Analytical Solutions and Optimization of the Exo

Analytical Solutions and Optimization of the Exo-Irreversible Schmidt Cycle with Imperfect

Regeneration for the 3 Classical Typesof Stirling Engine

P. Rochelle and L. Grosu

Laboratoire d'Énergétique, de Mécanique et d'Électromagnétisme, Université Paris Ouest, 50 rue de Sèvres, 92410 Ville d’Avray - Francee-mail: [email protected] - [email protected]

Résumé — Solutions analytiques et optimisation du cycle de Schmidt irréversible à régénérationimparfaite appliquées aux 3 types classiques de moteur Stirling — Le “vieux” moteur Stirling estl’un des moteurs à sources multiples d’énergie les plus prometteurs pour le futur. Des modèlesélémentaires simples et réalistes sont utiles pour faciliter l’optimisation de configurations préliminairesdu moteur. En plus de nouvelles solutions analytiques qui réduisent fortement le temps de calcul, cetteétude du cycle moteur de Schmidt-Stirling modifié est entreprise avec le point de vue de l’ingénieur enintroduisant les exo-irréversibilités dues aux transferts thermiques. Les paramètres de référence sont descontraintes technologiques ou physiques : la pression maximum, le volume maximum, les températuresde paroi extrêmes et la conductance totale, alors que les paramètres d’optimisation ajustables sont lerapport volumétrique de compression, les rapports de volume mort, le déphasage des volumes balayés,les caractéristiques du gaz, le rapport des conductances “chaude” et “froide” et l’efficacité durégénérateur. Des expressions analytiques nouvelles pour les caractéristiques de fonctionnement dumoteur : puissance, travail, rendement, pression moyenne, vitesse maximale, sont établies et quelquesnombres de références adimensionnels ou pas sont présentés ainsi que des exemples d’optimisation de lapuissance en fonction de la vitesse réduite (adimensionnelle), du rapport des volumes et de l’angle dedéphasage.

Abstract — Analytical Solutions and Optimization of the Exo-Irreversible Schmidt Cycle withImperfect Regeneration for the 3 Classical Types of Stirling Engine — The “old” Stirling engine is oneof the most promising multi-heat source engines for the future. Simple and realistic basic models areuseful to aid in optimizing a preliminary engine configuration. In addition to new proper analyticalsolutions for regeneration that dramatically reduce computing time, this study of the Schmidt-Stirlingengine cycle is carried out from an engineer-friendly viewpoint introducing exo-irreversible heattransfers. The reference parameters are the technological or physical constraints: the maximumpressure, the maximum volume, the extreme wall temperatures and the overall thermal conductance,while the adjustable optimization variables are the volumetric compression ratio, the dead volume ratios,the volume phase-lag, the gas characteristics, the hot-to-cold conductance ratio and the regeneratorefficiency. The new normalized analytical expressions for the operating characteristics of the engine:power, work, efficiency, mean pressure, maximum speed of revolution are derived, and somedimensionless and dimensional reference numbers are presented as well as power optimization exampleswith respect to non-dimensional speed, volume ratio and volume phase-lag angle.analytical solutions.

Oil & Gas Science and Technology – Rev. IFP Energies nouvelles, Vol. 66 (2011), No. 5, pp. 747-758Copyright © 2011, IFP Energies nouvellesDOI: 10.2516/ogst/2011127

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Oil & Gas Science and Technology – Rev. IFP Energies nouvelles, Vol. 66 (2011), No. 5748

NOMENCLATURE

Variables

cv Specific heat at constant volume (J.kg-1.K-1)E Energy (J)h Specific enthalpy (J/kg)k Regeneration loss factor (-)K Conductance (W.K-1)m Mass of working gas in ideal cycle (kg)n Speed of revolution (rps)p Pressure (Pa)P* Normalized mechanical power (-)Q Heat (J)Q.

Thermal power (W)r Gas constant (J.kg-1.K-1)T Temperature (K)U Internal energy (J)V Volume (m3)W Work (J)W.

Mechanical power (W)

Greek symbols

α Conductance ratio (-)ε Volumetric compression ratio (-)γ Adiabatic exponent (-)η Efficiency (-)τ Temperature ratio (-)

INTRODUCTION

To study machine cycles in a more realistic way than basicclassical thermodynamics do, one introduces the exo-irre-versibilities due to the finite heat transfer rate between thewall source (or sink or regenerator) and the working fluid and,sometimes, those due to internal and/or external frictions andthermal losses. At constant heat conductance, heat flows andwork, as well as theoretical power and thermal efficiency, aredetermined by the temperature gap – the lower the cycleperiod, the higher the gap; the higher the heat flows, the lowerthe work. However, often, an increase in heat flows is associ-ated with a decrease of efficiency, thus the point of maximumpower is not the point of maximum efficiency. Moreover, inaddition to the use of source and sink temperatures (TH andTL) as obvious given parameters, power optimization is gener-ally carried out using the working gas mass (m) as a referenceparameter. For engineers, though, the working gas mass is notthe preferred parameter to refer to because practical problemsare mainly constrained by technical and physical considera-tions such as material mechanical- and thermal resistance,

bulk volume, and heat exchanger conductance and efficiency.Consequently, it would be desirable to introduce, and substi-tute for the mass, parameters such as the maximum allowedpressure (pmax), maximum allowed volume (Vmax), and maxi-mum allowed exchanger area or conductance (KT). Usingspeed of revolution instead of time as the main variable is alsoof prime interest because heat and mass transfers, as well asfluid and mechanical frictions, are directly speed-dependantand thus should be naturally expressed with respect to it.To date, the engineer-friendly finite-time perspective hasbeen given only slight consideration (see the well-documented study of Durmayaz et al. [1]). In the followingsections we develop analytical solutions to show that newconclusions and propositions arise from this more practicalapproach and that analytical solutions for the exchangedenergies lead to a significant improvement in computingtime for initial-optimization procedures.

