Stirling Cycle Machine Analysis (updated 3/21/2016)

Russ College of Engineering and Technology

Stirling Cycle Machine Analysis

by Israel Urieli ( latest update September 2022 )

dedicated to William T. Beale (1928 – 2016), inventor of the Free Piston Stirling Engine,
Mentor and Friend

This web resource is intended to be totally self contained learning resource for the analysis and development of computer simulation of single phase, piston/cylinder Stirling cycle machines. It includes thermodynamic, heat transfer and fluid flow friction analysis, and until 2012 it was used as resource material for an advanced course for Mechanical Engineering majors. The course structure was based on the book by I.Urieli & D.M.Berchowitz 'Stirling Cycle Engine Analysis' (Adam Hilger, 1984). The computer simulation program modules (originally written in FORTRAN) have all been updated and rewritten in MATLAB, a convenient interactive language which allows direct graphical output - essential for Stirling cycle analysis. A complete set of all the m-files are developed and provided, and they can be augmented and adapted as needed for specific engine/refrigerator configurations.

This web resource is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States license and as such is freely available. Comments and constructive criticism are welcomed by the author.

Chapter 1: Background and Introduction
Chapter 2: Basic Engine Configurations a) Alpha Type Engines b) Beta Type Engines c) Gamma Type Engines
Chapter 3: Ideal Isothermal Analysis We define and analyze the Ideal Isothermal model of a Stirling engine, including the Schmidt Analysis, and discuss its limitations. One obviously incorrect conclusion of this analysis is that all three heat exchangers are redundant, and only contribute dead space, since all required heat transfer processes occur in the isothermal compression and expansion spaces. Nevertheless we can obtain a better understanding of a specific design, particularly when we augment the solution with Allan Organ's particle mass flow analysis. a) Ideal Isothermal Analysis Regenerator mean effective temperature Energy Analysis - Ideal Isothermal Model b) The Schmidt Closed Form Solution Beta and Gamma type sinusoidal volume variations Schmidt Analysis - Equation Summary c) Function set 'define'
Chapter 4: Ideal Adiabatic Analysis We find that the Ideal Isothermal analysis predicts that the heat exchangers of a Stirling engine are redundant, thus we cannot seriously use this model to predict the ideal performance of an actual machine. We thus turn to an alternative model in which the compression and expansion spaces are adiabatic. We find that there is no closed form solution to this model and we have to resort to computer simulation. We gain various insights from using this model in particular with regards to the importance of the regenerator, which was not understood for a significant period. a) Development of the Ideal Adiabatic Equation Set b) Equation Summary and Method of Solution c) Function set 'adiabatic' d) Case Study - D-90 Ross Yoke-drive Engine
Chapter 5: Simple Analysis This analysis approach uses the Ideal Adiabatic model as a basis to predict the real performance of the three heat exchanger sections, particularly with regards to heat transfer and pressure drop. The name Simple Analysis is to indicate that this is a simplification of the actual non-steady flow heat exchange, however it enables a parametric analysis of a specific machine. a) Scaling Parameters (Flow Friction and Convective Heat Transfer) b) Regenerator Simple Analysis c) Heater and Cooler Simple Analysis d) Pumping Loss Simple Analysis e) Function set 'simple'

Chapter 1: Background and Introduction

Chapter 2: Basic Engine Configurations

a) Alpha Type Engines

b) Beta Type Engines

c) Gamma Type Engines

Chapter 3: Ideal Isothermal Analysis

We define and analyze the Ideal Isothermal model of a Stirling engine, including the Schmidt Analysis, and discuss its limitations. One obviously incorrect conclusion of this analysis is that all three heat exchangers are redundant, and only contribute dead space, since all required heat transfer processes occur in the isothermal compression and expansion spaces. Nevertheless we can obtain a better understanding of a specific design, particularly when we augment the solution with Allan Organ's particle mass flow analysis.

a) Ideal Isothermal Analysis

Regenerator mean effective temperature

Energy Analysis - Ideal Isothermal Model

b) The Schmidt Closed Form Solution

Beta and Gamma type sinusoidal volume variations

Schmidt Analysis - Equation Summary

c) Function set 'define'

Chapter 4: Ideal Adiabatic Analysis

We find that the Ideal Isothermal analysis predicts that the heat exchangers of a Stirling engine are redundant, thus we cannot seriously use this model to predict the ideal performance of an actual machine. We thus turn to an alternative model in which the compression and expansion spaces are adiabatic. We find that there is no closed form solution to this model and we have to resort to computer simulation. We gain various insights from using this model in particular with regards to the importance of the regenerator, which was not understood for a significant period.

a) Development of the Ideal Adiabatic Equation Set

b) Equation Summary and Method of Solution

c) Function set 'adiabatic'

d) Case Study - D-90 Ross Yoke-drive Engine

Chapter 5: Simple Analysis

This analysis approach uses the Ideal Adiabatic model as a basis to predict the real performance of the three heat exchanger sections, particularly with regards to heat transfer and pressure drop. The name Simple Analysis is to indicate that this is a simplification of the actual non-steady flow heat exchange, however it enables a parametric analysis of a specific machine.

a) Scaling Parameters (Flow Friction and Convective Heat Transfer)

b) Regenerator Simple Analysis

c) Heater and Cooler Simple Analysis

d) Pumping Loss Simple Analysis

e) Function set 'simple'

This learning resource includes a set of tutorial MATLAB computer program modules for simulating specific Stirling engine configurations. The complete set of m-files can be downloaded in compressed format sea.zip ( sea = s tirling e ngine a nalysis). These modules can be augmented and adapted as required to simulate a specific engine design. Currently the engine modules are for Alpha machines, including a Sinusoidal drive, a Ross Yoke-drive and a Ross Rocker-V engine. The heat exchanger types include tubular, annular gap, and slot heat exchangers, and the regenerator matrix types include screen mesh and rolled foil matrices. Working gas types include air, helium, and hydrogen.

Note that the purpose of this learning resource is to develop an appreciation and understanding of the complexity of practical Stirling cycle machine performance simulation, mainly due to the heat transfer processes. It is not intended as an alternative to the Sage Software for engineering modeling and optimization of Stirling cycle machines.

Except for the Ohio University header and footer, including their graphics, Stirling Cycle Machine Analysis by Israel Urieli is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License

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