7/30/2019 144_0067 (1)

1/4

FREE PI STON STI RLING ENGINE DESIGN USING SIMILITUDE THEORY

Fabien Formosa 1, J ean-J acques Chaillout 2 1SYMME - Universit de Savoie BP 80439- 74944 ANNECY LE VIEUX CEDEX FRANCE

2LCARE, CEA-LETI-MINATEC, Grenoble, France

Abstract: The free piston Stirling engine shows some advantages for microminiature machine design: externalcombustion, closed cycle as well as the opportunity to use the converse cooling cycle. Based on non-dimensionalanalysis, the effect of the miniaturization for the design of such micro-heat engines can be studied. A similitudeapproach for Stirling engine is developed. Design charts are then set out so the engine parameters can be easilyhandled in relation with the working fluid, technological requirements and constraints. Some guidelines are thenoutlined for Stirling micro machine design.

K eywords: micro heat engine, Stirling, similitude theory, scaling

INTRODUCTION The effect of the miniaturization of micro heat

engines is crucial to asses the performanceprospective of a given technology. Based on non-dimensional analysis, turbines have been shown to bethe best promising technology for high specificpower. Indeed, they scaled as 1/L if the maximalvelocity is retained. Nevertheless, their high rotationalspeeds require specific developments to minimizewear and frictional losses. Moreover, the combustionresidence time must be large enough to obtain highefficiency [1-4].

The specific power of the Stirling cycle scales as: x pmean , in which p mean is the mean charge pressureand the operating frequency. Based on thiscriterion, the miniaturization process brings nointrinsic benefit for the Stirling cycle. Yet, it is knownfor its high efficiency and reliability especially forfree piston Stirling engines (FPSE) which appears tobe suitable for miniaturization [5]. From atechnological point of view the external combustionas well as the closed cycle are favourable advantagesof a miniaturized heat engine.

Second order models are effective tools to studythe global behaviour of classical Stirling engines [6].In the case of the FPSE, the complexity takes thingsup a notch. Indeed, in this case no mechanical linkagefixes the strokes and phase angle for the piston anddisplacer (Fig. 1). Hence, a global complex dynamicanalysis is required to predict the periodic steadyoperation. However, these approaches can not providean easy understanding of the main phenomena whichinfluence the global performances. In this frame, ananalytical model that underlines the effect of thegeometrical and thermal parameters on theperformances has been developed [7], but this modelrelies on set of parameters defined beforehand. As aconclusion, there is a need for a simplified model to

design a micro FPSE at a preliminary stage. Thesimilitude theory is an efficient alternative approachto effective prior design and appraisal of the objectiveperformances. It stems on the fundamental equationsthat describe the thermo-fluidic behaviour of theworking fluid within the machine. Organ hasproposed a set of similitude parameters dedicated tothe Stirling engine [8]. We propose here to extend thiswork to the study of the FPSE.

AA

BB

A-A

B-BBuffer spaces

Piston

Displacer

Regenerator

Heater

Cooler

Fig. 1: FPSE schematic architecture.

First, scale analysis brings out the non-dimensional groups (NDG) related to the FPSE. Therelevance of the NDG is then supported by acomparative study of ten Stirling engines, four of which are FPSE. Design charts are developed so theengine parameters (e.g. mean pressure, operatingfrequency, hydraulic radius, length, number of tubesfor the heat exchangers, piston and displacer masses,buffer spaces volumes) can be easily handled inrelation with the working fluid. Finally, the effects of the scaling down process are analyzed.

PowerMEMS 2009, Washington DC, USA, December 1-4, 20090-9743611-5-1/PMEMS2009/$202009TRF 562

7/30/2019 144_0067 (1)

2/4

THEORYK inematic similarity

From the Schmidt analysis, it is possible toanalytically express the instantaneous pressure and thepower of a given Stirling machine as:

p =M r

s 1

1 + cos ( - ) (1)

Power = pmean VE2 ( -1) sin( )

2 + 2 +2 cos( ) (2)

In which is related to geometric and kinematicparameters. Thus, the similitude process begins withthe conservation of the kinematic parameters some of them defined by Fig. 2:

= Vswc / Vswe the swept volume ratio = Tc / Te the temperature ratio the volume phase angle

Moreover, the dead volume ratio with respect tothe swept volume is kept constant during thesimilitude process. Consequently for each heatexchanger and regenerator, the following nondimensional parameter is defined:

DG x =4 rhx2 Lx n Tx

Vsw (3)

In which, n Tx is the number of tubes, rh the hydraulicradius, Lx the length of the considered heat exchangeror regenerator. x e, x k and x r for the heater,cooler and regenerator respectively.

