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THERMODYNAMIC ANALYSIS OF PHILIPS REFRIGERATOR
THERMODYNAMIC ANALYSIS OF PHILIPS REFRIGERATOR
Jin Xin and Chen Jianqing Department of Physics, Nanjing University, Nanjing, China
The coeficients of performance and the amounts of refri geration of the Philips refrigerator of different working gas have been discussed using an accurate three parameters equation of state. The results show that the coefficients of performance are £ N e > g ^ > £ f< for neon, hydrogen and helium as working gas respectively , but the amounts of refrigeration are Q--> Q N > Q_ · These give us a theore tical basis for choosing the best working gas of Philips refrigerator.
INTRODUCTION The Philips refrigerator which was developed by Philips Company of Eindhoven , Netherlands, based on principle of reverse Stirling cycle (Ref.1,2) has been got wide applications due to the advantages of small volume, light, efficient and convenience for operation. The pV diagram of idealized refrigeration cycle of Philips refrigerator is shown in Fig.1. Starting at a, the working gas is compressed isothermally at temperature T. to b with the evolution of heat, and the working gas is cooled in the regenerator at constant volume , V\, to point c where it is considered to have zero dead volume for gas in our theoretical analysis. Isothermal expansion then take place at low temperature, T-., until the working gas return to its original volume V at d. In this procces the working gas absorbs heat which is called the amound of refrigeration of this cycle of the Philips refrigerator. The working gas is then transferred through the same regenerator and at constant volume V to its original state at a. If the working gas is considered as ideal gas, the coefficient of performance is (Ref.^f)
£=
T
l
± V Tl
CD
The amount of refrigeration for one mole is V Q = RT ln-rpχ b
(2)
-1 —1 where R=8.31^1 J.mol .K is ideal gas constant. In Fig. 1, if the working gas is considered as a gas which is satisfied Van der Waals equation of state. For one mole, the amount of refrigeration is
Q=£
MS
544
rV
= Tn \
a -&P ( ) v dV D
V- b = R ^ l n - -*
V
(3)
b
where S, T, p are entropy(J/K), temperature(K), pressure(Pa),and b i s parameter of Van der Waals. For getting expression ( 3 ) , we have used f i r s t Tds equation TdS=CvdT+ T(
)ydV
W
Where C is specific heat at constant volume· We can get the amount of work done on the working gas for one cycle and for one mole· V -b W =- φ pdV = R(T.- T.)ln—(5) D
From (3) and (5)» the coefficient of performance of Philips refrigerator for the state of equation of working gas as Van der Waals equation may be written
£=
Q
W
=
T
i_ V Tj_
(6)
expression (6) is still the same with (1). This means that we can not get non-ideal gas effect of the reverse Stirling cycle of the working gas, even though the Van der Waals equation is used. It may be come from that T and p have the same linear relationship with ideal gas. For dealing with in detail the non-ideal gas effect of Philips refrigerator, we may need to find a new accurate equation of state about working gas· A NEW THREE PARAMETERS EQUATION OF STATE OF GAS For getting an accurate equation of state of gas in low temperature working range, the R-K equation (Ref.5) has been revised as following
V-b
TnV(V+b)
where V is molar volume, but a and b are little different with Van der Waals equation. From critical condition of equation (7)» we can get a = 0.k2?W
p
(8)
c
RT b = 0.0866^ 2( 9) p c where T and p are critical parameters, n = -0.06^ for helium, n = 0.28 for hydrogen and n = O.A-8 for neon. According the calculation of pressure results of equation (7) to compare
545
w i t h d a t a ( R e f . 6 , 7 , 8 ) , the r e l a t i v e e r r o r i s from 0.02% t o 2.5% f o r h e l i u m , 0.02% t o 3 . 5 # f o r hydrogen and 0.03% t o 1.5% for neon when temperature i s lower then 300K and p r e s s u r e i s lower then 10MPa. NON-IDEAL GAS EFFECT OF PHILIPS REFRIGERATOR Using e q u a t i o n ( 7 ) and ( Ό , we can g e t the amound of r e f r i g e r a t i o n of the P h i l i p s r e f r i g e r a t o r f o r one mole and f o r one c y c l e p r o c e s s .
