Enhancing and Multi-Objective Optimizing of the

International Journal of Ambient Energy

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Enhancing and Multi-Objective Optimizing of the Performance of Stirling Engine Using Third Order Thermodynamic Analysis Mazdak Hooshang, Somayeh Toghyani, Alibakhsh Kasaeian, Reza Askari Moghadam & Mohammadhossein Ahmadi To cite this article: Mazdak Hooshang, Somayeh Toghyani, Alibakhsh Kasaeian, Reza Askari Moghadam & Mohammadhossein Ahmadi (2017): Enhancing and Multi-Objective Optimizing of the Performance of Stirling Engine Using Third Order Thermodynamic Analysis, International Journal of Ambient Energy, DOI: 10.1080/01430750.2017.1303638 To link to this article: http://dx.doi.org/10.1080/01430750.2017.1303638

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Date: 08 March 2017, At: 09:09

1

Publisher: Taylor & Francis & Informa UK Limited, trading as Taylor & Francis Group

2

Journal: International Journal of Ambient Energy

3

DOI: 10.1080/01430750.2017.1303638

4

Enhancing and Multi-Objective Optimizing of the Performance of Stirling Engine Using Third Order Thermodynamic Analysis

5 6 7 8 9

Mazdak Hooshang1, Somayeh Toghyani2, Alibakhsh Kasaeian3,

10

Reza Askari Moghadam4, Mohammadhossein Ahmadi5*

11 12 13 14 15

1,4

Department of Mechatronics Engineering, Faculty of New Science & Technologies, University of Tehran, Tehran, Iran. 2,3,5

Department of Renewable Energies, Faculty of New Science & Technologies, University of Tehran, Tehran, Iran.

16 17

Abstract - Stirling engine is an external combustion engine which uses eternal heat sources like

18

solar radiation for heating a compressible fluid inside cylinders. In the recent years, significant

19

attention is drawn to Stirling engines due to the clear advantages, high efficiency potential,

20

flexible fuel, lower nitrogen oxides, quiet and minimal vibration, high reliability and highest

21

specific output work for any closed regenerative cycle. The third order thermal analysis is one of

22

the analyses which has been applied in several studies which have been carried out on Stirling

23

engines. NSGA-II algorithm is applied to optimize the differential regenerator pressure (bar) and

24

the power output (kW) for a Stirling engine system. In this study, three decision-making

25

techniques are utilized to optimize the solutions, obtained of the results. At last, the employed

26

techniques are compared with the data of an experimental research work.

27

Keywords: Stirling; multi objective optimization; NSGA; third order thermodynamic.

28

1. Introduction

29

Needing to reduce fossil fuel consumption and using renewable energies, has caused increasing

30

attention to Stirling engines. Considering that, these engines are of external combustion engines,

31

and they can use any heat source like biomass, convectional fuel, nuclear, and solar as heat

32

sources which lead to less pollution [1-2]. In comparison with other reciprocating engines,

33

Stirling engine can present an appropriate efficiency, which can be nearer to the efficiency of the

34

Carnot cycle [3-7].

35

The complexity of the analysis and the simulation of the Stirling engines have always been

36

obstacle in the way of developing these engines. The fundamental problem in the analysis of

37

Stirling engine is that various molecules of the working fluid pass different thermodynamic

38

cycles. In the attempts for modeling Stirling engines which have been carried out over many

39

years, several methods have been proposed that include: the zero-order models, the first-order

40

models, the second-order models, the third-order models, and the multi-dimensional modeling.

41

The zero-order models or the empirical models include explicit mathematical relationships that

42

estimate the output power of a Stirling engine according to its general performance [8]. The first

43

order models of Stirling engine was presented by Schmidt in 1871 [9]. One of the assumptions,

44

considered in the Schmidt's model, it is that the behavior of gas is isothermal in the compression

45

and expansion situations. At high speeds, this assumption could not be true in practice; because

46

the expansion and compression processes at high speeds are close to the adiabatic state. For this

47

reason, Finkelstein proposed the second order model of Stirling engine in 1960. [10].

