_{1}

The reported discrepancy between theory and experiment for external combustion Stirling engines is explained by the addition of thermal resistance of the combustion gasses to the standard Carnot model. In these cases, the Stirling engine ideal efficiency is not as is normally reported equal to the Carnot cycle efficiency but is significantly lower. A new equation for ideal Stirling engine efficiency when the heat is obtained through external combustion without pre-heating the air, is presented and results for various fuels tabulated. The results show that petrol and diesel, internal combustion engines (Otto cycle) have a higher ideal efficiency than the Stirling engine. When comparing thermoacoustic engines heated by wood, efficiency should not be quoted as a percentage of the Carnot efficiency, but against a figure 48% lower than Carnot. The effect is not seen with electrically heated rigs, solar or nuclear fission heated engines.

The maximum theoretical efficiency of the Stirling cycle is often quoted as the same as the Carnot efficiency [

The rationale behind writing this paper is to provide a new upper limit to the efficiency of external combustion Stirling engines and hence an explanation for the often reported low measured efficiency of Stirling engines in real-world external-combustion situations, compared with theoretical or rig results. Hopefully researchers and lecturers will include this paper’s finding in their teaching of Stirling engines.

There are many references to a discrepancy between theoretical and experimental efficiency with the Stirling Cycle. Thermoacoustic rig test results using electrical heating on the hot heat exchanger have shown impressive thermal efficiencies of over 18% [

However, work on a swash plate V4X2 Stirling engine for automotive use [

Other automotive applications such as the Philips engine [

More recently, modelling of an alpha Stirling engine for automotive [

The apparent discrepancy between theory and experiment has also been attributed to pressure gradients in the thermoacoustic regenerator, and much work undertaken to mathematically model regenerator performance [

By modelling the entire system, including combustion, this paper presents a more accurate upper bound for external combustion Stirling engine cycle efficiency and better explains the discrepancies. When heating of the hot heat exchanger is by external combustion, efficiency is much less than the Carnot efficiency even in the idealised case.

In this paper, we have augmented the maximum theoretical Carnot efficiency for specific fuel types using the model in

In the ideal case, with perfect heat exchangers:

It should be noted that Rθ_{c} is a function of the combustion_{ }gas itself and is independent of the heat transfer mechanism, i.e. it applies equally to radiation, conduction or convection mechanisms and where:

The flame at a temperature T_{f} passes heat through the thermal resistances to the surrounding temperature T_{a}. So that heat available from combustion is

And heat supplied to the Stirling Engine (SE) is:

The difference between

Inspecting Equations (5) and (1) show that when

For the general case where the fuel may contain water (for example wood), mass flow of the combustion products is the sum of its constituents:

and letting

The specific heat of the combustion products is a function of fuel, air and fuel moisture content given by:

The heat available from combustion is reduced due to the evaporation of water content in the fuel, so:

Assuming no dissociation of component gasses, the idealised flame temperature is:

Making substitutions from (6) and dividing by fuel mass flow rate gives:

Maximum work output from the Stirling engine is:

Defining overall efficiency as

And substituting from Equations (4), (5) with

gives:

Differentiating and setting:

Gives the maximum efficiency condition when:

The results for different fuels, using Equations (11), (15) and (17) are shown on

The column η_{SE}, shows the idealised efficiency of an external combustion Stirling engine assuming perfect heat exchangers for the hot and ambient side.

η_{SE} is the maximum theoretical efficiency of the fuels shown and can be seen that it varies between 39.3% for wood to 53.3% for hydrogen, which represents a reduction from the normally quoted maximum (Carnot) SE efficiency (shown as “Ratio” on the table) of 48.2% and 58.8% respectively.

All the conditions are for use with air as the oxidising agent at standard pressure and without any pre-heating.

Wood combustion particularly when using cut branches is highly variable and heat produced depends on how dry the wood is and how well the fire is made. The values of 50% excess air (Xs = 1.5) and 15% moisture content are considered the best achievable conditions.

The Diesel figures show a 50% excess air value to compare with the Otto cycle discussed in the next section. Without the excess air, results are very similar to the Petrol figures.

Fuel | Efficiency result | |||||||
---|---|---|---|---|---|---|---|---|

η_{SE} | Carnot | Ratio | X_{s} | A_{f} | C_{vf} | C_{pf} | R_{w} | |

Wood | 39.3% | 81.5% | 48.2% | 1.5 | 6 | 15,000 | 1.8 | 15.0% |

Petrol | 51.7% | 89.8% | 57.6% | 1 | 14.7 | 45,000 | 2.22 | 0.0% |

Diesel | 46.2% | 86.5% | 53.4% | 1.5 | 14.5 | 46,000 | 2.22 | 0.0% |

Methane | 52.9% | 90.5% | 58.5% | 1 | 17.19 | 55,500 | 2.22 | 0.0% |

Propane | 53.1% | 90.6% | 58.6% | 1 | 15.67 | 50,300 | 1.67 | 0.0% |

Methanol | 52.2% | 90.1% | 57.9% | 1 | 6.47 | 23,000 | 1.93 | 0.0% |

Ethanol | 52.3% | 90.2% | 58.0% | 1 | 9 | 30,000 | 1.88 | 0.0% |

Hydrogen | 53.3% | 90.7% | 58.8 | 1 | 34 | 141,790 | 14.3 | 0.0% |

Fuel type | Efficiency for given compression ratio | ||
---|---|---|---|

η_{SE} | Otto cycle | r | |

Petrol | 51.7% | 60.2% | 10 |

Diesel | 46.2% | 69.8% | 20 |

The maximum efficiency of an ideal Otto cycle engine, is given by[

From

The results in the previous section go some way to explaining the discrepancy between the observed theoretical and experimental results for external combustion. However, there are cases where Equation (1) does hold. Many Stirling or thermoacoustic engine experiments use electrical heating to simulate the external combustion process. In such cases heat transfer from the electrical heater to the HHX is by a combination of radiation, convection and conduction, with the proportion of each depending on relative temperatures. The temperature of the heating coil will be higher than T_{e} so as to transfer the heat. However, this temperature should not be confused with T_{f,}_{ }which is a fuel flame temperature or Rθ_{c} which is thermal resistance based on combustion products. The difference in perspective arises from heat losses. With electrical heating, coil temperature increases do not increase losses, as the heat input is dependent on the product of coil voltage and current. There is still an Rθ value but this is a method of determining coil temperature rise and does not directly affect losses. There may be other secondary and parasitic losses due to ineffective insulation, where increased coil temperature increases heat to the environment that bypasses the engine, but in the ideal case these losses are zero.

Equation (1) is still valid for any heat source that retains its temperature when the SE absorbs heat. One typical example is solar radiation, where the temperature of the sun’s surface does not change no matter how much power earth-based Stirling engines absorb. Another is in heating with nuclear fission, where the concept of T_{f} is not valid.

The ideal Stirling cycle for external combustion engines that use non-pre-heated air as the oxidiser is always significantly lower than the equivalent Carnot cycle engine and for commonly used fuels is lower than the ideal Otto cycle efficiency.

The η_{SE} measure gives a more accurate representation of the idealised efficiency of the external combustion Stirling engine cycle.

When modelling the efficiency of external combustion Stirling cycles, the effective thermal resistance of the combustion products should be included in the calculation.

The ideal Stirling engine gives lower efficiency for automotive applications than conventional Otto cycle engines.

This work was sponsored by EPSRC grant EP/J013986/1.

Paul H.Riley, (2015) The Myth of the High-Efficiency External-Combustion Stirling Engine. Engineering,07,789-795. doi: 10.4236/eng.2015.712068