Turbomolecular Pump

Turbomolecular pumps (TMPs) are widely used in vacuum system design. A TMP is a bladed molecular turbine that compresses gases by momentum transfer from the rapidly rotating blades of the rotor to the gas molecules. In the free molecular flow range (typically less than 10-1 Pa), the particles collide primarily with the rotor, rather than each other, resulting in an efficient pumping process. The following assumptions are made:

•

Intermolecular collisions are negligible,

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The molecules follow a Maxwellian velocity distribution,

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The gas temperature is constant, and

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The particles experience diffuse scattering at the blade walls, following the distribution of velocity direction called Lambert’s cosine law or Knudsen’s cosine law.

This 3D model uses the Mathematical Particle Tracing interface to evaluate the performance of a simple turbomolecular pump in the free molecular flow regime. While the Free Molecular Flow interface provides an efficient and deterministic calculation of gas density in the free molecular flow regime, it uses the angular coefficient method which is only applicable for molecules that are moving much faster than any object in the geometry. For TMPs, where the blades move over similar time scales as the molecules themselves, this requirement is not satisfied, so a Monte Carlo approach is used instead.

The particle trajectories are computed in a rotating frame of reference, attached to the rotor, in which the fictitious centrifugal and Coriolis forces are automatically taken into account. The model shows the effect of the blade velocity ratio on the pumping characteristics, including the transmission probability, maximum compression ratio, and maximum speed factor.

Model Definition

A TMP may include a large number of stages, or rows of blades. For the TMP design considered in this example, each row of blades is arranged in a ring, and each blade is held at an angle. By rotating these rings of blades, a change in gas density in the axial direction is produced because the inclined blades are more likely to permit a gas molecule to cross the stage in one direction than in the opposite direction. The stage is called a rotor if it is allowed to rotate, or a stator if it is held in a fixed position. A single-stage rotor is illustrated in Figure 1; if the blades rotate counterclockwise as indicated by the red arrows, molecules are more likely to cross the stage going downward (as indicated by the blue arrows) than they are to go upward. Due to sector symmetry, the geometry can be simplified by considering only the gap between two blades, such as the region highlighted in Figure 2.

Figure 1: Full geometry of a turbomolecular pump stage.

Figure 2: Geometry of a turbomolecular pump stage, where a single unit cell and the molecules entering and leaving it have been highlighted.

As in Ref. 1, the pump stage consists of 36 blades, each inclined at a 30 degree angle. The ratio of the root radius to the tip radius is 0.8. Note that if the root-to-tip ratio is very close to unity, the centrifugal and Coriolis forces can be neglected to some extent, allowing a less computationally intensive quasi-2D model to be used instead.

The relevant pump characteristics are based on the transmission probability of argon atoms from the inlet to the outlet M12 and the transmission probability from the outlet back to the inlet M21 (both dimensionless). In order to compute these transmission probabilities, the model uses two features to release particles on either side of the pump and two features to capture particles that reach the boundaries. The can be used to compute the number of particles from each that reach each .

Each releases argon atoms with a Lambertian distribution of initial directions and a Maxwellian distribution of initial speeds. The most probable speed of particles in the Maxwellian distribution vp (SI unit: m/s) is

where

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kB = 1.380649 × 10-23 J/K is the Boltzmann constant,

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T = 300 K is the ambient temperature,

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M = 39.948 g/mol is the molar mass of argon, and

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NA = 6.02214076 × 1023 1/mol is the Avogadro constant.

This gives a most probable speed of approximately vp = 353 m/s.

The root wall, tip wall, and blade surfaces use the dedicated boundary condition, in which incident molecules are adsorbed onto the wall and then immediately released back into the modeling domain with a random velocity based on the surface temperature. The root wall and blades are stationary with respect to the rotating frame of reference, while the tip wall is stationary with respect to the inertial (laboratory) frame, and therefore moves relative to the rotating frame.

Modern turbomolecular pumps can operate at very high rotor speeds reaching 90,000 revolutions per minute. In this model, the angular velocity of the pump is specified in terms of the dimensionless blade velocity ratio C, defined as

(1)

where vb (SI unit: m/s) is the blade velocity, or the rms velocity magnitude of the rotating frame. The transmission probability is computed for different values of C from 0 to 4 using a ; the maximum value C = 4 corresponds to a blade velocity of approximately vb = 1415 m/s.

