George C. Soulas

Glenn Research Center, Cleveland, Ohio

Gas Flux and Density Surrounding a Cylindrical

Aperture in the Free Molecular Flow Regime

NASA/TM—2011-216970

March 2011
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George C. Soulas

Glenn Research Center, Cleveland, Ohio

Gas Flux and Density Surrounding a Cylindrical

Aperture in the Free Molecular Flow Regime

NASA/TM—2011-216970

March 2011

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135
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Level of Review: This material has been technically reviewed by technical management. 
NASA/TM—2011-216970 1 

Gas Flux and Density Surrounding a Cylindrical Aperture 

in the Free Molecular Flow Regime 

George C. Soulas 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Abstract 

The equations for rigorously calculating the particle flux and density surrounding a cylindrical aperture in the 

free molecular flow regime are developed and presented. The fundamental equations for particle flux and density 

from a reservoir and a diffusely reflecting surface will initially be developed. Assumptions will include a MaxwellBoltzmann speed distribution, equal particle and wall temperatures, and a linear flux distribution along the 

cylindrical aperture walls. With this information, the equations for axial flux and density surrounding a cylindrical 

aperture will be developed. The cylindrical aperture will be divided into multiple volumes and regions to rigorously 

determine the surrounding axial flux and density, and appropriate limits of integration will be determined. The 

results of these equations will then be evaluated. The linear wall flux distribution assumption will be assessed. The 

axial flux and density surrounding a cylindrical aperture with a thickness-to-radius ratio of 1.25 will be presented. 

Finally, the equations determined in this study will be verified using multiple methods. 

Nomenclature 

a y-intercept in non-dimensional wall flux equation 

A area 

b slope in non-dimensional wall flux equation 

c mean thermal speed of particles within the reservoir 

f(v) Maxwell-Boltzmann space speed distribution function 

n particle density 

L distance between the surface normals of dAo and dAc

N particle flow rate 

P particle transmission probability through a cylindrical aperture 

R aperture radius 

Rp projected circle radius 

r radial location 

roII radial limit of integration for the reservoir’s contribution to the downstream flux and density 

roIII radial limit of integration for the reservoir’s contribution to the downstream flux and density 

t aperture plate thickness 

T far-field angular distribution function for particles downstream of a cylindrical aperture 

v particle speed 

z axial location 

zwII axial limit of integration for the wall’s contribution to the downstream flux and density 

zwIIn axial limit of integration for the wall’s contribution to the upstream flux and density 

Greek Symbols: 

α angle from dAo surface normal to L 

β angle 
NASA/TM—2011-216970 2 

γ angle 

Γ particle flux 

o ˆ reservoir or wall flux per steradian 

δ angle from dAc surface normal to L 

θ polar angle in a spherical coordinate system 

κ far-field angular distribution function variable 

φ azimuthal angle 

φomax angular limit of integration for the reservoir’s contribution to the downstream flux and density 

φoII angular limit of integration for the reservoir’s contribution to the downstream flux and density 

φwII angular limit of integration for the wall’s contribution to the downstream flux and density 

φwIIn angular limit of integration for the wall’s contribution to the upstream flux and density 

φwt angular limit of integration for the wall’s contribution to the flux and density 

ω solid angle 

Subscripts and Superscripts: 

- non-dimensionalized value (dimensions are divided by R, densities by no, and fluxes by Γo) 

c collection site 

f final location for the limit of integration 

far far-field variable 

i initial location for the limit of integration 

o reservoir 

up upstream reservoir 

w aperture wall 

I. Introduction 

Accurately predicting the density and flux of particles emanating from a circular aperture of finite thickness 

(i.e., a cylindrical aperture) is of significant interest for ion thrusters. Ion thrusters create ions within a discharge 

chamber and accelerate these ions through perforated grids composed of thousands of cylindrical apertures to 

produce thrust (Refs. 1 and 2). Xenon is the propellant used for present spacecraft ion thruster applications. One 

consequence of beam ion generation is that unionized gas can escape from the thruster discharge chamber through 

the cylindrical apertures of the grids. Predicting the downstream unionized gas density distribution is important for 

determining its impact on the thruster and surrounding spacecraft environment. For example, slow-moving ions are 

created with beam ions and the escaping unionized gas due to a process known as charge-exchange. These slowmoving ions can either impact the thruster’s accelerator grid, causing erosion and limiting thruster service life, or 

they can impact spacecraft surfaces. 