1 CASE OF ENDO-REVERSIBLE EXO-IRREVERSIBLEIDEAL CARNOT-LIKE CYCLE WITH IMPERFECTREGENERATION

1.1 General Case

This endo-reversible cycle with (Stirling-, Ericsson-, ...,cycles) or without (Carnot cycle) regeneration is assumed toevolve between two reservoirs at constant wall temperaturesTH and TL (overall temperature ratio τ = TL/TH), with anisothermal heat delivery Qinrev to the hot gas at temperatureTh, an isothermal heat release Qoutrev from the cold gas at Tland a delivered work W. In case of an endo-reversible cyclewith imperfect regeneration, this is revealed by a differencebetween the inflow and outflow temperatures (resp. specificenthalpies) at each end of the regenerator. To preserve theideal pressure/temperature history in the swept volumes, theimperfect regeneration must be continuously compensatedwith an added-heat delivery ΔQreg from the hot source to thegas issuing from the hot outlet of the regenerator into theexpansion volume (Fig. 1, 2) to rise the outlet temperaturelevel to the one in the volume.

The same amount of heat is assumed to be lost to the lowtemperature sink from the gas issuing from the cold outlet ofthe regenerator into the compression volume. From the endo-reversibility assumption, it comes that the ratio of the isother-mally transferred heats is equal to the “internal” ratio τi of thetemperatures of the hot- and cold isothermal volume gases:

(1)

Hence, the total heat Qin delivered to the gas in the hot(expansion) volume is the sum of the isothermally deliveredheat Qinrev added to the imperfect-regeneration compensatingheat ΔQreg and the total heat Qout released from the gas in thecold (compression) volume is the sum of the isothermally

Q

Q

T

Toutrev

inrev

l

h

i= = τ .


P Rochelle and L Grosu / Analytical Solutions and Optimization of the Exo-Irreversible Schmidt Cyclewith Imperfect Regeneration for the 3 Classical Types of Stirling Engine

749

released heat Qoutrev added to the imperfect-regeneration heatloss –ΔQreg:

Qin = Qinrev + ΔQreg (2)

Qout = Qoutrev – ΔQreg . (3)

ΔQreg is a part of the heat Q+reg which is reversibly released

and caught by the regenerator matrix in the case of perfectregeneration. Let ηreg be the regenerator efficiency, then:

ΔQreg = (1 – ηreg) · Q+reg. (4)

Q+reg and Qinrev are dependant on the reference pressure,

the geometry and kinematics of the engine as well as on thetemperature ratio τi.

The work W is the sum of the delivered heat (+) andreleased heat (–):

|W | = Qin + Qout = Qinrev + Qoutrev = Qinrev · (1 – τi). (5)

The internal Carnot efficiency ηi is the cycle efficiency inthe case of perfect regeneration:

(6)

In case of exo-reversibility (classical thermodynamics),and τi = τ and ηi = ηCarnot = 1 – τ.

1.2 Effect of Exo-Irreversibility and ImperfectRegeneration

Assuming KT as the total convective heat conductance of thegas, which is the sum of the cycle time-averaged hot and coldwall/fluid conductances, α as the relative part of conductance

η τi

inrev

i

W

Q= = −1 .

involved in the heat transfer at hot source and n as the speedof revolution. Hence, energies could be written as follows:

(7)

(8)

(9)

and, noting that τl = τh · τi, it gives, first with Equations (5)and (9), second with Equations (2), (4), (7) and (8):

(10a)

(10b)

Combining Equations (10a) and (10b) and assuming noexplicit dependence of the various parameters on n exceptedthe one given by (11):

(11)

where has dimension of a speed of revolution (in

rps) or an inverse of time (in 1/s).

The expression of n (Eq. 11) could be re-introduced intoEquation (10) to get τh with respect to τi and then intoEquation (6) to get Qin.

The cycle efficiency is:

(12)

and, from Equations (5) and (11), with respect to τi the poweris:

(13)P n W K TT H

i i

i i

= ⋅ = ⋅ ⋅⋅ − ⋅ − ⋅ −

+ + ⋅

α α τ τ τ

τ α τ

( ) ( ) ( )

(

1 1

1−−[ ] ⋅ − ⋅⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

+

α η) ( )

.

1 regreg

inrev

Q

Q

ητ

η

= =−

+ − ⋅+

W

Q Q

Qin

i

regreg

inrev

( )

( )

1

1 1

K T

QT H

inrev

⋅

nK T

QT H

inrev

i

i i

=⋅⋅

⋅ − ⋅ −

+ + ⋅ −[ ]

α α τ τ

τ α τ α

( ) ( )

( )

1

1 ⋅⋅ − ⋅⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

+

( )1 ηregreg

inrev

Q

Q

τα α τ

τ

α αh

inrev i

T H

n Q

K T=+ − ⋅[ ] − ⋅ ⋅ −

⋅+ − ⋅

( )( )

( )

11

1 ττ

τα

η

i

hinrev

T H

regren Q

K T

Q

[ ]

= −⋅⋅ ⋅

⋅ + − ⋅

and

1 1 1( ) gg

inrevQ

+⎡

⎣⎢

⎤

⎦⎥.

