Modeling Investigation on Gas Backflow Performances in Screw Vacuum Pump

Abstract

Rotor structure has a great influence on the gas backflow in a screw vacuum pump. The characteristics of the gas main flow along the spiral groove of the screw rotor and the gas reverse flow along the tooth-shaped, tooth side, radial, and circumferential clearances are investigated. A new mathematical model of the pumping flow and backflow involved in a flow balance model is proposed to investigate the actions of the shearing force and pressure difference force. The calculated backflow is verified by comparing the experimental measured results. The relationships of the structural parameters of the screw rotor are established. The effects of the rotor parameters, such as pitch, diameter, and compression ratio, on backflow are revealed. The results show that the rotor diameter and compression ratio remain constant and that the influence of pitch on the backflow is slightly weak, with backflow variations of less than 3%, whereas the pitch, rotor length, and compression ratio are constant and the rotor addendum diameter is directly proportional to the backflow. The addendum diameter of rotor #4 is the largest, and its backflow is about 1.5 times larger than that of rotor #1. When the rotor radial sizes and the pitch of the suction end are constant, the compression ratio is inversely proportional to the backflow in the low-pressure region and proportional to the backflow in the high-pressure regions. Therefore, for a vacuum pump operating in low-pressure areas, the use of the compression ratio of 2.2 or higher is favorable for the reduction in backflow.

1. Introduction

Vacuum acquisition equipment is very important and has a great effect on the stability, reliability, and other performances of the entire vacuum environment. The screw vacuum pump is the typical power source used in vacuum acquisition equipment, and its core components consist of a pair of screw rotors meshed with each other and two rotors driven by a pair of synchronous gears [1,2,3,4]. The surface of the high-speed rotating rotors carries gas molecules moving together, causing the gas to be continuously extracted from the pump body.

Rabiger et al. [5] set up a finite volume model to numerically simulate the temperature, pressure, and velocity behaviors in screw vacuum pump clearances. Li et al. [6] proposed a physical model to analyze the conical rotors and the influence of basic parameters on the performance of the rotors. Utri et al. [7] set up an optimized approach by using the dual-lead rotors for an application of a screw compressor and a wrap rotor by using the Nelder–Mead algorithm. Utri et al. [8] established optimization in terms of efficiency and rotor geometry. Straalsund et al. [9] discussed the bench-scale evaluations and investigated the performances of a type of screws. Zöllig [10] utilized a kind of dry screw vacuum pump and Roots pumps to look into the degassing process, and lower energy consumption for a variable-pitch cylindrical rotor was observed. Rane et al. [11] established a design method to predict the rotor performances of a non-conventional rotor by using SCORG V5.5 commercial software. The speed of suction based on reference pressure and the smallest clearance for gas flow in the pump were modeled by Dirk et al. and Salikeev et al. [12,13]. Abhay et al. [14] studied the influence of heat generation on performance and potential mitigation in the process of wet gas compression. Burmistrov et al. [15] made use of a kind of circular–angular coefficient to predict channel conductivity in complexed profiles, i.e., the molecular flow regime in a Roots pump. On the basis of the thermal network of a resistance model of a screw vacuum pump, part thermal deformation, the process of the heat transfer of parts, and the performances and behaviors of load and power in the screw vacuum pump were investigated [16,17]. Li et al. [18] obtained novel profiles of the conical rotors and revealed the smaller contact line length. Gan et al. [19] proposed a transformation approach for plot rotor profiles at different sections of a screw vacuum pump. Meanwhile, relative formulas and transformation steps were achieved.

Some optimized strategies regarding the thermodynamic and exhausted process, the predefined sealing line, algorithms of meshing clearance simulation, the simplified profiles of the minimum distance of two normal racks, the largest spherical and the smallest cross-sectional area methods, the suitable clearance methods, the real-time balance for the screw rotor, and the helix profiles in the conical rotor are widely applied [20,21,22,23,24,25,26,27,28]. With respect to the modeling and numerical simulation of gas flow in a pump, the Reynolds time-averaged Navier–Stokes approach and large eddy simulation are generally utilized in the field of the industrial applications [29,30]^. Khoshkalam et al. [31] investigated the loss mechanisms and established the fluid stream numerical model in a compressor. Wei et al. [32] introduced a winglet and a composite blade tip into the axial tip clearance of a liquid ring pump blade and analyzed their mechanism on the clearance leakage flow field using a numerical simulation method.

