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Research of leakage characteristics of single screw refrigeration compressors with the Multicolumn Envelope Meshing Pair
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Research of leakage characteristics of single screw refrigeration compressors with the Multicolumn Envelope Meshing Pair Zengli Wang, Zhan Liu, Weifeng Wu, Quanke Feng* School of Energy and Power Engineering, Xi'an Jiaotong University, Xi'an 710049, China
article info
abstract
Article history:
The Multicolumn Envelope Meshing Pair (MEMP) was proposed and has been applied for the
Received 29 May 2014
single screw refrigeration compressor (SSRC) to reduce the wear of the meshing pair. However,
Received in revised form
the geometric model shows its changed shape of the leakage paths compared with the existing
2 September 2014
straight line envelope meshing pair (LEMP). It is necessary to research the leakage character-
Accepted 6 September 2014
istics of the SSRC with MEMP and LEMP, to evaluate the value of the proposed MEMP. In this
Available online 28 September 2014
paper, the geometric model of the leakage paths between star-wheel and screw rotor is established. A two-phase leakage mathematical model for gas-oil flow is presented to predict
Keywords:
the gas leakage rate of the SSRC with MEMP. The experiments of the performance of a SSRC
Leakage characteristic
with MEMP were conducted to verify the leakage mathematical model. Obtained results show
Single screw refrigeration
the leakage of the SSRC with MEMP is a little bit smaller than that of the SSRC with LEMP.
compressor
© 2014 Elsevier Ltd and IIR. All rights reserved.
Multicolumn Envelope Meshing Pair Geometric model
ristiques de fuite de compresseurs Recherche des caracte monovis avec couple satellite-rotor d'enveloppe frigorifiques a colonne multiple a ristiques de fuite ; Compresseur frigorifique a monovis ; Couple satellite-rotor d'enveloppe de colonne multiple ; Mots-cles : Caracte le ge ome trique Mode
1.
Introduction
Single screw refrigeration compressors (SSRC) developed by Zimmern in the 1960s (Zimmern and Patel, 1972) are widely used in air compression systems and refrigeration systems
* Corresponding author. Tel./fax: þ86 029 82663783. E-mail address: [email protected] (Z. Wang). http://dx.doi.org/10.1016/j.ijrefrig.2014.09.005 0140-7007/© 2014 Elsevier Ltd and IIR. All rights reserved.
(Wang, 1978). The performance of a newly produced SSRC is superior to that of the single piston compressor and the screw compressor (Bein, 1991; Zhang, 2007), but the discharge capacity decreases sharply several hundred hours after initial operation (Zimmern, 1990). This is because that the starwheel tooth flank of the SSRC wears rapidly and the
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Nomenclature u P A d b l Dp U m_ x S d/D h
Height of the envelope column (m) Teeth number ratio between the star-wheel and the screw Central distance between the star-wheel and the screw rotor (m) Diameter of the envelope column (m) Width of the leakage path (m) Length of the star-wheel tooth meshing into the screw groove (m) Pressure difference (Pa) Wall velocity (m s1) Leakage flow rate of the mixture (kg s1) Dryness of the mixture Area of the leakage path (m2) Diameter ratio of the orifice to pipe Axial leakage clearance (m)
clearance between the star-wheel and the screw increases. In the existing design of the SSRC, the tooth flank contacts with the groove flank at its sharp straight edge and wore at this sharp straight at first. Structure of the tooth and the groove meshing pair of existing SSRC is called as with straight line envelope meshing pair (LEMP). In order to enhance the wearresistance of the meshing pair, its profile must be improved. On the basis of column envelope profile proposed by Zimmern and multi-straight line envelope profile proposed by Feng (Feng et al., 2005), a new Multicolumn Envelope Meshing Pair MEMP was deduced by our group in 2009 (Wu and Feng, 2009). This MEMP could significantly improve the wear resistance of the compressor (Li et al., 2013). However, leakage path shapes of the LEMP and the MEMP compared by Wu show that the leakage paths are changed (Wu, 2010). Through the qualitative analysis, the leakage characteristics of the leakage paths between the star-wheel and the screw are very different for the SSRC with LEMP and MEMP. Further more, the leakage characteristics are a very important features in the process of performance evaluation of the SSRC. Thus, it is necessary to investigate the leakage characteristic of the compressor with the MEMP. Several scholars had investigated the leakage characteristics of the SSRC with the LEMP (Jin and Lin, 1986; Li, 1994; Wu, 1996; Zhou, 1999). In there study, the parallel plate model and the wedge plate model were used to calculate the leakage rate. Because shapes of the leakage paths of the SSRC with MEMP are very different from the existing design meshing pair LEMP, the leakage characteristics of this new meshing pair MEMP must be investigated and to evaluate the performance of the compressor with the MEMP is good or bad. In this paper, the geometrical model of the leakage path in the SSRC with MEMP is established. A two-phase leakage mathematical model for gas-oil flow is presented to predict the gas leakage rate through the leakage paths of the SSRC. In order to verify the model, an oil-injection SSRC with MEMP has been developed and its performance has been tested by the experiment. Obtained results are compared with that of the SSRC with LEMP. It is found that the leakage rate of the SSRC with MEMP is smaller than that of a single screw refrigeration compressor with LEMP.
Subscripts g Gas l Lubricating oil m Mixture i Number of the envelope column Greeks q a b x m ε k l r
Envelope angle (rad) Star-wheel rotation angle (rad) Inclination angle of the column, (rad) Circumferential Angle of the boundary point. (rad) Viscosity of the fluid (Pa.s) Expansion coefficient Flow coefficient Correction coefficient Density of the flow (kg m3)
2.
Leakage path geometrical model
2.1.
The feature of the leakage path with MEMP
2.1.1.
Leakage path
As show in Fig. 1, there are 9 leakage paths in the SSRC. Path 1 is the clearance between the star-wheel tooth top and the bottom of the screw groove. Path 2 and path 4 are the axial clearance between the star-wheel tooth flank and the screw groove, path 3 and path 5 are the radial clearance between the star-wheel tooth flank and the screw groove. Path 6 is the clearance between the upper surface of the star-wheel tooth and the casing. Path 7 and path 8 are the clearance between the outer surface of the screw and the inside surface of the casing. Path 9 is the clearance between the outer surface of the screw and the inside surface of the casing at the discharge end. Leakage rate through the leakage path is affected by the area and the shape of the clearance. So the leakage path shape should be discussed before the leakage analysis. The
Fig. 1 e Leakage paths of the SSRC. This figure shows all of the leakage paths of the single screw refrigeration compressor.
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Fig. 2 e Leakage path shape. Shows the leakage path shape between the tooth flank and the screw groove in the SSRC with MEMP.
Fig. 3 e Leakage path shape of the SSRC with MEMP. The star-wheel tooth flank which influences the geometrical model of the leakage path between the tooth flank and the screw groove is shown in Fig. 3.
qualitative comparative analysis presented by Wu (Wu, 2010) showed that only the axial and radial clearance between the tooth flank and the screw groove in the SSRC with MEMP was quite different from that in the SSRC with LEMP. So the leakage characteristics of these axial and radial paths need to be analyzed in this study. Fig. 2 shows the leakage path shape between the tooth flank and the screw groove in the SSRC with MEMP (Fig. 2(a)) and with LEMP (Fig. 2(b)). As shown in this figure the radial leakage path area in the SSRC with MEMP is larger than that with LEMP, and the axial leakage path shapes are different from each other. But the leakage characteristics of the axial and radial paths in the SSRC with MEMP have not been studied. Thus it is necessary to investigate the geometric characteristics of axial clearance (path 2 and path 4) and the radial clearance (path 3 and path 5) firstly.
2.1.2.
