Question

Air enters the turbine of a gas turbine at \(1200 \mathrm{kPa}, 1200 \mathrm{~K}\), and expands to \(100 \mathrm{kPa}\) in two stages. Between the stages, the air is reheated at a constant pressure of \(350 \mathrm{kPa}\) to \(1200 \mathrm{~K}\). The expansion through each turbine stage is isentropic. Determine, in \(\mathrm{kJ}\) per \(\mathrm{kg}\) of air flowing (a) the work developed by each stage. (b) the heat transfer for the reheat process. (c) the increase in net work as compared to a single stage of expansion with no reheat.

Short answer

Work in each stage: Calculate first via isentropic relations; heat added during reheat: CP×ΔT; net work comparison: single-stage work vs two-stage work.

Step-by-step solution

Analyze Given Data

Identify and list all given data. Initial state of air entering the turbine is at 1200 kPa and 1200 K. Air expands to 100 kPa in two stages. Between the stages, air is reheated at 350 kPa to 1200 K. Each stage of expansion is isentropic.

Use Isentropic Relations

For an isentropic process, the relation between temperatures and pressures is given by \[ \frac{T_2}{T_1} = \left( \frac{P_2}{P_1} \right)^{\frac{\gamma-1}{\gamma}} \]where \( \gamma \) (ratio of specific heats for air) is approximately 1.4. Apply this for each stage.

First Expansion Stage Calculation

For the first stage expansion from 1200 kPa to 350 kPa:\[ \frac{T_2}{1200} = \left( \frac{350}{1200} \right)^{\frac{1.4-1}{1.4}} \] Calculate the exit temperature \( T_2 \).

Reheat Process

During reheating, pressure remains constant at 350 kPa, and temperature returns to 1200 K. Reheating does not affect the pressure but resets the temperature for the next stage.

Second Expansion Stage Calculation

For the second stage expansion from 350 kPa to 100 kPa:\[ \frac{T_4}{1200} = \left( \frac{100}{350} \right)^{\frac{1.4-1}{1.4}} \] Calculate the exit temperature \( T_4 \).

Work Done by Each Turbine Stage

The work done in each stage can be calculated using \[ W = c_p (T_1 - T_2) \] where \( c_p \) is the specific heat at constant pressure, approximately \( 1.005 \text{ kJ/kg·K} \). Calculate the work done in both stages using temperatures obtained in steps 3 and 5.

Heat Transfer in Reheat Process

Heat added during reheating can be found by \[ Q = c_p (T_3 - T_2) \]. Since \( T_2 \) is the temperature after the first stage and \( T_3 \) is 1200 K, calculate \( Q \).

Compare with Single Stage Expansion

For a single-stage expansion from 1200 kPa to 100 kPa: \[ \frac{T_{single}}{1200} = \left( \frac{100}{1200} \right)^{\frac{1.4-1}{1.4}} \] Calculate the exit temperature and the work done in a single stage using similar steps. Compare with the two-stage process to find the net increase in work.

Key concepts

gas turbine

A gas turbine is a type of internal combustion engine that converts natural gas or other liquid fuels into mechanical energy. This energy can then be used to generate electricity through a generator. The process involves several key steps: compression of air, mixing it with fuel, ignition, and expansion through a turbine. The basic principle is to harness the energy produced by these expanding gases to drive a turbine, spinning it at high speeds.

• Air is compressed and then heated by burning fuel in a combustion chamber.
• The high-pressure, high-temperature gases expand through the turbine, producing work.
• Finally, the turbine rotates, driving a generator that produces electricity.

Gas turbines are common in power plants, aircraft engines, and natural gas facilities.

thermodynamic process

In a thermodynamic process, the state of a system changes. The processes can include heat transfer, work done, changes in internal energy, and phase transitions. The laws of thermodynamics govern these processes, ensuring energy conservation and determining the direction and feasibility of reactions.
An important type of thermodynamic process is an isentropic process, where entropy remains constant. This implies no heat exchange, and all changes are due to work done.

• Isentropic processes are idealized and help simplify calculations in problems like turbine expansions.
• The efficiency of real systems can be estimated by comparing them to idealized isentropic systems.

In the exercise, the expansions in the turbine are considered isentropic. This simplification allows us to use specific relations between pressures and temperatures.

reheat process

The reheat process involves heating the working fluid between stages of expansion to increase efficiency. After the first stage of expansion, the fluid's temperature lowers, reducing its ability to do work. By reheating the fluid before it undergoes further expansion, you can achieve more work output.

• This is particularly useful in power cycles where multiple stages of expansion occur, like in a multi-stage gas turbine or steam turbine.
• Reheating restores the fluid's temperature without increasing its pressure, setting it up for more efficient subsequent expansion stages.

In the given exercise, air is reheated to 1200 K at a constant pressure of 350 kPa between the stages. This reheating step ensures that the air entering the second stage of the turbine remains at high efficiency.

work calculation

Work calculation in thermodynamic processes typically involves understanding the relation between pressure, volume, and temperature changes. For turbine work, it is often related to the change in the enthalpy of the working fluid.
In isentropic processes, the temperature and pressure relationships can help determine the exit temperatures. The specific heat at constant pressure (\text{c_p}) is used to calculate the work done.

• The relationship for work done in a turbine stage can be expressed as: \[ W = c_p (T_1 - T_2) \] where \(T_1\) is the initial temperature and \(T_2\) is the final temperature after expansion.
• Using the temperatures found in isentropic relations, you can then calculate the work done in each stage.

In the exercise, you calculate the work done in each turbine stage and compare them with the work done in a single-stage expansion. This helps illustrate the efficiency benefits of using a multi-stage process with reheating.