effects of leakage and friction on the miniaturization of a wankel compressor - yilin zhang

5/28/2018 Effects of Leakage and Friction on the Miniaturization of a Wankel Compressor - Yilin ZHANG

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RESEARCH ARTICLE

Yilin ZHANG, Wen WANG

Effects of leakage and friction on the miniaturization of aWankel compressor

Higher Education Press and Springer-Verlag Berlin Heidelberg 2010

Abstract This paper presents a numerical simulation ofthe performance of a meso-scale Wankel compressor anddiscusses the factors affecting its miniaturization. Thediscussion is related to the effect of leakage and friction onthe design limit (cooling capacity and dimension) of themeso Wankel compressor. In the simulation, the mainleakage comes from the gaps between the rotor and theendplates as well as between the seal apex and the cylinder.The largest friction originates from the clearance amongthe end face of the eccentric shaft, the end faces of therotor, and the endplates. The decreasing cooling capacityof the meso Wankel compressor increases the proportion ofleakage to displacement and causes the coefcient of

performance COP and the mechanical efciency todecrease. The rational design cooling capacity limit for

the meso-scale Wankel compressor is approximately 4 W.

Keywords meso-scale, Wankel compressor, leakage,friction

1 Introduction

Highly efcient mini and meso refrigeration systems arewidely needed in the eld of electronic cooling. TheWankel compressor has good potential for the meso vaporcompression refrigeration system because of its advan-

tages, such as simple structure, high ef

ciency, lowvibration, and low noise.The Wankel compressor is similar to the Wankel engine

and rotary compressor in terms of structure and operation;it consists of a rotor, a cylinder, a shaft, a pair of gears andapex seals, and three enveloped chambers (V1, V2, V3)(Fig. 1). The driving structure of a Wankel compressor is

made up of a gear pair and an eccentric shaft, which drivesthe rotor to make planetary motion in the cylinder of thecompressor. The angular speed of the eccentric shaft isthree times that of the rotor; thus, two refrigeration cyclesare accomplished when the eccentric shaft completes arevolution.

The Wankel machine is a kind of rotary machine. Greatprogress has been made in terms of research about Wankeand rotary machines in the last century. Numericalsimulations have contributed greatly in the research anddevelopment of Wankel and rotary machines. Pennock andBeard [1] derived equations for the radial and transversecomponents of the acceleration of an apex seal in the rotorof a Wankel rotary compressor and made a dynamic forceanalysis of the seal, including the friction forces between

the tips of the seal and the chamber as well as between theside of the seal and the rotor. Heppner et al. [24] analyzedthe leakage ow and the friction loss of rotary engine andcompressor and established design parameters for microengine sealing systems. Prater and William [5,6] describedthe methodology and results of an experiment to measurethe fundamental undamped natural frequency and dampingratio for the discharge reed valve in a rolling piston rotarycompressor. Hsiao et al. [710] presented the simulation ofa rotary compressor and its performance comparison withmeasured results.

Several factors, such as friction loss, leakage, the rotor

and cylinder pro

les, and the positions of the inlet and

Received May 20, 2010; accepted August 27, 2010

Yilin ZHANG, Wen WANG ()

School of Mechanical and Power Engineering, Shanghai Jiao TongUniversity, Shanghai 200240, China

E-mail: [email protected]

Fig. 1 Schematic diagram of a Wankel compressor

Front. Energy 2011, 5(1): 8392DOI 10.1007/s11708-010-0125-7


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exhaust ports, should be considered in designing a mesoWankel compressor. The introduction of computer simula-tion can greatly shorten the design period and reducedesign cost. Therefore, numerical simulation is necessaryto obtain the rational design dimensions. The inuences ofsome factors, such as leakage and the friction loss, on the

performance of a meso Wankel compressor differ from theeffects on a conventional Wankel compressor. Based oncurrent manufacturing and lubrication technologies, theinuences of leakage and friction loss increase continu-ously with the gradual decrease of the design dimensionsand the cooling capacity. The displacement and themechanical efciency drop greatly when the designdimensions and the cooling capacity reach certain values.Thus, the limit of rational design dimension should beanalyzed in designing a meso Wankel compressor.