Star-wheel tooth flank
The star-wheel tooth flank which influences the geometrical model of the leakage path between the tooth flank and the screw groove is shown in Fig. 3. The figure shows there are multiple column segments in the tooth flank. Along the direction B shown in Fig. 3(a), we divide the star-wheel tooth flank into the left tooth flank and the right tooth flank. Take the left tooth flank for example there are two column segments near the root position of the tooth flank, and one column segment near the tip position of the tooth flank. Between
envelope column and surface of the tooth is the tangent between them. In coordinate plane S1 of Fig. 3(b), the envelope column center position at the root position of the tooth flank and the inclination angle of the column axis are shown in Table 1. The inclination angle is the angle between the axis of the column and the Z axis which is perpendicular to the XeY plane and goes through the origin S1. Take the left tooth flank for example, the tooth flank shape in AeA section is shown in Fig. 3(b). In which the envelope column Ob1 is the benchmark column with a circular arc section and the rest envelope columns are ellipse sections. In the u height section of the benchmark column, the envelope column center positions are: 8 < Li1 ¼ Lr Benchmark column : Mi1 ¼ Mr : Ki1 ¼ Kr
(1)
Table 1 e The parameters of the envelope columns. The envelope column center position at the root position of the tooth flank and the inclination angle of the column are shown in Table 1 Envelope column Benchmark column The second envelope column
X axis
Y axis
Z axis
Angle
Lr Lri
Mr Mri
Kr Kri
0 bri
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8 Li2 ðuÞ ¼ Lri u tan bri > > < Mi2 ¼ Mri The second envelope column : K ¼ Kri > > : i2 bi2 ¼ bri
2.2.
at any star-wheel rotation angle when the height of the envelope column equal to ui is expressed as: (2)
Tooth flank profile analysis
See the Fig. 3(b), the tooth flank profile in this section is composed by five curves: the tangent 1e2 between the second envelope column and the upper surface of the tooth, elliptic arc 2e3, tangent 3e4 between envelope columns, circular arc 4e5 and the tangent 5e6 between benchmark column and the lower surface of the tooth. To analyze tooth flank profile and set up geometry model of the leakage path, the equations of the above curves and the spiral groove flank profile in the same section at any tooth flank height need to be got. The circular arc 4e5 is the intersecting line of the benchmark column and the Section AeA. Its parametric equation in coordinate plane S1 is that: 8 d > > < x ¼ Li1 þ cos f 2 (3) > > : y ¼ Mi1 þ d sin f 2 The parametric equation of the intersecting line of the second envelope column and the Section AeA (elliptic arc 2e3) is that: x ¼ Li2 þ d cos 4=ð2 cos bi2 Þ (4) y ¼ Mi2 þ d sin 4=2 Tangent 3e4 is the common tangent between the second envelope column and the benchmark column. By the Equations (3) and (4) the tangent of the elliptic arc 2e3 at point 3 and the tangent of the circular arc 4e5 at point 4 are shown as following: y0 Mi2 2 ½x Li2 þ 2 ðy Mi2 Þ ¼ 1 d d 2 cos bi2 2 ( x0 ¼ Li2 þ d cos 40 =ð2 cos bi2 Þ
1 ðui þ Ki cos bi Þ Li sin bi PMi sinðbi þ aÞ qða; ui Þ ¼ arctan P A Ki cos a Li sin a ui cosðbi þ aÞ
(8)
where a is the star-wheel rotation angle, ui is the height of the envelope column, bi is the inclination angle of the column, (Li,Ki,Mi) are the envelope column center position, P is the teeth number ratio between the star-wheel tooth and the screw groove usually chosen as 11/6, A is the central distance between the star-wheel and the screw rotor. Simultaneous Equations (5)e(8), the maximum envelope angle qmax(u) and the tangent between the second envelope column and the upper surface of the tooth (tangent 1e2) at any tooth height will be solved. The tangent between the envelope column and the lower surface of the tooth in the right tooth flank can also be obtained by using the same method. The tangent between benchmark column and the lower surface of the tooth (tangent 5e6) is the tangent at the contact point when the envelope Angle is the minimum. The contact point with the minimum envelope Angle is usually located at the position that the star-wheel tooth meshing out of the screw groove (Wu, 2010). The projection of the straight line which passes the contact point M and perpendiculars to the axis of the screw on the neutral surface of meshing pair is li1. The projection of the straight line passing the contact point M and perpendicular to the axis of the star-wheel on the surface which contains the axis of the star-wheel and perpendiculars to the axis of the screw is li2. As shown in Fig. 4, when li1 þ li2 ¼ A, the contact point is on the position with the minimum envelope Angle.
x0 Li2
(5)
y0 ¼ Mi2 þ d sin 40 =2 d d cos f0 sin f0 2 2 2 ðx Li1 Þ þ 2 ðy Mi1 Þ ¼ 1 d d 2 2
(6)
Simultaneous Equations (5) and (6) and make the coefficients equal to each other respectively, then the Angles 40 and f0 will be solved. Bringing 40 and f0 into the above formulas, the equation of tangent 3e4 will be obtained. The tangent between the second envelope column and the upper surface of the tooth (tangent 1e2) is the tangent at the contact point when the envelope Angle is the maximum. At that moment, the corresponding height of this contact point is: ui ðuÞ ¼
u d ± cos q tan bi cos bi 2
(7)
where q is the envelope angle, þ is used for the right tooth flank and e is used for the left tooth flank. The envelope Angle
Fig. 4 e The Geometric position relationship of the contact point. In this figure, the Geometric position relationship of the contact point was given out.
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d A ¼ Ki cos a þ Li sin a þ ui cosða þ bi Þ± cos q sinða þ bi Þ 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 d þ R22 Mi ± sin q 2
process. In the analysis of the axial and radial leakage characteristics for the SSRC with MEMP that follows, the following assumptions have been employed: (9)
In this equation þ is used for the right tooth flank and e is used for the left tooth flank. According to Equations (7)e(9), the minimum envelope Angle qmin(u) and the tangent between the benchmark column and the lower surface of the tooth (tangent 5e6) at any tooth height can be solved. The same method can be used to obtain the tangent between the envelope column and the upper surface of the tooth in the right tooth flank.
2.3.
Leakage path area
Fig. 2(a) shows the shape of the leakage path between the tooth flank and the screw groove, the shaded area is the radial leakage path area. In order to calculate this area, the tooth flank profile and the screw groove flank profile should be obtained. Through analyzing the tooth flank profile, the tooth flank profile equation according to the different height in Y direction can be calculated separately. The coordinates of the boundary points along the Y direction are as follows: d (10) ti ðuÞ ¼ Mi þ sin xi 2 where u is the tooth height, d is the diameter of the envelope column, xi is the circumferential Angle of the boundary point. Using the coordinates of the boundary points and the Equations (3)e(6), the tooth flank profile x1(t,u) can be obtained. In view of the tooth flank thickness is much smaller than the screw groove length the intersection curve of screw groove can be set as the tangent of the envelope column in the contact point. The tangent can be calculated by the Equations (5) and (6). Considering the existence of the meshing clearance, the equation of the screw groove flank profile x2(t,u,a) can be received by moving the tangent of the contact point outward a meshing clearance d. According to the above calculation, the radial leakage path area and the axial leakage clearance value can be easily calculated. The selection of plus or minus in these equations is the same as Equation (7). Z0 ðx1 ðt; uÞ x2 ðt; u; aÞÞdt
(11)
hðt; u; aÞ ¼ ±ðx1 ðt; uÞ x2 ðt; u; aÞÞcosðqða; uÞÞ
(12)
SðaÞ ¼ ± tu
3.
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(1) The influences of the body-forces of gas are negligible. (2) Gas-oil mixture flow through the leakage paths is taken to be isentropic and compressible since the velocity is high. (3) The gas-oil mixture is homogeneous in the axial and radial leakage paths.
3.1.
The axial leakage model
Specific to the axial leakage path, the shapes of the leakage path and the wall motion feature are similar to the radial leakage path in rotary compressors, so the research result in the leakage analysis of rotary compressors can be used for reference in our study. The visualization experiments on the oil distributed in the rotary refrigeration compressors showed that the radial clearance was sealed by the mixture of the lubricating oil and the gas (Costa, 1990; Mitsuhiro, 1996). At the inlet region, the flow is a single-phase mixture flow, but after some extension of the flow from the inlet the tiny bubbles formed, and the flow is a two-phase flow. Based on the visualization experimental results, the oil-gas mixture flowing through the leakage path assumes to be two phase fluid homogeneous flow in this article.