48

For analyzing of third order, the form of partial differential equations of mass, momentum and

49

energy for several control volumes of engine are written. Dyson et al. [9] and Anderson [11]

50

studied some examples of such models. Gedeon in 1986 developed the GLIMPS simulation code

51

based on the third-order model. He developed a new code named Sage in which the used method

52

was based on the same method that was used in the GLIMPS code. In this code, two-dimensional

53

finite difference grid in terms of time and space is used for solving the partial equations of mass,

54

momentum and energy balances. Tew et al. carried out a simulation on Stirling engines using

55

Sage code in a NASA’s project [12]. Huang developed the H-FAST analysis code in order to

56

analyze Stirling engine using the harmonic analysis method for solving the equations of mass,

57

energy and momentum [9]. In the fourth order model, all the flow passing chambers are meshed

58

as multi-dimensions. Hence, the changes of pressure and velocity in whole directions are

59

obtained. The existing simulation tools in this field like Fluent and ANSYS are utilized for this

60

purpose [13].

61

In the mentioned studies, a gap is observed on the optimization methods. The optimization with

62

neural networks covers few variables and we need to optimize more variables. For considering to

63

the content provided above, the main objective of this study is to conduct two objective functions

64

including the output power (kW), and the differential regenerator pressure (bar) of the considered

65

Stirling engine by using the third-order model of Stirling engine.

66

The Skyline computation and vehicle routing issues are of the problems in which, the multi-

67

objective optimization has been utilized [14-15]. During the 18th century, the evolutionary

68

algorithms (EA) were used in an effort to stochastically solve the problems of this generic class

69

[16]. A sensible solving method to a multi-objective quandary is applying a set of solutions of

70

which each satisfies the objectives at an satisfactory level without being overcome by any other

71

solution [17]. During the current years, many reserchers have paid attention to the multi-

72

objective optimization of several themodynamic systems [18-44].

73

In the current study, six decision variables including the heater diameter, total regenerator mass,

74

the regenerator matrix diameter, the regenerator length, the engine frequency, and the engine

75

pressure charge are considered for the multi-objective optimization.

76

2. Stirling System

77

Stirling engine is a kind of an external combustion engine that produces mechanical or electrical

78

output power, by receiving and rejecting heat [44-51]. At higher temperatures, heat enters in the

79

engine and a section of its exergy turns into mechanical or electrical output power while the

80

remainder of the heat is rejected at lower temperatures. Stirling engine has two hot and cold

81

chambers which are called as the expansion and the compression spaces in where; the working

82

fluid at relatively high pressure is placed. The fluctuating in the space of the two chambers

83

causes receiving thermal energy from a heat source, creating mechanical output power, and

84

releasing heat in a thermal rejecter.

85

A Stirling engine thermodynamic cycle consists of four thermodynamic processes: isothermal

86

compression, heat receiving at constant volume, isothermal expansion, and heat rejecting at

87

constant volume, which are shown in Fig.1. The fluctuation in the spaces of the hot and cold

88

chambers is done in such a way that the above-mentioned four-step process would operate [52].

89

During the stable performance of a Stirling engine, at the end of each cycle, the gas returns to the

90

initial conditions of the cycle.

91

92

[Insert Figure 1 here]

93

In order to perform each of the four step processes, a set of mechanical parts with simultaneous

94

and arranged are applied to cause the working fluid performing the procedures. The arrangement

95

and lay-out of the components constitute several structures of Stirling engines that include the

96

Alpha, Beta, and Gamma types.

97

[Insert Figure 2 here]

98

In the Stirling engine with Gamma structure, the engine has been studied in this research, the

99

power piston and displacer are located in two separate cylinders (Fig 2). The compression ratio of

100

the Gamma Stirling engine is less than the Alpha and Beta Stirling engines but in the mechanical

101

point of view, it has simplest arrangement among the other structures. The specifications of the

102

applied engine for this study (ST500) are presented in Table (1).

103

[Insert Table 1 here]

104 105

3. Theoretical Model

106

The Nlog analyzer was applied in order to optimization of Stirling engine. Nlog code is a third-order

107

analyzer of Stirling engine that the generated power and rejected heat of Stirling engine can be calculated

108

by it. The Nlog analyzer is applied for solving the partial equations of mass, momentum and

109

energy [53]. In the Nlog code, the input variables include:

110

-

The type of Stirling engine.

