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Volumetric efficiency and experimental errors of rotary compressors
Volumetric efficiency and experimental errors of rotary compressors T. Shimizu, M. Kobayashi and T. Yanagisawa
On a effectual des recherches thdoriques et exp~rimenta/es sur les facteurs ayant une influence sur le rendement vo/um#trique d'un compresseur ~ piston
rotatif. Les r#su/tats ont montr# qu'une tendance du rendement vo/um#trique thdorique concordait assez bien avec /a tendancelexpdrimentale et que les fuites de gaz par les jeux du piston ainsi que I'#l#vation de temperature du gaz d'aspiration nuisaient fortement au rendement vo/urn#trique. De plus, on a estimd quantitativement /'incidence des erreurs de mesure sur le rendement vo/umdtrique ca/cutd.
Theoretical and experimental investigations of t h e factors w h i c h a f f e c t t h e v o l u m e t r i c efficiency of a rolling piston compressor have been made. The results s h o w t h a t a trend in t h e theoretical v o l u m e t r i c efficiency agrees fairly w e l l w i t h the experimental one, and
t h a t gas leakage t h r o u g h clearances, as well as t h e t e m p e r a t u r e rise of the suction gas, strongly affects t h e v o l u m e t r i c efficiency. In addition, t h e e f f e c t o f m e a s u r e m e n t errors on t h e calculated v o l u m e t r i c efficiency w a s estimated q u a n t i t a t i v e l y .
Rendement volum6trique des compresseurs rotatifs et erreurs exp6rimentales
Nomenclature Cp specific heat at constant pressure g gravitational acceleration hm average heat transfer coefficient / cylinder width Lm motor input N number of motor revolutions Pb pressure at suction chamber Po pressure at compression chamber Pd discharge pressure Ps suction pressure (Pr) Prandtl's number Q reading of liquid refrigerant volume flow by flow meter r piston radius R cylinder radius SH degree of superheat Tb temperature at suction chamber -/-bBI, temperatures at suction chamber in Fig. 2 YbM Tbc J fs suction temperature Vo displacement volume by rolJing piston
As compressors for room air-conditioners, rotary compressors have recently become widely used in place of reciprocating ones because of their compactness, low noise output and high efficiency. These compressors are classified into sliding vane and rolling piston types. There is some published
W discharge mass per rotation Wo theoretical suction mass per rotation c~ revised coefficient of specific weight of flow meter 7b specific weight of gas refrigerant in suction chamber 7c specific weight of gas refrigerant in compression chamber just before gas expands in it 7/ specific weight of liquid at calibration of flow meter 7 specific weight of suction gas A error of mean square 5Pb pressure loss in suction 5Pc pressure loss in discharge 5Tb temperature rise of suction gas heated 5W leakage at some channels of gaps top clearance ratio ~/v experimental volumetric efficiency I/vth theoretical volumetric efficiency rotative angle of crankshaft specific heat ratio ~' kinematic viscosity work 1'2, on the former type, however, unti recently 3 very few papers had been published on the latter.
Faculty of Engineering, Shizuoka University, Hamamatsu, Japan and MK isat the Peripheral Devices Laboratory, Fujitsu Laboratories Ltd, Kawasaki, Japan. Paper received 8 January 1980
One criterion of performance of compressors is the volumetric efficiency, and to improve the performance this efficiency has to be elucidated. Accurate, volumetric efficiency values are difficult to obtain and, as has been pointed out 4.5, measurement errors affect the calculated efficiency. Therefore, careful analytical and experimental studies of the volumetric efficiency of a roiling piston compressor (Fig. 1), operated under various conditions, have been made.
Volume 3 Num6ro 4 Juillet 1980
01 40-7007/80/04021 9-07502.00 © 1980 IPC Business Press Ltd and IIR
TS and TY are with the Department of Mechanical Engineering,
219
Furthermore, an error analysis has also been carried out to ascertain the reliability of this volumetric efficiency.
(At/v) 2 = (c3~/Ap/) 2 +
Mass flow of the discharge gas from the compressor is less than the theoretical value based on the suction gas entering the compressor due to heating in the cylinder wall, gas leakage through channels and gaps, re-expansion of the gas in the top clearance, etc. Actual mass flow of the discharge gas, which is normally measured by a flow. meter for liquid refrigerant at the outlet of the condenser at steady state conditions, is given by:
G =~ylO
(1)
~r/v
where, Vo=~(R2-r2)/ and outlet ports).
2
4-
(5)
where the symbol A expresses an amount of uncertainty. (5) shows the effect of the measurement errors on the calculation error of the volumetric efficiency. In (3) Vo is regarded as constant and involving no error. Then, "Calculation accuracy of the volumetric efficiency' is derived from (3) and
(5):
The theoretical mass flow of the suction gas is given by using a specific weight of gas refrigerant in the suction line, y~, and is as follows: G O= 60mVoY~
(>o) 2 -,- &/,,
+
Definition of the experimental volumetric efficiency
ar/v
tG) _¢8(~yi)AP,'~ 2
(2)
/0 (~y/) AT,'~ 2
(ignoring the effect of inlet [AQ\ 2 [ d N AL~\ 2
The volumetric efficiency of the practical compressor is defined as the ratio of (1) and (2): G o~y/Q r/~- Go - 60mVoys
(3)
(6)
Error analysis of the volumetric efficiency Because of difficulties in making accurate, experimental measurements, the measuring volumetric efficiency expressed by (3) is usually calculated roughly, and shows fluctuations. Therefore, to ascertain the reliability of the experimental value, an error analysis of the experimental volumetric efficiency has been carried out. ~, in (3), is a function of the specific weight of the supercooled liquid refrigerant, Y/, entering the flow meter, as will be discussed later. Furthermore, y/is also a function of the measured pressure, P/, and temperature, 7-/, of the liquid refrigerant. Also, N depends on the measured motor input, L~, and Ys depends on the measured suction pressure, Ps, and suction temperature, Ts. These relationships are summarized as follows: o~=o~(y,) =~(P/,T/);
N=N(Lm);
y/=
~,s j -t, aPs >,~ j
(APsV+/ATs%2
=\ 77J=\7(J \W)
<7)
Theoretical volumetric efficiency Derivation of the theoretical volumetric efficiency
yI(P/,T/); (4)
~ = ~(Ps, T~) Each directly measured quantity in (4) contains. random and independent errors. Applying the Law of Propagation of Errors to (3) and taking (4) into consideration:
220
Provided that the suction gas is approximately an ideal gas throughout this analysis, the accuracy of the specific weight of the suction gas, Ys, the fifth and the sixth terms in (4), is as follows:
The volumetric efficiency is defined as the ratio of the theoretical mass flow of the suction gas and the actual mass flow of the discharge gas as mentioned earlier. The factors which affect the volumetric efficiency are: the top clearance Volume of a discharge portion (~Vo); the pressure losses in the suction and discharge ports (aP b, 8Pc); gas leakage (aW); and the temperature rise of the suction gas
(aTb).
