (expansion process), and horizontal lines connecting Points 2 and 3 (constant pressure heat addition) and Points 4 and 1 (constant pressure heat rejection). Label the points with the given temperature and pressure values. 2. Isentropic efficiency of the turbine: Follow Steps 4-6 in the solution to find the actual and isentropic power outputs of the turbine and then calculate the isentropic efficiency: \(\eta_{\text{isentropic}} = \frac{\dot{W}_{\text{turbine, actual}}}{\dot{W}_{\text{turbine, isentropic}}}\). 3. Cycle thermal efficiency: Use the values obtained from Step 7 in the solution to calculate the cycle thermal efficiency: \(\eta_{\text{thermal}} = \frac{h_3 - h_4}{h_3 - h_2}\).

Step-by-step solution

Points in the cycle

In a simple Brayton cycle, there are four main points: 1. Inlet of the compressor 2. Outlet of the compressor 3. Inlet of the turbine 4. Outlet of the turbine **Step 2: Find the temperature and pressure at each point**

Temperature and pressure at each point

We are given some of the values: - \(T_1 = 30^\circ C = 303.15 K\) - \(T_2 = 330^\circ C = 603.15 K\) - \(P_1 = 100 kPa\) - \(P_2 = 800 kPa\) - \(T_3 = 1400 K\) Using the given pressure ratio, we can find the pressure at points 3 and 4: \(P_4 = P_3 = P_2 = 800 kPa\) **Step 3: Find the enthalpy at points 2 and 4**

Enthalpy at Points 2 and 4

From property for constant specific heat: \(h_2 - h_1 = c_p(T_2 - T_1) \Rightarrow h_2 = c_p(T_2 - T_1) + h_1\) \(h_4 - h_3 = c_p(T_4 - T_3) \Rightarrow h_4 = c_p(T_4 - T_3) + h_3\) In order to find \(h_1\) and \(h_3\), we need to know the specific volume at points 1 and 3. For this, we use the ideal gas equation: \(v_1 = \frac{RT_1}{P_1}, v_3 = \frac{RT_3}{P_2}\) Now, we can use the specific volume and temperature at points 1 and 3 to find the enthalpies: \(h_1 = c_pT_1, h_3 = c_pT_3\) With all the enthalpy values, we can now find \(h_2\) and \(h_4\) as mentioned in the above equations. **Step 4: Calculate the actual power output of the turbine**

Actual power output of the turbine

The actual power output of the turbine is given by the equation: \(\dot{W}_{\text{turbine, actual}} = m\dot{(h_3 - h_4)}\) We are given the mass flow rate, \(m\dot{ = 200 kg/s}\), and have obtained the values of \(h_3\) and \(h_4\). Plug in the values to get the actual power output of the turbine. **Step 5: Calculate the isentropic power output of the turbine**

Isentropic power output of the turbine

The isentropic power output can be found by using the isentropic temperature relationship: \(\frac{T_{4s}}{T_3} = \left(\frac{P_4}{P_3}\right)^{\frac{k-1}{k}} \Rightarrow T_{4s} = T_3\left(\frac{P_4}{P_3}\right)^{\frac{k-1}{k}}\) Now find \(h_{4s}\): \(h_{4s} = c_pT_{4s}\) The isentropic power output of the turbine is given by: \(\dot{W}_{\text{turbine, isentropic}} = m\dot{(h_3 - h_{4s})}\) Plug in the values to get the isentropic power output of the turbine. **Step 6: Determine the isentropic efficiency of the turbine**

Isentropic efficiency of the turbine

The isentropic efficiency of the turbine is given by the equation: \(\eta_{\text{isentropic}} = \frac{\dot{W}_{\text{turbine, actual}}}{\dot{W}_{\text{turbine, isentropic}}}\) Using the values obtained in Steps 4 and 5, we can calculate the isentropic efficiency of the turbine. **Step 7: Determine the cycle thermal efficiency**

Cycle thermal efficiency

The cycle thermal efficiency is given by the equation: \(\eta_{\text{thermal}} = \frac{\dot{W}_{\text{net}}}{\dot{Q}_{\text{in}}} = \frac{m\dot{(h_3 - h_4)}}{m\dot{(h_3 - h_2)}} = \frac{h_3 - h_4}{h_3 - h_2}\) Using the values obtained in Steps 3 and 4, we can calculate the cycle thermal efficiency. **Step 8: Sketch the T-s diagram**

T-s diagram

Now that we have found all the relevant temperatures and enthalpies, we can sketch the \(T\)-s diagram for the cycle. The diagram should consist of: - A vertical line connecting Points 1 and 2, representing the compression process in the compressor. - A horizontal line connecting Points 2 and 3, representing the constant pressure heat addition process in the combustion chamber. - A vertical line connecting Points 3 and 4, representing the expansion process in the turbine. - A horizontal line connecting Points 4 and 1, representing the constant pressure heat rejection process in the heat exchanger. Label the lines with the process numbers and the points with the temperatures and pressures.