Benchmark Model of Molecular Flow Through an S-Bend

Computing molecular flows in arbitrary geometries produces complex integral equations that are very difficult to compute analytically. There are two widely used methods for modeling molecular flows, the Monte Carlo method (which computes the trajectories of large numbers of randomized particles through the system) and the angular coefficient method. The Free Molecular Flow interfaces uses the angular coefficient method, which computes the molecular flow by summing the flux arriving at a surface from all other surfaces in its line of sight. The macroscopic variables in the vicinity of the surface can be derived from kinetic theory. The Particle Tracing Module can be used to implement the Monte Carlo method for molecular flows.

This application requires the Particle Tracing Module.

Model Definition

This model compares the transmission probability, computed by two different methods of a gas through an s-bend geometry. The model geometry is shown in Figure 1.

Figure 1: Model geometry for the s-bend. The dimensions are in centimeters.

Particle trajectories are computed using Newton’s law of motion:

The particle velocity is given by:

When the particles encounter a boundary, they scatter according to Knudsen’s law.

where the subscript t denotes the tangential component of the velocity and n the normal. In addition, c is the particle speed and θ is the angle between the wall normal and the particle velocity. The particle speed is sampled from the following probability density function:

This is accomplished using the normally distributed random function, which is used to assign a new normally distributed random speed to each particle at each time step. The angle, θ is a random angle:

where Γ is a uniformly distributed random number between 0 and 1. Furthermore, it is unique for each particle at each time step. In all 10,000 particles are released into the modeling domain. The number of particles that reaches the outlet boundary determines the transmission probability.

The transmission probability for the molecular flow interface is given by the following:

(1)

One of the most important features of the Free Molecular Flow interface is its ability to reconstruct the number density on domains. This is done by evaluating an integral equation for each mesh element. Denoting the distance vector between the boundary and the centroid of each mesh element in the domain by r the number density can be defined as:

(2)

where the integration is carried out over all boundaries in the model. The expression changes slightly for the 3D case. The quantity

is the rate of emission of molecules per unit area

is the boundary normal, and

is the inverse mean speed of the gas molecules. For more information, see the “Theory for the Free Molecular Flow Interface” in the Molecular Flow Module User’s Guide.

Results and Discussion

The incident molecular flux on the exterior boundaries is shown in Figure 2. Using Equation 2 the reconstructed number density is shown in Figure 3.

Figure 2: Plot of the incident molecular flux in the s-bend.

Figure 3: Plot of the reconstructed number density inside the s-bend.

The transmission probability for the molecular flow interface is 0.19676 and 0.1992 for the particle tracing. The two computed values are in close agreement. Note that slightly different results may be obtained for the particle tracing model each time the model is solved because random numbers are used so heavily in the computation.

When a Monte Carlo method is used to estimate the number density in the modeling domain, the results are in agreement with the reconstructed number density of Figure 3, as shown in Figure 4.

Figure 4: Comparison of the number density from the Molecular Flow and Particle Tracing Modules.

Notes About the COMSOL Implementation

The model is solved in two stages. First the Free Molecular Flow interface is used to compute the transmission probability using a Stationary study. Subsequently, the Mathematical Particle Tracing interface is used to compute the transmission probability with a Time Dependent study using a Monte Carlo approach. An Accumulator feature is applied to the domain to estimate the number density of particles.

Molecular_Flow_Module/Benchmarks/s_bend_benchmark

Modeling Instructions

From the menu, choose .

New

In the window, click .

Model Wizard

In the window, click .

In the tree, select .

Click .

In the tree, select .

Click .

Geometry 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Create the bends in the geometry by taking the difference of concentric circles

Rectangle 1 (r1)

On the toolbar, click and choose .

In the window for , locate the section.

In the text field, type 2.

Right-click and choose .

Circle 1 (c1)

On the toolbar, click and choose .

In the window for , locate the section.

In the text field, type 90.

Locate the section. In the text field, type 2.

In the text field, type 2.

Locate the section. In the text field, type 270.

Right-click and choose .

Click the button on the toolbar.

Circle 2 (c2)

Right-click and choose .

In the window for , locate the section.

In the text field, type 2.

Right-click and choose .

Click the button on the toolbar.

Difference 1 (dif1)

On the toolbar, click and choose .

Select the object only.

In the window for , locate the section.

Find the subsection. Select the toggle button.

Select the object only.

Right-click and choose .

Circle 3 (c3)

On the toolbar, click and choose .

In the window for , locate the section.

In the text field, type 90.

Locate the section. In the text field, type 5.

In the text field, type 2.

Locate the section. In the text field, type 90.

Right-click and choose .

Circle 4 (c4)

Right-click and choose .

In the window for , locate the section.

In the text field, type 2.

Right-click and choose .

Click the button on the toolbar.

Difference 2 (dif2)

On the toolbar, click and choose .

Select the object only.

In the window for , locate the section.

Find the subsection. Select the toggle button.

Select the object only.

Click .

Rectangle 2 (r2)

On the toolbar, click and choose .

In the window for , locate the section.

In the text field, type 2.

Locate the section. In the text field, type 5.

In the text field, type 3.

Click .

Click the button on the toolbar.

