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A computer simulation of a rotary compressor for household refrigerators
Applied Thermal EngineeringVol. 17, No. I, pp. 65-18. 1997 Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved PII: s1359-4311(%)00013-0 1359-431l/97 $15.00 + 0.00
Pergamon
A COMPUTER SIMULATION OF A ROTARY COMPRESSOR FOR HOUSEHOLD REFRIGERATORS K. T. Ooi and T. N. Wong School
of Mechanical
and Production Avenue,
Engineering, Nanyang Technological Singapore 639798, Singapore
University,
Nanyang
(Received 8 February 1996) Abstract-This paper presents analytical studies of a fractional horse-power rotary refrigeration compressor and its performance comparison with measured results. The study employed a general-purpose performance model by considering the thermodynamic cycle of the compressor and its mechanical losses. The real gas equation of state is used to describe the changes of state of the refrigerant. The study reveals that for this quarter horse-power compressor unit the energy consumption in performing the compression cycle was 8&90% of the compressor shaft energy input, and the other l&15% of the energy was dissipated as mechanical friction. The results also show that the performance predictions are satisfactory when compared with measured results. The model has been used in assisting the design of new compressors for use with new environmentally friendly refrigerants. This study also paves the way for more comprehensive simulation studies and for possible overall computerised optimisation design study in the future. Copyright 0 1996 Elsevier Science Ltd. Keywords-Refrigeration,
compressor,
rotary,
simulation,
modelling.
NOMENCLATURE A A, a B
d
E e F.
area, m2 valve flow area, m* radius ratio, R,/R, effective valve force area, m* coefficient of discharge force area coefficient valve body width, m Young’s modulus, N/m’ eccentricity, m net pressure force across the vane in the x direction, N net pressure force across the vane in the y direction, N vane spring force, N inertia force of vane, N tangential force at vane tip, N normal force at vane tip, N tangential forces at vane-side contact points, N normal forces at vane side contact points, N blade height, m, or specific enthalpy, J/kg upstream enthalpy, kJ/kg downstream enthalpy assuming isentropic expansion, kJ/kg area moment of inertia, m4 moment of inertia of roller, kg m’ vane spring constant N/m cylinder length, effective valve length, m friction loss at vane tip, W friction loss at vane side, W friction loss between eccentric and cylinder head face, W friction loss between roller and cylinder head face, W friction loss between the roller and eccentric, W mass of fluid, kg mass flow rate, kg/s friction moment from eccentric to cylinder head face, Nm friction moment from eccentric to roller, Nm friction moment from roller to cylinder head face, Nm mass of vane, kg pressure, N/m’ pressure in suction and compression chambers, N/m’ 65
66
K. T. Ooi and T. N. Wong
P Id P over PR P,, P*
indicated power, W over compression power loss, W pressure ratio, P/PC suction and discharge pressures, N/m’ heat transfer, J mode participation factor gas constant, J/(kgK) cylinder radius, m roller outer radius, m radius of eccentric, m shaft radius, m vane tip radius, m valve tip radius, m temperature, K temperature ratio, T/T, valve thickness, m blade thickness, m specific internal energy, J/kg volume, m3 velocity, m/s sliding velocity at vane tip, m/s specific volume, ml/kg vane extension or distance from valve free end, m work, J deflection of valve, m critical compressibility, P,/(Rp,T,)
Q q(x)
R RC R, R, R, R, r T TR t lb ” yc3 V V Vt 0s X w
Y(X) Z Greek
letters
angular position of rotor, radian radial clearance between roller and cylinder at 0 = 0, m clearance between roller and cylinder head face, m radial clearance between roller and eccentric, m vane protrusion, m clearance between eccentric and cylinder head face, m viscosity of lubricating oil, Ns/m2 angular velocity of eccentric or natural frequency, rad/s angular velocity of rolling piston, rad/s valve shape function plate density, kg/m’ damping coefficient coefficient of friction at vane side coefficient of friction at vane tip density ratio, p/pC time differential
INTRODUCTION
Over the last two decades, computer simulation studies have benefitted compressor research and development. In the late sixties and early seventies, the introduction of high-speed digital computer technology made a major difference to the analysis of refrigeration compressors. Today, simulation studies which account for real gas property simulation, interaction of valve dynamics with gas pulsation in the plenum chambers, heat transfer, flow simulation using two-dimensional representation and valve stress analysis using finite element techniques are also incorporated. In general, compressor modelling studies can be broadly classified into two categories [l], which are general performance modelling and special-purpose modelling. The general-purpose model includes thermodynamic simulation of working chamber processes, flow-through valves and valve dynamics. More comprehensive models may include pulsation effects, friction, leakage and heat transfer. Special-purpose models, in contrast, are formulated to look at a specific area of interest in a more detailed manner, such as valve stress analysis and its relation with fluid flow through valve ports. Literature shows that much work has been done on compressors. On the research work on rolling piston compressors, previous studies [2-51 employing mathematical simulation were aimed at studying the behaviour of the machine and no attempt was made to compare predictions with the experimental results. Other studies [68] concentrated on extensive experimental work. This paper presents a comparison of the performance prediction of a general-purpose model of a rolling piston refrigeration compressor with measured results. The compressor is rated as quarter
67
Computer simulation of a rotary compressor Electric motor assembly
Pump assembly --
WorkingcAi
7
,A,z
stator
slhaft
Discharge tube
Suction tubk Upper be Motor rotor
Roll& piston Lower bearing
Fig. 1. Cut-away view of a compressor used in a household refrigerator.
The model incorporated basic working horse-power and is used in household refrigerators. principles of the compressor, thermodynamic simulation of the working chamber, valve flow and valve dynamics, as well as the mechanical losses of the moving components. MATHEMATICAL
MODELLING
Introduction Rolling piston compressors are commonly found in air compression and refrigeration industries. They are geometrically simple machines with one roller, a cylinder, a spring and a vane, as shown in Fig. 1. Figure 2 shows a schematic configuration of the compressor. During the operation, the piston rolls along the inner wall of the cylinder. Within the roller and the cylinder wall there exists two chambers which are separated by the vane and the roller-cylinder contact. While one chamber undergoes compression/delivery process the other undergoes suction. The process is cyclic and it takes two revolutions to complete a cycle. The existence of the compression spring is to ensure that the vane tip-roller contact is maintained throughout the operational process. The overall 820
Fig. 2. Rolling piston compressor schematic.
