Stirling Thermodynamics using Phasor Notation

Phasor mathematics is used to develop the isothermal Stirling cycle and extended to the ideal adiabatic Stirling cycle. The results are developed for piston – piston (alpha) machines and displacer – piston (beta and gamma) machines. The effect of non-ideal regeneration is handled by defining a regenerator effectiveness ratio. The importance of the amplitude pressure ratio (pressure amplitude to the mean pressure) is developed and shown to be a useful parameter when evaluating the effect of dead volume or when applying simple cycle analyses. The analysis is developed for both power producing and cooling engines. The utility of these analyses is discussed with respect to calibrated results of real machines. Background Much like the Otto and Diesel cycles are the classic cycles used to describe the spark and compression ignition engines, the cycle described by Schmidt in 1871[1] has become the classic Stirling cycle. This is more about the Schmidt analysis being mathematically tractable rather than its reflection of the real cycle. The major assumption of Schmidt's analysis is that the gas in the working spaces and attendant heat exchangers is at their respective constant upper and lower cycle temperatures, and that regeneration is perfect. The gas is therefore isothermal everywhere but with spatial temperature distribution as indicated in Fig. 1. The assumption of isothermal working spaces is the most limiting aspect of the analysis. Schmidt also assumed that the volumes of the working spaces vary sinusoidally as is shown in a diagram from his original analysis of Lehmann’s machine (Fig. 2). Fig. 1. Isothermal and adiabatic assumptions * dberchowitz@globalcooling.com T! T! T" T" 〈V#〉 〈V$〉 V! V% V" Isothermal Assumption: Working spaces are at constant adjacent heat exchanger temperature Adiabatic Assumption: Working spaces are adiabatic and heat exchangers isothermal © The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). E3S Web of Conferences 313, 12003 (2021) https://doi.org/10.1051/e3sconf/202131312003 19° International Stirling Engine Conference


Background
Much like the Otto and Diesel cycles are the classic cycles used to describe the spark and compression ignition engines, the cycle described by Schmidt in 1871 [1] has become the classic Stirling cycle. This is more about the Schmidt analysis being mathematically tractable rather than its reflection of the real cycle. The major assumption of Schmidt's analysis is that the gas in the working spaces and attendant heat exchangers is at their respective constant upper and lower cycle temperatures, and that regeneration is perfect. The gas is therefore isothermal everywhere but with spatial temperature distribution as indicated in Fig. 1. The assumption of isothermal working spaces is the most limiting aspect of the analysis. Schmidt also assumed that the volumes of the working spaces vary sinusoidally as is shown in a diagram from his original analysis of Lehmann's machine (Fig. 2).  In what follows, an ideal analysis will be developed using phasor notation that includes the effect of regeneration and adiabatic working spaces. In all other respects, the general assumptions are identical to those used by Schmidt.
Following the Schmidt analysis, the total mass of gas in the engine is given by the sum of the gas masses in the various spaces Substituting for the varying volumes and temperatures, (1) and (2), and solving for the pressure gives: Noting that . << 1, we use the Binomial theorem and rewrite as follows And, ignoring second order small terms (  678,:,;  67<,=   and is the pressure amplitude ratio. We now need a result that relates mt to the mean pressure 〈 〉. This is done by defining the mean cyclic pressure (also referred to as the charge pressure) as follows: Which, from standard integral tables [9] gives: Substituting into (5), the pressure may now be written: This has the same form as the Schmidt isothermal result except that the s would be zero.
The amplitude of the first harmonic is therefore given by: 6 = 7 <* %1 − ?1 − 7 * and has a phase of .
For beta or gamma configurations (Fig. 3), it is convenient to express the volume variations in terms of the piston and displacer motions relative to the piston.
The volume, motion and pressure phasors are shown in Fig. 4. For cooling engines, the pressure leads piston motions and for power engines, the pressure lags piston motions. This is explained in [12]. The presence of the displacer rod results in the total volume variation not being slightly off anti-phase to the piston motions.
Taking Vc as the reference phasor, a phasor diagram for the pressures and volumes would look as in Fig. 5. When the total volume is maximum, the pressure would be close to minimum (depending on temperatures). We will proceed further assuming Vc as reference phasor.   By inspection of a simple control volume containing either the rejector or acceptor, there is no net enthalpy transferred across the heat exchanger / regenerator interface. The heat transfer must therefore equal the work in the associated working space: Substitution of (7) and the derivative of V from (1) into the above, yields: The solution of (13) involves a certain amount of manipulation and has been solved many times in the literature; in approximate form [2], and exact [3,4] and [5]. Here, the final exact solution is given: Each working space is therefore: The solution to obtain the pressure is iterative and proceeds as follows: 1) Assume initial values of 4 and 5 equal to zero.
2) Solve for the pressure (7) . On first pass, this will be the Schmidt result 3) Solve for the mass flow amplitudes and the flow reversal points, (A.20). 4) Solve for the constants B1 and C2 for each space (A.28) and (A.31). 7 is almost equal to 6 and can be assumed so for most situations. 5) The temperature terms, 〈 〉 and can now be found from (A.23) and (A.24) after evaluating the various B constants from (A.14) and (A.18). 6) Iterate on the mean temperatures 〈 4 〉 and 〈 5 〉 back to (2) until desired tolerance is obtained. Convergence of better than 1% on the mean temperatures is usually obtained in less than eight iterations. 7) Finally the cyclic works Wc and We are obtained from Equations (15) and (16). 8) Harmonics of the pressure may be obtained from the Fourier series (8).
Pressure, temperature and working space mass flow profiles are shown in Figs. 6, 7 and 8 for a representative machine. Fig. 9 shows the phasor diagram of these quantities.

