lable at ScienceDirect

Applied Thermal Engineering 89 (2015) 978e989

Contents lists avai

Applied Thermal Engineering

journal homepage: www.elsevier .com/locate/apthermeng

Research paper

Preliminary design criteria of Stirling engines taking into account realgas effects

Fernando Sala a, Costante Invernizzi a, David Garcia b, Miguel-Angel Gonzalez b,Jesús-Ignacio Prieto b, *

a University of Brescia, Via Branze 38, 25123 Brescia, Italyb University of Oviedo, Campus de Viesques, 33204 Gij�on, Spain

h i g h l i g h t s

� The theoretical limit of the gas circuit performance considers real gas effects.� The influences of physical gas properties are analysed.� Simulations predict the feasibility of operating under real gas effects.� Equations and correlations are proposed as preliminary design criteria.� The procedure is supported by an experimental database.

a r t i c l e i n f o

Article history:Received 15 December 2014Accepted 26 June 2015Available online 6 July 2015

Keywords:Stirling enginePreliminary designReal gas effectDimensional analysisPerformance correlations

* Corresponding author.E-mail address: jprieto@uniovi.es (J.-I. Prieto).

http://dx.doi.org/10.1016/j.applthermaleng.2015.06.071359-4311/© 2015 Elsevier Ltd. All rights reserved.

a b s t r a c t

This article deals with the main geometric parameters, operating variables and experimental results ofengines with different size and characteristics, as well as simulation predictions at operating conditionsfor which the working fluid may evidence real gas effects. The concept of dimensionless quasi-staticindicated work, introduced previously, is computed assuming both ideal and real gas models. Bothvalues are compared with the Schmidt's model prediction, to evaluate separately how the mechanismsimplification and the equation of state affect. The influence of physical gas properties are also analysedregarding the gas circuit performance and mechanical efficiency. Pending of experimental corroboration,some simulations show the possibility of obtaining interesting operating conditions under real gaseffects.

Semi-empirical equations and experimental correlations are proposed as preliminary design criteria.The ratio between the dimensionless values of the maximum indicated power (a sort of ‘indicated’ Bealenumber) and the quasi-static indicated work is analysed. The procedure to estimate the maximumindicated power is completed by analysing the characteristic Mach number (dimensionless rotationfrequency). Analogous procedure is applied for the maximum brake power and corresponding enginespeed. Use of dimensionless variables facilitates the generalization of analyses by means of dynamicsimilarity criteria.

© 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Potential benefits justify that the Stirling engine has competedwith other technologies over the past three decades for applica-tions such as cogeneration at industrial and domestic scales, un-derwater systems and air independent propulsion of submarines,

3

solar thermal energy conversion, and space vehicles. The com-mercial battle is full of ups and downs, since the cost of the productis mainly determined by the volume of series production of itscomponents, and by fluctuations in the price of fossil fuels. So far,the Stirling cycle has achieved greater commercial success in theirapplications as reverse cycle machine, an option that other thermalengines do not have.

The gas circuit and the drive mechanism are the main sub-systems of a Stirling engine. The numerical computing powercurrently available allows the development of models to study

Nomenclature

Acx cross-sectional area of space x (m2)Cf local, instantaneous friction factor

¼ Dp$2rðAxx= _mÞ2ðrhx=LxÞcp constant pressure heat capacity (J/kg K)cv constant volume heat capacity (J/kg K)h convective heat transfer coefficient (W/m2 K)k thermal conductivity (W/m K)Lx length of space x (m)_m mass flow rate (kg/s)m1 reference mass of drive mechanism moving parts (kg)NB Beale numberNm characteristic drive mechanism number ¼ m1RTC/

(pmVsw)

NMA characteristic Mach number ¼ nsV1=3Sw =

ffiffiffiffiffiffiffiffiRTC

pNma local, instantaneous Mach number ¼ _m

ffiffiffiffiffiffiffiffiRTC

p=ðpAxxÞ

NMA,max Mach number corresponding to the maximum

indicated power ¼ ns;maxV1=3Sw =

ffiffiffiffiffiffiffiffiRTC

pN*MA;max Mach number corresponding to the maximum brake

power ¼ n*s;maxV1=3Sw =

ffiffiffiffiffiffiffiffiRTC

pNma,max local Mach number corresponding to the maximum

mass flow rate

Np characteristic pressure number ¼ pmV1=3Sw =ðm ffiffiffiffiffiffiffiffi

RTCp Þ

NPR characteristic Prandtl number ¼ mcp/kNpr local, instantaneous Prandtl numberNre local, instantaneous Reynolds number ¼ 4 _mrhx=ðmAxxÞNre,max local Reynolds number corresponding to the

maximum mass flow rateNSG characteristic Stirling number ¼ pm/(mns)ns engine speed (rev/s)ns,max engine speed at maximum indicated power (rev/s)n*s;max engine speed at maximum brake power (rev/s)Nst local, instantaneous Stanton number ¼ hAxx=ð _mcpÞNTCR regenerator thermal capacity ratio ¼ rRcR/

(rcp) ¼ rRcRTC/pm$(g � 1)/gNW West numberNa characteristic regenerator thermal diffusivity number

¼ aR=ðV1=3sw

ffiffiffiffiffiffiffiffiRTC

p Þp pressure (Pa)PB brake power (W)pcr pressure at the critical point (Pa)Pind indicated power (W)pm mean pressure (Pa)Pmec mechanical power losses ¼ Pind � PB (W)pr reduced pressure ¼ p/pcrprm reduced mean pressure ¼ pm/pcrR specific gas constant (J/kg K)rhx hydraulic radius of space x (m)Tcr temperature at the critical point (K)Trx reduced temperature at space x ¼ Tx/Tcr

Tx working gas temperature at space x (K)V volume (m3)VC swept volume of the compression piston (m3)Vdx dead volume of space x (m3)VE swept volume of the expansion piston (m3)Vsw net swept volume ¼ Vmax � Vmin (m3)W0 quasi-static indicated work per cycle (J)a, b, c, d dimensionless coefficients of mechanical power lossesa phase angle (rad)aR regenerator material thermal diffusivity (m2/s)acx dimensionless cross-sectional area of space

x ¼ Axx=V2=3sw

d1 … dn dimensionless geometrical parameters, includingthose characteristic of the drive mechanism

hmec mechanical efficiency ¼ PB/Pindhmec,max mechanical efficiency at n*s;max

F dimensionless factor of linear losses of indicatedpower

g adiabatic coefficient ¼ cp/cvk swept volume ratio ¼ VC/VE

ks swept volume ratio according to Iwamoto et al. ¼ Vsw/VE

lhx dimensionless hydraulic radius of space x ¼ rhx=V1=3sw

m working fluid viscosity (Pa s)mdx dead volume ratio of space x ¼ Vdx/Vsw

mL lubricant viscosity (Pa s)¶V Regenerator volumetric porosityq crank angle (rad)J dimensionless factor of quadratic losses of indicated

powerr density (kg/m3)rRcR regenerator material volumetric specific heat capacity