Q =
RT
V -b in—2—
Vb
+
Vu+b -2£-ln(-J2
Vb
K
V S-)
(10)
V
b
The work done on working g a s of one c y c l e for one mole W= H(V h
V -b 3 · ) In—ä b
*
V
+
„ V +b Λ Ä(_L__-2-.)la(—S b τ^ τ£ v a+ b
V *-)
(11)
vb
From (10) and (11), can get the coeficient of performance of Philips refri gerator. V -b HT in -* 1 V.-b b R(T - T ) l n — S h
λ
v.-b D
V,+b + -üä_in(_fe bT? V +b 1 a^ +
| (_L
Έ
V
*-) Vu b
(12)
!_)m(_S
τ?
τ"
1
h
S)
v +b v a
b
Because b/V, or b/V < < 0 , (11) and (12) may be expanded in a series. We can clearly show the correct terms comparing with ideal gas as following.
Q = HT
l n
V V Λ _ S _ + (RT.b + £ * - ) ( — * - - 1 ) - ! — 11 v T v v v b 1 b a
(13)
V V W = R(T.- T n ) l n — — + [Bb(T,- T. ) + a ( — — ) 1 (— - Ό — (1*0 h 1 U h 1 j v Tn Tn b 1 h b a It is very clear that the first terms on the right of (13), (1*0 are the ideal gas expression of Q and W, but the second terms of two above equations come from non-ideal gas effect. DISCUSSION Based on expressions (10) to (12), and a,b parameters which can be calculated from (8) and (9), we can discuss the behaviours of different real gas as working gas of Philips refrigerator, and choose which one is the best working gas. For convenience, we assume T =300K, T,=70K, V =0.001m5/mol, and*y=V /V, as variable parameter , Fig.2 shows the relationship of the difference of real gas amount of refrigeration, Q , minusing ideal gas of it, Q., with Y · Fig.2 shows clearly that the amounts of refrigeration are Q > Q 1 > Q for hydrogen, neon and helium. But the coefficients of performance are £ N > £ H >
546
showing in Fig.3· So that, if εgeration of Philips refrigerator Ηβ
one wants to increas the amount of refri the best working gas may be hydrogen, but if one wants to increas the coefficient of performance the best one may be neon« REFERENCES 1 Bailey, C. A. Advanced Cryogenics, Plenum Press, (1971)» PP· 199-212. 2 Jin, X., OuYang, R. B·, Chen, W. M., •Stirling refrigeration cycle of para magnetic salt1. Cryogenics and Superconductivity (China), Vol.15 (1987)» PP· 15-18. 3 Chen, G. B., 'Progress in cryogenerator - Introduction to the 4th Inter national Cryogenarator·. Cryogenics (China), No. 2 (1987)» ρρ· 55-59· k Barron, R., Cryogenic Systems, McGraw-Hill Book Company,(1966), pp. 218-321. 5 Redlich, 0. and Kwong, J. N. S.,· On the thermodynamics of solutions'. Chem. Rev.. Vol.kk (19^9)· pp. 233-238. 6 McCarty, R. D., 'Thermophysical properties of helium-^ from ^ to 3000R with pressure to 15000 PSIA·. NBS Technical Note 622. 7 McCarty, R. D., 'Hydrogen technological survey- thermophysical properties'. NASA, Report SP-3089. 8 Vargaftik, H. B., 'The Tabales On The Thermophsical Properties Of Liquids And Gases, John Wiley & Sons. Inc., Second Edition(1975).
Fig.1 Thermodynamic cycle for the Philips refrigerator
547
100
8ο
5
60
•H I
kO L
20 U
0
2
k
r
6
8
10
af b F i g . 2 The r e l e t i o n s h i p s of (Q - Q. ) with_Y^ V a /V H for h y d r o g e n , neon and h e l i u m , where Τ χ = 70Κ / v ^ O . O O I n T
0.316
0.312 l· H3
0.308
He
0.30^
0.300
I
I
i
I
I
I
I
L 10
r F i g . 3 Coefficients of r e f r i g e r a t i o n as f u n c t i o n ofJ^V /V, f o r neon, hydrogen and h e l i u m , where T = 300K, T =70K, ®Υ = 0.001m
548
Jin Xin and Chen Jianqing Department of Physics, Nanjing University, Nanjing, China
The coeficients of performance and the amounts of refri geration of the Philips refrigerator of different working gas have been discussed using an accurate three parameters equation of state. The results show that the coefficients of performance are £ N e > g ^ > £ f< for neon, hydrogen and helium as working gas respectively , but the amounts of refrigeration are Q--> Q N > Q_ · These give us a theore tical basis for choosing the best working gas of Philips refrigerator.
