Geometric characteristics analysis for inner surface of working chamber in single screw compressor with multicolumn envelope meshing pair

International Journal of Refrigeration 108 (2019) 347–357

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International Journal of Refrigeration journal homepage: www.elsevier.com/locate/ijrefrig

Geometric characteristics analysis for inner surface of working chamber in single screw compressor with multicolumn envelope meshing pair Zengli Wang a,∗, Zhan Liu b, Hao Wang a, Jun Wang a, Quanke Feng c, Qiang Li a a b c

College of New Energy, China University of Petroleum (East China), Qingdao 266580, China College of Electromechanical Engineering, Qingdao University of Science and Technology, Qingdao 266061, China School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an 710049, China

a r t i c l e

i n f o

Article history: Received 7 March 2019 Revised 1 August 2019 Accepted 27 August 2019 Available online 29 August 2019 Keywords: Single screw compressor Geometric characteristics Inner surface area Multicolumn envelope meshing pair

a b s t r a c t In single screw compressor (SSC), the inner surface of the working chamber has always been a key issue that affects oil application and heat transfer. However, the geometric characteristics for the inner surface of the working chamber in SSC with Multicolumn Envelope Meshing Pair (MEMP) have not been carried out yet. Therefore, in this paper, a geometric model of working chamber inner surface for the SSC with MEMP is established to predict the geometric characteristics of the working process. The presented model is verified as a powerful tool for geometric characteristics analysis by the simulation results. With the validated model, the geometric characteristics of the working process and the effect of MEMP structural parameters on the geometric characteristics and the working performance for the SSC with MEMP have been analyzed. Through the comparison of the inner surface area with different structural parameters, optimization analysis of the structural parameters can be obtained. All of this work can provide the basis for the later heat transfer analysis and optimization structural design of the SSC. © 2019 Elsevier Ltd and IIR. All rights reserved.

Analyse des caractéristiques géométriques de la surface interne de la chambre de travail dans un compresseur monovis avec un couple satellite-rotor à enveloppe à colonnes multiples Mots-clés: Compresseur monovis; Caractéristiques géométriques; Surface interne; Couple satellite-rotor à enveloppe à colonnes multiples

1. Introduction Single screw compressor (SSC) which was firstly developed in 1960s by Zimmern and Patel (1972) is the mostly used gas pressurization equipment in refrigeration, air conditioning systems, petrochemical and other industries at present (Wang, 1978; Chen, 2010) with the advantages of simple structure, good stress balance, long bearing life, small vibration, and low noise (Wu and Jin, 1996). In addition, SSC is the major energy consumption equipment of the industrial production, and its energy consumption accounts for 30% ∗

Corresponding author. E-mail address: [email protected] (Z. Wang).

https://doi.org/10.1016/j.ijrefrig.2019.08.029 0140-7007/© 2019 Elsevier Ltd and IIR. All rights reserved.

or more of the total energy consumption. Therefore, the efficiency and energy consumption of the SSC has become a hot research topic so as to save energy and protect the environment. For SSC which is composed by one screw rotor and two starwheels, the gas pressurization process was realized by the meshing movement between the star-wheel and the screw rotor to form a periodically changed working chamber. During the working process, oil injection characteristic and heat transfer performance are the main factor affecting the efficiency and energy consumption of the SSC. Therefore, one possible way to improve the overall energy efficiency of the SSC is to optimize the heat transfer process. In order to achieve the above purpose, oil injection technology was first put forward into the SSC in 1982 (Zimmern, 1984). Soon the

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Nomenclature A b d K L l M P R s t u Y

Central distance between the star-wheel and the screw (m) Width (m) Diameter (m) Coordinate of Z axis Coordinate of X axis Length (m) Coordinate of Y axis Teeth number ratio between the star-wheel and the screw Radius (m) Circumferential coordinate Time step Height of the envelope column (m) Axial coordinate

Greek symbols α Star-wheel rotation angle (rad) β Inclination angle of the column (rad) γ Half tooth width angle (rad) δ Tooth width angle (rad) η Coordinates in the tooth width direction θ Envelope angle (rad) φ Inclination angle of milling cutter (rad) Subscripts 1 Screw rotor 2 Star- wheel a Front side b Back side c Casing inner wall i Number of the envelope column in State of medium enter the system mid Top end of front-side star-wheel tooth exactly right breaks away from the screw groove N Milling cutter head out Whole star-wheel tooth exactly right breaks away from the screw groove P1 profile VF P2 Lubricating oil r State of medium leave the system s The indirect measurement result T Root of the milling cutter