W Q QK T

nout inT H

h l= + =⋅⋅ ⋅ − + − ⋅ −[ ]α τ α τ τ( ) ( ) ( )1 1

=⋅⋅ + − ⋅[ ] − ⋅ + − ⋅[ ]{ }K T

nT H

h iα α τ τ α α τ( ) ( )1 1

QK

nT T

K T

n

T

T

Tout

TL l

T H L

H

l= − ⋅ ⋅ − = − ⋅⋅⋅ −( ) ( ) ( )1 1α α

TT

K T

n

H

T Hl

⎛

⎝⎜

⎞

⎠⎟

= − ⋅⋅⋅ − ( ) ( )1 α τ τ

QK

nT T

K T

ninT

H hT H

h= ⋅ ⋅ − = ⋅⋅⋅ −α α τ( ) ( )1

Qinrev THΔQreg

Qoutrev ΔQreg

Th

Tl

TL

W

Figure 1

Balance of energy transfers in an endo-reversible exo-irreversible cycle with imperfect regeneration.



With a given KT · TH, derivations of Equation (13) andequalization to zero of the derivatives give the deduced opti-mum values ηreg = 1, α = 0.5 (Feidt et al. [2]), τi = √

—τ and,

then, the (overall) maximum maximorum power:

(14)

τi evolves from τ at very low speed, to 1 (Th = Tl), at thespeed limit nlim. At the speed limit, with the same assumptionas for Equation (11) and, as an addition, with the obviousassumption of no heat transfer for regeneration-loss compen-sation at Th = Tl, then Q+

reg = 0 and Qinrev(τi) = Qinrev(1) =Qinrev1, giving:

(15)

It gives also:

τhlim = τlim = α+ (1 –α) · τ (16)

and, using α optimum value:

(17)

(18)

2 THE SCHMIDT-STIRLING EXO-IRREVERSIBLEENDO-REVERSIBLE CYCLE WITH IMPERFECTREGENERATION

Q+reg and Qinrev depend on the type of reversible cycle. Let us

examine the case of the Schmidt-Stirling cycle.There are 3 basic configurations for the classical Stirling

engine (Fig. 2). A classical way to model this engine withsome realism is to use the Schmidt model. Its main assump-tions, slightly completed, are:– same instantaneous pressure throughout the engine;– use of an ideal gas as the working fluid;– constant working fluid mass (no leakage, no delivery)

during a cycle;– constant cylinder wall temperature;– harmonic/sinusoidal movement of the pistons (idealized

crankshaft);– constant temperature of gas in the hot and cold volumes.

This is nearly verified in LTD (Low TemperatureDifferential) Stirling machines with a low speed of revolu-tion and heat exchanging cylinder head and wall;

– constant speed of revolution;– perfect regeneration.

�Q K Tin T Hlim min .= ⋅ ⋅

−( )1

4

τ

nK T

QT H

inrev

lim min =⋅⋅−( )

1

1

4

τ

nK T

QT H

inrev

lim ( ) .=⋅⋅ ⋅ − ⋅ −( )

1

1 1α α τ

P K TT Hmax max .= ⋅ ⋅−( )1

4

2τ

This last assumption implies that the entirety of the heatreleased to the regenerator material during the gas flow fromthe hot volume to the cold volume is reversibly restituted tothe gas during the back flow, at the same levels of tempera-ture. In our case of imperfect regeneration, it will be assumedthat the gas pressure/temperature history will remain thesame but the part of regeneration heat lost (to the cold sink,by conduction or other transport) will be continuously com-pensated by a supplement of heat ΔQreg provided by the hotsource during each cycle as seen before (Sect. 1.1).

REGSH

Expansion sidepiston

Compression sidepiston

SC

Alpha

Beta

Gamma

Displacer

Displacer

Working piston

Working piston

SH

REG

SC

SH

REG

SC

Figure 2

The 3 classical types of Stirling engine configuration.



751

2.1 Instantaneous Volume Expressions

In the 3 types of engine, the expansion (hot) space variationhas a unique expression:

(19)

where ϕ is the rotation angle of the idealized crankshaft andVE0 is the “swept expansion volume”; in the case of beta- andgamma-type engines, this is the displacer swept volume.

For the compression space, in the case of beta- or gamma-type engines, that geometrically differ by either a not common-or a common cylinder and an overlapping swept volume),there is a combination of volume variations; it could beexpressed as:

(20)

where aj and bj values, displayed in Table 1, depend on thetype of engine, ϕ0 is the phase lag angle of the piston move-ments and VC0 is the “swept compression volume”. In fact, inour case of beta- and gamma-type engines, it only is the work-ing-piston swept volume. Vol is the overlapping volume in thecase of a beta-type engine and is due to the intrusion of thedisplacer piston into the working piston swept volume. Here,this volume equation is obtained by assuming there is one andonly one contact point between the displacer piston and the

working piston (VC = 0 and = 0 for ϕ = ϕcontact), duringtheir cyclical movement.

TABLE 1

Engine-type Alpha Beta Gamma

aj 0 1 1

bj 0 1 0

Dead volumes, due to the heat exchangers and the imper-fect geometry of the cylinder volumes must be taken intoaccount. Let VES, VCS, VR be the 3 dead volumes respectivelyrelated to the expansion and hot exchanger volumes (VES), tothe compression and cold exchanger volumes (VCS) and tothe regenerator volume (VR); the sum of these will be thetotal dead volume VS.

