Theory
The ideal Stirling cycle is represented in the figure below and consists of four processes which combine to form a closed cycle: two isothermal and two isochoric processes. The processes are shown on both a pressure-volume (P-v) diagram and a temperature-entropy (T-s) diagram. The area under the process path of the P-v diagram is the work and the area under the process path of the T-s diagram is the heat. Depending on the direction of integration the work and heat will either be added to or subtracted from the system. Work is produced by the cycle only during the isothermal processes. To facilitate the exchange of work to and from the system a flywheel must be integrated into the design which serves as an energy exchange hub or storage device. Heat must be transferred during all processes. See Table for a description of the 4 processes of the ideal Stirling cycle.
The net work produced by the closed ideal Stirling cycle is represented by the area 1-2-3-4 on the P-v diagram. From the first law of thermodynamics the net work output must equal the net heat input represented by the area 1-2-3-4 on the T-s diagram. The Stirling cycle can best approximate the Carnot cycle out of all gas powered engine cycles by integrating a regenerator into the design. The regenerator can be used to take heat from the working gas in process 4-1 and return the heat in process 2-3. Recall that the Carnot cycle represents the maximum theoretical efficiency of a thermodynamic cycle. Cycle efficiency is of prime importance for a solar powered engine for reasons that the size of the solar collector can be reduced and thus the cost to power output ratio can be decreased.
Process 1-2 : Isothermal compression
Heat rejection to low temperature heat sink
1Q2 = area 1-2-b-a on T-s diagram
Work is done on the working fluid (energy exchange from flywheel)
1W2 = area 1-2-b-a on P-v diagram
Process 2-3 : Isochoric heat addition
Heat addition (energy exchange from regenerator)
2Q3 = area 2-3-c-b on T-s diagram
No work is done
1W2 = 0
Process 3-4 : Isothermal expansion
Heat addition from high temperature heat sink
3Q4 = area 3-4-d-c on T-s diagram
Work is done by the working fluid (energy exchange to flywheel)
3W4 = area 3-4-a-b on P-v diagram
Process 4-1 : Isochoric heat rejection
Heat rejection (energy exchange to regenerator)
4Q1 = area 1-4-d-a on T-s diagram
No work is done
4W1 = 0
Real Stirling Cycle Engine
The real Stirling engine cycle is represented in Fig. 3 below. As can be seen there is work being done during processes 2-3 and 4-1 unlike the prediction of zero work in the ideal cycle. One of the major causes for inefficiency of the real Stirling cycle involves the regenerator. The addition of a regenerator adds friction to the flow of the working gas. In order for the real cycle to approximate the Carnot cycle the regenerator would have to reach the temperature of the high temperature thermal sink so that TR=TH. A measure of the regenerator effectiveness is given by Equation 1, with the value of e=1 being ideal.
Another major cause for inefficiencies of the real Stirling cycle engine is that not all of the working gas participates in the cycle, i.e. dead volume. The dead volume involves the volume that does not participate in the swept volume of the piston stroke. Martini (2004) states that the relationship between the percentage of dead volume in the system to the decrease in work done per cycle is linear. Therefore, if the engine has 20% dead volume then the power output would be 80% of the power that would be produced with zero dead volume. In actuality, dead space will always be present because the addition of internal heat exchangers, clearances, transfer tubes, and regenerators are required to enhance the heat exchange of the real system.
Though the ideal Stirling cycle can be analyzed using known thermodynamic principles, the analysis exists as an approximation of the real Stirling engine. Team 4 took this into consideration in the final design of the Stirling engine so that certain design parameters such as the stroke length, temperature differential, and flywheel mass could be altered during the testing phase to optimize the Stirling engine.