K. T. Ooi and T. N. Wong
68
performance of a given machine depends on the balance of compression, frictional effects, leakage effects, suction and discharge effects. The compressor is of hermetic type and lubricating oil is used to seal and to lubricate. Today, with the availability of high-speed computers, a complete modelling of the machine that caters for geometrical, fluid flow, heat transfer, dynamic and stress analysis may be incorporated with comparative ease. This section briefs the simulation of the machine. The model accounts for geometric, thermodynamic, dynamics and frictional effects in the machine, and a real gas equation was used to describe the state of the refrigerant. The model allows the variation of refrigerating fluid pressure, temperature and mass flow to be evaluated in relation to the geometrical configuration and the operating conditions of the machine. The performance of the machine in terms of power input as well as cooling capacity may be evaluated. Geometry
model
The geometry model of the compressor accounts for all important physical parameters which are within the working chamber of the compressor that may affect the performance of the compressor. Referring back to Fig. 2, the chamber volume trapped within the roller, cylinder and the vane may be given by
V(e)
=
$$
(
(1 _
a’)()
_
Q-j-@ sin20
_
a2
sin- ’((i
- 1)sinO)
- a(I -L)sinDJw).
To account the volume
for the vane thickness, the volume given by equation (1):
occupied
v’(e) = v(e) -
by the vane must be subtracted
(1)
away from
F ,
where 6, = R,(l - (1 - a) case - J(1
- a)2 cos2e + 2a - 1) .
(3)
Valve jlow model In the present simulation, the suction port is considered as a simple orifice without a valve. The effective flow area may be accounted for by introducing a discharge coefficient on the calculated geometrical area. The flow through the suction port is assumed to be one-dimensional adiabatic flow. The isentropic efficiency is accounted for by lumping it into the discharge coefficient. Reversed flows are taken into consideration where the lumped discharge coefficient is assumed constant throughout the suction process. The same value of the discharge coefficient is also used for reversed flows. The discharge port is covered with a reed valve. The valve flow area may be the valve port area or the valve lift annulus area. The latter must be determined from the instantaneous valve deflection. The effective flow area is taken as the area which imposes more restrictions to the flow. Assumptions as in suction flow are applicable. By first assuming that the flow through the valve is isentropic, the flow velocity may be calculated:
V=J_.
(4)
Computer simulation of a rotary compressor
69
Fig. 3. Geometry of the value reed.
In the above equation, subscripts 1 and 2 denote the conditions of the fluid upstream and downstream of the valve port, respectively. The mass flow rate can be found by dm -=-----_ C,AP df9
(5)
vs2
where h: is the downstream enthalpy assuming an isentropic expansion. To obtain this enthalpy, the knowledge of the downstream specific volume and the temperature must be known. This information may be obtained by solving the equation of state, as shown in the Appendix using the Newton-Raphson iterative method; C, is the discharge coefficient which accounts for the effective flow area and the isentropic efficiency, for actual non-isentropic flow. The choke flow condition is taken into consideration by taking the maximum flow velocity to be equal to the sonic velocity at the throat of the port. Valve dynamics
In the discharge side of the machine, there exists a discharge reed valve. Figure 3 shows the geometry of the valve plate. Its geometry may be simplified by assuming the valve free end as a circular disc that is connected to a rectangular valve plate. The valve displacement is constrained by the valve backing plate which is placed a small distance above the valve plate. The function of this backing plate is to prevent the valve plate from over stressing due to unnecessarily large deflection. The dynamic behaviour of the valve is characterised by the vibration of the valve plate. Assuming the valve as a beam with varying width, neglecting shear and rotary motion, the beam equation describing the beam vibration [4] is
g
(EZ(x@+
&A(X)
2 =P(x,t)
>
and assuming the valve deflection is given by
Y = i
4”(X)4”W>
II=,
(7)
where d”(x) is the valve shape function which may be determined from the free vibration analysis. There exists an infinite number of combinations of mode shapes since n ranges from 1 to infinity, but in many cases, especially for the valve flow area calculation, the first mode consideration alone may be sufficient [2]; qn(t) is the generalised coordinate or mode participation factor which may be obtained by integrating the valve vibration governing equation using an initial value integration technique.
IO
K. T. Ooi and T. N. Wong
The mode participation
factor is obtained from
4. + 210&k + O;q” =
J
wWx)t, dx
’
(8)
where in is the damping coefficient of the valve reed due to the air cushioning. For a single valve port configuration and to avoid considerations of the flow in the valve gap, the gas force acting on the valve may be taken to be equal to the product of the effective valve force area and the pressure drop across the valve [2,4],
s
4dx)P dx = ~,(xP(x)Wt) ,
(9)
where B(x) is the effective force area and AZ’(t) is the pressure difference across the valve. Generally, the term B(x) has to be obtained from experiments. It is a function of valve lift. Substituting equation (9) into equation (8) the following expression may be obtained:
In the above equation, the natural frequency of the valve may be obtained approximately Rayleigh’s quotient, i.e. -x Y’
w;=
from
(11)
where
(12) and Y = pp
x’A(~)c$(~)2 dx + ppA ‘4(x)’ dx ,
s0
(13)
s XI
the values of ,4(x) and Z(x) are given below: o
- x2
A = dt Z(x) =
t’ 6
z=
J
2rx
-
x2
dt’ 12.
In the above solution, the instantaneous value of valve lift is monitored to determine if the valve hits the backing plate and the correct boundary conditions are imposed. Two cases are considered: one is when the valve leaves the seat, the other is when the valve hits the valve backing plate [4]. In the present model, only the first mode of valve deflection is considered, the model, however, takes into consideration different boundary conditions, as well as valve mode shapes, when the valve departs from the valve seat, and when it hits and departs from the valve backing plate. Figure 4 shows the two different valve mode shapes.
Computer
simulation
71
of a rotary compressor
0.7 0.6 z
-
Shape 2
.......
Shape
.5 0.5E
. ..’ . . . .
1
.... _:’ _’
:
:
:
:
:
:
:
:
:
,:’ .“.
After hits valve stop
Before
hits valve stop
Distance
Fig. 4. Different
I 0.020
0.015
0.010
0.005
In the present seat is obtained
. . . . . ..’
from clamped end (mm)
valve shape
functions.
analysis, the shape function for the case when the valve is departed from a standard polynomial function for a cantilever beam [4]:
from the valve
(14)
rn(x)=($-4(f)+3.
In equation
(14) the natural
and geometric &x=0,-=
boundary
W(O)
o,
conditions
d’440) -=
o
dx3
dx*
at x = 1, 4(1) = 0, $$ When the valve hits the backing
for both ends are satisfied:
plate, the following
standard
’
= 0. polynomial
function
is applicable
[4]:
(15)
Again in equation (15) the natural and geometry boundary conditions are satisfied: at x = 0, 4(O) = 0, d*@(O)/dx* = 0 and at x = I, $(I) = 0, d4/dx = 0. The mode shape has been verified by comparing the deflected shape of the valve reed with a more comprehensive finite element model employing the actual geometry of the valve reed. Figure 5 shows that good agreement was obtained. 0.7 0.6
-
Simplified
model model
. . . . . . . Finite element
z e
0.5
Distance
Fig. 5. Comparison
from clamped end (mm)
of valve displacements.
