Stirling Engine 03

El Hassani et al. International Journal of Energy andEnvironmental Engineering 2013, 4:40http://www.journal-ijeee.com/content/4/1/40

ORIGINAL RESEARCH Open Access

Study of a low-temperature Stirling engine drivenby a rhombic drive mechanismHind El Hassani1*, Nour-eddine Boutammachte2, Juergen Knorr3 and Mohamed El Hannaoui1

Abstract

In this paper, a parametric study of the rhombic drive mechanism of a solar low-temperature Stirling engine wasconducted. The goal is to find out the relationship between the rhombic drive mechanism parameters and theperformance of a low-temperature differential Stirling engine. The results indicate that the offset distance from thegear center and gear radius of the rhombic drive mechanism have to be maximized, in order to increase theindicated power, while other rhombic drive parameters should be reduced for the same reason. Another result is thatfor realizing the optimized phase angle for beta-type Stirling engines, one should act on working piston-related barsand rhombic drive gear radius.

Keywords: Rhombic drive; Stirling engine; Indicated power; Phase angle

BackgroundThe Stirling engine is an externally heated machinewhich runs according to a reversible closed cycle. It wasinvented in 1816 by the Stirling brothers [1]. This engineis known by high heat conversion efficiency, reliability,low noise operation, and ability to use many fuels. Thesecharacteristics make the Stirling engines adapted to thedemand of the effective use of energy and environmentalpreservation [2,3]. In fact, reducing environmental im-pacts of conventional energy resources and meeting thegrowing energy demand of the global population hadmotivated considerable research attention in a widerange of environmental and engineering application ofrenewable form of energy [4], and among all possiblealternative energy options, solar energy is becomingmore popular in the world. This is mainly due to theavailability of plenty of sunlight in many countries [5].The low-temperature differential Stirling engine (LTD-

SE) is a kind of Stirling engine that can run with a smalltemperature difference between the hot and the coldsources. In 1983, Kolin [6] demonstrated the first LTD-SE. Senft [6,7] developed the Ringbom engine usingKolin's conclusions. Many other prototypes of LTD-SE

* Correspondence: [email protected] of Physics, Faculty of Sciences, Moulay Ismail University, Meknès50050, MoroccoFull list of author information is available at the end of the article

© El Hassani et al. This is an open access(http://creativecommons.org/licenses/by/2.0), wprovided the original work is properly cited.

2013

were manufactured and tested. In 2006, Kongtragool andWongwises [8] constructed twin power pistons and fourpower pistons, gamma-configuration LTD Stirling engines.In 2007, Martaj et al. [9] presented a thermodynamic ana-lysis of a low-temperature Stirling engine at steady stateoperation. In 2012, Boutammachte and Knorr [10] studieda γ-type LTD-SE. They presented several experimentalmeasurements made under laboratory and field-test con-ditions in Morocco [11]. Measurements with flat platecooler and discontinuous motion of the displacer wereconducted to verify some recommendations of Kolin inexpectation of power output improvement. They con-cluded that the assumption of Kolin related to the propor-tionality of speed to temperature difference is justified, butthe expectation about improvement of the machine per-formance with discontinuous motion of the two pistonsneeds to be reviewed.As a result of a scientific cooperation between Moulay

Ismail University in Morocco and the University ofTechnology of Dresden in Germany, an LTD γ-type Stir-ling engine was manufactured. This engine was testedand studied under real conditions [11]. Based on theconclusions of this study, a new prototype of LTD-SEwith rhombic drive mechanism is being developed. Thisnew prototype is designed to be simple, robust, cheap,and stable to be used for water pumping in poor ruralAfrican areas. In fact, this engine is manufactured more

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than 90% from mechanical cheap parts available in poorareas, so that it could be easily manufactured and main-tained right there. Besides, it is a solar engine, fact thatlowers significantly its use price. In addition to that, thenew prototype is a beta type which has a symmetric geom-etry that gives the engine more stability compared to γ-type Stirling engines, and the drive mechanism used, whichis the rhombic drive, allows the coaxial position of the en-gine pistons, fact that improves also the engine stability.Since this machine is developed to be as low expensive

as possible, it was designed without any concentratormechanism; consequently, its power output is relativelylow. Therefore, a special attention should be kept foreach construction parameter in order to optimize themachine performance.In this optic, this paper aims to study the drive mech-

anism adopted in the studied machine, which is therhombic drive mechanism. Thus, a parametric studybased on Schmidt theory is conducted to find out the re-lationship between the rhombic drive parameters andthe performance of low-temperature Stirling engines. Athermodynamic analysis of the prototype is also con-ducted. Moreover, as the relationship between rhombicdrive mechanism parameters is not enough studied andavailable for researchers, a study of these parameters isconducted, and a methodology of choosing their valuesis suggested.

