FINDING TUTOR WITH ENGINEERING BACKGROUND

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The working process of an oil-free сlaw vacuum pump A mathematical model for analysis of the pumping characteristics

A. Raykov, S. Salikeev, A. Burmistrov, V. Alyayev, M. Fomina

Mathematical model of oil free claw vacuum pump is presented. The model is based on energy balance differential equations of thermodynamic system of variable mass working body. Using the equations of coordinates transformation and contact line of ro- tors continuity condition, equations for geometry of сlaw pump working chamber on condition of rotors point connection are obtained. To evaluate the leakage through the rotor mecha- nism channels, their existence graphs are plotted and geometric parameters of the channels depending on rotors position are determined.

As a result of modeling, depen- dence of pressure and temperature in

suction and compression-discharge chambers on rotation angle at differ- ent rotary speeds and on different inlet pressures and relationship between pumping speed and inlet pressure are obtained.

The comparison between calcu- lated values and experimental data ob- tained for one-stage claw pump with identical rotors is carried out. The maxi- mal difference between the calculated and experimental values does not exceed 15 %. The developed mathe- matical model is recommended for analysis of influence of rotors geome- try on the working process parameters and pumping characteristics of an oil free claw pump.

SUMMARY

∆α

δ RR

δ S1 δ

δ RS

R S

FIGURE 1: Main dimensions of a claw vacuum pump.

1 Introduction Non-contact vacuum pumps are the most promising for oil free rough pump- ing. One of them is claw type pump. It is a non-contact oil free rotating dis- placement pump where gas moves by periodic changing of closed chamber volume formed between rotors teeth, housing boring, and end caps.

The development of claw type pumps is possible only on the basis of the reliable mathematical model which makes it possible to predict its charac- teristics when any constructive param- eter is being varied. That is why the aim of this work is the development of such a model and its testing by comparison of calculated and experimental results.

Theoretical studies of claw pumps are presented in a number of works [1-5]. In one of the first works [1] calcu- lation of power consumption at differ- ent inlet pressures is given. The model used in this work is based on differential equations describing pressure and tem- perature variation in the pump working volume. Drawing of claw pump rotors profile and its optimization is presented in [2]. The technique of drawing of pump geometry with identical rotors and of working chamber volume calculation is given in [3]. Influence of geometri- cal parameters on claw vacuum pump characteristics is considered in [4]. The commercial CFD (computational fluid dynamics) code, Fluent, is used there for pumping process modeling, and distri- bution of temperature and pressure in the pump working chamber is obtained. The similar approach is used in [5] where the study of the improved rotor geome- try is carried out.

In the present work integral relation- ships between pumping speed and in- let pressure, between power consump- tion and inlet pressure, pressure and temperature in the working chamber

as a function of rotation angle are ob- tained. A mathematical model of a claw pump is described, and comparison be- tween the calculation results and exper- imental data [6] for the pump is carried out.

2 Mathematical model and assumptions

The following assumptions are used in the present mathematical model of claw vacuum pump working process which had reliable verification in theo-

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Der Arbeitsprozess trockenlaufender Klauenvakuumpumpen – Ein mathematisches Modell zur Analyse des Einflusses der Rotorgeometrie auf die Pumpcharakteristik Auf Grundlage von energiebilanzbe- trachtenden Differentialgleichungen unter Berücksichtigung veränderli- cher Massen wird ein mathematisches Modell zur Beschreibung ölfreier kur- vengesteuerter Klauenpumpen en- twickelt. Basierend auf der Stetigkeit der Linie des Kämmeingriffes erhält man mittels Koordinatentransforma- tion Gleichungen, die die Geometrie der Arbeitskammer unter der Bedin- gung der Punkteinbindung des Rotors beschreiben. Die Rückströmung durch die Kanäle des kontaktfreien Rotor- mechanismus wurde in Abhängigkeit von der Rotorposition bestimmt.

Temperatur- und Druckabhän- gigkeiten in Ansaug- und Kompres- sions-Entladungs-Kammer wurden für verschiedene Drehgeschwindigkeiten und Einlassdrücke ermittelt. Ein Ver- gleich der mittels Modell berechneten Daten mit experimentell bestimmten Werten ergab Abweichungen von höchstens 15 %, die das Modell als zuverlässige Basis zur Bewertung geo- metrischer Einflüsse auf die Pump- charakteristik für diesen Pumpenart empfehlen.