To date, there are few publication reports on modeling studies regarding the screw vacuum pump. The screw vacuum pump has a kind of non-contact transmission existing in the clearances between male and female rotors and the body, which lead to the formation of complex flow channels. During the process of transporting gas, the extracted gas inevitably enters in the opposite direction along the clearances. Gas backflow has a significant influence on the pump’s pumping speed, ultimate pressure, and power. As for the structure of the vacuum pumps, the sizes of the screw rotor are closely related to the flow channels. The performances of the overall pumping in the screw vacuum pump are primarily dependent on the spatial shapes and lengths of the channels. Therefore, it is of great importance to investigate the gas flow characteristics in the pump cavity under the combined actions of the shearing force and pressure difference force of the screw rotor. In this work, a mathematical model for both pumping flow and backflow is proposed, and the calculation results are validated by experimental results. The effects of the rotor diameter, pitch, compression ratio, and rotor type on backflow are discussed. The relationships between the rotor structural parameters and backflow are built up for the optimization of the performance design of the vacuum pump.

2. Gas Flow Analysis of Screw Vacuum Pump

The illustrations of two kinds of gas flow in a screw type of vacuum pump are provided in Figure 1. When the rotors rotate in the opposite direction, the gas pumping flow (Q_M) implies that the extracted gas went through an axial transport migration from the suction end to the exhaust end due to shearing stresses (see Figure 2), in which there is low pressure at the suction end and high pressure at the exhaust end. Gas backflow (Q_F) implies that a small amount of gas reverse flows went through the high-pressure exhaust end to the low-pressure suction end due to pressure difference. Meanwhile, the gas pumping flow is the 8-shaped clearance channel between the addendum circle of the rotor and the body, and gas backflow is the backflow channel along the spiral groove of the rotor. Pumping flow and backflow are functions of the gas flow channel structure, rotor speed, and pressure:

$$Q_M=f(n,structural parameters of screw rotor,P_{in})$$

(1)

$$Q_F=f(\Delta P,T,n,structural parameters of pump)$$

(2)

They interact with each other under the combined actions of the shearing force and pressure difference force in the process of operations for the screw vacuum pump. For these two flow rates reaching to achieve a dynamic balance, a stable value of gas pressure in the pump cavity is obtained, indicating that they are equivalent for this equilibrium pressure. Gas backflow has a significant influence on the performances of the vacuum pump, and a smaller one is favorable for reaching the higher vacuum limitation quickly, as well as reducing loss of power. The flow balance equation is given based on the corresponding assumptions: (1) gas is considered to be ideal and incompressible; (2) heat transfer between the cavity and body is neglected.

$$Q_M=Q_F$$

(3)

2.1. Gas Pumping Flow

Gas flow in the pump cavity is simplified as a kind of relative flow between two walls. Figure 3 shows a simplified model of the rotor volume chamber. The pumping flow speed at any point in the volume chamber is

$$d{u}_M={\displaystyle \frac{v_M}{h}}dz={\displaystyle \frac{v\cos \beta }{h}}dz$$

(4)

The rotating motion of the rotor drives the flow of the gas boundary layer attached to the cavity surface with the same rotor speed and drives the inner layer gas to the spiral groove, resulting in the formation of gas pumping flow due to shear force. The pumping flow is closely affected by the structural parameters of the rotor, pressure at the suction end, rotor speed, etc. The flow equation is

$$d{Q}_M=k{A}_c{P}_{in}d{u}_M$$

(5)

2.2. Gas Backflow

According to the Bernoulli energy conservation equation [33], when incompressible ideal fluid flow remains steady under the action of gravity, the total sum of the potential, pressure potential, and kinetic energies of the entire flow field are constant on the basis of the energy conservation law. The energy conservation equation of gas flow in the screw vacuum pump is

$$z_{out}+{\displaystyle \frac{p_{out}}{\rho g}}+{\displaystyle \frac{v_{out}^2}{2g}}=z_{in}+{\displaystyle \frac{p_{in}}{\rho g}}+{\displaystyle \frac{v_{in}^2}{2g}}+\Delta E$$

(6)

The potential energy and kinetic energy at the suction and exhaust ends of a vacuum pump are the same: z_out = z_in, v²_out/2g = v²_in/2g. This is due to the fact that friction resistance is caused by gas viscosity as a result of energy loss ΔE. Pressure drop occurs during the gas flow process based on Equation (6), it can be expressed as

$$\Delta P=\rho g\Delta E$$

(7)

Here, Couette backflow and orifice backflow can be generated because of the complicated clearance channels and different shapes of the parts in the screw vacuum pump.