This paper presents a numerical simulation of theperformance of a meso-scale Wankel compressor and

discusses the design limit of the cooling capacity anddimensions for a meso Wankel compressor. The simulationmodel is focused on the cycle performance. The mechan-ical optimization simulation is performed to search for theoptimum dimensions of a meso Wankel compressor. Thedesign limit has been obtained based on the analysis ofleakage and friction loss of the meso Wankel compressor.

2 Mechanical optimization

2.1 Friction in a Wankel compressor

Friction loss usually plays an important role in compressorperformance. In meso and mini machines, the ratio ofsurface by volume to the contact surfaces with friction arelarge and cannot be lubricated adequately as normal scalemachines. In this simulation model, eight friction pairswere taken into account and listed as follows:

1) The friction loss between the main shaft and the mainbearing,L1,

L122R3s lm

cm: (1)

2) The friction loss between the main shaft and theaccessory bearing, L2,

L222R3s l

m

cm: (2)

3) The friction loss between the eccentric bearing andthe eccentric shaft, L3,

L32r2R3e le

3: (3)

4) The friction loss between the seal apex and the

internal surface of the cylinder, L4[A.10],

L4 z

540$

X2701

Ft2vTsin

sinX540

271Ft1

vTsin

sin

" #:

(4)

5) The friction loss between the apex seal and the sealing

groove,L5[A.13],

L5 z

540

X5401

f2pgsFTvR: (5)

6) The friction loss among the end face of the eccentricshaft, the end faces of the rotor, and the endplates, L6,

L6 2rR4rR4ge R4rR4e

6a

22R4eR4s 6b

( ):

(6)

7) The friction loss between the gears, L7[A.17].

L7f7lzz1Me

3

2sin : (7)

In this work, the friction losses ofL4, L5, and L7 werederived on the basis of friction principle. Detailedderivation is depicted in the Appendix, and the frictionloss ofL1,L2,L3, and L6are cited from [3,4,911].

2.2 Optimization to reduce friction in a Wankel compressor

Given that friction is unavoidable in a Wankel compressor,it is necessary to search for the optimum parameters basedon the minimum friction loss at given operationalconditions (Table 1). The complex optimization methodwas used in this study. The objective function is to searchfor the minimum total friction loss of the Wankelcompressor.

Here, nine design variables were chosen as optimalparameters based on the analysis of friction losses (L1L7)in Eqs. (1)(7). The design variables included the thicknessand height of apex seal, the shape factor, the offset, theradius of the main shaft, the radius and height of theeccentric shaft, the number of tooth of the external gear,and the modulus of the gear.

Table 1 Operational conditions of a Wankel compressor with

refrigerant R134a

No. parameter description value

1 Q cooling capacity/ W 300

2 n motor speed/(r$min1

) 1800

3 Teva

evaporation temperature/C 5

4 Tcon condensation temperature/C 46

5 pin intake pressure/MPa 0.35

6 pin back pressure/MPa 1.20

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The optimization can be described asThe objective function is

F FminL1L2L3L4L5L6L7f bs,hs,K,a,Rr,Re,he,z1,mg and (8)

is subjected to the following constraints:Explicit constraints:

ajxjbjj1,2,:::,9: (9)Implicit constraints:

giX0i1,2,:::,5: (10)In Eq. (9), the xj(j= 1,2,,9) are the explicit constrains

represented by the parameters bs,hs,K,a,Rs,Re,he,z1, andmg, respectively. The values ofajand bj(Table 2) are the

range limits. In Eq. (10), gi (i= 1,2,,5) is the implicitconstraint, and i is the number of implicit constraints.Table 3 lists the limit values of the implicit constraints.

The optimum results in terms of dimensions and frictionlosses obtained from the optimization are shown inTables 4 and 5, respectively. Results show a total frictionloss of 11.03 W; the largest friction loss comes from L6,which is equivalent to approximately 63.29%.