The total instantaneous working gas volume Vt is the sumof the previous ones:

(21)

The maximum global volume is:

VT = VE0 + VC0 + VS – bjVol. (22)

V V V VV

at E C SE

j= + + = ⋅ −[ ] + ⋅ +[ ]{ }0

21 1cos( ) cos( )ϕ ϕ

+ ⋅ − −[ ] − ⋅ +V

b V VCj ol S

002

1 cos( ) .ϕ ϕ

∂∂VC

ϕ

V aV V

bC jE C

j= ⋅ ⋅ +[ ] + ⋅ − −[ ] −0 002

12

1cos( ) cos( )ϕ ϕ ϕ ⋅⋅Vol

VV

EE= ⋅ −[ ]0

21 cos( )ϕ

Normalizing the volumes with respect to VT, Vt isexpressed as a function of expansion-, compression-, dead-and overlapping volume ratios (εE, εC, εS and εol) and,even, of ω which is the compression to expansion swept

volume ratio ( ):

(23)

Using a classical trigonometric relation (see set of Eq. A1in Appendix) gives:

V*t = BV – AV · cos(ϕ–ϕV) (24)

with BV, AV and ϕV expressed as:

Let , then one gets the normalized volume

V*t = BV · [1–δV · cos(ϕ–ϕV)] and the volumetric compression

ratio:

(25)

2.2 Instantaneous Pressure Expression

Assuming a constant working gas mass in the engine, whichis the sum of the masses in each volume, this total mass ofgas is expressed as a function of the instantaneous pressureand volumes:

(26)

where, remembering that , the regenerator mean temperature could be either:

TT T

TRh l

hi=

+= ⋅

+2

1

2

τ

τ il

h

T

T=

mp V

r T

p V

r T

p V

r T

p V

r T

pT

E

h

ES

h

C

l

CS

l

=⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅VV

r TR

R⋅

εδδ

δεε

= = =+−↔ =

−+

V

V

V

Vt

t

t

t

V

V

Vmax

min

max*

min*

1

1

1

11.

δVV

V

A

B=

B a b

A

VE

jS

E

jol

E

VE

= ⋅ + + + ⋅ − ⋅ ⋅⎡

⎣⎢

⎤

⎦⎥

=

εω

εε

εε

ε

21 2 2 ,

221 0

2

0

2⋅ − + ⋅⎡⎣ ⎤⎦ + ⋅[ ]

=

aj

V

ω ϕ ω ϕ

ϕ

cos( ) sin( )

sin( )εε

ω ϕ

ϕε

ω

E

V

VE

V

j

A

Aa

2

21

0⋅⋅ ⋅ ( )

=⋅⋅ − + ⋅

sin

cos( )

and

ccos .ϕ0( )⎡⎣ ⎤⎦

⎧

⎨

⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

V V V V

a

t E C S

Ej

* * * *

cos( ) co

= + +

= ⋅ −[ ] + ⋅ + ε

ϕ2

1 1 ss( )

cos( ) .

ϕ

εϕ ϕ ε ε

[ ]{ }

+ ⋅ − −[ ] − ⋅ + Cj ol Sb

21 0

ωωω

= =V

VC

E

C

E

0

0



or, assuming a linear temperature profile in the regenerator asUrieli and Berchowitz [3] did:

then . For p, Equation (26) gives:

(27a)

(see Eq. A2).After passing Th into the numerator and normalizing the

volumes with respect to VT, with the same trigonometricmethod as before, the new denominator D (Eq. A3), havingdimension of a volume, could be written as:

VT · [Bp – Ap · cos(ϕ –ϕp)].with the set of equations obtained by identification of theparameters:

The maximum pressure is obtained for ϕ=ϕp:

Normalizing p with respect to pmax, it becomes:

(27b)

where .The maximum to minimum pressure ratio ϖ is:

(28)

2.3 Expressions of the Work and IsothermallyDelivered Heat

Since the heat Qinrev delivered isothermally at hot temperatureTh during a complete cycle equals the opposite value of the

ϖδ

δ= =

+

−=+

−p

p

B A

B Ap p

p p

p

p

max

min

.1

1

δ pp

p

A

B=

p p

p p

*

cos( )=

−

− ⋅ −

1

1

δ

δ ϕ ϕ

pm r T

V B AT h

T p p

max .=⋅ ⋅⋅ −⎡⎣ ⎤⎦

Ba b

pE j

i i

ES

E

CS j ol= ⋅ +⎛

⎝⎜

⎞

⎠⎟+ +

⋅+

− ⋅(ετ

ωτ

εε

ε ε

21

2 ))⋅ +

⋅⋅

⎡

⎣⎢⎢

⎤

⎦

= − +

ε τεε

ετωτ

E i

R

E

h

R

pE j

i

T

T

Aa

2 2

21

ii i

p

cos sin

sin( )

ϕωτ

ϕ

ϕε

0

2

0

2

( )⎡

⎣⎢

⎤

⎦⎥ + ( )⎡

⎣⎢

⎤

⎦⎥

= EE

p i

pE

p

j

i

A

A

a

2

21

0⋅⋅ ⋅

=⋅

− +

ωτ

ϕ

ϕε

τωτ

sin( )

cos( ) ( )ii

⋅⎡

⎣⎢

⎤

⎦⎥

⎧

⎨

⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

cos( ) .ϕ0

pm r

V V

T

V V

T

V

T

m r T

V

T

E ES

h

C CS

l

R

R

T h

=⋅

+++

+

=⋅ ⋅

( EE ES

h

l

C CSh

R

RVT

TV V

T

TV

N

D+ + ⋅ + + ⋅=

) ( )

T

T

T

Th

R i

h

R

i

i

=+

=−−

2

1 1τττ

or ln( )

TT T

T

T

TRh l

h

l

hi

i

=−= ⋅

−

ln( )

( )

ln( )