3.1.1.
The leakage model
Considering the clearance is generally far less than the size of every component in the compressor, the axial leakage path is simplified as shown in Fig. 5. The figure shows that the clearance is filled with oil-gas mixture. As a result of that the wall velocity is close to the oil-gas mixture flow through the clearance, the influence of the wall velocity can't be ignored due to the influence of viscosity. So in the research of the leakage through the axial leakage path should not only consider the leakage caused by pressure difference but also consider the leakage caused by wall velocity. In addition to the physical model, the boundary conditions of the flow through the axial leakage path are very close to the Couette-Poiseuille flow. The Y axis of the coordinate system as shown in Fig. 5 along the edge of a narrow slit, the X axis in accordance with the rules of the right hand is perpendicular to the Y axis. The minimum clearance of leakage path is set as d. The boundary conditions of the axial leakage model are that: the inlet pressure is P, the outlet pressure is Ps, the wall velocity equals to U. Because of d << l and d << b, the velocity uy and uz can be set as zero. So the flow through the axial leakage path can be
Leakage mathematical model
Besides the leak path geometry model, another important factor for the research of the leakage characteristics is the fluid flow pattern. For the SSRC with injection, the lubricating oil in compressor which is essential to ensure the normal operation of compressor makes the fluid flow pattern more complicated. So for the SSRC with MEMP, a two phase leakage model need to be come up with a consideration of lubricating oil to analyze the influence of the leakage loss in the working
Fig. 5 e The simplify model of the axial leakage path. Fig. 5 shows the axial leakage path.
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simplified as one-dimensional steady stationary flow. Under the condition of laminar flow and without considering the quality force, the continuity equations and the NaviereStocks equations can be reduced to: (13) (14)
Boundary conditions of the axial leakage flow are that: 8 X ¼ 0; u ¼ U > > < X ¼ hðxÞ; u ¼ 0 > Y ¼ 0; P ¼ PðaÞ > : Y ¼ l; P ¼ Ps
(15)
The flow velocity at any flow section of the axial leakage flow can be derived by Equations (13)e(15), the expression is: 1 dp 2 U 1 dp x þ h x 2m dy h 2m dy
(16)
Then the leakage flow rate can be calculated by flow velocity integral along the X direction: m_ ¼
Zh rubdx ¼
rbhU rbh3 dp 2 12m dy
(17)
0
Simultaneous Equations (12) and (17), the leakage flow rate of the whole axial leakage path is expressed as: m_ a ¼ rm
ZbðaÞ 0
mg ml > > > : m ¼ ð1 xÞm þ xm g
vm ¼0 vx 1 vp m v2 ux ¼ r vx r vy2
u¼
8 rg rl > > > rm ¼ ð1 xÞr þ xr < g l
Z
1 12m
l
Dp 6mU hðt; u; aÞ2 dy 0 db Z l hðt; u; aÞ3 dy
(18)
0
where rm is the density of the mixture, b(a) is the width of the leakage path, l is the length of the star-wheel tooth meshing into the screw groove, Dp is the pressure difference, m is the viscosity of the fluid, U is the wall velocity. In this analysis, the mixture has a single density and viscosity. Then the thermal properties of leakage fluid can be calculated by the pure thermal properties of oil and gas:
(19)
l
where rg and rl are the density of gas and oil respectively, mg and ml are the viscosity of gas and oil respectively.
3.2.
The radial leakage model
The shadows showed in Fig. 2 (a) expressed the mixture of the gas and oil leakage through the radial leakage path. The flow in this clearance can be simplified as orifice flow model. The flow state of the leakage flow through the orifice related to the type of orifice. Reason for the structure of the SSRC, leakage flow through the radial leakage path is almost line contact with the wall of the orifice. So the leakage flow through the radial leakage path can be analyzed by thinwalled orifice plate model in which the fluid is fully turbulent flow. Then the oil-gas mixture flow through the leakage path can assume to be two phase fluid homogeneous flow. Using the continuity equations and the NaviereStocks equations, the leakage flow rate of the whole axial leakage path is expressed as: pffiffiffiffiffiffiffiffiffiffiffiffiffi kεS 2Dprl qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m_ ¼ rg 4 lþx l 1 ðd=DÞ rl 3 2 rg rg rg 60:6150 þ 44:6954 l ¼1:48625 9:26541 rl rl rl 4 5 rg rg 5:12966 26:5743 rl rl
(20)
where m_ is the leakage flow rate of the mixture, x is the dryness of the mixture, ε is the expansion coefficient, k is the flow coefficient which can be determined by the experimental data of cooling capacity and shaft power, S is the area of the leakage path which can be calculated by Equation (11), Dp is
Fig. 6 e Measured and calculated cooling capacity of the SSRC with MEMP. This figure shows the measured and the calculated cooling capacity of the SSRC under conditions with different evaporation and condensation temperatures.
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Table 2 e The structure and operation parameters of SSRC. The structure parameters of a typical SSRC used in refrigeration and air conditioning field are shown in Table 2. Parameters P R1 (mm) R2 (mm) A (mm) b (mm) n (r min1)
Value
Parameters
Value
11/6 181 181 144.8 26.5 2880
Compression medium Lubricating oil Suction temperature (K) Discharge temperature (K) Suction pressure (MPa) Discharge pressure (MPa)
R22 Suniso 4 GS 283.15 311.15 0.584 1.46
the pressure difference, rg is the density of the gas, rl is the density of the oil, l is the correction coefficient, d/D is the diameter ratio of the orifice to pipe. After the calculation of the leakage rate of the mixture using Equations (18) and (20), the gas leakage rate can be obtained as following: m_ g ¼
4.
_ 12xmn 60
(21)
through the gas meter to measure the flow rate. By using this test rig, the performances of the SSRC such as cooling capacity and shaft power consumption were measured. In the experiment, several pressure sensor with the error less than ±0.25% and thermocouples with an uncertainty of ±0.1 K were used to measure the pressure and temperature. The accuracy of the gas flow rate measurement and the liquid flow rate measurement were with in ±0.75% and ±0.5% respectively. The error of the digital power meter used for measuring the shaft power was less than ±0.1%.
Experimental research 5.
In order to verify the theoretical results, an oil-injection SSRC with MEMP was developed and its performance was tested with a test rig shown in Fig. 6. The oil-gas mixture was compressed by the SSRC and discharged into an oil-gas separator. After oil-gas separating, Part of the gas entered into the condenser to cool gas into liquid, and to measure the flow rate through the liquid flow meter, the other part of the gas directly into the gas cooler. After being cooled, a part of the liquid refrigerant re-injected into the compressor and the rest of the liquid refrigerant flowed into a gas cooler. In the gas cooler the heat of the gas was transferred to the liquid, which lead to liquid evaporation. Then all of the gas flowed
Results and discussion
The models are applied to predict the leakage characteristics of the SSRC with this new meshing pair MEMP. A typical SSRC used in refrigeration and air conditioning field with the structure parameters as shown in Table 2 was developed for this analysis.
5.1.
Mathematical model validation
Fig. 6 shows the measured and the calculated cooling capacity of the SSRC under conditions with different evaporation and condensation temperatures. The contrast of the measured
Fig. 7 e The leakage area of the radial leakage path. This figure shows the leakage area of the radial leakage path in the SSRC with LEMP and MEMP, in which Fig. 6(a) refer to the left tooth flank and Fig. 6(a) refer to the right tooth flank.
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Fig. 8 e The radial leakage flow rate. The radial leakage flow rate in the SSRC with the LEMP and the MEMP at the left side of the tooth flank and the right side of the tooth flank are calculated and shown in Fig. 7(a) and (b) respectively.
and the calculated results in these figures show that the calculated cooling capacity is in good agreement with the measured data. This indicates that the leakage models presented by this paper are reasonable to estimate the leakage characteristics of the SSRC with MEMP.
5.2.
Geometry features of the leakage path
Fig. 7 shows the leakage area of the radial leakage path in the SSRC with LEMP and MEMP, in which Fig. 7(a) refer to the left tooth flank and Fig. 7(a) refer to the right tooth flank. By the contrast analysis of these two curves in each figure, we can see that the leakage areas of the radial leakage path with MEMP are larger than that with LEMP. Because of that the leakage flow rate through these paths will become large, so the leakage characteristic research of SSRC with MEMP is necessary.