ZUSAMMENFASSUNG

influence of external factors. ` Gas-dynamic losses defined by fric-

tion of gas on the working chamber walls are not taken into account.

` Gas parameters before the suction port and behind the discharge port are constant. Pressure oscillations in the suction and the discharge ports are not taken into account in the course of pump operation.

` Under vacuum conditions heat transfer between the working gas, rotors, pump housing and atmo- sphere is negligible. Differential equations based on en-

ergy balance of thermodynamic system of variable mass [7] are used for working process modeling (1), where V is the current working chamber volume; QT is external heat; ω is the angular veloc- ity; MIN, MOUT are mass flow rates of gas flowing in and gas flowing out; hIN, hOUT are enthalpies of gas flowing in and gas flowing out.

Using enthalpy value h = cPT and taking into account leakage through groups of the channels (external heat being negligible) we’ll obtain:

` for the suction chamber (2), where MPV-SC is the mass flow rate from the pumped volume to the suction chamber; MPO-SC is the mass flow rate from the pump outlet to the suction chamber; MDC-SC is the mass flow rate from the compression-discharge chamber to the suction chamber;

MSC is the mass flow rate from the suction chamber;

` for the compression-discharge chamber (3), where MPV-DC is the mass flow rate from the pumped vol- ume to the compression-discharge chamber; MPO-DC is the mass flow rate from the pump outlet to the compression-discharge chamber; MSC-DC is the mass flow rate from the suction chamber to the compres- sion-discharge chamber; MDC is the mass flow rate from the compres- sion-discharge chamber. Subscripts SC and DC belonging to variables P, T, V designate suction chamber and compression-discharge chamber, respectively.

The equations (1–3) are first-order dif- ferential equations and they are solved by LSODE numerical method (Liver- more Solver for Ordinary Differential Equations) with automatic switch from Adams method to Gear method in soft- ware Wolfram Mathematica [8].

The initial conditions are gas pres- sure PIN and gas temperature TIN in the pumped volume; gas pressure POUT and gas temperature TOUT in the pump out- let.

The solution of equation systems (2) and (3) is pressure dependence and temperature dependence in the suction chamber and in the compres- sion-discharge chamber on rotation an- gle {PSC [φ], TSC [φ], PDC [φ], TDC [φ]}.The pumping of one portion of gas is real- ized in two rotor revolutions (Fig. 2): the beginning of one portion of gas suction (Fig. 2, b); the end of the suction (Fig. 2, c); the beginning of one portion of gas compression (Fig. 2, d); the beginning of the discharge process (Fig. 2, e); the end of the discharge process and formation of the dead volume passing to the fol- lowing pumping cycle (Fig. 2, f).

The calculation is carried out by method of successive approximations. At the first approximation , the calcu- lation of pressure and temperature in the suction chamber {PSC[φ], TSC[φ]} is carried out by solution of equation (2) at rotation angles in the range from 0 to 2π. The initial pressure PSC INIT and the initial temperature TSC INIT in the suc- tion chamber are taken equal to pres- sure and temperature in the pumped volume (PSC INIT = PIN , TSC INIT = T IN). As pressure and temperature dependence

dP ___ dφ = k – 1 ____ ωV ( ω dQT ___ dφ + MINhIN – MOUThOUT – k ___ k–1 ωP dV ___ dφ ) ,

dT ___ dφ = (k – 1)T

______ ωP V ( ω dQT ___ dφ + k–1 ____ k (MIN – MOUT) hOUT + MIN (hIN – hOUT) – ω P

dV ___ dφ ) (1) for the suction chamber

dPSC ____ dφ =

k – 1 ____ ωVSC ( MPV–SCTPVcP + MPO–SCTOUTcP + MDC–SCTDCcP –

–MSCTSCcP – k ____ k – 1 ωPSC

dVSC ____ dφ ) ,

dTSC ____ dφ =

(k –1) TSC _______ ωPSCVSC ( k –1 ____ k (MPV-SC + MPO-SC+ MDC-SC – MSC) TSCcP

+ MPV-SC (TPV – TSC)cP + MPO-SC (TOUT – TSC)cP + MDC-SC (TDC

–TSC)cP – ωPSC dVSC ____ dφ ) , (2)

for the compression – discharge chamber

dPDC ____ dφ =

k–1 ____ ωVDC ( MPV-DC TPVcP + MPO-SCTOUTcP + MSC-DCTSCcP

– MDC TDCcP – k ___ k–1 ωPDC

dVDC ____ dφ ) , dTDC ____ dφ =

(k–1) TDC _______ ωPDCVDC ( k–1 ___ k (MPV-DC + MPO-DC + MSC-DC – MDC)

× TDCcP + MPV-DC (TPV – TDC)cP + MPO-DC (TOUT –TDC)cP

+ MSC-DC (TSC – TDC)cP – ωPDC dVDC _____ dφ ) , (3)

retical and experimental studies of rotor pumps and compressors.