Ideal Efficiency and COP
Irreversibilities introduced by the adiabatic processes cause the performance of the cycle to be less than Carnot.
Substituting (15) and (16) into (17) gives: Typically, the phases of the working space temperatures are very close to the pressure phase. Making this assumption on (7), ̂ may be approximated.
Solving for and substituting into (18), gives the following rather simple result: Showing that the ideal adiabatic efficiency is only a function of the mean working space temperatures.
Substituting for the 〈 〉s from Equation (A.23) and ignoring second order small terms, the adiabatic efficiency may be represented by the following machine parameters: This result shows directly that the lower the pressure ratio the closer the efficiency will approach Carnot. Since the adiabatic temperature swings are a direct consequence of the pressure swing, this conclusion may well have been expected. Fig. 10 shows the ideal adiabatic efficiency versus pressure amplitude ratio for typical power engine parameters. The air charged machine has higher ideal performance because the adiabatic temperatures deviate less from the source and sink temperatures. For cooling engines, the following results are found similarly.
Since heat pumps operate over a large temperature range it is instructive to see the effect of both temperature and pressure ratios on available COP. Fig.11 shows the fraction of Carnot versus temperature and pressure ratio for typical machine parameters. For near ambient cooling, the effect of pressure ratio is particularly important.

Ideal Regenerator Heat Flow
The ideal regenerator heat flow for the Schmidt analysis was originally developed by Creswick [6]. From the First Law, it can be shown that the rate of heat transfer to the regenerator gas is equal to the change in internal energy minus the net enthalpy convected into the gas.
where wkr and wra are respectively the mass flows across the rejector/regenerator and regenerator/acceptor interfaces. By applying mass balances over the heat exchanger and associated working space, we have: Since the adiabatic working space temperature variations are typically small with respect to their mean value, we will approximate the ideal gas law from (A.6) as follows: Substituting (28) into the interface mass flows, (25) becomes: Since the second harmonics do not contribute to the net uni-directional regenerator heat flow: The heat transfer ratio is defined.
The regenerator effectiveness is given by [7]: where i is the ideal and act is the actual unidirectional heat flow.
For a machine having a non-ideal regenerator, the working gas exits the expansion-side of the regenerator at a different temperature than the acceptor temperature. Thus, energy needs to be transferred at the acceptor to maintain its temperature. The acceptor heat transfer is therefore The indicated efficiency and COP with regerator effectiveness are now It is interesting to note is that (31) shows that increasing the pressure amplitude ratio will reduce the heat transfer ratio which reveals a contradiction for the cycle. Referring to the adiabatic cycle efficiency (21) or the COP (24), energy performance is optimized by reducing the p term while improving regenerator performance requires just the opposite. This implies an optimum pressure amplitude ratio which is clearly obvious in Figs. 12 and 13. The isothermal (Schmidt) results, compared here, do not show this optimum. This trend has been identified before by simple cycle analysis [5,8].  Fig. 10 with regenerator effectiveness of 0.95. This completely changes the character of the curves. No longer is the efficiency optimized for lower pressure ratios. Also interesting is that higher pressure amplitude ratios improve aircharged engines more significantly than helium-charged engines. The effect of regenerator effectiveness on cooling engine performance is shown in Fig. 15. In this case he regenerator effectiveness is taken at 0.99. The consequence is quite dramatic when compared to Fig. 11.