About Particle Tracing in Rotating Domains

To model the motion of the argon atoms in a rotating frame, a set of second-order differential equations for the components of the particle position q (SI unit: m) are solved:

where

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Fcen is the centrifugal force,

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Fcor is the Coriolis force,

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Feul is the Euler force,

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Ω (SI unit: rad/s) is the angular velocity of the rotating frame, and

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r (SI unit: m) is the displacement from the center of rotation to the atom’s position.

All of these fictitious forces are automatically defined by the feature. In this example, the frame rotates at constant angular velocity, so the Euler force is zero.

Results and Discussion

The over the velocity ratio C begins at C = 0 and ends at C = 4 with intervals of 0.5. The final particle positions for six of these values are shown in Figure 3. The green particles begin at the top boundary while the red particles are released at the bottom boundary. As the velocity ratio increases, more green particles are able to cross from the top to the bottom, but fewer red particles can cross from the bottom to the top. Both effects cause the compression ratio of the turbopump stage to increase.

Figure 3: Final particle positions for different values of the blade velocity ratio C.

The transmission probabilities M12 and M21 for propagation in the forward and reverse directions are shown in Figure 4 and Figure 5, respectively. As expected, the transmission probabilities are almost equal when the blades are at rest (C = 0) because there is no distinction between the forward and reverse directions. As the blades begin to rotate faster, particles are more likely to be transmitted in the forward direction due to the momentum transferred to the argon atoms from the rotating walls.

Figure 4: Fraction of particles transmitted from the inlet to the outlet M12, as a function of the blade velocity ratio C.

Figure 5: Fraction of particles transmitted from the outlet to the inlet M21, as a function of the blade velocity ratio C.

Multiple-bladed structures with several disks provide adequate compression and speed to make a functional pump. The blades nearest the inlet are usually designed to have a high pumping speed and a low compression ratio. The blades nearest the outlet are designed to have a high compression ratio and a low pumping speed.

The maximum compression ratio Kmax is defined by

(2)

and the maximum speed factor Wmax by

(3)

Figure 6 and Figure 7 show the variation of these values as the blade velocity is increased. Compare these results to the corresponding plots in Ref. 1. The compression ratio is expected to increase monotonically. If C is further increased, some statistical noise is expected as the denominator M21 becomes very small. In such cases, consider increasing the number of model particles for a more statistically converged solution.

Figure 6: Maximum compression ratio Kmax, as a function of the blade velocity ratio C.

Figure 7: Maximum speed factor Wmax, as a function of the blade velocity ratio C.

Reference

1. Y. Li and others, “Numerical investigation of three turbomolecular pump models in the free molecular flow range,” Vacuum, vol. 101, pp. 337–344, 2014.

Particle_Tracing_Module/Vacuum_Systems/turbomolecular_pump

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Root

Insert the prepared geometry sequence from file. You can read the instructions for creating the geometry in the appendix.

Geometry 1

In the toolbar, click and choose .

Browse to the model’s Application Libraries folder and double-click the file turbomolecular_pump_geom_sequence.mph.

In the toolbar, click

Click the

button in the toolbar.

Component 1 (comp1)

In the window, click .

In the window for , locate the section.

From the list, choose . Linear geometry shape order is the most robust option for modeling diffuse scattering at convex curved surfaces, such as the tip wall in this geometry.

Global Definitions

Geometry Parameters

In the window, under click .

In the window for , type Geometry Parameters in the text field.

Physics Parameters

In the toolbar, click

and choose .

In the window for , type Physics Parameters in the text field.

Locate the section. Click

Browse to the model’s Application Libraries folder and double-click the file turbomolecular_pump_parameters.txt.

Definitions

Variables 1

In the window, under right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
M12	pt.pcnt1.alpha		Transmission probability, forward
M21	pt.pcnt2.alpha		Transmission probability, backward
Kmax	M12/M21		Maximum Compression Ratio
Wmax	M12-M21		Maximum Speed Factor