(J/m3 K)t temperature ratio ¼ TC/TEzB dimensionless brake power ¼ PB/(pm Vsw ns)zind dimensionless indicated power ¼ Pind/(pm Vsw ns)zmec dimensionless mechanical power losses ¼ Pmec/

(pm Vsw ns)z0 dimensionless quasi-static work per cycle ¼ W0/

(pm Vsw )

SubscriptsC cooler or compression spacecc compression cylinderE heater or expansion spaceec expansion cylinderR regeneratorx generic space

SuperscriptsR considering real gas effectsS according to Schmidt cycle

F. Sala et al. / Applied Thermal Engineering 89 (2015) 978e989 979

these subsystems with fewer simplifying assumptions than waspossible decades ago. Thus, the development of Stirling enginemodels capable of being integrated into schemes for the overalltechno-economic analysis of applications is still a topic of currentinterest [1e6]. Because all models must be validated experimen-tally, the benefit of the advanced models is not greater accuracy butthe ability to analyse physical phenomena dependent on variableswhose experimental measurement is practically impossible. For

now, simple models are needed at the preliminary design stage,while the advanced models can be appropriate for optimisationtasks.

Beale and West numbers were probably the simplest and mostcommonly used criteria for sizing of engines at the preliminarydesign stage. However, without prejudice to its historical impor-tance, it should be recognized that both concepts are experimentalcorrelations where great simplifications have been made. So far it

F. Sala et al. / Applied Thermal Engineering 89 (2015) 978e989980

has hardly been highlighted that these correlations have twoimportant limitations, namely: (a) do not provide criteria of inde-pendent design for the gas circuit and the drive mechanism,because they refer to brake power; and (b) do not allow to accu-rately estimate the maximum brake power, since there is no cor-relation or design criteria generally accepted to estimate the speedcorresponding to the maximum brake power. In previous publica-tions, more complete preliminary design criteria have been pro-posed bymeans of dimensional analysis, considering separately thegas circuit and the drive mechanism. In this method, the concept ofquasi-static simulation and characteristic Mach number areessential for dimensioning the gas circuit, while the characteristicStirling number is important from the standpoint of mechanicalefficiency. In the present article that procedure is revised. Sincerecent investigations claim that Stirling engine can be interesting atoperating conditions for which theworking fluid may evidence realgas effects [7,8], the influence of critical pressure and temperatureare taken into account. However, a comprehensive study of oper-ation in conditions where effects of real gases can be producedexceeds the objectives of the present work, seeking only to identifysome points of operation where such effects can be at leastacceptable and thus guide future research that may extend thescope of application of the Stirling engine. The development ofStirling engines operating at very low temperatures could befacilitated by the experience proven for applications as reversecycle machine [9].

2. Theoretical limits of the gas circuit performance

The concept of quasi-static dimensionless indicated work z0 wasintroduced in previous publications to express the theoretical limitof the Stirling engine gas circuit performance [10]. It can be iden-tified as the value obtained from a simulation assuming an idealcycle without any losses caused by thermal or mechanical irre-versibilities. Consequently, gas processes should be isothermal atthe temperatures of the heat reservoirs, the regenerator efficiencyshould be ideal (100%) and the pressure drop along the gas circuitshould be negligible. The additional hypotheses of heat reservoirsof infinite heat capacity, perfect mixing of the gas contained in eachspace, absence of gas leakage and heat conduction losses in theregenerator (thermal short cuts) allow the quasi-static Stirling cycleto reach the Carnot efficiency regardless the drive mechanismconsidered.

The quasi-static indicated work per cycle z0 depends on thetemperatures, the dead volumes, and the drive mechanism pa-rameters, according to the following functional relationship:

z0 ¼ f ðt; k; d1;…; dn;mdec;mdE;mdR;mdcc;mdCÞ (1)

which leads to results not too different from Schmidt's modelpredictions for engines with near-harmonic drive mechanism.

The assumptions of simple harmonic drive and ideal EOS(equation of state) allow closed analytical solutions to be obtained,which is useful to describe the thermodynamic cycle of theworkingfluid with sufficient accuracy at the preliminary design stage [11].Table 1 shows the difference between values of zS0 and z0 for threebenchmark prototypes. This difference is relatively high for therhombic-drive GPU3 engine if zS0 is computed assuming the phaseangle between pistons that maximise zS0. The difference may bemore significant for Rosseyoke mechanisms [12].

On the other hand, the possibility of realizing Stirling enginespressurized with working fluids presenting real gas effects hasbeen analysed in recent work [7,8]. The usual working fluids inStirling engines are hydrogen, helium and air (or nitrogen), oper-ating at conditions where their thermodynamic behaviour can be

approximated like an ideal gas. Changing the working condition ofthe engine or changing the working fluid under the same operatingcondition can make the assumption of ideal gas invalid. Therefore,to set the general theoretical limit of the gas circuit performance,the functional relationship (1) must be modified to consider an EOStaking into account real gas effects, for example introducing asparameters the reduced cooler temperature and mean pressure:

zR0 ¼ f ðt; k; d1;…; dn;mdce;mdE;mdR;mdcc;mdC ; TrC ; prmÞ (2)

Fig. 1 shows the effects on zR0 due to variations in reducedtemperature and pressure for the P40 engine operating with ni-trogen and the GPU3 engine operating with hydrogen at the sametemperature ratio of t ¼ 0.5. The calculations were made throughthe PengeRobinson EOS [17]. It is observed that the trend of zR0 issimilar for both cases. This can be explained thanks to the principleof corresponding states which shows that all gases, whencompared at the same reduced condition, have approximately thesame deviation from the ideal gas behaviour. In this figure it is alsoseen that z0 is an asymptote of zR0 lines. In particular it was foundthat for TrC > 3 the difference between z0 and zR0 regarding z0 is lessthan 5% irrespective of the value of prm. It is also interesting to notethat zR0 can result in more than double that z0 if the minimum cycletemperature is diminished (TrC / 1). This effect is due to thereduction of fluid compressibility under thermodynamic condi-tions close to the critical point temperature. Though it is difficult toimagine an engine working with a heat sink near the nitrogen orhydrogen critical points (critical temperatures of these fluids arerespectively �146.95 �C and �240.17 �C [18]), it is possible lookingfor other working fluids having a critical point close to operatingtemperatures of usual Stirling engine applications. A candidate tobe investigated in future works could be carbon dioxide that has acritical temperature of about 31 �C [18].