From the expression of the power, it is possible to

define the Beale number:NBe =

Power pmean Vsw

(4)

VHC

VCC

Vh

VR

Ve

Vc

Vk

Te

Tc

800 1000

Vtot

Vc

Ve

V

Vswe

Vswc

Fig. 2: Generic scheme for the Schmidt analysis.

Energetic similarityWe assume a 1D behaviour of the gas within the

heat exchangers which is the usual assumption forStirling engine modelling. Fundamental equationsrelated to the fluid behaviour are the following:

Continuity equation

t+

z

m

Aff =0 (5)

Momentum equationt

mAff +

z

m

2

Aff 2=-

p z -

2dh

f F m |m| Aff 2

(6)

Energy equation for gas

Aff t Cv T +1/2

m

Aff

2

+ h AL ( T - Ts)

+

z Cp m T+1/2

m

Aff

2

=0

(7)

The previous equations are then modified in order

to get NDG. The reference parameters are Tc, , pmean and L =

3 Vsw which are significant for a given

engine [8]. From the result of the process, two newgroups are defined for each heat exchanger andregenerator:

NMx = Lx

r Tc (8)

DGRe =pmean rh Lx

(9)

Thermal similarity

Another important parameter is the thermaltransfer coefficient of the exchangers. Based on theNTU experimental correlations new groups aredefined therefore:

N Tx ==

Lxrhx

1.2

pmean

-0.2

(10)

For a tubes heat exchangers

N Tx ==

Lxrhx

1.5

pmean

-0.5

(11)

For a mesh grid regenerator

Dynamic similarityDynamical equilibrium equations for the moving

parts are obtained from the Newtons second law. Thus, four more non dimensional groups are derived. They account for the stiffness behaviour of the bufferspring chambers as well as the balance betweenpressure and inertia effects:

Nbu = AbVb0

(12)

NF pd =(Adcs Adhc ) pmean

md 2 L (13)

563

7/30/2019 144_0067 (1)

3/4

NF pp =Ap pmeanmp 2 L

(14)

It is worthy of note that the essential role of thedissipative effect from the pressure drop is handled bythe DGRe group. Indeed, the friction factor is related tothe Reynolds number and the geometrical aspect of

the exchangers both being kept constant by previousrequirements.

DISCUSSIONAnalysis of documented engines

In order to support the approach, the NDG havebeen evaluated for fully documented real enginesnamely: GPU-3, PD46, V160, MP1002CA, USSP40,400HP, RE-1000, CTPC. The last two of them areFPSE.

10 -3 p

10

100

50

20

30

15

70

V160

400 HP

P40

GPU3

MP1002

CTPC

10 5020 3015 70

10 -3 (L/rh)3/2

RE1000

PD46

Fig. 3: NTU group for the regenerator.

Figure 3 shows that the chosen NDG NTx (see Eq10) is indicative. For the particular case of FPSE, thedynamic groups NF pd and NF pp (Eq. 13-14) areevaluated with two more FPSE engines: B10 [9] andDFPSE [10].

20 40 60 80

f [Hz]

200

400

600

800

1000CTPC

DFPSE B10100

200

300

400

500

RE1000

p AVsw m

piston

displacer

Fig. 4: Dynamic groups for piston and displacer..

Again, it appears that the dynamic characteristicsof FPSE can defined with respect to a given nondimensional parameter.

Design charts for scaled enginesWe postulate that the swept volume Vsw is

representative of the Stirling engine size. With the

additional previous choices of the extremetemperatures Te and Tc and required power, it ispossible to define the remaining parameters. Based onthe set of NDG for the RE-1000 engine, designscharts are drawn.