(1) The Couette backflow is incurred by the relative movements located between the rotors and the body. Circumferential, radial, and tooth-shaped clearance channels are found between two rotors and the body. These channels are classified into two parallel and relatively moving plates, in which the clearance between plates is much smaller than the length and width of the plates. The formation of the laminar gas flow between two plates can be considered as the backflow passing through this clearance; it is defined as

$$d{Q}_{F1}=v\cos \beta l{P}_{in}d\delta$$

(8)

(2) The orifice backflow is incurred by gas going through the small hole of the mounted thin wall, as shown in Figure 4. The complex gas flow is attributed to the significant shrinkage or expansion changes in the cross-sectional areas when gas flows into and exits toward the circumferential, radial, tooth side, and tooth-shaped clearance channels. It can be affected by multiple factors, such as the clearance channel area, operation temperature, and pressure between the rotor interstage and structural parameters. It is defined as

$$d{Q}_{F2}=\sqrt{{\displaystyle \frac{2K}{K-1}}{\displaystyle \frac{RT}{\mu }}(1-x^{\frac{K-1}{K}})}{x}^{{\displaystyle \frac{1}{K}}}{P}_{in}ld\delta$$

(9)

Thus, the total backflow is the sum of the Couette backflow and orifice backflow; it is

$$Q_F=Q_{F1}+Q_{F2}$$

(10)

3. Experimental Validation

Figure 5 shows the experimental setup with the same rotor length L = 645 mm, radius of addendum circle R_a = 123 mm, radius of dedendum circle R_f = 45 mm, and different compression ratios ε = 1.8 and ε = 2.1. The compression ratio is defined by the pitch ratios between the suction end and exhaust end. Figure 6 shows the setup of the screw vacuum pump consisting of the vacuum pump and cover, the pressure sensor, the flow regulating valve, and the motor.

Regarding the operation of the vacuum pump, when the flow regulating valve is opened, gas enters the vacuum cover. At this time, the testing system simultaneously begins to extract and input gas. By adjusting the opening of the valve, the amount of gas entering the vacuum cover can be controlled, thereby keeping the pressure inside the vacuum cover at different values. The gas pressure entering the flow regulating valve is atmospheric pressure P₁; the pressure inside the cover is P₂ that is measured by the pressure sensor; then, gas flow velocity S_v entering the valve can be directly read. The range of the pressure sensor is from 0.1 Pa to 100,000 Pa; reading accuracy is 0.25%. The flow regulating valve ranges from 0.6 to 6 m³/h, 2.5 to 25 m³/h, 6 to 60 m³/h, 50 to 250 m³/h, and 80 to 400 m³/h, with a reading accuracy of 2.5%. The test result is shown in Figure 7.

Based on the ideal gas law in Equation (11) [34], the pumping speed S_p of the vacuum pump was obtained in Equation (12) (see Figure 8). As soon as the pressure acting on the cover reaches a stable state, the total backflow and pumping speed of the vacuum pump are considered to be the same; this is expressed by Equation (13).

$${\displaystyle \frac{P_1{V}_1}{T_1}}={\displaystyle \frac{P_2{V}_2}{T_2}}$$

(11)

$$P_1{S}_v=P_2{S}_p$$

(12)

$$Q_F=P_2{S}_p$$

(13)

The experimental data can be converted into the backflow based on the synthetic uncertainty method. Figure 9 shows that the simulations agree well with the experimental data and that the errors are acceptable. The reason is that the generation of heat changed the thermal expansion of the rotors and bodies, leading to the variation in gas backflow during the operations of the vacuum pump; this proves that the established model is reliable and reasonable for this task.

