Specific Power Estimations for Free-Piston Stirling Engines

Seon-Young Kim* Sunpower, Inc., Athens, Ohio, 45701

David M. Berchowitz † Global Cooling Manufacturing, Inc., Athens, Ohio, 45701

NASA seeks high-power conversion systems for space applications and their requirements emphasize high efficiency and low mass. This paper presents preliminary designs and specific power estimates of high efficiency, free-piston Stirling machines for 5 kWe , 10 kWe and 25 kWe outputs. In each layout, several engine configurations have been considered: single and dual opposed beta and three, four and six cylinder alphas. A critical factor in obtaining high specific power is related to the specific power optimization of the alternator. This is shown to be a strong function of the piston amplitude. Monocoque finned heads are found to be compromised by temperature gradients across the pressure-containing walls. These temperature gradients become significant at power levels above 2 kW. A stepped piston three-cylinder alpha arrangement is shown to have favorable dynamics for high efficiencies and good specific powers.

I. Introduction

F ree-piston Stirling engines (FPSE) have high potential for space power applications due to not only their characteristics of high efficiency, low mass and long life but also their adaptability to various heat sources: solar,

radioisotope and nuclear reactor. A report was done by MTI 1 to study engines ranging from 25 to 150 kWe in 1989 and a second report by MTI 2 describes a 50 kWe FPSE development program from 1988 through 1993.

In addition, for many decades, NASA has studied the use of nuclear power for lunar and Mars surface applications. A lunar surface mission is anticipated to occur in the early 2020’s and power requirements for human-tended surface outposts and bases are expected to range from 25 to 100 kW during the early build-up phases. Later power requirements may approach 1 MW as the base becomes fully operational 7. For successful operation in space, the FPSE is required to have a high efficiency, since it will most likely be coupled with a massive reactor and radiator. The specific power of the FPSE is also of concern owing to the difficulties associated with space transportation of heavy items.

Sunpower carried out a number of projects under a NASA Small Business Innovative Research (SBIR) award including 35 We 3 and 80 We 4 free-piston Stirling converters and is developing an “Advanced Stirling Converter” (ASC, 80 We) 5, 8 under a three-year NASA program. Performance characterization based on recent experiences with hardware and analysis 6 show that significantly higher performances are achievable compared to that of previous studies. In February 2006, Sunpower, assisted by Global Cooling, began to pursue an investigation of high power FPSEs under a NASA SBIR Phase 1 award. The goal of the program is to provide preliminary design information for high specific power FPSEs for space applications. To this end, various free-piston configurations are being investigated, including the so-called alpha arrangement. For a given power level, the alpha configuration benefits from its multiple smaller cylinders having thinner wall thickness and higher surface area to volume ratio as well as one moving part per cylinder.

This paper presents some preliminary layouts and specific power estimates of high efficiency FPSEs for 5 kWe, 10 kWe and 25 kWe outputs. In each layout, several engine configurations have been considered: single and dual opposed beta as well as multiple-cylinder alphas (three, four and six cylinders). A stepped piston three-cylinder alpha arrangement, proposed by Dr. Berchowitz, has been found to have favorable dynamics for high efficiencies and good specific powers.

* Project Engineer, member AIAA, ASME † Chief Engineer and Vice President, member ASME, ASHRAE

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II. System requirements and Engine design points As mentioned, lunar outpost power requirements are expected to be over 25 kW. Based on various system

studies for nuclear-reactor power systems, a heater head temperature of 1050 K is considered to be the maximum allowable for super alloy-based Stirling engines. For space applications, the heat reject temperature is optimized for a minimum radiator specific mass, which leads to a temperature much higher than that for terrestrial systems. Based on a previous study, a heat reject temperature of 500 K is considered reasonable for an acceptable radiator mass 7.