Ion thrusters operate at neutral densities that are so low that particles travel from surface to surface without 

colliding. For example, the neutral-to-neutral mean free paths are greater than 34 cm within a state-of-the-art 

discharge chamber, which is greater than the radius of the chamber (Ref. 3). This rarified gas flow regime is often 

referred to as free molecular flow. Solving for the downstream particle density distribution requires an 

understanding of how the thickness of a cylindrical aperture can affect this downstream distribution. 

Many ion thrusters utilize two grids to accelerate ions, and these include an upstream screen grid and a 

downstream accelerator grid. So, for a unionized gas to escape the discharge chamber, it must travel through two 

apertures, which presents a difficult problem to solve. Fortunately, though, the upstream screen grid aperture is 

typically much larger and thinner than the accelerator grid aperture. As a result, the angular distribution of particles 

exiting the accelerator aperture is well modeled by assuming a single accelerator aperture (Ref. 3). 

Many past studies have considered this problem of the free molecular flow through a cylindrical aperture. 

Clausing was the first to determine the downstream angular distribution pattern of particles emanating from a thin 

cylindrical tube under free molecular flow conditions (Refs. 4 and 5). However, his distribution equation was only 
NASA/TM—2011-216970 3 

appropriate for far-field calculations of particle fluxes and densities, as explained by Reynolds and Richley (Ref. 6). 

Richley and Reynolds determined the flux of particles within converging and diverging cylindrical tubes (Refs. 6 

and 7). And Robertson conducted similar studies to determine particle densities as well (Ref. 8). The results of both 

investigations can be extended axially downstream of the cylindrical aperture to determine near- and far-field 

densities and fluxes. Unfortunately, the resulting equations are not accurate at radial locations greater than the radius 

of the aperture, which imposes a severe limitation on their use. Another method of determining particle density and 

flux distributions is to use a Monte Carlo simulation technique (Ref. 9). Such a technique allows for the quick 

analysis of complex geometries (see, for example, Ref. 10). However, accurately analyzing downstream regions 

where particle densities and fluxes have decreased significantly is computationally difficult. In addition, this 

technique does not lend itself well to ion thruster grids which have thousands of cylindrical apertures. 

In this study, equations will be developed for rigorously calculating the flux and density of particles 

surrounding a cylindrical aperture in the free molecular flow regime. The fundamental equations and their 

assumptions necessary to determine particle densities and fluxes will be presented first. These equations will then be 

applied to a cylindrical aperture to determine the particle flux distribution from the cylindrical aperture walls. With 

this information, the equations for particle density and flux surrounding a cylindrical aperture will be developed. 

Finally, the resulting equations will be applied to a typical ion thruster aperture and compared to that of Clausing in 

the far-field (Ref. 4). 

II. Fundamental Equations for Particle Flux and Density 

The particle flux and density distributions surrounding a cylindrical aperture were determined using the 

following assumptions: 

1. Particles upstream of a cylindrical aperture reside within a reservoir and exhaust into a vacuum through the 

cylindrical aperture; 

2. Equilibrium conditions exist, so that the rate of particles leaving the reservoir is equal to the rate of 

particles fed into the reservoir; 

3. The particle density is sufficiently low that free molecular flow conditions exist within the aperture; 

4. All wall reflections are diffuse; 

5. Particle adsorption onto the walls and surface diffusion are negligible; 

6. Wall temperatures are equal to the particle temperature; and 

7. Particles within the reservoir have a Maxwell-Boltzmann speed distribution. 

The geometric relation between a differential element from a particle emission site (i.e., a wall or the upstream 

reservoir) and a differential element from a downstream collection site is shown in Figure 1. 

