j. basic. appl. sci. res., 2(2)1395-1406, 2012

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J. Basic. Appl. Sci. Res. , 2(2)1395-1406, 2012

2012, TextRoad Publication

ISSN 2090-4304 Journal of Basic and Applied

Scientific Research www.textroad.com

*Corresponding Author: Mostafa mahmoodi, Faculty of Mechanical Engineering, K. N. Toosi University of Technology, Tehran, Iran.P.O.Box: 1999-19395. Email: [email protected].

Numerical Solution of Beta-type Stirling Engine by Optimizing Heat Regenerator forIncreasing Output Power and Efficiency

Masoud Ziabasharhagh 1 Mostafa Mahmoodi 2

1 Associate professor, Faculty of Mechanical Engineering, K. N. Toosi University of Technology, Tehran, Iran.2 Ph.D. candidate, Faculty of Mechanical Engineering, K. N. Toosi University of Technology, Tehran, Iran.

ABSTRACT

In this paper, an appropriate thermodynamic model was developed for the beta-type Stirling engine. Thismodel is capable of predicting and optimizing the output power and operational parameters of the Stirlingengine based on the geographical and physical conditions of all its components. To this end, the numericalsolution of the beta-type Stirling engine was conducted by the five-volume method and using the adiabatic

model. To confirm the obtained results, the operational and geographical specifications of the Stirling enginemade in the General Motors Corporation were used and the obtained results were compared with the published results. These results showed the increase in the energy absorption by the regenerator in highoperational pressure and bigger length-to-diameter ratio of the regenerator. Moreover, considering theachieved results, an optimum point can be determined for the length of the regenerator in order to reach themost efficiency and highest output power.KEY WORDS: Striling engine, Numerical solution, Adiabatic model, Regenerator.

NomenclatureA Area, m 2 Greek symbols Cp specific heat at constant pressure, J/(kg K) Ideal gas specific heat ratioCv specific heat at constant volume, J/(kg K) Crank rotational angle, radians

D diameter, m Subscriptsdm wire diameter, m c Compression spacee Eccentricity, m ck Interface between compression space and coolerL Connecting rod length, m cl Clearance space in engineM mass of working gas in the engine, kg e Expansion spacem mass of gas in different components, kg h HeaterP Pressure, Pascal he Interface between heater and expansion spaceQ Heat, J k Coolerr Connecting rod length, mm kr Interface between cooler and regeneratorR gas constant, J/(kg K) r Regenerator

T Temperature, K rh Interface between regenerator and heater

1. INTRODUCTION

Many attempts are being made all over the world in order to make power-generator engines withappropriate efficiency and with the capability of using waste heat recovery and renewable energy sources at thesame level as the fossil fuels. Increase of energy cost and environmental and noise pollutions have led to moreserious studies on the new power generation engines. Use of non-renewable energy sources of the earth including

petroleum, gas and coal destroys the societies public wealth and produces one-fourth of the total carbon dioxide ofthe world. Stirling engine is one of the ideas which have attracted the attention of many interested researchers inrecent years. In physical terms, the Stirling engine is an external combustion engine and can use any kinds ofexternal heat sources (waste heat from industrial machinery, combustion and solar energy) for producingmechanical energy. Recently, researchers have reached good results in designing and using Stirling engines. Stirlingtechnology development for making solar engines which generate the axial power of 10 KW is one of the mostimportant achievements. Also, the combined heat and power (CHP) generation plan is one of the new ideas whichhave been commercialized by the generation companies of this technology and are used in power generation bases.At present, new methods of using Stirling methods are studied which include application of Stirling engines for

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Ziabasharhagh and mahmoodi, 2012

providing the required power for satellites and using Stirling engines as an alternative for the steam turbines ofnuclear power plants.