K. T. Ooi and T. N. Wong
72
THERMODYNAMIC
MODEL
In the thermodynamic model [5] the relationships within pressure, temperature and the mass will be formulated. These three unknowns were related using the energy conservation relation, mass conservation relation and the equation of state of a real gas. A brief account of the model is given below. Pressure, temperature and mass relationships
To obtain the relationship of the state of the working medium in the compressor, applying the First Law of Thermodynamics to the control volume by taking the boundary of the chamber as the boundary of the control volume and neglecting the kinetic and potential energy:
Q de
+Cshi=
$
+C%ho+
-&(muJ,
(16)
where subscripts i and o denote inlet and outlet. Taking the work term as the mechanical work: dw= PdV,,
(17)
h=u+PV,,
(18)
from the relationship of the enthalpy:
the pressure is a function of the temperature
and the specific volume:
p = P(T,V,) 9
(19)
V, = Vc/mc .
(20)
where the specific volume is given by
After manipulation,
the following expressions may be obtained: 1
!!!L=
v,
+b%~68)~
de
- $
C2
(h, - h,) - C 2
_ r @/aT) vs (aPclaT,i
(21)
(ho - hi)
and
The mass in the working chamber can be obtained from the continuity equation, i.e. dm, dmi -- dm, de - dB dmO ’
(23)
where the two main sources of the mi and the m, are from suction and discharge flows through the respective ports, The partial derivatives for refrigerant property in the above equations can be obtained by differentiating equations (A. l)-(A.6), detailed simulation of the refrigerant properties is depicted in the Appendix.
Computer
simulation
of a rotary
compressor
13
e=o
Compression chamber
Fig. 6. Roller friction
FRICTION
moments.
MODEL
To determine the friction losses of the compressor, the vane and the roller dynamics must be considered. The information about the vane sliding velocity and roller velocity must be obtained. There are six areas where friction losses may take place [3, 5, 8-101, as follows (Figs 6 and 7): Eccentric
and the inner
surface
of the roller, L
Roller
face and the cylinder
= (w - WPf,,
.
head face, LC = M,CW,
Eccentric
(25)
face and the cylinder
(26)
head face, L,, = Me,0 .
(27)
L,. = V,F,, .
(28)
Vane tip and roller,
FS
Fig. 7. Vane force balance.
K. T. Ooi and T. N. Wong
14
Table I. Operating pressures for various freons at Tcvap= - 23.3”C and r,, Suction pressure (bar) Gas temperature outside the suction valve (“C) Discharge pressure (bar) Gas temperature after the discharge valve (“C) Pressure ratio Rotational speed (rev/min)
= 54.4”C
RI2
R22
Rl34a
1.324 64.3 13.46 120.8 10.17 3487
2. I51 68.0 21.47 134.7 9.95 3395
I.151 61.3 14.68 128.0 12.75 3350
where V, = R,o, + eocose/cosa
.
(29)
5. Vane sides and vane slot, (30) 6. Outer roller surface and the inner cylinder surface. Item (4) can be ignored if it is assumed that there are no contact forces between an outer roller surface and the cylinder. In the above equations other terms are defined as follows:
F,=(P,b-P,(4
F,, =
A4, = 27~w,r/(R,4 - R:)/d,
(31)
M, = ~rrp(2R,4 - R:)/6,
(32)
M,, = 27ty1(0- o,)lR@,
(33)
F, = (PC - P,)xl
(34)
F, = k(xc,,x, - x)
(35)
F, = - M,i
(36)
+R,sina)-P,(?
-R.sina))l
(37)
p,F,(h + 6,~) + (Fy + F, + Fd(x - h) (cos a + pv sin a)(x - h - 2pSR, sin a) + @(sin a - pycos a)[fbps + h + x - 2r,.(l -
cos a)] (38)
Fv, = ,G.F,,,
(39)
F,, = p,F,,
(40)
F,, = IAF,, ,
(41)
where the last unknown o, may be obtained from equation (41) if steady roller rotation is assumed: (21rplR,3/62 - R,F,,,)&d, Or = 2q(IR,d,
If angular acceleration is taken into account, equation (43) numerically as follows [8-lo]:
+ (R,d - R&5,)
.
the value of o, can be obtained
I$& = Me, - R,F,,, - Mx .
(42)
by solving
(43)
Computer
simulation
of a rotary
compressor
16.0-
c
12.8 -
d e & 9.6: “a k 6.4% 5
3.2 -
L 0
I
120
I
I
I
I
240
360
480
600
Vane angular Fig. 8. Variation
of chamber
RESULTS
presssure
AND
position
I 720
(“)
with vane angular
position.
DISCUSSIONS
Table 1 shows the operational conditions of the compressor for various refrigerants. The variation in the operating pressure for different refrigerant are caused by the need to maintain the evaporating and condensing temperatures at - 23.3 and 54.4”C. The latter are the operating conditions of the compressor with R12 as the refrigerant. The predicted results on the variations of pressure, temperature and mass in the chamber with angular position, as well as variation of pressure-volume diagrams are shown. Typical simulation results at the above operational conditions with R12 as the refrigerant are presented in Figs 8 - 11. Figure 8 shows the variation of pressure-angle history. The fluctuation of the pressure during the discharge process is caused by the operational characteristics of the discharge valve reed [4]. Figure 9 shows the pressure-volume diagram. The area of the diagram is proportional to the indicated power of the machine. The temperature angle history is shown in Fig. 10. It is shown that the temperature variation exhibits the same characteristics as that of the pressure. Figure 11 shows the variation of the refrigerant mass in the working chamber of the compressor. It shows that the refrigerant mass increases slowly during the suction process and the process lasts for about 380” angular span. During the compression process, since the perfectly sealed model is assumed, the refrigerant mass remains constant. The discharge port is uncovered at the angular position of about 621”, where the mass of the refrigerant is discharged. The mass angle curve shows a steeper gradient at the early stage of the discharge process, due to a high-pressure differential across the valve port at that moment. This phenomenon is confirmed by the pressure angle history shown in Fig. 8. Table 2 shows that the indicated power caters for about 85% of the shaft power input, whereas the frictional power loss caters for about 13%. 16r
I O.zoEO 1.30E-6 I
1
I
2.60E-6 Column
volume
Fig. 9. Pressure-volume
3.90E-6 (m3) diagram.
I
5.20E-6
K. T. Ooi and T. N. Wong
76 45Or
3201 0
I 120
I I I 480 240 360 Vane angular position (“)
I 600
I 720
Fig. 10. Variation of temperature with vane angular position.