water and refrigerant injection technology was proposed and applied in single screw refrigeration compressor (Dong et al., 1996; Zha et al., 1997). So far, liquid injection technology has played an important role of cooling in SSC which will affect its efficiency and energy consumption performance. At present, several scholars at home and abroad had investigated the heat transfer process of the screw compressor with liquid injection. Hundy analyzed the effect of oil-injection quantity on the efficiency of single screw compressor (Hundy, 1982). Blaise investigated the influence of oil-injection parameters on the adiabatic efficiency and volumetric efficiency of the single-screw compressor (Blaise and Dutto, 1988). The oil atomization process was also hot issue. The thermodynamic model of the working process for the oil injection compressor was established by Zhou et al. and Paepe et al., by using these models, the influence of oil injection parameters and oil droplet diameter on the compressor performance were researched (Zhou and Zhu, 1987; De Paepe et al., 2005). The effect of the atomization device structure parameters on the oil droplets atomization and the compressor performance were analyzed by Li et al. to improve the heat

transfer process between gas and lubricating oil (Li et al., 1992). The oil droplets motion law in the working chamber and its impact on internal heat transfer process were studied by Zhou and Jin (1998). Zhao et al. analyzed the heat transfer of SSC based on fuzzy random wavelet finite element method. Through this study the effect of atomization on heat transfer is obtained (Zhao et al., 2016). Although a series of studies have been done on the lubricating oil injection process and the laws of the oil droplets atomization, in the actual compression process, affected by the structure of oil injection hole, oil droplets cannot achieve good atomization in SSC. So the above models that assume a good atomization of oil droplets are not accurate in the calculation of heat transfer process for SSC. For the SSC with no good atomization oil droplets, most of the heat is exchanged between the gas and the liquid film on the wall surface of the working chamber. Based on this reason, the heat transfer between the lubricating oil and the gas is not sufficient in the compression process (Xing, 20 0 0). For the SSC, the main heat transfer process of gas and lubricating oil occurs in the exhaust process and exhaust pipe. So the inner surface area of the working chamber becomes an important parameter in the study of the heat transfer process. But the calculation model of inner surface area in the above analysis is also based on the SSC with the straight line envelope meshing pair (LEMP), rather than the SSC with MEMP. As mentioned in the studies of Wang et al. (2015, 2016, 2018a), the structural characteristics of the MEMP are very different from that of the LEMP, which will seriously impact the shape and size of the inner surface of the working chamber in the SSC, so the investigation method based on LEMP is no longer applicable. For the MEMP which was deduced to further improve the wear resistance of meshing pairs (Wu et al., 2007), several studies have been carried out by the finite element method, which is the most common method used to describe the thermal-mechanical coupling process (Jiang et al., 2012, 2018; Xie et al., 2017) to research it’s design method (Wu et al., 2011; Huang et al., 2014), process ability (Wu et al., 2009), leakage characteristics (Wang et al., 2015, 2018a), injection process (Wang et al., 2018b), lubrication characteristics (Huang et al., 2015; Li et al., 2016a; 2016b) and wear resistance (Li et al., 2013). But no research has been carried out to analyze the geometric characteristics of working chamber inner surface and the influence of the meshing pair structural parameters on the geometric characteristics in the SSC with MEMP. Thus, it is very important to put forward the inner surface area geometrical model for the compressor with MEMP, so as to provide the foundation for the analysis of the heat transfer process of the SSC. In our study, geometric models of inner surface area of the working chamber which can be used in the heat transfer analysis of the SSC with MEMP are established based on the profile features of MEMP. A three-dimensional model of the inner surface area for the SSC with MEMP used to verify the theoretical calculation model is also established by the 3d direct numerical modeling method. With the validated model, the geometric characteristics of inner surface area can be analyzed and the influence of the meshing pair structural parameters on the inner surface area of the SSC with MEMP also can be investigated. All the results of this research can useful for the later heat transfer analysis and optimization structural design of the SSC. 2. Basic structure and meshing pair characteristics 2.1. Basic structure As shown in Fig. 1, the SSC depend on the meshing movement of a screw and two star-wheels to form a periodically changing working chamber to realize suction, compression and exhaust process. The periodically changing working chamber used in the

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face area of the working chamber. All of these results can useful for the analysis of oil film calculation, oil application and heat transfer. 2.2. Profile features of MEMP

Fig. 1. The schematic of the SSC.