International Journal of Refrigeration
The effects of circulation of the lubricating oil, variation in the rotational velocity, etc. are regarded as relatively small, and are disregarded in the present analysis. Theoretical suction mass per revolution is given by: Go
W° - 60-m - V°7~
(8)
considered narrow enough to be neglected: The leakages at a and b are assumed to be as a one dimensional adiabatic gas flow, as Stein et al. 1 considered, and the formula for a convergent nozzle with compressible fluid has been used. For the leakages from the gaps at a and b in Fig. 1 the following three conditions have to be taken into account as pressure rises. For gas flow up to the critical (acoustic) velocity, the mass flow is:
On the other hand, the actual mass per revolution is given by: G W = 60N = V°(1
(9)
+s)%-sVoYc-SW
Thus the theoretical volumetric efficiency is given by:
( k F/po\ (k-1)lk Ti1/2 Wl=~2g~Pb%Lt~b ) -1~
r After reaching the critical velocity, the pressure in the compression chamber, Pc, continues to rise, and the critical velocity also increases. Then, the mass flow is:
w2=F W
/~vth= d o =
I !--cS
--1
7b ~W do
~
(15a)
/
2
~(k+l)/(k-1)
\PU
(10)
J 15b)
A
while, 1c/?b is expressed as follows:
7c_(Pd+&Pc~ Ilk
(11)
Furthermore, provided that the gas refrigerant temperature at the suction Ts is raised to Tb=Ts+ST b during the suction stroke, Yb/Ts is expressed by: 7b P~-&Pb
7s
Ps
Ts
(12)
T~+&Tb
I
Therefore (10) is given by:
\
&W ~vth = ~vp~vt d o
(1 3)
C
a ~[[Pd-FSPc~
1/k
b
~]}Ps--~Pb
vp= 1-OLb s < >W -,]j> Ps(14)
Fig. la
Position of leakage paths
F/~7. la
Emp/acement des passages de fuite
Ts r/vt- Ts+ &Tb
Calculation of the internal leakage A rolling piston compressor has to minimize the gas leakage by using precisely manufactured components. However, reduction of clearances is limited by problems of lubrication, fabrication techniques, frictional losses, etc. As a result, gas leakage th,rough gaps has a rather large effect on volumetric efficiency. As shown in Fig. la and Fig. l b possible, leakage paths are as follows:a - across the edges of the dividing valve plate; b - between the piston and the cylinder wall; c and d - between the end faces of the piston and cylinder. Since c and d are normally filled with lubricating oil, these gaps are
Volume 3 Number 4 July 1980
Fig. l b Section showing theleakage paths: 1 -piston; 2-crankshaft; 3 cylinder;4 dividing vane; 5-cylinder head F/~7. I b Coupe rnontrant les passages de fuite. I - piston; 2 - vi/ebrequin; 3 cy/mdre; 4 - aube de sbparation; 5 t#te de cy/indre
221
Finally the pressure in the compression chamber, Po=Pd+SPc, becomes constant. Then, the mass flow can be taken as constant: F
/
2 \(,+w(k-~)
(1 5c)
J
Where W~, W 2 and W 3 are all mass flows per unit area.
By taking 5~ as the dimension of the leakage gap between the dividing vane and the cylinder head on each side, the leaked mass at a of Fig. 1 is as follows: ~p
t;(
=
(
~p
(Re) -
fsd~m v
-
4G
%dsrn~sV
(20)
where fs is an equivalent inlet velocity.
dOf
xw 1
xW2dO
0
receives a great deal of heat during the suction stroke, and the resulting temperature rise has a rather large effect on the volumetric efficiency. This heat is received as the gas flows between the piston and the cylinder wall. The suction route at the maximum suction volume is from A, B, M, to C in Fig. 2a, so the gas temperatures at points B, M, C are naturally different because the gas at each point is inhaled into the cylinder at different times. The temperature at the centre, M, is assumed to represent the average temperature. Next, using an average heating area Sin, an equivalent diameter of pipe dsm is expressed as 4Vo/S m, and Reynolds' number is as follows:
Using (20), Nusselt's number is given by Dittus and Boelter's formula as follows:
8c
2=
(Nu) = 0.0023 (Re)O.S (pr)O-4
A\
(16)
Op
(21)
SimilarLy, by taking 5b as the dimension of the leakage gap between the piston and the cylinder wall, the leaked mass as b in Fig. 1 is as follows: ec
wb=;(f bWld0 @ ~P
2=
(17) where co is an angular velocity of a crankshaft, and / is a width of cylinder. 0c and 8p are the rotational angles of a crankshaft when the gas flow reaches the critical velocity and when the discharge va!ve starts to open, respectively, x, the displacement of the dividing vane, can be expressed as follows:
,[1 -(R-r)
a
Fig. 2a Flow passage of suction gas F/~7.2a Passagede /'6cou/ement du gaz d'aspiration
s,n cos 0
Tcr =const.
