Introduction to Air-Standard Cycles
0:0011:19
Engineering

Air Standard Cycles in Thermal Engineering

A comprehensive guide to air-standard cycles, covering thermodynamic classification, ideal assumptions, and the mechanics of Otto, Diesel, and Dual cycles.

May 4, 202618 min listen5 chapters
What you'll learn
  • Understand the core assumptions of air-standard thermodynamic models
  • Differentiate between power, gas, and combustion cycles
  • Analyze the four stages of Otto, Diesel, and Dual cycles
  • Compare thermal efficiency based on compression and cut-off ratios

Introduction to Air-Standard Cycles

note

Air-standard cycle

A simplified thermodynamic model of an engine cycle where:

  • the working fluid is treated as air
  • air behaves as an ideal gas
  • the cycle is analyzed as closed and reversible in the idealized sense

Used to study real engines without all the messy details.

diagram
note

Thermodynamic cycle classification

1) By purpose

  • Power cycles: produce net work
  • Refrigeration / heat pump cycles: consume work to move heat

2) By working fluid

  • Gas cycles: working fluid remains a gas
  • Vapor cycles: working fluid undergoes phase change
diagram
equation
Wnet=Wout−WinW_{net} = W_{out} - W_{in}

System Boundaries and Assumptions

note

Open vs Closed cycle

  • Closed cycle: same mass of working fluid remains inside the system
  • Open cycle: mass enters and leaves the system boundary

Key idea

  • Closed → analyze as a fixed mass system
  • Open → analyze as a control volume
diagram
note

Air-standard assumptions

  1. The working fluid is air.
  2. Air behaves as an ideal/perfect gas.
  3. The cycle is internally reversible.
  4. Combustion is replaced by an external heat-addition process.
  5. Exhaust is replaced by an external heat-rejection process.

These assumptions make the cycle easy to analyze.

equation
PV=nRTPV = nRT
note

Perfect gas / ideal gas

For air-standard analysis, perfect gas usually means:

  • equation of state: (PV = nRT)
  • properties depend mainly on temperature
  • specific heats are often treated as constant in the simplest model
equation
PV=mRTPV = mRT

The Carnot Cycle Reference

illustration
P-V diagram of a Carnot cycle labeled 1-2-3-4 with the four stages: 1-2 isothermal expansion, 2-3 adiabatic expansion, 3-4 isothermal compression, 4-1 adiabatic compression. Axes labeled Pressure P and Volume V, closed loop clearly marked, stage arrows and labels visible.
note

Carnot cycle stages

  1. Isothermal expansion at high temperature (T_H)
  2. Adiabatic expansion
  3. Isothermal compression at low temperature (T_L)
  4. Adiabatic compression

Meaning

  • Isothermal: heat transfer occurs to keep temperature constant
  • Adiabatic: no heat transfer
equation
ΔU=0for an isothermal ideal-gas process\Delta U = 0 \quad \text{for an isothermal ideal-gas process}
note

Carnot cycle: four steps in detail

1) Isothermal expansion at (T_H)

  • Gas expands
  • It does work on the surroundings
  • Heat enters to keep temperature constant

2) Adiabatic expansion

  • Gas keeps expanding
  • No heat transfer
  • Temperature drops from (T_H) to (T_L)

3) Isothermal compression at (T_L)

  • Surroundings compress the gas
  • Heat leaves the gas
  • Temperature stays constant

4) Adiabatic compression

  • Gas is compressed with no heat transfer
  • Temperature rises from (T_L) back to (T_H)

Result

  • The cycle returns to its initial state
  • Net work equals the area enclosed by the P–V loop
equation
ΔU=Q−W\Delta U = Q - W
note

State-point notation

  • (U_1, U_2, U_3, U_4): internal energy at states 1, 2, 3, 4
  • (P_1, P_2, P_3, P_4): pressure at states 1, 2, 3, 4
  • (T_1, T_2, T_3, T_4): temperature at states 1, 2, 3, 4
  • (W_t): total work output of the cycle

Otto Cycle Analysis

note

Ideal cycle processes on a P–V diagram

Common process types

  • Isothermal: temperature constant
  • Adiabatic: no heat transfer
  • Isobaric: pressure constant
  • Isochoric: volume constant

For many air-standard cycles

  • Compression is often adiabatic
  • Expansion is often adiabatic
  • Heat addition/rejection can be isochoric or isobaric depending on the cycle
diagram
equation
Q=ΔU+WQ = \Delta U + W
diagram
equation
ηOtto=1−1rγ−1\eta_{Otto} = 1 - \frac{1}{r^{\gamma - 1}}
diagram
note

