study of a single screw compressor with a conical teeth gate rotor

STUDY OF A SINGLE SCREW COMPRESSOR WITH A

CONICAL TEETH GATE ROTOR

*Yang,Shyue-Cheng ** Tsang-Lang Liang

*Professor ** Associate Professor

Department of Industrial Education and TechnologyNational Changhua University of Education,

Bao-Shan Campus: Number 2, Shi-Da Road, Changhua, 500, Taiwan, R.O.C.

Corresponding author. E-mail:[email protected]

Received September 2005, Accepted November 2008No. 05-CSME-55, E.LC. Accession 2907

ABSTRACT:

From a geometric viewpoint, a mathematical model of a single screw compressor with a conjugate

pair of meshing conical teeth gate rotor is a conjugate problem. Coordinate transformation and envelope

theory are applied to determine the sets of spatial points of the contacting surfaces that define the main

rotor of a single screw compressor. Envelope theory and analytical procedure are used to derive

mathematical models of a gate rotor and a main rotor. Stress analysis for the single screw compressor

mechanism is performed. PowerMILL software package is used to simulate the manufacture of a main

rotor. A numerical example with a compressor ratio of 11:6 is presented to demonstrate the application of

the mathematical models developed in this paper.

ETUDE D'UN COMPRESSEUR SIMPLE DE VIS AVEC UN ROTOR CONIQUE

DE PORTE DE DENTS

D'un point de vue geometrique, un modele mathematique d'un compresseur simple de vis avec une

paire conjuguee d'engrener Ie rotor conique de porte de dents est un probleme conjugue. La theorie du

meme rang de transformation et d'enveloppe sont appliquees pour determiner les ensembles de points

spatiaux des surfaces de contact qui definissent Ie rotor principal d'un compresseur simple de vis. La

theorie d'enveloppe et Ie procede analytique sont employes pour deriver les modeles mathematiques d'un

rotor de porte et d'un rotor principal. L'analyse des contraintes pour Ie mecanisme simple de compresseur

de vis est executee. Le progiciel de PowerMILL est employe pour simuler la fabrication d'un rotor

principal. Un exemple numerique avec un rapport de compresseur de 11:6 est presente pour demontrer

I'application des modeles mathematiques developpes en ce document.

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1. INTRODUCTION

Twin-screw compressors are being so widely used that the term 'screw compressor' is

generally associated with 'twin-screw compressor'. The advantages of a twin-screw compressor

include a lower production cost because it allows the use of both male and female rotors, its

relative simplicity, general reliability and high availability. However, the disadvantage of a

twin-screw compressor is the poor balance of force of its compressive gas. This is because the

axes between the two rotors are parallel, thus keeping the wearing forces high [1, 2]. To

compensate this disadvantage, a single screw compressor can be used. In factor, Haselden [3]

proposed that the single-screw compressor has the potential to become the dominant form of

compressor in the 50 to 150 kw power range. It is capable of high energy efficiency because

losses due to leakage, frictional effects and heat transfer factors can be reduced to a low level.

The gate rotor design for such a single-screw compressor is described. Such a gate rotor is less

expensive to manufacture, its bearings simpler, and its casing becoming a simple pressure vessel

[3]. In addition, the compression ratio of a single-screw compressor is higher than a twin-screw

compressor [4, 5]. Because with two gate rotors, compression occurs simultaneously on both

sides of the main rotor. Thus compressive forces are radially balanced. Moreover, thrust forces

are minimal since suction pressure is ported to both ends of the screw. In addition, since the axes

of a gate rotor are perpendicular to the axis of its main rotor, the frictional forces of the main

rotor are lower than that of a twin-screw compressor.

Hence a single screw compressor is a very important device in oil-injected refrigeration and

air conditioning compressors. This important machine was invented in the early 1960s. Its

primary components comprise a main rotor, two gate rotors, and a main rotor housing as shown

in Fig. 1. The two gate rotors are located on both sides of the main rotor, which is usually a

multiple-threaded screw. The gate rotor generally has eleven teeth. The main rotor is driven, and

rotates about its central axis while the gate rotor, which has at least one tooth in meshing

engagement with the main rotor threads is typically driven by the main rotor.

Zimmem [6] was the first to report the design and operating characteristics of the Zimmem

single screw compressor. Later, he presented a slide-proof method for processing fluids under

high differential pressure and liquid lock for U.S patent [7, 8]. His design consists of a screw

rotor meshing with one pair of gate-rotors. Others, such as Aquino et al. [9] (U.S. Patent),

presented a method for making a screw rotor, and Jensen [10, 11] presented a method for

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manufacturing compressor rotors. Based on these patents and papers, the operation of a single

screw compressor can be divided into four-phases: suction, sealing, compressing and discharging.