The design dimensions (Table 4) shows the creationof a Wankel compressor prototype and the experimentconducted on friction loss under normal atmosphere.Friction losses were obtained by detaching the friction

parts of the Wankel compressor prototype through a

step-by-step process, after which the corresponding outputpower of the motor was measured. The predicted and theexperimental results are compared in Table 6. The relativeerror of total friction loss is about 5.53%, indicatingthat the rationality of the friction loss equations andthe optimization model of Wankel compressor are

feasible.

3 Thermodynamic analysis of a mesoWankel compressor

3.1 Leakage of a Wankel compressor

Gas leakage occurs in all the gaps connected to thechamber (Fig. 2). The leakage in a Wankel compressor can

be divided into internal and external leakages. The internalleakage in a Wankel compressor may occur in the four gaps

between the seal apex and the cylinder (mscy), the seal sides

and the endplates (msc), the seal and the sealing groove(msg), and the rotor and the endplates (mrc). The externalleakage mainly occurs in the gap between the main bearingand the main shaft (mbs) and can be neglected by therational shaft seal design.

The gaps of leakage mscy are very short; therefore, theleakage process is simplied as a gas ow through theconvergent nozzle. The leakage can be calculated by

mfAgpi,i,po, (11)where

gpi,i,po

ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffi2k

k1ipi

po

pi

2k

1 po

pi

k

1k

24

35,

vuuut popi

2

k1 k

k1,

ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiipi

2

k1 k1

k1,

vuut popi

>>>>>>>>>>>>>>>>:

(12)

The other three leakage gaps employ long narrowpassages compared with their height and are simplied as aconvergent nozzle with equal section straight pipe. Themass ow can be expressed as [11]:

mHpeve=RgTe: (13)

3.2 Analysis of compression process

The optimum dimensions and the operation condition arepresented in Tables 4 and 1, respectively. Predicted results,such as the variation of volume, pressure, temperature, andmass in three chambers in a Wankel compressor withdifferent eccentric angle, were obtained by numericalsimulation. Other performance parameters, such as cooling

capacity, leakage, intake mass, exhaust mass, and so on,were obtained as well. The typical simulation results aredemonstrated in Fig. 3. The variations of parameters in thethree chambers (V1, V2, and V3) are the same. The V3

chamber was taken in this study as a sample to be analyzedin detail.

Figure 3(a) shows the variation of pressure in threechambers of the Wankel compressor. The pressure (p_V3)remained almost unchanged during the intake process. Thecompression process commenced when the volume startedto increase until the volume reached the maximum. The

pressure increased continually with the volume decreaseuntil the pressure reached the back pressure (the eccentricangle is approximately 360 in this paper). The pressurewas almost stable at a certain level in the exhaust processuntil the exhaust valve shut off (at this point, the eccentric

Yilin ZHANG et al. Effects of leakage and friction on the miniaturization of a Wankel compressor 85


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angle was approximately 450). The operation processtransitioned to the expansion process at the closing of theexhaust valve and ended at the opening of the intake valve,thereby completing the whole cycle.

The variation of gas temperature shown in Fig. 3(b) isproportional to the gas pressure in the chamber. In theintake process, the intake gas was heated by the high-

temperature gas leaked from the other two chambers andthe high-temperature wall of the rotor, cylinder, andendplates, resulting in a slight increase of gas temperaturein the intake process. In the compression process, thetemperature increased rapidly with the increase of

pressure, and the gas temperature decreased continuallyin the exhaust and the expansion process.

Figure 3(c) illustrates the variation of the gas mass inthree chambers of the Wankel compressor. The gas massmaintained constant increase in the intake process anddecreased continually in the exhaust process. The gas mass

Table 2 Explicit constraints used in optimization design

No. explicit constraints xj description lower limit higher limit

1 bs thickness of apex seal/mm 0.5 3

2 hs height of apex seal/mm 4 10

3 Ka) shape factor 5.5 7.5

4 a offset/mm 0.5 1.5

5 Rs radius of main shaft/mm 2 5

6 Re radius of eccentric shaft/mm 3 15

7 he height of eccentric shaft/mm 3 15

8 z1 number of tooth of external gear 18 30

9 mg modulus of gear/mm 0.4 1

a) K is the shape factor of the Wankel compressor, and it is equal to R/e.

