Question

Using EES (or other) software, investigate the effect of reheat pressure on the performance of an ideal Rankine cycle. The maximum and minimum pressures in the cycle are \(15 \mathrm{MPa}\) and \(10 \mathrm{kPa}\) respectively, and steam enters both stages of the turbine at \(500^{\circ} \mathrm{C}\). The reheat pressure is varied from 12.5 to 0.5 MPa. Determine the thermal efficiency of the cycle and plot it against the reheat pressure, and discuss the results.

Short answer

Answer: The reheat pressure affects the thermal efficiency of the ideal Rankine cycle by changing the work and heat transfer in the cycle components. By varying the reheat pressure, we can calculate the thermal efficiency for different reheat pressures. By plotting these values, it's possible to analyze and discuss the impact of reheat pressure on the performance of the Rankine cycle.

Step-by-step solution

Understand the Ideal Rankine Cycle with Reheat

An ideal Rankine cycle with reheat consists of four components: a boiler, a turbine, a condenser, and a pump. The working fluid undergoes a series of thermodynamic processes as it flows through these components. The reheat process is an additional step that aims to improve the cycle efficiency by reheating the working fluid before it enters the second stage of the turbine.

Determine the States of the Working Fluid

In order to analyze the Rankine cycle, we need to determine the state of the working fluid at each point in the cycle. The following conditions are given: 1. Maximum Pressure: \(P_{1} = 15 \mathrm{MPa}\) 2. Minimum Pressure: \(P_{4} = 10 \mathrm{kPa}\) 3. Temperature at turbine stages: \(T_{1} = 500^{\circ} \mathrm{C} = T_{2}\) We can find the following points in the cycle: 1. State 1: At the exit of the boiler and entrance to the high-pressure turbine, the fluid is at maximum pressure and temperature (\(T_{1}\) and \(P_{1}\)). 2. State 2: At the exit of the high-pressure turbine and entrance to the reheater, the fluid is at the reheat pressure \(P_{2}\) and temperature \(T_{2} = 500^{\circ} \mathrm{C}\). 3. State 3: At the exit of reheater and entrance to the low-pressure turbine, the fluid is at temperature \(T_{3} = T_{2}\) and pressure \(P_{3} = P_{2}\). 4. State 4: At the exit of the low-pressure turbine and entrance to the condenser, the fluid is at minimum pressure (\(P_{4}\)). We can assume it to be a saturated liquid-vapor mixture (\(x_{4}\), quality). Finding enthalpy at these points will help us find the work and heat of the cycle.

Calculate the Work and Heat of the Cycle

We can use the following equations to find the work and heat of the cycle: 1. Turbine Work (\(W_{T}\)): \(W_{T} = W_{T,1} + W_{T,2} = (h_{1} - h_{2}) + (h_{3} - h_{4})\) 2. Pump Work (\(W_{P}\)): \(W_{P} = v_{f} \cdot (P_{1} - P_{4})\) 3. Boiler Heat (\(Q_{B}\)): \(Q_{B} = Q_{B,1} + Q_{B,2} = (h_{1} - h_{f_{4}}) + (h_{3} - h_{2})\) Here, \(h_{i}\) represents the enthalpy at state \(i\), \(v_{f}\) is the specific volume of the saturated liquid at \(P_{4}\), and \(x_{4}\) is the quality of the working fluid at state 4.

Calculate the Thermal Efficiency of the Cycle

Thermal efficiency (\(\eta\)) of the cycle can be calculated using the following formula: \(\eta = \frac{W_{net}}{Q_{B}} = \frac{W_{T} - W_{P}}{Q_{B}}\)

Vary the Reheat Pressure and Plot the Results

By varying the reheat pressure (\(P_{2}\)) from 12.5 MPa to 0.5 MPa, we can calculate the thermal efficiency of the Rankine cycle for different reheat pressures. By plotting these values, we can analyze the impact of reheat pressure on thermal efficiency and discuss the results. To summarize, we have analyzed the ideal Rankine cycle with reheat, determined the states of the working fluid, calculated the work and heat of the cycle, and found the thermal efficiency. By plotting the thermal efficiency against various reheat pressures, we can discuss the impact of reheat pressure on the performance of the Rankine cycle.