INTRODUCTION The Philips refrigerator which was developed by Philips Company of Eindhoven , Netherlands, based on principle of reverse Stirling cycle (Ref.1,2) has been got wide applications due to the advantages of small volume, light, efficient and convenience for operation. The pV diagram of idealized refrigeration cycle of Philips refrigerator is shown in Fig.1. Starting at a, the working gas is compressed isothermally at temperature T. to b with the evolution of heat, and the working gas is cooled in the regenerator at constant volume , V\, to point c where it is considered to have zero dead volume for gas in our theoretical analysis. Isothermal expansion then take place at low temperature, T-., until the working gas return to its original volume V at d. In this procces the working gas absorbs heat which is called the amound of refrigeration of this cycle of the Philips refrigerator. The working gas is then transferred through the same regenerator and at constant volume V to its original state at a. If the working gas is considered as ideal gas, the coefficient of performance is (Ref.^f)
£=
T
l
± V Tl
CD
The amount of refrigeration for one mole is V Q = RT ln-rpχ b
(2)
-1 —1 where R=8.31^1 J.mol .K is ideal gas constant. In Fig. 1, if the working gas is considered as a gas which is satisfied Van der Waals equation of state. For one mole, the amount of refrigeration is
Q=£
MS
544
rV
= Tn \
a -&P ( ) v dV D
V- b = R ^ l n - -*
V
(3)
b
where S, T, p are entropy(J/K), temperature(K), pressure(Pa),and b i s parameter of Van der Waals. For getting expression ( 3 ) , we have used f i r s t Tds equation TdS=CvdT+ T(
)ydV
W
Where C is specific heat at constant volume· We can get the amount of work done on the working gas for one cycle and for one mole· V -b W =- φ pdV = R(T.- T.)ln—(5) D
From (3) and (5)» the coefficient of performance of Philips refrigerator for the state of equation of working gas as Van der Waals equation may be written
£=
Q
W
=
T
i_ V Tj_
(6)
expression (6) is still the same with (1). This means that we can not get non-ideal gas effect of the reverse Stirling cycle of the working gas, even though the Van der Waals equation is used. It may be come from that T and p have the same linear relationship with ideal gas. For dealing with in detail the non-ideal gas effect of Philips refrigerator, we may need to find a new accurate equation of state about working gas· A NEW THREE PARAMETERS EQUATION OF STATE OF GAS For getting an accurate equation of state of gas in low temperature working range, the R-K equation (Ref.5) has been revised as following
V-b
TnV(V+b)
where V is molar volume, but a and b are little different with Van der Waals equation. From critical condition of equation (7)» we can get a = 0.k2?W
p
(8)
c
RT b = 0.0866^ 2( 9) p c where T and p are critical parameters, n = -0.06^ for helium, n = 0.28 for hydrogen and n = O.A-8 for neon. According the calculation of pressure results of equation (7) to compare
545
w i t h d a t a ( R e f . 6 , 7 , 8 ) , the r e l a t i v e e r r o r i s from 0.02% t o 2.5% f o r h e l i u m , 0.02% t o 3 . 5 # f o r hydrogen and 0.03% t o 1.5% for neon when temperature i s lower then 300K and p r e s s u r e i s lower then 10MPa. NON-IDEAL GAS EFFECT OF PHILIPS REFRIGERATOR Using e q u a t i o n ( 7 ) and ( Ό , we can g e t the amound of r e f r i g e r a t i o n of the P h i l i p s r e f r i g e r a t o r f o r one mole and f o r one c y c l e p r o c e s s .