Many studies have been done on Stirling engines since their invention by Robert Stirling. The first

mathematical analysis which was acceptable for analyzing Stirling cycle was presented by Schmidt fifty years afterits invention [1]. Schmidts analysis was done based on the theories of isotherm compression chamber and isothermexpansion chamber. Using the Schmidts assumption, the thermodynamic equations were linear and the initialcalculations were easily done for measuring the engines output efficiency and power. Today, Schmidts analysis iswidely used for the initial analysis of Stirling engines. Schmidts cycle assumes expansion and compression

processes as isotherm processes. However, in practice, this assumption is not true for the engines with 1000 rpm ormore engine cycles since, as proved by Rankin, heating or cooling does not occur exactly at the constanttemperature or constant volume and expansion and compression processes of the cylinders of the Stirling engine arecloser to the adiabatic. Therefore, more appropriate assumptions should be used for thermodynamic modelling inorder to get closer to the actual efficiency of the engine by these models.

If attempts for modelling Stirling engines are not within the isotherm solution, the equations will not beclear and can only be solved differentially and using numerical methods. In the adiabatic cycle, heat efficiency isturned to a function which depends not only on temperature but also on swept volume ratio, phase angle and deadvolume ratio. In fact, the output power is a function of all the mentioned parameters either in the isotherm oradiabatic cycles. In 1975, Finkelstein [2] improved Schmidts thermodynamic analysis and presented the initialadiabatic analyses. While solving equations in the adiabatic way, compression chamber and expansion chamber areconsidered adiabatic. Considering the adiabatic assumption, the equations become non-linear and numericalmethods should be used for solving them. Since the Finkelstein model was presented, the conducted analyses have

been done based on different thermodynamic models (isotherm and adiabatic), use of d ifferent heat sources (heatwaste, solar and combustion) and different forms of Stirling engine (gamma-, beta- and alpha-type engines); thestudies done by Urieli and Berchowitz [3] which used the adiabatic thermodynamic model for obtaining the Stirlingengines output efficiency and power can be referred to. Kongtragool and Wongwises [4] conducted the Stirlingengines modelling and optimization using the isotherm model and Timoumi al. [5, 6 and 7] investigated its lossesand irreversibility using the Stirling engines adiabatic modelling. In recent studies done by Tlili et al.[8, 9], theStirling engine was modelled using the solar energy as the heat source. Thombare and, Verma [10] gathered theavailable technologies and obtained achievements with regard to the analysis of Stirling engines and, at the end,

presented some suggestions for their applications. Furthermore, Alireza Tavakkolpour et al. [11] analyzed thegamma-type Stirling engine using the Schmidts theory, solving the equations in the isotherm way and using flat

panels for absorbing solar heat as the hot temperature source. After modelling the Stirling engine, Invernizzi [12]investigated the effect of using different gasses on the engines output power and efficiency. Formosa and Despesse[13] conducted the modelling by means of the isotherm model in order to investigate the effects of dead volumes onthe engines output power and efficiency.

In the present study, the thermodynamic relations of all the engine components were applied, the numericalcode written in the Matlab software was used and, as a result, the beta-type Stirling engine was modelled in the five-volume method using the adiabatic thermodynamic model. To increase the accuracy of simulation, actual relationswere used for the changes of volume of the pistons inside the cylinders and no simplification was used in thisregard. At the end, the obtained results were compared with the results presented for the engine made by GeneralMotors Corporation (GPU-3), which geometrical and operational specifications are present. The obtained resultsshowed that energy flow was very important for heat regenerator and its optimized design can prevent the waste of alarge amount of the received energy from the high-temperature heat source during the engine work cycle. Finally,

the effects of different geometrical and physical conditions of heat regenerator on the engines output power andefficiency were studies and the optimized values were suggested.