3. Influence of the physical properties on the gas circuitperformance

The description of the actual operation of the gas circuit requirestaking into account irreversibilities inherent to the circulation ofthe working fluid through the different engine spaces, particularlythe heat exchangers. Phenomena like pressure drop and heattransmission are modelled by simulation programs using empiricalcorrelations in which the influence of mass flow rate, physicalproperties of the working fluid and geometry are consideredthrough dimensionless numbers (Reynolds, Prandtl, Mach, etc.)that must be computed at instantaneous and local scales. The in-fluence of reduced temperature and pressure on fluid physicalproperties (density, heat capacities, dynamic viscosity, etc.) cancause the corresponding dimensionless numbers are outside therange of validity of the correlations.

Fig. 2 shows the values of nitrogen physical propertiescomputed using Aspen Plus® [19]. It is seen that the influence of thereduced pressure on the dynamic viscosity, heat capacity ratio(adiabatic coefficient) and heat capacities is almost negligible forreduced temperature values greater than 3. The asymptotic ten-dency of viscosity is justified by the kinetic theory of gases whichpredicts that m is independent on pressure and proportional to thesquare root of temperature. The same tendency applies for thethermal conductivity, while the heat capacities tend to the constantvalues of ideal gases. Regarding the variation of gwith Tr and pr, it ismainly caused by cp variations since cv is almost independent onthe reduced conditions.

In order to carry out a preliminary evaluation of real gas effectsfor reduced temperatures lower than 3, Table 2 has been con-structed for the P40 engine, which develops approximately the

Table 1Influence of drive mechanism on the quasi-static simulation.

Fig. 1. Influence of reduced pressure and temperature on the quasi-static simulation.

Fig. 2. Influence of reduced pressure and temperature on nitrogen physical properties.

F. Sala et al. / Applied Thermal Engineering 89 (2015) 978e989 981

maximum brake power of 19,500 W at 1700 rpm, operating withnitrogen at the mean pressure of 150 bar and temperatures ofTE ¼ 1023 K and TC ¼ 333 K [14]. The temperature values have beenmodified to obtain the ratio t¼ 0.5. Low cooler temperatures can beinteresting for practical applications like Stirling engines with coldsink temperature regulated by liquid nitrogen or cold energyrecovering in LNG (liquefied natural gas) terminals [20]. The

pressure has not been changed regarding the original level, whichcombined with lower temperatures may imply a requirementreduction interesting from the standpoint of strength of materials.

It is seen that differences between zR0 and z0 increase as TrCvalues tend to unity, but this thermodynamic improvement isaccompanied by fluid dynamic effects that require additionalevaluation. For example, the values of the Reynolds number

Table 2Influence of real gas effects on zR0 and local, instantaneous dimensionless numbers for the P40 engine gas circuit.

Gas Nitrogen Nitrogen Nitrogen Nitrogen Nitrogen Nitrogen

Vsw (cm3) 134.40 134.40 134.40 134.40 134.40 134.40pm (bar) 150 150 150 150 150 150TE (K) 757.2 631.0 504.8 378.6 315.6 277.6TC (K) 378.6 315.5 252.4 189.3 157.8 138.8ns (rpm) 1700 1700 1700 1700 1700 1700t 0.5 0.5 0.5 0.5 0.5 0.5Trc 3 2.5 2 1.5 1.25 1.1prm 4.47 4.47 4.47 4.47 4.47 4.47z0 0.113 0.113 0.113 0.113 0.113 0.113zR0 0.122 0.128 0.140 0.177 0.234 0.290Nma,max 0.0043 0.0047 0.0055 0.0072 0.0091 0.0102Nre,max 730 983 1400 2293 3084 3369NTCR 44.93 36.72 28.14 18.00 12.10 8.58Npr 0.74 0.77 0.86 1.11 1.37 1.43

F. Sala et al. / Applied Thermal Engineering 89 (2015) 978e989982

calculated at the regenerator middle section for the maximummass flow rate increase markedly despite the same rotational fre-quency is maintained. The values of the local, instantaneous Machnumber calculated for the same conditions also increase, but theyremain below the limits for which according to Organ [21]compressibility effects could lead to unpredictable results. Thevalues of the local Prandtl number calculated at the same sectionfor the mean pressure are mainly influenced by the tendency of thespecific heat capacity at constant pressure, as Fig. 3 shows. On theother hand it is seen that, assuming the value rRcR ¼ 4.32$106 J/(m3K), the regenerator thermal capacity ratio NTCR, calculated forthe mean pressure, decreases to values for which new materials ornew geometry could be required to maintain the regeneratorefficiency.

From the analyses of Stirling engines with very different size andcharacteristics, an equation has been proposed to explicitly statethe influence of the velocity (and the consequent irreversibilities)on the indicated power [10]. Taking into account the possibility ofreal gas effects, such an approach is now updated according to thefollowing expression:

zind ¼ zR0 � FRNMA �JRN2MA (3)

Fig. 3. Influence of reduced pressure and temperature on Prandtl number.

The coefficients FR and JR are macroscopic representations ofthe indicated power losses and can be estimated by integratingconservation laws along the gas circuit and the cycle. In the pro-cedure, the influence of physical properties of the working fluid isconsidered through abovementioned local, instantaneous dimen-sionless numbers which are included in experimental correlationsof the following type:

Cf ;NstN2=3pr ¼ a

Nreþ b (4)

where a and b depend on geometric parameters. The additionalinfluence of Nma on a and b is claimed by some authors forparticular gases and operating conditions [21].