` The working gas is considered to be continuous medium.

` Air is considered to be ideal gas un- der rough vacuum conditions.

` In each chamber gas parameters changes occur instantly under the

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on rotation angle in the compres- sion-discharge chamber PDC[φ] and TDC[φ] which are necessary for the first approximation calculation are unknown, they are taken arbitrarily. The obtained dependence PSC[φ] and TSC[φ] are put into equation (3) at the rotation angle in the range from 2π to 4π. The initial pressure PDC INIT and the initial temperature TDC INIT in the com- pression-discharge chamber are taken equal to pressure and temperature in the suction chamber at the end of the previous revolution {PDC INIT = PSC[2π], TDC INIT = TSC[2π]}.

At the next approximation, the gas parameters obtained at the previous iteration are put into the equation system as the initial conditions {PSC = PSC[2π], TSC = TSC[2π]}. For leakage cal- culation, the functions of gas parame- ters obtained at the first approximation PDC[φ] and TDC[φ] are used. The convergence condition of calcu- lation is the difference less than 1 Pa between the previous approximation pressure diagram and the following ap- proximation pressure diagram and the difference no more than 0.5 K between the previous approximation tempera- ture diagram and the following approx- imation temperature diagram Max(|PSC[φ]i– PSC[φ]i-1|) ≤ 1, Max(|TSC[φ]i – TSC[φ]i-1|)≤ 0.5 in the range from 0 to 2π and Max(|PDC[φ]i – PDC[φ]i-1|) ≤ 1, Max(|TDC[φ]i– TSC[φ]i-1|) ≤ 0.5 in the range from 2π to 4π.

The drawback of using CFD in [4, 5] is considerable calculation time, and as a result, it is time consuming to obtain such integral characteristics as depen- dence of pumping speed and power consumption on pressure. The model presented in this work does not have such a drawback. Here the working chamber is considered as a whole and the only equation for the whole cham- ber is solved instead of many differen- tial equations for each cell.

Geometrical pumping speed is cal- culated as a product of the maximal suction chamber volume (at the mo- ment of suction port closing) and rotary speed n

SG = VSC [β1CL] · n , (4)

where β1 CL is the rotor angle coordinate when the suction port is closing.

Using pressure and temperature val- ues obtained by solution of differential equations systems (2) and (3), the differ- ence between the mass of gas entering the pump and the mass of gas flowing back to the pumped volume during one revolution is the following

mOUT = ∫ 0 2π

(MDC-PO – MPO-DC – MPO-SC) dφ. (5)

The pumping speed is calculated as

SIN = mOUTRGTIN ________ PIN

, (6)

where RG is the gas constant.

3 Geometry of the working cham- ber

Dependence of the working chamber volume on rotation angle and leak- ages values through clearances are necessary for pumping characteristics calculation. For this purpose, the draw- ing of the working chamber geometry defined by parametric equations is nec- essary.

Equations (and their boundary con- ditions), determining uniquely the working chamber geometry on the condition of rotors point conjugation, are derived with the help of coordinates transformation equations and on the basis of contact line of rotors continuity. In Fig. 3 the housing geometry is limited by curves FK1, FK2 which represent circle arcs with intersection points PTK, PBK. Cy- lindrical part of the left and the right ro- tors is defined by curves F16, F26, which are the arcs of the initial circle. The claws

of rotors are formed by elongated epi- cycloids F11a, F14, F21, F24 and circle arcs F15, F25. It may be useful in some cases to draw asymmetric rotors. Asymmetry results in appearance of the additional curve F11 on the rotor (in the larger housing boring) which is the extension of the epicycloid F21 located on the right rotor. Borings in rotors are formed by epicycloids F13, F23, which are drawn on the condition of engagement with points P22 and P12 , respectively. Circle arcs F12 and F22 , connecting claw pro- file with rotor boring, gear with curves

1 VSC

V DC

V DEAD

e) f )

c) d)

a) b)

FIGURE 2: Rotors positions in the pumping process; 1: suction port; 2: discharge port; VSC: suction chamber volume; VDC: compres- sion-discharge chamber volume; VDEAD: dead volume.