Sagara et al. [12] developed oil-free single-screw compressors and used liquid-refrigerant

injection to cool the compressor. To support this technology, compressor-performance analysis

tools have been developed that model the flow of mixed liquid- and gas-phase refrigerant in the

compressor chamber. For example, Yang [13] used the inversed envelope concept to determine

the cutting-edge curve for machining the screw rotor where the teeth of the gate rotor are planar

and the grooves of the screw rotor are directly generated by the straight edge of the gate rotor. In

this process, the direct envelope concept and equation of meshing are not applied to obtain the

main rotor. Bein et al. [14] presented a computer modeling of an oil flooded single screw air

compressor.

In summary, past patents and papers have presented single screw compressors, and Yang [13]

have studied gate rotors with straight-line teeth. However, a mathematical model of a single

screw compressor with conical teeth gate rotors has yet to be presented. Fig. 1 shows the

mechanism of a single screw compressor. ~l is the gate rotor, and ~2 the main rotor. The gate

rotors include a number of conical teeth. The main rotor is produced by multiple-threaded

grooves. A pair of gate rotors is placed on both sides of the main rotor. When the main rotor is

driven by a motor, the meshing side ofthe gate rotor is driven. The teeth on the gate rotor can be

linear or conical. In this paper, conical teeth gate rotors are studied and the direct envelope

concept is used. Three-dimensional stress analysis of the proposed single screw compressor

mechanism is also investigated. A tool path generation method for determining the cutter location

file for CNC five-axis machining of a single screw compressor with conical teeth is implemented.

In general, the compressor ratio of a single screw compressor is 11 :6.

This work presents a direct envelope method to determine the mathematical models of

single screw compressors and the contact lines of the engagement pairs. According to gear theory

[15, 16], the mathematical model of the main rotor is regarded as an envelope to a one-parameter

family of gate rotor's surfaces. The geometric model of a single screw compressor with

compression ratio 11:6 is illustrated by an example design. In paper [13], the mathematical

model of the main rotor is obtained by the family of gate-rotor's curves. The draw of the main

rotor with gate rotors is given to show the contour of a single screw compressor. Essentially, the

basic idea for the single screw compressor is developed from the gear theory, in that it consists of

a screw rotor meshing with one pair of gate rotors.

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According to developed mathematical models, the Turbo C ++ programming language is

applied to evaluate the spatial set of contact points between the gate rotor and the main rotor. The

ASCI file form is used to save the result of the evaluation. Based on the PowerMILL software

package, the post-process file of a main rotor is applied to generate the cutting path of its

geometrical model. Finally, the MSC. visualNastran Desktop software is used to develop the

animation of a single screw compression mechanism. The results for this study can be used to

manufacture the main rotor.

Gate Rotor LI

Fig. 1 Single screw compressor rotors.

2. THE DESIGN OF THE GATE ROTOR PROFILE

A screw compressor with a gate rotor and a main rotor is chosen. As shown in Fig. 1, the

shape of conical teeth gate rotors is studied. The radius of pitch of the gate rotors is re • The- - -

profile of the gate rotors is designed to consist of three regions AB, BD, and AI, as shown inFig. 2. The equation for the gate rotors can be expressed as coordinate system SI:

2-1 REGION AB OF GATE ROTOR

The region AB of the gate rotor is used to generate the different sides of the main rotor.Fig. 2 shows 81 , the curvilinear parameter of the generating (gate rotor) surface. The equation

for region AB, represented in coordinate system SI' can be written as follows:

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(d + 01 sin~)cos f3 COSe - (re COSe' + (eo - O,)cos~)sine o< 8 < e 0 < f3 < 2;rR1AS = -(d+0Isin~)cosf3sine-(recose'+(eO-01)cOS~)cose' I 0'

(d + 01 sin~) sin f31

(1)

where Fig. 2 is a normal section of the gate rotor. The range of f3 is [0, 27!] in order to

create the gate rotor conical teeth surfaces. re is the pitch radius of the gate rotor. The position

vector, RJAB , generates the conical surface of a tooth, as shown in Fig. 2. The tooth thickness is

2d. According to the geometry ofFig.2, the data ~ =sin-I ((re sinO' -d)/ ea) and d=re sine' /77

can be determined listed in Table 1. Form parameters of the gate rotor are

c;=sin-1((re sinO'-d)/eJ and O'=~, where TJ is the design coefficient, N J is theTJNJ

number of convex teeth of the gate rotor, ea is the tooth length, and c; is the tape angle of

tooth. The fist tooth of the gate rotor is at 0=0. Based on Eq. (1), Table! and Turbo C++

programming language and SolidWorks software, a tooth of the gate rotor is obtained. Other teeth

can be replicated using parameter 0 in a radial array. For example, the second tooth of the gate

rotor is at 0 = 7!/ N J •

Fig. 2 A profile of a gate rotor is designed to consist of three regions.