2. Thermodynamic ModellingThe Stirling engine works in a closed thermodynamic cycle and converts thermal energy to mechanical

motion. Considering the physical structure of this engine, it has five main subsystems; each subsystem is considereda control volume in modelling. There are two spaces with variable volumes which are called expansion space andcompression space and there are three heat exchanger with the constant volume which are called heater, cooler andregenerator (Figure 1). Moreover, the engine has a drift mechanism which controls volume changes during the workcycle and transfers the alternating linear motion of pistons as the angular momentum to the driving shaft. Differenttypes of the Stirling engine are identified by the names alpha, beta and gamma. All of them are similar in terms ofthermodynamic cycle; however, there are fundamental differences as far as mechanical mechanisms are concerned.Alpha-type Stirling engine has two pistons in two separate cylinders. A heater is built in one cylinder and a cooler in

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4Heater

3

Regenerator

2

Cooler1

another cylinder. The operating gas starts its going and coming movement from the heater and enters the cooler bythe regenerator. In the beta-type Stirling engine, there are two pistons inside one cylinder which are called displacer

piston and power piston. The displacement piston moves the operating fluid between the cold space and hot space

through the heater, cooler and regenerator and leads to the movement of the power piston (Figure 2). The gamma-type Stirling engine is a combination of alpha and beta types. In theoretical terms, the efficiency of the Stirlingengine is equal to the efficiency of the Carnot cycle. The transferred heat from the operating gas to the regeneratorduring the 4 to 1 process is absorbed by the operating gas during the 2 to 3 process. A source for heat sink andanother source for heat source are required to be used during the 1 to 2 and 3 to 4 processes (Figure 1).

Figure 1: The components of Stirling engine and the operation of their cycle

Figure 2: Beta-type Stirling engine

Heat regenerator is probably the most impotent component of the Stirling engine which has a fundamentalrole in increasing its efficiency. As far as the physical structure is concerned, the heat regenerator is made ofstainless steel as mesh sheets or stainless steel rods which are stacked (Figure 3). During the half of the engineswork cycle, the heat regenerator acts like a sponge and leads to heat absorption from the operating cycle. In anotherhalf of the cycle, the regenerator returns the heat to the operating gas; therefore, there is less amount of heat in thecold area of the engine for dissipation, which leads to the increase of engine efficiency. Thus, the use of theregenerator in the Stirling engine decreases the heat waste and, finally, increases the engine efficiency.

Compression space

Power piston

Displacer piston

Cooler

Heater

Expansion space

Regenerator

Carter

Rod

Crank

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c

c

c

V

T

m

p

ck

ck

T

m

k

k

k

V

T

m

kr

kr

T

m

r

r

r

V

T

m

rh

rh

T

m

he

he

T

m

h

h

h

V

T

m

c

c

c

V

T

m

W W

k Q hQ

e

e

e

V

T

m

Figure 3: The structure of heat regenerator

The modeling conducted in this paper was for the beta-type engine using the five-volume method. In thismethod, the components of the Stirling engine were divided to five separate sections and thermodynamic equationswere extracted for each unit. Then, appropriate boundary conditions were considered for the movement of the fluidflow inside the components of the engine at the interfaces between cells. Finally, the numerically obtained equationswere solved using the method mentioned below. Figure 4 shows the schematic model and the distribution ofdifferent temperatures at different parts of the engine.

Figure 5 demonstrates the rate of volume change using the motion mechanism and its obtained relations.

Reliable input specifications are required for testing the capabilities of the conducted modeling.

Figure 4: The five-volume model for simulating the Stirling engine

Figure 5: Operation mechanism and volume changes in the beta-type Stirling engine

To this end, exact specifications of the engine made by General Motors Corporation(Ground Power Unit-3) were used; in addition to its physical and geometrical characteristics, its operationalspecifications are available, which have been used by many scholars for modeling validation. In Table 1, theengines geometrical parameters including the exact geometrical specifications and the swept volume by the power

piston and displacement piston, looseness amount of the power piston and displacement piston, geometricalcharacteristics and the number of pipes of heat exchangers are given. In Table 2, the engines operationalspecifications including the operating gas, temperature of cold and hot areas, average pressure of the operating fluid,mass of gas and motion frequency of the engine are mentioned. Table 3 presents the specifications of theregenerator cell with porosity and wires with different diameters in order to observe the engines output power andefficiency variation compared with their changes. Other input parameters required for the analysis based on theengines conditions and form were calculated using the available equations during the numerical solution.