SSC is formed by the screw groove, the upper plane of star-wheel tooth and the main case inner surface. There are 12 working chambers realize gas compression process respectively in the SSC. The working process in each working chamber under the same starwheel angle is basically the same, so one of the working chambers was selected as the control volume in the geometric characteristics analysis of inner surface area for the SSC with MEMP. For one working chamber of SSC, since the oil injection nozzle hole is a round hole directly opened to the wall (Wang et al., 2018b), oil droplets cannot achieve good atomization. So most injected oil forms a liquid film on the wall surface of the working chamber. In this condition, most of the heat transfer process during the compression process occurs between the liquid film on the wall surface of the working chamber and the compressed gas. As a result, the geometric characteristics of inner surface area for the SSC with MEMP a key issue that affects oil application and heat transfer. Then the geometric model of the working chamber inner surface area must be built. According to the analysis of working process, inner surface area of the working chamber is the base of heat transfer modeling in the study of internal working process of SSC. So the geometrical model of the inner surface area must be established firstly. In this paper, a simplified inner surface area algorithm is proposed for the SSC with MEMP to calculate the surface area of the screw groove, the upper plane of star-wheel tooth and the wall of casing that make up the working chamber, so as to obtain the total inner sur-

In the SSC with MEMP, the inner surface area becomes more complex due to the influence of the profile features of MEMP. So the profile features of MEMP must be researched. Taking the MEMP with three envelope columns as shown in Fig. 2 for example, there are three envelope columns and several transition sections between the envelope columns in the front and back sides of the star-wheel tooth flank. The column segments on the profile of the star-wheel tooth are respectively meshing with the screw groove of the screw rotor. A section A-A perpendicular to the upper surface of the starwheel tooth as shown in Fig. 2(a) is selected as analytical model. The relative position of the envelope columns on the front and back of the tooth flank in section A-A is shown in Fig. 2(b), in which columnOa1 and column Ob1 is the standard column with the inclination angle equal to 0, The rest envelope columns 2 and 3 are the normal columns with a inclination angle of the column axis equal to a constant β ai . Each envelope column center position at the root cross section and the inclination angle of the column axis are shown in Table 1. The inclination angle had been defined as the angle between the axis of the column and the Z axis which is perpendicular to the X–Y plane and goes through the cross origin of the star-wheel tooth flank. Due to the influence of inclination angle, the central coordinates of normal columns at different tooth heights are different. In the u height section of the benchmark column, the envelope column center positions are: Benchmark column:

⎧ ⎨Lau = La

Mau = Ma

⎩

(1)

Kau = Ka Envelope column:

⎧ Laui (u ) = Lai − u tan βai ⎪ ⎪ ⎨M = M aui

ai

K = Kai ⎪ ⎪ ⎩ aui βaui = βai

(2)

For the back side envelope column, the center positions in the u height section of the benchmark column follow the same rules as the front side envelope column. In the working process of the SSC with MEMP, the column segments on the profile at different star-wheel rotation angle are different from each other, which lead to the actual tooth width of the star-wheel tooth engaged into the screw groove changing

Fig. 2. MEMP with three envelope columns.

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Z. Wang, Z. Liu and H. Wang et al. / International Journal of Refrigeration 108 (2019) 347–357 Table 1 The parameters of the envelope columns. Envelope column Root Root Root Root

position position position position

of of of of

front side standard column front side column back side standard column back side column

X axis

Y axis

Z axis

La Lai Lb Lbi

Ma Mai Mb Mbi

Ka Kai Kb Kbi

Deflection angle 0

β ai 0

β bi

with the envelope column. So it is necessary to establish the tooth width calculation model for the star-wheel tooth engaged into the screw groove to put forward the geometrical model of the inner surface area. Based on the meshing characteristics of the MEMP, the calculation model of the actual star-wheel tooth width can be established. The tooth width of the front half tooth at any star-wheel rotation Angle is:



ba ( α ) =

d sin (θai (α , u ) ) − Maui 2



tan (θai (α , u ) )

d cos (θai (α , u ) ) − Laui (u ) 2

+

(3)

The tooth width of the back half tooth at any star-wheel rotation Angle is:



bb ( α ) =

d sin (θbi (α , u ) ) + Mbui 2

+



tan (θbi (α , u ) )

d cos (θbi (α , u ) ) + Lbui (u ) 2

(4)

Where b is the star-wheel tooth width, d is the diameter of the envelope column, α is the star-wheel rotation angle, u is the height of the envelope column, θ ai (α , u) is the envelope angle, the envelope angle at any star-wheel rotation angle when the height of the envelope column equal to u is expressed as:

θai (α , u) = arctan



1 (u + Kaui cos βai ) − Laui (u ) sin βai − P Maui sin (βai + α ) P A − Kaui cos α − Laui (u ) sin α − u cos (βai + α )



Fig. 3. The upper plane area calculation model.

