Gas Cycles Cheat Sheet - Otto, Diesel, and Brayton Cycles

Gas Cycles - Gas cycles form the backbone of thermodynamic systems used in engines, turbines, and power plants. They describe the processes through which energy is transferred in cyclic operations, enabling the conversion of heat into useful work or vice versa. This comprehensive cheat sheet explores various gas cycles, their key assumptions, efficiency, and applications.

Key Gas Cycles Overview

1. Air Standard Cycle Assumptions

The air standard cycle is an idealization used to analyze gas cycles. The key assumptions include:

• The working fluid is treated as an ideal gas.
• The gas circulates in a closed loop.
• Heat input and rejection replace combustion and exhaust processes.
• Constant heat capacities are assumed.

The Otto Cycle

The Otto cycle is the thermodynamic model for spark-ignition engines, such as gasoline engines. It consists of four strokes:

1. Isentropic Compression: Work is done on the gas, increasing its pressure and temperature.
2. Isochoric Heat Addition: Combustion adds heat at constant volume.
3. Isentropic Expansion: The gas expands, performing work on the piston.
4. Isochoric Heat Rejection: Heat is released at constant volume.

Efficiency of the Otto Cycle: $\eta = 1 - \frac{1}{r^{\gamma - 1}}$
Where:

• $r$: Compression ratio
• $\gamma$: Specific heat ratio

The Diesel Cycle

The Diesel cycle represents compression-ignition engines, which are more efficient at higher compression ratios compared to Otto engines. The cycle involves:

1. Isentropic Compression: Air is compressed without fuel injection.
2. Isobaric Heat Addition: Fuel is injected and burns at constant pressure.
3. Isentropic Expansion: Work is extracted as the gas expands.
4. Isochoric Heat Rejection: Heat is released at constant volume.

Efficiency of the Diesel Cycle: $\eta = 1 - \frac{1}{r^{\gamma - 1}} \cdot \frac{\rho^\gamma - 1}{\gamma(\rho - 1)}$
Where:

$\rho$: Cut-off ratio (volume after combustion to volume before combustion)

The Brayton (Joule) Cycle

The Brayton cycle models gas turbines, including jet engines and power plants. It consists of:

1. Isentropic Compression: Air is compressed in a turbine.
2. Isochoric Heat Addition: Combustion or heat addition occurs at constant pressure.
3. Isentropic Expansion: The gas expands, driving the turbine and producing work.
4. Isochoric Heat Rejection: Heat is rejected at constant pressure.

Efficiency of the Brayton Cycle: $\eta = 1 - \frac{1}{r_p^{(\gamma - 1)/\gamma}}$
Where:

$r_p$: Pressure ratio across the compressor

The Turbojet Cycle

The turbojet cycle is a specific application of the Brayton cycle used in jet engines to generate thrust:

• Air enters through a diffuser and is compressed.
• Heat is added in the combustion chamber.
• The high-pressure exhaust gases expand through a turbine and nozzle to produce thrust.

Thrust Formula: $F = \dot{m} (v_{exit} - v_{inlet})$
Where:

• $\dot{m}$: Mass flow rate
• $v_{exit}$: Exit velocity
• $v_{inlet}$: Inlet velocity

Comparing Otto, Diesel, and Brayton Cycles

Cycle	Process Phases	Heat Addition	Key Application
Otto	Isentropic, Isochoric	At constant volume	Gasoline engines
Diesel	Isentropic, Isochoric, Isobaric	At constant pressure	Diesel engines
Brayton	Isentropic, Isochoric	At constant pressure	Gas turbines, jet engines

Efficiency Considerations

Otto Cycle: Efficiency depends primarily on the compression ratio (

$r$). Higher compression ratios result in better efficiency but risk knocking in real engines.

Diesel Cycle: Higher efficiency than the Otto cycle due to higher compression ratios. Cut-off ratios also impact efficiency.

Brayton Cycle: Efficiency is a function of the pressure ratio. Increasing the pressure ratio improves efficiency, but practical limits exist due to material strength and temperature constraints.

Applications of Gas Cycles

• Otto Cycle: Used in spark-ignition engines like those in cars and motorcycles.
• Diesel Cycle: Common in trucks, buses, and industrial machinery.
• Brayton Cycle: Found in jet engines, gas turbines, and power plants.
• Turbojet Cycle: Utilized in aircraft for propulsion.

Summary Table of Key Equations

Concept	Equation
Otto cycle efficiency	$\eta = 1 - \frac{1}{r^{\gamma - 1}}$
Diesel cycle efficiency	$\eta = 1 - \frac{1}{r^{\gamma - 1}} \cdot \frac{\rho^\gamma - 1}{\gamma(\rho - 1)}$
Brayton cycle efficiency	$\eta = 1 - \frac{1}{r_p^{(\gamma - 1)/\gamma}}$
Thrust	$F = \dot{m} (v_{exit} - v_{inlet})$

FAQs About Gas Cycles

Q1: What is the primary difference between the Otto and Diesel cycles?

The Otto cycle uses constant volume heat addition, while the Diesel cycle employs constant pressure heat addition, allowing for higher compression ratios and efficiency in Diesel engines.

Q2: Why is the Brayton cycle important for gas turbines?

The Brayton cycle models the thermodynamic processes in gas turbines, providing insights into efficiency and work output.

Q3: How does the turbojet cycle produce thrust?

The turbojet cycle uses high-pressure exhaust gases to accelerate through a nozzle, generating thrust based on the momentum difference.

Q4: What factors affect the efficiency of gas cycles?

Efficiency depends on parameters such as compression ratio (Otto, Diesel), pressure ratio (Brayton), and cut-off ratio (Diesel).

Q5: Can real engines achieve ideal cycle efficiency?

No, real engines are less efficient due to irreversibilities like friction, heat losses, and non-ideal fluid behavior.