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2-2 REGION BD OF THE GATE ROTOR

Region BD of the gate rotor is used to generate the bottomland of the helical grooves of

the main rotor. 62 is the curvilinear parameter of the gate rotor. The position vector, R iBD , for

Region BD of the gate rotor can be obtained using:

62 cos fJ cos B - (recos B' + ea cos~) sin B

RIBD

= - 62 cos fJsinB - (recosB' + ea cos~)cosB , 0 < f3 < 2;r, -d < 02 < d (2)62 sinfJ

1

where the range of fJ is [0, 2:r] in order to create the gate rotor conical teeth surfaces.

2-3 REGION AI OF THE GATE ROTOR

Originally, Region AI of the gate rotor is used to generate the top lands of a main rotor.

However, in generating the main rotor, the generated main rotor is required to be a part of the

original cylindrical surface, which is a critical sealing surface with the casing from which it is

cut. Thus, Region AI of the gate rotor is only used to generate the top lands of the main rotor.

The position vector for Region AI of the gate rotor can be expressed as follows:

re sin Br

RIA! = - re cos Br

63

I

B'< B, r < 3B', -d < 03 < d. (3)

where Br and 63 are the design parameters of the gate rotor. Based on Eqs. (1)-(3), Table

1 and Turbo C++ programming language and SolidWorks software, the complete profile of the

gate rotor is obtained and shown in Fig. 3.


Fig. 3 A 3D model of a gate rotor.

3. COORDINATE TRANSFORMATION MATRICES

Fig. 4 shows the coordinate systems S/(OI,XI,~,Z/) and S2(02,X2,r;,Z2) of the

gate-rotor's rotary center and the main-rotor's rotary center, respectively. Coordinate systems SI

and S are allowed to rotate about the Z -axis and Z2 -axis, respectively. Coordinate system2 1

Sj(Xj,Yj,Zj) shares the same origin as the coordinate system SI' but is fixed. The ZI' Z2'

and 2 j axes are determined by the right-hand co-ordinate system. When the gate-rotor rotates

along the 2/ axis with the rotary angle ¢I' the main-rotor rotates along the Z2 axis with the

angle ¢2' The tooth length of the gate rotor is ea' The distance between 01

and 02 is a.

According to envelope theory, the profile of a main rotor is the envelope to the family of the

gate-rotor's surfaces. In such a condition, the main-rotor is assumed to be stationary while the

gate-rotor rotates at the angle ¢1 about the 21

axis in the clockwise direction. At the same

time, the main-rotor rotates at the angle ¢2 about the Z2 axis in the clockwise direction. The

relationship between ¢1 and ¢2 is N1¢1 =N 2 ¢2' where N

1is number of teeth of the gate

rotor, and Nz is the number ofhelical grooves of the main rotor.

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Fig. 4 The coordinate systems of a single screw compressor.

In Fig. 4, the position vector of the contact point P between the gate-rotor's surface and themain-rotor's surface is represented in the 8

1coordinate system. It is fixed in the gate-rotor's

surface, and is denoted as RJi(f3, 8). The subscript i is i = AB, BD, and AI. The

subscript j is 1, 2, and 3. The equation of the gate-rotor's surface, '2/, in the 81

coordinate

system may then be expressed as Rli (fJ,8j ) = lXli(fJ,8j)'~i(fJ,8j),Zli(fJ,8)J, where fJ and

8. are the surface parameters of the gate rotor (~1). According to gear theory [15, 16] andJ

differential geometry [17], the contact point must lie in the parameter family of the main-rotor's

surfaces. Therefore, the family of gate-rotor's surfaces can be expressed by R2JfJ, 8j

, tP2) in

the 82

coordinate system. By applying the homogeneous coordinate transformation matrix, the

relationship between the coordinate systems 82

and 81

can be expressed as

sintP2 sintP, sintP2 COStPl -COStP2 asintP2

M21

= COStP2 sintPl COStP2 COStPl sintP2 acostP2 (4)

COS¢1 - sin¢l 0 0

o 0 0 1

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The relationship between the angles tP, and tP2 is tP,(tP2)' given that tPI = (N2/ N1)tP2'

where ¢i is the rotary angle of the gate-rotor and ¢2 is the rotary angle of the main-rotor.