MethodsPrototype descriptionAs a result of a scientific cooperation between DresdenUniversity of Technology and Moulay Ismail University inMorocco, an LTD-SE (beta type) is under construction.

Figure 1 Schematic illustration of the prototype.

The mechanical arrangement of the manufactured LTD-SEand its specifications are shown in Figure 1 and Table 1.

Kinematic relationsFigure 2 shows the parameters of the rhombic drivemechanism. The position and velocity vectors of eachlink are expressed as follows:

oa→¼ rg þ ecosO=

� �i→ þesinO= j

→ ð1Þ

va→¼ d oa

→

dt¼ −ew sinO= i

→ þew cosO= j→ ð2Þ

ob→¼ rg þ e cosO=þ d1 cosO= 1

� �i→ þ e sinO=þ d1 sinO= 1ð Þ j

→

ð3Þ

vb→¼ d ob

→

dt¼ −ew sinO=−d1w1 sinO= 1ð Þ i

→

þ ew cosO=þ d1w1 cosO= 1ð Þ j→

ð4Þ

ob0→

¼oa→ þ ab

0→

¼ rg þ e cosO=þ d2 cosO= 2

� �i→

þ e sinO=þ d2 sinO= 2ð Þ j→

ð5Þ

vb0→¼ d ob0

→

dt¼ −ew sinO=−d2w2 sinO= 2ð Þ i

→

þ ew cosO=þ d2w2 cosO= 2ð Þ j→:

ð6Þ

From the geometrical constraints,

rg þ e cosO=þ d1 cosO= 1 ¼ d3=2 ð7Þ

rg þ e cosO=þ d2 cosO= 2 ¼ d4=2 ð8Þ

Absorber

Displacer

Cooler

Working piston

Fly wheel

Rhombic drive mechanism


Table 1 Technical specifications of the prototype(baseline case)

Parameters Description/value

Engine type β

Working fluid Air

Cooling system Water cooled

Displacer diameter 1.35 m

Working piston diameter 0.29 m

Absorber diameter 1.51 m

Speed 30 rpm

Absorber temperature 70°C

Cooler temperature 20°C

Regenerator volume 0.063 m3


O= 1 ¼ cos−11d1

d3

2−rg−e cosO=

� �� ð9Þ

O= 2 ¼ cos−11d2

d4

2−rg−e cosO=

� �� : ð10Þ

Then, the y position of the working piston and the dis-placer are as follows:

Y working pistonð Þ ¼ Ldþ e sinO=þ d1 sinO= 1 ð11Þ

Y displacerð Þ ¼ Lpþ e sinO=þ d2 sinO= 2: ð12Þ

rgy

x

d3

c'

c

Working Piston

Displacer

d

Figure 2 Schematic illustration of the rhombic drive mechanism para

Based on the prototype data (baseline case), availablein Tables 1 and 2, one can get the vertical positions ofthe working piston and displacer (Figure 3) and variationof engine spaces versus crank angle (Figure 4).

Engine work and powerSchmidt [12] presented a mathematically exact expressionfor calculating the indicated work of Stirling engines. TheSchmidt theory adopts many ideal assumptions, like per-fect regeneration, isothermal compression and expansion,and harmonic motion of the displacer and the working pis-ton. The Schmidt theory is widely used due to its mathem-atical simplicity, and it is considered by many researchers,suitable for studying Stirling engines in designing stage, asit is the case for this study.Schmidt first calculated the gas mass in different

parts of the machine and used the perfect gas equation.Applied on our prototype (Figure 5, Table 3), we obtainthe following equations:

p ¼ mrVcTc

þ V f

Tfþ VTr

Tfþ Vr

Trþ Vmc

Tcþ Vmf

Tfþ VmTr

Tf

� �−1

:

ð13Þ

Schmidt adopted the assumption of harmonic motion:

Vc ¼ V SC

21þ cos θ þ αð Þð Þ ð14Þ

Ld

Lp

1

O

b

e

a

b'

d2

d1

o‘

2

4

meters.