α

δ

dAo, Γo, no

Emission Site

Collection Site 

dA, Γ, n

Surface 

Normal

Surface 

Normal

L 

Figure 1.—Geometric relation 

between a particle emission 

site and the downstream 

collection site.
NASA/TM—2011-216970 4 

The variable α is the angle between the surface normal of the emission site to the collecting site’s differential surface 

area, dA, and δ is the angle between the surface normal of dA to the differential emitting area, dAo. The variable L is 

the distance between the emitting and collecting differential areas. The particle flow rate, N , from a differential 

emission area to a downstream differential collection area is given by: 

d N  ˆ

o Ao  dAo  cos d 2  (1)

Here, o ˆ is the emitted particle flux per steradian from differential area dAo and is a function of Ao. 

The downstream solid angle, dω, is given by: 

 

2 L

dA cos d  

  (2)

By substituting Equation (2) into Equation (1) and dividing both sides by dA, the following equation is obtained: 

      o 2 o o dA

L

cos cos d ˆ A

dA

dN d          













  (3)

Here, Γ is the flux to the collection site. The emitted particle flux, Γo, can be evaluated as a function of the emitted 

particle flux per steradian by noting that for diffuse surface emission, o ˆ is independent of emission angle, so that: 

    o

2

0

2

0

o o ˆ cos sin d d   ˆ           





(4)

Note that particles from the upstream reservoir and reflected from a wall can both be assumed to be leave 

differential area dAo diffusely. Substituting this result into Equation (3) and integrating both sides yields our 

fundamental equation for particle fluxes: 

              

 

Ao

o 2 o o dA

L

cos cos A 1 (5)

Note that if the emission site is the reservoir, Γo is independent of Ao and equal to: 

4

n c o o

   (6)

where c is the mean speed of the particles within the reservoir and no is the reservoir particle density. 

The downstream particle density, n, from a reservoir of particles can be solved by starting with the MaxwellBoltzmann phase space speed distribution function (Ref. 11). Because equilibrium conditions were assumed, the 

speed distribution function, f(v), is independent of position (i.e., particle velocity is isotropic). So, the particle 

density downstream of a reservoir is: 

       



 

 

             

o o

o

0

o

0

o o 3 o n n f v d v n f v d dv n f v dv d (7)

Here, no is the particle density within the reservoir and the differential solid angle, dωo, is given by: 

 

2

o o L

dA cos d  

  (8)
NASA/TM—2011-216970 5 

The first integral on the right-hand side of Equation (7) can be shown to be 1/(4·π), so substituting this and 

Equation (8) into Equation (7) yields: 

       

Ao

o 2

o dA

L

cos

4

n

n (9)

This equation is in a form that is useful for solving for the downstream particle density when the emission site is the 

reservoir. When the emission site is a wall, it is more useful to re-express Equation (9) using a particle flux. This is 

done by solving Equation (6) for nw and substituting this into Equation (7) to obtain: 

    







         

o

o w

0

w 3 d

c

4 f v d v f v dv

c

4

n (10)

Note that Γw appears with the second integral because the wall flux will be shown to be a function of the location on 

the wall. Substituting Equation (8) into Equation (10) yields: 

           

Aw

w w 2 dAw L

cos A

c

1

n (11)

which is in a form that is useful for solving the particle density when a reflecting wall is the emission site. 

III. Equation for the Particle Flux From the Aperture Walls 

The particle fluxes and densities surrounding a cylindrical aperture are a function of particle fluxes from the 

reservoir and cylindrical aperture walls. While the particle flux from a reservoir is given by Equation (6), the wall 

flux has yet to be determined. The following section will review the development of the wall flux equation used in 

this study. 

Figure 2 is a schematic of the cylindrical aperture and the coordinate system used in this study. A cylindrical 

coordinate system will be used for the cylindrical aperture of radius R and thickness t. Also illustrated in Figure 1 

are the labels for the relevant surfaces and volumes. These labels will be used as subscripts within this study. It is 

convenient to non-dimensionalize the resulting equations, as was done in References 6 and 7. All dimensions will, 

therefore, be non-dimensionalized with the aperture radius, R, all particle fluxes will be non-dimensionalized with 

reservoir flux, Γo, and all particle densities will be non-dimensionalized with reservoir density, no. Nondimensionalized values will be denoted with a bar above the variable throughout this paper. 