Table 3 shows a comparison between measured and predicted results. All values shown are normalised with respect to the measured data. The measured results were obtained using the industrial standard compressor performance test chamber. The results for cooling capacity are predicted by taking the refrigeration effect at the evaporating temperature of - 23.3”C. The values of the coefficient of performance (COP) were taken as the cooling capacity divided by the compressor power input. The effects of suction heating were accounted for by measuring the refrigerant gas temperature as close as possible to the inlet port of the compressor, and using it as the suction temperature in the simulation model. Generally speaking, the predictions are in good comparison with the measured results. The prediction for the performance parameters (e.g. cooling capacity, input power, mass flow rate and COP) when employing R12 as the refrigerant are below 10% discrepancy, when compared with experiment results. When R22 was used as the refrigerant, the results show that the compressor shaft power was underpredicted by 6.5%, which resulted in 8.3% overprediction in the value of COP. The discrepancy between the prediction and the measured results was greater when R134a was used. The results show that a 10.5% underprediction in the power input can result in 11.1% overprediction of the COP. The discrepancy may be due to the use of similar lubricant properties in all the predictions. CONCLUSION
The study revealed that for this quarter horse-power model of a compressor the energy consumed in the compression cycle was BO-90% of the compressor shaft energy input and lO-15% of the energy was dissipated as mechanical friction. 4.OE-5-
G 3.OE-5-Y 3 z 2.OE-5 8 % r l.OE-5 -
O.OEO 0
120
480 240 360 Vane angular position (“)
600
Fig. 11. Variation of chamber mass with vane angular position.
720
Computer
simulation
of a rotary
compressor
Table 2. Compressor cower ratios for various refriaerants
(Indicated power)/(Shaft power) (Friction power loss)/(Shaft power)
RI2
R22
Rl34a
85.9% 14.12%
90.3% 963%
84.8% 15.15%
Table 3. Comparison between the experimental and predicted results at r,,,, = - 23.3”C and ‘I&., = 54.4”C and T. = 32.2”C. Results are normalised relative to measured results
Normalised Normal&d Normal&d Normalised
RI2
R22
Rl34a
shaft power, TV,,,/W.,, refrigerant Row rate, wi&+~.~~ cooling capacity, C,,,/C.,,
0.976 ( - 2.4%) 1.004( + 0.4%) I.013 (+ 1.3%)
0.935 ( - 6.5%) 1.006( + 0.6%) 1.013 (+ 1.3%)
0.895 (- 10.5%) 0.986( - 1.4%) 0.995 ( - 0.05%)
COP, COPtAc/COPexn
1.038 ( + 3.8%)
1.083 ( + 8.3%)
I.111 (+
11.1%)
The study showed that when compared with the currently available experimental data, the model is capable of predicting the performance of the compressor with about a 10% discrepancy. The model shows that the power input is always underpredicted, while the mass flow rate and the cooling capacity are always overpredicted. As a result of these discrepancies, the COP is always overpredicted. The result also showed that Coulomb-type friction may be used to describe rubbing at the vane tip and vane sides. The frictional coefficient of 0.02 has been assigned to describe rubbing at these regions. The model, with its simplicity, has been used in assisting in the design of new compressors for use with new environmentally friendly refrigerants. It also paves the way for more comprehensive simulation studies and for possible overall computer optimisation design study. Acknowledgements-The
authors
wish to thank Matsushita
Refrigeration
Industries
(S) Pte Ltd for sponsoring
this project.
REFERENCES of reciprocating compressors: the state of the art. Inrernational Conf. on I. G. Phang and K. T. Ooi, Simulations Mathematical Modelling, Invited Lectures and Extended Abstracls, Brunei, pp. 334338 (1995). 2. J. F. Hamilton, Extension of mathematical modelling of positive displacement type compressors, short course text, Purdue University (1974). 3. P. N. Pandeya and W. Soedel, Rolling-piston-type rotary compressors with special attention to friction and leakage. Proc. 1978 Purdue University Compressor Technology Conf., pp. 2099218 (1978). 4. K. T. Ooi, G. B. Chai and E. C. Kwek, A simple valve model to study the performance of a small compressor. Proc. International Compressor Engineering Conf., Purdue University, Vol. I, pp. 147-156 (1992). 5. K. T. Ooi, T. N. Wong and E. C. Kwek, A real gas simulation of a refrigeration compressor and its performance comparison for CFCs and non-CFCs. Proc. Internafional Compressor Engineering Conf.. Purdue University, Vol. III, pp. 797-808 (1992). in a rotary compressor. Proc. 1982 Purdue University Compressor Technology 6. W. Hsiao et al., Analysis of performance Conf., pp. 14&147 (1982). I. Chu Itsuo et al., Analysis of the rolling-piston-type rotary compressor. Proc. 1978 Purdue University Compressor Technology Conf., pp. 219-225 (1978). 8. T. Yanagiswa and T. Shimizu, Friction losses in rolling-piston-type rotary compressors III. Int. J. Refrig. 8(3), 159-165 (1985). 9. T. Yanagiswa, T. Shimizu, I. Chu and K. Ishijima, Motion analysis of rolling piston in rotary compressor. Proc. 1982 Purdue University Compressor Technology Conf., Purdue University (1982). 10. Z. Zhou and Y. Gong, The estimation of the frictional losses of rolling-piston-type refrigerant compressors, Proc. 1988 Inrernafional Compressor Engineering Conf. at Purdue, Purdue University (1988). II. R. C. Downing. Fluorocarbon Refrigeranfs Handbook. Prentice Hall, New Jersey (1988). 12. Japanese Association of Refrigeration and Japan Flon Gas Association, Thermophysical Properties of Environmentally Acceptable Fluorocarbans HFC-134a and HCFC-123. Japanese Association of Refrigeration and Japan Flon Gas Association ( 1990).
APPENDIX SIMULATION
OF
REFRIGERANT
PROPERTIES
Refrigerant properties used in the analysis are simulated using real gas equations. The Martin-Downing-type equation (11) for refrigerant vapour is used for R12. Where the equation of state is given in the form,
of
78
K. T. Ooi and T. N. Wong
the enthalpy is given as
the entropy is given as S = a(lnl0) log T + bT + q
+ F
-
$
+ JR(ln10) log (VS- b) - ii5 ,_>(z-
l)(;-
64.3)
b)‘-’ ’
For the properties of R134a (which is not available in ref. [I l]), the Piao et al.-type of equations [12] are used. These equations cater for a wider range of applicability. It covers up to compressed liquid-state properties. The equation of state is given in the form P=Pc(F
+:y(.,,+
2
+ 2
+ z
+ @Pk),
64.4)
the enthalpy is given as
Pi-’ + 2
c
+ 2
E
(cl - l)ln 3
+ c2(TR - PC) + Q(
Ti - 2(fi)?
+ c (T:) - (pR)’ + c 4 3
(A.3
where
_ cl (TRY- ~(TRo)*TR+ ~(TRo)‘) _ cI (Tr,d - 4(TRo)‘TR + 3(T,o)3 - (TT+- TROC, - cr 6 12
(A.6)
The entropy is given as Os=RZ
+2+
TR
+3% ++ TR
R
$-C (cl -
l)ln 2
- cz(TR - pR) - c, iTa -$fi)*)
where a2,-as5, Q-C,, c, and c, are coefficient values depending on freon properties.