Union 1 (uni1)

On the toolbar, click and choose .

Click in the window and then press Ctrl+A to select all objects.

In the window for , locate the section.

Clear the check box.

Click .

The built geometry should appear as in Figure 1.

Click the button on the toolbar.

Now add some parameters for the gas temperature and molecular weight.

Global Definitions

Parameters

On the toolbar, click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
T0	293.15[K]	293.15 K	Temperature
Mw	0.028[kg/mol]	0.028 kg/mol	Molecular weight
p_in	1E-3[Pa]	0.001 Pa	Inlet pressure

Free Molecular Flow (fmf)

Molecular Flow 1

In the window, under click .

In the window for , locate the section.

In the Mn, G text field, type Mw.

Surface Temperature 1

In the window, under click .

In the window for , locate the section.

In the T text field, type T0.

Reservoir 1

In the window, right-click and choose .

Select Boundary 1 only.

In the window for , locate the section.

In the p0,G text field, type p_in.

Total Vacuum 1

Right-click and choose .

Select Boundary 6 only.

The number density inside the domain can be reconstructed by adding the feature.

Number Density Reconstruction 1

Right-click and choose the domain setting .

Click in the window and then press Ctrl+A to select all domains.

In order to compute the transmission probability, only the molecular flux needs to be computed. To save some computation time, specify in the physics interface not to compute the pressure.

In the window, click .

In the window for , locate the section.

Clear the check box.

Locate the section. From the list, choose .

Add integration component couplings on the inlet and outlet.

Definitions

Integration 1 (intop1)

On the toolbar, click and choose .

In the window for , locate the section.

From the list, choose .

Select Boundary 1 only.

Integration 2 (intop2)

On the toolbar, click and choose .

In the window for , locate the section.

From the list, choose .

Select Boundary 6 only.

Add expressions to compute the transmission probability using Equation 1

Variables 1

On the toolbar, click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
Jin	intop1(fmf.J_G)	1/(m·s)	Influx
Jout	intop2(G)	1/(m·s)	Outflux
alpha	Jout/Jin		Transmission probability

Mesh 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Click .

Click the button on the toolbar.

Study 1

On the toolbar, click .

Results

Incident Molecular Flux (fmf)

Click the button on the toolbar.

2D Plot Group 3

On the toolbar, click and choose .

In the window for , type Number Density (fmf) in the text field.

Surface 1

Right-click and choose .

In the window for , click in the upper-right corner of the section. From the menu, choose .

On the toolbar, click .

Click the button on the toolbar.

Now evaluate the transmission probability.

Global Evaluation 1

On the toolbar, click .

In the window for , click in the upper-right corner of the section. From the menu, choose .

Click .

Now add the Charged Particle Tracing interface to compute the transmission probability and number density using a Monte Carlo approach.

Add Physics

On the toolbar, click to open the window.

Go to the window.

In the tree, select .

It is possible to compute the transmission probability using the Mathematical Particle Tracing interface. However, the Charged Particle Tracing interface can also be used to estimate the number density within the domain.

Find the subsection. In the table, clear the check box for .

Click in the window toolbar.

On the toolbar, click to close the window.

Add Study

On the toolbar, click to open the window.

Go to the window.

Find the subsection. In the tree, select .

Find the subsection. In the table, clear the check box for the interface.

Click in the window toolbar.

On the toolbar, click to close the window.

Mathematical Particle Tracing (pt)

On the toolbar, click and choose .

In the window, under click .

In the window for , locate the section.

In the text field, type 0.

Global Definitions

Parameters

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
M	Mw/N_A_const	4.6495E-26 kg	Particle Mass
cs	sqrt(k_B_const*T0/M)	295.04 m/s	Sound Speed
J_in	p_inN_A_const/sqrt(2R_constT0Mw*pi)	2.9082E19 1/(m²·s)	Emitted molecular flux
L	0.01[m]	0.01 m	Inlet length
Np	10000	10000	Number of particles

Definitions

Create a normally distributed random number with zero mean and standard deviation corresponding to the thermal velocity of the particles.

Random 1 (rn1)

On the toolbar, click and choose .

In the window for , locate the section.

From the list, choose .

In the text field, type 3.

In the text field, type cs.

Now define expressions for the particle velocity after it makes contact with the wall. The function random is a built in function to generate a uniformly distributed random number between -0.5 and 0.5 for an arbitrary number of input arguments. The variable pt.pidx is a built-in variable and is unique for each particle. The expression random(pt.pidx,t[1/s],1) will create a unique random number for each particle at each timestep.

Variables 2

On the toolbar, click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
theta	asin(2*random(pt.pidx,t[1/s],1))	rad	Particle emission angle
wall_speed	sqrt(rn1(pt.pidx,t[1/s],1)^2+rn1(pt.pidx,t[1/s],2)^2+rn1(pt.pidx,t[1/s],3)^2)		Particle speed on the wall
wall_t_angle	sin(theta)		Wall angle in the tangential direction
wall_n_angle	cos(theta)		Wall angle in the normal direction
wall_t	wall_speed*wall_t_angle		Particle boundary condition in tangential direction
wall_n	wall_speed*wall_n_angle		Particle boundary condition in the normal direction