5.3. 5.3.1.
Leakage rate
tooth flank are calculated and shown in Fig. 8(a) and Fig. 8(b) respectively. As shown in Fig. 8(a) the leakage flow rates at the left side in the SSRC with LEMP and MEMP are all increase with the rotation of star-wheel. Fig. 8(b) shows the leakage flow rate at the right side in the SSRC with MEMP is fluctuation at the beginning, but also increase with the rotation of star-wheel generally. Comparing the two curves in Fig. 8(a) and Fig. 8(b), the leakage flow rates through the radial leakage path with MEMP are larger than that with LEMP. The surround area of the curve and abscissa axis shown in this figure is the total leakage in a working period. Through the curve integral, the total leakages through the radial leakage path with LEMP and MEMP in a working period are listed in Table 3. The results presented in Table 3 can be used to show the difference of leakage rate of these two kinds of meshing pairs. Leakage rate of the radial leakage path in the SSRC with MEMP is nearing about 10 times of the leakage rate in the SSRC with LEMP.
5.3.2.
The radial leakage flow rate
The radial leakage flow rate in the SSRC with the LEMP and the MEMP at the left side of the tooth flank and the right side of the
The axial leakage flow rate
The axial leakage flow rate in the SSRC with the LEMP and the MEMP at the left side of the tooth flank and the right side of the tooth flank are shown in Fig. 9(a) and Fig. 9(b) respectively.
Table 3 e The leakage rate through the radial and axial leakage paths. The total leakages through the side radial leakage path with LEMP and MEMP in a working period are listed in Table 3. Meshing pair type The radial leakage paths The axial leakage paths Total leakage
leakage leakage leakage leakage leakage
rate through path rate through path rate through path rate through path rate through path
1
3 (kg s ) 5 (kg s1) 2 (kg s1) 4 (kg s1) 1e9 (kg s1)
LEMP
MEMP
1.20E-03 3.91E-03 2.51E-01 2.63E-01 3.88E-01
1.16E-02 4.23E-02 1.64E-01 2.62E-01 3.76E-01
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Fig. 9 e The axial leakage flow rate. The axial leakage flow rate in the SSRC with the LEMP and the MEMP at the left side of the tooth flank and the right side of the tooth flank are shown in Fig. 8(a) and (b) respectively.
As shown in Fig. 9 the axial leakage flow rates in the SSRC with LEMP and MEMP are all increase with the rotation of starwheel. Comparing the two curves in Fig. 9(a) and Fig. 9(b), the axial leakage flow rate at the left side of the tooth flank with
MEMP is smaller than that with LEMP. But the axial leakage flow rates at the right side of the tooth flank with MEMP and with LEMP are almost the same to each other. The data of Table 3 shows that the leakage rate of the left axial leakage path in the SSRC with MEMP is about 65% of the leakage rate in the SSRC with LEMP. And the leakage rate of the right axial leakage path in the SSRC with MEMP is about 99.6% of the leakage rate in the SSRC with LEMP.
5.3.3.
The leakage characteristics
Fig. 10 shows the proportion of the leakage rate through each leakage path in SSRC with MEMP, the value greater than zero
Fig. 10 e The Leakage characteristics of the SSRC with MEMP. shows the proportion of the leakage through each leakage path in SSRC with MEMP, the value greater than zero indicates the gas leakage into the working chamber, the value less than zero indicates the gas leakage out of the working chamber.
Fig. 11 e The total leakage flow rate. The total leakage flow rates in SSRC with LEMP and MEMP are shown in Fig. 10.
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indicates the gas leakage into the working chamber, the value less than zero indicates the gas leakage out of the working chamber. As shown in this figure the leakage through paths 1,2,3,4,5,6,8 and 9 is out of the working chamber, but the leakage through path 7 is into the working path. Through the compression of the leakage rates through these paths, the leakage rates through paths 2,4,7,8 and 9 are lager than that through paths 1,3,5,6. So the increase in leakage through path 3 and path 5 causes by the use of this new meshing pair MEMP can be ignored. The total leakage flow rates in SSRC with LEMP and MEMP are shown in Fig. 11. The figure shows that at the beginning of the compression process, the gas leakage into the working chamber from the high pressure chamber is larger than the gas leakage out of the working chamber. But as the star-wheel rotating the leakage flow rates out of the working chamber increase and become larger than the gas leakage into the chamber. Another result is that the leakage rate of all the paths in SSRC with LEMP and MEMP is all most the same with each other. The total leakage flow rate can be calculated from the integral of the surround areas of the cures and the abscissa axis and listed in Table 3. The results show that the total leakage rate of these paths in SSRC with MEMP is a little decrease than that with LEMP due to the decrease of the leakage through path 2 and path 4.
6.
Conclusion
Research of leakage characteristics of SSRC with MEMP and LEMP are carried out by theoretical and experimental method in this paper. The results are obtained as following. (1) The calculated cooling capacity is in good agreement with the measured data which means the leakage models presented by this paper is reasonable to estimate the leakage characteristics of SSRC with MEMP. (2) The leakage areas of the radial leakage path at the left side of the tooth flank in SSRC with MEMP are larger than that with LEMP, which will lead to larger leakage through these paths. (3) The leakage flow rate at the both side of the tooth flank in the SSRC with LEMP and MEMP are all increased with the rotation of star-wheel. The radial leakage flow rate with MEMP is nearing about 10 times of the radial leakage rate with LEMP. (4) The axial leakage flow rate in the SSRC with LEMP and MEMP are all increased with the rotation of star-wheel. The leakage flow rate of the left axial leakage path in the SSRC with MEMP is about 65% of the leakage flow rate in the SSRC with LEMP. And the leakage flow rate of the right axial leakage path in the SSRC with MEMP is about 99.6% of the leakage flow rate in the SSRC with LEMP. (5) The leakage through paths 1,2,3,4,5,6,8 and 9 is out of the working chamber, but the leakage through path 7 is into the working chamber. Through the compression,
the leakage flow rate through paths 2,4,7,8 and 9 is lager than that through paths 1,3,5,6. So the leakage increase through path 3 and path 5 by the use of MEMP can be ignored. The total leakage rate of these paths in SSRC with MEMP is a little decrease than that with LEMP due to the leakage decrease in path 2 and path 4. According to obtained results, design of the Multicolumn Envelope Meshing Pair in a single screw refrigeration compressor will not lead to an increase of leakage. Because of its considerable increase of wear resistance, replace the straight line envelope meshing pair by the Multicolumn Envelope Meshing Pair in a single screw refrigeration compressor is suggested in this paper.
references
Bein, T.W., 1991. High Pressure Single Screw Compressors. US Patent No. 4981424. Costa, C.M.N.F., Ferreira, R.T.S., Prata, A.T., 1990. Considerations about the leakage through the minimal clearance in a rolling piston compressor. In: 1990 International Compressor Engineering Conference at Purdue, pp. 853e862. Feng, Q.K., Guo, B., Zhao, C., et al., 2005. A design method of meshing pair profile of single screw compressor enveloped by multi straight line. Compress. Technol. 19 (3), 1e6. Jin, G.X., Lin, Q., 1986. Calculation of the leakage path and leakage rate in single screw compressor. J. Fluids Eng. 01, 19e24. Li, J., Feng, Q.K., Liu, F.L., et al., 2013. Experimental studies of the tooth wear resistance with different profilesin single screw compressor. Tribol. Int. 57, 210e215. Li, H.Q., 1994. The Working Process of Oil-flooded Single Screw Compressor. PhD thesis. Xi'an Jiaotong University. Mitsuhiro, F., Tadashi, Y., Takashi, S., Toru, M., 1996. Observation of oil film condition in a cylinder of rotary compressor. In: 1996 International Compressor Engineering Conference at Purdue, pp. 401e406. Wang, S.H., 1978. Single screw refrigeration compressor. Chem. Ind. Gen. Mach. 07, 50e52. Wu, J.H., Jin, G.X., Shu, P.C., 1996. The research of the leakage characteristics and its influence on the performance of single screw compressor. J. Xi'an Jiaot. Univ. 30 (4), 66e71. Wu, W.F., Feng, Q.K., 2009. A Multicolumn Envelope Meshing Pair for single screw compressors. J. Mech. Des. 131 (74505), 1e3. Wu, W.F., 2010. Research of the Multicolumn Envelope Meshing Pair and its Mashing Technology for Single Screw Compressor. PhD thesis. Xi’an Jiaotong University. Zimmern, B., Patel, G.J., 1972. Design and operating characteristics of the zimmern single screw compressor. In: 1972 Purdue Compressor Technology Conference, pp. 96e99. Zimmern, B., 1990. Method and a Screw Machine for Processing Fluid Under High Pressures, with Liquid Injection Between a Sealing Portion and a Support Portion of the Gate-Rotor. US Patent No. 4900239. Zhang, S., 2007. The comparison of twin and single screw refrigerating compressor. Low Temp. Spec. Gases 25 (03), 1e4 (in Chinese). Zhou, L., 1999. Study on Thermo-elastic Deformation and Dynamic Characteristics of Engagement Pair in Single Screw Compressors. PhD thesis. Xi'an Jiaotong University.