_ c4 (Tit -3(fi)‘)
+ c,
, >>
(4.7)
Pergamon
A COMPUTER SIMULATION OF A ROTARY COMPRESSOR FOR HOUSEHOLD REFRIGERATORS K. T. Ooi and T. N. Wong School
of Mechanical
and Production Avenue,
Engineering, Nanyang Technological Singapore 639798, Singapore
University,
Nanyang
(Received 8 February 1996) Abstract-This paper presents analytical studies of a fractional horse-power rotary refrigeration compressor and its performance comparison with measured results. The study employed a general-purpose performance model by considering the thermodynamic cycle of the compressor and its mechanical losses. The real gas equation of state is used to describe the changes of state of the refrigerant. The study reveals that for this quarter horse-power compressor unit the energy consumption in performing the compression cycle was 8&90% of the compressor shaft energy input, and the other l&15% of the energy was dissipated as mechanical friction. The results also show that the performance predictions are satisfactory when compared with measured results. The model has been used in assisting the design of new compressors for use with new environmentally friendly refrigerants. This study also paves the way for more comprehensive simulation studies and for possible overall computerised optimisation design study in the future. Copyright 0 1996 Elsevier Science Ltd. Keywords-Refrigeration,
compressor,
rotary,
simulation,
modelling.
NOMENCLATURE A A, a B
d
E e F.
area, m2 valve flow area, m* radius ratio, R,/R, effective valve force area, m* coefficient of discharge force area coefficient valve body width, m Young’s modulus, N/m’ eccentricity, m net pressure force across the vane in the x direction, N net pressure force across the vane in the y direction, N vane spring force, N inertia force of vane, N tangential force at vane tip, N normal force at vane tip, N tangential forces at vane-side contact points, N normal forces at vane side contact points, N blade height, m, or specific enthalpy, J/kg upstream enthalpy, kJ/kg downstream enthalpy assuming isentropic expansion, kJ/kg area moment of inertia, m4 moment of inertia of roller, kg m’ vane spring constant N/m cylinder length, effective valve length, m friction loss at vane tip, W friction loss at vane side, W friction loss between eccentric and cylinder head face, W friction loss between roller and cylinder head face, W friction loss between the roller and eccentric, W mass of fluid, kg mass flow rate, kg/s friction moment from eccentric to cylinder head face, Nm friction moment from eccentric to roller, Nm friction moment from roller to cylinder head face, Nm mass of vane, kg pressure, N/m’ pressure in suction and compression chambers, N/m’ 65
66
K. T. Ooi and T. N. Wong
P Id P over PR P,, P*
indicated power, W over compression power loss, W pressure ratio, P/PC suction and discharge pressures, N/m’ heat transfer, J mode participation factor gas constant, J/(kgK) cylinder radius, m roller outer radius, m radius of eccentric, m shaft radius, m vane tip radius, m valve tip radius, m temperature, K temperature ratio, T/T, valve thickness, m blade thickness, m specific internal energy, J/kg volume, m3 velocity, m/s sliding velocity at vane tip, m/s specific volume, ml/kg vane extension or distance from valve free end, m work, J deflection of valve, m critical compressibility, P,/(Rp,T,)
Q q(x)
R RC R, R, R, R, r T TR t lb ” yc3 V V Vt 0s X w
Y(X) Z Greek
letters
angular position of rotor, radian radial clearance between roller and cylinder at 0 = 0, m clearance between roller and cylinder head face, m radial clearance between roller and eccentric, m vane protrusion, m clearance between eccentric and cylinder head face, m viscosity of lubricating oil, Ns/m2 angular velocity of eccentric or natural frequency, rad/s angular velocity of rolling piston, rad/s valve shape function plate density, kg/m’ damping coefficient coefficient of friction at vane side coefficient of friction at vane tip density ratio, p/pC time differential
INTRODUCTION
Over the last two decades, computer simulation studies have benefitted compressor research and development. In the late sixties and early seventies, the introduction of high-speed digital computer technology made a major difference to the analysis of refrigeration compressors. Today, simulation studies which account for real gas property simulation, interaction of valve dynamics with gas pulsation in the plenum chambers, heat transfer, flow simulation using two-dimensional representation and valve stress analysis using finite element techniques are also incorporated. In general, compressor modelling studies can be broadly classified into two categories [l], which are general performance modelling and special-purpose modelling. The general-purpose model includes thermodynamic simulation of working chamber processes, flow-through valves and valve dynamics. More comprehensive models may include pulsation effects, friction, leakage and heat transfer. Special-purpose models, in contrast, are formulated to look at a specific area of interest in a more detailed manner, such as valve stress analysis and its relation with fluid flow through valve ports. Literature shows that much work has been done on compressors. On the research work on rolling piston compressors, previous studies [2-51 employing mathematical simulation were aimed at studying the behaviour of the machine and no attempt was made to compare predictions with the experimental results. Other studies [68] concentrated on extensive experimental work. This paper presents a comparison of the performance prediction of a general-purpose model of a rolling piston refrigeration compressor with measured results. The compressor is rated as quarter
67
Computer simulation of a rotary compressor Electric motor assembly
Pump assembly --
WorkingcAi
7
,A,z
stator
slhaft
Discharge tube
Suction tubk Upper be Motor rotor
Roll& piston Lower bearing
Fig. 1. Cut-away view of a compressor used in a household refrigerator.
The model incorporated basic working horse-power and is used in household refrigerators. principles of the compressor, thermodynamic simulation of the working chamber, valve flow and valve dynamics, as well as the mechanical losses of the moving components. MATHEMATICAL
MODELLING
Introduction Rolling piston compressors are commonly found in air compression and refrigeration industries. They are geometrically simple machines with one roller, a cylinder, a spring and a vane, as shown in Fig. 1. Figure 2 shows a schematic configuration of the compressor. During the operation, the piston rolls along the inner wall of the cylinder. Within the roller and the cylinder wall there exists two chambers which are separated by the vane and the roller-cylinder contact. While one chamber undergoes compression/delivery process the other undergoes suction. The process is cyclic and it takes two revolutions to complete a cycle. The existence of the compression spring is to ensure that the vane tip-roller contact is maintained throughout the operational process. The overall 820
Fig. 2. Rolling piston compressor schematic.
K. T. Ooi and T. N. Wong
68
performance of a given machine depends on the balance of compression, frictional effects, leakage effects, suction and discharge effects. The compressor is of hermetic type and lubricating oil is used to seal and to lubricate. Today, with the availability of high-speed computers, a complete modelling of the machine that caters for geometrical, fluid flow, heat transfer, dynamic and stress analysis may be incorporated with comparative ease. This section briefs the simulation of the machine. The model accounts for geometric, thermodynamic, dynamics and frictional effects in the machine, and a real gas equation was used to describe the state of the refrigerant. The model allows the variation of refrigerating fluid pressure, temperature and mass flow to be evaluated in relation to the geometrical configuration and the operating conditions of the machine. The performance of the machine in terms of power input as well as cooling capacity may be evaluated. Geometry
model
The geometry model of the compressor accounts for all important physical parameters which are within the working chamber of the compressor that may affect the performance of the compressor. Referring back to Fig. 2, the chamber volume trapped within the roller, cylinder and the vane may be given by
V(e)
=
$$
(
(1 _
a’)()
_
Q-j-@ sin20
_
a2
sin- ’((i
- 1)sinO)
- a(I -L)sinDJw).