Figure 2.—Schematic of the cylindrical aperture for this investigation. 

t 

Exit Plane 

Inlet Plane r

z

R φ

Downstream 

Volume, “c” 

Upstream Volume 

or Reservoir, “o”

Cylindrical Wall, 

“w” 

Interior 

Volume 
NASA/TM—2011-216970 6 

The particle flux to the aperture walls has been solved many times before in literature (see, for example, Refs. 6, 

7, and 12). The following derivation can be found in References 6 and 7. The particle flux from the wall (i.e., 

surface “w”) of the aperture consists of the particle flux from upstream reservoir (i.e., surface “o”) arriving directly 

onto the wall and the particle flux from the other wall regions arriving back onto itself. Because of the symmetry of 

the problem, the particle flux from the wall is independent of φ. So, by applying Equation (5) to the reservoir and 

wall, the flux from the wall at any location z becomes: 

                       

      



 



 

Aw

2 w

w w

w w w w w

Ao

o 2

w o

o o w w o w dA

L

cos cos z 1 dA

L

cos cos z (12)

Note here that a two digit subscript denotes two different surfaces. So, for example, Lo-w refers to the distance L 

between elemental areas on the wall and inlet plane (see Fig. 1). 

To solve this equation, cos(α), cos(δ), L, and A for each integral must be solved as functions of z and the 

respective integral variable. The resulting solution in non-dimensional form is given by (Refs. 6 and 7): 

     

   

 





















 

      

















 



   

t

0

w

2

3

2 w

2 w w w w

2

2

w dz

z z 4

z z 6 1 z z 2z z

z 4

z 2

2

1 z (13)

This equation has the form of a Fredholm integral equation of the second kind. This integral equation was solved 

numerically using an algorithm identical to that described in Reference 7. The resulting non-dimensional particle 

fluxes from the aperture wall are plotted as a function of the normalized wall axial location for various aperture 

thickness-to-diameter ratios in Figure 3. The figure illustrates a critical characteristic of free molecular flow through 

an aperture: the particle flux from an aperture wall as a function of wall axial location is linear over a broad range 

aperture thickness-to-diameter ratios. Indeed, as the aperture thickness-to-radius decreases, the functional relation 

becomes more linear. Over the range of thickness-to-radius ratios examined by this author in Figure 3, a linear 

assumption accurately models the wall particle flux as a function of wall axial location. Other characteristics 

include: 

1. The wall particle flux as a function of wall axial location has a negative slope. This is because the source of 

particles is the upstream reservoir. 

2. The non-dimensional particle flux approaches 1 at the entrance and 0 at the exit as the thickness-to-radius 

ratio increases. Again, this is because the source of particles is the upstream reservoir. 

3. Over the range of thickness-to-radius ratios examined, the non-dimensional wall particle flux was 1/2 at the 

middle of the wall. 

Given the linear behavior of wall particle flux as a function of axial location, the integral equation of 

Equation (13) can be solved by assuming a linear equation for the solution. Clausing was the first to do this, and 

found the solution to be (Ref. 4): 

w z  a  b  z (14)

where a and b are given by: 

t 4 2

2

t 1 t 4

2

t

a

2

2

2

 

   





 









 (15) 

and: 

 





 





  

     

t t 4 2

t 2 t 1 t 4 b

2

2 2

 (16) 

respectively. 
NASA/TM—2011-216970 7 

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Non‐dimensionalWall Particle Flux, Γˉw

Normalized Wall Axial Location, z/t

32

16

8

4

2

1

0.5

Thickness‐to‐Radius Ratio:

Figure 3.—Non-dimensional particle flux from the aperture walls as a function of 

the normalized wall axial location for various aperture thickness-to-radius ratios. 

Aperture locations were normalized with the aperture thickness, t. 

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Non‐dimensionalWall Particle Flux, Γˉw

Normalized Wall Axial Location, z/t

32

8

2

0.5

Thickness Thickness‐to‐Radius Ratio: ‐to‐Radius Ratio:

Figure 4.—Comparison particle fluxes from the aperture walls as a function of 

the normalized wall axial location for various aperture thickness-to-radius 

ratios. Solid lines are numerically-determined and the dotted red lines are the 

respective values determined assuming a linear distribution. The wall axial 

location was normalized to the aperture thickness, t. 