11τ

τ

,

gas work done in the expansion volume VE, therefore (withthe “continental” convention that the work, as well as theheat, produced and lost by the gas is negative):

(29)

It gives, following Meijer [4], Finkelstein [5], Walker [6],Rochelle and Andrzjewski [7] (pp. 745-746) and applyingthe properties of the finite trigonometric integrals (Dwight[8]) (Eq. A4):

(30a)

and, from the endo-reversibility assumption: Qoutrev =–τi·Qinrev

then:

(30b)

the same ones are given under their normalized form withrespect to pmax · VT, as follows:

(31a)

(31b)

2.4 Analytical Expression of the Perfect-Regeneration Heat

In case of perfect regeneration, the gas temperature at eachextremity of the regenerator equals the gas temperature in theadjacent volume; hence, the elementary energy balance in theconstant-volume regenerator is given by the following equa-tion (developed in Eq. A5):

the elementary masses dmE and dmC being considered aspositive when issuing from the regenerator. From theisothermal assumption in the adjacent volumes:

dmd p V

r Tdm

d p V

r TEE

h

CC

l

=⋅⋅

=⋅⋅

+ +( ) ( ) and

where VV V V V V V

V

E E ES C C CS

E

+ += + = + and

Noting that

.

++ ++ + = −−⋅ =V

VV V VC

Rt R tRγγγ

1

dQ dU h dmV dp r

T dmreg R j j

j

Rh E= − ⋅ =

⋅−+⋅−⋅ ⋅ +∑

γγγ1 1

( TT dml C⋅ )

and πW i Ep

p p

* ( )( )

= − ⋅ ⋅−

⋅−−

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

11 1

11

2τ ε

δ

δ δ⋅⋅ sin( ).ϕ p

Qinrev Ep

p p

* ( )sin(= ⋅

−⋅−−

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥⋅π ε

δ

δ δϕ

1 1

11

2 pp )

W p VT i Ep

p p

= ⋅ ⋅ − ⋅ ⋅−

⋅−−

⎡

⎣

⎢⎢

⎤

max ( )( )

11 1

11

2τ ε

δ

δ δπ

⎦⎦

⎥⎥⋅ sin( )ϕ p

Q p Vinrev T Ep

p p

= ⋅ ⋅ ⋅−

⋅−−

⎡

⎣

⎢⎢

⎤

⎦

⎥max

( )π ε

δ

δ δ

1 1

11

2 ⎥⎥⋅ sin( )ϕ p

W Q

Q W p dV p

i inrev

inrev E E

= − ⋅

= − = ⋅ =∫( )1 τ , where:

� mmax* *

max ( )s

⋅ ⋅ ⋅

= ⋅ ⋅ − ⋅ ⋅

∫V p dV

p V

T E

T pE

� 1

2δε iin( )

cos( ).

ϕ ϕδ ϕ ϕ

⋅− ⋅ −∫ d

p p1�



753

dQreg can be rewritten as:

(32a)

with VtR used as a provisional volume for the demonstrationpurpose. Developed under its normalized form and letting

and it leads to the normalized form of dQreg:

(32b)

The perfect regeneration heat Qreg is null on a completecycle, resulting from the sum of 2 equal and opposed parts.The angles corresponding to the change of sign dQreg ordQ*

reg are obtained for d(p* ·V*tR) = 0 hence, in this case, (from

Eq. A6):

(33)

From Equation (32b), Q+*reg is given by the expression of

the definite integral:

(34)

The 2 solutions of Equation (33) are obtained, after itsdecomposition (set of Eq. A7 and A8) and the use of theprevious trigonometric method, as:

(35)

where:A p VR VR' cos( ) sin( )

'

= − ⋅⎡⎣ ⎤⎦ + ⋅[ ]

=

δ δ δ

δ

Φ Φ2 2

and AA

VR p

'

sin( ).

δ δ⋅ ⋅ Φ

cos( )

'sin( ) cos( )θ

δδ

δi iVR

A= =⋅⋅ − ⋅ −1 2

2 1or ''Φ Φ∓

δδ

δp

VR

⎡

⎣⎢

⎤

⎦⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

Q Bregp

VR

VR

+ =⋅ −

−⋅

⋅− ⋅ −

* ( )

( )

cos(

γ δ

γ

δ θ

1

1

1 2 ΦΦ Φ)

cos( )

cos( )

cos( )1

1

12

1

1− ⋅−− ⋅ −− ⋅δ θδ θδ θp

VR

p

⎡⎡

⎣⎢⎢

⎤

⎦⎥⎥.

1− ⋅⎡⎣ ⎤⎦⋅ −δ θ θp VAcos( ) sin( )Φ

wher

− − ⋅ −[ ] ⋅ ⋅ =B AVR V pcos( ) sin( )θ δ θΦ 0

ee and (= cst). θ ϕ ϕ ϕ ϕ= − = −p V pΦ

dQ d p V

B

reg tR

p

* * *( )

( )

( )

=−⋅ ⋅

=⋅ −

−⋅

γγ

γ δ

γ

1

1

1 VVR

VR V

p p

d⋅− ⋅ −− ⋅ −

⎡

⎣⎢⎢

⎤

⎦⎥⎥

1

1

δ ϕ ϕδ ϕ ϕ

cos( )

cos( ).