To account the volume
for the vane thickness, the volume given by equation (1):
occupied
v’(e) = v(e) -
by the vane must be subtracted
(1)
away from
F ,
where 6, = R,(l - (1 - a) case - J(1
- a)2 cos2e + 2a - 1) .
(3)
Valve jlow model In the present simulation, the suction port is considered as a simple orifice without a valve. The effective flow area may be accounted for by introducing a discharge coefficient on the calculated geometrical area. The flow through the suction port is assumed to be one-dimensional adiabatic flow. The isentropic efficiency is accounted for by lumping it into the discharge coefficient. Reversed flows are taken into consideration where the lumped discharge coefficient is assumed constant throughout the suction process. The same value of the discharge coefficient is also used for reversed flows. The discharge port is covered with a reed valve. The valve flow area may be the valve port area or the valve lift annulus area. The latter must be determined from the instantaneous valve deflection. The effective flow area is taken as the area which imposes more restrictions to the flow. Assumptions as in suction flow are applicable. By first assuming that the flow through the valve is isentropic, the flow velocity may be calculated:
V=J_.
(4)
Computer simulation of a rotary compressor
69
Fig. 3. Geometry of the value reed.
In the above equation, subscripts 1 and 2 denote the conditions of the fluid upstream and downstream of the valve port, respectively. The mass flow rate can be found by dm -=-----_ C,AP df9
(5)
vs2
where h: is the downstream enthalpy assuming an isentropic expansion. To obtain this enthalpy, the knowledge of the downstream specific volume and the temperature must be known. This information may be obtained by solving the equation of state, as shown in the Appendix using the Newton-Raphson iterative method; C, is the discharge coefficient which accounts for the effective flow area and the isentropic efficiency, for actual non-isentropic flow. The choke flow condition is taken into consideration by taking the maximum flow velocity to be equal to the sonic velocity at the throat of the port. Valve dynamics
In the discharge side of the machine, there exists a discharge reed valve. Figure 3 shows the geometry of the valve plate. Its geometry may be simplified by assuming the valve free end as a circular disc that is connected to a rectangular valve plate. The valve displacement is constrained by the valve backing plate which is placed a small distance above the valve plate. The function of this backing plate is to prevent the valve plate from over stressing due to unnecessarily large deflection. The dynamic behaviour of the valve is characterised by the vibration of the valve plate. Assuming the valve as a beam with varying width, neglecting shear and rotary motion, the beam equation describing the beam vibration [4] is
g
(EZ(x@+
&A(X)
2 =P(x,t)
>
and assuming the valve deflection is given by
Y = i
4”(X)4”W>
II=,
(7)
where d”(x) is the valve shape function which may be determined from the free vibration analysis. There exists an infinite number of combinations of mode shapes since n ranges from 1 to infinity, but in many cases, especially for the valve flow area calculation, the first mode consideration alone may be sufficient [2]; qn(t) is the generalised coordinate or mode participation factor which may be obtained by integrating the valve vibration governing equation using an initial value integration technique.
IO
K. T. Ooi and T. N. Wong
The mode participation
factor is obtained from
4. + 210&k + O;q” =
J
wWx)t, dx
’
(8)
where in is the damping coefficient of the valve reed due to the air cushioning. For a single valve port configuration and to avoid considerations of the flow in the valve gap, the gas force acting on the valve may be taken to be equal to the product of the effective valve force area and the pressure drop across the valve [2,4],
s
4dx)P dx = ~,(xP(x)Wt) ,
(9)
where B(x) is the effective force area and AZ’(t) is the pressure difference across the valve. Generally, the term B(x) has to be obtained from experiments. It is a function of valve lift. Substituting equation (9) into equation (8) the following expression may be obtained:
In the above equation, the natural frequency of the valve may be obtained approximately Rayleigh’s quotient, i.e. -x Y’
w;=
from
(11)
where
(12) and Y = pp
x’A(~)c$(~)2 dx + ppA ‘4(x)’ dx ,
s0
(13)
s XI
the values of ,4(x) and Z(x) are given below: o
- x2
A = dt Z(x) =
t’ 6
z=
J
2rx
-
x2
dt’ 12.
In the above solution, the instantaneous value of valve lift is monitored to determine if the valve hits the backing plate and the correct boundary conditions are imposed. Two cases are considered: one is when the valve leaves the seat, the other is when the valve hits the valve backing plate [4]. In the present model, only the first mode of valve deflection is considered, the model, however, takes into consideration different boundary conditions, as well as valve mode shapes, when the valve departs from the valve seat, and when it hits and departs from the valve backing plate. Figure 4 shows the two different valve mode shapes.
Computer
simulation
71
of a rotary compressor
0.7 0.6 z
-
Shape 2
.......
Shape
.5 0.5E
. ..’ . . . .
1
.... _:’ _’
:
:
:
:
:
:
:
:
:
,:’ .“.
After hits valve stop
Before
hits valve stop
Distance
Fig. 4. Different
I 0.020
0.015
0.010
0.005
In the present seat is obtained
. . . . . ..’
from clamped end (mm)
valve shape
functions.
analysis, the shape function for the case when the valve is departed from a standard polynomial function for a cantilever beam [4]:
from the valve
(14)
rn(x)=($-4(f)+3.
In equation
(14) the natural
and geometric &x=0,-=
boundary
W(O)
o,
conditions
d’440) -=
o
dx3
dx*
at x = 1, 4(1) = 0, $$ When the valve hits the backing
for both ends are satisfied:
plate, the following
standard
’
= 0. polynomial
function
is applicable
[4]:
(15)
Again in equation (15) the natural and geometry boundary conditions are satisfied: at x = 0, 4(O) = 0, d*@(O)/dx* = 0 and at x = I, $(I) = 0, d4/dx = 0. The mode shape has been verified by comparing the deflected shape of the valve reed with a more comprehensive finite element model employing the actual geometry of the valve reed. Figure 5 shows that good agreement was obtained. 0.7 0.6
-
Simplified
model model
. . . . . . . Finite element
z e
0.5
Distance
Fig. 5. Comparison
from clamped end (mm)
of valve displacements.