Where Y is the axial coordinate, S is the circumferential coordinate, R1p is the equivalent radius at the meshing position, R1 is the radius of the screw rotor, α in is the suction angle, α δ is the angle of the star-wheel at the meshing position. All of these Eqs. (1)–(7) describe the profile features of MEMP. On basis of these profile features, the inner surface area calculation model of the SSC with MEMP can be obtained as follows.

(5) Where β ai is the inclination angle of the envelope column, 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. Based on the calculation model of the actual star-wheel tooth width, the equations of the screw groove profile used for building the inner surface area calculation model can be obtained. The equations of the screw groove profile in the rectangular plane coordinate system are shown as follows. In this rectangular plane coordinate system, the horizontal axis refers to the axial direction of the screw rotor, and the vertical axis refers to the circumferential direction of the screw rotor. The equation of the screw rotor front groove along the direction of star-wheel rotation VF:

⎧

 a (α ) α ) = A − R1p tan α + 2bcos ⎪ α ⎨Yp1 (

 a (αin ) − A − R1p tan αin − 2bcos ⎪

 αin ⎩ Sp1 (α ) = P R1 α − αin − αPδ

(6)

The equation of the screw rotor behind groove along the direction of star-wheel rotation VB:

⎧

 b (α ) α ) = A − R1p tan α − 2bcos ⎪ α ⎨Yp2 (



b (αin ) − A − R1p tan αin − 2bcos ⎪

 αin ⎩ Sp2 (α ) = P R1 α − αin − αPδ

(7)

3. Inner surface area calculation model 3.1. Upper plane area of star-wheel tooth As mentioned in Section 2, the periodically changing working chamber used in the SSC is formed by the screw groove, the upper plane of star-wheel tooth and the main case inner surface. According to the geometric relationship shown in Fig. 3, the upper plane area of star-wheel tooth can be obtained by the infinitesimal calculus of the shaded micro element area shown in Fig. 3 along the direction of the tooth width. In order to facilitate the calculation, the upper plane area of star-wheel tooth can be divided into two parts for integral calculation. The first part of the upper plane area is the area of the star-wheel tooth when the star-wheel rotation angle in the interval of α in ≤ α ≤ α mid , the second part of the upper plane area is the area of the star-wheel tooth when the starwheel rotation angle in the interval of α mid ≤ α ≤ α out . In which α in is the rotation angle under which the screw groove, star-wheel teeth, casing just form closing volume, α mid is the rotation angle under which the top end of front-side star-wheel tooth exactly right breaks away from the screw groove, α out is the rotation angle under which the whole star-wheel tooth exactly right breaks away from the screw groove. According to the geometric relationship of the micro element area in these two parts, the upper plane area of the star-wheel can

Z. Wang, Z. Liu and H. Wang et al. / International Journal of Refrigeration 108 (2019) 347–357 Table 2 Structure and operation parameters of the SSC with MEMP.

be gained as follows.

 ⎧ ba (α )  A − R1 ⎪ 2 − η 2 − η tan α − ⎪ R dη ⎪ 2 ⎪ cos α −bb (α ) ⎪ ⎪ ⎨ αin ≤ α ≤ αmid As ( α ) =    R2 sin γ  ⎪ A − R1 ⎪ 2 2 ⎪ R2 − η − η tan α − dη ⎪ ⎪ cos α ⎪ ⎩ −bb (α ) αmid < α ≤ αout

(8)

Where η is the coordinates in the tooth width direction, γ is the half tooth width angle between the symmetric line of star-wheel tooth and the radius thru the junction point of the star-wheel top tooth and the outer edge of the screw rotor, it can be calculated by 1 γ = arccos( A−R ) − α. R 2

3.2. Area of the screw groove bottom The screw groove bottom can be considered to be a surfaces formed by the top line of the star-wheel tooth rotating along the screw rotor shaft. So the area of the screw groove bottom can be obtained by the top line of the star-wheel tooth and the rotation radius of its center around the screw rotor shaft. As known from the geometrical relationship between the star-wheel and the screw rotor, the top line length of the star-wheel tooth can be expressed as:



lr ( α ) =

2R2 δ



R2 arccos

A−R1  R2

−α+δ



αin ≤ α ≤ αmid αmid < α ≤ αout

(9)

(10)

( b ( α )+ b ( α ) )

Where δ is the tooth width angle, δ = arcsin( a 2R b ), l is the 2 length of the side line of the star-wheel tooth. Then the area of the screw groove bottom can be obtained by the product of the top line length of the star-wheel tooth and its rotation radius:

⎧ αmid  αout ⎪ lr (α )Rr (α )P dα + lr (α )Rr (α )P dα ⎪ ⎪ ⎨ α αmid αin ≤ α ≤ αmid Ar ( α ) =  αout ⎪ ⎪ ⎪ ⎩ lr (α )Rr (α )P dα αmid < α ≤ αout