B BA

BVR V R VRV

VR

= −−⋅ =

γγε δ

1 and , it gives:

V B A

B

tR V R V V

VR

* cos( )= −−⋅ − ⋅ −

= ⋅

γγε ϕ ϕ

1

1−− ⋅ −⎡⎣ ⎤⎦δ ϕ ϕVR Vcos( )

dQ p dV V dp d p Vreg tR tR tR=−⋅ ⋅ + ⋅{ } =

−⋅ ⋅

γγ

γγ1 1

( )

Then, after further similar developments, we get Equation(36):

(36)

Hence, after using Equations (35) and (36) in (34) (see Eq.A9), and after further simplifications and factorizations (seeEq. A10), we get:

(37)

With this result, efficiency η (Eq. 12) and power P (Eq. 13)could be expressed:– with the ratio of the developed expressions of isothermally

delivered heat Qinrev and positive exchanged heat of regen-eration Q+

reg or;– with the ratio of their normalized expressions (Eq. 30b and

37) given by Equation (40) (see Eq. A11). They are, underthere completely developed form, functions of seven para-meters (εE, εC, εES, εCS, εR, γ, ϕ0) and one variable (τi).

2.5 Speed, Power and Efficiency Expressions

Introducing a reference speed of revolution

into Equation (11), the normalized speed is given by:

(38)

the normalized power is:

(39)

and the efficiency is:

(12’)

could be substituted into the above equations

with the help of expressions respectively given by Equations

(30b) and (37): see Equation (40).

QQ

Qinrevreg

inrev

**

* and

+

ητ

η

= =+

=−

+ − ⋅+

W

Q

W

Q Q Qin inrev reg

i

regr

*

* *

( )

( )

1

1 1 eeg

inrevQ

+*

*

.

P i i

i i

* ( ) ( ) ( )

( ) (

=⋅ − ⋅ − ⋅ −

+ + ⋅ −[ ] ⋅ −

α α τ τ τ

τ α τ α

1 1

1 1 ηηregreg

inrev

Q

Q)

*

*⋅

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

+

nQinrev

i

i i

**

.( ) ( )

( ) (

=⋅ − ⋅ −

+ + ⋅ −[ ] ⋅

1 1

1

α α τ τ

τ α τ α 11− ⋅⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

+

ηregreg

inrev

Q

Q)

*

*

nK T

p VrefT H

T

=⋅⋅max

QB

regVR

p

p VR V

+ =−⋅+

⋅ ⋅ ⋅

*

( ) ( )

cos(

2

γγ δ

δ δ ϕ

1 1

−− −⎡⎣ ⎤⎦ − − ⋅ −ϕ δ δp p VR) ( ) ( )1 1 12 2 2 .

cos( – )'

sin( ) –θδ

δδ

δδi i

p VR

pAΦ Φ= =

⋅⋅ − ⋅1 2

2 1or ''∓ ccos( ) .Φ

⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

Q

Q

Q

Q

Breg

inrev

reg

inrev

p VR p VR+ +

= =⋅ ⋅ ⋅ ⋅ ⋅*

*

2 γ δ δ δ ⋅⋅ − −⎡⎣ ⎤⎦ − − ⋅ −

− ⋅

cos( ) ( ) ( )

( )

ϕ ϕ δ δ

γ π

V p p VR1 1 1

1

2 2 2

⋅⋅ ⋅ − − −⎡⎣⎢

⎤⎦⎥⋅ε δ δ ϕE p p p1 12 2( ) sin( )

(40)



3 APPLICATIONS OF THE ANALYTICAL SOLUTIONSTO THE PARTIAL CYCLE OPTIMIZATION

As an example application, the previous equations are usedhere to describe the influence of the compression to expansionvolume ratio ω and of the speed n* on the main operatingparameters W*, P* and η (Fig. 3).

For the 3 engine types at the same time, computation anddisplay of these lines and surfaces (and much more), wereobtained within few seconds, with Matlab software.

In this first example, the engine is of alpha type, the phaselag angle ϕ0 is π/2, the gas has a specific heat ratio γ of 1.4,the dead volume ratios εES, εCS, εR are respectively 0.06, 0.06,0.08, the heat conductance ratio α equals 0.5, the temperatureratio τ is 0.5 and the regeneratio efficiency ηreg is 1, or 0(lower right quadrant).

With perfect regeneration, both the work (upper leftquadrant) and the efficiency (lower left quadrant) are maxi-mum at very low speed. The work at its overall maximum is

obtained for a value of the volume ratio ω slightly lower than1, as previously stated by Walker [6]; the efficiency is con-stant (= 0.5) at n* = 0 whatever the volume ratio is (basicthermodynamics case). The power representing-surface(upper right quadrant) shows a bended crest of constantheight, indicating that maximum power could be obtained fora particular value of n* whatever the volume ratio is, butspeed is at a minimum for a value of volume ratio ω slightlylower than 1. Moreover, a high power could be also obtainedwithin a large range of high speed in a narrow band of low tovery low volume ratios, at the cost of high optimum speeds.

Without regeneration (Fig. 3, lower right quadrant), thepower is more than halved compared to the value obtainedwith perfect regeneration and it increases with volume ratio.Moreover, the optimum speed for power is lower, and theband of high power at low volume ratios doesn’t really exist.

