rus
Issue #1/2022
A.A.Isaev, A.A.Raykov, A.V.Burmistov, S.I.Salikeev
CONDUCTIVITY OF THE ROOTS TYPE DOUBLE ROTOR VACUUM PUMP CHANNELS IN THE MOLECULAR GAS FLOW MODE
CONDUCTIVITY OF THE ROOTS TYPE DOUBLE ROTOR VACUUM PUMP CHANNELS IN THE MOLECULAR GAS FLOW MODE
10.22184/1993-8578.2022.15.1.58.63
INTRODUCTION
Nowadays it is difficult to imagine a pumping system ensuring a medium or high vacuum without booster vacuum pumps, which considerably reduce the pumping time and the residual pressure in the working chamber. The most commonly used vacuum pumps are of the Roots-type double-rotor vacuum pumps, which are practically indispensable because of obtaining oil-free vacuum.
As it is known, the operation process in a double-rotor vacuum pump (DRVP) (Fig.1) consists of two components: transfer by rotors of cut-off volumes, from inlet to outlet and backflows from outlet to inlet due to pressure and temperature differences, through slotted channels of the rotor mechanism. Accordingly, the efficiency of the DRVP depends on the ratio of these two processes.
Values of the volumes pumped by the rotors per a revolution and the amount of backflow through the slotted channels depend on the rotor profile. When selecting the profile type and its geometric parameters, the aim should be to increase the cut-off volume pumped by the rotors, i.e. to increase the geometric speed of actuation and reduce the backflow (conductivity) of the slotted channels.
The geometric speed of the DRVP is determined by the formula:
, (1)
where R and L are the radius and length of the rotor; n is the rotor speed; χ is the volume utilisation factor, determined from the ratio:
, (2)
where fP is the rotor cross section area.
Thus, for a given R and L, one should aim to increase the coefficient χ.
The return flow through the slotted channels is characterised by their conductivity. Since the slotted channels of the DRVP are operated in parallel, the total conductivity is defined as [1, 2]:
, (3)
where UPK1 and UPK2 are radial channel conductances (between rotor head and cylindrical body), UPP are the inter-rotor channel conductivity, UT1 and UT2 are end channel conductivity.
From the viewpoint of minimizing overflows, the optimal flow of the DRVP is the molecular mode of flow in slotted channels, since here the conductivity is minimal. It is known that the conductivity of an arbitrarily slotted channel is calculated by the formula [1–3]:
, (4)
where C is the arithmetic mean velocity of the gas molecules, К3 is the conductivity coefficient for the corresponding channel.
RESEARCH METHODS
In order to find conductivity of channels in the molecular mode, the Monte Carlo method (MCM) or angular coefficient method (ECM) are most commonly used. In [3, 4] a method of calculating conductivity of the channels having minimal clearance in some cross-section has been developed. The radial and inter-rotor channels of an DRVP are just of such kind. For such channels the resistance is determined by the section in the place of minimum clearance, and walls can be replaced by arcs of circle with radii R1 and R2. The blowdown [2] of two-rotor DVN-50 pump manufactured by "Vakuummash" JSC [5] at different rotor rotation angles was carried out. The resulting air conductivity values and the corresponding gap values are shown in Table 1. The measurements were taken in 15° steps. The gaps were measured using feeler gauges.
In the same table the results of calculations according to the method of works [3, 4] are presented. The conductivity of the end channels was calculated using the formula for a long flat slot [6]. The end clearances on both sides were the same and equaled 0.11 mm each. The average deviation of the calculation results from the experiment is 13.8% according to the methodology presented in [3, 4].
In the present paper, the total conductivity of the CVD channels was calculated using the COMSOL Multiphysics software package [7], which implements the MMK and ICC methods. In spite of the fact that the MMK method is probably a more universal one, in the present work the conductivity calculation is performed using the ICC method. Such choice is explained by the specificity of the channels under study – their considerable length and low probability of passage of molecules, which requires a considerably long machine time.
As it is well known, the ICC is based on the analogy between gas flow in channels with diffuse reflection from the walls and radiant heat transfer in diathermic confined media. The channel walls and inlet and outlet surfaces are split into elementary areas. The quality of the surface splitting determines the accuracy of the channel walls curvature. For each area, an angular coefficient is calculated, which is the fraction of the flow of molecules coming from one elementary surface and falling on the other. The resulting conductivity coefficient is determined by integrating the fluxes from all elementary sites relative to the inlet and outlet surfaces. The best convergence of the experiment data and the calculation data was shown by the Hemicube integration method with a resolution of 1,024 dpi.
RESULTS AND DISCUSSION
The data in the table show that the deviation of the calculation results in COMSOL Multiphysics has two times less deviation from the experiment. This is due to replacement of real geometry of channel walls with arcs of circles used in [3, 4]. The deviation of calculations from the experiment is caused by impossibility to accurately determine the clearances in the rotor mechanism.
CONCLUSIONS
The calculations have shown good prospects to use the COMSOL Multiphysics software package for calculations of gas flow in the curved channels of vacuum pumps in the molecular mode. It is worth noting that the calculations were not carried out for each channel separately but for the whole rotor mechanism of the DRVP assembly.
