Purdue University
Purdue e-Pubs International Compressor Engineering Conference
School of Mechanical Engineering
2000
Calculations for Compression Efficiency Caused by Heat Transfer in Compact Rotary Compressors N. Ishii Osaka Electro-Communication University
N. Morita Osaka Electro-Communication University
M. Kurimoto Matsushita Electric Industrial Co.
S. Yamamoto Matsushita Electric Industrial Co.
K. Sano Matsushita Electric Industrial Co.
Follow this and additional works at: http://docs.lib.purdue.edu/icec Ishii, N.; Morita, N.; Kurimoto, M.; Yamamoto, S.; and Sano, K., "Calculations for Compression Efficiency Caused by Heat Transfer in Compact Rotary Compressors" (2000). International Compressor Engineering Conference. Paper 1423. http://docs.lib.purdue.edu/icec/1423
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CALCULATIONS FOR COMPRESSION EFFICIENCY CAUSED BY HEAT TRANSFER IN COMPACT ROTARY COMPRESSORS by Noriaki Ishiil, Noriyuki Morita2 , Mitsuru Kurimoto3, Shuichi Yamamoto4 , Kiyoshi Sano5 and Kiyoshi Sawai6 1 Professor,
Faculty of Engineering, Osaka Electro-Communication University, 18-8, Neyagawa City, Osaka 572-8530, JAPAN Tel: +81-728-20-4561, Fax~ +81-728-20-4577, E-mail:
[email protected] 2 Graduate Student, Osaka Electro-Communication University 3 Engineer, Compressor Division, 4 Manager, Compressor Division, 5 General Manager, Air Conditioning Research Laboratory, 6 Manager, Air Conditioning Research Laboratory, Matsushita Electric Industrial Co., Ltd. (Panasonic) 3-1-1 Nojihigashi, 2 Chome, Kusatsu City, Shiga 525-8520, JAPAN Tel; +81-77-567-9801, Fax; +81-77-561-3201 ABSTRACT
This paper presents calculations for the compression efficiency of the rolling-piston type rotary compressors, which is significantly determined by the heat transfer from the high temperature circumstances outside the compression mechanism into the compression chamber, through the thrust plates. The heat transfer was calculated by the forced turbulent convection heat transfer theory, and the compressed gas pressure was calculated to reveal the compression efficiency. Calculations were made for a representative combination of the major dimensions, such as the cylinder inner radius , the rolling-piston radius and the cylinder depth. INTRODUCTION
The suction volume of rolling-piston type rotary compressors is determined on the basis of the major dimensions, such as the rolling-piston diameter, the cylinder depth and the cylinder bore. It becomes clear that there are many combinations of the major dimensions that yield a rolling-piston rotary compressor with the same suction volume. It must be quite significant to examine which combination is the best in performance for the compressors. The frictional losses and the refrigerant gas leakage change, depending upon the combination of the major dimensions, thus significantly affecting upon the mechanical and volumetric efficiencies. Based on such viewpoint, the computer simulations were made for the optimum combination of the major dimensions, which yields the highest performance in mechanical and volumetric efficiencies, by Ishii et al. (1990 Ill; 1998 /2/). Ishii et al. (1990 /3/; 1992 /4/; 1994 /5/; 1996 161; 1996 17/) has made similar calculations for the scroll compressors. Of quite significance in addition to the mechanical and volumetric efficiencies is the compression efficiency that is caused by the heat losses. Especially the heat loss caused by the heat transfer from the high temperature circumstances outside the compression mechanism into the compression chamber, through the thrust plates, is quite significant. The cylinder wall is fairly thick, compared with the thrust plates. Therefore, it was assumed that the heat transfer through the cylinder wall is less significant. The present paper assumes the flow pattern of the refrigerant gas in the compressed chamber to be a well-developed turbulent flow. Thereby the heat transfer trough the thrust plates was calculated by the forced turbulent convection heat transfer theory for a flat plate, which is based on the Colburn's analogy. Caused by the heat
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transfer into the compression chamber, the compressed gas pressure significantly increases, thus lowering the compression efficiency. The present study summarizes the basic theory to calculate the heat transfer coefficient, the heat transfer quantity and the pressure, first. Secondly the calculated results for the compression efficiency in addition to for the heat transfer and the resulting pressure is presented. The calculations are made for a representative combination of the major dimensions, such as the rolling-piston diameter, the cylinder depth and the cylinder bore. The calculation method presented in this study is quite significant for the next study by Ishii et al. (2000 /8/), where the net efficiency of the rotary compressors is ultimately calculated for its optimal design.