Table 2 Geometrical variables of the rhombic drivemechanism (baseline case)

Value (m)

e 0.035

d1 0.125

d2 0.127

d3 0.228

d4 0.226

rg 0.18

Ld 0.808

Lp 0.112

00.10.20.30.40.50.60.70.8

0 100 200 300

y (m

)

Crank angle (°)

Displacer position Working piston position

Figure 3 Working piston and displacer positions versuscrank angle.


Vf ¼ V SC

21− cos θ þ αð Þð Þ ð15Þ

VTR ¼ V STR

21þ cosθð Þ: ð16Þ

Table 3 Nomenclature and indices

Description

Nomenclature

Cp Specific heat at p = cst (J/kg K)

Cv Specific heat at v = cst (J/kg K)

e Offset distance from the gear center (m)

h Specific enthalpy (J/kg)

Ld Displacer length (m)

Lp Working piston length (m)

n Engine speed (Hz)

P Pressure (Pa)

Q Heat (J)

r Specific gas constant (J/kg K)

rg Rhombic drive gear radius (m)

T Temperature (K)

V Volume (m3)

VSC Maximal volume of hot space (m3)

VSTR Maximal volume of working space (m3)

W Work (J)

α Phase angle

Φ Crank angle

η Efficiency

Indices

c Hot space

f Cold space

m Dead (volume)

r Regenerator

th Thermal

TR Working space

The pressure is expressed as follows:

P ¼ mr=k

1þ Ak

� �cos θ−βð Þ ð17Þ

k ¼ V SC

2Tcþ V SC

2Tfþ V STR

2Tfþ 2Vr

Tf þ Tcþ Vmc

Tcþ Vmf

Tfþ VmTR

Tf

ð18Þ

A ¼ 12

"V STR

Tf

� �2

þ V SC

Tc−V SC

Tf

� �2

−2V STR

Tf1Tf

−1Tc

� �V SC � cosα

#12

ð19Þ

tanβ ¼1Tf −

1Tc

� sinα

V STRTf�V SC

− 1Tf −

1Tc

� cosα

: ð20Þ

Cyclic work and power are expressed as follows:

W tot ¼ WTR ¼ ∫2π

0 PdVTR ¼ −12V STR ∫

2π

0 Psinθ dθ

ð21Þ

W tot ¼ V STRπmRA

1ffiffiffiffiffiffiffiffiffiffiffiffi1−k12

p −1

!sinβ ð22Þ

P ¼ n�W tot

2π¼ V STR

nmr2A

1ffiffiffiffiffiffiffiffiffiffiffiffi1−k12

p −1

!sinβ; ð23Þ

where n is the engine speed and m is the total gas mass.

Thermodynamic analysisApplying the thermodynamic analysis proposed by Martajet al. [9] to the particular studied prototype, we obtain thefollowing results.


00.020.040.060.080.1

0.120.14

0 100 200 300

Vo

lum

e (m

3)

Crank angle (°)

Hot volume Cold volume Total volume

Figure 4 Variation of engine spaces versus crank angle.

VR VR

Vf

Vc

Vmtr

Vmf

Vmc

Displacer

Vtr

W. piston

Figure 5 Different prototype spaces.


Regenerator analysisThe regenerator is used to store and release the heatexchanged with the fluid during its movement from thecold cell towards the hot cell and conversely. Schmidtassumes that the regeneration is perfect. But in practice,it is difficult to realize a perfect regenerator. In thisthermodynamic analysis, it is supposed an imperfect re-generation. Thus, the fluid temperature at the exit of theregenerator towards the cold cell Tf′ is higher than Tf,and the fluid temperature at the exit of the regeneratortowards the hot cell Tc′ is lower than Tc (Figure 6).The regenerator efficiency is defined by

ηr ¼Tc−Tf ′Tc−Tf

¼ Tc′−TfTc−Tf

¼ 1−ΔTr

Tc−Tf; ð24Þ

where ΔTr represents the temperature pinch in the re-generator, assumed identical at the two extreme orificesof the regenerator:

ΔTr ¼ Tf ′−Tf ¼ Tc−Tc′: ð25Þ

Since the regenerator volume is constant, the work ex-changed is null, and the average temperature Tr is sup-posed to be constant.