Another example application concerns the optimization ofthe power with respect to the phase angle and to the compres-sion-to-expansion volume ratio. With help of Equation (13),

W* = f (omega, n*), etareg = 1 P* = f (omega, n*), etareg = 1

Eta = f (omega, n*), etareg = 1 P* = f (omega, n*), etareg = 0

Omegan* n*

n* n*

Omega

Omega Omega

W*

P*

Eta P*

0.10

0.05

0

0.1

0.2

0.3

0.4

0

00

1

2 6 2 64 4

2 20

1

2 64

20

0

0.02

0.01

0.02

0.01

00

1

2 64

20

00

1

Figure 3

Work, power and efficiency versus volume ratio and speed in case of perfect regeneration and, lower right quadrant, power withoutregeneration.


giving expression of power to be normalized by KT · TH, andEquation (40), we get the results illustrated by Figure 4 afterfew minutes of iterative calculations to find the set of maxi-mum power values and associated values of the other operat-ing parameters. The chosen engine, of which results are dis-played here, is of alpha type and the fixed parameters are thesame as in the previous example except the regeneration effi-ciency at 0.5 and the phase lag angle ϕ0 considered as a vari-able. Maximum power increases with ω increase and ϕ0decrease, but the criterion of maximum power is not the onlyone to consider as Figure 4 shows: in fact the correspondingwork (or torque), the efficiency and the speed of revolutionmust be examined too. A compromise solution could befound through a high work (or a low speed) or a high effi-ciency is privileged in addition to maximum power. In thisparticular case, we see (higher right quadrant) that a maxi-mum work (or torque) is obtained for a phase lag angleapproximately equal to 1.6 radian and a volume ratio of

approximately 0.8. This point corresponds nearly to theminimum of speed of revolution (lower left quadrant) with anot-too-much reduced value of efficiency (lower right quad-rant). Favoring the efficiency could be done choosing a lowervalue of phase lag angle and a higher value of volume ratio atthe price of a higher speed and lower work (or torque).

These examples show that a first approximate optimiza-tion, which, however, neglects conduction- and frictionlosses, is possible without large efforts.

CONCLUSION

In this paper, we have studied the exo-irreversible, endo-reversible Schmidt-Stirling engine cycle. Analytical expressionswere derived for the phase angles at gas flow inversion withinthe regenerator and for the positive or negative perfect-regeneration heat. Adding these ones to other previously


755

Pmax* = f (omega, phi0) WPmax* = f (omega, phi0)

nPmax* = f (omega, phi0) EtaPmax = f (omega, phi0)

phi0 Omega phi0 Omega

phi0 Omega phi0 Omega

Pm

ax*

WP

max

*E

taP

max

nP

max

*

0 64

202

1

0

0.5

1.0

1.5

2.0

2.5

0 64

202

1

0

0.01

0.02

0.03

0 64

202

1

0

0.03

0.02

0.01

0.04

0.05

0 64

202

1

0

0.1

0.2

0.3

0.4

0.5

Figure 4

Maximum-power surface and corresponding work, speed and efficiency surfaces versus phase lag angle ϕ0 and volume ratio ω.



obtained analytical expressions (expansion and compressionvolumes, pressure, work, isothermally delivered heat) allowedanalytical calculations of each cycle-averaged energy transferand of efficiency with respect to geometrical and physicalparameters (e.g. regenerator efficiency and overall heatconductance) without step-by-step numerical computation ofthe cycle. An example of cycle optimization with a givenphase angle was described; it illustrated the versatility of thisset of equations.

Moreover, a second example using this set of equationswith an iterative calculation allowed the choice of near-optimum phase angle and volume ratio to obtain a “good”compromise between maximum power and maximum work(or torque) at minimum speed.

Nevertheless, to be closer to reality, a more elaborateprocedure could be followed which takes into account thespeed-dependence of physical phenomena such as convectiveheat transfers, conductive heat losses, gas friction andmechanical friction as, for instance, Senft [9,10] and Petrescuet al. [11] did. Moreover, the normalization could be donewith respect to more representative and “absolute” con-straint-parameter combinations. For instance, we found ear-lier [12] that, for an exo-irreversible ideal Stirling cycle, themaximum attainable theoretical work is given by:

which could

be used instead of pmax · VT. It can be established, too, that themaximum heat delivered per unit time (which has dimension

of power) is which could be used

instead of KT · TH.This study could be extended to the exergy balance, to

improve the energy use and optimize the cycle by irre-versibility localizations (Martaj et al. [13]).

Using this set of equations, a preliminary design of aStirling engine, to be used in a solar power plant at mediumsource temperature, is under progress (Nov. 2009).

REFERENCES

1 Durmayaz A. et al. (2004) Optimization of thermal systemsbased on finite-time thermodynamics and thermoeconomics,Progr. Energ. Combust. Sci. 30, 175-217.

2 Feidt M. et al. (2002) Optimal allocation of Heat Exchangerinventory associated with fixed power output or fixed heat trans-fer rate input, Int. J. Appl. Thermo. 5, 1, 25-36.

3 Urieli I., Berchowitz D.M. (1984) Stirling cycle engine analysis,Adam Hilger, Bristol, UK.

4 Meijer R.J. (1960) The Philips Stirling thermal engine, PhDThesis, Delft Technical University.

5 Finkelstein T. (1961) Generalized thermodynamic analysis ofStirling engines (paper 118B) and Optimization of phase angleand volume ratios in Stirling engines (paper 118C), SAE AnnualWinter Meeting, Detroit, USA.

6 Walker G. (1973) Stirling-cycle machines, Oxford UniversityPress, Oxford, UK.

7 Rochelle P., Andrzjewski J. (1974) Optimisation des cycles àrendement maximal, Revue de l’Institut Français du Pétrole 29,731-749.

8 Dwight H.B. (1971) Tables of integrals and other mathematicaldata, 4th edition, MacMillan Company, New York, USA.

9 Senft J.R. (1998) Theoretical limits on the performance ofStirling engines, Int. J. Energy Res. 22, 991-1000.

10 Senft J.R. (2002) Optimum Stirling engine geometry, Int. J.Energy Res. 26, 1087-1101.

11 Petrescu S., Costea M. et al. (2002) Application of the DirectMethod to irreversible Stirling cycles with finite speed, Int. J.Energy Res. 26, 589-609.