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Research of leakage characteristics of single screw refrigeration compressors with the Multicolumn Envelope Meshing Pair Zengli Wang, Zhan Liu, Weifeng Wu, Quanke Feng* School of Energy and Power Engineering, Xi'an Jiaotong University, Xi'an 710049, China
article info
abstract
Article history:
The Multicolumn Envelope Meshing Pair (MEMP) was proposed and has been applied for the
Received 29 May 2014
single screw refrigeration compressor (SSRC) to reduce the wear of the meshing pair. However,
Received in revised form
the geometric model shows its changed shape of the leakage paths compared with the existing
2 September 2014
straight line envelope meshing pair (LEMP). It is necessary to research the leakage character-
Accepted 6 September 2014
istics of the SSRC with MEMP and LEMP, to evaluate the value of the proposed MEMP. In this
Available online 28 September 2014
paper, the geometric model of the leakage paths between star-wheel and screw rotor is established. A two-phase leakage mathematical model for gas-oil flow is presented to predict
Keywords:
the gas leakage rate of the SSRC with MEMP. The experiments of the performance of a SSRC
Leakage characteristic
with MEMP were conducted to verify the leakage mathematical model. Obtained results show
Single screw refrigeration
the leakage of the SSRC with MEMP is a little bit smaller than that of the SSRC with LEMP.
compressor
© 2014 Elsevier Ltd and IIR. All rights reserved.
Multicolumn Envelope Meshing Pair Geometric model
ristiques de fuite de compresseurs Recherche des caracte monovis avec couple satellite-rotor d'enveloppe frigorifiques a colonne multiple a ristiques de fuite ; Compresseur frigorifique a monovis ; Couple satellite-rotor d'enveloppe de colonne multiple ; Mots-cles : Caracte le ge ome trique Mode
1.
Introduction
Single screw refrigeration compressors (SSRC) developed by Zimmern in the 1960s (Zimmern and Patel, 1972) are widely used in air compression systems and refrigeration systems
* Corresponding author. Tel./fax: þ86 029 82663783. E-mail address: [email protected] (Z. Wang). http://dx.doi.org/10.1016/j.ijrefrig.2014.09.005 0140-7007/© 2014 Elsevier Ltd and IIR. All rights reserved.
(Wang, 1978). The performance of a newly produced SSRC is superior to that of the single piston compressor and the screw compressor (Bein, 1991; Zhang, 2007), but the discharge capacity decreases sharply several hundred hours after initial operation (Zimmern, 1990). This is because that the starwheel tooth flank of the SSRC wears rapidly and the
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Nomenclature u P A d b l Dp U m_ x S d/D h
Height of the envelope column (m) Teeth number ratio between the star-wheel and the screw Central distance between the star-wheel and the screw rotor (m) Diameter of the envelope column (m) Width of the leakage path (m) Length of the star-wheel tooth meshing into the screw groove (m) Pressure difference (Pa) Wall velocity (m s1) Leakage flow rate of the mixture (kg s1) Dryness of the mixture Area of the leakage path (m2) Diameter ratio of the orifice to pipe Axial leakage clearance (m)
clearance between the star-wheel and the screw increases. In the existing design of the SSRC, the tooth flank contacts with the groove flank at its sharp straight edge and wore at this sharp straight at first. Structure of the tooth and the groove meshing pair of existing SSRC is called as with straight line envelope meshing pair (LEMP). In order to enhance the wearresistance of the meshing pair, its profile must be improved. On the basis of column envelope profile proposed by Zimmern and multi-straight line envelope profile proposed by Feng (Feng et al., 2005), a new Multicolumn Envelope Meshing Pair MEMP was deduced by our group in 2009 (Wu and Feng, 2009). This MEMP could significantly improve the wear resistance of the compressor (Li et al., 2013). However, leakage path shapes of the LEMP and the MEMP compared by Wu show that the leakage paths are changed (Wu, 2010). Through the qualitative analysis, the leakage characteristics of the leakage paths between the star-wheel and the screw are very different for the SSRC with LEMP and MEMP. Further more, the leakage characteristics are a very important features in the process of performance evaluation of the SSRC. Thus, it is necessary to investigate the leakage characteristic of the compressor with the MEMP. Several scholars had investigated the leakage characteristics of the SSRC with the LEMP (Jin and Lin, 1986; Li, 1994; Wu, 1996; Zhou, 1999). In there study, the parallel plate model and the wedge plate model were used to calculate the leakage rate. Because shapes of the leakage paths of the SSRC with MEMP are very different from the existing design meshing pair LEMP, the leakage characteristics of this new meshing pair MEMP must be investigated and to evaluate the performance of the compressor with the MEMP is good or bad. In this paper, the geometrical model of the leakage path in the SSRC with MEMP is established. A two-phase leakage mathematical model for gas-oil flow is presented to predict the gas leakage rate through the leakage paths of the SSRC. In order to verify the model, an oil-injection SSRC with MEMP has been developed and its performance has been tested by the experiment. Obtained results are compared with that of the SSRC with LEMP. It is found that the leakage rate of the SSRC with MEMP is smaller than that of a single screw refrigeration compressor with LEMP.
Subscripts g Gas l Lubricating oil m Mixture i Number of the envelope column Greeks q a b x m ε k l r
Envelope angle (rad) Star-wheel rotation angle (rad) Inclination angle of the column, (rad) Circumferential Angle of the boundary point. (rad) Viscosity of the fluid (Pa.s) Expansion coefficient Flow coefficient Correction coefficient Density of the flow (kg m3)
2.
Leakage path geometrical model
2.1.
The feature of the leakage path with MEMP
2.1.1.
Leakage path
As show in Fig. 1, there are 9 leakage paths in the SSRC. Path 1 is the clearance between the star-wheel tooth top and the bottom of the screw groove. Path 2 and path 4 are the axial clearance between the star-wheel tooth flank and the screw groove, path 3 and path 5 are the radial clearance between the star-wheel tooth flank and the screw groove. Path 6 is the clearance between the upper surface of the star-wheel tooth and the casing. Path 7 and path 8 are the clearance between the outer surface of the screw and the inside surface of the casing. Path 9 is the clearance between the outer surface of the screw and the inside surface of the casing at the discharge end. Leakage rate through the leakage path is affected by the area and the shape of the clearance. So the leakage path shape should be discussed before the leakage analysis. The
Fig. 1 e Leakage paths of the SSRC. This figure shows all of the leakage paths of the single screw refrigeration compressor.
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Fig. 2 e Leakage path shape. Shows the leakage path shape between the tooth flank and the screw groove in the SSRC with MEMP.