Tr ¼ Tf þ Tc2

: ð26Þ

The first thermodynamic principle for an open systemis written as

δQþ δW þX

hidmi ¼ Cv d mTð Þ: ð27Þ

In the regenerator cell, Equation 27 can be written as

δQr ¼ CpTrf dmf þ CpTrcdmc þ Cv

rV rdp; ð28Þ

where dmi is positive when it enters the volume i. Twocases arise for each opening:

– Flow from cold side towards the regenerator: dmf < 0,then Trf = Tf,Trc = Tc′.

– Flow from hot side towards the regenerator: dmc < 0,then Trc = Tc, Trf = Tf′.

The mass in this cell is

mr ¼ PV r

rTr: ð29Þ

The mass variation in the regenerator is

dmr ¼ −dmc−dmh ¼ Vr

rTrdp: ð30Þ


Tf

TcTc’

Tc

Tf

Tf’

R RDisplacer

W. piston

Figure 6 Schematic illustration of the prototype temperatures.


Heating cellThe heating cell has a single communication orifice;Equation 27 could be written in the following form:

δQc þ δWc þ CpTrcdmc ¼ Cv⋅d mTð Þc: ð31Þ

As stated before, if elementary mass is entering thehot volume (dmc > 0), then Trc = Tc′, else Trc = Tc.The differential form of the perfect gas state equation is

dpp

þ dV c

V c¼ dmc

mcþ dTc

Tc: ð32Þ

The hot cell is supposed to be isothermal; thus, Equa-tion 33 becomes

dpp

þ dVc

V c¼ dmc

mc: ð33Þ

According to the perfect gas state equation,

mc ¼ PV c

rTc

and

dmc ¼ Vc

Tc

dprþ PTc

dV c

r¼ 1

rTcV cdpþ PdVcð Þ:

ð34Þ

Equation 31 becomes

δQc ¼ −CpTrcdmc þ Cv

rV cdpþ Cp

rPdV c ð35Þ

δQc ¼ VcCp

r1γ−Trc

Tc

� �dP þ P

Cp

r1−

Trc

Tc

� �dVc:

ð36Þ

Cooling cellUsing the same reasoning as for the previous cell,Equation 27 becomes

δQf ¼ −Cp Trf dmf þ Cv

rV f dpþ Cp

rP dV f : ð37Þ

If dmf > 0, then Trf = Tf′, else Trf = Tf.As for the heating cell, we obtain also

dmf ¼ V f

Tf

dprþ PTf

dV f

r¼ 1

rT fV f dpþ PdV f� �

:

ð38Þ

Thus, Equation 37 becomes

δQf ¼ V fCp

r1γ−Trf

T f

� �dP þ P

Cp

r1−

Trf

T f

� �dV f :

ð39Þ

Engine balanceDuring a crankshaft rotation, the heat exchanged in thecooling and heating cells are Qc = ∮ δQc and Qf = ∮ δQf ob-tained by integration of Equations 36 and 39. Since the re-generator is supposed imperfect, an additional quantity ofheat Qr, to bring by the hot source, is essential:

∮δQr ¼ ∮CpTrf dmf þ ∮CpTrcdmc: ð40Þ

The overall engine balance is written as

W þ Qf þ Qc þ Qr ¼ 0

W ¼ − Qf þ Qc þ Qr

� :

ð41Þ

The work is carried out by the working piston duringcompression and expansion. It can also be calculated bythe following expression:

W ¼ Wf þWc: ð42Þ

This work is also

W ¼ −∮PdV ; ð43Þ

where dV is the total fluid variation and P is the internalpressure. The thermal engine efficiency is expressed by

ηth ¼W

Qc þ Qr: ð45Þ



Rhombic drive studyThe rhombic drive mechanism was proposed by thePhilips Company [13] and widely used for beta-typeStirling engines [14]. The major advantage of the rhom-bic drive mechanism is that the coaxial movement ofthe piston and the displacer is quiet and requires noserious lubrication [15].As the relationship between rhombic drive mechanism

parameters is not studied enough and available for re-searchers, a study of these parameters was conducted.Here are some conditions for good functioning of therhombic drive mechanism:

1. At Φ = 0°, the rigid rods d1 and d2 should be tallenough in order to reach the most far point duringthe cycle:

d1 > rg−d32þ e ð46Þ

0

5

10

15

20

25

30

0.005 0.015 0.025 0.035

Po

wer

(w

)

Offset distance from gear center e (m)

