Thermodynamic and Dynamic Analysis of Free Piston Stirling Engine Using Numerical Techniques

1. Thermodynamic and Dynamic Analysis of Free Piston Stirling Engine Using Numerical Techniques ME 535 – Spring 2014 Nitish Sanjay Hirve Amrit Om Nayak

2. Free Piston Stirling engine Stirling engine ideal cycle

3. Isothermal analysis Methodology- Assume sinusoidal volume variation Use ideal gas equation to determine instantaneous pressure Calculate instantaneous PV work, dW Iterate for entire cycle and keep adding instantaneous work dW to get work output per cycle Matlab output- Work per cycle in J by numerical integration : 17.2093 Work per cycle in J by Mayers relation : 13.5294 Work per cycle in J as per Senft equation : 18.1755 Work per cycle in J by Cooke-Yarborough relation : 17.5579

4. Adiabatic analysis Methodology- Assume sinusoidal volume variation Use approximate solution for change in instantaneous pressure Reiterate the entire cycle till stable pressure profile is obtained Calculate instantaneous PV work, dW Iterate for entire cycle and keep adding instantaneous work dW to get work output per cycle Matlab Output- Enter the crank angle increment in degrees : 0.01 Enter number of iterations : 9 Compression ratio: 1.3859 Work output in J/cycle: 16.6654 Heat input in J/cycle: 29.6967 Heat rejected in J/cycle: -15.5192 Regenerator heat in J/cycle: -2.1517e-04 Efficiency of engine: 56.1189

6. DYNAMIC ANALYSIS Displacer’s equation of motion Working Piston’s equation of motion

8. Operating Frequency General Oscillation Criterion

10. NUMERICAL ANALYSIS USING THE FOLLOWING: Iterated parameters like pressure, impulse on displacer head, & working piston, mass of working fluid in compression and expansion spaces, gas displacement etc. from thermodynamic analysis Explicit approach driven by heuristic inferences since there is no analytical solution to the complex energy equation or the dynamic mathematical model of the system We express the desired variables in the form of an explicit equation, and then eliminate the extra variables using thermodynamics and dynamic relations Taylor/Mclaurin series expansions with varying orders

11. MATLAB OUTPUT Enter the crank angle increment in degrees : 0.1 Enter number of iterations : 6 Enter the required taylor series order : 8 Average Power from engine in Watts = 57.320961565215846 Average operating frequency of engine = 13.148354491987975 Stroke-ratio of engine = 1.093457321442433 Optimal Phase difference between displacer and working piston in degrees = 82.212576286029801 ENGINE PARAMETERS USED: Operating temperature = 550oC = 823 K Operating frequency (design) = 15 Hz Engine starting pressure = 3 bar

14. Observations & Inferences • Average power output by adiabatic analysis is 249.98 Watts. However, dynamic analysis shows an average power output of just 57.321 Watts- Power lost! • The operating frequency after iteration is predicted by the dynamic analysis to be 8 - 13.15 Hz – Design frequency of 15 Hz • The general oscillation criterion is violated between 130 to 200 degrees and 270 to 325 degrees - no net power in these regions - damping - system continues motion by virtue of momentum and energy stored in the springs. • Inference - viscous damping, sliding friction and spring damping need to be reduced while spring stiffness needs to be increased

15. Error % plotted against step size and Taylor series order – No significant impact on results - Inherent heuristic iterative approach is efficient! • At very small step size (<=0.01 crank angle increment), the power (=118 Watts) and operating frequency (=8 Hz) are realistic. • Inference - large number of iterations (>50000) for stable solutions due to the complexity of the mathematical model. • Low step sizes require > 2000 seconds - time required increases drastically with reduction in step size - process is expensive and more powerful computing resources are required to evaluate the model more accurately.