Q =
RT
V -b in—2—
Vb
+
Vu+b -2£-ln(-J2
Vb
K
V S-)
(10)
V
b
The work done on working g a s of one c y c l e for one mole W= H(V h
V -b 3 · ) In—ä b
*
V
+
„ V +b Λ Ä(_L__-2-.)la(—S b τ^ τ£ v a+ b
V *-)
(11)
vb
From (10) and (11), can get the coeficient of performance of Philips refri gerator. V -b HT in -* 1 V.-b b R(T - T ) l n — S h
λ
v.-b D
V,+b + -üä_in(_fe bT? V +b 1 a^ +
| (_L
Έ
V
*-) Vu b
(12)
!_)m(_S
τ?
τ"
1
h
S)
v +b v a
b
Because b/V, or b/V < < 0 , (11) and (12) may be expanded in a series. We can clearly show the correct terms comparing with ideal gas as following.
Q = HT
l n
V V Λ _ S _ + (RT.b + £ * - ) ( — * - - 1 ) - ! — 11 v T v v v b 1 b a
(13)
V V W = R(T.- T n ) l n — — + [Bb(T,- T. ) + a ( — — ) 1 (— - Ό — (1*0 h 1 U h 1 j v Tn Tn b 1 h b a It is very clear that the first terms on the right of (13), (1*0 are the ideal gas expression of Q and W, but the second terms of two above equations come from non-ideal gas effect. DISCUSSION Based on expressions (10) to (12), and a,b parameters which can be calculated from (8) and (9), we can discuss the behaviours of different real gas as working gas of Philips refrigerator, and choose which one is the best working gas. For convenience, we assume T =300K, T,=70K, V =0.001m5/mol, and*y=V /V, as variable parameter , Fig.2 shows the relationship of the difference of real gas amount of refrigeration, Q , minusing ideal gas of it, Q., with Y · Fig.2 shows clearly that the amounts of refrigeration are Q > Q 1 > Q for hydrogen, neon and helium. But the coefficients of performance are £ N > £ H >
546
showing in Fig.3· So that, if εgeration of Philips refrigerator Ηβ
one wants to increas the amount of refri the best working gas may be hydrogen, but if one wants to increas the coefficient of performance the best one may be neon« REFERENCES 1 Bailey, C. A. Advanced Cryogenics, Plenum Press, (1971)» PP· 199-212. 2 Jin, X., OuYang, R. B·, Chen, W. M., •Stirling refrigeration cycle of para magnetic salt1. Cryogenics and Superconductivity (China), Vol.15 (1987)» PP· 15-18. 3 Chen, G. B., 'Progress in cryogenerator - Introduction to the 4th Inter national Cryogenarator·. Cryogenics (China), No. 2 (1987)» ρρ· 55-59· k Barron, R., Cryogenic Systems, McGraw-Hill Book Company,(1966), pp. 218-321. 5 Redlich, 0. and Kwong, J. N. S.,· On the thermodynamics of solutions'. Chem. Rev.. Vol.kk (19^9)· pp. 233-238. 6 McCarty, R. D., 'Thermophysical properties of helium-^ from ^ to 3000R with pressure to 15000 PSIA·. NBS Technical Note 622. 7 McCarty, R. D., 'Hydrogen technological survey- thermophysical properties'. NASA, Report SP-3089. 8 Vargaftik, H. B., 'The Tabales On The Thermophsical Properties Of Liquids And Gases, John Wiley & Sons. Inc., Second Edition(1975).
Fig.1 Thermodynamic cycle for the Philips refrigerator
547
100
8ο
5
60
•H I
kO L
20 U
0
2
k
r
6
8
10
af b F i g . 2 The r e l e t i o n s h i p s of (Q - Q. ) with_Y^ V a /V H for h y d r o g e n , neon and h e l i u m , where Τ χ = 70Κ / v ^ O . O O I n T
0.316
0.312 l· H3
0.308
He
0.30^
0.300
I
I
i
I
I
I
I
L 10
r F i g . 3 Coefficients of r e f r i g e r a t i o n as f u n c t i o n ofJ^V /V, f o r neon, hydrogen and h e l i u m , where T = 300K, T =70K, ®Υ = 0.001m
548