Baseline case parameters

14

19

24

29

34

0.112 0.212 0.312

Po

wer

(w

)

d1 (m)

e =0.04m, rg =0.185m

9

14

19

24

29

34

39

44

0.05 0.15 0.25

Po

wer

(w

)

d3 (m)

e =0.04m, rg =0.185m, d1=0.2m d2=0.2m

a

c

e

Figure 7 Effect of rhombic drive parameters on power.

d2 > rg−d42þ e: ð47Þ

2. At Φ = 180°, gears should be far enough from eachother:

rg > eþ d32

ð48Þ

rg > eþ d42: ð49Þ

As a consequence, during the design of the rhombicdrive mechanism, we propose the following order tochoose the rhombic drive parameters:Choosing the crossbar lengths d3 and d4 and the offset

distance from the gear center e

1517192123252729

0.16 0.17 0.18

Po

wer

(w

)

Gear radius rg (m)

e =0.04m

2627282930313233

0.12 0.17 0.22 0.27 0.32 0.37

Po

wer

(W

)

d2 (m)

e =0.04m, rg=0.185m, d1=0.112m

17

19

21

23

25

0.15 0.2 0.25 0.3

Po

wer

(w

)

d4 (m)

e =0.04m, rg=0.185 m, d1=0.2m d2=0.2m, d3=0.15m

b

d

f



1. Calculating the interval of values allowed to gearradius as follows:

min eþ d3

2; eþ d3

2

� �< rg < max d2 þ d4

2−e; d1 þ d3

2−e

� �:

ð50Þ

2. Choosing the connecting bar lengths d1 and d2, withrespect to the conditions below:

d1 > rg−d32þ e ð51Þ

d2 > rg−d42þ e: ð52Þ

A parametric study based on the theoretical analysis isperformed in order to evaluate the effect of each rhom-bic drive parameter on the machine characteristics. Note

0

0.05

0.1

0.005 0.015 0.025 0.035

Str

oke

(m

)


Displacer strokeWorking piston stroke

Baseline case parameters

0.08

0.1

0.12

0.14

0.112 0.212 0.312

Str

oke

(m

)

d1 (m)


e =0.04mrg =0.185m

0.07

0.12

0.17

0.05 0.15 0.25

Str

oke

(m

)

d3 (m)

Working piston strokerDisplacer stroke

e =0.04mrg =0.185md1=0.2m d2=0.2m

a

c

e

Figure 8 Effect of rhombic drive parameters on piston strokes.

that Schmidt calculating assumptions are adopted. Ateach study, only one parameter is changed in order tofind out its single effect. Results are found using a com-puter program on the Microsoft EXCEL. For a completerevolution of the crank and at a step of 1°, the programcomputes the instantaneous positions of the displacerand the working piston, the volume of each space of themachine, and the total mass, work, and power of thecycle. The program has the ability to change power cal-culation parameters in order to show their single effect.The geometrical and the operating variables which are

regarded as the baseline case in this study are listed inTables 1 and 2. In this case, the rotational speed is set at30 rpm; the absorber and the cooler temperatures areset at 70°C and 20°C, respectively.

Results and discussionFigure 7 displays the effect of rhombic drive parameterson indicated power. Figure 7a,b shows that any increasein offset distance from the gear center (e) or gear radius

0.085

0.09

0.095

0.1

0.105

0.16 0.17 0.18

Str

oke

(m

)

Gear radius rg (m)


e =0.04m

0.08

0.1

0.12

0.14

0.12 0.17 0.22 0.27 0.32 0.37

Str

oke

(m

)

d2 (m)


e =0.04mrg=0.185md1=0.112m

0.08

0.085

0.09

0.095

0.1

0.15 0.175 0.2 0.225 0.25 0.275

Str

oke

(m

)

d4 (m)

Working piston strokeDisplacer stroke

e =0.04 mrg=0.185md1=0.2md2=0.2md3=0.15m

b

d

f



(rg) has a positive influence on power. Figure 8a,b givesthe explanation of the former fact; when increasing eand rg, strokes of displacer and also working piston in-crease, then dead volumes are reduced, and total volumevariation is increased. All of these have direct positiveinfluence on indicated power. It is noticed that for thesame variation of e and rg (Δe ≈ Δrg ≈ 0.025 m), the off-set distance from gear e has larger positive effect on theindicated power than rg (ΔP(e) = 18 W, ΔP(rg) = 13 W).Then, considering only power, the optimal values of eand rg are their maxima values, but one should keep inmind that we can increase the stroke and power conse-quently - just as much as the machine volume allows it.Maximum value of rg is

rg ¼ min d2 þ d4

2−e; d1 þ d3

2−e

� �: ð53Þ

It is noticed that strokes of displacer and working pis-ton are equal and vary in the same way while changing eand rg. Shown in Figure 7c,d,e,f are the respective effects