HEAT TRANSFER IN ROTARY COMPRESSORS A cross-sectional view of the rolling-piston rotary compressor is shown in Figure la, cylinder inner radius is represented by R and the rolling piston radius is by r. The the where suction volume Vs is given by V s = 1t (R2 - r2) L
or
_H_ -
V5/R 2
-
1t{ 1-{r1R)2}
'
(1)
where H is the cylinder depth. The characteristic curve for combinations of the major dimensions, R, r and L, is shown in Figure 1b, where the ordinate is the reduced cylinder depth.
Case(l) : r/R.=O. 5
(a) Cross-sectional view
Case(2) : r/R.=O. 7
4.0 . - - - - - - - - - - - , - - - - - ,
Case(3) : r/R.=O. 8
0
Top view 0.5 r/R
(b) Characteristic curves
Side view
1.0
(c) Various design and heat transfer
Figure 1. Rolling-piston rotary compressor and heat transfer through the thrust plates. Fifteenth International Compressor Engineering Conference at Purdue University, West Lafayette, IN, USA- July 25-28, 2000
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Schematic explanation for 3 cases of combination is given in Figure lc, where the suction volume was fixed at 10.26 cc and the cylinder radius at 20 mm. As the piston radius r increases, the suction area between the cylinder and the rolling piston decreases, thus increasing the cylinder depth L. Of quite significance is that the heat transfer quantity changes depending upon the combinations. As shown by the white arrows in Figure lc, Case (1) has a large heat-transfer area, thus inducing a large heat transfer into the compression chamber, through the thrust plates. Case (2) induces a medium heat transfer, and Case (3) induces a small. Thus, the compression efficiency caused by the heat transfer loss definitely changes, depending upon the combination of the major dimensions.
BASIC THEORY Heat Transfer Coefficient and Heat Transfer Rate The rolling piston forces the refrigerant gas in the compression chamber to move relative to the thrust plates. Therefore, the heat transfer from the thrust plates into the L refrigerant gas should be treated as the forced convection heat transfer from the heated flat Thrust plate plate. The problem is how the pattern of the flow in the compression chamber consists, here. Piston Blade Regarding the flow pattern in the compression chamber, few studies have been presented, and one may assume that the refrigerant gas compressed at a high speed results in a well-developed turbulent flow pattern, as shown in Figure 2. The representative length of the heat transfer area Figure 2. Well-developed turbulent flow pattern is indicated by L, and the main flow velocity in the compression chamber. is by Ua,. The heat transfer coefficient can be derived from the Colburn's analogy: N u =9:u 2 ..'e
p} r '
(2)
which prescribes the relation between the skin friction coefficient for the velocity boundary layer, C f• and the Nusselt number for the thermal boundary layer, N u· The Reynolds number is represented by Re and the Prandtl number is by P r· N u, Re and P rare respectively defined by _ _ax
Nu =
A
=
'Re-
u""x
v
=
'Pr-
J.LCp ' A
( ) 3
where a. is the local heat transfer coefficient, A. is the thermal conductivity, v is the kinematic viscosity, J.L is the viscosity coefficient and ~ is the specific heat at constant pressure. As shown in Figure 2, the distance from the piston Is represented by x. The Nusselt number represents the heat quantity ratio of the heat transfer to the heat conduction. The Prandtl number represents the ratio of the kinematic viscosity to the thermal
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diffusivity defined by A/pep (p: specific mass), which takes nearly on a constant value for gas, independent of temperature and pressure. It has been well studied that the Colburn's analogy is valid for the fluid of the Prandtl number from 0.5 to 50, even if the fluid is under a welldeveloped turbulent flow. If the refrigerant gas flow on the thrust plates is so well-developed turbulent that the Reynolds number is from 5 X 1cY to 107, the skin friction coefficient C r is empirically given by _l
Ct = 0.0592 Re
(see Johnson & Rubesin, 1948/9/)
5 •
(4)
Substituting this expression into the Colburn's analogy, the Nusselt number is given by 4_
_l
Nu = 0.0296 R 5 P e
r
3
or
u
)4-
1
a.= 0.0296 A (~ 5pJ X -1/5.
(5)
The average heat transfer coefficient over the thrust plate with the length of L , a , is derived as follows:
(6)
where the mean Reynolds number ReL is defined by
ucxL
Rec=-. v
(7)
The heat transfer from the heated solid surface into the fluid is essentially subject to the Newton's law of cooling. Representing the thrust plate temperature by Tw and the compressed gas temperature by T, the heat transfer rate Q is given by Q=a.(Tw- T) S (8) where S represents the heat transfer area, as shown in Figure 1a.