The charge pressure and operating frequency canthus be easily handled in relation with the working

fluid (Fig. 5-6).

p r e s s u r e

[ M P a ]

S w e p

t v o l u m e

[ c c ]

p o w e r

[ W ]

0.1

1.

10.

0.1

1.

10.

H2 air He

1.

2.

3.

4.5.6.

8.

1.

2.

3.

4.5.6.

8.

1.

2.

3.

4.5.6.

8.

Linear scalefactor =1/10

Fig. 5: Example data plot.

Figure 5 shows the evaluation of the meanpressure of the engine with given swept volume andpower. As the pressure increases higher power can beproduced by the engine. Moreover, the same powercan be reached using a lower pressure and a suitableworking fluid. These results are in agreement with the

classical literature of Stirling engines.

20.

30.

40.

50.60.

80.

100.120.140.

20.

30.

40.

50.60.

80.

100.

120.

140.

0.1

1.

10.

0.1

1.

10.

0.01

f r e q u e n c y

[ H z ]

S w e p

t v o l u m e

[ c c ]

p o w e r

[ W ]

H2 air He

20.

30.

40.

50.

60.

80.

100.

120.140.

Fig. 6: Example data plot.

Figure 6 shows the evaluation of the operatingfrequency. Once again, the classical result that asmaller size is balanced by a higher frequency isobtained. The numerical evaluation of the relation can

564

7/30/2019 144_0067 (1)

4/4

be useful for preliminary design regardingtechnological constraints. Contrary to a geometricalscaling, it can be seen in Fig. 7 that the smaller thesize of the engine, the higher are the relative movingparts masses with respect to the whole mass of theengine.

CONCLUSIONA simple preliminary FPSE design method hasbeen developed. Scaling effect on the Stirling engineparameters can be studied. Moreover, additionalconstraints such as maximum pressure or powerrequirements can be added to easily validate asuggested design. The preliminary defined engineparameters can then be used in a refined model tooptimize the design.

REFERENCES[1] N. Mller and L.G. Frchette, Performance

analysis of Brayton and Rankine cycle Microsystemsfor portable power generation, Pro. IMECE2002,November 17-22 2002, New Orleans, Louisiana .[2] S. Tanaka, K. Hikichi1, S. Togo and al.,Worlds smallest gas turbine establishing Braytoncycle, Proc. PowerMEMS 2007, Nov 28 - 29,Freiburg, Germany , pp. 359-362.[3] J . Peirs, D. Reynaerts and F. Verplaetsen, Amicroturbine for electric power generation, Sensorsand Actuators A: Physical, Vol. 113 (1) (2004), pp.86-93.

[4] F.X. Nicoul, J. Guidez, O. Dessornes, Y.Ribaud, Two stage ultra micro turbine:thermodynamic and performance study, Proc.PowerMEMS 2007, Nov 28 - 29, Freiburg, Germany ,pp. 301-304.[5] L. Bowman, D.M. Berchowitz and al.,Microminiature Stirling cycle cryocoolers and

engines, US Patent 05749226 (1994).[6] Y . Timoumi, I. Tlili, S. Ben Nasrallah, Designand performance optimization of GPU-3 Stirlingengines, Energy 33 7 (2008) , pp. 1100-1114.[7] F. Formosa, J.J . Chaillout and O. Dessornes,Size effects on Stirling cycle micro engine, Proc.PowerMEMS 2008, 9-12 November 2008 Sendai,

J apan , pp. 105-108.[8] A.J . Organ, The Regenerator and the StirlingEngine, Mechanical Engineering Publications,London , (1997).[9] J. G. M. Saturno, Some mathematical models

to describe the dynamic behavior of the B-10 FPSE,PhD thesis of The Faculty of the Russ College of Engineering and Technology Ohio University,London , (1994).[10] J . Boucher, F. Lanzetta, P. Nika, Optimizationof a dual free piston Stirling engine, Applied ThermalEngineering 27 (2007), pp. 802811).

1001010.1 Vsw [cc]

1/1.041/2.251/4.851/10.45 scale [-]

10

100

1

0.1

1000

geometric scaling

m [g] Air

HeH2

d i s

p l a

c e

p i s t o

n

initial design

Fig. 7: Example data plot.

565