Figure 4 compares the wall particle flux predicted by Equations (14) to (16) to that determined numerically. The 

worst-case difference between the two methods for aperture thickness-to-radius ratios up to 32 was less than 0.01 (in 

non-dimensional form). 

The remainder of this study will therefore assume that the wall particle flux as a function of wall axial location 

is given by Equations (14) to (16) because the assumption used to develop these equations is valid over the aperture 

thickness-to-radius ratios of interest to ion thruster grids (i.e., an aperture thickness-to-radius ratio of 1.25). The 

efficacy of this linear assumption will be assessed later in the results section. 
NASA/TM—2011-216970 8 

IV. Equations for the Particle Flux and Densities 

Surrounding a Cylindrical Aperture 

In this section, the equations for determining particle fluxes and densities surrounding a cylindrical aperture will 

be determined. The first section develops the base equations. The second section divides the surrounding aperture in 

volumes, some of which are further divided into separate regions, and develops the limits of integration for these 

different cases. The last section presents the final equations for particle flux and density, as well as the transmission 

probability. 

A. Development of the Base Equations for Particle Flux and 

Density Surrounding a Cylindrical Aperture 

In this section, the base equations for determining the flux and density surrounding a cylindrical aperture are 

determined. Particle fluxes and densities surrounding an aperture are made up of contributions from the upstream 

reservoir and the aperture walls. Using Equation (5), the flux surrounding an aperture can be shown to be given by: 

      





















                  

 



 

o Aw

2 w

w c

w c w c w w

A

o 2

o c

o c o c o dA

L

cos cos dA A

L

1 cos cos (17)

Likewise, using Equations (9) and (11), the density surrounding an aperture can be shown to be given by: 

     





















              







o Aw

2 w

w c

w c w w

A

o 2

o c

o o c dA

L

cos A

c

1 dA

L

cos

4

1 n

n (18) 

Figure 5 shows a sketch of the geometric variables used in the equations above. For both equations, the left-hand 

integral is the reservoir’s contribution while the right-hand is the aperture wall’s contribution. Note that the reservoir 

flux and density are constants in Equations (17) and (18), respectively, while the wall contributions of both 

equations are a function of the wall location. And the origin of the coordinate system is the upstream center of the 

cylindrical aperture. Finally note that as shown in Figure 5, the flux determined in this study is in the axially 

downstream direction. 

To solve Equations (17) and (18), the variables αo-c, δo-c, Lo-c, αw-c, δw-c, and Lw-c and must be defined as 

functions of R, t, ro, rw, φ, r, and z. For the reservoir’s contribution, it can be shown that: 

L   z  r  r  2 r ro  cos   2

o 2 2 o c (19)

and: 













   



 

o c

o c c o L

z

arccos (20)

For the aperture wall’s contribution, it can be shown that: 

L      z  z  r  R  2 r R  cos 

2 2 2 w c w (21)

 











     





w c

w c L

R r cos

arccos (22)

and: 











   





w c

w c w L

z z

arccos (23)
NASA/TM—2011-216970 9 

Equations (19) to (23) can be substituted into Equations (17) and (18) to yield the equations for the flux and 

density surrounding a cylindrical aperture. Note that there is symmetry about φ, so numerical integration can be 

simplified by integrating only half of the φ integral and multiplying the result by 2. The resulting flux is given in 

non-dimensional form by: 

     

       

      



















  

      

          

      

        









w,f

w,i

w,f

w,i

o,f

o,i

o,f

o,i

z

z

2 w 2 2 w

w w w

r

r

o 2

o 2o 2 2

o 2

dz d

z z r 1 2 r cos

z z z 1 r cos dr d

z r r 2 r r cos

2 z r r, z (24)

while the particle density is given by: 

 

   

     

      



















  

      

        

      

        









w,f

w,i

w,f

w,i

o,f

o,i

o,f

o,i

z

z

w

2

3

2 2 w

w w

r

r

o

2

3

o 2o 2 2

o dz d

z z r 1 2 r cos

z 1 r cos dr d

z r r 2 r r cos

z r

2

1 n r, z (25)