Optimization
Walker [2,4] showed that it is possible to non-dimensionalize the Schmidt and adiabatic results by defining three independent variable groups. The groups here are slightly different to those used by Walker owing to the inclusion of the cooler and heater volumes. Also, the temperature ratio used here is the inverse of Walker's. The independent variable groups are therefore: and is referred to as the reduced dead volume where for the purposes of optimization, the working space clearance volumes are taken as zero. Therefore each = 1.
The s and 〈 〉s are obtained as in Appendix A.
W # indicates the cyclic work for a particular engine size and maximum working gas pressure. The maximum working gas pressure gives an indication of the mass of the engine since it determines the thickness of the pressure vessel walls. Consequently, optimizing W # for k and a given r and t would give the highest cylic work for working gas volume and engine mass. Fig.14 and 15 show the isothermal and adiabatic results respectively. Regeneration is taken as perfect and for the adiabatic case, helium is assumed. The optimisation is not accurate for the adiabatic case when and begin to approach unity. The isothermal results are not affected by this because the equations are exact. Also, the assumption of zero clearance spaces will introduce error for the adiabatic results due to the linearization (A.17) which again, does not apply to the isothermal result.

Conclusions and Observations
Ideal analyses should provide useful insights for design. In this regard, the phasor analysis indicates: 1. For adiabatic working spaces, there is a clear optimum pressure amplitude ratio where efficiency or COP is optimized, and this depends strongly on regenerator effectiveness. 2. The pressure amplitude ratio is a fundamental property of the cycle and includes the effects of dead volume. Depending on the location on the thermal performance curve, increasing dead volume may reduce or improve thermal performance insofar as it alters the pressure amplitude ratio (Figs. 12 and 13). This has been noted by Gschwendtner and Bell [14]. 3. The ideal adiabatic efficiency or COP depends, in the first order, only on the mean working spaces temperatures. 4. Optimum work or lift for a given maximum pressure and volume requires somewhat different stroke ratios and volume phases for adiabatic and isothermal conditions. And, finally, the phasor analysis offers a generic approach to the condition of the working spaces in that the isothermal result is a simple consequence of the analysis.
Of course, correlation with actual engines will depend on the degree that irreversibilities and losses have been minimized. Table 1 lists some engines to which the analyses have been applied. Regenerator effectiveness and displacer drive power have been accounted for in each case. Calibrated results are taken from a much more extensive simulation that includes parasitic losses and that has been calibrated against actual data. The regenerator effectiveness numbers comes from the simulation and is based on a calculation of the thermal performance of the regenerator. All calculations are based on gas temperatures.
The pressure amplitude ratio and phase tend to be better predicted by the adiabatic analysis versus isothermal. This is important for free-piston dymanics calculations. Energy transfers are quite reasonable even for the Schmidt analysis.  Taking the first harmonic pressure variation and noting that < 1, (A.9) may be simplified by applying Taylor's expansion theorem around = 〈 〉: where C1 is now a new constant and higher order terms have been neglected.
Rewriting (A.13) for easier manipulation: The working spaces temperatures are now fully defined except for the original constants of integration, 6 and 7 , which must be solved from appropriate boundary conditions. From the energy equation (A.1), and noting that the internal energy over a cycle is zero: Which indicates that the net enthalpy flow into the working space is equal to the net pV work done by the piston.
The enthalpy flow into the working space may be evaluated as follows: x (d ) d + ∫ ( ) (d ) d where T is the working gas instantaneous temperature and is the associated constant heat exchanger temperature. The limits of integration occur at the zero flow points.
Substituting for T for dm ≤ 0 from (A.14) and for dm from (A.20), (A.25) may be integrated to yield where | | = 〈 〉 The pV work is given by (14), and combined with (10), gives This completes the necessary equations to solve the temperature variations in the working spaces.