4. Effects of Structural Parameters on Backflow

The partial structural parameters and clearance channels of the rotors are shown in Figure 10, where ①–④ represent the tooth-shaped, tooth side, radial, and circumferential clearance channels, respectively. The mathematical models of the structural parameters are given as follows:

$$l_1=\sqrt{{{\partial }^{″}}^2+k^{\prime }{(\lambda -2B)}^2}$$

(14)

$$l_2={\displaystyle \frac{{\partial }^{″}}{\cos \beta }}$$

(15)

$$l_3=2B{k}^{″}$$

(16)

$$l_4=2(1-{\displaystyle \frac{1}{\pi }}\arccos \vartheta )\sqrt{{(2k^{‴}\pi R_a)}^2+{\lambda }^2}$$

(17)

$$B={\displaystyle \frac{\lambda }{2}}(1-{\displaystyle \frac{\sqrt{{\partial }^2-1}-\sqrt{{{\partial }^{\prime }}^2-1}-\arctan \sqrt{{\partial }^2-1}+\arctan \sqrt{{{\partial }^{\prime }}^2-1}}{2\pi }})$$

(18)

$$V_i={\displaystyle \frac{A_c{\lambda }_i}{2\pi }}$$

(19)

$$\tan \beta ={\displaystyle \frac{\lambda }{2\pi R_a}}$$

(20)

$$\epsilon ={\displaystyle \frac{{\lambda }_{in}}{{\lambda }_{out}}}$$

(21)

Based on Equations (14)–(21), Figure 11 shows the relationships among the structural parameters of the rotor, i.e., the clearance channel length, pitch, tooth width, and working chamber volume. The clearance channel length has a significant effect on backflow in association with the pitch, pitch angle, tooth width, etc. The working chamber volume is a function of the addendum circle radius, dedendum circle radius, and pitch. Based on Equations (8) and (9), backflow is directly proportional to clearance length. Longer clearances induced much more backflow.

The corresponding rotor structures are shown in

Table 1. Screw rotor design cases.

Case	Ra (mm)	Rf (mm)	λin (mm)	L (mm)	ε	Geometric Models
#1	123	45	163	645	1.8
#2	123	45	190	645	1.8
#3	123	45	215	730	1.8
#4	174	61	163	645	1.8
#5	74	31	163	645	1.8
#6	123	45	163	585	2.2
#7	123	45	163	536	2.7

and are represented by addendum circle diameter, dedendum circle diameter, and pitch. The four clearances of rotors #1–7 are set to δ₁ = 0.5 mm, δ₂ = 0.15 mm, δ₃ = 0.45 mm, and δ₄ = 0.58 mm. Here, rotor #1 is set to the comparative reference object, and rotors #2 and #3 are achieved by using a different pitch and the same diameter and compression ratios. Rotors #4 and #5 can be obtained by the variations in diameter, constant compression ratio, and rotor stage. Rotors #6 and #7 are obtained by changing the compression ratio, constant diameter, and rotor stage.

Based on Equations (8) and (9), it can be inferred that the structural parameters may affect the backflow, including clearance channel length, pitch angle, and clearance. Four clearance channel lengths are functionally related to the structural parameters, such as rotor diameters and pitch. The influence of the parameters on the backflow is studied in detail.

4.1. Effects of Pitch on the Backflow

Rotors #1–3 have the same rotor diameters and different pitches. The pitches of rotors #2–3 are larger than those of rotor #1. The larger pitch has a wider tooth size. Figure 12 depicts the tooth width in rotor curves #1–3 along the rotor length. The trends of the tooth width change in rotors #1–3 are observed at the ends between the exhaust and suction because they have the same compression ratios, and rotor #3 has the largest tooth width. Tooth width is proportional to radial clearance, as shown in Equation (16). Thus, radial clearance length will increase proportionally with an increase in pitch. Although circumferential clearance length is proportional to pitch, the influence of pitch on the circumferential clearance length is much smaller than that of the addendum circle radius. Therefore, when the addendum circle radius remains constant, the influence of pitch on the circumferential clearance length is obscured. Figure 13 shows the trend of the length variation in the four clearances for rotors #1–3. The four clearance lengths of rotors #2,3 are larger than those of rotor #1. The most increased degrees of radial clearance length are for rotor #2, with 16.6%, and rotor #3, with 31.9%, even if the circumferential clearance length slightly increased with only 0.24% and 0.51%, respectively.