The reactor and radiator are so dominant in the system mass and size, that it becomes critical for the FPSE to have a high efficiency. In fact, the engine specific power is of secondary importance. So the question becomes: What is the best specific mass for the highest possible efficiency? In order to achieve this goal, we have investigated several engine power levels and configurations to attempt to estimate the relationship of size and mass to power level and configuration. Three different power levels have been investigated at 5, 10 and 25 kWe for five different physical configurations, namely single beta, dual opposed beta and three, four and six cylinder alpha. The 25 kWe single beta is excluded because such a study has been conducted by MTI 1 and is provided as a comparison point. Heater and cooler temperatures are set to be 1050 K and 500 K as suggested in the NASA study 7 for a refractory reactor, which has a higher temperature ratio than that of a stainless steel reactor (850 K/460 K). The working gas is Helium and charge pressure and operating frequency are determined depending on the dynamics of the thermodynamic cycle and the optimization of the alternator. In Table 1, the engine design points are presented.

Table 1 Engine Design Points

Single beta Dual opposed beta Multiple cylinder alphas No. of cylinder 1 2 3 4 6 Engine Power (kWe) 5, 10 5, 10, 25 ← Th (K) 1050 ← ← Tc (K) 500 ← ← Charge Pressure (bar)* 45 36 ~ 48 38 ~ 48 28 ~ 48 26 ~ 43.5 Operating Frequency (Hz)* 65 ~ 70 65 ~ 70 50 ~ 60 60 60 ~ 65

* Determined by ensuring that the sprung mass is greater than the magnet mass. These parameters are not arbitrary.

III. Engine layout designs

A. Engine Heat Exchanger Configuration The heaters are mostly assumed to be tubes and coolers are assumed to be tube-and-shell arrangement. The

number of tubes on the heads ranges from 120 to 270, relatively small compared to over 1000 of the SPED (Space Power Demonstrator Engine) designed by MTI 2. However, in the case of the 5 kWe four and six cylinder and 10 kWe six cylinder engines, finned monocoque construction is assumed. Finned monocoque heads are highly compromised in temperature gradients across the primary heat transport surfaces in high power applications. The calculations show that efficiency performance of monocoque designs begins to fall off at just above the 3 kW thermal input level and by 6 kW becomes severely compromised. The study therefore includes monocoque heads in cases where the acceptor thermal flux is sufficiently low (i.e., the four and six cylinder 5 kWe layouts and for the six cylinder 10 kWe layout). Of course, concepts such as the ‘starfish’ head proposed by MTI 2 will extend the practicality of the monocoque head, but those concepts are beyond the scope of this study. All the engines studied here have a foil regenerator placed annularly to the piston. Foil regenerators are assumed due to anticipated higher reliability and acceptable thermal performance. Inconel 718 is assumed for the heater tube material and stainless steel for the cooler tube and regenerator material.

B. Engine Head and Pressure Vessel Super-alloys like Mar-M-247 or Udimet 720 are taken to be appropriate for the head material. Inconel 625 is

taken for the pressure vessel material because of its superior allowable stress to density ratio compared to stainless steel.

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C. Thermodynamics All performance estimates were done utilizing coupled linear thermodynamic and dynamic calculations by way

of SAUCE‡. Estimates are made of all the known losses in the machines. These include displacer and piston blow-by, gas bearing effort, gas hysteresis, internal conduction losses, off-resonance alternator operation and so on. The calculations are well calibrated against a variety of machines and typically predict within 5 to 10% of actual performance.

D. Dynamics In each case, full free-piston dynamics have been calculated. This is necessary to determine the allowable sprung

mass for the piston which should be larger than the magnet mass in order to accommodate additional piston structural mass (See Appendix). For sake of simplifying the comparisons, the sprung mass is calculated only from the gas spring effect of the thermodynamic cycle. No consideration is taken of providing additional springing from mechanical, magnetic or separate gas springs. The sprung mass is set at some reasonable value above the magnet mass so that additional piston structural mass can be accounted for. Of course this is not precise but it should provide a suitable starting point for detailed design. In the case of the displacers, it has been assumed that the mass of displacer - spring assemblies will scale according to the ratio of the displacer diameter compared to the 1 kWe Sunpower FPSE (similar displacer-spring assemblies are assumed). Errors in estimation of the total structural masses of the displacer and piston assemblies are not likely to be significant since they contribute a small fraction of the total system mass. What is more important is to be reasonably sure that a particular operational frequency is achievable since this does have a first order effect on the mass of the system. This is the assurance that the dynamic calculations are attempting to provide.

