Jhen-Syuan Huang, Prof. Chin-Hsiang Cheng Date : 2016/08/25

(1)

Jhen-Syuan Huang, Prof. Chin-Hsiang Cheng Date : 2016/08/25

Department of Aeronautics and Astronautics,

National Cheng Kung University, Tainan, Taiwan

(2)

 Introduction

 Prototypes and experimental apparatus

 Theoretical model

 Results and discussion

 Conclusions

(3)

Ringbom Beta-type

Alpha-type Thermal-lag Free-piston

Stirling cooler

Stirling engine

(4)

Rotation Speed (r.p.m.)

Power(W) Temperature(°C)

1000 1500 2000

400 600 800 1000 1200 1400 1600

0 200 400 600 800 1000 1200 1400 1600

Power Heating Temp.

Expansion chamber Temp.

Reg. chamber entrance Temp.

Reg. chamber exit Temp.

Cooler Temp.

Solar tracker

10.75 kW@1000 W/m

²

Diameter：3.7m

1.5 kW beta-type Stirling engine

(5)

CFD analysis

(Temperature variation) Stress analysis

(Von-Mises stress)

Volume [cm³]

Pressure[atm]

0 100 200 300 400 500 600

6 8 10 12 14

16 ω = 600 rpm

ω = 800 rpm ω = 900 rpm ω = 950 rpm ω = 1000 rpm ω = 1050 rpm ω = 1200 rpm

Volume [cm³]

Pressure[atm]

0 100 200 300 400 500

6 8 10 12 14 16

ω = 600 rpm ω = 800 rpm ω = 900 rpm ω = 950 rpm ω = 1000 rpm ω = 1050 rpm ω = 1200 rpm

Time (s)

EquivalentStress(MPa)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0

100 200 300 400 500 600

700 Allowable Stress

Piston

(6)

90K

t (sec) Twe1,Twe2(K)

0 1000 2000 3000

50 100 150 200 250 300

Exp.

Num.

T_we2=82K Twe1=156K (ζ₁=12,ζ₂=17)

Two-stage beta-type Stirling cooler

One-stage beta-type Stirling cooler

(7)

t (sec)

T(K)

0 200 400 600 800 1000

100 150 200 250 300

Split Stirling cooler

(Helium、5atm、2000rpm)

103K

T_we(K)

C.O.P.

50 100 150 200 250

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

Exp.

Num.

Ikuda et al. [23] (FPSC)

Park et al. [24] (FPSC/12.5atm,50Hz) present (STSC/3atm,1600rpm)

(8)

Heating power

Engine

Cooling power

Cooler

Stirling engine Stirling cooler

Mechanical power

(9)

(10)

Engine speed (rpm)

Shaftpower(W)

0 500 1000 1500 2000 2500 3000 3500

0 100 200 300 400 500 600 700 800

T_h=800°C T_h=700°C T_h=600°C Helium, 8 atm

Parameter Value

Bore size × Stroke 35.22 mm × 70 mm Displacer swept volume 135 c.c.

Swept volume ratio 1 Dead volume ratio 1.478 Compression ratio 1.364

Cooling temperature 300 K(liquid cooling)

550 W

Experimental data of shaft power

(11)

time (s) Twe(K)

0 50 100 150 200 250 300

50 100 150 200 250 300

Exp.

Num.

Helium, 5atm, 1000rpm

73 K

Parameter Value

Stroke of displacer 20.3 mm Displacer swept volume 5.2 c.c.

Diameter of cold cylinder 19 mm Dead volume ratio 0.35 Compression ratio 2.7

Cooling temperature 300 K(liquid cooling)

(12)

Stirling cooler Stiling engine PID controller Vacuum pump

Thermocouple Electric heater

Water circuit

Cold head

(13)

(14)

Schematic diagram of beta-type Stirling engine Schematic diagram of beta-type Stirling cooler

Ld,c

L1,c

L2,c

Rc

L3,c

L4,c

Lc

Yd,c

Yp,c

θc

Regenerator

Cold head Expansion

chamber

Compression chamber

Dp,c

Dd,c

L_2,e

R_e L_3,e

L4,e

Le

Yd,e

Y_p,e

θe

Piston

Displacer L_d,e Expansion

chamber

Compression chamber Regenerator

Heater

Cooler

L_1,e D_d,e

Parameters of engine

θ_e 0 degree

Charged pressure, P_e 5 atm Cooling temperature 300 K Heating temperature 1100 K

Parameters of cooler

θ_e 0 degree

Charged pressure, P_c 1 atm Cooling temperature 300 K Initial speed 200 rpm

(15)

( )

1 2 2 4,

2

, , 3,

1 2 2 1,

2

, , 2,

Position of displacer :

sin cos

2 2

Position of piston :

sin cos

2 2

Volume of expansion chamber :

e e

d e dt e e e e e e

e e

p e pt e e e e e e

ex

L L

Y t L R L R

L L

Y t L R L R

V

θ θ

   