Thus, variations of local, instantaneous dimensionless numberscaused by real gas effects, like those listed in Table 2, are equivalentat macroscopic scale to the inclusion of the reduced temperatureand pressure in the list of variables influencing onFR andJR. In thecharacteristic dimensionless numbers, it is noted that physicalproperties of the working gas are calculated at the reference tem-perature of TC. The influence of the characteristic Prandtl number isusually neglected in the range of ideal gas performance but now isalso considered, that is to say:

FR;JR¼f�t;k;d1;…;dn;mdce;mdE;mdR;mdcc;mdC ;acE;acR;acC ;lhE;

lhR;lhC ;g;Np;NPR;Na;NTCR;TrC ;prm�

(5)

Imposing the condition dPind/dns ¼ 0 on the function Pindderived from the equation (3) allows to obtain the two essentialdata for the indicated power analysis [10]:

NMA;max ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiFR� �2 þ 3zR0J

Rq

�FR

3JR (6)

zind;max ¼ lþ 23ðlþ 1Þz

R0 (7)

where the parameter l is defined in the real interval [0, 1] bymeansof:

l ¼ FRffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�FR�2 þ 3zR0J

Rq (8)

At the moment there is no experimental data about Stirlingengines working with strong real gas effects, so in this article FR

and JR have been estimated for some cases by means of the

F. Sala et al. / Applied Thermal Engineering 89 (2015) 978e989 983

simulation program GGSISM® [22,23], that uses the Van der WaalsEOS and the Kays and London correlations [24]. The procedureconsists in computing the maximum indicated power and thecorresponding engine speed, from which FR and JR can be ob-tained taking into account that equations (6) and (7) lead to:

FR ¼ 2zR0 � 3zind;max

NMA;max(9)

JR ¼ 2zind;max � zR0

N2MA;max

(10)

For the time being it has not been possible to carry out simu-lations for a variety of engines, because operating conditions whereeffects of real gases may occur can lead to values of dimensionlessvariables that are outside the range of validity of the friction andheat transfer correlations implemented in existing programs,which frequently hinders the numerical convergence. Table 3shows the simulation results for the P40 engine with hydrogenand air as the working fluids operating at different thermodynamicconditions. The simulations evidence that FR and JR are stronglydependent on both reduced temperature and pressure. For bothgases, it is observed that real gas effects can lead to operating pointswhich are favourable from the thermodynamic viewpoint (zR0 > z0)but the advantages are compensated by increases of indicated po-wer losses, that is to say of FR and JR. Both thermodynamic andfluid-dynamic performances can be summarised by means of onlyone index (NMA,max) which expresses the development level of thegas circuit design [10]. In short, it is concluded that despite thecomputed NMA,max values are lower than those listed in Table 4 forthis engine at high temperatures, some operating points with aircould be applicable. This is not the case for hydrogen due to thevery low temperatures of questionable interest in practice. In anycase, the finding must be corroborated by future experimentalresearch.

4. Influence of the physical properties on the mechanicalpower losses

Themechanical power losses Pmec can be indirectly expressed bythe following model, developed from experimental data ofbenchmark engines [16]:

zmec ¼ a exp�� b N�1

SG

�þ cþ dN�1

SG (11)

The coefficients of this equation fulfil functional relationships ofthe following type:

Table 3Simulation predictions for the P40 engine operating with hydrogen and air at low reduc

Gas Hydrogen Hydrogen Air

Pind,max (W) 2876 1080 10,492ns,max (rpm) 2205 2058 1116pm (bar) 22.3 9.01 146TE (K) 132.8 132.8 504.8TC (K) 66.4 66.4 252.4t 0.50 0.50 0.50TrC 2 2 1.9prm 1.69 0.69 3.85z0 0.113 0.113 0.113zR0 0.126 0.118 0.140zind,max 0.066 0.065 0.072NMA,max 0.0036 0.0034 0.003FR 15.15 12.12 18.22JR 431 1094 300

a; b;c; d ¼f

d1; :::; ds; t;

pmVsw

m1RTC;m2

m1; :::;

mn

m1;

I1m1V

2=3sw

;I2I1; :::;

InI1;

g;mL

m;Na;NTCR;Np

!

(12)

where d1 … ds represent the list of dimensionless, geometrical pa-rameters of the whole engine, i.e. those related to gas circuit, drivemechanism, sealing, clearances, and other geometrical character-istics. For a given engine operating with different gases atapproximately constant TC, it can be written:

a; b;c; dzf�t;g;

mL

m;Nm;Np

�(13)

The following equation fits with acceptable accuracy the GPU3engine data operating with hydrogen and helium in the range0.295 � t � 0.347 and mean pressures from 13.8 to 69.6 bar (35operating points) [16]:

zmec ¼ 1:8772N0:6910m

N0:5540p

exp

� 9:7656$109

ðmL=mÞ0:0197N�1SG

!

þ 2:1469$107

N0:3890m N1:0430

p(14)

It can be noted that, according to the functional relationship(13), these numerical coefficients and exponents implicitly includethe influence of g and t variations.

For the V160 engine, operating with helium at TE ¼ 898K, TCbetween 308 and 338 K, and pm between 37 and 125 bar (11operating points), the experimental data are in agreement with thefollowing equation [13]:

zmec ¼ 0:5001

N0:139p

exp

� 7:250$1010

ðmL=mÞ0:020N�1SG

!þ 0:580N0:199p

þ 2:990$108N0:008p

NSG(15)

In this correlation, t, g and Nm were not considered as variablesdue to the small variation of t and the utilization of only oneworking fluid.

The P40 and M102C engines have been re-analysed through themodels of equations (3) and (11), since it has recently been evi-denced that both indicated power and brake power can be char-acterized if sufficient experimental brake power data are available

ed temperature.

Air Air Air

13,780 17,724 10,8721169 1229 1062178 213 113504.8 504.8 315.5252.4 252.4 157.75

0.50 0.50 0.501.9 1.9 1.254.69 5.61 3.330.113 0.113 0.1130.143 0.135 0.2220.074 0.076 0.101

5 0.0036 0.0038 0.004217.36 11.13 28.42

352 1065 0

Table 4Main geometric parameters, operating variables and experimental power and efficiency data used in preliminary design correlations.