Table 3 Implicit constraints used in the optimization design

No. implicit constraints gi description lower limit higher limit

1 mgz1/4 eccentric distance/mm 2 5

2 Re

Rs

e difference between eccentric shaft radius and sumof main shaft radius and eccentric distance/mm

0 6

3 Va=3 ffiffiffi3pKeae height of cylinder/mm 4 204 K$e3e2 depth of sealing groove/mm 5 155 Va=3 ffiffiffi3pKeaehe height of gear/mm 1 126 Ke5e thickness of thinnest wall of rotor/mm 3 10

a) V is the volume for each chamber of the Wankel compressor, and it is decided by the cooling capacity.

Table 4 Optimum dimensions of the Wankel compressor

No. parameter value No. parameter value

1 bs/ mm 1.64 6 Re/mm 5.82

2 hs/mm 5.56 7 he/mm 4.05

3 K 6 8 z1 20

4 a/mm 0.86 9 mg/mm 0.6

5 R/mm 2.82

Table 5 Mechanical analysis for the Wankel compressor under the aforementioned operational conditions and optimum dimensions

friction loss

L1 L2 L3 L4 L5 L6 L7 total

predicted/W 0.28 0.28 1.18 1.49 0.46 6.98 0.36 11.03

contribution/% 2.51 2.51 10.67 13.55 4.20 63.29 3.29 100.00

Table 6 Comparison between the predicted and experimental (under normal pressure andn = 1800 r/min) results of friction losses

friction loss

L1 L2 L3+L7 L4+L5 L6 total

predicted /W 0.28 0.28 1.54 2.09 6.98 11.17

experimental /W 0.23 0.23 1.27 1.76 7.06 10.55

relative deviation /% 16.77 16.77 17.22 15.76 1.14 5.53

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had a slight increase in the early stage of the compressionprocess and a slight decrease in the later stage of thecompression process. This condition was conversed in theexpansion process. This mechanism is attributed to the gas

pressure in chamber V3 being the lowest of the threechambers at the beginning of compression, causing the

increased mass to come out from the other two chambersthrough the leakage gaps. With the continued compression

process, the gas pressure in chamber V3 increasedcontinually, and the gas mass decreased when the massexchange with the other two chambers became negative.The mass variation in the expansion process is similar to

the compression process.The variation of the intake mass and the exhaust mass

are shown in Fig. 3(d). The intake mass ow increased atrst and then decreased in the intake process. This is a

phenomenon, which is related with the volume variation ofchambers. The exhaust mass ow decreased continuouslyin the exhaust process. The exhaust rate was higher thanthe intake rate because the gas density in the exhaustchamber was higher than that in the intake chamber.

Figures 3(e) and 3(f) show the variation of leakage. The

Fig. 2 Leakage of a Wankel compressor

Fig. 3 Variation of operation parameters of the Wankel compressor with eccentric angle

(a) Pressure; (b) gas temperature; (c) gas mass; (d) exhaust mass; (e), (f) leakage



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main leakage ways are shown by the gap between the rotorand endplates and that between the seal apex and cylinder,accounting for 42% and 28% of the total leakage,respectively.

4 Design limit of a meso Wankelcompressor

The impact of leakage and friction loss on compressorperformance gradually increases with the decrease of themeso Wankel compressor dimension. Accordingly, thereare two factors determining the design limit of the mesoWankel compressor. First, with an increasingly seriousleakage situation, lesser compressed gas is pushed out ofthe exhaust valve, making it difcult for the gas pressure inthe compression chamber to reach the back pressure.Therefore, one design limit for the meso Wankel

compressor was that the exhaust valve cannot be openedduring the entire operation. Friction loss served as anotherdeterminant of the design limit. The proportion of frictionloss to shaft power continuously increased, and themechanical efciency gradually decreased with thedecrease of compressor dimension. Another design limitwas set when the mechanical efciency reached below acertain value (50% in this paper).