The backflow of rotors #1–3 is calculated using Equations (8)–(10). Figure 14 shows the relation curves of the total backflow and inlet pressure of rotors #1–3, where the total backflow of three rotors is very close. The backflow of the circumferential clearance is much greater than that of the others, demonstrating that it is the factor that has the most impact on the total backflow [35]. Therefore, the total backflow of rotors #1–3 is very close, increasing by approximately 1.4% and 2.7% compared to rotor #1, respectively. So, it is simple to change the rotor pitch that does not have a significant influence on controlling the backflow in the vacuum pump.

4.2. Effects of Rotor Diameters on the Backflow

Rotors #1 and #4–5 have the same pitch, rotor length, and compression ratio, but their addendum circle and dedendum circle diameters are different. As the rotor diameters change, parameters such as pitch circle radius, tooth groove depth, tooth width, and pitch angle are also altered. The relationships are as follows:

$$R_d={\displaystyle \frac{R_a+R_f}{2}}$$

(22)

$$h=R_a-R_f$$

(23)

Figure 15 shows the trend of these structure parameters at the suction end for rotors #1, 4–5. The change in rotor diameter leads to the corresponding changes in the pitch circle and tooth groove depth. Compared to rotor #1, the pitch circle and tooth groove depth of rotor #4 increase with the increasing rotor diameter, while rotor #5 decreases relatively. The tooth groove depth increased by approximately 44.9% and decreased by 47.4%, respectively. Owing to the constant rotor pitch and base circle, the width of the tooth was inversely proportional to the addendum circle and directly proportional to the dedendum circle. Affected by both factors, the tooth width of rotor #5 increased by about 30.3% compared to rotor #1, and the relative decrease in rotor #4 was about 28.4%. The pitch angle was inversely proportional to the addendum circle, resulting in an increase in pitch angle of approximately 62.2% in rotor #5 compared to rotor #1, and a relative decrease of 28.8% in rotor #4.

Figure 16 shows the trend of the length variation in the four clearances for rotors #1, 4–5. Compared to rotor #1, rotor #5 only has an increase in radial clearance length, while the other three types of clearance length decrease, and rotor #4 is exactly the opposite. The circumferential clearance length is directly proportional to the addendum circle radius and pitch of the exhaust end. The influence of the addendum radius on the length is very significant. Therefore, the circumferential clearance length of rotor #4 increases, but it decreases for rotor #5. The radial clearance length is linearly related to tooth width and increases with the increasing tooth width. Furthermore, the radial clearance length of rotor #5 increases by about 30.5% compared to rotor #1. The tooth side clearance length is proportional to the tooth groove depth and pitch angle, but the influence of the pitch angle on length is not as significant as the tooth groove depth. Therefore, although the pitch angle of rotor #5 increases, the tooth side clearance length still decreases under the comprehensive influence of multiple parameters. The tooth-shaped clearance length is directly proportional to the tooth groove depth and inversely proportional to tooth width, and the tooth-shaped clearance length of rotor #4 increases while that of rotor #5 decreases.

The circumferential, tooth-shaped, and tooth side clearance lengths increased relatively by 40.4%, 47.8%, and 43.4% in comparison with rotor #1, but the radial clearance length decreased relatively by approximately 28.4%. We can see that the reduction in backflow caused by clearance is much smaller than its increase. For the whole process of pumping, the total backflow of rotor #4 is approximately 1.5 times larger than that of rotor #1. Under the same structural parameters and clearance conditions, the backflow deteriorated because of the increasing rotor diameter and lengthening circumferential clearance. The radial dimension of rotor #5 was smaller, and the circumferential tooth-shaped and tooth side clearance lengths were also relatively small. For the whole process of pumping, the total backflow of rotor #5 decreased by 39.3% for rotor #1. Figure 17 shows the relation curves between the total backflow and inlet pressure of rotors #1, 4–5. Thus, the purpose is to effectively control the backflow of the vacuum pump, and smaller radial sizes of the rotor should be used as much as possible while meeting the design requirements.

