Question

Using EES (or other) software, investigate the effect of the boiler pressure on the performance of a simple ideal Rankine cycle. Steam enters the turbine at \(500^{\circ} \mathrm{C}\) and exits at \(10 \mathrm{kPa}\). The boiler pressure is varied from 0.5 to 20 MPa. Determine the thermal efficiency of the cycle and plot it against the boiler pressure, and discuss the results.

Short answer

Answer: By increasing boiler pressure from 0.5 MPa to 20 MPa, the thermal efficiency of the simple ideal Rankine cycle increases. However, this increase in efficiency is not linear, and there may be an optimal boiler pressure value that maximizes the cycle's efficiency, which can be determined from the efficiency vs pressure plot.

Step-by-step solution

Determine the enthalpy values at each stage

First, we need to find the enthalpy values at each point in a simple ideal Rankine cycle: points 1, 2, 3, and 4 using EES or any other thermodynamics software. - Point 1: Boiler outlet, where steam enters the turbine. Temperature is given as \(500^{\circ} \mathrm{C}\). - Point 2: Turbine outlet, where steam exits the turbine. Pressure is given as \(10 \mathrm{kPa}\). - Point 3: Condenser outlet, where saturated liquid exits the condenser. Pressure is given as \(10 \mathrm{kPa}\). - Point 4: Pump inlet, where the liquid is pumped back into the boiler. Assume the pump process to be isentropic. Using the given conditions, find enthalpy values (\(h_1, h_2, h_3, h_4\)) by looking up the appropriate thermodynamic tables or using software.

Calculate work output from the turbine and work input for the pump

Next, we need to determine the work output from the turbine and the work input for the pump. - Turbine work output: \(W_T = m(h_1 - h_2)\), where m is the mass flow rate. - Pump work input: \(W_P = m(h_4 - h_3)\). Since the pump process is assumed isentropic, use \(h_4 = h_3 + v_3 (P_4 - P_3)\), where \(v_3\) is the specific volume of the liquid at point 3. Calculate \(W_T\) and \(W_P\) with the obtained values for enthalpies.

Calculate the heat input in the boiler and the cycle efficiency

Now, we need to calculate the heat input during the boiler process, which can be found by considering the enthalpy difference between points 1 and 4: \(Q_{in} = m(h_1 - h_4)\). Finally, we can determine the efficiency of the Rankine cycle: \(\eta = \frac{W_T - W_P}{Q_{in}}\).

Vary the boiler pressure and plot the efficiency against the pressure

Using EES or any other thermodynamics software, repeat Steps 1-3 for various boiler pressures between 0.5 MPa and 20 MPa. Compute the efficiency for each pressure and plot the efficiency against the boiler pressure.

Discuss the results

Analyze the plot created in Step 4 and discuss how the efficiency of the Rankine cycle changes with the boiler pressure. Notice any trends or optimal pressure values for higher efficiency. This information can be useful for designing practical steam power plants with better performance.

Key concepts

Boiler Pressure

When examining a Rankine cycle, understanding the role of boiler pressure is crucial. Higher boiler pressure leads to an increase in the temperature at which heat is added to the system, which generally increases the efficiency of the cycle. This is because a higher temperature difference between the heat source and the working fluid allows for more efficient heat transfer.

However, as the pressure rises, the materials and construction of the boiler must be designed to withstand these pressures, which can increase costs. As the pressure in the boiler increases, the thermal efficiency of the Rankine cycle may improve, but only up to a certain point, after which the efficiency gains may not justify the additional costs and complexities. It’s important to balance efficiency with practical considerations such as material strength and cost.

Thermal Efficiency

Thermal efficiency in the context of the Rankine cycle is a measure of how well the cycle converts heat into work. It's the ratio of the net work output of the cycle to the heat input at the boiler stage. The formula for thermal efficiency \( \eta \) for the Rankine cycle can be expressed as \( \eta = \frac{W_T - W_P}{Q_{in}} \) where \( W_T \) is the turbine work output, \( W_P \) is the pump work input, and \( Q_{in} \) is the heat input to the boiler.

Increasing the boiler pressure can increase \( \eta \) by allowing the steam to expand through a larger temperature range, which increases the potential work output \( W_T \) from the turbine. However, the relationship isn’t linear, and each system will have an optimal pressure that maximizes efficiency.

Enthalpy Values

Enthalpy values represent the total heat content of a system and are a central part of thermodynamics and the Rankine cycle analysis. In a Rankine cycle, these values are necessary to calculate the heat input in the boiler \( Q_{in} \) and the work done by the turbine \( W_T \) and required by the pump \( W_P \).

At each stage of the cycle, the enthalpy is determined for a specific state of the working fluid—superheated vapor, saturated vapor, saturated liquid, and subcooled liquid. Tools such as thermodynamic tables or software like EES help in finding these enthalpy values at various pressures and temperatures. By accurately determining these values, we can calculate the work and heat transfer processes occurring within the cycle, ultimately allowing us to assess the cycle's performance and efficiency.

Isentropic Process

An isentropic process is an idealized thermodynamic process that is both adiabatic and reversible, meaning there is no heat transfer with the surroundings, and no entropy is generated within the system. In the Rankine cycle, the pump and turbine are often assumed to operate in an isentropic manner to simplify the analysis.

During an isentropic compression in the pump, the fluid’s pressure increases without a change in entropy, leading to a minimal increase in enthalpy. Conversely, isentropic expansion in the turbine results in a drop in pressure and temperature without a change in entropy, leading to the production of work. Isentropic efficiency is a measure of how closely a real process approaches the ideal isentropic process and is a critical parameter in evaluating the performance of turbines and compressors in power generation systems.