(18)
Integration of (16) and (1 7) can be made using Simpson's rule. Then, the leakage, 5W, in (1 3) is:
aw=awo+awb Calculation of temperature
(19) rise o f s u c t i o n gas
Since the pressure inside the hermetic case of a rolling piston compressor is generally the same as the discharge value, the temperature of the cylinder becomes rather high, Therefore the suction gas
222
rbc
b Fig. 2b
Temperature rise curve of suction gas
Fig. 2b
Courbe de/'d/dvation de tempdrature du gaz d'aspbation
Revue Internationale du Froid
From (21), using thermal conductivity coefficient, 2, the average heat transfer coefficient is given as follows: hmi
(Nu)2 dsm
(22)
Provided that a temperature rise of the suction gas follows the same trend as the fluid flowing in a pipe w i t h constant wall temperature, Tc,, the temperatures at the points A, B, M and C in Fig. 2a can be reasonably assumed to change exponentially as Ts, TbB, fbM and Tbc in Fig. 2b. So the mutual relation between these temperatures is:
(To,-TbM)I(To,-Ts) = exp[ - (hmSm)/(CpG)]
(23)
Then 5Tb is given by:
5Tb= TbM-- Ts
(24)
addition, a supplementary heater was placed in the evaporator so that the cycle could be operated even in the high superheat range. Temperatures at respective points were measured by copper-constantan thermocouples, and a steady state of the cycle was confirmed by recording the pipe wall temperatures at a suction plenum and at the outlet of the compressor. Pressures were measured by Bourdon gauges. Volume flow of a liquid refrigerant, R22, was measured by a float type, area flow meter, and the input to a compressor motor was measured by a single phase, watt meter. Experiments were carried out to examine the performance of a compressor under the following two operational conditions: condensing pressure was varied while keeping the evaporating pressure and superheat constant in order to examine the effect of compression ratio and discharge pressure Qn the performance: and the degree of superheat was varied while keeping the evaporating and condensing pressure constant in order to examine the effect of suction temperature on the performance.
E x p e r i m e n t a l a p p a r a t u s and p r o c e d u r e The flow diagram of the experimental apparatus is shown in Fig. 3. The hermetic rolling piston compressor of 1 kW class was used. Constant temperature, water supply systems to the condenser and evaporator are shown as broken lines in Fig. 3. In
D e t e r m i n a t i o n of a f u n c t i o n a l f o r m used for error c a l c u l a t i o n The revised coefficient of the specific weight of the flow meter, ~, was calculated as follows: 0~= [ ( T f - - ~ / ) 7 0 / { ( ' ~ f - - 7 0 ) ~ / }
As 7f and 7o are constant, ~ is a function of the specific weight of the liquid refrigerant entering the flow meter, 7/- And 7/is the value determined by measuring the pressure, P/, and the temperature, T/, at the flow meter. Therefore, ~ and 7/become the functions used in (4). Furthermore, the degree of subcooling of the liquid refrigerant entering the flow meter has an effect on P/in this experiment, so 7/ can be treated as a function of 7-/. By means of a least squares fit the experimental data yield:
__
] 2
I
I I
I
r
I
i/ f~m ii I I 9
(25)
] 1/2
~ = 1 . 5 5 0 - 5.510 x 10-3Tl +1.246 x 10-STj 2
I
1
where 286 K < T S 3 2 8
Io
5
(26)
K
y / : 1 . 1 1 4 - 1 . 0 5 4 x 10-2T/
I I
1
I ,2 I
I
li
Fig. 3 F l o w d i a g r a m of e x p e r i m e n t a l apparatus: 1 - c o m p r e s s o r ; 2 - c o n d e n s e r ; 3 - f l o w meter; 4 - hand e x p a n s i o n valve; 5 - e v a p o r a t o r ; 6 - s u c t i o n muffler; 7 electric p o w e r ; 8 - w a t t meter; 9 - f l o w c o n t r o l valve; 10 - w a t e r pump; 11 - heater; 12 pen recorder Fig. 3 D i a g r a m m e de p r i n c i p e d e / ' a p p a r e i / e x p # r i m e n t a / . 1 - compresseur, 2 - condenseur, 3 f/uxm#tre; 4 - d#tendeur a main, 5 - dvaporateur; 6 - s//encieux d'aspirat/on; 7 - courant d/ectrique; 8 - wattm#tre; 9 - rob/net de rdgu/at/on de / ' d c o u / e m e n t ; 10 - p o m p e ~ eau, 11 - rdchauffeur,- 12 - e n r e g/streur ~ p/ume
Volume 3 Num6ro 4 Juillet 1980
- 1 . 3 3 5 x 10-5T/2
(27)
w h e r e 286 K_
- -
7.28 x 1 0-2Lm
- 2 . 0 x lO-SL~
(28)
where 790 W__
223
The calculation accuracy of the volumetric efficiency is given by substituting (7), (26), (27) and (28) into (6).
Estimation of measurement
errors
The measurement errors are evaluated from the calibrations of measuring instruments, their grades and accuracy, deviation and fluctuation of the readings, etc.