Value

Structure parameters

Value

Star-wheel diameter (m) Screw rotor diameter (m) Central distance (m) Teeth number ratio

0.181 0.181 0.1448 11/6

Thickness of star-wheel (m) Length of the screw (m) Star-wheel tooth width (m) Suction seal angle α in (rad)

0.00725 0.1557 0.0265 −0.681

3.4. The front and back wall area of the screw groove The front and back wall of the screw groove can be approximately regarded as the surface obtained by the front and back edges of the star-wheel tooth rotating along the screw axis. So it is necessary to obtain the length of the star-wheel tooth edge and the rotation radius of the tooth edge center relative to the screw axis at any star-wheel rotation angle to calculate the front and back wall area of the screw groove. According to the meshing principle of MEMP and the relative position relation between the star-wheel and the screw rotor, the length of the star-wheel tooth edge and the rotation radius of the tooth edge center relative to the screw axis can be obtained as follows. The length of the front star-wheel tooth edge:

(11)

A−R1 



R2 −

la ( α ) =

cos α

αin ≤ α ≤ αmid αmid < α ≤ αout

(13)

The rotation radius of the front tooth edge center relative to the screw axis:

R1 − la (α ) cos2 α 0

αin ≤ α ≤ αmid αmid < α ≤ αout

(14)

Then the front wall area of the screw groove can be calculated by the integration of the tooth edge length and the rotation radius:

Aa ( α ) =

 αmid α

la (α )Ra (α )P dα

αin ≤ α ≤ αmid αmid < α ≤ αout

0

(15)

The length of the back front star-wheel tooth edge:



lb ( α ) =

(α )

A−R1 + bb tan cos α arccos((A−R1 )/R2 )−α −δ R2 tan α

R2 −

α

αin ≤ α ≤ αmid αmid < α ≤ αout

(16)

The rotation radius of the back tooth edge center relative to the screw axis:

Rb ( α ) = R1 − lb ( α )

α

− ba (α ) tan α

0

Ra ( α ) =

αin ≤ α ≤ αmid αmid < α ≤ αout

Structure parameters



The rotation radius can be expressed as:

⎧ ⎨A − R2 cos α     A−R1 Rr ( α ) = arccos −α + δ R2 ⎩A − R2 cos 2

351

cos α 2

(17)

The back wall area of the screw groove:

3.3. Area of the casing inner surface The inner surface of the casing is the area of the casing surrounded by the outer edge of the front and back screw groove. So this area can be calculated by the equations of the screw groove profile VF and VB in the rectangular plane coordinate system. Subtract the axial coordinates of the VF and VB, and integrate them in the circular direction, the area of the casing inner surface can be obtained:

⎧ αmid

 ⎪ Yp1 (α ) − Yp2 (α ) P R1 dα ⎪ ⎪ ⎪ ⎨ α  αout

 + Yp1 (αmid ) − Yp2 (α ) P R1 dα αin ≤ α ≤ αmid Ac ( α ) = αmid ⎪ ⎪ ⎪ αout

 ⎪ ⎩ Yp1 (αmid ) − Yp2 (α ) P R1 dα αmid < α ≤ αout α

(12)

Ab ( α ) =

 αout α

lb (α )Rb (α )P dα

(18)

Then the total heat transfer area can be expressed as follows:

Atot (α ) = As (α ) + Ar (α ) + Ac (α ) + Af (α ) + Ab (α )

(19)

All of these equations are the geometric models of inner surface area for the SSC with MEMP which will be used for the investigation of the heat transfer characteristics. 4. Results and discussion In order to solve and verified the geometric model of inner surface area for the SSC with MEMP, a typical SSC with the structure parameters as shown in Table 2 was developed in this paper as an example for the theoretical study. According to the geometric mod-

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Fig. 4. Measured and calculated total inner surface area of the SSC with MEMP.

els of inner surface area, the geometric characteristics of working chamber for the SSC with MEMP were investigated and the influence of the meshing pair structural parameters on the geometric characteristics are also discussed in this section. 4.1. Model validation As mentioned in Section 2.1, the screw rotor and the star-wheel maintain the conjugate meshing relationship in the working process of SSC. Based on the meshing principle and structural characteristics, a 3d direct numerical modeling method were proposed to establish the three-dimensional model of the working chamber for the SSC with MEMP. In this method, the screw rotor work blank is fixed, and the milling process is simulated by transferring the screw rotor movement process into the movement process of the milling cutter. For simulating the milling process, the coordinates and movement path of the milling cutter must be determined. According to the meshing principle and the geometrical structure characteristics, the coordinates and the motion path of the milling cutter can be expressed by the following equation:

⎧ xi,N = −Ri,N · sin (P π t ) · sin (φi,N + π t ) + Mi · cos (P π t ) ⎪ ⎪ ⎪ ⎪ + A · sin (P π t ) ⎪ ⎨ yi,N = −Ri,N · cos (P π t ) · sin (φi,N + π t ) − Mi · sin (P π t ) ⎪ ⎪ ⎪ + A · cos (P π t ) ⎪ ⎪ ⎩ zi,N = Ri,N · cos (φi,N + π t ) ⎧ xi,T = −Ri,T · sin (P π t ) · sin (φi,T + π t ) + Mi · cos (P π t ) ⎪ ⎪ ⎪ ⎪ + A · sin (P π t ) ⎪ ⎨ yi,T = −Ri,T · cos (P π t ) · sin (φi,T + π t ) − Mi · sin (P π t ) ⎪ ⎪ ⎪ + A · cos (P π t ) ⎪ ⎪ ⎩ zi,T = Ri,T · cos (φi,T + π t )

(20)

(21)

Where i is the number of the envelope column, N represents the milling cutter head, T represents the root of the milling cutter, Mi is the basic structural parameters of an envelope column listed in Table 1, t is the processing time step, φ i, N and Ri, T are the structure parameters of milling cutter. Based on the three-dimensional model of the inner surface area, the total wall area of the working chamber can be tested by the three-dimensional modeling software. The measured and calculated total inner surface areas of working chamber under different star-wheel rotation angle are shown in Fig. 4.

Fig. 5. Inner surface area of the SSC with MEMP.

As shown as the curves of the measured and the calculated results in this figure, the calculated total inner surface area under different star-wheel rotation angle are all in good agreement with the measured data, the maximum error of the calculated total inner surface area is 4.9%. This indicates that the presented geometric model of inner surface area for the SSC with MEMP is reasonable to estimate the geometric characteristics of the compressor and be used to research the influence of the meshing pair structural parameters on the heat transfer characteristics. 4.2. Geometric characteristics for working process With the help of the validated mathematical model presented in this paper, the geometric characteristics for working chamber were studied. In order to analysis the influence of the meshing pair structural parameters on the geometric characteristics, further research is proposed. The central distance between the star-wheel and the screw rotor, the diameter ratio of the star-wheel to the screw rotor, the star-wheel diameter and the screw rotor diameter were selected as the main profile parameters to be examined. The inner surface areas calculated by the validated mathematical model are shown in Fig. 5. It can be seen from the changing tendency of the curves in Fig. 5(a) that the surface area of screw groove bottom, the surface area for the main casing inner wall and the area for the surface of front and back screw groove decrease

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353

Fig. 6. Effect of meshing pair structural parameters under constant volume condition.

with the increase of the star-wheel rotation Angle. Among these areas, the change curve of surface area for the main casing inner wall with the star-wheel rotation angle is of type ‘S’, and the change rate first decreases and then increases. In addition to these areas, the upper plane surface area of star-wheel tooth increases first and then decrease with the increase of the star-wheel rotation Angle. As mentioned in Section 2, the total inner surface area is the sum of the areas shown in Fig. 5(a), so according to the calculation results as shown in Fig. 5(a), the total inner surface area of the SSC with MEMP can obtained and shown in Fig. 5(b). As can be seen in

this figure, the total inner surface area decreases linearly with the increase of the star-wheel rotation angle. 4.3. Effect of meshing pair structural parameters In the SSC with MEMP, the meshing pair structural parameters have great influence on the shape and size of the working chamber wall area. Therefore, in order to study and optimize the heat transfer process of the SSC with MEMP, it is necessary to study the influence of meshing pair structural parameters on the geometric characteristics of the working chamber.

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4.3.1. Constant volume condition The influences of the meshing pair structural parameters on geometric characteristics for the working chamber were investigated under the condition that the maximum working chamber volume of compressor is constant. In the studies, the effects of structural parameters including the central distance between the star-wheel and the screw rotor and the diameter ratio of the star-wheel to the screw rotor were considered. Research results are shown in Fig. 6(a) and 6(b). Fig. 6(a) shows the effect of the diameter ratio of the starwheel to the screw rotor on the total inner surface area. In this study, the maximum working chamber volume remained constant. So the tooth width of the star-wheel decreased with the increase of the diameter ratio of the star-wheel to the screw rotor. Meanwhile, through the comparison of curves in the figure, it can be seen that the total inner surface area will gradually increase with the increase of the diameter ratio. But the change of the total inner surface area with the diameter ratio is small. The effects of the central distance between the star-wheel and the screw rotor on the geometric characteristic of the working chamber were analyzed with the maximum working chamber volume remained constant. During this analysis, the diameter of the star-wheel and the screw rotor remains constant, therefore, the tooth width of the star-wheel increases with the increase of central distance between the star-wheel and the screw rotor. The analysis results obtained based on this condition are shown in Fig. 6(b). The contrast of the curves under different central distance shows that the total inner surface area decreases with the increase of the central distance. After comprehensive consideration of the influence of central distance, diameter ratio and tooth width on geometric characteristic of the working chamber, it can be known that when the working chamber volume is certain, the design of the star-wheel tooth into a narrow and long shape can have a large inner surface area, which is conducive to the heat transfer process.