In the calculation, the meso Wankel compressor has thesame operation conditions except the cooling capacity (as

presented in Table 1) and machining tolerance (5 m). Inthis work, the initial cooling capacity of the Wankelcompressor was estimated on the basis of mathematicalmodel without considering leakage and friction loss; inaddition, the volumetric efciency of the Wankel com-

pressor was assumed as 80%. Table 7 lists a set ofoptimization dimensions obtained by the optimizationmodel of the meso Wankel compressor. The corresponding

performance parameters obtained by the simulation modelare respectively presented in Tables 8 and 9.

Table 8 shows that the proportion of the leakage to the gasdisplacement increases with the decrease of the coolingcapacity of the meso Wankel compressor. The deviation

between the actual cooling capacity and the initial design

value is magnied gradually. Under 5 m machiningtolerance, the leakage between the three chambers is higherthan the displacement when the initial cooling capacityreaches 10 W, and the actual cooling capacity is onlyapproximately 4 W. The compressed gas cannot reach the

back pressure, and the exhaust valve cannot be opened whenthe initial cooling capacity is approximately 5 W; otherwise,the meso Wankel compressor cannot work normally.

Table 9 shows the variation of friction loss, shaft power,coefcient of performance (COP), and mechanical ef-ciency with different cooling capacities of the mesoWankel compressor. The COP and the mechanical

efciency gradually decrease with the decrease of thecooling capacity; both parameters drop to 2.1 and 44.6%,respectively, when the initial cooling capacity of the mesoWankel compressor reaches approximately 10 W.

From the above analysis, the rational cooling capacitylimitation obtained for the meso Wankel compressor isapproximately 4 W.

5 Conclusion

Based on the predicted results of the model for the mesoWankel compressor, the variation of pressure, temperature,mass, and leakage in chambers are analyzed in detail in this

paper. The analysis has shown that the main leakage stemsfrom the gap between the rotor and the endplates and that

between the seal apex and the cylinder.The optimization for the meso Wankel compressor is

Table 7 A set of the optimization dimensions for the Wankel compressor with different cooling capacities

cooling capacity/W

300 250 200 150 100 50 40 30 20 10 5

bs/mm 1.6 1.5 1.4 1.2 1 1 0.9 0.9 0.8 0.7 0.6

hs/mm 5.6 5.3 5.1 4.8 4.6 4.3 4 3.8 3.6 3.2 2.5

mg/mm 0.6 0.6 0.6 0.5 0.5 0.5 0.5 0.4 0.4 0.4 0.3K 6 6 6 6 6 6 6 6 6 6 6

z1 20 20 18 20 18 16 14 16 16 14 14

Re/mm 5.8 5.5 5 4.6 4.25 3.8 3.25 2.9 2.7 2.4 2.05

Rs/mm 2.8 2.5 2.3 2.1 2 1.8 1.5 1.3 1.1 1 1

V/mm3 3388 2767 2112 1704 1123 524 450 336 252 129 74

a/mm 0.86 0.82 0.75 0.7 0.65 0.6 0.5 0.4 0.4 0.4 0.3

he/mm 4.1 3.6 3.2 3 2.6 2 2 1.8 1.6 1.4 1.2

R/mm 18 18 16.2 15 13.5 12 10.5 9.6 9.6 8.4 6.3

e/mm 3 3 2.7 2.5 2.25 2 1.75 1.6 1.6 1.4 1.05

H/mm 11.5 9.4 8.9 8.4 6.8 4 4.5 4.0 3.0 2.0 2.1

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discussed based on the friction loss analysis. There areseven kinds of friction loss (L1L7) discussed in this paper.The greatest friction loss comes fromL6, which contributesto approximately 63.29% of the total friction loss.