Subsequently, using Eqs. (1)-(3) and the coordinate transformation matrix M 2" the family

of the gate-rotor's surfaces Ii can be obtained by

(13 6 ,/,) {M21~ABR2i , j' 'r2 = M »_.21~"1BD

(5)

According to gear theory, the geometric model of the main rotor must simultaneously

satisfy Eq. (5) and

N~ •Vp = [Xli (Xli.oYliOj - XliOjYJi.o) - Zli (ZliOjYli.o - YliOjZli.o)] sin tPl +

[YJi( XJi.oY'U5j - YliPxJi(jj) - Zli(Zli.oXliOj - Xli.oZliOj )]cos ¢l + (6)

[xJi (xliOjZJi.o - XJi.oz liOj ) - Yli (YJi.oZliOj - YliOjZli.o)] :~: - a(XJi.oYliOj - YJi.oXliOj) = 0

where Eq. (6) is called equation of meshing. The values Xli' Yli and Zli are shown in Section

2. The subscript i represents AB, BD and AI. The symbols x lifJ ' YlifJ and ZlifJ are

8X1i /813, 8y1i /813 and 8zIi /813, respectively. Similarly, the symbols x lio ., Ylio' and Zlio'} } }

are 8X1i /86j , 8y1i /86j and 8ZIi /86j , respectively. N~ is normal to the gate-rotor's surface

L', and is represented in the fixed coordinate system Sf' Vector Vj2 is the relative velocity

between the gate-rotor's surface L' and the main-rotor's surface L 2, and is represented in the

Sf co-ordinate system which is rigidly connected to the compressor house. R2i (f3, 6j , tP2)

is the family of the gate-rotor's surfaces L', and is represented in the co-ordinate system S2

which is rigidly connected to the main-rotor's surface. An envelope to the family of gate rotor's

surfaces must simultaneously determine Eqs. (5) and (6).


4. GENERATION OF MAIN ROTOR SURFACE

The generated main rotor has to be a part of the original cylindrical surface from which it is

cut. The top surface of the main rotor is a part of the original cylindrical surface from which it is

cut. The cylindrical surface is a critical sealing surface with the casing. In fact, the addendum

surface of a main rotor may take the shape of the cylindrical flank. Therefore, Region AI of

Fig. 2 cannot be considered in the generating process of a screw rotor. The main rotor is

generated by the relative motion between the gate rotor and the main rotor as determined by the

coordinate setting graphics. The profile of the gate rotor is a surface of revolution, the teeth of

the gate rotor are the conical surface, and j3 and 6j are the defined curvilinear parameters.

Substituting Eqs. (1) to (3) and their partial derivative with respect to surface parameters j3 and

6j into Eq. (6) and rearranging these terms, the parameter f3 must be replaced by 6j and

¢2 . In other words, the surface parameters of the main rotor (generated surface) are 6j and ¢2.

The parameter ¢2 is used to generate the helical grooves of the main rotor. As a rule, one can- --

obtain the meshing equations of the strips AB, BD, AI are obtained as:

The envelope of the region AB of gate rotor is

jR2Ai/fJ' cS" rPJ = M21R1AB

fJ = tan -, ( (re cos(J' + (eo - cSJcos~) - ((d + cS1 sin~)tan~)(orP, / 0rP2) ) .(a - (d + cS, sin~) tan ~ cos( (J - rP1 » - (re cos (J' + (eo - cS1) cos~ cos((J - rP, »

o~ cS, ~ ea

The envelope of the region BD of gate rotor is

R 2BD (j3, 62, ¢2) =M21RIBD

sine 8¢1

j3 = tan -I ( 8¢2 ), - d < 62< dcos(e - ¢I)

The envelope of the region AI of gate rotor is

(7)

(8)

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{

R 2Aj (j3, 03' ¢2) = Mz1R1Aj

03 =0(9)

where the subscript 2 refers to the generated surface (main rotor). The dimensional parameters of

the geometric surface of a single screw compressor with a gate rotor are listed in Table 1. In

general, the relationship between the angles ¢I and ¢2 is given by ¢l = (Nz / N1)¢2' Based on

Eqs. (7) - (9), Turbo C++ programming language is used to generate the spatial set of meshing

points for the contacting surfaces that define the main rotor. The geometric model of the main

rotor can then be described using the SolidWorks software package. To prove the mathematical

model, a rapid prototyping technology is used to assemble the screw rotor and the gate rotors, as

shown in Fig. 5. For simplicity, the gate rotors in Fig. 5 were manufactured using plastic planar

teeth.

