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Mathematical model of working process of oil free scroll vacuum pump Influence of leakage and heat transfer on pumping characteristics

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

A mathematical model of oil free scroll machine working in vacuum conditions is considered. Differential equations and main assumptions in the model are presented. Four types of control volumes are specified. The model takes into consideration heat exchange between the pump working

elements, change of clearances due to thermal deformations, and mobility of the scroll element wall in calculation of leakage. The comparison of the cal- culated values of the pumping speed with experimental values obtained at different rotation speed of the drive- shaft is carried out.

SUMMARY

FIGURE 1: Scroll vacuum pump engineering design; 1 – orbiting scroll; 2 – fixed scroll; 3 – inlet chan- nel; 4 – discharge port; 5 – conventional surface of dividing Vsc and VIN

Introduction

Mechanical oil free rough and medium vacuum systems are few and include such non-contact pumps as scroll, screw, claw pumps, and units on the ba- sis of Roots pumps.

Scroll vacuum pumps have the fol- lowing advantages in comparison with other oil free pumps: high pressure ratio due to low flow rate between working chambers which are numerous, low suction losses, and low power con- sumption. That is why scroll machines are widely used. Oil free units where scroll pumps work as fore pumps are of high interest as well.

Problem formulation and assump- tions

Development of new scroll pumps and uprating of the existing products are possible only on the basis of the reliable mathematical model of the working process which makes it possible to com- pare the influence of different factors on pump characteristics.

The most part of such studies in- volves development of mathematical models of a scroll mechanism working in regime of a scroll compressor [1–5]. Scroll vacuum pumps studies [6–8] are considerably fewer. That is why the development of mathematical model of scroll pump working process is top- ical. It enables prediction of pumping parameters when geometrical sizes of scrolls and clearances between them, rotary speed of the driveshaft, pressure, temperature, and molecular weight of the gas vary.

The assumptions in the mathemat- ical model of the scroll pump working process are the following: working me- dium is considered to be ideal gas; gas parameters changes in each chamber under the influence of external factors

occur instantly; gas-dynamic losses de- fined by friction of gas against the work- ing chamber walls are not taken into account; gas parameters before the suc- tion channel and behind the discharge channel are constant; pressure oscilla- tions in the suction and the discharge channels are not taken into account in the course of pump operation.

Modeling of working process is car- ried out by control volumes method. In the presented mathematical model of a scroll vacuum pump four types of con- trol volumes (Fig. 1) are specified: inter- nal volume of the scroll pump, suction volume, compression volume, and dis- charge volume.

The specification of internal volume of the scroll pump where gas flows to

through suction channel makes it pos- sible to take into account the heating of gas during suction and the resistance of the inlet channel.

When orbiting scroll element is mov- ing, two crescent shaped chambers between the scrolls are formed where gas moves from periphery to the center. This process begins with the formation of suction volumes which increase with scroll motion. The specification of these volumes makes it possible to model the suction process and take into account the resistance of the channel formed by initial scrolls parts in the neighborhood of surface 5 (Fig. 1a).

The first (internal) cut off volume VCUT1 (Fig. 1a) is formed when the cham- ber between the concave side of the

VAKUUM

Mathematisches Modell für das Entleeren ölfreier Scrollpumpen

Im Artikel wird ein mathematisches Modell einer ölfreien im luftleeren Raum arbeitenden Maschine be- handelt. Die im Modell gebrauch- ten Differenzialgleichungen und Vereinfachungen sind dargestellt. Vier Kontrollvolumina werden her- vorgehoben. Im Modell werden sol- che Aspekte, wie der Wärmeumsatz zwischen Funktionsbauteilen der Pumpe, die Beweglichkeit der Ge- triebschnecke bei der Berechnung des Überströmens sowie die Verän- derung der Klaffstellungen aufgrund des Wärmeverzugs berücksichtigt. Abschließend werden die berech- neten für die Pumpgeschwindigkeit Werte mit den experimentell für ver- schiedene Rotationsgeschwindigkei- ten der Antriebswelle bestimmten Daten verglichen.

