Numerical Modelling

It was originally intended to optimise the design parameters using a computer software package called SAGE™. This software allows the user to input a configuration of Stirling engine components such as variable volume spaces, heat inputs, conduction paths, pressure sources and pistons and displacers, and then assign numerical values to certain geometries and temperatures as well as alter the gas type. The model is then able to be solved to find user defined variables which would be most importantly power output and efficiency. In early stages of modelling however it was found that, for some reason as yet unknown, the validity of the SAGE™ model would break down when larger geometries were used. This meant that in effect the software was no help in designing the engine due to its large size.

It was found however that by scaling down the geometry of the model that some meaningful results could be obtained. The configuration pictured in Figure 91 was used to model a smaller version of the prototype engine, with the results produced shown in Figure 92 through to Figure 96.

Numerical Modelling

Figure 91: Screenshot of the SAGE™ model used to simulate the prototype engine

Numerical Modelling

Figure 92: Simulated results for phase angle from a SAGE™ model of a smaller engine

Simulated Power and Efficiency for Various Gases









Power (W) Efficiency (%)


Numerical Modelling









Figure 93: Simulated results for gas type from a SAGE™ model of a smaller engine

Numerical Modelling


Power And Efficiency vs. Speed and Mean Engine Pressure

Power (W) @ 1.0 MPa Power (W) @ 1.5 MPa Power (W) @ 2.0 MPa Efficiency (%) @ 1.0 MPa Efficiency (%) @ 1.5 MPa Efficiency (%) @ 2.0 MPa

Figure 95: Simulated results for engine speed and pressure from a SAGE™ model of a smaller engine

Numerical Modelling

These results are very much in agreement with both the theory and results recorded from real Stirling engines. Comparing the graph of power vs. phase angle in Figure 92 to that shown in Figure 46 (Section 3.1.1), it is apparent that the shape of the graph is nearly identical, though the peak is centred about a lower value of approximately 80° for the simulated results vs. approximately 100° for the documented results. Another difference is that the simulated results show power produced between a phase difference of 0° and 160° whereas in the documented results the power production extends all the way through to 180°.

The simulated graph of power and efficiency for different gases (Figure 93) shows the expected increase improvements of the progressively lighter gases which is comparable to results available in literature and is in agreement with theory presented in Section 2.I.5.3.

It is interesting to note that in Figure 94 and Figure 95 the power and efficiency begin to drop at high values of engine pressure or operating speed. The main reason for this is the increase in pumping losses through the regenerator and heat exchangers. These losses increase according to a squared relationship with fluid mass flow rate (proportional to engine speed), as stipulated in Section 2.3.3.I. This means that even though the engine is producing more power at higher speeds (linear relationship due to power being a product of work per cycle and cycle speed), a greater and greater portion of that power goes into keeping the engine running. Of course, this is somewhat irrelevant to the operation of the prototype engine as its pumping losses are all fielded by the electric motors actuating the displacer which does not draw directly upon the engine power. The shapes of the curves in Figure 95 are in close agreement with those in Figure 28, Section 2.I.5.3.

The shape of the efficiency curve in Figure 96 is also somewhat of a surprise, as one would expect this to increase in a near linear fashion as per the formula for Carnot efficiency. However, at higher temperatures due to the way this engine is modelled there is a large amount of heat conduction loss leading to the efficiency curve being almost flat whilst the power curve is close to linear. This phenomenon was confirmed in SAGE™ by keeping the component marked as ‘parasitic source’ at a constant value equal to the ambient temperature, while varying the heater temperature. This resulted in a much more linear relationship between temperature difference and efficiency.

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