4.3.2. Non-uniform volume condition In this section the effects of different meshing pair structural parameters on geometric characteristics for the working chamber are analyzed under non-uniform maximum working chamber volume condition. In the first condition, the central distance and the star-wheel tooth width remain unchanged when the diameter ratio changes. Thus the maximum working chamber volume increases with the increase of the diameter ratio of the star-wheel to the screw rotor as can be seen in Fig. 7(b). As the volume increases, the total inner surface area of the working chamber also increases remarkably with the increase of the diameter ratio which can be seen in Fig. 7(a). In order to further analyze the relationship between the increase of volume and the increase of maximum inner face area caused by the change of diameter ratio, the curves in Fig. 7(b) were compared and analyzed. It can be seen from the changing trend of the curves in Fig. 7(b) that the maximum working chamber volume and maximum inner surface area increase linearly with the increase of the diameter ratio. In addition, the increase rate of maximum working chamber volume with diameter ratio is bigger than the increase rate of maximum inner surface area with diameter ratio. That is to say that if the tooth width and central distance remain unchanged, increasing the working chamber volume by increasing the diameter ratio will reduce the heat transfer performance in the working process. Furthermore, in order to express and evaluate the influence of meshing profile parameters on working chamber volume and heat transfer area, a dimensionless equation related to the total heat transfer area and working chamber

Fig. 7. Effect of the diameter ratio under non-uniform volume condition.

volume was put forward as follows:

AOV (α ) =

d(Atot (α ) ) dj

d V α ( ( )) dj

(22)

Where j is any one of the meshing pair parameters. Then this dimensionless equation can be used to analyze the influence of meshing profile parameters on the heat transfer performance in future. In the second condition, the diameter of screw rotor and starwheel and the star-wheel tooth width remain unchanged when the central distance between the star-wheel and the screw rotor changes. Thus the maximum working chamber volume decreases with the increase of the central distance between the star-wheel and the screw rotor as can be seen in Fig. 8(b). As the maximum working chamber volume decreases, the total inner surface area of the working chamber also decreases remarkably with the increase of the central distance between the star-wheel and the screw rotor as shown in Fig. 8(a). The increase rate of maximum working chamber volume and the increase rate of maximum inner surface area caused by the change of central distance were analyzed and described in Fig. 8(b). The cures in Fig. 8(b) were compared and analyzed to research the effect of the central distance on the geometric

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Fig. 8. Effect of the central distance under non-uniform volume condition.

Fig. 9. Effect of the diameter with no constant central distance.

characteristic of the working chamber and the heat transfer process of the SSC. By contrast of the curves in Fig. 8(b), the maximum working chamber volume and the maximum inner surface area decrease linearly with the increase of the central distance. However, the decrease rate of the maximum working chamber volume with central distance is bigger than the decrease rate of maximum inner surface area with central distance. Through the above analysis, the following conclusions can be drawn, by reducing the central distance, the maximum working chamber volume can be improved, which will also reduce the inner surface area between the gas and the wall during the working process.

volume increases with the increase of the screw rotor/star-wheel diameter as shown in Fig. 9(b). Meanwhile, the total inner surface area of the working chamber also increases remarkably with the increase of the screw rotor/star-wheel diameter which can be seen in Fig. 9(a). Fig. 9(b) shows the growth trend of the maximum working chamber volume and maximum inner surface area with the increase of the screw rotor/star-wheel diameter when the central distance changes with the screw rotor diameter. Through the contrast of the curves in this figure, the increase rate of maximum working chamber volume and maximum inner surface area caused by the change of screw rotor/star-wheel diameter were analyzed to research the effect of the screw rotor/star-wheel diameter on the geometric characteristic of the working chamber and the heat transfer process of the SSC. Analysis results show that the maximum working chamber volume and the maximum inner surface area increase linearly with the increase of the diameter. The increase rate of the maximum working chamber volume with diameter is smaller than the increase rate of maximum inner surface area with diameter. This shows that the heat exchange performance between the gas and the wall during the working process will be strengthened slightly when the maximum working chamber volume is increased by increasing the diameter of screw rotor and star-wheel in an equivalent amount.