Engine GPU3 GPU3 M102C M102C M102C M102C M102C M102C M102C P40 P40 P40 V161 V160 V160 V160 V160 V160 Ecoboy Ya-1 Ya-2

[Ref.] [10] [16] [10] [16] [15] [15] [15] [15] [15] [15] [15] [14] [15] [14] [15] [14] [15] [13] [13] [13] [13] [13] [13] [16] [10] [16]

Type [25] Beta Beta Beta Beta Beta Beta Beta Beta Beta Alpha Alpha Alpha Alpha Alpha Alpha Alpha Alpha Alpha Gamma Gamma AlphaGas H2 He Air Air Air Air Air Air Air H2 He N2 H2 He He He He He Air Air AirVsw (cm3) 119.7 119.7 64.2 64.2 64.2 64.2 64.2 64.2 64.2 134.4 134.4 134.4 194.5 194.5 194.5 194.5 194.5 194.5 81.4 25,132.7 20,106.2R (J/kg.K) 4120 2080 287 287 287 287 287 287 287 4120 2080 297 4120 2080 2080 2080 2080 2080 287 287 287g 1.41 1.67 1.40 1.40 1.40 1.40 1.40 1.40 1.40 1.40 1.67 1.40 1.41 1.67 1.67 1.67 1.67 1.67 1.40 1.40 1.40TE (K) 977 977 1073 1073 1073 1073 1073 1073 1073 1023 1023 1023 1005 898 898 898 898 898 703 403 368TC (K) 288 286 333 333 333 333 333 333 333 333 333 333 328 336 329.8 323 316.4 309 323 313 300pm (bar) 27.6 27.6 12.41 11.03 9.66 8.28 6.90 5.52 4.14 150 150 150 112.2 120 100 80 60 40 8 1 7Pind,max (W) 3850 2480 539 503 458 418 351 276 188 77,365 57,150 27,755 28,182 13,428 11,443 9375 7200 4925 115 233 1142ns,max (rpm) 4500 2900 1507 1580 1645 1749 1764 1733 1576 5775 4150 1930 7120 3430 3445 3460 3480 3500 1840 165 284PB,max (W) 2700 1600 415 392 361 333 282 222 153 45,245 29,695 19,525 17,860 8355 6835 5270 3675 2115 60 146 670n*s;max (rpm) 3600 2350 1425 1490 1545 1635 1650 1615 1475 4835 3265 1670 5150 2460 2365 2235 2045 1750 1150 135 186t 0.295 0.293 0.310 0.310 0.310 0.310 0.310 0.310 0.310 0.326 0.326 0.326 0.326 0.374 0.367 0.360 0.352 0.344 0.459 0.777 0.815ks 0.991 0.991 1.081 1.081 1.081 1.081 1.081 1.081 1.081 1.414 1.414 1.414 1.218 1.218 1.218 1.218 1.218 1.218 1.000 0.625 1.000rhR/LR 0.0012 0.0012 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0006 0.0006 0.0006 0.0009 0.0006 0.0006 0.0006 0.0006 0.0006 0.0013 0.0020a 0.0025aP

mdx 1.661 1.661 0.776 0.776 0.776 0.776 0.776 0.776 0.776 1.794 1.794 1.794 2.005 1.729 1.729 1.729 1.729 1.729 3.439 2.480 3.730Vmax/Vmin 1.497 1.497 1.728 1.728 1.728 1.728 1.728 1.728 1.728 1.491 1.493 1.493 1.420 1.475 1.475 1.475 1.475 1.475 1.223 1.245 1.236NTCR 451 447 1159 1303 1488 1736 2084 2605 3473 96 96 96 126 121 142 174 228 334 1743 13,514 1850Np 1.43Eþ7 1.02Eþ7 8.06Eþ6 7.16Eþ6 6.27Eþ6 5.38Eþ6 4.48Eþ6 3.58Eþ6 2.69Eþ6 6.76Eþ7 4.81Eþ7 1.26Eþ8 5.82Eþ7 4.31Eþ7 3.67Eþ7 3.01Eþ7 2.31Eþ7 1.59Eþ7 5.84Eþ6 5.12Eþ6 3.51Eþ7zind,max 0.155 0.155 0.269 0.270 0.269 0.270 0.270 0.270 0.269 0.100 0.103 0.107 0.109 0.101 0.102 0.105 0.106 0.109 0.058 0.034 0.017NMA,max 0.0034 0.0031 0.0032 0.0034 0.0035 0.0038 0.0038 0.0037 0.0034 0.0042 0.0043 0.0052 0.0059 0.0040 0.0040 0.0041 0.0041 0.0042 0.0044 0.0027 0.0044zR0 0.312 0.314 0.404 0.404 0.404 0.404 0.404 0.404 0.404 0.183 0.183 0.183 0.202 0.191 0.194 0.198 0.201 0.205 0.114 0.068 0.033FR 46.25 51.40 0.00 0.00 0.00 0.00 0.00 0.00 0.00 16.05 13.93 8.72 13.10 20.10 20.10 20.10 20.10 20.20 12.70 12.60 3.10JR 0 0 12,775 11,575 10,715 9450 9300 9625 11,675 910 1200 1120 455 675 675 675 675 665 60 0 95zB,max 0.136 0.124 0.219 0.223 0.226 0.230 0.232 0.233 0.234 0.070 0.068 0.087 0.095 0.087 0.089 0.091 0.092 0.093 0.048 0.026 0.015N*MA;max 0.0027 0.0025 0.0031 0.0032 0.0033 0.0035 0.0036 0.0035 0.0032 0.0035 0.0033 0.0045 0.0043 0.0028 0.0028 0.0026 0.0024 0.0021 0.0027 0.0022 0.0029

hmec,max 0.729 0.667 0.774 0.783 0.793 0.802 0.808 0.810 0.819 0.602 0.549 0.720 0.692 0.681 0.667 0.649 0.623 0.583 0.609 0.647 0.672

a Estimated value.

F.Salaet

al./Applied

Thermal

Engineering89

(2015)978

e989

984

Fig. 4. Comparison between quasi-static pressures for the GPU-3 engine operatingwith hydrogen at different reduced temperature.

F. Sala et al. / Applied Thermal Engineering 89 (2015) 978e989 985

[15]. For the P40 engine, with hydrogen, helium and nitrogen as theworking fluids at TE ¼ 1023 K, TC ¼ 333 K and pm ¼ 150 bar (23operating points), it has been obtained the following equation:

zmec ¼ 4:780N0:723m

N0:528p

exp�� 1:120$1010N�1

SG

�þ 3:990$107

N0:375m N1:050

p

þ 6:400$107N�1SG

(16)

For the M102C engine, with air at TE¼ 1073 K, TC ¼ 333 K and pmbetween 4.14 and 12.41 bar (28 operating points), it has been ob-tained the following equation:

zmec¼30:000N0:500m

N0:507p

exp��9:400$109N�1

SG

�þ 137:000N0:389m N0:316

p(17)

In equations (16) and (17), the ratio mL/mwas excluded due to thelow exponent obtained for both the GPU3 and V160 cases.