IOC
Ps, MN m-2
90--
~>
Ts,
• 0.376-0.:595 o 0.572-0.591
K
In the temperature measurements, iATJ is estimated to be 3 K, because the suction temperature was not measured for the gas refrigerant directly but on the pipe wall of the suction plenum. And IALI is evaluated to be 1 K, because the temperature of the liquid refrigerant entering the flow meter was measured directly. In the pressure measurement, IAPsl is evaluated to be 0.05 kN m -2. In the volume flow measurement of a liquid refrigerant, }AQ} is found to be 0.5 I h -1, and in the motor input measurement, }ALto} is 15 W. After the error analysis
SH, °
271- 274 5-8 285-288 5-10
• o
~
I00
7£
9O
Q584-O.386 0.578-0.584
1.57- 1.58 2.1:5
5.55-5.57 5.65-5.68
-
275-521 279-515
i
"% 6(-
o
o
~"
7O
o o o
o o. o
o
--50
o o
8C
a:
-50
7O
7C 0
0
--40 ~
0
0
0
0
~.c
G --50
50
--
2000--
0
-- 60
--40 1500-
o
o
o
o
150C
~:
--
o
Lm
~jE
o
o
O OO I000--
5OO-
o
0
•
•
I
3
a
• •
•
•
@
•
•
• ••
•
o
I
I
4
5O(
I
5
6
I
I0
7
pd/p,
o
o
@
•
o
Lm
•
I
I
20
30
SH,
b
O
30
I 40
5O
60
o
Fig. 4a Experimental volumetric efficiency, mass f l o w of discharge gas and motor input vs compression ratio
Fig. 4b Experimental volumetric efficiency, mass f l o w of discharge gas and motor input vs degree of superheat
Fig. 4a Rendemem volum#trique experimental, d#bi~-masse du gaz de refou/ement et consommat/on du moteur par rapport au taux de compress~on
Fig. 4b Rendement volum6trique exp6rimenta/, d#bibmasse du gaz de refou/ement et consommat/on du moteur par rapport au degr# de surchauffe
Table 1. A sample of calculating process for t h e calculation error of t h e v o l u m e t r i c efficiency Tab/eau 1. Echant/I/on de mdthode de ca/cu/ de /'erreur sur /e rendement vo/um~rique Measured quantities
Values of the terms in (6)
Ps, MN m -2
Pd, MN m -2
Ts, K
T1, K
Q, I h -1
Lm, W
0.579
2.05
286
308
53,6
1250
2.1 51 x 10 -6 Accuracy and error
22.4
(dN ALq2 / vsAPs 2 (VsATq2
AT 1
8.702 x 10 -5
2.843 x 10 -7
7.182 x 10 -5
1.099x 10 -4
A~/v/r/v, %
(fly)e, %
A1/v, %
1.6
78.9
1.3
International Journal of Refrigeration
50 L 20
I00
MN m-2
•
0 ;76-0
o
0.572-0.591
-
-____~ } vp
l-% %,
_
/
9O-
% 30
o
~ 0"~
oc3
~
t~/.~ 7
O--
i -~'f
"
~
-
>_0
----
I
I
l
I
5
4
5
6
F
70-
MN m-2 0 0.385 0.582
Ps,
--
W
~" 8 0 -
~ "
"qv,h
o
,o
60~-
~/v
I
5
o
~=0.582 MN m-2, Ts=275 K "o..... -2 Ps=0.576-0.595 MN m , Ts=271-274 K
J
4
I
5
I
6
P~/C Fig. 5 Mass ratio of leakage and temperature risevscompression ratio
Fig. 6 Comparisonof respectivemagnitudes of factors affecting theoretical volumetric efficiency vs compression ratio
F/g. 5 Faux de masse de fuite$, et d/dvat/on de M tempdrature p a r rapport au taux de compression
Fig. 6 Compara/sondes grandeurs respect/ves des facteurs ayant une inf/uence sur /e rendement vo/umdtrique en fonction du taux de compress~on
a more exact volumetric efficiency is defined using the calculated value from measurements, (r/v)e, and adding the term to indicate the calculation accuracy as follows:
(29)
; / v = (//v)e( 1 =}zA;/v//Tv)
Bands marked I in Fig. 4a and b illustrate upper and
lower limits of the added term.
Experimental
and
calculated
results
Experimental results for the volumetric efficiency, the mass flow of the discharge gas and motor input are shown in Fig. 4a and b. In Fig. 4a it is shown that the volumetric efficiency decreases as the compression ratio or the suction pressure increase when the degree of superheat is constant. It can also be seen that the calculation error of the volumetric efficiency becomes larger as the compression ratio or the suction pressure decreases. Fig. 4b shows that both the volumetric efficiency and its calculation error are nearly independent of the degree of superheat. From the bands marked I i n Fig. 4a and b which indicate the range of calculation error of the volumetric efficiency, the calculating volumetric efficiency can be expressed with a proper characteristic curve which depends on the volumetric efficiency. In addition, an example of the calculation error of the volumetric efficiency by using (6) for a standard refrigeration cycle for air conditioning (marked * in Fig. 4a) is shown in Table 1. Table 1 shows that,
irrespective of the different experimental conditions, the measurement errors in the suction temperature, the suction pressure and the volume flow of a refrigerant have relatively the largest effects. The effects of the mass ratio of the leakage, ~W/W, and the temperature rise of the suction gas, 6T b, by the above calculating method, are shown in Fig. 5. As for the leakage, ~ W / W increases as the compression ratio and the suction pressure increase. As for the temperature rise, 6T b increases as the compression ratio becomes larger, and is nearly independent of the variations of the suction pressure. The magnitudes of the factors affecting ;/vth in (1 3), r/vp, r/vT and 6 W / W o are compared in Fig. 6. It is shown that r/vp, t/vT an(] ( ~ d / d O increase as the compression ratio becomes larger. In Fig. 6, the experimental values, r/v, are also plotted and they agree fairly well with the theoretical values, f/vth.
References
1 Bransford, E. 0., Stein, R. A, Design controJ of overcompression in rotary-vanecompressorsJourna/of Engineering for Power, TransASME (1960) 221 2 Stein, R. A., Beck, W. D., Eibling, J. A. Design for minimum leakage in rotary-vanerefrigeration compressors TransAS/-/R,4E71 1 (1965) 192 3 Shimizu, T., Nagasaku, E. Volumetric efficiency of rolling piston compressorRefrigerat/on JapaneseAssociat(on of Refrigeration (1975) 806 Kamata, Y., Tashiro, M., Inoue, M, Effects of measuring accuracies of temperaturesexerted on caJcuMtion of compressor efficiency TurbomachTneryJapan (1974) 458 Bohanon, H. R. Laboratoryfan test: error analysis Trans ASNRAE 81 1 (1976) 83
Volume 3 Number 4 July 1980 225
On a effectual des recherches thdoriques et exp~rimenta/es sur les facteurs ayant une influence sur le rendement vo/um#trique d'un compresseur ~ piston
rotatif. Les r#su/tats ont montr# qu'une tendance du rendement vo/um#trique thdorique concordait assez bien avec /a tendancelexpdrimentale et que les fuites de gaz par les jeux du piston ainsi que I'#l#vation de temperature du gaz d'aspiration nuisaient fortement au rendement vo/urn#trique. De plus, on a estimd quantitativement /'incidence des erreurs de mesure sur le rendement vo/umdtrique ca/cutd.