12 Grosu L., Rochelle P., Martaj N. (2008) Thermodynamique àéchelle finie : optimisation du cycle moteur de Stirling pourl’ingénieur, COFRET’08, June 2008, Nantes, France.

13 Martaj N., Grosu L.,, Rochelle P. (2007) Thermodynamic studyof a low temperature difference Stirling engine at steady stateoperation, Int. J. Thermo. 10, 4, 165-176.

Final manuscript received in February 2011Published online in November 2011

�Q K Tin T Hmax max

( )= ⋅ ⋅

−1

4

τ

Wp V

eeT

max maxmax ( ) ( exp( ) . )=⋅

− = ≅1 1 2 72τ

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757

APPENDIX

(A1)

(A2)

(A3)

(A4)

(A5)

(A6)

(A7)

(A8)

(A9)

Q B

A

regp

VR

VR p

+ =⋅ −

−⋅

⋅

−⋅

⋅

* ( )

( )

' '

γ δ

γ

δ δ

δ

1

1

1

⋅⋅ + − ⋅ −⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

sin( ) ' cos( )Φ Φδδδ

2 1 VR

p ⎭⎭⎪

−⋅

⋅⋅ − − ⋅ −

⎡1 12δ δ

δδ

δ

δp VR p

VRA' 'sin( ) ' cos( )Φ Φ

⎣⎣⎢

⎤

⎦⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

−

−⋅

⋅⋅ − −1 12δ δ

δδVR p

A' 'sin( ) 'Φ ⋅⋅ −

⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

−⋅

⋅

δδ

δ δ

δ

VR

p

p VR

cos( )

'

Φ

1AA

p

VR'sin( ) ' cos( )⋅ + − ⋅ −

⎡

⎣⎢

⎤

⎦⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

Φ Φδδ

δ2 1

⎭⎭⎪

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

cos( )'

sin( ) cos( ') cosθ δ θ δ δ− = ⋅ ⋅ ⋅ − + − ⋅Φ Φ Φ1

A p VR p (( ) sin( ')Φ Φ⎡⎣ ⎤⎦⋅ −{ }θ

cos( ') cos( )'

sin( ) sin( )'

Φ Φ Φ Φ Φ− = ⋅ ⋅ + ⋅ ⋅δ

δVRpA A

1−− ⋅⎡⎣ ⎤⎦ = ⋅ ⋅

− =

δ δ

δ

VR p

VR

Acos( )

'sin( )

sin( ')

Φ Φ

Φ Φ

1

AA A Ap

VR p' 'cos( )

'cos( )− ⋅

⎡

⎣⎢

⎤

⎦⎥ = ⋅ − ⋅⎡⎣ ⎤⎦

δδ δΦ Φ

1

⎧⎧

⎨⎪⎪

⎩⎪⎪

1− ⋅ −⎡⎣ ⎤⎦⋅ ⋅ − − − ⋅δ ϕ ϕ ϕ ϕp p V V VR VA B Acos( ) sin( ) cos(ϕϕ ϕ δ ϕ ϕ−[ ] ⋅ ⋅ − =V p p) sin( ) 0

dQV dp r

T dm T dm pregR

h E l C=⋅−+⋅−⋅ ⋅ + ⋅ =

−γγγ

γγ1 1 1

( ) ⋅⋅ +⎡⎣ ⎤⎦+ + +⎡

⎣⎢

⎤

⎦⎥⋅

⎧⎨⎩⎪

⎫⎬⎭

+ + + +dV dV V VV

dpE C E CR

γ ⎪⎪

D V aTE E

i

iC= ⋅ ⋅ −[ ] +

⋅⋅ ⋅ +[ ] +ε

ϕετ

ϕε

21

21cos( ) cos( )

221

10⋅

⋅ − −[ ] + + − ⋅ ⋅ +τ

ϕ ϕ ε ε ετε

i

ES CS i ol

i

Rbcos( ) ( ) ⋅⋅⎧⎨⎩

⎫⎬⎭

T

Th

R

pm r

V

T

V aT

E

h

E i

=⋅

⋅ −[ ]⋅

+⋅ ⋅ +[ ] +0 01

2

1cos( ) cos( )ϕ ϕ VV

T

V

T

V

T

V b VC

l

R

R

ES

h

CS i o0 01

2

⋅ − −[ ]⋅

+ + +− ⋅cos( )ϕ ϕ

ll

lT

a b c B A

A

I− ⋅ − ⋅ + = − ⋅ −cos( ) cos( ) cos( )ϕ ϕ ϕ ϕ ϕ0

where == + + ⋅ ⋅ ⋅ =

=+

b c b c B a

bI

2 202 cos( )

cos( )

ϕ

ϕ

, ,

cc

A

c

AI

⋅=⋅

⎧

⎨

⎪⎪

⎩

⎪⎪ cos( )

sin( )sin( )ϕ

ϕϕ0 0,



(A10)

(A11)Q

Q

Q

Q

Breg

inrev

reg

inrev

p VR p VR+ +

= =⋅ ⋅ ⋅ ⋅ ⋅*

*

2 γ δ δ δ ⋅⋅ − −⎡⎣ ⎤⎦ − − ⋅ −

− ⋅

cos( ) ( ) ( )

( )

ϕ ϕ δ δ

γ

V p p VR1 1 1

1

2 2 2

π ⋅⋅ ⋅ − − −⎡⎣⎢

⎤⎦⎥⋅ε δ δ ϕE p p p1 12 2( ) sin( )


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Analytical Solutions and Optimization of the Exo