E. Linear Alternator The alternator is required to operate at a temperature higher than 500 K reject temperature to account for the

losses in the alternator. 513K has been chosen somewhat arbitrarily. This temperature level is above the capability of most standard materials and therefore Samarium Cobalt (Sm2Co17) magnets have been assumed. A second effect of the alternator temperature is the higher resistivity of the coil windings leading to higher coil mass. For minimum size and mass, materials with the highest possible saturation flux density should be employed and therefore Hyperco is selected as the core material. The inner iron assumes tapered laminations for maximum packing density.

0

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5kW at 80 Hz

0.25kW at 120 Hz

(a) Specific power versus Frequency (b) Specific power versus piston amplitude Figure 1. Specific power results of Linear Alternator. A number of linear alternator studies were completed and it is clear that larger powers and higher frequencies

produce better specific power densities (Figure 1 (a)). Also there is a distinct optimum point for amplitude depending on the power and frequency (Figure 1 (b)). However, optimum linear alternator power densities appear to occur at impractically high amplitudes. Therefore a critical factor in obtaining high specific power is related to the ‡ SAUCE is a proprietary free-piston simulation code.

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interaction of the dynamics of the thermodynamic cycle and the optimization of the alternator. For example, increasing the piston amplitude is favorable to the alternator but, for a given power, leads to a smaller piston diameter. This, in turn, leads to a smaller springing effect. Following this process soon leads the design to a point where the magnet mass cannot be sprung by the engine. Typically, the optimum point for minimum mass of the linear alternator and the engine do not coincide.

380 357

445

500 858

384 383 686 464

880

577 492572

1181

CSSE Approx. Scale

Figure 2. Engine layouts of the single beta, dual opposed beta and multiple-cylinder alphas. (Length [mm])

F. Engine Layouts In Figure 2, all the engine layouts are presented. The 25 kWe single beta engine (Conceptual Stirling Space

Engine, CSSE) is taken from the MTI scaling study 1 and is shown for reference of a more mature design. In the dual opposed arrangements, the expansion space of one side is connected to that of the opposite side to ensure a single dynamic operating point. The 10 kWe dual opposed is based on two 5 kWe single betas. In the multiple-cylinder alphas, the tube heads have the inter-cycle connections at the hot end and the finned head engines have the

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inter-cycle connections at the cold end. In order to keep reasonable tube lengths and smaller diameter pressure vessels, some alpha layouts employ tandem or triple alternators per cylinder.

The MTI Component Test Power Converter (CTPC), a single side test unit, is shown in Figure 3 as a mature example, which was meant to be an opposed beta. The CTPC tests were performed with the heater temperature at 800 K and the cooler temperature at 400 K and showed above 12.5 kWe alternator power output and 22% engine efficiency or 44% of Carnot. This machine demonstrated a specific power of 142 W/kg 2. The machines investigated in this study show fractions of Carnot of between 50% and 60%. Obviously the level of design maturity will adjust these numbers although the ASC program has already demonstrated a fraction of Carnot as high as 54% and is expected to eventually reach 60% 8.

IV. Specific Power Estimates SAUCE simulations and mass calculations were conducted for

each design point. In the mass calculations, the allowable stress for the head and pressure vessel as suggested by the ASC program has been used. This reflects the best thinking for reliable life calculations. The thickness of tubes was assumed to be 10% of inner diameter, which is much thicker than the minimum thickness obtained from hoop stress calculation. This approximation does not add any significant error to our overall estimates. Since the head and pressure vessel was assumed to be of a simple cup shape and structural optimization was not considered, some additional mass reduction would be possible.