111

-

The geometric characteristics of pipes, chambers, and all channels.

112

-

Geometry and porosity and the type of regenerator.

113

-

The area and roughness of the wetted surfaces.

114

-

The equations of fluid thermal conductivity, fluid viscosity, and compressibility factor.

115

-

The local temperatures and the initial charge pressure of engine.

116

-

The temperature gradient of the heat exchangers walls.

117

The output of Nlog code includes:

118

- The output power and efficiency of engine.

119

- Pressure versus total volume.

120

- The rate of heat transfer.

121

- The velocity of working gas in channels and pipes.

122

- The velocity of rigid bodies against time.

123

- Working gas temperature versus time.

124 125 126 127

The instant state variables of the engine are as followings: 1) The geometric characteristics of pipes, compression and expansion chambers, and all gas transferring channels.

128

2) The geometric characteristics of linkages between moving parts of engine.

129

3) The local temperatures and the initial charge pressure of engine.

130

4) The heat exchangers wall’s temperature.

131

For determining the thermodynamic and dynamic parameters of the gas for each control

132

volume, the Nlog code divides the entire gas flow to several control volumes and conducts

133

simultaneous solving of all the balance equations for each portion. The equations of balance

134

are presented in Eqs. (1)- (3), respectively [53]. These equation are simplified to one-

135

dimensional form that use in the third-order analysis. ρv ndA +

∂ ( ∂t

ρdV) = 0

(1)

∮ ρv ndA + τ A ∂ ( ∂t

(

ρdV) = 0

(

ρv dV) +

P ndA +

(v ρ)v ndA +

(2)

=0 (ρu + ρ

v )dV) + 2

ρu + ρ

v + P v ndA + 2

P

dx dQ ndA − =0 dt dt

(3)

136

Eq. (1) is converted to Eq. (4) by assuming constant free flow area for the working fluid

137

transformation for each control volume. Considering that the area during time is constant, so Eq.

138

(5) is achieved with this assumption.

ρv ndA +

∂ (ρ A x ) = 0 ∂t

ρv ndA + A

(4)

∂ (ρ x ) = 0 ∂t

(5)

139

The final continuity equation that used in the code with the assumption of constant density is

140

achieved according to the following equation: ∑

(6)

ρ v n A + A (ρ x ) = 0

141

By assuming constant amounts for the gas density, velocity, and the cross sectional area, the

142

summarized forms of the balance equations are read as Eq. (7) and Eq. (8): ∂ ρ A x v ∂t

+

Pn A +

(v ρ )v n A + τ A

(7)

=0

143 ρ A x

144

u +

+∑

(ρ u + ρ

+ P )v n A + A P

−

=0

By using Eq. (9), one can calculate the overall heat transferring into each control volume:

(8)

dQ Nu K A = dt D 145

T

(9)

−T

For each control volume, the changes on the wall temperature can be stated as followings: dT 1 + c m dt



dQ dQ − dt dt

(10)

=0

146 147

The specific internal energy of helium as working fluid is obtained as according to Eq. (11) [53-

148

55]: u = c T , (c = 3.11 kJ/kg. K)

149

The conductivity of gas thermal can be calculated as Eq. (12) for helium [53-55]. K = 2.6810

150

(11)

× (1 + 1.123P 10 )(T

.

×

)

(12)

Helium is considered as ideal gas and it can be stated as followings [53-55]. P = ρ rT , (r = 2.08kJ/kg. K)

(13)

151

The assumptions for isothermal wall and isentropic regenerative wall are presented in Eq. (14-a)

152

and Eq. (14-b).

153

dQ =0 dt

(14-a)

dT =0 dt

(14-b)

The Reynolds (Re) and kinetic Reynolds (Re ) numbers may be calculated by Eqs. (15) - (18).

Re =

ρ v D μ

(15)

Re

=

ρ D θ μ

μ = 1.475 × 10 D 154

=

(16)

.

(17)

4 A A

(18)

A schematic of the engine is shown in Fig.3.