ZUSAMMENFASSUNG

orbiting scroll and convex side of the fixed scroll closes. In a half-turn the sec- ond (external) cut off volume VCUT2 (Fig. 1b) is formed. Let’s mention that VCUT1 < VCUT2 . The summarized volume of these chambers defines the geometrical pumping speed of the pump

SG = (VCUT1 + VCUT2)n, (1)

where n is the rotation speed of the driveshaft.

In the following moment the cut off chambers are transformed into com- pression chambers Vc1 and Vc2 which gradually decrease their volumes when they move to the center.

At the end of the compression pro- cess, the joining of the pair chambers and the internal chamber in the neigh- borhood of the discharge port 4 (Fig. 1a) takes place, and the integrated dis- charge chamber VD is formed. The dis- charge process begins when the outer surface of the orbiting scroll breaks away from the inner surface of the fixed scroll and the cut off chamber connects with the discharge port. This process has the angular duration of 2π.

The system of differential equations describing gas pressure and gas tem- perature variation with respect to the

driveshaft rotation angle (which was presented in [9, 11] and was used for volumetric machines working in vac- uum conditions [11]) is now used for modeling of pumping process.

Its own system of differential equa- tions is written for each chamber. For example, differential equations for compression volume are written in the form above (2), where V is the current chamber volume; ω is the angular ve- locity; QT is external heat; k is adiabatic coefficient; φ is the rotation angle of the driveshaft; M is mass flow rate of gas flowing in and gas flowing out; h is gas enthalpy. Here subscripts correspond to the chambers between which mass transfer takes place: D is discharge vol- ume; IN is internal volume; C is com- pression volume; C + 2π, C – 2π are the following and preceding volumes.

The position of the scroll one turn before the formation of the cut off chamber is accepted as the initial angle φ = 0. Thus, the equation system (2) de- scribes the state of gas in three consec- utive chambers: VSC at 0 ≤ φ < 2π, VCUT at φ =2π and VC at 2π < φ < φD, where φD is the angle of discharge beginning.

Mass rates due to leakage between the chambers exist depending on the rotation angle of the driveshaft: MIN↔C at 0 ≤ φ < 2π, MC–2π ↔ C at 2π ≤ φ < φD, MC+2π↔C at 0 ≤ φ < φD – 2π, MDC at φD – 2π ≤ φ < φD.

Volume of each chamber is calcu- lated as product of scroll height and area of this chamber obtained by nu- merical method with the help of Green formula.

The similar systems of equations are written for suction and discharge chambers. The calculation is carried out by the method of successive approxi- mations. The convergence condition of the calculation is discrepancy of pres- sure and temperature diagrams of the preceding and the following approxi- mations less than 1 Pa and 1 K, respec- tively.

Leakage and heat exchange

The main problem in pumping process modeling is the calculation of gas leak- age through the slot channels between the chambers. The channel between the tip face of one scroll and face disc of the other scroll (face channel) is sealed by face seal made of PTFE-based graph- itized material and that is why it is not taken into consideration in calculation.

Radial channel is formed by side sur- faces of scrolls one of which executes orbital motion. Due to this fact, the cut off chamber moves from the scroll pe- riphery to the center with velocity 2π r n, where r is a radius of curvature in the contact point. The contact point exe- cutes the same motion, and wall veloc- ity here may reach 20 – 30 m/s.

As numerical and natural exper- iments showed, at pressures below 10 kPa mobility of channel walls cannot but be taken into account [12]. This can be seen in Fig. 2 where ratio of gas mass rate through the channel with moving walls mw to the mass rate through the channel with fixed walls mw=0 is pre- sented (w is the wall velocity). Calcula- tions are carried out in ANSYS-FLUENT [13]. It can be seen that gas rate through

dPc ___ dφ =

k–1 _____ ω · Vc ·

QT + MIN chIN – Mc INhc + MDchD – McDhc + Mc+2πchc+2π –

– Mc+2π hc + Mc–2πchc–2π – Mcc–2πhc – ω k ____ k –1

· Pc dVc ___ dφ ( (

dTc ___ dφ =

(k – 1) Tc _______ ωPcVc ·

QT + k – 1 ____ k (MIN c + MD c + Mc+2πc – Mcc+2π+ Mc–2π c– Mcc-2π –

(McIN – McD) hc + MIN c (hIN – hc) + MDc (hD –hc) +

+ Mc+2πc (hc+2π – hc) + Mc-2πc (hc-2π – hc) – ω PcdVc _____ dφ

( ( (2)

