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Edited by: Guillermo Rein, Imperial College London, United Kingdom

Reviewed by: Alexander S. Rattner, Pennsylvania State University, United States; Wei Tang, University of Maryland, College Park, United States

Specialty section: This article was submitted to Thermal and Mass Transport, a section of the journal Frontiers in Mechanical Engineering

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

This paper presents the exergy analysis and optimization of the Stirling engine, which has enormous potential for use in the renewable energy industry as it is quiet, efficient, and can operate with a variety of different heat sources and, therefore, has multi-fuel capabilities. This work aims to present a method that can be used by a Stirling engine designer to quickly and efficiently find near-optimal or optimal Stirling engine geometry and operating conditions. The model applies the exergy analysis methodology to the ideal-adiabatic Stirling engine model. In the past, this analysis technique has only been applied to highly idealized Stirling cycle models and this study shows its use in the realm of Stirling cycle optimization when applied to a more complex model. The implicit filtering optimization algorithm is used to optimize the engine as it quickly and efficiently computes the optimal geometry and operating frequency that gives maximum net-work output at a fixed energy input. A numerical example of a 1,000 cm^{3} engine is presented, where the geometry and operating frequency of the engine are optimized for four different regenerator mesh types, varying heater inlet temperature and a fixed energy input of 15 kW. The WN200 mesh is seen to perform best of the four mesh types analyzed, giving the greatest net-work output and efficiency. The optimal values of several different engine parameters are presented in the work. It is shown that the net-work output and efficiency increase with increasing heater inlet temperature. The optimal dead-volume ratio, swept volume ratio, operating frequency, and phase angle are all shown to decrease with increasing heater inlet temperature. In terms of the heat exchanger geometry, the heater and cooler tubes are seen to decrease in size and the cooler and heater effectiveness is seen to decrease with increasing heater temperature, whereas the regenerator is seen to increase in size and effectiveness. In terms of the regenerator mesh type, it was found that the WN200 mesh gave a shorter regenerator with greater cross-sectional flow area which gave a smaller pressure drop.

The Stirling engine was invented by Rev. Robert Stirling some 200 years ago, at the time the engine received some attention and saw commercial use (Stirling,

There are a variety of different approaches to Stirling engine modeling and there exist several different orders of models (Dyson et al.,

There have been several studies that have applied the exergy analysis methodology to ideal Stirling cycle models. The study conducted by Martaj et al. (

There have been many studies that have optimized power cycles using exergy analysis methodology. Where exergy is defined as the energy that is available to do work. The Gouy–Stodola theorem, which describes the relationship between reversible work _{0} (Bejan,

The development of this equation was a major advancement in the thermodynamics of the time, and the expression shows that the rate of entropy generation is directly proportional to the rate at which work is destroyed. While using this methodology, it has been emphasized that it is crucial to optimize the system in its entirety, rather than as individual components (Bejan,

This paper presents a novel approach to modeling the losses and optimizing the alpha type Stirling engine, which involves the application of exergy analysis methodology to the ideal adiabatic model of the Stirling cycle. The model incorporates the irreversibility due to heat transfer through a finite temperature difference, pressure drops and conductive thermal bridging loss. The model presented is used with the implicit filtering algorithm to optimize a 1,000 cm^{3} Stirling engine for maximum power production with four different regenerator mesh types and a fixed energy input. In the analysis, the working fluid is assumed to be an ideal gas and a finite heat capacity rate is assumed in the heater and cooler, the number of heater and cooler tubes are also fixed.

The methodology used to optimize the 1,000 cm^{3} alpha type Stirling engine for maximum work output with a fixed energy input is presented in this section. The working fluid is assumed to be pressurized air which behaves as an ideal gas. The heater and cooler external fluids are assumed to have finite heat capacity rates, and four different regenerator mesh types are used in this analysis.

A diagram of the alpha type Stirling engine used in the analysis can be seen as Figure

Diagram of the alpha type Stirling engine.

The expressions for the compression and expansion space volumes are Eqs _{c} using the clearance volume _{ccl}, swept volume _{c,swept}, and the crank angle θ.

Similarly, Eq. _{ecl}, swept volume _{e,swept}, crank angle θ, and phase difference α.

Equations are required to determine the volumes from the geometric variables for the heat exchangers. Equations

Equations _{k}_{k}_{k}_{k}_{k}

Equations _{h}_{h}_{h}_{h}_{h}

Equations _{r}_{r}_{r}_{r}_{hyd}.

Figure

Table of fixed parameters.

