# A Theoretical and Experimental Study of Moderate Temperature Alfa Type Stirling Engines

^{*}

## Abstract

**:**

## 1. Introduction

^{3}was calculated, which corresponded to the operating frequency of 75 Hz and the power of 250 W.

^{3}and higher, the electrical efficiency of 29% at 38 kW can be achieved.

## 2. Prototype Stirling Engine Type Alpha

## 3. Experimental Setup and Test Procedure

_{seni}is pressure of the working gas measured by the sensor, ΔV

_{ei}is change of volume in the expansion cylinder, ΔV

_{ci}is change of volume in the compression cylinder and i

_{max}is the number of samples (360 for assumed resolution 1 sample per 1 crankshaft rotation degree).

## 4. Engine Performance

## 5. Theoretical Model of the Stirling Engine

_{r}. Analogically, the gas coming into the heater from the regenerator will be cooler than temperature of gas in the heater by 2εT

_{r}.

_{L}is the loss coefficient, $\mathsf{\varrho}$ is the working gas density in the pipeline (Figure 1) and w is the instantaneous working gas velocity in the pipeline (Figure 1).

## 6. Results and Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

A_{g} | heater area of the heat exchange [m^{2}], |

A_{w} | cooler area of the heat exchange [m^{2}], |

c_{p} | specific heat capacity of the working fluid at constant pressure [J/(kg·K)], |

c_{v} | specific heat capacity of the working fluid at constant volume [J/(kg·K)], |

${\dot{\mathrm{E}}}_{\mathrm{x}}$ | rate of increase of the internal energy of the working gas in the space x [W], |

K_{L} | loss coefficient [-], |

m | total mass of gas in the engine [kg], |

m_{x} | mass of the working gas in the space x [kg], |

${\dot{\mathrm{m}}}_{\mathrm{x}}$ | mass accumulation speed of the working gas in the space x [kg/s], |

${\dot{\mathrm{m}}}_{\mathrm{xy}}$ | mass flow of the working gas between the spaces x and y [kg/s], |

n | rotational speed of the crankshaft [rev/s], |

p | pressure of the working gas [Pa], |

P_{ind} | indicated power of the engine [W], |

p_{sen} | pressure of the working gas measured by the sensor [Pa], |

Q_{in} | energy delivered to the Stirling engine [J], |

Q_{out} | energy loss during cooling of the Stirling engine [J], |

${\dot{\mathrm{Q}}}_{\mathrm{x}}$ | rate of heat transferred into the space x [W], |

${\overline{\dot{\mathrm{Q}}}}_{\mathrm{h}}$ | average rate of heat delivered to the working gas in the heater [W], |

${\overline{\dot{\mathrm{Q}}}}_{\mathrm{k}}$ | average rate of heat received from the working gas in the cooler [W], |

R | gas constant [J/(kg·K)], |

Re | Reynolds number [-], |

T_{x} | temperature of the working gas in the space x [K], |

T_{xy} | temperature of the working gas flowing between the spaces x and y [K], |

T_{in(out)} | temperature of the working gas coming into (out of) the analysed space [K], |

w | instantaneous working gas velocity in the pipeline [m/s] |

W | work produced by the Stirling engine [J], |

W_{ind} | indicated work of the engine [J], |

${\dot{\mathrm{W}}}_{\mathrm{x}}$ | rate of work done on the surroundings in the space x [W], |

V_{SC.c} | volume of the compression space swept capacity [m^{3}], |

V_{SC.e} | volume of the expansion space swept capacity [m^{3}], |

V_{x} | volume of the space x [m^{3}], |

## Greek Symbols

α | angle of the phase shift between the spaces of expansion and compression, |

ΔT_{r} | regenerator temperature difference, |

ΔV_{c} | change of volume in the compression cylinder, |

ΔV_{e} | change of volume in the expansion cylinder, |

ε | regenerator effectiveness, |

η^{net}_{a} | net adiabatic efficiency of the Stirling engine, |

ϕ | angle of rotation of the crankshaft shaft with respect to the cylinder of the expansion space, |

$\mathsf{\kappa}$ | isentropic exponent, |

τ_{c} | time of the cycle, |

## Subscripts

ad | adiabatic, |

c | compression space, |

ch | charge pressure, |

e | expansion space, |

g | gas supplying the heat to the engine, |

h | heater, |

k | cooler, |

pump | work pumping |

r | regenerator, |

w | cooling water |

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**Figure 2.**Test stand equipped with the prototype Stirling engine, starting motor and exhaust gas delivery system (from a spark ignition internal combustion engine).