FIGURE 2: Influence of pressure and wall velocity on mass rate through radial channel: δ = 0.1 mm; r1 = 60 mm; r2 = 64 mm; T = 330 K (blue lines – air, red lines – helium).

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the channel may increase considerably due to the wall motion. Influence of walls mobility on gas mass rate through the channel declines with pressure in- crease and intensifies with clearance decrease, increase of walls curvature radius and increase of gas molecular weight.

On the basis of above-stated, data array for different pressures at the inlet and the outlet of the channel, curvature radii and walls velocity was obtained with the help of ANSYS-FLUENT. These data were used for leakage calculation in the mathematical model.

At pressures higher than 10 kPa the influence of wall velocity on mass rate is slight that is why methods of conduc- tance calculation [14–16] are used in this range.

In the mathematical model losses at the inlet and the outlet are taken into consideration by introducing of the rate coefficient which was obtained by modeling of gas flow with the help of ANSYS-FLUENT. Partial overlap of the discharge port by the end part of the or- biting scroll occurs during its motion, i.e. the geometry of the discharge channel varies depending on the rotation angle φ of the driveshaft. That is why calcula- tions for the discharge channel are car- ried out for several positions of the scroll at different rotation angles [17]. It was found out that outlet losses do not influ- ence the pumping speed considerably (less than 1 %), and inlet channel losses influence the pumping speed only at low pressures (less than 10 %).

The mathematical model takes into account heat exchange of the working body with scroll elements. For taking

into consideration heat exchange with working chamber walls the following well-known equation is used

dQ = αFW(TW – TG)dτ , (3)

where TW, TG are temperature of the wall and gas, respectively; α is heat transfer coefficient; FW is the surface of working chamber; τ is time.

Heat transfer coefficient between the gas in the working chamber and the scroll tip is determined according to the empirical formula presented in [8, 18]

α = λ __ dE . 0.023 . Re0.8 . Pr0.43 (1+1.77 dE __ R ) , (4)

where λ is gas heat conductivity; R is curvature radius of the scroll wall at the given point; dE is characteristic dimen- sion, i.e. hydraulic diameter of the chan- nel; Re is Reynolds number; Pr is Prandtl number.

To take into consideration the influ- ence of rarefaction on heat exchange, the correction into the gas dynamic viscosity coefficient is introduced anal- ogous to [8]

μ = μATM ________ (1 + β . Kn) , (5)

where Kn is Knudsen number; µATM is gas dynamic viscosity at atmospheric pressure.

Parameters Kn and β are connected in the following way

Kn = t – A ______________ (2 –c2) (3 –c2) t

2–c2 ,

β = c1t c2 + A/(c1c2 (2 – c2) (3 – c2) t) (6)

where t is parametric variable; А = 0.15; с1 = 1.479952; с2 = 0.1551753.

For heat exchange calculation in slot channels an assumption was taken that the gas temperature after the slot becomes equal to the average tem- perature of the walls. This assumption is confirmed by numerical modeling in ANSYS-FLUENT and by natural exper- iments [14]. To take into account this effect, the additional heat flow between gas and walls in the clearances is calcu- lated

dq = M . c(t2 – t1)dτ, (7)

where M is mass of gas flowing through the clearance; c is heat capacity of gas; t1, t2 are temperatures of gas at the in- let and the outlet of the clearance, re- spectively. This heat flow is apportioned between the walls according to the dif- ferences of temperature of the wall and temperature of gas at the channel inlet.