155 156

[Insert Figure 3 here] Eqs. (19) and (20) relate these parameters to b and b . l

=r

+b

+ 2r b cos (θ)

(19)

l

=r

+b

+ 2r b cos (θ + φ)

(20)

157

By solving the above equations for b and b , these are described as functions of crank angle in

158

Eqs. (21) and Eq. (22). Thus, according to Eqs. (23) - (25), x to x may be stated as functions of

159

crank angle. b = (r cos θ + l − r ) − r cosθ

(21)

r cos(2φ + 2θ) + l b = 2

(22)

r − 2

+ r cos(φ + θ)

x = c − a − b

(23)

x = c − a − b

(24)

x = d− a − x

(25)

160

The values of τ and Nu are stated based on the experimental models, suggested by Kays and

161

London [56], De Monte et al. [57,58], and S. Y. Zheng [59].

162

The output power of engine and the rejected heat of Stirling engine can be calculated using Eqs.

163

(26) and (27).

p=

A P

:

Q =

dx dt

(26)

(27)

Q

:

164

4. Multi-Objective Optimization

165 166

In the issues of multi-objective optimization, there are multiple objective functions for each set

167

of input variables. In such a space, it is difficult to find the answer based on gradient methods.

168

NSGA-II optimization algorithm is one of ways that has been used for solving many multi-

169

objective problems [60].

170

The general form of multi-objective problems is as followings [61]: Minimize / Maximize f m (x) Subject to gj (x)≥ 0 hk (x)=0 X

171

( )

≤Xi≤ X

( )



m = 1,2,...,M

(28)

j=1,2,…,J

(29)

k=1,2,…,K

(28)

i=1,2,…,n

(31)

An x solution is a vector of n decision variables: X= [X X … X ]

(32)

172

In the method, each decision variable is limited between the amounts of the lower limit (X

173

and the upper limit (X

174

vector of m objective functions.

( )

( )

)

). These limits make the space of the decision variable. Also, f is a

f (x) = [f1(x) f2(x) … fn(x)]

(29)

175

Where gj (x) and hk (x) are constraint functions and J and K are number of the inequality

176

constraints and number of the equality constraints, respectively. The mapping between decision

177

variable space and objective function space is shown in Fig. 4.

178

[Insert Figure 4 here]

179

4.1. NSGA-II

180

Multi-objective optimization has many differences with single-objective optimization. The best

181

decision is usually the absolute extreme of the objective function in single-objective

182

optimization. However, in multi-objective optimization, the fundamental problem is the conflict

183

between objectives. Namely, there is no a unique solution that can optimize all the objectives

184

simultaneously. So, the best solution covers the spots which have best distance from the spots

185

which optimize every objective. The solutions are called Pareto optimal solution or non-

186

dominated solution [62-64]. In the Pareto optimal solution, none of the solutions dominates other

187

solutions. Selecting one of them requires recognizing the problem and its correlated factors.

188

Also, it depends on the opinion of designer and the design conditions. Before a complete

189

description of the algorithm, it is necessary to define the concept of dominance and crowding

190

distance.

191

4.1.1. The Concept of Dominance

192

In a minimization problem with more than one objective, we can say solution X1 dominates the

193

solution X2, if and only if the solution X2 would not be better than the solution X1 as the first

194

condition. The second condition is that the solution X1 would be better than the solution X2, at

195

least for one time.

196 197

4.1.2. Crowding Distance

198

The concept of crowding distance is used to remove some solutions in the high density space. In

199

order to determine the population density around a particular point, the criterion to adjust the

200

population variety is achieved which is called as the crowding distance. For this purpose, the

201

average distance from the two solutions located in two sides of the ith solution is calculated for

202

each of the m objective functions (Fig. 5).

203 204

[Insert Figure 5 here] This concept is mathematically defined as: d = d =

f x(

)

f f x( f

− f x(

)

(33)

)

(34)

−f )

− f x( −f

d =d + d

(35)

205

where x , d , d , and d are the th solution, the crowding distance of the th solution in the first

206

objective function, the crowding distance of the th solution in the second objective function and

207

the crowding distance of the th solution, respectively.