r

cV

eV

b L

e

L b

e y

c y

p y

d y max d y

b

)cosr e( sinr A

d dV pc 2

cosr A A

Ad dV

d dV

d p

d c

e

2

222 er Lb

22 r e Lb

221 r e Lb

223 r e Lb

) sinr bb( AV V d clee 2 )bb( AV V pclcc 12

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Table 1: The geometrical parameters of the GPU-3 engineClearancevolumes

Compression space 28.68 cm 3

Cooler

Tubes number 312Expansion space 30.52 cm 3 Interns tube 1.08mm

Sweptvolumes

Compression space 113.14 cm 3 Diameter 46.1 mmExpansion space 120.82cm 3 Tube length 46.1mm

Connecting rod length 46 mm Void volume 13.8 cm 3 Eccentricity 20.8 mm Regenerator

Power piston diameter 69.9 mmThe regenerator body is pipe-like in which metal wires are

accumulated on each other (Figure 3)Displacer diameter 69 mm Diameter 22.6 mm

Exchanger piston conductivity 15 W/mK Length 22.6 mmExchanger piston stroke 46 mm Wire diameter 40 m

Heater

Tubes number 40 Porosity 0.697Tube inside diameter 3.02 mm Units numbers/cylinder 8

Tube length 245.3 mm Thermal conductivity 15 W/mKVoid volume 70.88 cm 3 Void volume 50.55 cm 3

Table 2: Operational parameters of the GPU-3 engine

Kpa4130 Mean pressureheliumGas grm1.135 Total mass of gas977 KHot space temperature(Th)Hz41.7 Frequency288 KCold space temperature(Tk)

Table 3: Specifications of the regenerator cell with different wire diameter and porosityPorosity Wire diameter (d m) mm

M1 0.9122 0.0035M2 0.8359 0.0065M3 0.7508 0.007M4 0.7221 0.007M5 0.6655 0.008M6 0.6112 0.008

3. Governing Equations and the Solution Method

The thermodynamic modeling of the engine is done in two sections. In the first section, the modeling isdone in the isotherm method using the Schmidts model. The results obtained from this section aree used as theinitial values for the second section.3.1 Section One: Analyzing the Stirling Engine in the Isotherm Mode

The aim of isotherm analysis is to obtain the conducted work as a result of the pressure changes andtemperature of the operating gas using the heat transfer to the inside of the engine. The main attraction of theisotherm analysis is the closed solution method which appears in its equations. The fundamental assumption in thisanalysis is that the gas in the expansion chamber and heater and the gas in the compression chamber and cooler arekept at the heaters heat degree and coolers heat degree, respectively. The thermodynamic cycle of isotherm iscomposed of two isotherm processes and two constant volume processes. In addition, it iss assumed that theexpansion and compression processes inside the engine are isotherm and the effects of non-ideal regenerator and

pressure drop are not considered. The starting point of the analysis is to hold the total mass constant in all thevolume occupied by the gas [10].

ehr k c mmmmm M (1) Ideal gas law is used and the equation of state is inserted in Equation (1):

RT V T V T V T V T V p M hehhr r k k k c /)/////( (2)

Based on the Schmidts assumption, the level of T r is shown linearly and by means of the following relation [3]:

)/ln(/)( k hk hr T T T T T (3)

Finally, the pressure equation is obtained in the following way by putting Equation (3) in the equation of state:

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1/ln

h

e

h

h

k h

k hr

k

k

k

c

T V

T V

T T T T V

T V

T V

MR p (4)

There is a relationship between the volume of the expansion and compression chambers at any moment andthe engines kinematics mechanism movement at different angles of crank. The relationship between the volumechanges and kinematics mechanisms of the engine is g iven in Figure 5.

3.2 Section Two: Analyzing the Stirling Engine in the Adiabatic ModeThe adiabatic model is based on these assumptions that the cooler and heater had infinite heat transfer and

the isotherm condition is true in them. Therefore, the fluid in the heat exchanger is always at the highesttemperature, i.e. T max, or at the lowest temperature, i.e. T min .The temperature of the operating fluid in the cylinderscan be less or more than T max in the expansion space or T min in the compression space during the cycle. Figure 6shows the temperature in different sections of the engine and the temperature gradient in 5 sections of the engine.