Table 1 Major design parameters of the single screw compressor.

Parameters

Number ofTeeth N 1 11

Number ofhelical grooves Nz 6

re 108mm

Length ofTooth ea 50

Center-Distance a 2re

fJ O~27r

()' ;r/(7]N1)

Tape angle of tooth ~ sin-I «re sin B' - d) / eJ

()r [()', 3()']

Tooth Thickness d re sin ()' / 7]

1. 2(main rotor)

Coefficient of design 7] 1.3 (gate rotor)


Fig. 5 The assembly of the screw rotor and the gate rotors.

5. CONTACT LINE

The mating surfaces LJ and L2 contact each other at every instant along a line called

the contact line or characteristic line. The location ofthe instantaneous contact line on the mating

surface depends on the parameter of motion, ¢2' The contact lines on the gate-rotor's surface

'L/ may be expressed as:

(10)

where ¢y) are constant values when the parameter of motion, ¢2' is given constant values.

Using Eq. (6), E(P, OJ' ¢?)) is the meshing equation.

A similar procedure can be applied to determine the instantaneous contact lines on the

surface 2:2, which is the envelope to the family of gate-rotor surfaces. The instantaneous

contact line on surface 2: 2 can be expressed as:

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{RZi = R2i (/3,e,t/JY»)

E(/3,e,t/JY») = ° (i= AB, BD, AI, k=0, 1,2,3,...) (11)

where t/J Y) are constant values when the parameter ofmotion ¢2 is given constant values.

Based on Eqs. (1)~(3) and (10), the instantaneous contact lines on the generating surface

(gate rotor) are plotted (Fig. 6a). According to Eqs. (7)~(9) and (11), the instantaneous contact

lines on the generated surface (main rotor) can be drawn (Fig. 6b). From Figs. 6a and 6b, the

instantaneous contact line between the gate rotor and main rotor is made up of lines. The contact

line ensures that the gate rotor and the generated main rotor are engaged at every point along the

contact lines. The dimensional parameters of the geometrical surface of the single screw

compressor listed in Table 1 and Eqs. (7)-(9) , yield a tight seal and lubrication that reduce the

effect of leakage on compressor performance. The geometric model of the main rotor and gate

rotors are assembled using SolidWorks software package. The contour of the main rotor and gate

rotors is shown in Fig. 7.

Contact Lines

Fig. 6a The instantaneous contact lines on a gate rotor's surface.


Contact Lines

Fig. 6b The instantaneous contact lines on a main rotor's surface.

Fig. 7 Finite element model of two meshing gate and main rotors

6. STRESS ANALYSIS

This study assumed zero assembly error. The geometrical model of the single screw

compressor mechanism is developed using Turbo C++ programming language and a computer

aided design (CAD) program. Applying finite methodology [18] and a general proposed

computer program [19], stress analysis for the single screw compressor is performed. A

computer program running on a Windows XP operating system is used to evaluate the numerical

solution for the contact stress. The material is assumed to be isotropic and homogeneous with a

Young's modulus of E = 207GPa and a Poisson ratio of v = 0.3.

A CAD package SolidWorks 2006 is used to develop a geometric model for the single screw

compressor. The models are developed by the CAD computer program and then transferred to


the FE package ANSYS Workbench for stress analysis. Fig. 7 shows a total of 23941 elements

with 36010 nodes for the left gate rotor, and a total of 24609 elements with 36938 nodes for the

right gate rotor, and a total of 18487 elements with 29414 nodes for the main rotor. Two

additional cylindrical supports within the gate rotors were referenced to its own center. A torque

of 10Nm was applied to the single screw rotational axis to create surface contact between the

gate rotors and the main rotor. The gate rotors were free at the tangent to the outer cylindrical

surface. In the contact stress analysis, the "contact pair" was defined as the "contact bodies" and

the "target bodies". The contact bodies were identified as the gate rotors and the target bodies

were identified as the single screw rotor. The interaction option between the contact surfaces was

defined as zero seperation. The stress distribution of the single screw compressor mechanism

was evaluated and the results shown in Fig. 8. The maximum stress of von-Mises in Fig. 8 was

8.73 x 105 N / m2• As shown in Fig. 8, the maximum von-Mises stress is on the root of the gate

rotor. Therefore, a small circular arc curve can be used to the design of the gate rotor on root and

to reduce the maximum stress on the root of gate rotor.