Otto cycle on a P–V diagram

  • 1→2: compression curve rises as volume decreases
  • 2→3: vertical line up, because volume is constant
  • 3→4: expansion curve falls as volume increases
  • 4→1: vertical line down, because volume is constant
equation
η=WnetQin=1−QoutQin\eta = \frac{W_{net}}{Q_{in}} = 1 - \frac{Q_{out}}{Q_{in}}
equation
ηOtto=1−T4−T1T3−T2\eta_{Otto} = 1 - \frac{T_4 - T_1}{T_3 - T_2}
equation
T2T1=T3T4=rγ−1\frac{T_2}{T_1} = \frac{T_3}{T_4} = r^{\gamma - 1}
equation
ηOtto=1−1rγ−1\eta_{Otto} = 1 - \frac{1}{r^{\gamma - 1}}

Diesel and Dual Cycle Comparison

diagram
equation
ηDiesel=1−1rγ−1 ργ−1γ(ρ−1)\eta_{Diesel} = 1 - \frac{1}{r^{\gamma-1}}\,\frac{\rho^{\gamma}-1}{\gamma(\rho-1)}
note

Diesel cycle key ratios

  • Compression ratio: (r = \dfrac{V_1}{V_2})
  • Cut-off ratio: (\rho = \dfrac{V_3}{V_2})
  • (\gamma = \dfrac{c_p}{c_v})
illustration
T-V diagram of the ideal Diesel cycle with states 1-2-3-4 labeled. Show 1→2 isentropic compression curve, 2→3 constant-pressure heat addition curve rising to the right, 3→4 isentropic expansion curve, and 4→1 constant-volume heat rejection vertical line. Axes labeled Temperature T and Volume V. Clear arrows and state points visible.
diagram
note

Diesel cycle on a P–V diagram

  • 1→2: compression curve goes up-left
  • 2→3: constant-pressure line goes to the right
  • 3→4: expansion curve goes down-right
  • 4→1: constant-volume line goes straight down
illustration
P-V diagram of the ideal Diesel cycle with Pressure P on the x-axis and Volume V on the y-axis. Show states 1-2-3-4 and arrows: 1→2 isentropic compression curve, 2→3 constant-pressure heat addition horizontal line, 3→4 isentropic expansion curve, 4→1 constant-volume heat rejection vertical line. Clear labels and axes visible.
note

Diesel cycle working

  1. Compression stroke (1→2)

    • Air is compressed strongly
    • Pressure and temperature rise
    • Idealized as isentropic

  2. Heat addition at constant pressure (2→3)

    • Fuel is injected and burns
    • Pressure stays constant
    • Volume increases
    • This is the cut-off period

  3. Expansion stroke (3→4)

    • Hot gases expand and do work on the piston
    • Pressure and temperature fall
    • Idealized as isentropic

  4. Heat rejection at constant volume (4→1)

    • Exhaust heat is rejected in the ideal model
    • Volume stays fixed
    • Pressure drops back to the start state
equation
r=V1V2,ρ=V3V2r = \frac{V_1}{V_2}, \qquad \rho = \frac{V_3}{V_2}
diagram
note

Dual cycle

  • 1→2: isentropic compression
  • 2→3: constant-volume heat addition
  • 3→4: constant-pressure heat addition
  • 4→5: isentropic expansion
  • 5→1: constant-volume heat rejection
equation
ηDual=1−QoutQin\eta_{Dual} = 1 - \frac{Q_{out}}{Q_{in}}
equation
Qin=mcv(T3−T2)+mcp(T4−T3)Q_{in} = m c_v (T_3 - T_2) + m c_p (T_4 - T_3)
equation
Qout=mcv(T5−T1)Q_{out} = m c_v (T_5 - T_1)
note

Otto vs Diesel vs Dual

CycleHeat additionCompressionExpansion
OttoConstant volumeIsentropicIsentropic
DieselConstant pressureIsentropicIsentropic
DualBoth constant volume and constant pressureIsentropicIsentropic

Exam cue

  • Otto: spark ignition
  • Diesel: compression ignition
  • Dual: mixed ideal model
diagram
equation
ηOtto=1−1rγ−1\eta_{Otto} = 1 - \frac{1}{r^{\gamma-1}}
equation
ηDiesel=1−1rγ−1 ργ−1γ(ρ−1)\eta_{Diesel} = 1 - \frac{1}{r^{\gamma-1}}\,\frac{\rho^{\gamma}-1}{\gamma(\rho-1)}
note

Efficiency trend

  • Higher compression ratio generally means higher efficiency
  • For the same compression ratio, Otto is usually more efficient than Diesel
  • Dual lies between Otto and Diesel depending on how heat is split
diagram
note

Same compression ratio, same heat input

  • Otto: heat added at constant volume → larger pressure rise
  • Diesel: heat added at constant pressure → volume rises during heat addition
  • Therefore, for the same (r) and same (Q_{in}), Otto usually has higher thermal efficiency
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