Fig. 3 e Leakage path shape of the SSRC with MEMP. The star-wheel tooth flank which influences the geometrical model of the leakage path between the tooth flank and the screw groove is shown in Fig. 3.
qualitative comparative analysis presented by Wu (Wu, 2010) showed that only the axial and radial clearance between the tooth flank and the screw groove in the SSRC with MEMP was quite different from that in the SSRC with LEMP. So the leakage characteristics of these axial and radial paths need to be analyzed in this study. Fig. 2 shows the leakage path shape between the tooth flank and the screw groove in the SSRC with MEMP (Fig. 2(a)) and with LEMP (Fig. 2(b)). As shown in this figure the radial leakage path area in the SSRC with MEMP is larger than that with LEMP, and the axial leakage path shapes are different from each other. But the leakage characteristics of the axial and radial paths in the SSRC with MEMP have not been studied. Thus it is necessary to investigate the geometric characteristics of axial clearance (path 2 and path 4) and the radial clearance (path 3 and path 5) firstly.
2.1.2.
Star-wheel tooth flank
The star-wheel tooth flank which influences the geometrical model of the leakage path between the tooth flank and the screw groove is shown in Fig. 3. The figure shows there are multiple column segments in the tooth flank. Along the direction B shown in Fig. 3(a), we divide the star-wheel tooth flank into the left tooth flank and the right tooth flank. Take the left tooth flank for example there are two column segments near the root position of the tooth flank, and one column segment near the tip position of the tooth flank. Between
envelope column and surface of the tooth is the tangent between them. In coordinate plane S1 of Fig. 3(b), the envelope column center position at the root position of the tooth flank and the inclination angle of the column axis are shown in Table 1. The inclination angle is the angle between the axis of the column and the Z axis which is perpendicular to the XeY plane and goes through the origin S1. Take the left tooth flank for example, the tooth flank shape in AeA section is shown in Fig. 3(b). In which the envelope column Ob1 is the benchmark column with a circular arc section and the rest envelope columns are ellipse sections. In the u height section of the benchmark column, the envelope column center positions are: 8 < Li1 ¼ Lr Benchmark column : Mi1 ¼ Mr : Ki1 ¼ Kr
(1)
Table 1 e The parameters of the envelope columns. The envelope column center position at the root position of the tooth flank and the inclination angle of the column are shown in Table 1 Envelope column Benchmark column The second envelope column
X axis
Y axis
Z axis
Angle
Lr Lri
Mr Mri
Kr Kri
0 bri
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8 Li2 ðuÞ ¼ Lri u tan bri > > < Mi2 ¼ Mri The second envelope column : K ¼ Kri > > : i2 bi2 ¼ bri
2.2.
at any star-wheel rotation angle when the height of the envelope column equal to ui is expressed as: (2)
Tooth flank profile analysis
See the Fig. 3(b), the tooth flank profile in this section is composed by five curves: the tangent 1e2 between the second envelope column and the upper surface of the tooth, elliptic arc 2e3, tangent 3e4 between envelope columns, circular arc 4e5 and the tangent 5e6 between benchmark column and the lower surface of the tooth. To analyze tooth flank profile and set up geometry model of the leakage path, the equations of the above curves and the spiral groove flank profile in the same section at any tooth flank height need to be got. The circular arc 4e5 is the intersecting line of the benchmark column and the Section AeA. Its parametric equation in coordinate plane S1 is that: 8 d > > < x ¼ Li1 þ cos f 2 (3) > > : y ¼ Mi1 þ d sin f 2 The parametric equation of the intersecting line of the second envelope column and the Section AeA (elliptic arc 2e3) is that: x ¼ Li2 þ d cos 4=ð2 cos bi2 Þ (4) y ¼ Mi2 þ d sin 4=2 Tangent 3e4 is the common tangent between the second envelope column and the benchmark column. By the Equations (3) and (4) the tangent of the elliptic arc 2e3 at point 3 and the tangent of the circular arc 4e5 at point 4 are shown as following: y0 Mi2 2 ½x Li2 þ 2 ðy Mi2 Þ ¼ 1 d d 2 cos bi2 2 ( x0 ¼ Li2 þ d cos 40 =ð2 cos bi2 Þ
1 ðui þ Ki cos bi Þ Li sin bi PMi sinðbi þ aÞ qða; ui Þ ¼ arctan P A Ki cos a Li sin a ui cosðbi þ aÞ
(8)
where a is the star-wheel rotation angle, ui is the height of the envelope column, bi is the inclination angle of the column, (Li,Ki,Mi) are the envelope column center position, P is the teeth number ratio between the star-wheel tooth and the screw groove usually chosen as 11/6, A is the central distance between the star-wheel and the screw rotor. Simultaneous Equations (5)e(8), the maximum envelope angle qmax(u) and the tangent between the second envelope column and the upper surface of the tooth (tangent 1e2) at any tooth height will be solved. The tangent between the envelope column and the lower surface of the tooth in the right tooth flank can also be obtained by using the same method. The tangent between benchmark column and the lower surface of the tooth (tangent 5e6) is the tangent at the contact point when the envelope Angle is the minimum. The contact point with the minimum envelope Angle is usually located at the position that the star-wheel tooth meshing out of the screw groove (Wu, 2010). The projection of the straight line which passes the contact point M and perpendiculars to the axis of the screw on the neutral surface of meshing pair is li1. The projection of the straight line passing the contact point M and perpendicular to the axis of the star-wheel on the surface which contains the axis of the star-wheel and perpendiculars to the axis of the screw is li2. As shown in Fig. 4, when li1 þ li2 ¼ A, the contact point is on the position with the minimum envelope Angle.
x0 Li2
(5)
y0 ¼ Mi2 þ d sin 40 =2 d d cos f0 sin f0 2 2 2 ðx Li1 Þ þ 2 ðy Mi1 Þ ¼ 1 d d 2 2
(6)
Simultaneous Equations (5) and (6) and make the coefficients equal to each other respectively, then the Angles 40 and f0 will be solved. Bringing 40 and f0 into the above formulas, the equation of tangent 3e4 will be obtained. The tangent between the second envelope column and the upper surface of the tooth (tangent 1e2) is the tangent at the contact point when the envelope Angle is the maximum. At that moment, the corresponding height of this contact point is: ui ðuÞ ¼
u d ± cos q tan bi cos bi 2
(7)
where q is the envelope angle, þ is used for the right tooth flank and e is used for the left tooth flank. The envelope Angle
Fig. 4 e The Geometric position relationship of the contact point. In this figure, the Geometric position relationship of the contact point was given out.
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 4 9 ( 2 0 1 5 ) 1 e1 0
d A ¼ Ki cos a þ Li sin a þ ui cosða þ bi Þ± cos q sinða þ bi Þ 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 d þ R22 Mi ± sin q 2
process. In the analysis of the axial and radial leakage characteristics for the SSRC with MEMP that follows, the following assumptions have been employed: (9)
In this equation þ is used for the right tooth flank and e is used for the left tooth flank. According to Equations (7)e(9), the minimum envelope Angle qmin(u) and the tangent between the benchmark column and the lower surface of the tooth (tangent 5e6) at any tooth height can be solved. The same method can be used to obtain the tangent between the envelope column and the upper surface of the tooth in the right tooth flank.
2.3.
Leakage path area
Fig. 2(a) shows the shape of the leakage path between the tooth flank and the screw groove, the shaded area is the radial leakage path area. In order to calculate this area, the tooth flank profile and the screw groove flank profile should be obtained. Through analyzing the tooth flank profile, the tooth flank profile equation according to the different height in Y direction can be calculated separately. The coordinates of the boundary points along the Y direction are as follows: d (10) ti ðuÞ ¼ Mi þ sin xi 2 where u is the tooth height, d is the diameter of the envelope column, xi is the circumferential Angle of the boundary point. Using the coordinates of the boundary points and the Equations (3)e(6), the tooth flank profile x1(t,u) can be obtained. In view of the tooth flank thickness is much smaller than the screw groove length the intersection curve of screw groove can be set as the tangent of the envelope column in the contact point. The tangent can be calculated by the Equations (5) and (6). Considering the existence of the meshing clearance, the equation of the screw groove flank profile x2(t,u,a) can be received by moving the tangent of the contact point outward a meshing clearance d. According to the above calculation, the radial leakage path area and the axial leakage clearance value can be easily calculated. The selection of plus or minus in these equations is the same as Equation (7). Z0 ðx1 ðt; uÞ x2 ðt; u; aÞÞdt
(11)
hðt; u; aÞ ¼ ±ðx1 ðt; uÞ x2 ðt; u; aÞÞcosðqða; uÞÞ
(12)
SðaÞ ¼ ± tu
3.
