BRAYTON CYCLE: PROCESS, EFFICIENCY, APPLICATIONS, EXERCISES - PHYSICAL - 2024

The Brayton cycle is a thermodynamic cycle that consists of four processes and is applied to a compressible thermodynamic fluid such as a gas. Its first mention dates from the late 18th century, although it was some time before it was first raised by James Joule. This is why it is also known as the Joule cycle.

It consists of the following stages, which are conveniently illustrated in the pressure-volume diagram in Figure 1: adiabatic compression (no heat is exchanged), isobaric expansion (occurs at constant pressure), adiabatic expansion (no heat is exchanged) and isobaric compression (occurs at constant pressure).

Figure 1. Brayton cycle. Source: self made.

Process and description

The Brayton cycle is the ideal thermodynamic cycle that is best applied to explain the thermodynamic operation of gas turbines and air-fuel mixture, used for the generation of electrical energy and in aircraft engines.

Figure 2. Turbine diagram and flow stages. Source: self made.

For example, in the operation of a turbine there are several stages in the operating gas flow, which we will see below.

Admission

It consists of the entry of air at ambient temperature and pressure through the inlet opening of the turbine.

Compression

Air is compressed by rotating blades against fixed blades in the compressor section of the turbine. This compression is so rapid that there is practically no heat exchange, so it is modeled by the adiabatic process AB of the Brayton cycle. The air leaving the compressor has increased its pressure and temperature.

Combustion

The air is mixed with propane gas or pulverized fuel that is introduced through the injectors of the combustion chamber. The mixture produces a chemical reaction of combustion.

This reaction is what provides the heat that increases the temperature and the kinetic energy of the gas particles that expand in the combustion chamber at constant pressure. In the Brayton cycle this step is modeled with the BC process that occurs at constant pressure.

Expansion

In the section of the turbine itself, the air continues to expand against the turbine blades, causing it to rotate and producing mechanical work. In this step, the air lowers its temperature but without practically exchanging heat with the environment.

In the Brayton cycle this step is simulated as a CD adiabatic expansion process. Part of the turbine's work is transferred to the compressor and the other is used to drive a generator or propeller.

Escape

The outgoing air is at a constant pressure equal to the ambient pressure and transfers heat to the enormous mass of external air, so that in a short time it takes on the same temperature as the inlet air. In the Brayton cycle this step is simulated with the constant pressure DA process, closing the thermodynamic cycle.

Efficiency as a function of temperature, heat and pressure

We propose to calculate the efficiency of the Brayton cycle, for which we start from the definition of it.

In a heat engine, efficiency is defined as the net work done by the machine divided by the heat energy supplied.

The first principle of thermodynamics states that the net heat contributed to a gas in a thermodynamic process is equal to the change in the internal energy of the gas plus the work done by it.

But in a complete cycle the variation of the internal energy is zero, so the net heat contributed in the cycle is equal to the net work done.

Incoming heat, outgoing heat and efficiency

The previous expression allows us to write the efficiency as a function of the absorbed or incoming heat Qe (positive) and the transferred or outgoing heat Qs (negative).

Heat and pressure in the Brayton cycle

In the Brayton cycle, heat enters the isobaric process BC and exits in the isobaric process DA.

Assuming that n moles of gas at constant pressure that sensible heat Qe is supplied to it in process BC, then its temperature increases from Tb to Tc according to the following relationship:

The outgoing heat Qs can be calculated in a similar way by the following relation that applies to the constant pressure process DA:

Substituting these expressions in the expression that gives us the efficiency as a function of the incoming heat and outgoing heat, making the pertinent simplifications, the following relation for the efficiency is obtained:

Simplified result

It is possible to simplify the previous result if we take into account that Pa = Pd and that Pb = Pc given that the processes AD and BC are isobaric, that is, at the same pressure.

Furthermore, since the processes AB and CD are adiabatic, the Poisson's ratio is fulfilled for both processes:

Where gamma represents the adiabatic quotient, that is, the quotient between the heat capacity at constant pressure and the heat capacity at constant volume.

Using these relationships and the relationship from the ideal gas equation of state we can obtain an alternative expression for Poisson's ratio:

As we know that Pa = Pd and that Pb = Pc, substituting and dividing member by member, the following relationship between temperatures is obtained:

If each member of the previous equation is subtracted by unity, the difference is solved and the terms are arranged, it can be shown that:

Performance as a function of the pressure ratio

The expression obtained for the efficiency of the Brayton cycle as a function of temperatures can be rewritten to be formulated as a function of the pressure ratio at the compressor outlet and inlet.