4.3. Effects of Compression Ratio on the Backflow

Rotors #6–7 and rotor #1 have the same suction end diameter and pitch, but they have larger compression ratios, which is the running chamber ratio at the ends of the suction and exhaust. As shown in Figure 18, the larger compression ratio is favorable for the more significant variation in chamber volume starting from the section end to the variable-pitch section. The chamber volumes remain unchanged after the rotor angle of 6π since the exhaust section adopts the fixed-pitch design, and there is no internal compression.

The higher compression ratios of rotors #6 and #7 lead to larger gas internal compression, resulting in higher pressure. The number of rotor stages is arranged along the axial direction, and the suction section of the rotor is set as the 1st stage, and so on. The pressure ratio variation curve distributed in each cavity of the rotor is shown in Figure 19; here, the ratio of pressure is defined as the ratio of the previous one to the subsequent pressures. The pressure in the variable-pitch section increases due to the compression of the gas. When the inlet pressure remains constant, the pressure ratio of sections 1–2 of rotor #6–7 is inversely proportional to the compression ratio, and it is smaller than in rotor #1. In the variable-pitch section, higher pressure is caused by the compressed gas, resulting in the blockage of gas flow to the suction end, and the relatively high pressure in the third and fourth stages. Therefore, the pressure ratio of the 2–3 sections to rotor #6–7 continues to increase gradually and is always inversely proportional to the compression ratio. After the third stage of the rotor, there is no internal compression in the working chamber. Closing the exhaust end leads to a larger influence on the outlet pressure. The pressure in the exhaust sections of the rotor approaches the atmospheric pressure. The trend of the pressure ratio of rotors #1, 6–7 after the fixed-pitch section is reversed, and the pressure ratio is proportional to the compression ratio.

Figure 20 illustrates the curve locations between the total backflow and entrance pressures of rotors #1, 6–7. Based on Equation (9), it can be seen that the backflow is related to the two factors of the pressure ratio, χ^1/K and 1 − χ^K−1/K. When the inlet pressure is lower, the χ^1/K factor plays a dominant role, that is, the backflow is proportional to the pressure ratio. A larger pressure ratio led to a greater backflow. According to the above analysis, the pressure ratio at the suction end is inversely proportional to the compression ratio. Therefore, the backflow of rotors #1, 6–7 decreases with an increasing compression ratio. As the inlet pressure increases, the effect of 1 − χ^K−1/K becomes more significant. While the pressure ratio increases, the backflow decreases accordingly. Therefore, in the high-pressure stage, the backflow of rotors #6–7 increases. However, for the low-pressure stage, the backflow of rotors #6–7 decreases. Regarding vacuum pumps operating in low-pressure areas, a large compression ratio can reduce gas backflow, which can effectively improve the ultimate vacuum degree.

5. Conclusions and Remark

In this study, the behaviors and characteristics of gas flow in a screw vacuum pump were analyzed elaborately. Under the combined actions of the shearing force and pressure difference force that were generated by the rotor rotation, the gas flow was divided into, the pumping flow and the backflow. A novel mathematical model was proposed to analyze and numerically simulate their performances. In addition, the effect of the rotor structural parameters on the optimization of the geometric features and backflow behavior in the vacuum pump was revealed. Differences in the trapezoidal and rectangular rotors were investigated, and the primary conclusions are given as follows.

(1) Pumping flow along the spiral groove of rotor and the pumping backflow were investigated. The specific behaviors of the pumping flow, backflow, and flow balance in a screw vacuum pump, as well as the influence on the rotor structural parameters on the backflow, were disclosed. The circumferential clearance has the greatest impact on backflow.

(2) Under the same rotor diameter and compression ratio conditions, rotor pitch is directly proportional to the backflow, but the variation in pitch has a slightly weak effect on the backflow.

(3) Under the same pitch, rotor length, and compression ratio, the rotor addendum diameter and tooth groove depth are proportional to the backflow. The influence of the addendum diameter on the circumferential clearance length is very significant. A smaller rotor addendum diameter can effectively control the backflow of the vacuum pump.

(4) Under the same rotor radial size and pitch of the suction ends, the compression ratio is inversely proportional to the backflows in a relatively low-pressure region and proportional to the backflow in a high-pressure region. Therefore, for vacuum pump operating in low-pressure areas, a large compression ratio can reduce backflow.

(5) This study provides an important strategy for the reduction in backflow through the optimization of the structure of the screw rotor.