5
(1) The influences of the body-forces of gas are negligible. (2) Gas-oil mixture flow through the leakage paths is taken to be isentropic and compressible since the velocity is high. (3) The gas-oil mixture is homogeneous in the axial and radial leakage paths.
3.1.
The axial leakage model
Specific to the axial leakage path, the shapes of the leakage path and the wall motion feature are similar to the radial leakage path in rotary compressors, so the research result in the leakage analysis of rotary compressors can be used for reference in our study. The visualization experiments on the oil distributed in the rotary refrigeration compressors showed that the radial clearance was sealed by the mixture of the lubricating oil and the gas (Costa, 1990; Mitsuhiro, 1996). At the inlet region, the flow is a single-phase mixture flow, but after some extension of the flow from the inlet the tiny bubbles formed, and the flow is a two-phase flow. Based on the visualization experimental results, the oil-gas mixture flowing through the leakage path assumes to be two phase fluid homogeneous flow in this article.
3.1.1.
The leakage model
Considering the clearance is generally far less than the size of every component in the compressor, the axial leakage path is simplified as shown in Fig. 5. The figure shows that the clearance is filled with oil-gas mixture. As a result of that the wall velocity is close to the oil-gas mixture flow through the clearance, the influence of the wall velocity can't be ignored due to the influence of viscosity. So in the research of the leakage through the axial leakage path should not only consider the leakage caused by pressure difference but also consider the leakage caused by wall velocity. In addition to the physical model, the boundary conditions of the flow through the axial leakage path are very close to the Couette-Poiseuille flow. The Y axis of the coordinate system as shown in Fig. 5 along the edge of a narrow slit, the X axis in accordance with the rules of the right hand is perpendicular to the Y axis. The minimum clearance of leakage path is set as d. The boundary conditions of the axial leakage model are that: the inlet pressure is P, the outlet pressure is Ps, the wall velocity equals to U. Because of d << l and d << b, the velocity uy and uz can be set as zero. So the flow through the axial leakage path can be
Leakage mathematical model
Besides the leak path geometry model, another important factor for the research of the leakage characteristics is the fluid flow pattern. For the SSRC with injection, the lubricating oil in compressor which is essential to ensure the normal operation of compressor makes the fluid flow pattern more complicated. So for the SSRC with MEMP, a two phase leakage model need to be come up with a consideration of lubricating oil to analyze the influence of the leakage loss in the working
Fig. 5 e The simplify model of the axial leakage path. Fig. 5 shows the axial leakage path.
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simplified as one-dimensional steady stationary flow. Under the condition of laminar flow and without considering the quality force, the continuity equations and the NaviereStocks equations can be reduced to: (13) (14)
Boundary conditions of the axial leakage flow are that: 8 X ¼ 0; u ¼ U > > < X ¼ hðxÞ; u ¼ 0 > Y ¼ 0; P ¼ PðaÞ > : Y ¼ l; P ¼ Ps
(15)
The flow velocity at any flow section of the axial leakage flow can be derived by Equations (13)e(15), the expression is: 1 dp 2 U 1 dp x þ h x 2m dy h 2m dy
(16)
Then the leakage flow rate can be calculated by flow velocity integral along the X direction: m_ ¼
Zh rubdx ¼
rbhU rbh3 dp 2 12m dy
(17)
0
Simultaneous Equations (12) and (17), the leakage flow rate of the whole axial leakage path is expressed as: m_ a ¼ rm
ZbðaÞ 0
mg ml > > > : m ¼ ð1 xÞm þ xm g
vm ¼0 vx 1 vp m v2 ux ¼ r vx r vy2
u¼
8 rg rl > > > rm ¼ ð1 xÞr þ xr < g l
Z
1 12m
l
Dp 6mU hðt; u; aÞ2 dy 0 db Z l hðt; u; aÞ3 dy
(18)
0
where rm is the density of the mixture, b(a) is the width of the leakage path, l is the length of the star-wheel tooth meshing into the screw groove, Dp is the pressure difference, m is the viscosity of the fluid, U is the wall velocity. In this analysis, the mixture has a single density and viscosity. Then the thermal properties of leakage fluid can be calculated by the pure thermal properties of oil and gas:
(19)
l
where rg and rl are the density of gas and oil respectively, mg and ml are the viscosity of gas and oil respectively.
3.2.
The radial leakage model
The shadows showed in Fig. 2 (a) expressed the mixture of the gas and oil leakage through the radial leakage path. The flow in this clearance can be simplified as orifice flow model. The flow state of the leakage flow through the orifice related to the type of orifice. Reason for the structure of the SSRC, leakage flow through the radial leakage path is almost line contact with the wall of the orifice. So the leakage flow through the radial leakage path can be analyzed by thinwalled orifice plate model in which the fluid is fully turbulent flow. Then the oil-gas mixture flow through the leakage path can assume to be two phase fluid homogeneous flow. Using the continuity equations and the NaviereStocks equations, the leakage flow rate of the whole axial leakage path is expressed as: pffiffiffiffiffiffiffiffiffiffiffiffiffi kεS 2Dprl qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m_ ¼ rg 4 lþx l 1 ðd=DÞ rl 3 2 rg rg rg 60:6150 þ 44:6954 l ¼1:48625 9:26541 rl rl rl 4 5 rg rg 5:12966 26:5743 rl rl
(20)
where m_ is the leakage flow rate of the mixture, x is the dryness of the mixture, ε is the expansion coefficient, k is the flow coefficient which can be determined by the experimental data of cooling capacity and shaft power, S is the area of the leakage path which can be calculated by Equation (11), Dp is
Fig. 6 e Measured and calculated cooling capacity of the SSRC with MEMP. This figure shows the measured and the calculated cooling capacity of the SSRC under conditions with different evaporation and condensation temperatures.
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Table 2 e The structure and operation parameters of SSRC. The structure parameters of a typical SSRC used in refrigeration and air conditioning field are shown in Table 2. Parameters P R1 (mm) R2 (mm) A (mm) b (mm) n (r min1)
Value
Parameters
Value
11/6 181 181 144.8 26.5 2880
Compression medium Lubricating oil Suction temperature (K) Discharge temperature (K) Suction pressure (MPa) Discharge pressure (MPa)
R22 Suniso 4 GS 283.15 311.15 0.584 1.46
the pressure difference, rg is the density of the gas, rl is the density of the oil, l is the correction coefficient, d/D is the diameter ratio of the orifice to pipe. After the calculation of the leakage rate of the mixture using Equations (18) and (20), the gas leakage rate can be obtained as following: m_ g ¼
4.
_ 12xmn 60
(21)
through the gas meter to measure the flow rate. By using this test rig, the performances of the SSRC such as cooling capacity and shaft power consumption were measured. In the experiment, several pressure sensor with the error less than ±0.25% and thermocouples with an uncertainty of ±0.1 K were used to measure the pressure and temperature. The accuracy of the gas flow rate measurement and the liquid flow rate measurement were with in ±0.75% and ±0.5% respectively. The error of the digital power meter used for measuring the shaft power was less than ±0.1%.
Experimental research 5.
In order to verify the theoretical results, an oil-injection SSRC with MEMP was developed and its performance was tested with a test rig shown in Fig. 6. The oil-gas mixture was compressed by the SSRC and discharged into an oil-gas separator. After oil-gas separating, Part of the gas entered into the condenser to cool gas into liquid, and to measure the flow rate through the liquid flow meter, the other part of the gas directly into the gas cooler. After being cooled, a part of the liquid refrigerant re-injected into the compressor and the rest of the liquid refrigerant flowed into a gas cooler. In the gas cooler the heat of the gas was transferred to the liquid, which lead to liquid evaporation. Then all of the gas flowed
Results and discussion
The models are applied to predict the leakage characteristics of the SSRC with this new meshing pair MEMP. A typical SSRC used in refrigeration and air conditioning field with the structure parameters as shown in Table 2 was developed for this analysis.