Figure 6: The schematics of the temperature in different sections of the beta-type Stirling engine

To solve it in the adiabatic mode, first, mass is considered constant in the total system and then, using theenergy equations and the equation of state of the perfect gas, the equations required for measuring the level of heattransfer to the engine and the conducted work and, finally, the engines efficiency are obtained.According to the agreement, the individual suffixes in Figure 4 indicate the five cells of the engine and the dualsuffixes indicate the juncture of cells with each other. Considering the system of equations defined for the model, itis specified that there are 22 variables and 15 numbers of the differential equation for solving the engines cycle.The extracted equations are as follows [9]:

Equation of pressure

heehhr r k k ck eheeck c

T V T V T V T V T V T DV T DV p

Dp/)///(/

)//(

(5)

Equations of mass

)/()/( ck ccc RT DpV pDV Dm (6)Equations related to mass changes

(7)

Equations of temperature

)(

)/(

)/(

)/(

r hk ce

hhh

r r r

k k k

mmmm M m

RT pV m

RT pV m

RT pV m

hrhhe

r kr rh

k ck kr

cck

Dmmm

Dmmm

Dmmm

Dmm

p Dpm Dm

p Dpm Dm

p Dpm Dm

hh

r r

k k

/

/

/

Compression Cooler Regenerator Heater Expansion space

T c T k

T r

T h T e

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)/(

)/(

eee

ccc

Rm pV T

Rm pV T

(8)

Border conditions in the juncture of cells [8]:

hheeheeh

ckck kckc

T T OtherwiseT T Thenmif

T T OtherwiseT T Thenmif

0

0

(9)And, finally, the equations obtained for the amount of work and the heat given to the heater and the heat taken fromthe cooler and, at the end, the exchanged heat in the heat regenerator are obtained using the following relations [9]:

)(/

)(/

)(/

)(

heherhrh P V hh

rhrhkr kr P V r r

kr kr ck ck P V k k

ce

mT mT C R DpcV DQ

mT mT C R DpcV DQ

mT mT C R DpcV DQ

DV DV p DW

(10)For the numerical solution of the equations obtained from the adiabatic model, the equations of pressure

and mass changes should be simultaneously solved in the compression space along with the energy equations. The best method for the numerical solution is to use the initial values method. In this method, the initial values of all thevariables are clear in the zero point at the beginning and the final values of the equations are solved using theseinitial values so that they obtained functions include all the available variables along with the functions related tothe volume changes of the engine at different angles of the crank. To solve it, the y vector was defined as a functionof available variables. For example, y p shows the gas pressure of the system, y mc indicates the gas mass in thecompression chamber and so on. If the initial values of the variables are available, the y vector is defined as y(t 0)=y 0 . According to that, the value of y(t) was determined from the differential equations of Dy=F(t,y) and thedetermined values satisfied differential and initial equations simultaneously. In fact, in this numerical method, first,the initial values were calculated at time t 0 and, after that, new values were calculated at time t 1=t 0+t by a smallincrement of time. Therefore, there was a wide set of direct circles at different times which obtained the correctvalues of y(t). In the adiabatic solution method, initial values of mass changes and pressure in the compressionchamber should enter the equations along with other information of the problem such as volume changes. Thesevalues were obtained from the first section, i.e. solving the equations in the isotherm mode. By inserting theoperational and geometrical specifications of the GPU-3 engine and the initial values obtained from isothermsolution, the heat values were obtained for the heater, cooler, conducted work and engines efficiency. Thenumerical code was able to enter the considered changes in different parts in terms of physical and geometricalaspects and show the value changes of the operational power and engine efficiency by applying the new conditions.

4. DISCUSSING THE RESULTS

All the values required during the cycle were obtained by applying the input specifications required for thecode from Tables 1 and 2 which include the operational and geometrical specifications of the GPU-3 engine and b ysolving the equations using the numerical method. The comparison of the results obtained from the numericalsolution using the specifications of the GPU-3 engine with the values published in the papers is given in Table 4.