Compressed Gas Pressure The heat quantity which transfers into the compressed gas during a small time of dt, is given by Qdt. Therefore, the first law of thermodynamics gives the following equation: Qdt = G CvdT + pdV ,
(9)
where G is the gas mass, cv is the constant-volume specific heat, dT is the temperature increase, p is the pressure and dV is the volume increase. The right-hand side first term of (9) represents the internal energy increase. The equation of state is given by pV = GRT ,
(10)
from which the following relation can be derived.
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GdT = j_ (Vdp + pdV)- TdG ,
(11)
R
where R is the gas constant. Substituting (11) into (9); the pressure increase dp is given by dp = j_ {( 1C- 1)Qdt- 1C pdV + RTdG} ,
(12)
v
where K represents the specific heat ratio. dT =
GR
From (11), the temperature increase dT is given by
(Vdp + pdV)-,... dG .
(13)
u
Therefore, pressure p(t+dt) and temperature T(t+dt) at time (t+dt) can be calculated from those at time t, as follows: p(,t+dt)=p(t)+dp; T(t+dt)==T(t)+dT.
(14)
CALCULATED RESULTS Representative Length and Main Flow Velocity Of quite difficulty is to determine the representative length of the heat transfer area, L, and the main flow velocity of the compressed gas, 11oo , which were shown in figure 2. Many discussions could be made for the difficulty, and the present study roughly assumes that the representative length L is given by the average of the cylinder inner length and the piston outer length:
L == (2n: - e) (R+r) /2 ,
(15)
where 8 is the rotating angle of the piston, as shown in Figure 1a. Additionally, since the tangential velocity of the compressed gas changes from 2xfR (f: crankshaft rotating speed Hz) at its maximum value to zero, the present study roughly assumes that the main flow velocity 11oo is given by its average: UO()
== 1t f R .
(16)
Major Specifications Regarding the heat transfer, the viscosity coefficient, the kinematic viscosity, the heat conductivity, the constant-pressure and constant-volume specific heats are shown in Figure 3, which vary depending upon the pressure and the temperature of the thermal film near the thrust plate. The thermal film temperature Tr is given by the average of the thrust plate temperature Tw and the compressed gas temperature T. Other specifications are shown in Table 1. The refrigerant gas is compressed from the suction pressure Ps with the temperature Ts of 18°C, to the discharge pressure Pd of 2.05 MPa. The crankshaft rotating speed f was assumed 56.4 Hz. The thrust plate temperature Tw was assumed 180 °C . The leakage height was assumed 10 mm both for the radial and axial clearances. Fifteenth International Compressor Engineering Conference at Purdue University, West Lafayette, IN, USA- July 25-28, 2000
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17 ~
. "'
...!!!. E
liD
D.
'b
b
16
~
X
~
:::t. 15
0.5
1.5
2
~5
15
1.5
2
(b) Kinematic viscosity
(a) Viscosity coefficient
2
P [MPa]
P[MPa]
P[MPa]
(c) Heat conductivity
1300
Table I Specifications for catculations
1200
.
...,
g1ooo
Ps Ts Tw
~
0
56.4 Hz 2.05 MPa 0.62 MPa 18 oc 180 oc 10 JJ.m
f pd
~1100
900
1.5
2
P[MPa]
(d) constant-pressure specific heat
0.5
1.5
2
!3_
P[MPa]
(e) Constant-volume specific heat
Figure 3. Specifications for heat transfer. Temperature, Pressure and Compression Efficiency Here assume the suction volume v. to be a small value of 2.5 cm3 , where the cylinder radius R is 22 mm, the piston radius r is 14 mm and the cylinder height H is 2.8mm. Calculated results are shown in Figure 4, in which the abscissa is the crankshaft-rotating angle e for its one revolution. As shown in Diagram (a), the Prandtl number Pr is from 0.8 to 0.9 and the Reynolds number ~ is larger than 5 X 1fY except for near the end of compression. Thus it is ensured that the Colburn's analogy given by (2) is well valid and the skin friction coefficient Cf given by (4) is also valid. When the crankshaft rotation exceeds about 200 degrees, the compressed gas starts to be discharged to make a sudden change in Re.
a,
The Nusselt number Nu and the average heat transfer coefficient given by (5), are shown in Diagram (b). As the gas is compressed, the specific mass increases and the heat becomes easier to transfer into the gas, thus rapidly increasing the heat transfer coefficient. Nu naturally exhibits the similar curve as for ~Diagram (c) shows the heat transfer area S on the thrust plates, which decreases with the crankshaft rotation. Depending significantly upon this heat transfer area, the heat transfer rate Q given by (9) decreases, as shown in Diagram (d). As a result, the temperature T and the pressure p of the compressed gas, given by (13) to (15), can be calculated, as shown in Diagrams (e) and (f), where the solid line is for with the heat transfer and the broken line is for without the heat transfer. Due to the heat transfer, both the Fifteenth International Compressor Engineering Conference at Purdue University, West Lafayette, IN, USA- July 25-28, 2000
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temperature and the pressure significantly show a higher value, compared with those for without the heat transfer. The compression efficiency is defined by the ratio of compression energy for without the heat transfer to with the heat transfer. Diagram (f) resulted in the compression efficiency of 93.1%. 1-5.-----------, Re
0.1.-------------,
~
3000
..