PEER REVIEW INFO
Editorial board thanks the anonymous reviewer(s) for their contribution to the peer review of this work. It is also grateful for their consent to publish papers on the journal’s website and SEL eLibrary eLIBRARY.RU.
Declaration of Competing Interest. The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Nowadays it is difficult to imagine a pumping system ensuring a medium or high vacuum without booster vacuum pumps, which considerably reduce the pumping time and the residual pressure in the working chamber. The most commonly used vacuum pumps are of the Roots-type double-rotor vacuum pumps, which are practically indispensable because of obtaining oil-free vacuum.
As it is known, the operation process in a double-rotor vacuum pump (DRVP) (Fig.1) consists of two components: transfer by rotors of cut-off volumes, from inlet to outlet and backflows from outlet to inlet due to pressure and temperature differences, through slotted channels of the rotor mechanism. Accordingly, the efficiency of the DRVP depends on the ratio of these two processes.
Values of the volumes pumped by the rotors per a revolution and the amount of backflow through the slotted channels depend on the rotor profile. When selecting the profile type and its geometric parameters, the aim should be to increase the cut-off volume pumped by the rotors, i.e. to increase the geometric speed of actuation and reduce the backflow (conductivity) of the slotted channels.
The geometric speed of the DRVP is determined by the formula:
, (1)
where R and L are the radius and length of the rotor; n is the rotor speed; χ is the volume utilisation factor, determined from the ratio:
, (2)
where fP is the rotor cross section area.
Thus, for a given R and L, one should aim to increase the coefficient χ.
The return flow through the slotted channels is characterised by their conductivity. Since the slotted channels of the DRVP are operated in parallel, the total conductivity is defined as [1, 2]:
, (3)
where UPK1 and UPK2 are radial channel conductances (between rotor head and cylindrical body), UPP are the inter-rotor channel conductivity, UT1 and UT2 are end channel conductivity.
From the viewpoint of minimizing overflows, the optimal flow of the DRVP is the molecular mode of flow in slotted channels, since here the conductivity is minimal. It is known that the conductivity of an arbitrarily slotted channel is calculated by the formula [1–3]:
, (4)
where C is the arithmetic mean velocity of the gas molecules, К3 is the conductivity coefficient for the corresponding channel.
RESEARCH METHODS
In order to find conductivity of channels in the molecular mode, the Monte Carlo method (MCM) or angular coefficient method (ECM) are most commonly used. In [3, 4] a method of calculating conductivity of the channels having minimal clearance in some cross-section has been developed. The radial and inter-rotor channels of an DRVP are just of such kind. For such channels the resistance is determined by the section in the place of minimum clearance, and walls can be replaced by arcs of circle with radii R1 and R2. The blowdown [2] of two-rotor DVN-50 pump manufactured by "Vakuummash" JSC [5] at different rotor rotation angles was carried out. The resulting air conductivity values and the corresponding gap values are shown in Table 1. The measurements were taken in 15° steps. The gaps were measured using feeler gauges.
In the same table the results of calculations according to the method of works [3, 4] are presented. The conductivity of the end channels was calculated using the formula for a long flat slot [6]. The end clearances on both sides were the same and equaled 0.11 mm each. The average deviation of the calculation results from the experiment is 13.8% according to the methodology presented in [3, 4].
In the present paper, the total conductivity of the CVD channels was calculated using the COMSOL Multiphysics software package [7], which implements the MMK and ICC methods. In spite of the fact that the MMK method is probably a more universal one, in the present work the conductivity calculation is performed using the ICC method. Such choice is explained by the specificity of the channels under study – their considerable length and low probability of passage of molecules, which requires a considerably long machine time.
As it is well known, the ICC is based on the analogy between gas flow in channels with diffuse reflection from the walls and radiant heat transfer in diathermic confined media. The channel walls and inlet and outlet surfaces are split into elementary areas. The quality of the surface splitting determines the accuracy of the channel walls curvature. For each area, an angular coefficient is calculated, which is the fraction of the flow of molecules coming from one elementary surface and falling on the other. The resulting conductivity coefficient is determined by integrating the fluxes from all elementary sites relative to the inlet and outlet surfaces. The best convergence of the experiment data and the calculation data was shown by the Hemicube integration method with a resolution of 1,024 dpi.
RESULTS AND DISCUSSION
The data in the table show that the deviation of the calculation results in COMSOL Multiphysics has two times less deviation from the experiment. This is due to replacement of real geometry of channel walls with arcs of circles used in [3, 4]. The deviation of calculations from the experiment is caused by impossibility to accurately determine the clearances in the rotor mechanism.
CONCLUSIONS
The calculations have shown good prospects to use the COMSOL Multiphysics software package for calculations of gas flow in the curved channels of vacuum pumps in the molecular mode. It is worth noting that the calculations were not carried out for each channel separately but for the whole rotor mechanism of the DRVP assembly.
PEER REVIEW INFO
Editorial board thanks the anonymous reviewer(s) for their contribution to the peer review of this work. It is also grateful for their consent to publish papers on the journal’s website and SEL eLibrary eLIBRARY.RU.
Declaration of Competing Interest. The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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