The feasibility of the system model and the optimizationmodel of the meso Wankel compressor has been proven bythe friction loss experiment at normal pressure on a mesoWankel compressor prototype. The relative error of thetotal friction loss between the predicted and the experi-mental results is approximately 5.53%.

The impacts of leakage and friction loss are mainlyconsidered in analyzing the design limit of the meso Wankel

compressor. With the decrease of the cooling capacity, theproportion of leakage to displacement has been found togradually increase, and the COP and the mechanicalefciency gradually decreased. The rational coolingcapacity limit for the meso Wankel compressor has beenfound to be approximately 4 W, while those for COP andmechanical efciency are 2.1% and 44.6%, respectively.

The simulation model has been used in assisting thedesign of the meso Wankel compressor for microsystems.It also provides a way for more comprehensive simulationstudies and for possible overall computer optimizationdesign study.

Acknowledgements This work was supported by the National Natura

Science Foundation of China (Grant No. 50976067).

Appendices: The derivation of friction loss

1 The friction loss between the seal apexand the internal surface of cylinder,L4(Fig. A1)

The total force of the apex seal [12] is expressed by:

F pgBFRFs

pgTf2pgs

FT: (A:1)The gas pressure in the gap between the bottom of the

seal and the bottom of the sealing groove is

pgBlsbskBph: (A:2)The gas pressure in the gap between the side of the seal

and the side of the sealing groove is given by:

pgslshsksphlscpl: (A:3)The friction between the front seal and the cylinder is

expressed by:

Table 8 Parameters related to the Wankel compressor with different cooling capacity

cooling capacity/W

300 250 200 150 100 50 40 30 20 10 5

leakage/(104

kg$s1

) 2.66 2.40 2.18 2.01 1.68 1.16 1.11 0.97 0.83 0.58 0.41

displacement/(104

kg$s1

) 24.8 20.2 15.1 12.0 7.5 3.0 2.5 1.7 1.1 0.3 0

contribution/% 10.7 11.9 14.4 16.9 22.5 38.7 45.2 58.6 75.2 193

actual cooling capacity/W 332 270 202 160 100 40 33 22 15 4 0

Table 9 Parameters related to the Wankel compressor with different cooling capacities

cooling capacity/W

300 250 200 150 100 50 40 30 20 10 5

friction loss/W 11.0 11.1 7.7 5.9 4.2 2.6 1.8 1.4 1.3 1.1 0.7

shaft power/W 75.5 63.4 46.9 37.0 23.7 10.4 8.2 5.7 4.1 1.9 0.8

COP 4.4 4.3 4.3 4.4 4.2 3.9 4.1 4.0 3.6 2.1 0.0

mechanical efciency/% 85.4 82.6 83.6 83.9 82.1 75.2 77.9 75.0 69.3 44.6 17.9

Fig. A1Friction between seal and cylinder



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Ft1

F

cos

sin:

(A:4)

The friction between the post apex seal and the cylinderis expressed by:

Ft2 F

cossin: (A:5)

The friction angle is represented by:

arctan f1: (A:6)The tangential velocity of the seal is expressed by:

vT

vxsin

3 vycos 3 : (A:7)

The friction powers between the front seal and thecylinder and between the post seal and the cylinder aregiven by:

P1Ft1vTsin

sin (A:8)

and

P2Ft2vTsin

sin: (A:9)

Thus, the friction loss between the seal and the internalsurface of the cylinder can be shown as:

L4 z

540

X2701

Ft2vTsin

sinX540

271Ft1

vTsin

sin

" #:

(A:10)

2 The friction loss between the seal and thesealing groove, L5

The friction loss between the seal and the sealing groovefor a degree eccentric angle [12] is:

w 1

f2pgsFTvR, (A:11)

wherevR is the radial velocity of the seal expressed by:

vR

vxcos

3

vysin

3 : (A:12)

Thus,L5can be shown as:

L5 z

540

X5401

f2pgsFTvR: (A:13)

3 The friction loss between gear pairs,L7(Fig. A2)

The forces on the gear pairs come from the radial force andthe circumferential force. However, the impact of circum-ferential force is so small that it can be neglected byrational design in this paper.