This is achieved if the Poisson's ratio between points A and B is known as a function of pressure and temperature, obtaining that the efficiency of the cycle is expressed as follows:

A typical pressure ratio is 8. In this case the Brayton cycle has a theoretical yield of 45%.

Applications

The Brayton cycle as a model is applied to gas turbines used in thermoelectric plants in order to drive generators that produce electricity.

It is also a theoretical model that fits well with the operation of turboprop engines used in airplanes, but it is not applicable at all in airplane turbojets.

When you want to maximize the work produced by the turbine to drive the generators or propellers of an airplane, then the Brayton cycle is applied.

Figure 3. Turbofan engine more efficient than the turbojet. Source: Pixabay

In airplane turbojets, on the other hand, there is no interest in converting the kinetic energy of the combustion gases to produce work, which would be just enough to recharge the turbocharger.

On the contrary, it is interesting to obtain the highest possible kinetic energy of the expelled gas, so that according to the principle of action and reaction, the momentum of the aircraft is obtained.

Solved exercises

-Exercise 1

A gas turbine of the type used in thermoelectric plants has a pressure at the compressor outlet of 800 kPa. The incoming gas temperature is ambient and is 25 Celsius, and the pressure is 100 kPa.

In the combustion chamber the temperature rises to 1027 Celsius to enter the turbine.

Determine the cycle efficiency, the gas temperature at the compressor outlet, and the gas temperature at the turbine outlet.

Solution

As we have the pressure of the gas at the outlet of the compressor and we know that the inlet pressure is atmospheric pressure, then it is possible to obtain the pressure ratio:

r = Pb / Pa = 800 kPa / 100 KPa = 8

As the gas with which the turbine operates is a mixture of air and propane gas, the adiabatic coefficient is then applied for a diatomic ideal gas, that is, a gamma of 1.4.

The efficiency would then be calculated like this:

Where we have applied the relationship that gives the efficiency of the Brayton cycle as a function of the pressure ratio in the compressor.

Calculation of temperature

To determine the temperature at the outlet of the compressor, or what is the same the temperature with which the gas enters the combustion chamber, we apply the relationship of the efficiency with the inlet and outlet temperatures of the compressor.

If we solve for the temperature Tb from that expression, we obtain:

As data for the exercise we have that after combustion the temperature rises to 1027 Celsius, to enter the turbine. Part of the thermal energy of the gas is used to move the turbine, so the temperature at its outlet must be lower.

To calculate the temperature at the outlet of the turbine we will use a relationship between temperature obtained previously:

From there we solve for Td to obtain the temperature at the turbine outlet. After performing the calculations, the temperature obtained is:

Td = 143.05 Celsius.

-Exercise 2

A gas turbine follows the Brayton cycle. The pressure ratio between the compressor inlet and outlet is 12.

Assume the ambient temperature of 300 K. As additional data, it is known that the gas temperature after combustion (prior to entering the turbine) is 1000K.

Determine the temperature at the compressor outlet, and the temperature at the turbine outlet. Also determine how many kilograms of gas circulate through the turbine in each second, knowing that its power is 30 KW.

Assume the specific heat of the gas as constant and take its value at room temperature: Cp = 1.0035 J / (kg K).

Suppose also that the compression efficiency in the compressor and decompression efficiency in the turbine is 100%, which is an idealization because in practice losses always occur.

Solution

To determine the temperature at the compressor outlet, knowing the inlet temperature, we must remember that it is an adiabatic compression, so the Poisson's ratio can be applied for the AB process.

For any thermodynamic cycle, the net work will always equal the net heat exchanged in the cycle.

The net work per operating cycle can then be expressed as a function of the mass of gas that circulated in that cycle and the temperatures.

In this expression m is the mass of gas that circulated through the turbine in an operating cycle and Cp the specific heat.

If we take the derivative with respect to time of the previous expression, we obtain the net mean power as a function of the mass flow.

Solving for m point, and substituting the temperatures, power and heat capacity of the gas, we obtain a mass flow of 1578.4 kg / s.

References

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3. Sevilla University. Physics department. Brayton cycle. Recovered from: laplace.us.es.
4. National Experimental University of Táchira. Transportation Phenomena. Gas power cycles. Recovered from: unet.edu.ve.
5. Wikipedia. Brayton cycle. Recovered from: wikiwand.com
6. Wikipedia. Gas turbine. Recovered from: wikiwand.com.