60

65

70

75

80

0.005 0.015 0.025 0.035

Ph

ase

ang

le

(°)


36

56

76

96

0.112 0.212 0.312

Ph

ase

ang

le

(°)

d1 (m)

30

50

70

90

110

0.05 0.1 0.15 0.2 0.25 0.3

Ph

ase

ang

le

(°)

d3 (m)

a

c

e

αα

α

Figure 9 Effect of rhombic drive parameters on phase angle.

of the rhombic drive parameters d1, d2, d3, and d4 on in-dicated power. These figures indicate that increasing thevalue of any of these parameters has negative effect onpower. The same reasoning as for e and rg could bedone. Figure 8c,d,e,f gives the explanation of the formerfact; any increase in d1, d2, d3, and d4 values decreasesstrokes of both displacer and working piston, then deadvolumes will increase, and total volume variation will bedecreased, and consequently, the indicated power will bereduced. It is noticed that for the studied machine, forthe same variation of d1 and d2 (Δ(d1) ≈ Δ(d2) ≈0.273 m), the negative effect of d1 is greater three timesthan that of d2 (ΔP(d1) = −18 W, ΔP(d2) = −6 W). This isdue to the fact that the working piston is related to bard1; thus, its effect is much more important than that ofbar d2, which is related to the displacer which has noother role but moving gas from one side to another,whereas the working piston has an effect on total vol-ume variation, directly related to indicated work andpower. Also, for bars d3 and d4, similar remarks andconclusion could be done. For the same variation, the ef-fect of d3 is greater two times than that of d4 for the

40

50

60

70

80

90

0.16 0.17 0.18

Ph

ase

ang

le

(°)

Gear radius rg (m)

85

90

95

100

105

110

0.12 0.22 0.32 0.42

Ph

ase

ang

le

(°)

d2 (m)

50

55

60

65

70

75

0.15 0.175 0.2 0.225 0.25 0.275 0.3

Ph

ase

ang

le

(°)

d4 (m)

b

d

f

αα

α



studied prototype. Therefore, for increasing the machinepower, minima values of these parameters should beadopted:

d1 ¼ eþ rg−d3

2ð54Þ

d2 ¼ eþ rg−d4

2ð55Þ

d3 ¼ 2 eþ rg−d1ð Þ ð56Þ

d4 ¼ 2 eþ rg−d2ð Þ: ð57Þ

Figure 8 indicates that parameters d1 and d3 have aneffect only on working piston stroke, but no influenceon displacer stroke, while parameters d2 and d4 have aninfluence only on displacer stroke and no effect onworking piston stroke. For parameters e and rg, theyhave simultaneous equal effect on both displacer andworking piston strokes. This gives a directive aboutadjusting the stroke of one of the pistons (for example,for reducing dead volume) without changing the strokeof the other. Many investigations about optimal phaseangle for beta-type Stirling engines show that the opti-mal value is 90° [16].Figure 9 shows the influence of different parameters of

the rhombic drive mechanism on phase angle. As no-ticed for power, the increase in values of the offset dis-tance from the gear center (e) or gear radius (rg)increases the phase angle, whereas increasing the rest ofthe rhombic drive parameter values decreases the phaseangle. It is noticed that for the same variation of e andrg, the effect of gear radius rg is greater four times thanthe effect of e. Also, the effect of parameter d1 is greaterthree times than that of d2 on the phase angle of thestudied prototype, and the effect of d3 is greater twotimes than that of d4. This fact is due to the tightness ofworking space versus displacer-related space, which makesthe working piston parameters having more considerableinfluence on the phase angle than that of the displacer-related parameters. As a conclusion, for obtaining a spe-cial phase angle, one should pay a special attention toparameters rg, d1, and d3.

ConclusionsTheoretical analysis has been performed to study therhombic drive mechanism parameter effect on a low-temperature solar Stirling engine. The variation of themachine spaces was investigated, and a thermodynamicanalysis and a calculation of power based on Schmidttheory were conducted. A parametric study was thenperformed to evaluate the effect of each rhombic drive

mechanism parameter on some machine characteristics,like power, piston and displacer strokes, and phase angle.In accordance with the obtained results of this study,offset distance from the gear center and gear radius haveto be maximized in order to maximize the indicatedpower, while other rhombic drive parameters should bereduced for the same reason. Their minima values weregiven relatively to each other. Directives about realizingthe optimal phase angle of beta-type Stirling engineswere also given. As a conclusion, based only on rhombicdrive optimization, one can increase the studied proto-type power about 50%.