For both equations above, all dimensions (r, z, t, ro, and zw) are non-dimensionalized with the aperture radius, R, all 

fluxes are non-dimensionalized with the reservoir flux, Γo, and all densities are non-dimensionalized with the 

reservoir density, no. The “i” and “f” subscripts for the limits of integration denote initial and final locations. It is 

possible to integrate with respect to ro and zw (i.e., the inner integral of each double integral) in the equations above 

after substituting Equation (14) into Equations (24) and (25). So, the equation for particle flux can be rewritten as: 

r, z r, z w r, z   o   (26)

r

z

Upstream Reservoir

Downstream Volume 

r

φ

Lo-c 

δc-o

z 

Emission Site 

Collection Site 

ro

z

r

z

Upstream Reservoir

Downstream Volume 

φ

Lw-c 

αw-c

z 

Emission Site 

Collection Site

R

zw

r δc-w

αo-c

a. Emission from upstream reservoir. b. Emission from walls.

t t 

Figure 5.—Sketch of the aperture geometry for determining particle fluxes and densities from an upstream reservoir 

and wall to a collection surface. Although the collection site is shown to be in the downstream region, the resulting 

equations will be applicable to all regions. 
NASA/TM—2011-216970 10 

where the reservoir’s contribution is given by: 

   

       

 

   

 

   





































  

   

  

  

          

         





d

r sin z

r r cos arctan

r sin z

r cos

r sin z z r r 2 r r cos

z r r cos z r r, z

o,f

o,i

o,f

o,i

r

r

2 2 2

o

2

3

o 2 2 2 2o 2 2 2 2 2

2 2 o 2

o

   (27)

and the wall’s contribution is given by: 

              





 

























































    

 

    

       

         

w,f

w,i

w,f

w,i

d

r 1 2 r cos

z z arctan

r 1 2 r cos

b

z z r 1 2 r cos

a b z 1 r cos 1 r, z

z

z

2

w

2 2 2 w

w w

 (28)

Likewise, the equation for the particle density is given by: 

nr, z n r, z n w r, z  o  (29)

where the reservoir’s contribution is given by: 

     

       





















          

        





o,f

o,i

o,f

o,i

d

z r sin z r r 2 r r cos

r r cos z r

2

z n r, z

r

r o 2

o 2 2 2 2 2

2 2 o o (30)

and the wall’s contribution is given by: 

               

        





 





















            

                  

w,f

w,i

w,f

w,i

d

r 1 2 r cos z z r 1 2 r cos

r cos 1 a b z z z b r 1 2 r cos

2

1

n r, z

z

z

2 2 w 2

2 w w (31)

B. Development of Limits of Integration for the Flux and Density Base Equations 

With the base equations for particle flux and density developed, all that remains is defining the limits of 

integration. However, the limits of integration are not necessarily constant but can be variable (i.e., functions of r, z, 

and the aperture geometry). As shown in the prior section, the particle flux and density at any location surrounding a 

cylindrical aperture is made up of particles that have a direct line-of-sight from the reservoir and the aperture walls. 

The limits of integration are, therefore, a function of this desired location (i.e., r and z) as well as the aperture 

geometry. Specifically, the contribution of the reservoir and walls depends on which portion of each area has a 

direct line-of-sight to the location of interest. This is illustrated in Figures 6 and 7. As shown in the figures, all 

reservoir and cylindrical wall areas have a direct line-of-sight to downstream locations within region I. But only a 

portion of the reservoir and cylindrical wall areas have a direct line-of-sight to downstream locations within 

regions II and III. And no reservoir particles have a direct-line of sight to downstream locations in region IV. 

In addition to this, there are three volumes surrounding a cylindrical aperture where the particle flux 

and density are desired. These volumes include the downstream, aperture interior, and upstream volumes. 

The equations developed for the downstream volume will be applicable to the interior volume, but a 

separate set of equations will be required for the upstream volume. 

Though complicated, the division of the space surrounding a cylindrical aperture into separate volumes and 

regions with different limits of integration illustrated in Figures 6 and 7 will used to develop a set of equations that 

can rigorously determine the particle flux and density surrounding a cylindrical aperture. The following sections will 

determine the borders between the different regions and their respective limits of integration. 
NASA/TM—2011-216970 11 

Figure 6.—Definition of downstream regions for the development of the variable limits of 

integration for the particle flux from an upstream reservoir. 