Theoretical and experimental investigations of t h e factors w h i c h a f f e c t t h e v o l u m e t r i c efficiency of a rolling piston compressor have been made. The results s h o w t h a t a trend in t h e theoretical v o l u m e t r i c efficiency agrees fairly w e l l w i t h the experimental one, and
t h a t gas leakage t h r o u g h clearances, as well as t h e t e m p e r a t u r e rise of the suction gas, strongly affects t h e v o l u m e t r i c efficiency. In addition, t h e e f f e c t o f m e a s u r e m e n t errors on t h e calculated v o l u m e t r i c efficiency w a s estimated q u a n t i t a t i v e l y .
Rendement volum6trique des compresseurs rotatifs et erreurs exp6rimentales
Nomenclature Cp specific heat at constant pressure g gravitational acceleration hm average heat transfer coefficient / cylinder width Lm motor input N number of motor revolutions Pb pressure at suction chamber Po pressure at compression chamber Pd discharge pressure Ps suction pressure (Pr) Prandtl's number Q reading of liquid refrigerant volume flow by flow meter r piston radius R cylinder radius SH degree of superheat Tb temperature at suction chamber -/-bBI, temperatures at suction chamber in Fig. 2 YbM Tbc J fs suction temperature Vo displacement volume by rolJing piston
As compressors for room air-conditioners, rotary compressors have recently become widely used in place of reciprocating ones because of their compactness, low noise output and high efficiency. These compressors are classified into sliding vane and rolling piston types. There is some published
W discharge mass per rotation Wo theoretical suction mass per rotation c~ revised coefficient of specific weight of flow meter 7b specific weight of gas refrigerant in suction chamber 7c specific weight of gas refrigerant in compression chamber just before gas expands in it 7/ specific weight of liquid at calibration of flow meter 7 specific weight of suction gas A error of mean square 5Pb pressure loss in suction 5Pc pressure loss in discharge 5Tb temperature rise of suction gas heated 5W leakage at some channels of gaps top clearance ratio ~/v experimental volumetric efficiency I/vth theoretical volumetric efficiency rotative angle of crankshaft specific heat ratio ~' kinematic viscosity work 1'2, on the former type, however, unti recently 3 very few papers had been published on the latter.
Faculty of Engineering, Shizuoka University, Hamamatsu, Japan and MK isat the Peripheral Devices Laboratory, Fujitsu Laboratories Ltd, Kawasaki, Japan. Paper received 8 January 1980
One criterion of performance of compressors is the volumetric efficiency, and to improve the performance this efficiency has to be elucidated. Accurate, volumetric efficiency values are difficult to obtain and, as has been pointed out 4.5, measurement errors affect the calculated efficiency. Therefore, careful analytical and experimental studies of the volumetric efficiency of a roiling piston compressor (Fig. 1), operated under various conditions, have been made.
Volume 3 Num6ro 4 Juillet 1980
01 40-7007/80/04021 9-07502.00 © 1980 IPC Business Press Ltd and IIR
TS and TY are with the Department of Mechanical Engineering,
219
Furthermore, an error analysis has also been carried out to ascertain the reliability of this volumetric efficiency.
(At/v) 2 = (c3~/Ap/) 2 +
Mass flow of the discharge gas from the compressor is less than the theoretical value based on the suction gas entering the compressor due to heating in the cylinder wall, gas leakage through channels and gaps, re-expansion of the gas in the top clearance, etc. Actual mass flow of the discharge gas, which is normally measured by a flow. meter for liquid refrigerant at the outlet of the condenser at steady state conditions, is given by:
G =~ylO
(1)
~r/v
where, Vo=~(R2-r2)/ and outlet ports).
2
4-
(5)
where the symbol A expresses an amount of uncertainty. (5) shows the effect of the measurement errors on the calculation error of the volumetric efficiency. In (3) Vo is regarded as constant and involving no error. Then, "Calculation accuracy of the volumetric efficiency' is derived from (3) and
(5):
The theoretical mass flow of the suction gas is given by using a specific weight of gas refrigerant in the suction line, y~, and is as follows: G O= 60mVoY~
(>o) 2 -,- &/,,
+
Definition of the experimental volumetric efficiency
ar/v
tG) _¢8(~yi)AP,'~ 2
(2)
/0 (~y/) AT,'~ 2
(ignoring the effect of inlet [AQ\ 2 [ d N AL~\ 2
The volumetric efficiency of the practical compressor is defined as the ratio of (1) and (2): G o~y/Q r/~- Go - 60mVoys
(3)
(6)
Error analysis of the volumetric efficiency Because of difficulties in making accurate, experimental measurements, the measuring volumetric efficiency expressed by (3) is usually calculated roughly, and shows fluctuations. Therefore, to ascertain the reliability of the experimental value, an error analysis of the experimental volumetric efficiency has been carried out. ~, in (3), is a function of the specific weight of the supercooled liquid refrigerant, Y/, entering the flow meter, as will be discussed later. Furthermore, y/is also a function of the measured pressure, P/, and temperature, 7-/, of the liquid refrigerant. Also, N depends on the measured motor input, L~, and Ys depends on the measured suction pressure, Ps, and suction temperature, Ts. These relationships are summarized as follows: o~=o~(y,) =~(P/,T/);
N=N(Lm);
y/=
~,s j -t, aPs >,~ j
(APsV+/ATs%2
=\ 77J=\7(J \W)
<7)
Theoretical volumetric efficiency Derivation of the theoretical volumetric efficiency
yI(P/,T/); (4)
~ = ~(Ps, T~) Each directly measured quantity in (4) contains. random and independent errors. Applying the Law of Propagation of Errors to (3) and taking (4) into consideration:
220
Provided that the suction gas is approximately an ideal gas throughout this analysis, the accuracy of the specific weight of the suction gas, Ys, the fifth and the sixth terms in (4), is as follows:
The volumetric efficiency is defined as the ratio of the theoretical mass flow of the suction gas and the actual mass flow of the discharge gas as mentioned earlier. The factors which affect the volumetric efficiency are: the top clearance Volume of a discharge portion (~Vo); the pressure losses in the suction and discharge ports (aP b, 8Pc); gas leakage (aW); and the temperature rise of the suction gas
(aTb).