The centrifugal inertial force of the rotor [12] is givenby:

SrMe2: (A:14)The radial force is expressed by:

Fr Sr

2sin

: (A:15)

The contacting tooth number of the gear is three times

that of the external gear tooth with rotor rotating a cycle,and the total contacting length is 6z1lz. The relativevelocity is given as:

vz1lz

: (A:16)

So,L7is given as:

L7f7lzz1Me

3

2sin : (A:17)

Notations

Fig. A2Forces of gear pair

a offset/m

A effective ow area of the intake port or the exhaust

port/m2

bs thickness of apex seal/m

c rising height of seal in the sealing groove/m

cm, c

m radial clearance values of the main bearing and

accessory bearing/m

e eccentric distance/m

f1,f2,f7 friction coefcients between the seal apex and the

cylinder surface, between the seal and the sealing

groove and between gear tooth

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9. Ooi K T, Wong T N. A computer simulation of a rotary compressor

Fr radial force of gear/N

FR radial inertial force of seal/N

Fs spring force of seal bottom/N

FT tangential inertial force of seal/N

g mass ow of unit area / (kg$m2

$s1

)

he, hs heights of the eccentric shaft and apex seal/m

kB, ks the average pressure reduction coefcients of the

seal bottom and seal side

K shape factor

lm, l

m , le, ls lengths of the main bearing, accessory bearing,

eccentric bearing and seal/m

lz contact length of gear tooth/m

L1 friction loss between the main shaft and the main

bearing/W

L2 friction loss between the main shaft and the

accessory bearing/W

L3 friction loss between the eccentric bearing and the

eccentric shaft/W

L4 friction loss between the seal apex and the internal

surface of cylinder/W

L5 friction loss between the apex seal and the sealing

groove/W

L6 friction loss among the end face of the eccentric

shaft, the end faces of rotor, and the endplates/W

L7 friction loss between the gear pairs/W

m mass ow/(kg$s1

)

mg modulus of gear

mbs leakage between the bearing and the main shaft

/(kg$s1

)

mscy leakage between the seal apex and the cylinder/(kg$s1

)

msc leakage between the apex seal sides and the

endplates/(kg$s1

)

msg leakage between the apex seal and the sealing

groove/(kg$s1

)

mrc leakage between the rotor and the endplates/kg

M quality of the rotor/kg

n motor speed (r$min1

)

Pi,Po pressure levels at the valve section of the inlet port

and the exhaust port/Pa

PgB, PgT, Pgs gas pressure levels of the seal bottom, seal apex,

and seal side/Pa

Ph, P1 gas pressure levels in the high-pressure chamber

and the low-pressure chamber/Pa

Q cooling capacity/W

R generation radius of the rotor/m

Rs, Re, Rge radii of the main shaft, eccentric shaft, and internal

gear/m

Rr equivalent radius of rotor/m

Sr centrifugal inertial force of the rotor/N

Teva,Tcon evaporation temperature and condensation

temperature/C

Ve velocity of exhaust section/(m$s1

)

VR, VT radial velocity and tangential velocity of sea

/(m$s1

)

vx,vy x and y axis velocity/(m$s1

)

H width of leakage gap/m

z number of seal

z1 tooth number of external gearGreek letters

rotation angle of main shaft/()

pressure anger of tooth/rad

lubricant viscosity/(Pa$s)

swinging angle/rad

height of leakage gap/m

3 gap between eccentric bearing and eccentric

shaft/m

6a, 6b gaps between rotor and endplate and between

eccentric shaft and endplate/m

,r anger velocities of the main shaft and rotor

/(rad$

s1

)

friction anger/rad

i gas density of intake section/(kg$m3

)

f ow coefcient

k gas isentropic coefcient



10/10

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12. Lu F, Yu N B. Wankel Engine. Beijing: National Defense Industry

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effects of leakage and friction on the miniaturization of a wankel compressor - yilin zhang

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