= + − − − −  

   

 

   

= + + − − −  

   

 

2 ,

, , ,

2 ,

, , , ,

( )

4

Volume of compression chamber :

( )

4

d e

e t e d e

p e

co e d e d e p e

D L Y

V D Y L Y

π

= −

= − −

For Stirling engine For Stirling cooler

( )

1 2 2 4,

2

, , 3,

1 2 2 1,

2

, , 2,

Position of displacer :

sin cos

2 2

Position of piston :

sin cos

2 2

Volume of expansion chamber :

c c

d c dt c c c c c c

c c

p c pt c c c c c c

ex

L L

Y t L R L R

L L

Y t L R L R

V

θ θ

   

= + − − − −  

   

 

   

= + + − − −  

   

 

2 ,

, , ,

2 2

, ,

, , , , , ,

( )

4

Volume of compression chamber :

( ) ( )

4 4

d c

c t c d c

p c d c

co c cy c p c d c cy c d c

D L Y

D D

V L Y Y L L

π

π π

= −

= − + − −

(16)

( )

(

^, ^, ^, ^,

)

, , , , , , , , , ,

, , , , , , ,

Pressure variation :

Mass flow rate : , ,

e co e col e ex e hex e

e

co e col e l e l e r e r e h e h e ex e hex e

l e l e e const l e r e r e e const r e l e h

P dV T dV T

dP V T V T V T V T V T

dm V dP R T dm V dP R T dm V

γ γ

− +

= + + + +

= = =

( ) ( )

, ,

, , , , , , , ,

1

, , ,

,

Temperature of expansion chamber : (

e e const h e

ex e e ex e ex e e const hex e co e e co e co e e const col e

i i i e ex

ex e ex e ex e i

e

dP R T

dm P dV V dP R T dm P dV V dP R T

dP dV

T T T

P

γ γ

+

= + = +

= + +

^, ^,

, ,

1

, , ,

, ,

)

Temperature of compression chamber : ( )

e ex e

i i

ex e ex e

co e co e

i i i e

co e co e co e i i i

e co e co e

dm

V m

dV dm T T T dP

P V m

+

−

= + + −

(17)

, , ,

, ,

, , , , , ,

, , , , ,

, ,

,

Pressure drop across the regenerator :

39.52 , 0.01

2 Re

ex c c r c

tol c const c

co c c r c

ex c ex c r c r c co c co c

r c d c f c r c r c

r c r c

hy c r c

P P P

m R

P V T V T V T P P P

f L u u

P f

D ρ

= − ∆

= + + = + ∆

∆ = = +

( )( ) ( )

,

1 1

, , , ,

, ,

1

, 1 1 1 , ,

, , ,

, 0 Re 100

Temperature of expansion and compression chambers :

2

r c

i i i i

i i

ex c ex c ex c ex c

ex c ex c

i i i i i i i

ex c i i i p in j out ex c ex c sh

ex c ex c v ex c v

p p V V

m T dt

T c m T m T Q Q Q

m m c m c

+ +

+

+ + +

< ≤

+ −

= − +    −  +  +  + 

( )( ) ( )

( )

1 1

, , , ,

, ,

1

, 1 1 1 , ,

, , ,

1

2 Energy variation of cold head :

i i

pu p

i i i i

i i

co c co c co c co c

co c co c

i i i i i i

co c i i i p in co c out i co c

co c co c v co c v

i i i i

head head ex k

Q

p p V V

m T dt

T c m T m T Q

m m c m c

E E Q Q dt

+ +

+

+ + +

+

+  

+ −

 

= − +  − + 

= − −



  

 

1 1

Temperature of cold head :

i

i head

we

head head

T E

m c ζ

+

=

+

(18)

1

2 1

1

3 1 2

4 3

1

5 3 4

6 5

Angles :

cos ( cos )

sin ( cos )

e

z z

z z θ θ

θ π θ

θ θ

θ π θ

θ θ

θ π θ

−

=

= −

= +

= −

= +

= −

3

1 2

7

5

4

6 8

θ1

θ5

θ3

θ2

θ6

θ4

displacer

piston y

x

L1,e

L2,e

L3,e

L4,e

Re

Lpt,e

Ldt,e

Lp,e

Ld,e

Le

1 1, 2,

2 2,

3 4, 3,

4 3,

Geometrical variables :

( ) 2

e e e

e e

e e e

e e

z L L L

z R L

z L L L

z R L

= −

1 2

2 2

2 1 1 2 1 1 2 1 1 1 2 1

3 2 2 1.5

1 2 1 1 2 1

4 3

2

4 1 1

4 1 1 4 1 1

5 2

3 4 1

Angular acceleration : ,

cos( ) sin( ) ( sin( ) ) (z z cos( )) 1 (z cos( )) 1 (z z cos( ))