The presented results demonstrate the capability of the numerical code in predicting the Stirling engines efficiencyand power. The difference between the obtained results and the published results in Table 4 is caused by theapplication of the exact operational specification and motion mechanism of the engine used in this research(Figure 5); these changes are usually simplified in other numerical codes and equivalent relations or the sinuschanges of the engine volume are applied.

Figure 7 shows the pressure changes in relation to the volume of the engine. The obtained level shows theamount of work conducted by the engine during each cycle. There is a considerable difference between the resultsobtained by the Carnot cycle consisting of two processes with the constant volume and two processes with theconstant temperature which is caused by applying actual conditions in the numerical code compared with the idealCarnot diagram. Figure 8 indicates the mass changes of the operating gas inside the engine during the cycle. Usingthe equations of mass and energy for the five cells of the engine and the relations of the ideal gas, the massaccumulation inside each of the cells is obtained during the engines cycle. The obtained results show the masschanges in the compression chamber in more intensity compared with the changes in the expansion chamber.

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

50

100

150

200

250

300

350

Reduced volume

C r a n

k A n g

l e ( d e g

)

Particle mass plot

Table 4: Comparing the obtained results using the specifications of the GPU-3 engine with the results of the published studies

Heat transferred to the cooler Q k Heat transferred to theheater Q h

Poweroutput

P[Watt]

Thermalefficiency

[%][J/cycle] [Watt] [J/cycle] [Watt]Urieli [3] 119.43 4980.231 318.5 13281.45 8300 62.5

Timoumi [7] 124.06 5173.302 327 13635.9 8286.7 62.06Present work 118.59 4945.203 317.099 13223.028 8279.48 62.6

Figure 7: The values of volume pressure for the engines cycle

Figure 8: Changes of the passing mass flow in the engines cells

Temperature fluctuations during the engine work cycle in the compression and expansion chambers aregiven in Figure 9. Table 2 demonstrates the input temperature degrees of the program which include the gastemperature in the cold and hot sides of the engine. The gas temperature is considered constant in the heatexchangers. Figure 10 shows the energy flow in addition to the total conducted work in each work cycle of theengine. The values for the dissipated heat by the cooler and added heat by the heater in addition to the totalconducted work are given in this figure. The difference in the energy range of the regenerator, heater and cooler iscompletely evident. Also, this figure shows that the total energy of the cycle passing an ideal model through theregenerator is equal to zero at the end of the cycle.

220 240 260 280 300 320 340 36025

30

35

40

45

50

55

60

65

Volume (cc)

P r e s s u r e

( b a r

[ 1 b a r =

1 0 0 k P a ]

)

P-v diagram

Ve Vh V r Vk V c

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Figure 9: Temperature changes in the cells of the Stirling engine

Figure 10: Energy changes in the cells of the engine

Figure 11 shows the sensitivity of the regenerator to the engines function pressure. The obtained resultsshow the increase of the regenerator absorption in high pressures of the operating gas. The engines heat efficiencyincreases with more absorption of the heat by the regenerator. Figure 12 demonstrates the effect of the regeneratorslength-to-diameter ratio on the level of the absorbed energy in the regenerator. The obtained results show that theless the regenerators length-to-diameter ratio, the more the level of the energy absorption during the cycle.

Figure 11: Changes of the regenerators received energy caused by pressure change

0 50 100 150 200 250 300 350 400200

300

400

500

600

700

800

900

1000

1100

Crank angle (degrees)

T e m p e r a

t u r e

( K )

Temperature vs crank angle

TcTeTkTr Th

0 50 100 150 200 250 300 350-600

-400

-200

0

200

400

600

800

1000

1200

1400


E n e r g y

[ J o u

l e s

]

Energy vs crank angle

QkQr QhWW(Compression)W(Expansion)

0 50 100 150 200 250 300 350 400-500

0

500

1000

1500

2000


Energy vs crank angle

E n e r g y

[ J o u

l e s

]

P=5000 kPaP=6000 kPaP=4000 kPaP=3000 kPaP=2000 kPa

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Figure 12: Changes of the regenerators received energy caused by its change of length