International Journal of Refrigeration
The effects of circulation of the lubricating oil, variation in the rotational velocity, etc. are regarded as relatively small, and are disregarded in the present analysis. Theoretical suction mass per revolution is given by: Go
W° - 60-m - V°7~
(8)
considered narrow enough to be neglected: The leakages at a and b are assumed to be as a one dimensional adiabatic gas flow, as Stein et al. 1 considered, and the formula for a convergent nozzle with compressible fluid has been used. For the leakages from the gaps at a and b in Fig. 1 the following three conditions have to be taken into account as pressure rises. For gas flow up to the critical (acoustic) velocity, the mass flow is:
On the other hand, the actual mass per revolution is given by: G W = 60N = V°(1
(9)
+s)%-sVoYc-SW
Thus the theoretical volumetric efficiency is given by:
( k F/po\ (k-1)lk Ti1/2 Wl=~2g~Pb%Lt~b ) -1~
r After reaching the critical velocity, the pressure in the compression chamber, Pc, continues to rise, and the critical velocity also increases. Then, the mass flow is:
w2=F W
/~vth= d o =
I !--cS
--1
7b ~W do
~
(15a)
/
2
~(k+l)/(k-1)
\PU
(10)
J 15b)
A
while, 1c/?b is expressed as follows:
7c_(Pd+&Pc~ Ilk
(11)
Furthermore, provided that the gas refrigerant temperature at the suction Ts is raised to Tb=Ts+ST b during the suction stroke, Yb/Ts is expressed by: 7b P~-&Pb
7s
Ps
Ts
(12)
T~+&Tb
I
Therefore (10) is given by:
\
&W ~vth = ~vp~vt d o
(1 3)
C
a ~[[Pd-FSPc~
1/k
b
~]}Ps--~Pb
vp= 1-OLb s < >W -,]j> Ps(14)
Fig. la
Position of leakage paths
F/~7. la
Emp/acement des passages de fuite
Ts r/vt- Ts+ &Tb
Calculation of the internal leakage A rolling piston compressor has to minimize the gas leakage by using precisely manufactured components. However, reduction of clearances is limited by problems of lubrication, fabrication techniques, frictional losses, etc. As a result, gas leakage th,rough gaps has a rather large effect on volumetric efficiency. As shown in Fig. la and Fig. l b possible, leakage paths are as follows:a - across the edges of the dividing valve plate; b - between the piston and the cylinder wall; c and d - between the end faces of the piston and cylinder. Since c and d are normally filled with lubricating oil, these gaps are
Volume 3 Number 4 July 1980
Fig. l b Section showing theleakage paths: 1 -piston; 2-crankshaft; 3 cylinder;4 dividing vane; 5-cylinder head F/~7. I b Coupe rnontrant les passages de fuite. I - piston; 2 - vi/ebrequin; 3 cy/mdre; 4 - aube de sbparation; 5 t#te de cy/indre
221
Finally the pressure in the compression chamber, Po=Pd+SPc, becomes constant. Then, the mass flow can be taken as constant: F
/
2 \(,+w(k-~)
(1 5c)
J
Where W~, W 2 and W 3 are all mass flows per unit area.
By taking 5~ as the dimension of the leakage gap between the dividing vane and the cylinder head on each side, the leaked mass at a of Fig. 1 is as follows: ~p
t;(
=
(
~p
(Re) -
fsd~m v
-
4G
%dsrn~sV
(20)
where fs is an equivalent inlet velocity.
dOf
xw 1
xW2dO
0
receives a great deal of heat during the suction stroke, and the resulting temperature rise has a rather large effect on the volumetric efficiency. This heat is received as the gas flows between the piston and the cylinder wall. The suction route at the maximum suction volume is from A, B, M, to C in Fig. 2a, so the gas temperatures at points B, M, C are naturally different because the gas at each point is inhaled into the cylinder at different times. The temperature at the centre, M, is assumed to represent the average temperature. Next, using an average heating area Sin, an equivalent diameter of pipe dsm is expressed as 4Vo/S m, and Reynolds' number is as follows:
Using (20), Nusselt's number is given by Dittus and Boelter's formula as follows:
8c
2=
(Nu) = 0.0023 (Re)O.S (pr)O-4
A\
(16)
Op
(21)
SimilarLy, by taking 5b as the dimension of the leakage gap between the piston and the cylinder wall, the leaked mass as b in Fig. 1 is as follows: ec
wb=;(f bWld0 @ ~P
2=
(17) where co is an angular velocity of a crankshaft, and / is a width of cylinder. 0c and 8p are the rotational angles of a crankshaft when the gas flow reaches the critical velocity and when the discharge va!ve starts to open, respectively, x, the displacement of the dividing vane, can be expressed as follows:
,[1 -(R-r)
a
Fig. 2a Flow passage of suction gas F/~7.2a Passagede /'6cou/ement du gaz d'aspiration
s,n cos 0
Tcr =const.