K. T. Ooi and T. N. Wong
72
THERMODYNAMIC
MODEL
In the thermodynamic model [5] the relationships within pressure, temperature and the mass will be formulated. These three unknowns were related using the energy conservation relation, mass conservation relation and the equation of state of a real gas. A brief account of the model is given below. Pressure, temperature and mass relationships
To obtain the relationship of the state of the working medium in the compressor, applying the First Law of Thermodynamics to the control volume by taking the boundary of the chamber as the boundary of the control volume and neglecting the kinetic and potential energy:
Q de
+Cshi=
$
+C%ho+
-&(muJ,
(16)
where subscripts i and o denote inlet and outlet. Taking the work term as the mechanical work: dw= PdV,,
(17)
h=u+PV,,
(18)
from the relationship of the enthalpy:
the pressure is a function of the temperature
and the specific volume:
p = P(T,V,) 9
(19)
V, = Vc/mc .
(20)
where the specific volume is given by
After manipulation,
the following expressions may be obtained: 1
!!!L=
v,
+b%~68)~
de
- $
C2
(h, - h,) - C 2
_ r @/aT) vs (aPclaT,i
(21)
(ho - hi)
and
The mass in the working chamber can be obtained from the continuity equation, i.e. dm, dmi -- dm, de - dB dmO ’
(23)
where the two main sources of the mi and the m, are from suction and discharge flows through the respective ports, The partial derivatives for refrigerant property in the above equations can be obtained by differentiating equations (A. l)-(A.6), detailed simulation of the refrigerant properties is depicted in the Appendix.
Computer
simulation
of a rotary
compressor
13
e=o
Compression chamber
Fig. 6. Roller friction
FRICTION
moments.
MODEL
To determine the friction losses of the compressor, the vane and the roller dynamics must be considered. The information about the vane sliding velocity and roller velocity must be obtained. There are six areas where friction losses may take place [3, 5, 8-101, as follows (Figs 6 and 7): Eccentric
and the inner
surface
of the roller, L
Roller
face and the cylinder
= (w - WPf,,
.
head face, LC = M,CW,
Eccentric
(25)
face and the cylinder
(26)
head face, L,, = Me,0 .
(27)
L,. = V,F,, .
(28)
Vane tip and roller,
FS
Fig. 7. Vane force balance.
K. T. Ooi and T. N. Wong
14
Table I. Operating pressures for various freons at Tcvap= - 23.3”C and r,, Suction pressure (bar) Gas temperature outside the suction valve (“C) Discharge pressure (bar) Gas temperature after the discharge valve (“C) Pressure ratio Rotational speed (rev/min)
= 54.4”C
RI2
R22
Rl34a
1.324 64.3 13.46 120.8 10.17 3487
2. I51 68.0 21.47 134.7 9.95 3395
I.151 61.3 14.68 128.0 12.75 3350
where V, = R,o, + eocose/cosa
.
(29)
5. Vane sides and vane slot, (30) 6. Outer roller surface and the inner cylinder surface. Item (4) can be ignored if it is assumed that there are no contact forces between an outer roller surface and the cylinder. In the above equations other terms are defined as follows:
F,=(P,b-P,(4
F,, =
A4, = 27~w,r/(R,4 - R:)/d,
(31)
M, = ~rrp(2R,4 - R:)/6,
(32)
M,, = 27ty1(0- o,)lR@,
(33)
F, = (PC - P,)xl
(34)
F, = k(xc,,x, - x)
(35)
F, = - M,i
(36)
+R,sina)-P,(?
-R.sina))l
(37)
p,F,(h + 6,~) + (Fy + F, + Fd(x - h) (cos a + pv sin a)(x - h - 2pSR, sin a) + @(sin a - pycos a)[fbps + h + x - 2r,.(l -
cos a)] (38)
Fv, = ,G.F,,,
(39)
F,, = p,F,,
(40)
F,, = IAF,, ,
(41)
where the last unknown o, may be obtained from equation (41) if steady roller rotation is assumed: (21rplR,3/62 - R,F,,,)&d, Or = 2q(IR,d,
If angular acceleration is taken into account, equation (43) numerically as follows [8-lo]:
+ (R,d - R&5,)
.
the value of o, can be obtained
I$& = Me, - R,F,,, - Mx .
(42)
by solving
(43)
Computer
simulation
of a rotary
compressor
16.0-
c
12.8 -
d e & 9.6: “a k 6.4% 5
3.2 -
L 0
I
120
I
I
I
I
240
360
480
600
Vane angular Fig. 8. Variation
of chamber
RESULTS
presssure
AND
position
I 720
(“)
with vane angular
position.
DISCUSSIONS
Table 1 shows the operational conditions of the compressor for various refrigerants. The variation in the operating pressure for different refrigerant are caused by the need to maintain the evaporating and condensing temperatures at - 23.3 and 54.4”C. The latter are the operating conditions of the compressor with R12 as the refrigerant. The predicted results on the variations of pressure, temperature and mass in the chamber with angular position, as well as variation of pressure-volume diagrams are shown. Typical simulation results at the above operational conditions with R12 as the refrigerant are presented in Figs 8 - 11. Figure 8 shows the variation of pressure-angle history. The fluctuation of the pressure during the discharge process is caused by the operational characteristics of the discharge valve reed [4]. Figure 9 shows the pressure-volume diagram. The area of the diagram is proportional to the indicated power of the machine. The temperature angle history is shown in Fig. 10. It is shown that the temperature variation exhibits the same characteristics as that of the pressure. Figure 11 shows the variation of the refrigerant mass in the working chamber of the compressor. It shows that the refrigerant mass increases slowly during the suction process and the process lasts for about 380” angular span. During the compression process, since the perfectly sealed model is assumed, the refrigerant mass remains constant. The discharge port is uncovered at the angular position of about 621”, where the mass of the refrigerant is discharged. The mass angle curve shows a steeper gradient at the early stage of the discharge process, due to a high-pressure differential across the valve port at that moment. This phenomenon is confirmed by the pressure angle history shown in Fig. 8. Table 2 shows that the indicated power caters for about 85% of the shaft power input, whereas the frictional power loss caters for about 13%. 16r
I O.zoEO 1.30E-6 I
1
I
2.60E-6 Column
volume
Fig. 9. Pressure-volume
3.90E-6 (m3) diagram.
I
5.20E-6
K. T. Ooi and T. N. Wong
76 45Or
3201 0
I 120
I I I 480 240 360 Vane angular position (“)
I 600
I 720
Fig. 10. Variation of temperature with vane angular position.