208

As it can be seen in the Fig.6, the performance procedures of multi-objective optimization model

209

are described as:

210

1. Generating random parent generation (Pt) of size N.

211

2. Arranging the initial generation of parents based on none dominated solution.

212

3. Considering proportional rank with the level of non-dominated for each of the non-

213 214 215 216

dominated solution. 4. Generating children generation (Qt) of size N using the selection, crossover, and mutation operators. 5. According to the first generated generation that includes the parents and the children

217

chromosomes, the new generation is defined as followings:

218

-

219 220

population of size 2N. -

Arranging the (Rt) generation based on the non-dominated sorting method and recognizing non-dominated fronts (F1, F2, …, Fl).

221 222

Combining chromosomes of the parents (Pt) and the children (Qt) and generating (Rt)

-

Generating parent generation to the next iteration (Pt+1) of size N using non-

223

dominated fronts. According to the number of the chromosomes which are required

224

for generating the parents (N), initially, first front chromosomes are selected for the

225

parent generation. If this number does not satisfy the total requirements of the parent

226

generation, the second, third, fourth … front levels are taken respectively to achieve

227

the total amount of N. Regarding the last front, the solutions which have more

228

crowding distance, would be considered as the priority for filling the layers.

229

-

(Pt+1) and creating of children generation of size N (Qt+1).

230 231 232 233

Applying crossover and mutation operators on the new created generation of parents

-

Repeating the loop of step 5 until the total number of the required iterations is achieved. [Insert Figure 6 here]

234

4.2. Objective functions, decision variables and constraints

235

In this study, two objective functions including the differential regenerator pressure (bar) and the

236

output power (kW) are considered. Also, six decision variables are taken into account, as

237

followings:

238

Dh: Heater diameter

239

Mr: Total mass of regenerator

240

D

241

Lr: Regenerator length

242

P: Engine pressure charge

243

f: Engine frequency

244

The constraints that considered for optimization is stated as follow:

: Regenerator matrix diameter

0.004 ≤ Dh ≤ 0.008 m

(36)

0.08 ≤ Mr ≤ 0.3 kg

(37)

10 ≤ D

(38)

≤ 30

m

0.02 ≤ Lr ≤ 0.06 m

(39)

5 ≤ P ≤ 14 bar

(40)

12 ≤ f ≤ 22 Hz

(41)

245 246

All the above objective functions except D Powertrain Company (IPCO).

247

In this study, three well-known decision makers including LINMAP, TOPSIS, and Fuzzy are

248

applied to determine the final answer achieved by a multi-objective optimization. The mentioned

249

decision makers are well described in the literature [65-68].

250 251

are taken of the experimental data of IranKhodro

252

5. Results and discussion

253

5.1. First scenario results

254

In order to modify the final model to characterize the objective and criteria, a primary

255

investigation and modeling was performed, which is expressed here for the first time.

256

5.1.1. The effects of frequency

257

As it is depicted in Fig.7, the output power increases up to a specified position by raising

258

frequency then, the power is decreased. This process is shown in Fig. 8, which depicts the

259

increase of differential pressure drop with increasing frequency. Also, in a specific frequency,

260

by increasing the charge pressure, the regenerator differential pressure rises as well.

261

5.1.2. The effects of regenerator length

262

As it is shown in fig Fig.9, by increasing the regenerator length, the output power is decreased.

263

Also, in a fixed certain regenerator length, the output power is increased by increasing charge

264

pressure. On the other hand, by increasing the regenerator length, the regenerator differential

265

pressure is decreased (Fig.10). Additionally, with increasing P in specific regenerator length, the

266

regenerator differential pressure is increased.

267

5.1.3. The effects of regenerator matrix diameter

268

As it is demonstrated in Fig.11, by increasing the regenerator matrix diameter, the output power

269

is increased. Also, in a certain regenerator matrix diameter, the output power is raised by

270

increasing pressure charge. According to Fig.12, by increasing the regenerator matrix diameter,

271

the differential pressure regenerator is decreased. Also, in a specific regenerator matrix diameter,

272

the differential regenerator pressure is increased by increasing charge pressure because by

273

increasing the regenerator matrix diameter, the porosity of regenerator decrease and the working

274

fluid pass easily through regenerator.