Equ~lenj (voD-Mises) Stress. X.1e5 P.a

8,m• '1.760

'6,100· '5JlllJ,

'4,ll5IJ'

.. :~m, \.940• ,0,970:

0:00:i

Maximum stress

Fig.8 Stress distribution of the gate rotors and the main rotor is represented in Von-Mises

stress contours

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7. COMPUTER SIMULATION FOR PROFILE DESIGN AND MANUFACTURE

To minimize human error in the design and manufacture of the geometric surface of a main

rotor with conical teeth , the generated cutting path was verified before actual machining. The

compressor ratio was set at 11 :6. The gate rotor has eleven teeth, and the main rotor has six

helical grooves. The pitch radius of the gate rotor is re ; however, in an actual case, the radius

selection will depend on the application of the single screw compressor mechanism. In addition,

this cutting simulation process, assumes the main rotor to be made of aluminum.

Although mathematical models of the main rotor in Sections 2 and 4 have been obtained,

the cutting process is an unknown. Therefore, a cutting simulation is important to the making of

the main rotor. Using the obtained results and PowerMILL software, the cutting process for

machining the main rotor is shown in Fig. 9. The simulation results show that the machining

process works be more specific. Using CNC five-axis machine, post-process file of the main

rotor, and FANUC 18iMb5 controller, the complete main rotors can be manufactured. In general,

a compressor ratio of the developed single screw compressor is 11:6 in compressor industry. A

smaller element can be done by the CNC five axes machine in our Lab. Therefore, the size of the

developed single screw must be reduced. A real scale prototype of the complete assembly for

single screw compressor was shown in Fig. 10.

Fig. 9 The cutting process for machining helix grooves of main rotor by PowerMiIl.

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Fig. 10 The real scale prototype of the complete assembly for single screw compressor

8. CONCLUSIONS

The mathematical model for the meshing analysis of a single screw compressor, which

includes a main rotor and a gate rotor, was established. Based on this mathematical model, the

cutting path and animation of a single screw compressor mechanism were generated. A complete

geometric model of a single screw compressor, including a main rotor and a gate rotor, was then

developed. The three-dimensional stress analysis of the proposed single screw compressor

mechanism was investigated. A tool path generation method for determining the cutter location

file for CNC five-axis machining of single screw compressor with conical teeth was

implemented. The proposed mathematical model analysis for a single screw compressor

mechanism should be helpful in the design and production of a main rotor and a gate rotor.

Finally, a computer program was also developed to generate a main rotor and a gate rotor.

ACKNOWLEDGEMENT

The author would like to thank the support of the National Science Council, Taiwan, R.O.c.,

under Grant NSC 96-2516-S-018-002

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Keywords: single screw compressor; conical teeth

NOTATION

a

d

distance between the center ofmain rotor and the center of gate rotor

tooth thickness.

tooth length

radius ofpitch of gate rotor

co-ordinate transfonnation matrix from co-ordinate system Sj to

co-ordinate system Si

N~ unit nonnal to the gate-rotor's surface

Nj number of teeth of gate-rotor

N2 number ofhelical groove ofthe main rotor

R li position vector of gate-rotor's surface (i= AB, BD, AI )

R2i family of gate-rotor's surfaces (i= AB, BD, AI)

Si (Oi' Xi' Y;, ZJ co-ordinate system where subscript i=1, 2, f 1 denotes gate-rotor's

surface, 2 denotes main-rotor's surface, f is rigidly connected to the

frame of reference

v/ relative velocity between gate-rotor's surface II and main-rotor's

surface I2

represented in Sf co-ordinate system

bi-parameter definition of gate-rotor's surface, j=l, 2, 3.

rotary angle of gate-rotor's surface

rotary angle ofmain-rotor' s surface

Transactions ofthe CSME Ide La SCGM Vol. 32, No. 3-4, 2008 351

()

()'

constant values when the parameter of motion ¢2 IS given constant

values.

the angle of a tooth on the gate rotor

profile angle of one tooth

tape angle of tooth

coefficient design

gate-rotor's surface

main-rotor's surface


Documents

study of a single screw compressor with a conical teeth gate rotor