The equation (8) model is also fulfilled by experimental data ofthe Yamanokami-1 engine, with air at TE ¼ 373 K, TC ¼ 298 K andpm ¼ 1 bar (12 operating points) [3], as well as the Yamanokami-2engine, with air at TE ¼ 368 K, TC ¼ 300 K and pm ¼ 7 bar (12operating points) and the Ecoboy engine, with helium at TE¼ 703 K,TC ¼ 323 K and pm ¼ 8 bar (16 operating points) [16]. Other enginesevidence tendencies which seem in agreement with similarmodels, but there are not sufficient data to carry out rigorousanalyses.

These empirical equations have the feature of making explicitthe influence of engine speed and can be interpreted as the result ofintegrating local, instantaneous dimensionless mechanical powerlosses that can be expected having a dependence on velocityanalogous to Stribeck's curve [16]. The mechanical power lossesalways grow with engine speed at normal operating points, but therapidity of the increase is relatively lower for engines with d z 0.Local velocities of mechanical moving parts are proportional to nsand local resistance forces and torques actuating on those parts areproportional to pm. The geometrical characteristics, working gasproperties and boundary temperatures complete the set of pa-rameters necessary to define the instantaneous pressure variation,which therefore is implicitly considered in the abovementionedempirical correlations by means of the variables t, Np and NSG. Totake into account the influence of real gas effects on the instanta-neous pressure variation, TrC and prmmust be added to the variablesof the empirical model.

For a preliminary estimation of real gas effects on the instan-taneous pressure variation, it can be considered that the equation(7) leads to the following inequality:

12zR0 � zind;max � 2

3zR0 (18)

which allows irreversibilities to be interpreted as the cause of adamping without offset of the quasi-static pressure wave, being thedamping factor bracketed between 1/2 and 2/3.

These limits are in accordance with researchers' observationsabout the practical Stirling engine performance. For example,Walker [25] states that “a value in excess of 0.4 for the relative effi-ciency is evidence of a well-designed machine. The maximumachievable value is about 0.7”. The relative efficiency is the ratiobetween that of a real engine and the Carnot efficiency. Since theregenerator efficiency of a well-designed engine typically exceedsvalues of 90%, that ratio can be approximated to zind;max=z

R0.

Fig. 4 shows the quasi-static pressure diagram for the GPU-3engine operating with hydrogen at prm ¼ 3 and t ¼ 0.5, for twodifferent values of TrC among which real gas effects are enclosed.

Since the quasi-static pressure differences are approximately lowerthan 10%, it seems possible to find appropriate operation points inwhich the increase in the pressure wave amplitude caused by realgas effects could be compatible with acceptable mechanicalperformance.

5. Preliminary design

Preliminary design criteria to be described below refer to ki-nematic Stirling engines, which have been the most used so far. Theprocedure is based on experimental data and theoretical conceptsthat have been used to derive equations (3) and (11). The use ofdimensionless variables facilitates the generalization of the resultsfor engines with very different characteristics, bymeans of dynamicsimilarity criteria.

5.1. Indicated power performance

The equation (3) justifies the use of quasi-static or isothermalmodels, like that of Schmidt, for parametric optimizations of theindicated work at practical engine speeds, because the calculationof zR0 and the inequality (18) allow zind,max to be estimated, i.e.:

zind;max

.zR0 ¼ 0:58±0:08 (19)

The accuracy of this semi-empirical rule has been recentlyimproved by updating and re-analysing a database that containsthe main geometric parameters, operating variables and experi-mental results of power and efficiency corresponding to engines ofdifferent size and characteristics, listed in Table 4 [15].

To obtain a practical design tool, it has tried to correlate zind,maxwith the minimum number of variables involved in equations (2)and (5). Since pressure drop is usually higher in the regeneratorthan in other heat exchangers, the ratio rhR/LR ¼ lhRacR/mdR has beenselected as an index of indicated power losses.

The following correlation matches the experimental data withCV(RMSE) ¼ 6.6% and R2 ¼ 0.9863:

zind;maxz2:249 zR01:054

�rhRLR

�0:190

(20)

F. Sala et al. / Applied Thermal Engineering 89 (2015) 978e989986

Fig. 5 allows experimental data and correlation predictions to becompared. Simulation results listed in Table 3 for the P40 engineoperating with air, as well as measurements made on an enginedeveloped by PHYWE for teaching purposes [26] are also repre-sented in Fig. 6, using parametric lines for three values of the ratiorhR/LR. Despite this engine does not have a regenerator in the strictsense, it is deduced that its displacer provides a regenerative effectequivalent to a regenerator with rhR/LR z 0.0020.

To complete the calculation of the maximum indicated power,some design criterion about the corresponding speed is necessary,since this is not an independent variable. To this end, the followingcorrelation has been constructed using t and

Pmdx as the most

influential variables on zR0, while rhR/LR, g and Np are indices ofindicated power losses:

NMA;maxz0:001913 ð1�tÞ0:355�rhRLR

�0:223

g�0:220�X

mdx

�0:217N0:146p

(21)

This correlation matches the experimental data withCV(RMSE) ¼ 11.0% and R2 ¼ 0.6233 (Fig. 7). In the future it is ex-pected to improve the correlation accuracy by means of the in-clusion of additional experimental data and may be the revision ofthe influencing variables.

In short, the calculation of zR0 and FR and JR, by means ofequations (9) (10) (20) and (21), allows indicated power maps to beconstructed for series of constant temperatures andmean pressure.

5.2. Indicated efficiency

The lack of experimental data about heat power consumptionhinders the analysis of indicated efficiency. In recent work [13], ithas been observed that at very low engine speeds, the indicatedefficiency should tend to its well-known quasi-static value, namely:

limns/0

hind ¼ 1� t (22)

as well as the following constraint for the dimensionless heat po-wer supplied to the engine heater:

0.000

0.100

0.200

0.300

001.0000.0

DIM

ENSI

ON

LESS

MA

XIM

UM

IND

ICAT

ED P

OW

ER (E

XPE

RIM

ENTA

L)

DIMENSIONLESS MAXIMUM I

Fig. 5. Correlation of dimensionles

limns/0

_QE

pmVswns¼ z0

1� t(23)

Thus, the dimensionless heat power consumption _QE=ðpmVswnsÞcan be analysed based on dimensionless groups that are charac-teristic of the Stirling engine's performance. The following resultobtained for the V160 engine operating with helium at TE ¼ 898 K,TC between 308 and 338 K, and pm between 37 and 125 bar can beused for guiding further research on the subject:

_QE

pmVswnsz

z0

1� tþ 8:871N0:101

MA N�0:230p (24)

5.3. Maximum brake power

Practical preliminary design procedures have usually beendeveloped after simplifying the influence of a part of the largenumber of performance parameters related to the two main sub-systems, namely, the gas circuit and the drive mechanism. How-ever, the validity of equations or empirical correlations dependsrespectively on assumptionsmade to simplify theoretical models orranges taken into account in experimental measurements.