Competing interestsThe authors declare that they have no competing interests.

Authors' contributionsHEH performed the rhombic drive and the thermodynamic studies.JK designed and manufactured the studied prototype. NB supervised theresearch, performed the Schmidt calculation, and helped JK on constructingthe studied machine. MEH supervised the research. All authors read andapproved the final manuscript.

AcknowledgementsThe authors are grateful to the staff of the Energy Institute of the TechnicalUniversity of Dresden for their support.

Author details1Department of Physics, Faculty of Sciences, Moulay Ismail University, Meknès50050, Morocco. 2Department of Energy, Ecole Nationale Supérieure d'Arts etMétiers, Moulay Ismail University, Meknès 50050, Morocco. 3Institute ofEnergy, Faculty of Mechanical Engineering, University of Technology,Dresden 01062, Germany.

Received: 3 July 2013 Accepted: 1 November 2013Published:

References1. Kolin, I: Stirling Motor: History-Theory-Practice. Zagreb University Publications,

Zagreb (1991)2. Bahrami, M, Hamidi, A, Porkhial, S: Investigation of the effect of organic

working fluids on thermodynamic performance of combined cycle Stirling-ORC. Int. J. Energ. Environ. Eng. 4, 12 (2013)

3. Thombarea, DG, Verma, SK: Technological development in the Stirling cycleengines. Renew Sustain Energy Rev. 12, 1–38 (2008)

4. Sunday, O, Oyedepo, SO, Adaramola, MS, Paul, S: Analysis of wind speeddata and wind energy potential in three selected locations in south-eastNigeria. Int. J. Energ. Environ. Eng. 3, 7 (2012)

5. Sharma, P, Harinarayana, T: Enhancement of energy generation from twolayer solar panels. Int. J. Energ. Environ. Eng. 3, 12 (2012)

6. Senft, JR: An Introduction to low temperature differential Stirling engines.Moriya Press, River Falls (2004)

7. Robson, A, Grassie, T, Kubie, J: Modeling of a low-temperature differentialStirling engine. Proc. IMechE. Part C J. Mech. Eng. Sci. 221, 927–943 (2007)

8. Kongtragool, B, Wongwises, S: Performance of low-temperature differentialStirling engines. Renew. Energ. 32, 547–566 (2007)

9. Martaj, N, Grosu, L, Rochelle, P: Thermodynamic study of a low temperaturedifference Stirling engine at steady state operation. Int. J. Thermodynam.10, 165–176 (2007)

10. Boutammachte, N: Untersuchungen zur instationären Wärmeübergangs- undStrömungsvorgängen in einem Niedertemperatur-Stirlingmotor. PhD thesis.University of Technology of Dresden (2007)

11. Boutammachte, N, Knorr, J: Field-test of a solar low delta-T Stirling engine.Solar. Energ. 86, 1849–1856 (2012)

12. Schmidt, G: Theorie der lehmannschen calorischen maschine. Zeit desVereines deutsch Ing 15, 97–112 (1871)

13. Hargreaves, CM: The Philips Stirling Engines. Elsevier, New York (1991)

27 Nov 2013



14. Kerdchang, P, Win, M, Teekasap, S, Hirunlabh, J, Khedari, J, Zeghmati, B:Development of a new solar thermal engine system for circulating water foraeration. Sol. Energy 78, 518–527 (2005)

15. Cheng, C, Yu, Y: Combining dynamic and thermodynamic models fordynamic simulation of beta-type Stirling engine with rhombic-drivemechanism. Renew. Energy 37, 161–173 (2012)

16. El Hassani, H, Boutammachte, N, Knorr, Y, Hannaoui, M: Etudes desparamètres influençant la puissance d'un moteur Stirling solaire à bassetempérature. In abstracts of the 11th Mechanics Congress, University IbnZohr, Agadir, 23-26 (2013)

Cite this article as: El Hassani et al.: Study of a low-temperature Stirlingengine driven by a rhombic drive mechanism. International Journal ofEnergy and Environmental Engineering

10.1186/2251-6832-4-40

2013, 4:40

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