Table 3 shows a comparison between measured and predicted results. All values shown are normalised with respect to the measured data. The measured results were obtained using the industrial standard compressor performance test chamber. The results for cooling capacity are predicted by taking the refrigeration effect at the evaporating temperature of - 23.3”C. The values of the coefficient of performance (COP) were taken as the cooling capacity divided by the compressor power input. The effects of suction heating were accounted for by measuring the refrigerant gas temperature as close as possible to the inlet port of the compressor, and using it as the suction temperature in the simulation model. Generally speaking, the predictions are in good comparison with the measured results. The prediction for the performance parameters (e.g. cooling capacity, input power, mass flow rate and COP) when employing R12 as the refrigerant are below 10% discrepancy, when compared with experiment results. When R22 was used as the refrigerant, the results show that the compressor shaft power was underpredicted by 6.5%, which resulted in 8.3% overprediction in the value of COP. The discrepancy between the prediction and the measured results was greater when R134a was used. The results show that a 10.5% underprediction in the power input can result in 11.1% overprediction of the COP. The discrepancy may be due to the use of similar lubricant properties in all the predictions. CONCLUSION
The study revealed that for this quarter horse-power model of a compressor the energy consumed in the compression cycle was BO-90% of the compressor shaft energy input and lO-15% of the energy was dissipated as mechanical friction. 4.OE-5-
G 3.OE-5-Y 3 z 2.OE-5 8 % r l.OE-5 -
O.OEO 0
120
480 240 360 Vane angular position (“)
600
Fig. 11. Variation of chamber mass with vane angular position.
720
Computer
simulation
of a rotary
compressor
Table 2. Compressor cower ratios for various refriaerants
(Indicated power)/(Shaft power) (Friction power loss)/(Shaft power)
RI2
R22
Rl34a
85.9% 14.12%
90.3% 963%
84.8% 15.15%
Table 3. Comparison between the experimental and predicted results at r,,,, = - 23.3”C and ‘I&., = 54.4”C and T. = 32.2”C. Results are normalised relative to measured results
Normalised Normal&d Normal&d Normalised
RI2
R22
Rl34a
shaft power, TV,,,/W.,, refrigerant Row rate, wi&+~.~~ cooling capacity, C,,,/C.,,
0.976 ( - 2.4%) 1.004( + 0.4%) I.013 (+ 1.3%)
0.935 ( - 6.5%) 1.006( + 0.6%) 1.013 (+ 1.3%)
0.895 (- 10.5%) 0.986( - 1.4%) 0.995 ( - 0.05%)
COP, COPtAc/COPexn
1.038 ( + 3.8%)
1.083 ( + 8.3%)
I.111 (+
11.1%)
The study showed that when compared with the currently available experimental data, the model is capable of predicting the performance of the compressor with about a 10% discrepancy. The model shows that the power input is always underpredicted, while the mass flow rate and the cooling capacity are always overpredicted. As a result of these discrepancies, the COP is always overpredicted. The result also showed that Coulomb-type friction may be used to describe rubbing at the vane tip and vane sides. The frictional coefficient of 0.02 has been assigned to describe rubbing at these regions. The model, with its simplicity, has been used in assisting in the design of new compressors for use with new environmentally friendly refrigerants. It also paves the way for more comprehensive simulation studies and for possible overall computer optimisation design study. Acknowledgements-The
authors
wish to thank Matsushita
Refrigeration
Industries
(S) Pte Ltd for sponsoring
this project.
REFERENCES of reciprocating compressors: the state of the art. Inrernational Conf. on I. G. Phang and K. T. Ooi, Simulations Mathematical Modelling, Invited Lectures and Extended Abstracls, Brunei, pp. 334338 (1995). 2. J. F. Hamilton, Extension of mathematical modelling of positive displacement type compressors, short course text, Purdue University (1974). 3. P. N. Pandeya and W. Soedel, Rolling-piston-type rotary compressors with special attention to friction and leakage. Proc. 1978 Purdue University Compressor Technology Conf., pp. 2099218 (1978). 4. K. T. Ooi, G. B. Chai and E. C. Kwek, A simple valve model to study the performance of a small compressor. Proc. International Compressor Engineering Conf., Purdue University, Vol. I, pp. 147-156 (1992). 5. K. T. Ooi, T. N. Wong and E. C. Kwek, A real gas simulation of a refrigeration compressor and its performance comparison for CFCs and non-CFCs. Proc. Internafional Compressor Engineering Conf.. Purdue University, Vol. III, pp. 797-808 (1992). in a rotary compressor. Proc. 1982 Purdue University Compressor Technology 6. W. Hsiao et al., Analysis of performance Conf., pp. 14&147 (1982). I. Chu Itsuo et al., Analysis of the rolling-piston-type rotary compressor. Proc. 1978 Purdue University Compressor Technology Conf., pp. 219-225 (1978). 8. T. Yanagiswa and T. Shimizu, Friction losses in rolling-piston-type rotary compressors III. Int. J. Refrig. 8(3), 159-165 (1985). 9. T. Yanagiswa, T. Shimizu, I. Chu and K. Ishijima, Motion analysis of rolling piston in rotary compressor. Proc. 1982 Purdue University Compressor Technology Conf., Purdue University (1982). 10. Z. Zhou and Y. Gong, The estimation of the frictional losses of rolling-piston-type refrigerant compressors, Proc. 1988 Inrernafional Compressor Engineering Conf. at Purdue, Purdue University (1988). II. R. C. Downing. Fluorocarbon Refrigeranfs Handbook. Prentice Hall, New Jersey (1988). 12. Japanese Association of Refrigeration and Japan Flon Gas Association, Thermophysical Properties of Environmentally Acceptable Fluorocarbans HFC-134a and HCFC-123. Japanese Association of Refrigeration and Japan Flon Gas Association ( 1990).
APPENDIX SIMULATION
OF
REFRIGERANT
PROPERTIES
Refrigerant properties used in the analysis are simulated using real gas equations. The Martin-Downing-type equation (11) for refrigerant vapour is used for R12. Where the equation of state is given in the form,
of
78
K. T. Ooi and T. N. Wong
the enthalpy is given as
the entropy is given as S = a(lnl0) log T + bT + q
+ F
-
$
+ JR(ln10) log (VS- b) - ii5 ,_>(z-
l)(;-
64.3)
b)‘-’ ’
For the properties of R134a (which is not available in ref. [I l]), the Piao et al.-type of equations [12] are used. These equations cater for a wider range of applicability. It covers up to compressed liquid-state properties. The equation of state is given in the form P=Pc(F
+:y(.,,+
2
+ 2
+ z
+ @Pk),
64.4)
the enthalpy is given as
Pi-’ + 2
c
+ 2
E
(cl - l)ln 3
+ c2(TR - PC) + Q(
Ti - 2(fi)?
+ c (T:) - (pR)’ + c 4 3
(A.3
where
_ cl (TRY- ~(TRo)*TR+ ~(TRo)‘) _ cI (Tr,d - 4(TRo)‘TR + 3(T,o)3 - (TT+- TROC, - cr 6 12
(A.6)
The entropy is given as Os=RZ
+2+
TR
+3% ++ TR
R
$-C (cl -
l)ln 2
- cz(TR - pR) - c, iTa -$fi)*)
where a2,-as5, Q-C,, c, and c, are coefficient values depending on freon properties.
_ c4 (Tit -3(fi)‘)
+ c,
, >>
(4.7)