5.1.
Mathematical model validation
Fig. 6 shows the measured and the calculated cooling capacity of the SSRC under conditions with different evaporation and condensation temperatures. The contrast of the measured
Fig. 7 e The leakage area of the radial leakage path. This figure shows the leakage area of the radial leakage path in the SSRC with LEMP and MEMP, in which Fig. 6(a) refer to the left tooth flank and Fig. 6(a) refer to the right tooth flank.
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i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 4 9 ( 2 0 1 5 ) 1 e1 0
Fig. 8 e The radial leakage flow rate. The radial leakage flow rate in the SSRC with the LEMP and the MEMP at the left side of the tooth flank and the right side of the tooth flank are calculated and shown in Fig. 7(a) and (b) respectively.
and the calculated results in these figures show that the calculated cooling capacity is in good agreement with the measured data. This indicates that the leakage models presented by this paper are reasonable to estimate the leakage characteristics of the SSRC with MEMP.
5.2.
Geometry features of the leakage path
Fig. 7 shows the leakage area of the radial leakage path in the SSRC with LEMP and MEMP, in which Fig. 7(a) refer to the left tooth flank and Fig. 7(a) refer to the right tooth flank. By the contrast analysis of these two curves in each figure, we can see that the leakage areas of the radial leakage path with MEMP are larger than that with LEMP. Because of that the leakage flow rate through these paths will become large, so the leakage characteristic research of SSRC with MEMP is necessary.
5.3. 5.3.1.
Leakage rate
tooth flank are calculated and shown in Fig. 8(a) and Fig. 8(b) respectively. As shown in Fig. 8(a) the leakage flow rates at the left side in the SSRC with LEMP and MEMP are all increase with the rotation of star-wheel. Fig. 8(b) shows the leakage flow rate at the right side in the SSRC with MEMP is fluctuation at the beginning, but also increase with the rotation of star-wheel generally. Comparing the two curves in Fig. 8(a) and Fig. 8(b), the leakage flow rates through the radial leakage path with MEMP are larger than that with LEMP. The surround area of the curve and abscissa axis shown in this figure is the total leakage in a working period. Through the curve integral, the total leakages through the radial leakage path with LEMP and MEMP in a working period are listed in Table 3. The results presented in Table 3 can be used to show the difference of leakage rate of these two kinds of meshing pairs. Leakage rate of the radial leakage path in the SSRC with MEMP is nearing about 10 times of the leakage rate in the SSRC with LEMP.
5.3.2.
The radial leakage flow rate
The radial leakage flow rate in the SSRC with the LEMP and the MEMP at the left side of the tooth flank and the right side of the
The axial leakage flow rate
The axial leakage flow rate in the SSRC with the LEMP and the MEMP at the left side of the tooth flank and the right side of the tooth flank are shown in Fig. 9(a) and Fig. 9(b) respectively.
Table 3 e The leakage rate through the radial and axial leakage paths. The total leakages through the side radial leakage path with LEMP and MEMP in a working period are listed in Table 3. Meshing pair type The radial leakage paths The axial leakage paths Total leakage
leakage leakage leakage leakage leakage
rate through path rate through path rate through path rate through path rate through path
1
3 (kg s ) 5 (kg s1) 2 (kg s1) 4 (kg s1) 1e9 (kg s1)
LEMP
MEMP
1.20E-03 3.91E-03 2.51E-01 2.63E-01 3.88E-01
1.16E-02 4.23E-02 1.64E-01 2.62E-01 3.76E-01
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 4 9 ( 2 0 1 5 ) 1 e1 0
9
Fig. 9 e The axial leakage flow rate. The axial leakage flow rate in the SSRC with the LEMP and the MEMP at the left side of the tooth flank and the right side of the tooth flank are shown in Fig. 8(a) and (b) respectively.
As shown in Fig. 9 the axial leakage flow rates in the SSRC with LEMP and MEMP are all increase with the rotation of starwheel. Comparing the two curves in Fig. 9(a) and Fig. 9(b), the axial leakage flow rate at the left side of the tooth flank with
MEMP is smaller than that with LEMP. But the axial leakage flow rates at the right side of the tooth flank with MEMP and with LEMP are almost the same to each other. The data of Table 3 shows that the leakage rate of the left axial leakage path in the SSRC with MEMP is about 65% of the leakage rate in the SSRC with LEMP. And the leakage rate of the right axial leakage path in the SSRC with MEMP is about 99.6% of the leakage rate in the SSRC with LEMP.
5.3.3.
The leakage characteristics
Fig. 10 shows the proportion of the leakage rate through each leakage path in SSRC with MEMP, the value greater than zero
Fig. 10 e The Leakage characteristics of the SSRC with MEMP. shows the proportion of the leakage through each leakage path in SSRC with MEMP, the value greater than zero indicates the gas leakage into the working chamber, the value less than zero indicates the gas leakage out of the working chamber.
Fig. 11 e The total leakage flow rate. The total leakage flow rates in SSRC with LEMP and MEMP are shown in Fig. 10.
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i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 4 9 ( 2 0 1 5 ) 1 e1 0
indicates the gas leakage into the working chamber, the value less than zero indicates the gas leakage out of the working chamber. As shown in this figure the leakage through paths 1,2,3,4,5,6,8 and 9 is out of the working chamber, but the leakage through path 7 is into the working path. Through the compression of the leakage rates through these paths, the leakage rates through paths 2,4,7,8 and 9 are lager than that through paths 1,3,5,6. So the increase in leakage through path 3 and path 5 causes by the use of this new meshing pair MEMP can be ignored. The total leakage flow rates in SSRC with LEMP and MEMP are shown in Fig. 11. The figure shows that at the beginning of the compression process, the gas leakage into the working chamber from the high pressure chamber is larger than the gas leakage out of the working chamber. But as the star-wheel rotating the leakage flow rates out of the working chamber increase and become larger than the gas leakage into the chamber. Another result is that the leakage rate of all the paths in SSRC with LEMP and MEMP is all most the same with each other. The total leakage flow rate can be calculated from the integral of the surround areas of the cures and the abscissa axis and listed in Table 3. The results show that the total leakage rate of these paths in SSRC with MEMP is a little decrease than that with LEMP due to the decrease of the leakage through path 2 and path 4.
6.
Conclusion
Research of leakage characteristics of SSRC with MEMP and LEMP are carried out by theoretical and experimental method in this paper. The results are obtained as following. (1) The calculated cooling capacity is in good agreement with the measured data which means the leakage models presented by this paper is reasonable to estimate the leakage characteristics of SSRC with MEMP. (2) The leakage areas of the radial leakage path at the left side of the tooth flank in SSRC with MEMP are larger than that with LEMP, which will lead to larger leakage through these paths. (3) The leakage flow rate at the both side of the tooth flank in the SSRC with LEMP and MEMP are all increased with the rotation of star-wheel. The radial leakage flow rate with MEMP is nearing about 10 times of the radial leakage rate with LEMP. (4) The axial leakage flow rate in the SSRC with LEMP and MEMP are all increased with the rotation of star-wheel. The leakage flow rate of the left axial leakage path in the SSRC with MEMP is about 65% of the leakage flow rate in the SSRC with LEMP. And the leakage flow rate of the right axial leakage path in the SSRC with MEMP is about 99.6% of the leakage flow rate in the SSRC with LEMP. (5) The leakage through paths 1,2,3,4,5,6,8 and 9 is out of the working chamber, but the leakage through path 7 is into the working chamber. Through the compression,
the leakage flow rate through paths 2,4,7,8 and 9 is lager than that through paths 1,3,5,6. So the leakage increase through path 3 and path 5 by the use of MEMP can be ignored. The total leakage rate of these paths in SSRC with MEMP is a little decrease than that with LEMP due to the leakage decrease in path 2 and path 4. According to obtained results, design of the Multicolumn Envelope Meshing Pair in a single screw refrigeration compressor will not lead to an increase of leakage. Because of its considerable increase of wear resistance, replace the straight line envelope meshing pair by the Multicolumn Envelope Meshing Pair in a single screw refrigeration compressor is suggested in this paper.
references
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