FIGURE 3: Working chamber geometry with asymmetric rotors.

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F15, F25. All listed curves are defined by parametric equations, their indepen- dent parameter being defined by inter- section points of curves P11 – P16 and P21 – P26.

It is necessary to know pump work- ing chambers volumes at any instant in order to calculate gas state parameters in these chambers according to equa- tions (2) and (3). Since cross section area of rotors and housing is constant along their length, working chamber volume is calculated as the product of its cross section area and rotor length.

Cross section area of chambers is cal- culated as integral about a closed path

F =  l {...} dl

where parametric equations of curves limiting the corresponding area are substituted. The result of calculation is dependence of working chambers ar- eas on rotation angle presented in Fig. 4 and Fig. 5.

Calculation results presented in Fig. 4, 5 are obtained for the following geometrical parameters: rotors are identical; center to center spacing is A = 75 mm; housing boring radius is RK = 60 mm; claw tip angle is α = 31.6 grad.

4 Calculation of leakage through slot channels

The gas mass change in working cham- bers of a claw vacuum pump takes place due to leakage through the following

channels of the rotor mechanism: ` channel between the claw edge of

the left rotor and the surface of the right rotor;

` channel between the claw edge of the right rotor and the surface of the left rotor;

` channel between the left and the right rotors;

` channel between the left rotor and the housing boring;

` channel between the right rotor and the housing boring;

` face channel between the left rotor face and the housing cap;

` face channel between the right rotor face and the housing cap;

` suction port; ` discharge port; ` face channels formed by rotors and

housing caps, through which ex- change between the suction and compression-discharge chambers takes place.

It was shown in [9] that all the listed channels (with the exception of suc- tion and discharge ports) can be con- sidered as one of four equivalent slots (Fig. 6).

Formulae for conductance calcu- lation of these types of slots and the general purpose method of calculation are presented in [10]. Clearances values are determined in view of thermal de- formations of rotors and the housing according to the formulae presented in [11].

Values of leakage through rotor mechanism are taken into account by plotting graphs of their existence (Fig. 7a, b) (1: channel exists, 0: channel does not exist). Obtained dependence of mass flow rate on rotation angle is summed up over leakage direction and is used when solving differential equations (2) and (3).

Some channels of the rotor mecha- nism at different rotation angles have variable length or their existence is terminated. In Fig. 8 a, b lengths of these channels are presented depending on the rotation angle.

0.01 0.008 0.006 0.004 0.002

2 4 6 8 10 12 φ, rad

F SC

, m2

0.01 0.008 0.006 0.004 0.002

2 4 6 8 10 12 φ, rad

F DC

, m2

FIGURE 4: Cross section area of the suction chamber.

FIGURE 5: Cross section area of the compres- sion-discharge chamber.

R 1

R 2

δδ δ

Channel 1 Channel 2 Channel 3 Channel 4

FIGURE 6: Types of slot channels.

1 1

00 2 4 6 8 10 12 2 4 6 8 10 12φ, rad φ, rad

a) b)

FIGURE 7: Existence graphs; a) channel between the claw edge of the left rotor and the surface of the right rotor; b) channel between the rotors.

l, ml, m

2 4 6 8 10 12 2 4 6 8 10 12φ, rad φ, rad a) b)

0.03 0.025

0.02 0.015

0.01 0.005

0.14 0.12

0.1 0.08 0.06 0.04 0.02

FIGURE 8: Channels length; a) channel between the left rotor and the housing boring; b) face chan- nel between the left rotor and the housing cap.

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The opening areas of suction and discharge ports are variable over the ro- tation angle. The inlet and output chan- nels have intricate three-dimensional geometry. That is why numerical calcula- tions of gas flow in these channels were carried out with the help of CFD Ansys- Fluent. The obtained mass flow rate val- ues at different rotation angles and inlet and outlet pressures were transformed into interpolation function which was used in the mathematical model.

5 Comparison of experimental and calculation data

The indicator pressure diagrams in suction and compression-discharge chambers were calculated using the de- veloped mathematical model. The ex- ample is presented in Fig. 9. Experimen- tal data obtained in [6] are presented as well. The maximal difference between the calculated and experimental data does not exceed 15 %.