Temperature field and deforma- tions model

The temperatures of the scroll elements are determined with the help of the other mathematical model [15] based on the finite-element method. The ini- tial data for calculation of temperature fields are pressures and temperatures in each chamber of the pump determined with the help of mathematical model presented above and temperature dis- tribution on the surfaces of the working elements.

The problem adds up to the solution of three-dimensional equation of heat conductivity (Laplace equation)

ΔT(x) = 0, x∈Ω (8)

Here Ω is three-dimensional area occu- pied by the scroll; x = (x1, x2, x3) is a point of three-dimensional space;

ΔT(x) = ∂ 2T(x)

_____ ∂x 21

+ ∂ 2T(x)

_____ ∂x 22

+ ∂2T(x)

_____ ∂x 23

is three-dimensional Laplace operator. Equation (8) is solved by finite-el-

ement method. For this purpose three-dimensional grid describing the scroll geometry is built, and the approx- imating linear function is written for each element.

Different boundary conditions are specified at different parts of grid area boundary in accordance with working regimes. Due to complex geometry of the scroll vacuum pump housing and difficulties connected with describ- ing of heat exchange between it and environment, the temperature mea- surement at the point of contact of the fixed scroll element and vacuum pump housing and specification of Dirichlet boundary condition are rather justi- fied. On all other surfaces the third-type boundary condition describing heat ex- change with surrounding environment with specified temperature is set.

Taking into account different ap- proaches to building of thermodynamic model and model of working elements temperature field calculation, succes- sive iterative method of solution is only possible. At the first iteration of thermo- dynamic model parameters calculation, temperature of scroll elements over the whole volume is taken constant and is assumed to be the average tempera- ture between temperatures at the inlet and the outlet of the pump.FIGURE 3: Distribution of temperature on orbiting scroll.

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After calculation of scroll elements temperature field, these data are trans- mitted to the thermodynamic model again. For this purpose temperature field is transformed into the depen- dence of the working volume walls av- erage temperature on the shaft rotation angle. Then iteration calculations are carried out with control of convergence according to the scroll tip temperature transmitted to the thermodynamic model.

Temperature fields of orbiting and fixed scroll elements are obtained as the result of the calculation. For exam- ple, temperature distribution over the surfaces of scroll elements for inlet pres- sure PIN = 5 kPa is presented in Fig. 3.

After calculation, the temperature distribution of scroll tip is transmitted to the thermodynamic model again for improvement of calculation. Then itera- tion process of calculation is carried out.

It is known that characteristics of scroll machines are determined by value of clearances between scroll ele- ments. For example, increase of radial clearance by 0.05 mm results in increase of ultimate pressure approximately tenfold, and when clearance is more than 0.15 mm scroll vacuum pump is no longer a medium vacuum pump [20]. It was shown [21, 22] that temperature change of the pump details contrib- utes considerably into change of clear- ances between the scroll elements. The change of the radial clearance (when it is equal to 0.1 mm) due to thermal de- formations may make up to 25 %.

Finite-element method realized in ANSYS is used for determination of scroll elements deformations. Orbiting and fixed scroll elements are split into tetrahedral finite elements dimension of which is determined by condition of grid convergence. Temperature fields of scroll elements obtained in thermo- dynamic calculation are introduced into thermal deformation model.

Obtained clearances values are in- troduced into the mathematical model and the calculation is repeated accord- ing to the algorithm presented above. After several iterations, dependence of pressures and temperatures in working chambers on driveshaft rotation angle is calculated taking into account radial clearance changes due to thermal de- formations.

The result of the calculation is the difference between masses of gas en- tering the pump and of gas flowing back into evacuated volume in one rev- olution

mIN = ∫ 0 2π

(MINC – MCIN) . dφ (9)

The pumping speed of scroll vac- uum pump is calculated according to the following formula

SIN = mINRGTIN _______ PIN

n, (10)

where RG is gas constant (Fig. 4).

Conclusion

Thus, the developed mathematical model of the working process of the scroll vacuum pump can be recom- mended for calculation of pumping characteristics of scroll vacuum pumps.

References [1] Y. Chen, N. Halm, E. Grolland, J. Braun: International

Compressor Engineering Conference, Purdue Univer- sity (2000) h715-724.