(18)
Integration of (16) and (1 7) can be made using Simpson's rule. Then, the leakage, 5W, in (1 3) is:
aw=awo+awb Calculation of temperature
(19) rise o f s u c t i o n gas
Since the pressure inside the hermetic case of a rolling piston compressor is generally the same as the discharge value, the temperature of the cylinder becomes rather high, Therefore the suction gas
222
rbc
b Fig. 2b
Temperature rise curve of suction gas
Fig. 2b
Courbe de/'d/dvation de tempdrature du gaz d'aspbation
Revue Internationale du Froid
From (21), using thermal conductivity coefficient, 2, the average heat transfer coefficient is given as follows: hmi
(Nu)2 dsm
(22)
Provided that a temperature rise of the suction gas follows the same trend as the fluid flowing in a pipe w i t h constant wall temperature, Tc,, the temperatures at the points A, B, M and C in Fig. 2a can be reasonably assumed to change exponentially as Ts, TbB, fbM and Tbc in Fig. 2b. So the mutual relation between these temperatures is:
(To,-TbM)I(To,-Ts) = exp[ - (hmSm)/(CpG)]
(23)
Then 5Tb is given by:
5Tb= TbM-- Ts
(24)
addition, a supplementary heater was placed in the evaporator so that the cycle could be operated even in the high superheat range. Temperatures at respective points were measured by copper-constantan thermocouples, and a steady state of the cycle was confirmed by recording the pipe wall temperatures at a suction plenum and at the outlet of the compressor. Pressures were measured by Bourdon gauges. Volume flow of a liquid refrigerant, R22, was measured by a float type, area flow meter, and the input to a compressor motor was measured by a single phase, watt meter. Experiments were carried out to examine the performance of a compressor under the following two operational conditions: condensing pressure was varied while keeping the evaporating pressure and superheat constant in order to examine the effect of compression ratio and discharge pressure Qn the performance: and the degree of superheat was varied while keeping the evaporating and condensing pressure constant in order to examine the effect of suction temperature on the performance.
E x p e r i m e n t a l a p p a r a t u s and p r o c e d u r e The flow diagram of the experimental apparatus is shown in Fig. 3. The hermetic rolling piston compressor of 1 kW class was used. Constant temperature, water supply systems to the condenser and evaporator are shown as broken lines in Fig. 3. In
D e t e r m i n a t i o n of a f u n c t i o n a l f o r m used for error c a l c u l a t i o n The revised coefficient of the specific weight of the flow meter, ~, was calculated as follows: 0~= [ ( T f - - ~ / ) 7 0 / { ( ' ~ f - - 7 0 ) ~ / }
As 7f and 7o are constant, ~ is a function of the specific weight of the liquid refrigerant entering the flow meter, 7/- And 7/is the value determined by measuring the pressure, P/, and the temperature, T/, at the flow meter. Therefore, ~ and 7/become the functions used in (4). Furthermore, the degree of subcooling of the liquid refrigerant entering the flow meter has an effect on P/in this experiment, so 7/ can be treated as a function of 7-/. By means of a least squares fit the experimental data yield:
__
] 2
I
I I
I
r
I
i/ f~m ii I I 9
(25)
] 1/2
~ = 1 . 5 5 0 - 5.510 x 10-3Tl +1.246 x 10-STj 2
I
1
where 286 K < T S 3 2 8
Io
5
(26)
K
y / : 1 . 1 1 4 - 1 . 0 5 4 x 10-2T/
I I
1
I ,2 I
I
li
Fig. 3 F l o w d i a g r a m of e x p e r i m e n t a l apparatus: 1 - c o m p r e s s o r ; 2 - c o n d e n s e r ; 3 - f l o w meter; 4 - hand e x p a n s i o n valve; 5 - e v a p o r a t o r ; 6 - s u c t i o n muffler; 7 electric p o w e r ; 8 - w a t t meter; 9 - f l o w c o n t r o l valve; 10 - w a t e r pump; 11 - heater; 12 pen recorder Fig. 3 D i a g r a m m e de p r i n c i p e d e / ' a p p a r e i / e x p # r i m e n t a / . 1 - compresseur, 2 - condenseur, 3 f/uxm#tre; 4 - d#tendeur a main, 5 - dvaporateur; 6 - s//encieux d'aspirat/on; 7 - courant d/ectrique; 8 - wattm#tre; 9 - rob/net de rdgu/at/on de / ' d c o u / e m e n t ; 10 - p o m p e ~ eau, 11 - rdchauffeur,- 12 - e n r e g/streur ~ p/ume
Volume 3 Num6ro 4 Juillet 1980
- 1 . 3 3 5 x 10-5T/2
(27)
w h e r e 286 K_
- -
7.28 x 1 0-2Lm
- 2 . 0 x lO-SL~
(28)
where 790 W__
223
The calculation accuracy of the volumetric efficiency is given by substituting (7), (26), (27) and (28) into (6).
Estimation of measurement
errors
The measurement errors are evaluated from the calibrations of measuring instruments, their grades and accuracy, deviation and fluctuation of the readings, etc.