/ '~\
~
./'
~
......
....-""/
----
I~
X
~
\. \
/,/'''
~2000
"'o
/Nu
. \
E
\
a
:I
z
;;; 0.05 \
\\
1000
0
100
200
300
200
300
0
100
(b) Nusselt number and Heat transfer coefficient
(a) Prandtl number and Reynolds number
200
300
8 [degree]
9 (degree]
fJ[degree]
(c) Heat transfer area
~--------------, I
400
0.005
~ 0
,---------I
1-
350
0.003
0
___ ......../ '
300
0.001
100
9 [~]
(d) Heat transfer rate
o
/
too
I
1
/\
without heat transfer
/
..... -;....."'// 200
9 (degree]
300
(e) Gas temperature
100
I I I I I I
/\
without heat transfer
200
300
9 [degree]
(f) Gas pressure
Fiure 4. Temperature and pressure calculated for the suction volume of 2.5cm3 , the cylinder radius of 22mm and the piston radius of 14mm.
CONCLUSIONS The present study introduced a rough treatment for the gas flow pattern in the compression chamber, the main flow velocity and the representative flow length on the heated thrust plates. Thereby, however, the heat transfer into the compression chamber, through the thrust plates, could be first calculated, thus making it possible to calculate the compression efficiency. The representative calculations for the suction volume of 2.5 cm3 with the cylinder radius of 22 mm and the piston radius of 14 mm were presented to prove the rough treatment to be fairly reasonable. Furthermore, it was shown that the reasonable value of 93.1 %could be obtained for the compression efficiency. The method of calculation for the compression efficiency, here presented in this study, could be effectively used in the next study for the net efficiency simulations for the optimal performance of the rotary compressors.
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ACKNOWLEDGEMENTS The authors would like to express their sincere gratitude to Mr. Tomio Kawabe, President of Air Conditioning Department, Mr. Ikuo Miyamoto, President of the Compressor Department, and Dr. Nobuo Sonoda, Head of the Air Conditioning Research Laboratory, Matsushita Electric Industrial Co. Ltd., for their good understandings in carrying out this work and their permission to publish the present results. The authors would like to express their sincere thanks to Mr. Naoyuki Wakiyarna and Mr. Masayuki Seto for their help in performing computer simulations.
REFERENCES (1) Ishii, N. et al., Optimum Combination of Dimensions for High Mechanical Efficiency of a Rolling-Piston Rotary Compressor, Proc. International Compressor Engineering Conference at Purdue, July, 1990, pp.418-424. (2) Ishii, N. et al., A Fundamental Optimum Design For High Mechanical and Volumetric Efficiency of Compact Rotary Compressors, Proc. International Compressor Engineering Conference at Purdue, July, 1998, pp.649-654. (3) Ishii, N. et al., Mechanical Efficiency of a Variable Speed Scroll Compressor, Proc. International Compressor Engineering Conference at Purdue, Vol.1, July, 1990, pp.192-199. (4) Ishii, N. et al., Optimum Combination of Parameters for High Mechanical Efficiency of a Scroll Compressor, Proc. International Compressor Engineering Conference at Purdue, July, 1992, pp.l18a1-118a8. (5) Ishii, N. et al., A Study on High Mechanical Efficiency of a Scroll Compressor with Fixed Cylinder Diameter, Proc. International Compressor Engineering Conference at Purdue, July, 1994, pp.677-682. (6) Ishii, N. et al., Refrigerant Leakage Flow Evaluation for Scroll Compressors, Proc. International Compressor Engineering Conference at Purdue, July, 1996, pp. 633-638. (7) Ishii, N. et al., A Fundamental Optimum Design For High Mechanical And Compact Scroll Compressors, Proc. International Volumetric Efficiency Of Compressor Engineering Conference at Purdue, July, 1996, pp. 639-644. (8) Ishii, N. et al., Net Efficiency Simulations of Compact Rotary Compressors for Its Optimal Performance, Proc. International Compressor Engineering Conference at Purdue, July, 2000. (9) Johnson, H.A & Rubesin, M.W., Trans. ASME, 71, 1948, p.449.
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