Symbol | Description | Value | Units |
---|---|---|---|

Input energy | 15 | kW | |

_{total} |
Total engine volume | 1,000 | cm^{3} |

_{K}_{1} |
Coolant inlet temperature | 298 | K |

_{h} |
Number of cooler tubes | 80 | – |

_{k} |
Number of heater tubes | 240 | – |

Mass of working fluid | 50 | g | |

_{h} |
Heater fluid heat capacity rate | 1 | kW K^{−1} |

_{k} |
Cooler fluid heat capacity rate | 1 | kW K^{−1} |

_{r,}_{flow}_{HE}_{,}_{flow} |
Heat exchanger flow area ratio | 8 | – |

_{max} |
Maximum total heat exchanger length | 30 | cm |

_{cond} |
Regenerator thermal conductivity | 0.05 | kW m^{−1} K^{−1} |

Along with these fixed parameters, four different mesh types are used in the optimization. The mesh types and their properties can be seen in Table

Table of regenerator mesh dimensions (Tanaka et al.,

Symbol | Diameter (mm) | Porosity (−) |
---|---|---|

WN50 | 0.23 | 0.645 |

WN100 | 0.1 | 0.711 |

WN150 | 0.06 | 0.754 |

WN200 | 0.05 | 0.729 |

The following section presents and describes the equations used to model the alpha type Stirling engine which is optimized in this study. First, the model and the ideal adiabatic model of the Stirling cycle are presented. Following this the equations that describe the heat transfer, flow friction and thermal bridging loss are presented and explained. Finally, the exergy and rate of entropy generation equations are introduced and the method of solution is described.

The model outlined assumes finite heat capacity rates in the heater and cooler. The compartment temperature diagram shows the different thermodynamic properties and temperature in each compartment, seen as Figure

Serially connected component and temperature diagram of the Stirling cycle.

Figure _{p}_{V}

The ideal adiabatic model was developed by Urieli and Berchowitz as a means of more accurately modeling the real Stirling cycle. At the time of the development of these models the iterative schemes took too long to solve, to make the model useful in the optimization of Stirling engine geometry. However, due to advances in computing and better models the solutions are arrived at in seconds rather than minutes, making these numerical models suitable for optimization purposes.

The full derivation of the equations is not presented but the equations are listed and briefly explained. To see the complete derivation of the equations, see the book by Urieli and Berchowitz (Berchowitz and Urieli,

The ideal adiabatic model assumes that there is negligible pressure variation throughout the engine. Therefore, Eq.

Assuming that the total mass of working fluid in the device is the sum of the masses of working fluid in each component yields Eq.

Assuming, the mass of working fluid remains constant, yields Eq.

For the cooler, regenerator and heater the volume and temperature are assumed to be constant. Therefore, the mass differential is defined as Eqs

Substituting Eqs

Applying the mathematical expression for the first law to a generalized cell of working space yields Eq.

Rearranging Eq.

Substituting Eqs

Defining the temperature differentials in the compression and expansion spaces, yields Eqs

The mass flows through the compartment boundaries are defined as Eqs

The conditional temperatures which depend on the direction of fluid flow in the heater and cooler are Eqs

The energy equations that describe the heat absorbed and rejected in the cooler, regenerator, and heater are Eqs

The energy equations which describe the work output of the cycle are Eqs

When calculating the pressure drop Δ_{r}_{D}

Equation _{flow} and hydraulic diameter _{hyd} are used.

Equation _{D}_{max} (Tanaka et al.,

In the case of the cooler and the heater unidirectional smooth pipe flow relations are used to calculate the Darcy friction factor _{D}

Equations

Equation

Assuming, the overall heat transfer coefficient

This is used to compute the NTU in the heater and cooler, which in turn is used to compute the effectiveness

The conductive thermal bridging loss _{cond}, the conduction area _{cond}, the regenerator length _{r}_{h}_{k}

Defining the exergy of the engine yields Eq.

Defining the second law mathematically in terms of crank angle θ, yields Eq.

Therefore, defining the entropy generation per cycle _{gen} and assuming the mass flow

Defining the difference in entropy between the entering and leaving gas _{out} − _{in} yields Eq.

Substituting Eq.

Therefore, the rate of entropy generation

Since the flow is reversing the entropy generation due to pressure drop can be written as Eq.