Figure 3. CTPC Engine Layout. (MTI)

Figure 4 shows the performance estimates of engines including that of CSSE at 25 kWe. The performance analyses indicate engine efficiencies ranging from 27.1 to 32.5%, with the three-cylinder alpha engines showing particularly good efficiency. A stepped piston three-cylinder alpha has more favorable volume phase angle (120°) than four cylinder engine (90°) and appears to be more compact compared to the six-cylinder which has a 120° phase as well. The finned monocoque designs show lower efficiencies, particularly at the higher power levels. It is clear that the essential reason for the higher efficiency designs is the use of tubes for the acceptor. If tube-type construction is disallowed, then efficiency will suffer. This is likely even for advanced concepts like MTI’s ‘starfish’ design though this particular design does demonstrate excellent application of structural optimization.

The specific power estimates of engines including that of CSSE at 25 kWe are shown in figure 5. The mass calculations indicate that the dual opposed and three and four-cylinder alphas have excellent specific power densities. Specific powers over 10 kWe range from 182 to 220 W/kg. Since the specific power (and diameter) of the alternator is an important factor to obtaining high specific power of engines (Figure 6 and 7), the 25 kWe dual opposed shows the highest specific power due to its higher alternator specific power. CSSE specific power is exceptionally high because no balance mass was included. In addition, our calculations suggest that high frequency, high amplitudes and high powers would be needed to approach that high alternator specific power.

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Figure 4. Engine Performance Estimates. Figure 5. Specific power of engines. (*CSSE estimation 1) (*CSSE estimation 1)

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Figure 6. Specific power of linear alternators. Figure 7. Specific power of engines without alternator. (*CSSE estimation 1) (*CSSE estimation 1)

V. Consideration of Reliability Tube joints to Mar-M-247 head at 1050K heater temperature would be very difficult to implement due to

diffusion that will lead to reliability problems. One possible way of obtaining high reliability would be to lower the heater temperature to 850K so that a stainless steel reactor could be used. Even though the number of tubes in these designs are not particularly high, the general consensus has been that tube joints will be a source of reliability difficulties no matter what the materials of construction might be. This consensus should be reviewed because if it is possible to lower the head temperature slightly and at the same time find an acceptable means to employ tubes as the primary heat transfer surface, then higher efficiencies may be possible. As mentioned, an alternative would be to use a starfish heater head developed by MTI which has a minimum of hot end joints. The ‘starfish’ head could easily be adapted to the alpha arrangements shown here in much the same way as the monocoque heads have been configured. That is, the inter-cycle connections would be at the cold end of the engine.

VI. Conclusion We have provided several layouts of free-piston Stirling engines at different power levels along with their

performance and specific power estimates. For accurate estimations, the interaction of free-piston dynamics, thermodynamics and linear alternator performance has been accounted for. Optimization of the thermodynamics and the linear alternator do not usually lead to a common best point and therefore compromises are necessary. Tube type heaters offer distinct advantages and serious thought should be applied to finding a means to reliably implement such designs. Higher power levels appear to improve the specific power of the tube head configurations while finned monocoque heads appear to favor lower powers. A stepped piston three-cylinder alpha arrangement indicates excellent efficiencies and good specific powers.

Appendix. Relationship of Sprung Mass to Power of Free-Piston Engines (D. M. Berchowitz, June, 2006)

From the simple dynamics result§ for power:

P =ω2

Ap∂p∂xd

Xd X p sinφ [W] (A.1)

where: P is power [W] ω is frequency [rad/s]

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§ Redlich R W and Berchowitz D M, ‘Linear dynamics of free-piston Stirling Engines’, Proc Instn Mech Engrs, Vol 199, No A3, March 1985.