( sin( ) cos( ) sin( )

1 (z cos( ))

e e

z z z

z

z z

z

α α α α

θ ω θ α θ ω θ

α θ θ

α α

θ ω θ α θ ω

α θ

= = −

+ +

= −

− +  − + 

= −

= − + +

− +

2

3 4 1

2 1.5

3 4 1

6 5

) (z z cos( )) 1 (z z cos( ))

θ θ

α α

+

 − + 

 

= −

Kinematics :

(19)

3

1 2

7

5

4

6 8

θ1

θ5

θ3

θ2

θ6

θ4

displacer

piston y

x

L1,e

L2,e

L3,e

L4,e

Re

Lpt,e

Ldt,e

Lp,e

Ld,e

Le

, , 56 78

2 2

37 1 1 1 1 2, 3 3 3 3

2 2

35 1 1 1 1 3, 5 5 5 5

x-direction :

0 , 0 , 0 , 0

sin( ) cos( ) 1 sin( ) cos( )

2

sin( ) cos( ) 1 cos( ) sin( )

2

dis x pis x x x

x e e e

a a a a

a R R L

θ θ θ θ θ θ θ θ

= = = =

 

= − − −  + 

 

= − − − − + 

   

48 37 46 35

2 2

, 1 1 1 1 2, 3 3 2, 3 3

2 2

, 1 1 1 1 3, 5 5 3, 5 5

56 , 78

,

y-direction :

cos( ) sin( ) sin( ) cos( )

,

x x x x

pis y e e e e

dis y e e e e

y dis y y pi

a a a a

a R R L L

a a a a

= − = −

= − − +

= − + −

= =

   

,

2 2

37 1 1 1 1 2, 3 3 3 3

2 2

35 1 1 1 1 3, 5 5 5 5

48 37 46 35

cos( ) sin( ) 1 sin( ) cos( )

2

cos( ) sin( ) 1 sin( ) cos( )

2 ,

s y

y e e e

y y y y

a R R L

a a a a

 

= − + − + 

 

= − +  − 

= =

   

Mass center acceleration :

(20)

, , ,

Force balance in y-direction :

ex e co e dis dis dis dis y

P P Y m g m a

− + + − =

5 6 56 56

dis 5 6 56 56 56

4, 4,

5 6

Force balance in x-direction :

-X X

Y Y Y

Moment balance :

Y Y 0

2 2

x

y

e e

m a

m g m a

L L

+ =

− − − − =

− =

, , ,

co e b e pis pis pis pis y

P P Y m g m a

− + + − =

7 8

Y_pis

X7

Y₇

X8

Y₈ m₇₈g

5 6

Ydis

X5 X6

Y5 Y6

m56g

7 8 78 78

pis 7 8 78 78 78

1, 1,

7 8

-X X

Y Y Y

Moment balance :

Y Y 0

2 2

x

y

e e

m a

m g m a

L L

+ =

− − − − =

− =

P

_co,e

m

_pis

g P

_b,e

Y

_pis

Pex,e

Pco,e Ydis

mdisg

(21)

35 5 35 35

35 5 35 35 35

3, 5 5 3, 5 5

3,

5 35 35 35

X X

Y Y

Moment balance :

sin( ) X cos( ) Y

cos( ) 2

x

y

e e

e

m a

m g m a

L L

L m g I

π θ π θ

π θ α

+ =

+ − =

− + −

− − =

4 8

Y₈ X₈

Y₄₈

X₄₈ m₄₈g θ4

4

6

Y₄₆ X₄₆

Y₆ X₆

m46g θ6

3 7

Y₇ X7

Y37

X₃₇ m₃₇g

θ3

3

5

m₃₅g Y₃₅

X₃₅

Y₅ X₅ θ5

7 37 37 37

7 37 37 37 37

2, 3 7 2, 3 7

2,

3 37 37 37

X X

Y Y

Moment balance :

sin X cos Y

cos 2

x

y

e e

e

m a

m g m a

L L

L m g I

θ θ

θ α

+ =

+ − =

− +

− =

46 6 46 46

46 6 46 46 46

3, 6 6 3, 6 6

3,

6 46 46 46

X X

Y Y

Moment balance :

sin X cos Y

cos 2

x

y

e e

e

m a

m g m a

L L

L m g I

θ θ

θ α

− =

+ − =

− −

+ =

8 48 48 48

8 48 48 48 48

2, 4 8 2, 4 8

2,

4 48 48 48

X X

Y Y

Moment balance :

sin( ) X cos( ) Y

cos( ) 2

x

y

e e

e

m a

m g m a

L L

L m g I

π θ π θ

π θ α

− + =

+ − =

− − −

+ − =

(22)