275

[Insert Figure 7 to 12 here]

276 277 278

5.2. Second scenario results

279

The goal of the multi-objective optimization in this paper is minimizing the differential

280

regenerator pressure (bar) and maximizing the output power (kW). The procedures were

281

conducted by a class of an evolutionary algorithm, called the NSGA-II multi-objective. The third

282

order thermodynamic analysis was applied for the optimization, with the presented constraints of

283

Eqs. (36)-(41). Fig.13a depicts the Pareto frontier in the space of the suggested objectives by the

284

optimization. In this figure, the results of the final solutions by the LINMAP, Fuzzy Bellman-

285

Zadeh, and TOPSIS methods are presented. It can be found that the obtained points by TOPSIS

286

and LINMAP have tendency towards each other. The optimal amount of the differential pressure

287

is in the range from 0.005 bar to 0.045 bar and the optimal value of the output power is varied

288

from 0.380 kW to 0.9 kW. Based on the obtained curve fit illustrated in Fig13b, Eq. (42) is

289

presented. differential pressure = a × exp(b × output power ) + c × exp(d × output power)

(42)

290

The coefficients (with 95% confidence bounds) and goodness of fit are reported in Table (2).

291

Also, the comparison of the final optimal solutions with experimental results is presented in Table (3).

292

By the use of multi-objective optimization, the differential pressure and the output power could

293

be decreased and increased, respectively.

294

[Insert Table 2 here]

295

[Insert Figure 13 (a,b) here]

296

[Insert Table 3 here]

297

Conclusions

298

The third order thermodynamic analysis is considered for this study to find out the differential

299

regenerator pressure and the power output of a Stirling engine. The regenerator matrix diameter,

300

regenerator length, engine charge pressure, and the engine frequency were applied as the design

301

parameters. The results indicate that, by increasing the regenerator matrix diameter, partial

302

pressure is increased while the output power is raised, as well. On the other hand, by increasing

303

the regenerator length, both output power and differential pressure of regenerator are decreased.

304

In the final solution, it was found that the Fuzzy decision-making method generated higher

305

output power. At last, for pressure drop, TOPSIS and LINMAP with the pre-mentioned amount

306

(0.013572925) have better status among Fuzzy model.

307

Nomenclature d f(x) X θ φ l l a a a b b

Crowding distance Objectives function Vector of decision variables Crank angle (Radian) Piston-displacer phase angle (Radian) Piston rod length (m) Displacer rod length (m) Piston gudgeon pin & surface distance (m) Displacer gudgeon pin & surface distance (m) Displacer height (m) Piston gudgeon pin & crank axis distance (m) Displacer gudgeon pin & crank axis distance (m)

v ρ n x A S V T P τ u Q

Gas velocity (m/s) Gas density (kg/m3) Normal vector on surface Control-volume length (m) Wall area of control-volume (m2) Connecting surfaces between adjacent control-volumes (m2) Volume (m3) Gas temperature (K) Gas pressure (Pa) Gas shear stress (Pa) Gas specific internal energy (J/kg) Net absorbed heat of gas (J)

c c d r A f η p Q T T

308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326

Piston cylinder head & crank axis distance (m) Displacer cylinder head & crank axis distance (m) Displacer stroke (cm) Crank radius (m) Gas flow area (m2) Engine frequency (Hz) Engine efficiency Engine output power (W) Total heat rate rejected (W) Heat rejection temperature (K) Heat absorption temperature (K)

Q m c T r c Re Re μ Nu k K t

Net injected heat to control-volume (J) Mass of control-volume wall (kg) Specific heat of wall (kJ/kg) Wall temperature (K) Working gas constant (kJ/kg.K) Gas specific heat (kJ/kg.K) Reynolds number of gas Reynolds number of gas Dynamic viscosity of gas (Pa.s) Nusselt number Control volume index Working gas conductivity (W/m.K) Time (s)

327 328 329 330

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331

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490 491 492 493 494 495 496 497 498 499 500 501

Figures:

