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Home/Thermodynamics/Stirling Cycle

Stirling Cycle

The Stirling cycle simulator visualizes the thermodynamic processes of an ideal Stirling engine, a closed-cycle regenerative heat engine. It models the engine's operation as a sequence of four reversible processes applied to a fixed mass of an ideal gas: two isothermal (constant temperature) and two isochoric (constant volume) transformations. The cycle begins with an isothermal compression (process 1→2), where the gas is compressed at a low temperature, T_C, rejecting heat to a cold reservoir. This is followed by an isochoric heating (2→3), where volume is held constant and temperature increases from T_C to T_H via an internal regenerator. Next, an isothermal expansion (3→4) occurs at the high temperature, T_H, where the gas absorbs heat from a hot reservoir and does work. The cycle closes with an isochoric cooling (4→1), where volume is again constant as the gas cools from T_H back to T_C, transferring its heat to the regenerator for reuse. The simulator plots these processes on a Pressure-Volume (PV) diagram, allowing users to see the enclosed area representing the net work output per cycle. The thermal efficiency for an ideal Stirling cycle with a perfect regenerator is η = 1 - (T_C / T_H), identical to the Carnot efficiency. This model simplifies real engines by assuming reversible processes, an ideal gas, instantaneous heat transfer, and a 100% effective regenerator that stores and releases heat internally without loss. By manipulating parameters like reservoir temperatures and initial gas volume, students can explore how these variables affect the PV diagram shape, net work, and efficiency, reinforcing concepts from the first and second laws of thermodynamics.

Who it's for: Undergraduate engineering and physics students studying thermodynamics, particularly in courses covering heat engines, reversible cycles, and the limitations of real engines.

Key terms

  • Stirling Cycle
  • Isothermal Process
  • Isochoric Process
  • Thermodynamic Efficiency
  • Pressure-Volume Diagram
  • Ideal Gas Law
  • Heat Regenerator
  • Carnot Efficiency

Cycle

1.25
3.1
420 K
280 K
0.35×

Two isotherms and two isochores. With a perfect regenerator storing heat during the constant-volume steps, the ideal Stirling efficiency equals the Carnot limit η = 1 − T_C/T_H. Net work per cycle (nR=1 model): W = (T_H − T_C) ln(V₂/V₁).

Measured values

η (ideal, regenerator)33.3%
W per cycle (model)127.16
V₂/V₁2.48

How it works

The Stirling engine cycle alternates isothermal expansion and compression at two reservoirs with constant-volume transfer between temperatures. It is a reversible idealization; real machines have losses, but the diagram shows why regeneration makes the efficiency approach Carnot.

Key equations

η_Stirling = 1 − T_C/T_H (ideal regenerator) · W = nR(T_H − T_C) ln(V₂/V₁)

Frequently asked questions

Why is the Stirling cycle's ideal efficiency equal to the Carnot efficiency?
The equality arises from the perfect regenerator. In the ideal model, the heat absorbed during the constant-volume heating process (Q_in,reg) is exactly equal to the heat rejected during the constant-volume cooling process (Q_out,reg). These internal heat transfers cancel out in the efficiency calculation. Therefore, the only net heat input comes from the high-temperature isothermal expansion, and the only net heat rejection is from the low-temperature isothermal compression, creating the same heat exchange conditions as a Carnot cycle: η = 1 - Q_c / Q_h = 1 - T_C / T_H.
What is the purpose of the regenerator in a Stirling engine?
The regenerator is a critical internal component, often a matrix of metal wire, that acts as a temporary thermal storage device. During the constant-volume cooling stroke, it absorbs heat from the hot gas, cooling it efficiently. During the subsequent constant-volume heating stroke, it returns that stored heat to the now-cold gas. This recycling of thermal energy within the engine dramatically improves real-world efficiency by reducing the amount of external heat that must be supplied and rejected in each cycle.
How does this ideal model differ from a real Stirling engine?
Real engines deviate due to irreversibilities. Friction causes pressure losses, finite heat transfer rates require temperature differences (making the isotherms imperfect), and the regenerator is never 100% effective, leading to some heat loss. Furthermore, real gas properties and mechanical dead volume (space not swept by the piston) reduce the usable pressure swing and work output. The simulator's ideal cycle provides an upper-bound benchmark against which practical engine performance can be compared.
Can the Stirling cycle run in reverse? What would it be?
Yes. Running the cycle in reverse—following the PV diagram counterclockwise—creates a Stirling refrigerator or heat pump. In this mode, net work is input to the system. The isothermal expansion now occurs at the cold temperature, absorbing heat from a refrigerated space, and the isothermal compression occurs at a higher temperature, rejecting heat to the surroundings. The regenerator again improves performance by internally transferring heat between the two constant-volume processes.