Ap is piston area [m2] p is pressure [Pa] xd is displacer position [m] Xd is displacer amplitude [m] Xp is piston amplitude [m] φ is the phase angle between the piston and displacer motions. Assuming that in the first order, the frequency of the engine is approximately the resonance of the piston. Thus: ω = k mp (A.2) where k is the piston spring rate [N/m] and mp is the piston mass [kg]. In the specific case where all springing is provided for by the pressure changes of the engine, the spring rate k is given by:

k = Ap∂p∂xp

[N/m] (A.3)

Substituting (A.3) into (A.2) into (A.1) leads to the following:

P =ω2

Ap∂p∂xp

∂x p

∂xd

Xd X p sinφ ≈ω 3

2mp X p

2 sinφ (A.4)

Piston mass is thus proportional as follows:

mp ∝P

ω 3X p2 (A.5)

This result shows that increasing sprung mass is best done by smaller amplitudes and low frequencies.

The mass that needs to be sprung is at a minimum, the magnet mass. At the most elementary level, the power of a linear alternator can be shown to be given by: Pm =ω X p M t BI pm [W] (A.6) where: M is the magnetization of the magnets [A/m] t is the magnet thickness [m] pm is the magnet circumference [m] BI is the winding field [V s/m2] or [N/(A m)] For a fixed power factor, it is possible to show that BI is approximately proportional as follows: BI ∝ M t g (A.7) where g [m] is the magnet gap dimension.

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The mass of the magnet is proportional to: mm ∝ ρm pm t X p (A.8) where ρm [kg/m3] is the magnet density. Thus the magnet mass can be seen to be proportional as follows;

mm ∝Pm

ωρm

M 2gt

(A.9)

ρm M 2 is a constant for a particular choice of magnet. Thus (A.9) may be further simplified to:

mm ∝Pm

ωgt

(A.10)

Results of (A.5) and (A.10) represent the difficulties of matching linear alternators to free-piston Stirling engines. Increasing frequency is the best means to improve specific power but the sprung mass reduces by the inverse of the third power of frequency while the magnet mass reduces only by the inverse of frequency.

Acknowledgments We wish to acknowledge the support of NASA for the SBIR project “High Specific Power Multiple-Cylinder

Free-Piston Alpha Stirling”, Contract NNC06CA84C. In particular we would like to thank S. Oriti, J. Schreiber and R. Shaltens for their valuable assistance.

References 1 Jones, D., “Space Power Free-Piston Stirling Engine Scaling Study”, NASA/CR-182218, 1989 2 Dhar, M., “Stirling Space Engine Program, Volume 1-Final Report”, NASA/CR-1999-209164/VOL1, 1999. 3 Wood, G.W., and Lane, N.W., “Progress Update on the Sunpower Stirling Converter”, Proceedings of AIAA IECEC2005

conference, Providence, Rhode Island, 2004. 4 Wood, G.W., and Lane, N.W. “Advanced Small Free-Piston Converter for Space Power Application”, Proceedings of

Space Technology and Applications International Forum, edited by M.S. El-Genk, CP699, American Institute of Physics, 2004. 5 Wood, G.W., and Carroll, C. and Penswick, L.B., “The Sunpower/Boeing Adanced Stirling Convertor”, Proceedings of

Space Technology and Applications International Forum, edited by M.S. El-Genk, CP699, American Institute of Physics, 2005. 6 Kim, S.Y., Huth, J.J., and Wood, J.G. “Performance Characterization of Sunpower Free-Piston Stirling Engine,”

Proceedings of AIAA IECEC2005 conference, San Francisco CA, 2005 7 Mason, L.S., “A Comparison of Fission Power System Options for Lunar and Mars Surface Applications” Proceedings of

Space Technology and Applications International Forum, Albuquerque New Mexico, 2006. 8 Wood, G.W., et al, “Advanced Stirling Convertor Update” Proceedings of Space Technology and Applications

International Forum,Albuquerque New Mexico, 2006.

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