For many years, maximum brake power predictions have beensupported by the correlation based on the Beale number NB, thatapplies for high temperatures (t z 0.3) and can be considered adefinition of dimensionless maximum brake power:

NB ¼ zB;max ¼ PB;max

pmVswn*s;maxz0:15 (25)

More general and accurate predictions can be expected from theconcept of the West number NW since the influence of t on zB,max isevaluated by means of the following correlation:

NW ¼ PB;max

pmVswn*s;maxTE�TCTEþTC

z0:25 (26)

that can also be written as follows:

003.0002.0

NDICATED POWER (PREDICTED)

GPU-3 (He)GPU-3 (H2)M102C (Air)P-40 (H2)P-40 (He)P-40 (N2)V-161 (H2)V-160 (He)Ecoboy (Air)Yamanokami-1 (Air)Yamanokami-2 (Air)

s maximum indicated power.

Fig. 6. Correlation of the ratio zind,max/z0 as a function of z0 and rhR/LR.

0.0000

0.0010

0.0020

0.0030

0.0040

0.0050

0.0060

0.0070

0.0000 0.0010 0.0020 0.0030 0.0040 0.0050 0.0060 0.0070

MA

CH

NU

MB

ER A

T M

AX

IMU

M IN

DIC

ATED

PO

WER

(E

XPE

RIM

ENTA

L)

MACH NUMBER AT MAXIMUM INDICATED POWER (PREDICTED)

GPU-3 (H2)GPU-3 (He)M102C (Air)P-40 (H2)P-40 (He)P-40 (N2)V-161 (H2)V-160 (He)Ecoboy (Air)Yamanokami-1 (Air)Yamanokami-2 (Air)

Fig. 7. Correlation of characteristic Mach number corresponding to maximum indicated power.

F. Sala et al. / Applied Thermal Engineering 89 (2015) 978e989 987

zB;maxz0:251� t

1þ t(27)

The alternative correlation introduced by Iwamoto et al. [27] canbe converted to the following expression [28]:

zB;maxz0:24p*

ks

1� t

1þ t(28)

where p* ¼ pm/plim, and plim denotes the limited maximum meanpressure that is allowable by the engine. This equation may beconsidered a variation of the West number correlation [13], since itis equivalent to NW z 0.24p*/ks.

To improve these approaches the following correlation has beenobtained for the engines listed in Table 4 with CV(RMSE)¼ 7.6% andR2 ¼ 0.9825 (Fig. 8) [15]:

zB;maxz2:301z1:0870

�rhRLR

�0:119N�0:039p (29)

From the figure it is deduced that NB is close to the average ofvalues of zB,max, but this average value only fits approximately theGPU3 engine values. In addition, the fact that Np is involved in theequation (29) in contrast to the equation (20) is due to the markedinfluence of this variable in the mechanical power losses. Thisobservation is in agreement with the expected influence of the gascircuit performance on the mechanical efficiency, described bySenft with originality [29].

On the other hand, so far it has not been sufficiently emphasizedthat the use of equations (25)e(29) to estimate themaximum brakepower PB,max is subjected to the availability of an accurate predic-tion of n*s;max, i.e., the engine speed corresponding to PB,max. In thisway, the approach by Iwamoto et al. [27] is themost advanced since

Fig. 8. Correlation of dimensionless maximum brake power.

0.0000

0.0010

0.0020

0.0030

0.0040

0.0050

0.0060

0.0070

0.0000 0.0010 0.0020 0.0030 0.0040 0.0050 0.0060 0.0070

MA

CH

NU

MB

ER A

T M

AX

IMU

M B

RA

KE

POW

ER (E

XPE

RIM

ENTA

L)

MACH NUMBER AT MAXIMUM BRAKE POWER (PREDICTED)

GPU-3 (H2)GPU-3 (He)M102C (Air)P-40 (H2)P-40 (He)P-40 (N2)V-161 (H2)V-160 (He)Ecoboy (Air)Yamanokami-1 (Air)Yamanokami-2 (Air)Eq.(30) predictions

Fig. 9. Correlation of characteristic Mach number corresponding to maximum brake power.

F. Sala et al. / Applied Thermal Engineering 89 (2015) 978e989988

the equation (28) is complemented by an additional criterion thatcan be converted to the following expression [28]:

N*MA;maxz6:8$10�5t�0:40k0:33s N0:20

p (30)

where N*MA;max ¼ n*s;maxV

1=3sw =

ffiffiffiffiffiffiffiffiRTC

p.

To complete the equation (29), the following correlation hasbeen obtained for the engines listed in Table 4 withCV(RMSE) ¼ 10.0% and R2 ¼ 0.7389:

N*MA;maxz0:00202ð1�tÞ0:485

�rhRLR

�0:414

g�0:493�X

mdx

�0:029N0:220p

(31)

Fig. 9 shows that predictions based on the equation (31) are lessdispersed from the experimental data than those based on thecorrelation (30).

On the other hand, the preliminary design criteria arecompleted with the ratio N*

MA;max=NMA;max and values of hmec,maxwhich can be derived from the aforesaid correlations.

6. Conclusions

Simple simulation models are needed at the preliminary designstage of Stirling engine, but some simplifications may lead to errorsthat can be avoided without precluding the simplicity of the model.

The assumption of harmonic drive for an engine with differentkind of mechanism causes errors that may be acceptable at thepreliminary design stage. The same result has been evidenced

F. Sala et al. / Applied Thermal Engineering 89 (2015) 978e989 989

about the EOS assumed for the working fluid operating at usualpressure and temperatures for conventional applications, butchanging the operating conditions or working fluid can make thehypothesis of ideal gas equation invalid, because the quasi-staticwork per cycle can be twice the value computed assuming idealgas behaviour. This advantage is reduced by the increase of indi-cated power losses caused by irreversibilities, but it seems possibleto find a combination of working fluid and geometric parametersthat allows the Stirling engine operation under real gas effects.

