Question

Steam at \(10 \mathrm{MPa}, 600^{\circ} \mathrm{C}\) enters the first-stage turbine of an ideal Rankine cycle with reheat. The steam leaving the reheat section of the steam generator is at \(500^{\circ} \mathrm{C}\), and the condenser pressure is \(6 \mathrm{kPa}\). If the quality at the exit of the secondstage turbine is \(90 \%\), determine the cycle thermal efficiency.

Short answer

Calculate enthalpies at all states, determine reheat pressure using quality, find work outputs and heat inputs, and use them to calculate the thermal efficiency using \ ( \frac{(W_{t1} + W_{t2})}{(Q_{in1} + Q_{in2})} \).

Step-by-step solution

Determine the properties at each state point

First, identify all state points in the Rankine cycle and find their corresponding properties using steam tables or Mollier charts: 1. State 1 (at the exit of the first-stage turbine): Pressure is 10 MPa and temperature is 600°C. 2. State 2 (at the exit of the reheat stage): Pressure is reheat pressure and temperature is 500°C. 3. State 3 (at the exit of the second-stage turbine): Pressure is 6 kPa and quality (x) is 90%. 4. State 4 (at the condenser exit): Pressure is 6 kPa and a saturated liquid.

Calculate the enthalpies (h) at each state point

Using the steam tables or Mollier charts, find the following enthalpies: 1. Enthalpy at state 1: \(h_1\) 2. Enthalpy at state 2: \(h_2\) 3. Enthalpy at state 3: \(h_3\) 4. Enthalpy at state 4: \(h_4\)

Apply energy balance to find reheat pressure and related enthalpy

Use the given reheat temperature and turbine exit quality to iterate and find the reheat pressure for state 2: - The enthalpy at reheat temperature 500°C will help define an accurate reheat pressure.

Calculate the work output from each turbine stage

Use the enthalpies to compute the work done by each turbine stage: 1. Work done by the first turbine stage: \( W_{t1} = h_1 - h_2 \) 2. Work done by the second turbine stage: \( W_{t2} = h_2 - h_3 \)

Calculate the heat added in the boiler and reheat sections

Calculate the heat added (Q) in the cycle using the enthalpies: 1. Heat added in the boiler: \( Q_{in1} = h_1 - h_4 \) 2. Heat added in the reheat stage: \( Q_{in2} = h_2 - h_3 \)

Determine the cycle thermal efficiency

Calculate the thermal efficiency using the formula: \[ \text{Efficiency} = \frac{(W_{t1} + W_{t2})}{(Q_{in1} + Q_{in2})} \]

Key concepts

thermodynamics

Thermodynamics is the branch of physics that deals with heat, work, and energy, and how they interrelate. It helps us understand how energy is transferred and transformed in systems like the Rankine cycle used in power plants. In the Rankine cycle, we analyze different state points and energy exchanges, including heat addition and work done by the turbines. Key principles include the First Law of Thermodynamics, which states that energy cannot be created or destroyed, only transferred, and the Second Law of Thermodynamics, which introduces the concept of entropy or disorder.

steam turbine

A steam turbine is a mechanical device that extracts thermal energy from pressurized steam and converts it into mechanical work. In the Rankine cycle, the steam turbine consists of multiple stages, where the steam expands and performs work on the turbine blades. Here’s how it works:
* **High-Pressure Turbine Stage:** Steam at high pressure and temperature enters and expands, doing work and losing some enthalpy.
* **Reheat Stage:** The steam is reheated to raise its temperature without changing its pressure significantly, allowing for more efficient energy extraction.
* **Low-Pressure Turbine Stage:** The reheated steam expands further in a second stage turbine to finally exit with lower pressure and quality, doing additional work.
By optimizing these stages, we can achieve better efficiency in converting heat energy into mechanical work.

enthalpy calculations

Enthalpy is a measure of the total heat content of a system. In thermodynamics, particularly the Rankine cycle, calculating enthalpy at various state points is crucial. Here’s how to do it:
* **Identify State Points:** Use steam tables or Mollier charts to find the enthalpy values for given pressures and temperatures.
* **Calculate Enthalpies:** For each key point (e.g., before and after each turbine stage), find the enthalpy values (h):
\[ h_1, h_2, h_3, \text{ and } h_4 \]
* **Energy Balance:** Apply the energy balance to check consistency and perform necessary iterations to refine values.
* **Extraction and Addition Points:** Understand enthalpy changes due to work extraction in turbines and heat addition in the boiler and reheat sections. This is essential for computing work outputs and heat inputs accurately.
Precise enthalpy calculations ensure the accuracy of efficiency and performance assessments in thermal cycles.

thermal efficiency analysis

Thermal efficiency is a measure of how effectively a thermodynamic system converts heat into work. In the Rankine cycle, it’s essential to analyze the thermal efficiency to understand the performance of the power plant. Here’s how it works:
* **Work Output Calculation:** Compute the work done by each turbine stage using the enthalpies:
\[ W_{t1} = h_1 - h_2 \]
\[ W_{t2} = h_2 - h_3 \]
* **Heat Added Calculation:** Determine the heat added in the boiler and reheat section:
\[ Q_{in1} = h_1 - h_4 \]
\[ Q_{in2} = h_2 - h_3 \]
* **Efficiency Formula:** Use the formula for thermal efficiency:
\[ \text{Efficiency} = \frac{(W_{t1} + W_{t2})}{(Q_{in1} + Q_{in2})} \]
This simple ratio of work done to heat added provides a clear picture of how well the cycle is converting thermal energy into mechanical energy. Analyzing and improving this efficiency is key to better fuel economy and reduced emissions in power generation.