4.3.3. Constant diameter ratio condition In this section the effects of different meshing pair structural parameters on geometric characteristics of the working chamber are analyzed under constant diameter ratio condition. In this analysis condition, the diameter ratio of the star-wheel to the screw rotor is equal to 1.0. Firstly, the effects of the screw rotor/star-wheel diameter on the geometric characteristics of the working chamber were researched when the central distance changes with the screw rotor diameter. In this study, the star-wheel tooth width remains unchanged, and the central distance between screw rotor and star-wheel equal to 0.8 times of the screw rotor, so the maximum working chamber

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ing chamber volume with diameter is similar to the increase rate of maximum inner surface area with diameter when the screw rotor/star-wheel diameter increase ratio below 1.125. Beyond this value, the increase rate of the maximum working chamber volume with diameter is slightly larger than the increase rate of maximum inner surface area with diameter. Then the following conclusions can be drawn that heat transfer performance between the gas and the wall during the working process will be basically unaffected by the increase of the diameter when the diameter doesn’t increase much, but heat exchange performance between the gas and the wall will be decreased when the screw rotor/star-wheel diameter increase ratio above 1.125.

5. Conclusion Geometric model of the working chamber inner surface area for the SSC with MEMP is established and used to research the geometric characteristics of the working chamber and the influence of the meshing pair structural parameters on the geometric characteristics and heat transfer process of the SSC with MEMP. The three-dimensional model of the inner surface area for the SSC with MEMP is also established to verify the theoretical calculation model. The results are obtained as follows.

Fig. 10. Effect of the diameter with no constant central distance.

In addition, the effects of the screw rotor/star-wheel diameter on the heat transfer on geometric characteristics of the working chamber were researched when the central distance stays the same. In this study, the star-wheel tooth width also remains unchanged, and the maximum working chamber volume also increases with the increase of the screw rotor/star-wheel diameter as shown in Fig. 10(b). Due to the increase of the working chamber volume, the total inner surface area of the working chamber also increases remarkably with the increase of the screw rotor/starwheel diameter which can be seen in Fig. 10(a), and the increment value of the total inner surface area caused by increasing the diameter when the central distance remains unchanged is greater than that caused by increasing the diameter when the central distance changes. Fig. 10(b) shows the growth trend of the maximum working chamber volume and maximum inner surface area with the increase of the screw rotor/star-wheel diameter when the central distance stays the same. So as to research the effect of the screw rotor/star-wheel diameter to the growth trend of the maximum working chamber volume and the maximum inner surface area, the curves in this figure were compared. The comparison of curves in the figure shows that the maximum working chamber volume and the maximum inner surface area increase linearly with the increase of the diameter. The increase rate of the maximum work-

(1) The calculated total inner surface area of the SSC are all in good agreement with the measured dates by threedimensional model, the maximum error of the calculated total inner surface area is 4.9% which means the theoretical calculation model presented in this paper is reasonable to estimate the geometric characteristics of the working chamber for SSC with MEMP. (2) The surface area of screw groove bottom, the surface area for the main casing inner wall and the surface area of front and back screw groove decrease with the increase of the starwheel rotation Angle, but the upper plane surface area of star-wheel tooth increases first and then decrease with the increase of the star-wheel rotation Angle. As a result, the total inner surface area decreases linearly with the increase of the star-wheel rotation angle. (3) When the maximum working chamber volume remains constant, the total inner surface area will gradually increases with the increase of the diameter ratio (tooth width of the star-wheel decreased) and decreases with the increase of the central distance(tooth width of the star-wheel increased). It means that when the working chamber volume is certain, the design of the star-wheel tooth into a narrow and long shape can have a large inner surface area, which is conducive to the heat transfer process. (4) When the central distance and the star-wheel tooth width remain unchanged, increasing the maximum working chamber volume by increasing the diameter ratio will reduce the heat exchange performance in the working process; increasing the maximum working chamber volume by reducing the central distance will also reduce the heat exchange performance between the gas and the wall during the working process. (5) When the diameter ratio of the star-wheel to the screw rotor remains content, the heat exchange performance between the gas and the wall will be strengthened slightly when the maximum working chamber volume is increased by increasing the diameter of screw rotor and star-wheel in an equivalent amount(central distance varies with screw diameter); the heat exchange performance between the gas and the wall will be basically unaffected by the increase of the diameter when the diameter doesn’t increase much, but will

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