At the preliminary design stage of a Stirling engine, it is inter-esting to differentiate between the gas circuit and the mechanicalsubsystems of the machine in order to analyse separately the in-fluence of these subsystems on the whole machine performance. Acomplete set of semi-empirical equations and experimental cor-relations are proposed to characterize both indicated and brakepower of Stirling engines from parameters known at the pre-liminary design stage.

Although experimental data are required to increase the confi-dence of these correlations and extend them to engines workingunder real gas effects, the fact that pressure variation under real gasconditions is not far from ideal gas behaviour suggests thatacceptable values of mechanical efficiency can be expected forengines with selected working fluids operating under real gaseffects.

The introduced correlations must be understood as a work incontinuous updating so it is strongly recommended that re-searchers present their experimental results with sufficient infor-mation to calculate the dimensionless numbers that characterizethe engine operation.

References

[1] F. Miccio, On the integration between fluidized bed and Stirling engine formicro-generation, Appl. Therm. Eng. 52 (2013) 46e53.

[2] C. Ulloa, P. Eguía, J.L. Míguez, J. Porteiro, J.M. Pousada-Carballo, A. Cacabelos,Feasibility of using a Stirling engine-based micro-CHP to provide heat andelectricity to a recreational sailing boat in different European ports, Appl.Therm. Eng. 59 (2013) 414e424.

[3] J.A. Araoz, M. Salomon, L. Alejo, T.H. Fransson, Non-ideal Stirling enginethermodynamic model suitable for the integration into overall energy sys-tems, Appl. Therm. Eng. 73 (2014) 205e221.

[4] A. Cacabelos, P. Eguía, J.L. Míguez, G. Rey, M.E. Arce, Development of animproved dynamic model of a Stirling engine and a performance analysis of acogeneration plant, Appl. Therm. Eng. 73 (2014) 608e621.

[5] M. Res�endiz-Antonio, M. Santill�an, On the dynamical vs. thermodynamicalperformance of a a-type Stirling engine, Phys. A 409 (2014) 162e174.

[6] J.H. Shazly, A.Z. Hafez, E.T. El Shenawy, M.B. Eteiba, Simulation, design andthermal analysis of a solar Stirling engine using MATLAB, Energy Convers.Manag. 79 (2014) 626e639.

[7] C.M. Invernizzi, Stirling engines using working fluids with strong real gaseffects, Appl. Therm. Eng. 30 (13) (2010) 1703e1710.

[8] F. Sala, C.M. Invernizzi, Low temperature Stirling engines pressurised with realgas effects, Energy 75 (2014) 225e236.

[9] G. Walker, E.R. Bingham, Low-capacity Cryogenic Refrigeration, ClarendonPress, Oxford, UK, 1994.

[10] J.I. Prieto, M.A. Gonz�alez, C. Gonz�alez, J. Fano, A new equation representing theperformance of kinematic Stirling engines, Proc. Inst. Mech. Eng. C J. Mech.Eng. Sci. 214 (2000) 449e464.

[11] J.I. Prieto, Performance characteristics and preliminary design criteria ofStirling engines. Hydrogen and Other Technologies vol. 11, 2015, in: J.N. Govil,R. Prasad, S. Sivakumar, U.C. Sharma (Eds.), Energy Science & Technology, vol.12, Studium Press LLC, Houston, TX, USA, 2015, pp. 439e480.

[12] D. García, J.I. Prieto, A non-tubular Stirling engine heater for a micro solarpower unit, Renew. Energy 46 (2012) 127e136.

[13] D. García, M.A. Gonz�alez, J.I. Prieto, S. Herrero, S. L�opez, I. Mesonero,C. Villasante, Characterization of the power and efficiency of Stirling enginesubsystems, Appl. Energy 121 (2014) 51e63.

[14] C. Brat, Design characteristics and test results of the US P-40 engine, in:Proceedings of the 15th International Energy Conversion Engineering Con-ference. Seattle, WA, USA, 1980, 1964:6.

[15] J.I. Prieto, Indirect Estimation of Stirling Engine Indicated Power by Means ofQuasi-static Simulation and Brake Power Measurements, Internal report,University of Oviedo, SP, 2014.

[16] J.I. Prieto, A.B. Stefanovskiy, Dimensional analysis of leakage and mechanicalpower losses of kinematic Stirling engines, Proc. Inst. Mech. Eng. C J. Mech.Eng. Sci. 217 (2003) 917e934.

[17] D.Y. Peng, D.B. Robinson, A new two-constant equation of state, Ind. Eng.Chem. Fundam. 15 (1) (1976) 59e64.

[18] C.M. Invernizzi, Closed Power Cycles. Lecture Notes on Energy, vol. 11,Springer, 2013.

[19] Aspen Plus® v.8.0, ASPENTECH-Aspen Technology Inc, 2012.[20] X. Shi, D. Che, A combined power cycle utilizing low-temperature waste heat

and LNG cold energy, Energy Convers. Manag. 50 (2009) 567e575.[21] A.J. Organ, The Regenerator and the Stirling Engine, Mechanical Engineering

Publications, London, UK, 1997.[22] F.J. García-Granados, GGSISM: García-granados Simulation for Stirling Ma-

chines. User Manual, 2013. http://www.stirlinginternational.org/e-journal/.[23] F.J. García-Granados, M.A. Silva-P�erez, V. Ruiz-Hern�andez, Thermal model of

the Eurodish solar Stirling engine, J. Sol. Energy Eng. 130 (2008), 011014e1-011014-8.

[24] W.M. Kays, A.L. London, Compact Heat Exchangers, McGraw Hill, New York,USA, 1964.

[25] G. Walker, Stirling engines, Oxford University Press, New York, USA, 1980.[26] R. Lebredo, Obtenci�on experimental de las características de funcionamiento

en motores Stirling cinem�aticos de medio y bajo salto t�ermico, Memoria deinvestigaci�on, Universidad de Oviedo, SP, 1999.

[27] S. Iwamoto, K. Hirata, F. Toda, Performance of Stirling engines (arrangingmethod of experimental results and performance prediction), JSME Int. J. B 44(1) (2001) 140e147.

[28] J.I. Prieto, Discussion on performance of Stirling engines (arranging method ofexperimental results and performance prediction), JSME Int. J. B 46 (1) (2003)214e218.

[29] J.R. Senft, Mechanical Efficiency of Heat Engines, Cambridge University Press,New York, USA, 2007.