9.3 Stirling and Ericsson cycles

Stirling and Ericsson Cycles

The Stirling and Ericsson cycles are closed gas power cycles that achieve the highest possible theoretical efficiency for a heat engine operating between two temperature reservoirs. They accomplish this by combining isothermal heat transfer processes with a regenerator, a device that internally recycles heat within the cycle. Understanding how they work, and why they differ from Otto and Diesel cycles, is a core part of analyzing gas power cycles.

Both cycles share a key trait: all heat addition and rejection occur isothermally (at constant temperature). This is what allows their ideal efficiency to match the Carnot efficiency. The difference between them lies in how the regenerator transfers heat: at constant volume (Stirling) or at constant pressure (Ericsson).

Stirling and Ericsson Cycle Fundamentals

Working Principle

Both cycles use a working fluid (typically air, helium, or hydrogen) that stays sealed inside the system. Because the cycle is closed, the combustion process is external and physically separated from the working fluid. This means you can use almost any heat source: fossil fuels, solar concentrators, geothermal energy, or industrial waste heat.

The regenerator is the component that makes these cycles special. It acts as a thermal energy storage device sitting between the hot and cold sides of the engine. During one part of the cycle, the working fluid passes through the regenerator and deposits heat into it. During another part, the fluid passes back through and picks that heat up again. This internal heat exchange reduces how much external heat you need to add, which raises the thermal efficiency.

Engine Components

Stirling engine: cylinder, power piston, displacer piston, regenerator, and two heat exchangers (one for heat addition, one for rejection). The displacer piston shuttles the working fluid between the hot and cold spaces through the regenerator.

Ericsson engine: cylinder, piston, compressor, regenerator, and two heat exchangers. The Ericsson engine is often implemented with a turbine-compressor arrangement rather than reciprocating pistons, since its constant-pressure processes lend themselves to steady-flow machinery.

Thermodynamic Processes in Each Cycle

Stirling Cycle (Two Isothermal + Two Isochoric Processes)

1. Isothermal compression (Process 1→2): The working fluid is compressed at constant low temperature $T_C$. Heat is rejected to the cold reservoir during compression to keep the temperature constant.
2. Isochoric (constant-volume) heat addition (Process 2→3): The fluid passes through the regenerator at constant volume, absorbing stored heat. Its temperature rises from $T_C$ to $T_H$. No work is done because volume doesn't change.
3. Isothermal expansion (Process 3→4): The fluid expands at constant high temperature $T_H$, producing work. Heat is absorbed from the external hot source to maintain the temperature.
4. Isochoric (constant-volume) heat rejection (Process 4→1): The fluid passes back through the regenerator at constant volume, depositing heat for later reuse. Its temperature drops from $T_H$ back to $T_C$.

The regenerator handles the heat transfer in processes 2→3 and 4→1. In an ideal cycle, the heat stored during 4→1 exactly equals the heat retrieved during 2→3, so these two processes have no net effect on the cycle's external heat exchange.

Ericsson Cycle (Two Isothermal + Two Isobaric Processes)

1. Isothermal compression (Process 1→2): The working fluid is compressed at constant low temperature $T_C$, with heat rejected to the cold reservoir.
2. Isobaric (constant-pressure) heat addition (Process 2→3): The fluid flows through the regenerator at constant pressure, absorbing stored heat. Temperature rises from $T_C$ to $T_H$.
3. Isothermal expansion (Process 3→4): The fluid expands at constant high temperature $T_H$, producing work while absorbing heat from the external source.
4. Isobaric (constant-pressure) heat rejection (Process 4→1): The fluid passes back through the regenerator at constant pressure, depositing heat. Temperature drops from $T_H$ to $T_C$.

The same regenerator logic applies: in an ideal Ericsson cycle, the heat exchanged in the two isobaric processes cancels internally, leaving only the two isothermal processes to interact with the external reservoirs.

Comparison with Other Gas Power Cycles

Advantages of Stirling and Ericsson Cycles

• Carnot-equivalent ideal efficiency. Because all external heat transfer is isothermal, the ideal thermal efficiency equals the Carnot efficiency. Otto and Diesel cycles cannot achieve this, since their heat addition occurs over a range of temperatures.
• Fuel flexibility. The closed-cycle, external-combustion design means any heat source works. This is a significant advantage for renewable energy applications.
• Low noise and vibration. There's no explosive internal combustion, so these engines run much more quietly than Otto or Diesel engines.
• Lower emissions. External combustion can be controlled more precisely, and the working fluid never contacts combustion products.

Disadvantages

• Low power-to-weight ratio. Heat must transfer through engine walls rather than being released directly into the working fluid (as in internal combustion). This limits how compact and lightweight the engine can be, which is why you rarely see Stirling engines in cars or aircraft.
• Slow transient response. Changing the power output requires changing heat transfer rates, which is inherently slower than adjusting fuel injection in an internal combustion engine.
• Complexity and cost. The regenerator, sealed working fluid system, and multiple heat exchangers add manufacturing complexity compared to simpler Otto or Diesel engines.

Efficiency and Work Output

Thermal Efficiency

Thermal efficiency is defined as:

$\eta_{th} = \frac{W_{net}}{Q_{in}}$

For an ideal Stirling or Ericsson cycle with a perfect regenerator, the only external heat interactions are the two isothermal processes. This gives:

$\eta_{th} = 1 - \frac{T_C}{T_H}$

where $T_C$ is the absolute temperature of the cold reservoir and $T_H$ is the absolute temperature of the hot reservoir (both in Kelvin or Rankine). This is identical to the Carnot efficiency.

Why does this work? The regenerator internally handles all the heat transfer that would otherwise require external input at intermediate temperatures. The cycle only "sees" two thermal reservoirs, just like a Carnot cycle.

In practice, real Stirling and Ericsson engines fall short of this ideal due to:

• Imperfect regeneration (the regenerator can't store and return 100% of the heat)
• Friction and pressure drops in the fluid passages
• Finite temperature differences needed to drive heat transfer at practical rates

Work Output

The net work per cycle is the difference between expansion work and compression work:

$W_{net} = W_{exp} - W_{comp}$

For the Stirling cycle, the isothermal work expressions (assuming an ideal gas) are:

$W_{exp} = mRT_H \ln\left(\frac{V_4}{V_3}\right)$

$W_{comp} = mRT_C \ln\left(\frac{V_1}{V_2}\right)$

The isochoric processes involve no work ($\Delta V = 0$), so they don't contribute.

For the Ericsson cycle, the isothermal work expressions are similar, but the constant-pressure processes also involve boundary work. In the ideal case with a perfect regenerator, the net isobaric work contributions cancel, and the net work again comes from the difference between the two isothermal processes.

Power output is then:

$\dot{W} = W_{net} \times N$

where $N$ is the number of cycles completed per unit time (cycles per second).