[2] T. Yanagisawa, M.C. Cheng, M. Fukuta, T. Shimizu: Inter- national Compressor Engineering Conference, Purdue University (1990) 425-433.

[3] N. Halm: Mathematical Modeling of Scroll Compres- sors, Master Thesis, Purdue University, 1997.

[4] Y. Paranin, R. Yakupov, A. Burmistrov: Bulletin of Kazan Technological University 16 (2013) 19, 267-271. (in Russian)

[5] E.J. Moore et al.: Analysis of a Two Wrap Meso Scale Sc- roll Pump, Rarefied Gas Dynamics: 23th International Symposium (2003) 1033-1040.

[6] Z. Li, L. Li, Y. Zhao, G. Bu, P. Shu, J. Liu: Vacuum 85 (2010) 95-100.

[7] Y. Su, T. Sawada, J. Takemotob, S. Haga: Vacuum 47 (1996) 815-818.

[8] Z. Li, L. Li, Y. Zhao, G. Bu, P. Shu: Vacuum 84 (2010) 3, 415-421.

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

[10] B.Fotin: Working processes in piston compressors, Thesis, Leningrad, 1974. (in Russian)

[11] A. Raykov, S. Salikeev, A. Burmistrov, M. Fomina: Va- kuum in Forschung und Praxis 28 (2016) 3, 43-47.

[12] M. Bronshtein, A. Raykov, A. Burmistrov, S. Salikeev, R. Yakupov: Kompressornaya tehnika i pnevmatica (Compressors and pneumatics) 1 (2017) 30-34 (in Russian).

[13] Ansys, Inc. license file for Kazan National Research Technology University c/n 657938.

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

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

[16] A. Burmistrov, S. Salikeev, M. Bronshtein, M. Fomina, A. Raykov: Vakuum in Forschung und Praxis 27 (2015) 1, 36-40.

[17] A. Raykov,A. Burmistrov, S. Salikeev, A.Gimaltynov, R. Yakupov: Izvestiya vuzov. Mashinostroenie 5 (2017) 686, 45-51. (in Russian)

[18] K. Jand, S. Jeond: International Journal of Refrigera- tion 29 (2006) 744-753.

[19] M. Khamidullin: Development and study of internal compression rotor compressor on the basis of geome- trical analysis and modeling of processes in working chambers, Thesis, Kazan, 1992, 193 p. (in Russian)

[20] A. Tyrin, S. Puzankov, A. Burmistrov, V. Alyaev: XXIII Conference “Vakuumnaya nauka I tekhnika” (Vacuum science and technology) (2016) 56-59. (in Russian)

[21] A. Raykov, R. Yakupov, S. Salikeev, A. Burmistrov: Bulle- tin of the Bauman Moscow State Technical University, Mechanical Engineering 3 (2015) 102, 92-102. (in Russian)

[22] A. Raykov, A. Burmistrov, S. Salikeev, M. Fomina, E. Ka- pustin: Vakuum in Forschung und Praxis 27 (2015) 3, 42-46.

[23] A. Burmistrov, S. Salikeev, V. Alyaev, E. Kapustin: XI In- ternational Conference “Vacuum technique, materials and technology”, Moscow (2016) 12-16. (in Russian)

Dr. Sergey Salikeev 1978, mechanical engineer, Associate Professor, Department of Vacuum Equipment, Kazan Nati- onal Research Technological University, Russia

Dr. Aleksey Burmistrov 1963, mechanical engineer, Professor, Department of Vacuum Equipment, Kazan National Research Technological University, Russia

Dr. Alexey Raykov 1986, Education: mechanical engineer, Associate Professor, De- partment of Vacuum Equipment, Kazan National Research Tech- nological University, Russia

Dr. Marina Fomina 1959, mechanical engineer, Associate Professor, Department of Vacuum Equipment, Kazan National Research Technological Uni- versity, 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]

FIGURE 4: Calculated and experimental pumping speed of scroll vacuum pump NVSp-12 (points – experiment).