Question

An ideal Rankine cycle with reheat uses water as the working fluid. The conditions at the inlet to the first-stage turbine are \(14 \mathrm{MPa}, 600^{\circ} \mathrm{C}\) and the steam is reheated between the turbine stages to \(600^{\circ} \mathrm{C}\). For a condenser pressure of \(6 \mathrm{kPa}\), plot the cycle thermal efficiency versus reheat pressure for pressures ranging from 2 to \(12 \mathrm{MPa}\).

Short answer

Determine steam properties, calculate thermal efficiency for each reheat pressure, and plot the efficiency vs reheat pressure.

Step-by-step solution

Understand the Problem Statement

The problem is about the Rankine cycle with reheat using water as the working fluid. The cycle starts with steam at 14 MPa and 600°C, reheats to 600°C before entering the second-stage turbine, and finally condenses at 6 kPa. The task is to plot the cycle thermal efficiency versus reheat pressure for pressures ranging from 2 to 12 MPa.

Define the Rankine Cycle Stages

Identify and define the stages of the Rankine cycle: (1) isentropic expansion in the first-stage turbine, (2) isobaric heat addition during reheat, (3) isentropic expansion in the second-stage turbine, and (4) isobaric heat rejection in the condenser.

Determine Steam Properties

Use steam tables or software to determine the properties of steam (enthalpy and entropy) at key points: at the inlet (14 MPa, 600°C), after the first-stage turbine, after the reheat (600°C), after the second-stage turbine, and after the condenser.

Energy Balance for Each Component

Apply energy balances for each component of the cycle: turbines, reheat, and condenser. Use the formula for energy balance: \(W = m(\triangle h)\), where W is the work done, m is the mass flow rate, and \(\triangle h\) is the change in enthalpy.

Calculate Thermal Efficiency

Determine the thermal efficiency of the cycle using the formula: \(\eta = \frac{W_{\text{turbine}} - W_{\text{pump}}}{Q_{\text{boiler}}}\). Repeat this calculation for different reheat pressures ranging from 2 MPa to 12 MPa.

Plot Thermal Efficiency vs Reheat Pressure

Plot the calculated thermal efficiencies against the corresponding reheat pressures to visualize how efficiency changes with reheat pressure.

Key concepts

Thermodynamics

Thermodynamics is the branch of physics that deals with heat and temperature, and their relation to work and energy. It plays a critical role in understanding the Rankine cycle with reheat. Here, we look at the principles governing how heat energy is converted into mechanical work in a steam power plant using water as a working fluid.

In the Rankine cycle, water undergoes phase changes from liquid to vapor and back to liquid. This cycle involves four main processes:

• Isentropic expansion in turbines
• Isobaric heat addition in reheat stages
• Isentropic expansion again in turbines post-reheat
• Isobaric heat rejection in the condenser

Understanding these processes and how they interrelate is fundamental to mastering thermodynamics and its application in thermal power cycles.

The use of steam tables or software to determine the properties of steam at various stages of the cycle is crucial. These properties help in calculating the work done and the heat added or rejected during each phase.

Thermal Efficiency

Thermal efficiency (\text{η}) is a measure of how well an engine or cycle converts heat from fuel into mechanical energy. For the Rankine cycle with reheat, it indicates the fraction of heat converted into work. The basic formula is:
\[\begin{equation} \text{η} = \frac{W_{\text{turbine}} - W_{\text{pump}}}{Q_{\text{boiler}}} \end{equation}\] Here, \text{W}_{\text{turbine}} is the work done by the turbines, \text{W}_{\text{pump}} is the work input by the pumps, and Q_{\text{boiler}} is the heat added in the boiler.

Key points affecting thermal efficiency in a Rankine cycle with reheat include:

• Reheat temperature: Higher temperatures typically increase efficiency as more energy is converted to work, reducing heat losses.
• Boiler pressure: Higher pressures at the turbine inlet usually improve efficiency by increasing the thermal energy available for conversion to work.
• Condenser pressure: Lower pressures at the turbine exit can enhance efficiency by lowering the heat rejection temperature, allowing more work extraction from the cycle.

By plotting thermal efficiency against reheat pressure, engineers can optimize the cycle to achieve higher efficiencies.

Reheat Pressure

Reheat pressure is a crucial parameter in designing and evaluating the performance of a Rankine cycle with reheat. It refers to the pressure at which steam is reheated between the turbines' stages.

In the given problem, the steam is reheated to 600°C in between the turbine stages. The task is to explore how changing reheat pressures, ranging from 2 MPa to 12 MPa, affects the thermal efficiency of the cycle.

The process involves:

• Isentropic expansion in the first turbine stage where steam initially expands and does work.
• Isobaric heat addition during the reheat stage, where steam temperature is elevated back to 600°C at various reheat pressures.
• Isentropic expansion in the second turbine stage, wherein reheat pressure impacts the steam's enthalpy and entropy, altering how much work can be done.

By manipulating the reheat pressure, we aim to find the optimal balance where the cycle attains maximum thermal efficiency. Understanding this relationship helps in designing more efficient real-world thermal power systems using the Rankine cycle with reheat.