The effect of using different types of regenerators in terms of wire diameter and porosity is given in Figure13. The obtained results demonstrate the increase of efficiency and decrease of the output power of the engineagainst the decrease of the porosity level of the regenerator and increase of the diameter of its wires. In fact, as theregenerators porosity decreases, the flow of fluid inside the regenerator becomes easier and the engine can generatemore power. Instead, as the effective area of the regenerator decreases, the saving power of the heat energydecreases and more heat is used. Figure 14 shows the effect of the regenerator length on the engines output powerand efficiency. Considering the obtained results, the less the length of the regenerator, the less the enginesefficiency; however, the power of engine increases. The collision point of the power curve and efficiency curve inthis figure is the optimum design point in terms of having appropriate power and efficiency for the engine. Thisvalue is within the 40 mm length for the regenerator.

Figure 13: Changes of the engines efficiency and power caused by the change in the structure of the regenerator

0 50 100 150 200 250 300 350 400-200

0

200

400

600

800

1000

1200

1400


R e g e n e r a

t o r

E n e r g y

[ J o u

l e s

]

Regenerator Energy vs crank angle

L/D=0.5L/D=1L/D=1.5L/D=2

M1

M2

M3

M4

M5

M6M1

M2

M3

M4

M5

M6

62.3

62.4

62.5

62.6

62.7

62.8

62.9

63

63.1

63.2

7400

7500

7600

7700

7800

7900

8000

8100

8200

0 1 2 3 4 5 6 7

T h e r m a l E f f i c i e n c y [ % ]

P o w e r o u t p u t [ W ]

mesh model

Efficiency

Power output

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Figure 14: Changes of the engines efficiency and power caused by the regenerator

5. Conclusion

The results obtained from the numerical solution showed an appropriate correspondence with the publishedvalues. The regenerated power inside the heat regenerator using the written numerical code was measurable and thereaction of the heat regenerator and, as a result of it, the changes of the regenerators output power and efficiencywere obtained by entering different physical and geometrical specifications. The following statements can bementioned considering the results obtained from this analysis:

The ratio of the regenerated heat to the input heat can be calculated using the numerical code. Bigger

values of the regenerated heat have more effects on the increase in the engines efficiency. The energy flow of the regenerator is almost five times more than that of the heater and six times morethan that of the cooler.

As the pressure increases, the amount of regenerated energy in the heat regenerator increases noticeably. As the length-to-diameter ratio of the regenerator decreases, its absorbed energy increases. In the heat regenerator, porosity plays an important role in increasing the output power and efficiency. As

the porosity of the regenerator decreases, the output power increases and efficiency decreases. Consideringthe values shown in the figure, the regenerator type M3 is recommended for the highest power-to-efficiency ratio in the engine.

The most optimum dimensions of the heat regenerator can be obtained for the highest power and the mostefficiency using the calculation code. For instance, considering the efficiency and power curve in differentlengths of the heat regenerator, the best length of the regenerator for the engine has been calculated to be40 mm.

The obtained results showed that the energy flow in the heat regenerator is very important and its optimumdesign can prevent from the loss of a large amount of heat received from the high-temperature heat sourceduring the engine work cycle

REFERENCES

[1] Schmidt, G., 1871. The theory of Lehmanns Calorimetric Machine. Z.ver.Dtsch.ing, 15, part 1.

[2] Finkelstein, T., 1975. Analogue simulation of Stirling engine. Simulation, No.2, March.

[3] Urieli, I., D.M. Berchowitz, 1984. Stirling cycle engine analysis. Bristol: Adam Hilger.

60.5

61

61.5

62

62.5

63

63.5

64

64.5

65

0

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4000

5000

6000

7000

8000

9000

10000

0 10 20 30 40 50 60

T h e r m a l e f f i c i e n c y [ % ]

P o w e r o u t p u t [ W ]

L(regenerator) [mm]

Total power output Thermal efficiency

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j. basic. appl. sci. res., 2(2)1395-1406, 2012

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