IOC
Ps, MN m-2
90--
~>
Ts,
• 0.376-0.:595 o 0.572-0.591
K
In the temperature measurements, iATJ is estimated to be 3 K, because the suction temperature was not measured for the gas refrigerant directly but on the pipe wall of the suction plenum. And IALI is evaluated to be 1 K, because the temperature of the liquid refrigerant entering the flow meter was measured directly. In the pressure measurement, IAPsl is evaluated to be 0.05 kN m -2. In the volume flow measurement of a liquid refrigerant, }AQ} is found to be 0.5 I h -1, and in the motor input measurement, }ALto} is 15 W. After the error analysis
SH, °
271- 274 5-8 285-288 5-10
• o
~
I00
7£
9O
Q584-O.386 0.578-0.584
1.57- 1.58 2.1:5
5.55-5.57 5.65-5.68
-
275-521 279-515
i
"% 6(-
o
o
~"
7O
o o o
o o. o
o
--50
o o
8C
a:
-50
7O
7C 0
0
--40 ~
0
0
0
0
~.c
G --50
50
--
2000--
0
-- 60
--40 1500-
o
o
o
o
150C
~:
--
o
Lm
~jE
o
o
O OO I000--
5OO-
o
0
•
•
I
3
a
• •
•
•
@
•
•
• ••
•
o
I
I
4
5O(
I
5
6
I
I0
7
pd/p,
o
o
@
•
o
Lm
•
I
I
20
30
SH,
b
O
30
I 40
5O
60
o
Fig. 4a Experimental volumetric efficiency, mass f l o w of discharge gas and motor input vs compression ratio
Fig. 4b Experimental volumetric efficiency, mass f l o w of discharge gas and motor input vs degree of superheat
Fig. 4a Rendemem volum#trique experimental, d#bi~-masse du gaz de refou/ement et consommat/on du moteur par rapport au taux de compress~on
Fig. 4b Rendement volum6trique exp6rimenta/, d#bibmasse du gaz de refou/ement et consommat/on du moteur par rapport au degr# de surchauffe
Table 1. A sample of calculating process for t h e calculation error of t h e v o l u m e t r i c efficiency Tab/eau 1. Echant/I/on de mdthode de ca/cu/ de /'erreur sur /e rendement vo/um~rique Measured quantities
Values of the terms in (6)
Ps, MN m -2
Pd, MN m -2
Ts, K
T1, K
Q, I h -1
Lm, W
0.579
2.05
286
308
53,6
1250
2.1 51 x 10 -6 Accuracy and error
22.4
(dN ALq2 / vsAPs 2 (VsATq2
AT 1
8.702 x 10 -5
2.843 x 10 -7
7.182 x 10 -5
1.099x 10 -4
A~/v/r/v, %
(fly)e, %
A1/v, %
1.6
78.9
1.3
International Journal of Refrigeration
50 L 20
I00
MN m-2
•
0 ;76-0
o
0.572-0.591
-
-____~ } vp
l-% %,
_
/
9O-
% 30
o
~ 0"~
oc3
~
t~/.~ 7
O--
i -~'f
"
~
-
>_0
----
I
I
l
I
5
4
5
6
F
70-
MN m-2 0 0.385 0.582
Ps,
--
W
~" 8 0 -
~ "
"qv,h
o
,o
60~-
~/v
I
5
o
~=0.582 MN m-2, Ts=275 K "o..... -2 Ps=0.576-0.595 MN m , Ts=271-274 K
J
4
I
5
I
6
P~/C Fig. 5 Mass ratio of leakage and temperature risevscompression ratio
Fig. 6 Comparisonof respectivemagnitudes of factors affecting theoretical volumetric efficiency vs compression ratio
F/g. 5 Faux de masse de fuite$, et d/dvat/on de M tempdrature p a r rapport au taux de compression
Fig. 6 Compara/sondes grandeurs respect/ves des facteurs ayant une inf/uence sur /e rendement vo/umdtrique en fonction du taux de compress~on
a more exact volumetric efficiency is defined using the calculated value from measurements, (r/v)e, and adding the term to indicate the calculation accuracy as follows:
(29)
; / v = (//v)e( 1 =}zA;/v//Tv)
Bands marked I in Fig. 4a and b illustrate upper and
lower limits of the added term.
Experimental
and
calculated
results
Experimental results for the volumetric efficiency, the mass flow of the discharge gas and motor input are shown in Fig. 4a and b. In Fig. 4a it is shown that the volumetric efficiency decreases as the compression ratio or the suction pressure increase when the degree of superheat is constant. It can also be seen that the calculation error of the volumetric efficiency becomes larger as the compression ratio or the suction pressure decreases. Fig. 4b shows that both the volumetric efficiency and its calculation error are nearly independent of the degree of superheat. From the bands marked I i n Fig. 4a and b which indicate the range of calculation error of the volumetric efficiency, the calculating volumetric efficiency can be expressed with a proper characteristic curve which depends on the volumetric efficiency. In addition, an example of the calculation error of the volumetric efficiency by using (6) for a standard refrigeration cycle for air conditioning (marked * in Fig. 4a) is shown in Table 1. Table 1 shows that,
irrespective of the different experimental conditions, the measurement errors in the suction temperature, the suction pressure and the volume flow of a refrigerant have relatively the largest effects. The effects of the mass ratio of the leakage, ~W/W, and the temperature rise of the suction gas, 6T b, by the above calculating method, are shown in Fig. 5. As for the leakage, ~ W / W increases as the compression ratio and the suction pressure increase. As for the temperature rise, 6T b increases as the compression ratio becomes larger, and is nearly independent of the variations of the suction pressure. The magnitudes of the factors affecting ;/vth in (1 3), r/vp, r/vT and 6 W / W o are compared in Fig. 6. It is shown that r/vp, t/vT an(] ( ~ d / d O increase as the compression ratio becomes larger. In Fig. 6, the experimental values, r/v, are also plotted and they agree fairly well with the theoretical values, f/vth.
References
1 Bransford, E. 0., Stein, R. A, Design controJ of overcompression in rotary-vanecompressorsJourna/of Engineering for Power, TransASME (1960) 221 2 Stein, R. A., Beck, W. D., Eibling, J. A. Design for minimum leakage in rotary-vanerefrigeration compressors TransAS/-/R,4E71 1 (1965) 192 3 Shimizu, T., Nagasaku, E. Volumetric efficiency of rolling piston compressorRefrigerat/on JapaneseAssociat(on of Refrigeration (1975) 806 Kamata, Y., Tashiro, M., Inoue, M, Effects of measuring accuracies of temperaturesexerted on caJcuMtion of compressor efficiency TurbomachTneryJapan (1974) 458 Bohanon, H. R. Laboratoryfan test: error analysis Trans ASNRAE 81 1 (1976) 83
Volume 3 Number 4 July 1980 225