**Figure 9.**Comparison of indicated work calculated on the basis of experiment and adiabatic models (pumping loss included) for heating temperature of 300 °C.

**Figure 10.**Comparison of indicated work calculated on the basis of experiment and adiabatic model (pumping loss included) for heating temperature of 350 °C.

**Figure 11.**Comparison of indicated work calculated on the basis of experiment and adiabatic model (pumping loss included) for heating temperature of 400 °C.

**Figure 13.**Temperatures distribution in controlled spaces for one cycle; charge pressure 6 bar; regenerator effectiveness 60%.

**Figure 14.**Temperatures distribution between controlled spaces for one cycle; charge pressure 6 bar; regenerator effectiveness 60%.

**Figure 15.**Volume–pressure diagram; charge pressure 6 bar; regenerator effectiveness 60%; heating temperature 350 °C.

**Figure 16.**Influence of charge pressure on efficiency and indicated power of the Stirling engine; heating temperature 350 °C, regenerator effectiveness 60%.

**Figure 17.**Influence of regenerator effectiveness on efficiency and indicated power of the Stirling engine; heating temperature 350 °C, charge pressure 6 bar.

**Figure 18.**Volume–pressure diagram for different values of regenerator effectiveness and heater/cooler size; heating temperature 350 °C, charge pressure 6 bar.

Parameter | Value | Unit |
---|---|---|

Working gas | air | - |

Charge pressure (bar) | 2-6 | bar |

Crankshaft rotational speed | 550 | rpm |

Compression (cold) volume: swept capacity | 730 | cm^{3} |

Pipeline (cold part) | 427 | cm^{3} |

Cooler volume | 304 | cm^{3} |

Cooler exchange area | 0.46 | m^{2} |

Number of tubes in cooler | 121 | - |

Water flow rate in cooler | 2.78 × 10^{−3} | kg/s |

Regenerator volume | 289 | cm^{3} |

Heater volume | 1140 | cm^{3} |

Heater exchange area | 1.71 | m^{2} |

Number of tubes in heater | 121 | - |

Pipeline (hot part) | 427 | cm^{3} |

Expansion (hot) volume: swept capacity | 730 | cm^{3} |

Parameter | Measurement Uncertainty | Unit |
---|---|---|

Pressure | 2 | kPa |

Temperature | 1.5 | °C |

Position | 0.03 | degree |

Rotational speed | 2 | rpm |

**Table 3.**Operating parameters of the Stirling engine for different values of regenerator effectiveness and heater/cooler size.

Heater/Cooler Construction | Heater Exchange Area [m^{2}] | Cooler Exchange Area [m^{2}] | Regenerator Effectiveness [%] | Indicated Efficiency [%] | Indicated Power [W] |
---|---|---|---|---|---|

prototype (original, 4 mm tubes) | 1.71 | 0.46 | 60% | 5.5 | 114 |

Optimized (4 mm tubes) | 1.45 | 1.07 | 60% | 6.8 | 144 |

Optimized (1 mm tubes) | 2.25 | 1.53 | 90% | 19.5 | 369 |

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**MDPI and ACS Style**

Kropiwnicki, J.; Furmanek, M.
A Theoretical and Experimental Study of Moderate Temperature Alfa Type Stirling Engines. *Energies* **2020**, *13*, 1622.
https://doi.org/10.3390/en13071622

**AMA Style**

Kropiwnicki J, Furmanek M.
A Theoretical and Experimental Study of Moderate Temperature Alfa Type Stirling Engines. *Energies*. 2020; 13(7):1622.
https://doi.org/10.3390/en13071622

**Chicago/Turabian Style**

Kropiwnicki, Jacek, and Mariusz Furmanek.
2020. "A Theoretical and Experimental Study of Moderate Temperature Alfa Type Stirling Engines" *Energies* 13, no. 7: 1622.
https://doi.org/10.3390/en13071622