502 503

504 505 506

Fig.1. P-V diagram of Stirling engine cycle

507 508

Fig.2. Components of the ST500 Gamma type Stirling, by IPCO

509 510

511 512

Fig.3. ST500 linkage kinematics

513

514 515

Fig.4. A schematic of the decision variable space and objective function space [31]

516 517 518 519

Fig.5. Schematic of the crowding distance [33]

520 521

Fig.6. The structure of NSGA-II model [33]

522 523 524 525 526 0.8

Output Power (kW)

0.7 0.6 0.5

P=5 (bar) P=10 (bar)

0.4

P=14 (bar) 0.3 0.2 10

12

14

16 18 Frequency (Hz)

20

527 528

Fig.7. The power output versus frequency

22

24

529 530

Differential Regenerator Pressure (bar)

0.0022 P=5 (bar)

0.002

P=10 (bar)

0.0018

P=14 (bar)

0.0016 0.0014 0.0012 0.001 0.0008 0.0006 0.0004 0.0002 10

12

14

16 18 Frequency (Hz)

20

22

24

531 532

Fig.8. Variation of the differential regenerator pressure versus frequency

533 534 P=5 (bar)

Output Power (kW)

1 0.9

P=10 (bar)

0.8

P=14 (bar)

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.01

0.02

0.03 0.04 0.05 Regenerator Length (m)

0.06

535 536

Fig.9. Changes of the output power versus regenerator length

0.07

537 Differential Regenerator Pressure (bar)

0.006 P=5 (bar)

0.005

P=10 (bar) 0.004

P=14 (bar)

0.003 0.002 0.001 0 0.01

0.02

0.03 0.04 0.05 Regenerator Length (m)

0.06

0.07

538 539

Fig.10. Variation of the differential regenerator pressure versus regenerator length

540 541 542 0.9

Output Power (kW)

0.8 0.7

P=5 (bar)

0.6

P=10 (bar)

0.5

P=14 (bar)

0.4 0.3 0.2 0.1 0 0

10 20 30 Regenerator matrix diameter (µm)

40

543 544

Fig.11. Variation of the output power at different diameters of regenerator matrix

545

Differential Regenerator Pressure (bar)

0.14 P=5 (bar) 0.12

P=10 (bar)

0.1

P=14 (bar)

0.08 0.06 0.04 0.02 0 0

10 20 30 Regenerator matrix diameter (µm)

40

546 547 548 549 550

Fig.12. Variation of the differential regenerator pressure versus regenerator matrix diameter

551

a

552 553

b Fig.13 (a,b). Pareto optimal frontier in the objectives space

554

Tables:

555 556

Table1. ST500 engine specification

Specifications Stirling type Electrical power Overall efficiency Standard charge pressure Working fluid Working frequency Fuel Cooling agent Water Power piston stroke Displacer stroke Phase angle Heater type Cooler type ×144 Regenerator material Heat absorption temperature Heat rejection temperature

Values Gamma 500W 8.5% 8bars Helium 14Hz Natural gas 0.075m 0.075m 90 deg Pipe (6mm dia.) ×20 Duct (13mm 2 section area) Steel matrix (0.96 porosity) 350-420˚C 30 - 50˚C

557 558 559

Table 2. The coefficients and goodness of fit

Coefficient a = 0.004557

b = 1.572

c = 5.673e-18

d = 38.93

Goodness of fit SSE: 0.001187

R-square: 0.8951

Adjusted R-square: 0.8937

560 561 562

RMSE: 0.002322

563 564 565

Table 3. Comparison of the final optimal solutions with experimental results Decision variables Dh(m)

Mr(kg)

0.006

0.2

TOPSIS

0.005735

LINMAP

(µm)

Objective functions Output Power (W)

Differential Pressure (bar)

15.20954

0.7387148

0.013572925

11.56312

15.20954

0.7387148

0.013572925

0.054146

13.08954

16.23126

0.8001117

0.016273058

0.056

6.22

16.16

0.363

-

Lr(m)

P(bar)

f(Hz)

250

0.056

6.22

16.16

0.114732

29.53309

0.053057

11.56312

0.005735

0.114732

29.53309

0.053057

Fuzzy

0.007876

0.117437

27.82535

Experiment

0.006

0.2

250

566 567 568