The equation for rate of entropy generation in the heater

The equation for rate of entropy generation in the cooler

Using the definition of effectiveness

Therefore, the equation for the rate of entropy generation in the regenerator

Assuming the rate of entropy generation in the expansion and compression spaces is negligible compared to the entropy generation rate in the heat exchangers

Defining the objective function by substituting Eqs

Equation

The solution is obtained using three iterative loops. The outer loop computes the energy input; the middle loop computes the temperature difference for adequate heat transfer in the cooler, and the inner loop computes the solution to the ideal adiabatic model equations. The middle loop takes the values for mass flow rate and energy from the inner loop, and uses these to compute the gas temperature in the heater and the cooler. This value is then compared to the previous iteration and if it is within the specified tolerance convergence has been reached. The inner loop computes the solution to the ideal adiabatic model as a closed form solution does not exist and, therefore, an iterative method is required. To quickly and effectively find the solution, two different iterative methods are used in the analysis. The Runge–Kutta method is used for the first four iterations and following this the Adams–Bashforth method is used. This reduces the computation time as the Adams–Bashforth method does not require the computation of intermediate steps but only uses previously computed derivatives, resulting in greater computational efficiency (Faires and Burden,

To optimize the geometry and operating frequency of the Stirling engine, the problem is initially formulated as a bounded constraint minimization problem. The standard form of such a problem is Eq.

The function to be minimized is Eq.

The function is discontinuous.

The function is non-smooth.

There is some degree of numerical noise.

Little is known about the function space.

Each function evaluation is expensive.

The function described exhibits these characteristics, and therefore a specialized algorithm is required. The algorithm of choice is the implicit filtering scheme originally developed by Professor Kelley and colleagues (Kelley,

The analysis and optimization procedure found the optimal geometry and engine speed that gave maximum net-work output at the fixed heat input. The variables which have been optimized are the total heat exchanger length, dead-volume ratio, regenerator length, heater tube length, cooler tube length, compression space to expansion space volume ratio, phase difference, and the operating frequency.

Figure

Maximum network output, minimum total irreversibility rate, and absorbed energy versus heater inlet temperature for the WN50

Figure

Figures

Maximum net-work output

The maximum efficiency plot seen as Figure

Figure

Optimal dead-volume ratio versus heater inlet temperature.

Figure

Figure

Optimal swept-volume ratio versus heater inlet temperature.

Figure

The sizing of the heat exchangers has a significant effect on the performance of the Stirling engine. The following plots show the optimal values for heat exchanger geometry at the different heater inlet temperatures that correspond to optimal engine performance at a fixed heat input.

Figure

Optimal regenerator length versus heater inlet temperature.

Figure

Figure

Optimal regenerator effectiveness versus heater inlet temperature.

Figure

Figure

Optimal

Figure

Optimal heater tube length

Figure

Optimal heater effectiveness

Figures

Figure

When comparing Figures

This study differs from many other studies as the optimal engine operating frequency is calculated, whereas in other studies the operating frequency is specified. This approach allows for the operating frequency to be optimized with the other variables to give optimal engine performance.

Figure

Optimal operating frequency versus heater inlet temperature.

Figure

This result also indicates why bigger machines will perform better, as bigger machines will absorb more energy per cycle and, thus, allow for lower operating frequencies and, thus, a lower irreversibility rate in the regenerator. This effect would be especially pronounced in the case of the lower heater inlet temperatures, as here the operating frequency is high and decreasing this would drastically improve engine performance.

Figure

Optimal phase difference versus heater inlet temperature.

Figure

This analysis and optimization of a 1,000 cm^{3} alpha type Stirling engine, with finite heat capacity rates in the heater and cooler and a fixed energy input, represents a new analysis and optimization of Stirling engine geometry, using exergy analysis methodology with an implicit filtering algorithm. This model can be used by Stirling engine designers as an initial optimization procedure to find optimal or near-optimal design points before more complex modeling and experimentation. The analysis shows the significant effect that the choice of regenerator mesh has on engine performance and the size of the optimal regenerator given the specified mesh dimensions. The optimization shows that the WN200 mesh gives the best engine performance of the mesh types analyzed. The analysis has shown that the optimal regenerator length increases and that the optimal engine operating frequency decreases with increasing heater inlet temperature. The optimal volume ratios, optimal heat exchanger geometry, and phase angle are also presented along with a discussion of the trends seen in the optimal variables. In terms of future work, the exergy analysis approach needs to be used with more complex multi-dimensional Stirling engine models. Also, the external heat transfer effects need to be accounted for in the heater and cooler, these effects include the effects of tube thickness and heat transfer coefficient on the outside of the tubes. These effects should be considered as they would affect the heater and cooler working fluid temperatures and, thus, engine performance.

JW developed the model, carried out the analysis, and wrote the manuscript. TB-O conceived the analysis and revised the manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The authors would like to express their gratitude to the University of Cape Town and National Research foundation (NRF) for their financial assistance in completing this work. Opinions expressed and conclusions arrived at are those of the authors and not necessarily attributed to the NRF and UCT.