1 35 37 1 35 37 12 1 1

1 46 48 1 46 48 12 2 2

Moment balance :

sin (X X ) cos (Y Y ) Y

2 sin( )(X X ) cos( )(Y Y ) Y

2 Torques exerted on the engine :

Frictional torque :

e

e e

e

e e e

e c fri

fri f e

R R L I

c

θ θ α

π θ π θ α τ

τ τ τ

τ ω

+ − + − =

− + + − + − = =

− −

=

( )

2

Angular speed is updated as :

_eⁱ 1 _eⁱ

dt

_e _c _fri

ω

⁺

= ω + I τ τ τ − −

3 1

Y35

X35

Y37

X37

θ1

Y12 2

4

Y46

X46

Y48

X48

θ2

Y12

(23)

Start

Initial conditions

t = t + dt θe = θe + dθe

θc = θc + dθc

Thermodynamic model

P,m,T Converge?

Dynamic model

Torques (τe , τc , τfri)

dθ =ωⁱ⁺¹dt

t = t_max ?

End

No

Yes

No

(24)

(25)

Time (s)

Position(mm)

400 400.1 400.2 400.3 400.4 400.5 400.6 100

150 200 250 300

Y_d,e Y_p,e

Time (s)

Position(mm)

400 400.2 400.4 400.6

40 60 80 100 120 140 160 180

Yd,c

Y_p,c

Time (s)

Volume(c.c.)

400 400.1 400.2 400.3 400.4 400.5 400.6

0 10 20 30 40 50 60

V_co,c V_ex,c

Time (s)

Volume(c.c.)

400 400.1 400.2 400.3 400.4

0 20 40 60 80 100 120 140 160 180

V_co,e V_ex,e

Position variation of the engine Position variation of the cooler

Volume variation of the engine Volume variation of the cooler

(26)

Instantaneous rotation speed and cold head temperature

Temperature of expansion and compression chamber in engine and cooler

Time (s)

Instantaneousrotationspeed(rpm) T we(K)

0 100 200 300 400

200 400 600 800 1000

230 240 250 260 270 280 290 300

Instantaneous rotation speed T_we

Time (s)

Instaneousrotationspeed(rpm)

300 300.1 300.2 300.3 300.4 510

520 530 540 550 560

Time (s) Tex,e,Tco,e,Tco,c,Tex,c(K)

400 400.1 400.2 400.3 400.4 400.5 400.6

200 250 300 350 400 450 500 550 600 650 700 750 800 850

T_ex,e T_co,e T_co,c T_ex,c

(27)

Volume (c.c.)

Pressure(kPa)

0 20 40 60 80 100 120 140

600 700 800 900 1000 1100 1200 1300 1400

P_c,e- V_c,e P_e,e- V_e,e

Volume (c.c.)

Pressure(kPa)

0 10 20 30 40 50

50 60 70 80 90 100 110 120 130 140 150 160 170

P_c,c- V_c,c P_e,c- V_e,c

P-V diagram for engine P-V diagram for cooler

(28)

Time (s)

Averagerotationspeed(rpm)

0 10 20 30 40 50 60

100 200 300 400 500 600 700 800 900

c_f= 0.001 c_f= 0.0015 c_f= 0.002

Time (s) Twe(K)

0 100 200 300 400 500

220 240 260 280 300

c_f= 0.001 c_f= 0.0015 c_f= 0.002

Average rotation speed Cold head temperature

(29)

Time (s) Twe(K)

0 100 200 300 400 500

200 220 240 260 280 300 320

P_engine= 5 atm P_engine= 6 atm P_engine= 7 atm P_engine= 8 atm

Time (s) Twe(K)

0 100 200 300 400 500

100 120 140 160 180 200 220 240 260 280 300 320

P_cooler= 1 atm P_cooler= 2 atm P_cooler= 3 atm

Effect of charged pressure of Stirling engine Effect of charged pressure of Stirling cooler

(30)

Time (s) Twe(K)

0 100 200 300 400 500 600 700 800 900

50 100 150 200 250 300

P_cooler= 3atm (Num.) P_cooler= 4atm (Num.) P_cooler= 5atm (Num.) P_cooler= 3atm (Exp.) P_cooler= 4atm (Exp.) P_cooler= 5atm (Exp.)

Numerical and experimental results of temperature with different cooler pressure when the engine is charged with 8 atm and the heating temperature is 1100 K.

(31)

 A beta-type Stirling engine and a beta-type Stirling cooler are connected and tested.

 Thermodynamic and dynamic analysis are built to predict the temperature performance of thermal-driven cooler.

 According to parametric analysis, increasing the charged pressure of Stirling cooler will better increase the cooling power and result in a lower cold head temperature.