The difference at the initial and fi- nal portions of the curves is connected with the considerable influence of leak- age through the channel formed by the claw edge, the value of which is difficult to measure with sufficient accuracy. It should be taken into account that sharp oscillations of medium parameters occur at rotation angles in the vicinity of 1; 6.28; 12 rad (due to gauges transi- tion from the suction chamber to the compression-discharge chamber; due to perturbation from rotors, moving in immediate vicinity of gauges). As a result, additional difference may occur at these portions. The difference at the portion in the range from 7 to 12 rad on the experimental curves is the result of oscillations in the outlet channel which are not taken into account in the math- ematical model. It can be clearly seen at high rotary speed. The underrating of pressure in the vicinity of 12 rad on the experimental indicator diagrams in comparison with calculated values can be explained by the increase of channel volume connecting the gauge with the working chamber (because its volume decreases).

The comparison between the calcu- lation and experimental dependence of pumping speed on inlet pressure was carried out (Fig. 10). The difference does not exceed 15 %.

Thus, the developed mathematical model can be used for analyses of influ- ence of geometry on working process parameters and pumping characteris- tics of a pump.

References [I] I.V. Ioffe, V.A. Koss, M. Gray, R.G. Livesey: J. Vac. Sci.

Technol. A13 (1995) 3, 536-539.

[2] C.F. Hsieh, Y.W. Hwang, Z.H. Fong: Mech. Mach. Theory 43 (2008) 7, 812-828.

[3] C.F. Hsieh: Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. 223 (2009) 9, 2063-2069.

[4] J. Wang, Y. Song, X. Jiang, D. Cui, Y. Qu: Vacuum 114 (2015) 66-77.

[5] J. Wang, X. Jiang, Y. Cai: Vacuum 111 (2015) 102-109.

[6] S.Salikeev, A. Burmistrov, K.Panfilovich: Vakuumnaya tekhnika i tekhnologiya (Vacuum Technique and Tech- nology) 15 (2005) 1, 21-27 (in Russian).

[7] M. Mamontov: Voprosy termodinamiki tela peremen- noi massy (Questions of variable mass body ther- modynamics), Priokskoe knizhnoe izdatelstvo, Tula, 1970, 87 p.(in Russian)

[8] S. Wolfram: Mathematica: A System for Doing Ma- thematics by Computer, Addison-Wesley Publishing Company, 2nd edition, 1991, 961 p.

[9] S. Salikeev, A. Burmistrov, M. Bronshtein, M. Fomina, F. Raykov: Vacuum 107 (2014) 178-183.

[10] S. Salikeev, A. Burmistrov, M. Bronshtein, M. Fomina: Vakuum in Forschung und Praxis 26 (2014) 1, 40-44.

[11] S. Salikeev: Development and study of claw vacuum pump, Thesis, Kazan, 2005, 144 p. (in Russian)

Dr. Sergey Salikeev 1978, mechanical engi- neer, Associate Professor, Department of Vacuum Equipment, Kazan Natio- nal Research Technologi- cal University, Russia

Dr. Alexey Raykov 1986, Education: mechanical engineer, Asso- ciate Professor, Department of Vacuum Equip- ment, Kazan National Research Technological University, Russia

Dr. Aleksey Burmistrov 1963, mechanical engineer, Professor, Depart-

ment of Vacuum Equipment, Kazan National Research Technological University, Russia

Dr. Valeriy Alyayev 1959, mechanical engineer, Professor, Chair of the Department of Vacuum Equipment, Kazan National Research Technological University, Russia

Dr. Marina Fomina 1959, mechanical engineer, Associate Professor, Department of Vacuum Equipment, Kazan Nati- onal Research Technological University, Russia

AUTHORS

Sergey Salikeev, 68 Karl Marx Street, Kazan 420015 Russia, Tel. +7 843 273 15 85, Tel.: +7 937 614 03 00, E-mail: [email protected]

P, Pa 130000

120000

110000

100000

90000

80000

70000

60000

experimental calculation

P IN

= 96 kPa

P IN

= 64 kPa

2 4 6 8 10 φ, rad

FIGURE 9: Comparison of experimental and calculation indicator pressure diagrams at rotation speed 1200 rpm.

80000800008000080000 PIN, Pa

450 rpm 978 rpm

1200 rpm 1800 rpm 2400rpm

3000 rpm 0

16 S

IN , l/s

FIGURE 10: Comparison between the calculation (lines) and experimental (points) dependence of pumping speed on the inlet pressure.