Figure 7.—Definition of regions for the development of the variable limits of integration 

for the particle flux from the aperture walls. 

Upstream 

Reservoir, no 

Region I 

t

 Region II 

Region III 

Downstream 

Volume 

Shaded Wall 

Portions Have 

Direct Line-of-sight 

to Upstream Region 

R

r

z

Cylindrical 

Aperture 

Cross-section 

Upstream 

Reservoir, no 

Region I 

t

Region II 

Region III 

Downstream 

Volume Shaded Wall 

Portions Have 

Direct Line-of-sight 

to Downstream 

Region

R

r

z

Cylindrical 

Aperture 

Cross-section 

a. Downstream and Interior Volumes

b. Upstream Volume

Upstream 

Reservoir, no 

Region I 

R

t

Region II 

Region IIIa 

Region IIIb 

Region IV 

Cylindrical 

Aperture 

Cross-section 

Downstream 

Volume Shaded Reservoir 

Portions Have 

Direct Line-of-sight 

to Downstream 

Region 

r

z
NASA/TM—2011-216970 12 

1. Reservoir Contribution: Interior Volume and Downstream Volume—Region I 

As shown in Figure 6, there is a direct line of sight from any location to the entire reservoir surface as long as 

the following radial and axial border conditions are met: 

r  1 and z  0 (32)

As a result, the limits of integration for Equations (27) and (30) are constants (i.e., independent of r and z) and equal 

to 0 to π for φ and 0 to 1 for r o ̄ . 

2. Reservoir Contribution: Downstream Volume—Region II 

Figure 6 shows region II of the downstream volume. It can easily be shown that region II is defined by the 

following radial and axial border conditions: 

t

z 1  r  and z  t (33)

Figure 8 shows the geometry of region II in greater detail. The shaded region in the figure has two different radii: roII

when φ is between 0 and φoII; and R when φ is between φoII and π. As a result, integration for this case is slightly more 

complicated because the integrals of Equations (27) and (30) must be split into the summation of two integrals. For the 

first integral, it can be inferred from the figure that limits of integration along φ from 0 to φoII will be between 0 and roII

for ro. The second integral’s limits of integration along φ between φoII and π will be from 0 to R for ro. 

The limits roII and φoII must be defined as functions of r and z. The limit roII is also a function of φ, which can be 

inferred from Figure 8, so the double integrals of Equations (27) and (30) must be integrated with respect to ro first. 

The limit roII is determined by noting that triangles at axial locations t and z in Figure 8 share a common angle, γ. 

The law of cosines is applied to the angle φ for the triangle at z to determine tan(α). The law of cosines is applied 

again to both triangles for the angle γ and the cos(γ) terms of each equation are equated to determine roII. It can be 

shown that roII is given in non-dimensional for by: 

 

   

z

t 1

sin φ z

t r cos φ 1 z

t r

r r, z,φ

2

2

oII



 













      

 (34) 

Figure 8.—Development of the reservoir’s 

contribution to region II of the downstream 

volume. The shaded reservoir region 

contributes to the downstream flux and 

density. 

Upstream Reservoir 

Downstream Volume 

r 

-φ

z 

α

t 

R 

roII -φ

γ

γ

φoII

R 

roII

roII z·tan(α)

t·tan(α)
NASA/TM—2011-216970 13 

To solve for φoII, the equation above is set equal to R (or 1 in non-dimensional form) and solved for φ. The result in 

non-dimensional form is given by: 

 



































 







   







 





   





 





  

 

z

t 1 zt r 2

z

t r

z

t 1 1

r, z arccos

2 2

oII (35)

3. Reservoir Contribution: Downstream Volume—Region III 

Figure 6 shows regions IIIa and IIIb of the downstream volume. Region III is similar to region II, with the only 

difference being that shaded region in Figure 9 does not include the center of the aperture as it does in Figure 8 for 

region II. In fact, the equations for roII and φoII are still applicable in region III. However, the geometry of this region 

is more complicated, requiring that it be further divided into two separate regions (i.e., IIIa and IIIb). The reason for 

this can be better understood by examining Figure 10, which shows the reservoir’s contribution at the aperture 

entrance. In this figure, the projected area is the circular area formed when viewing upstream from a downstream 

location through the aperture exit plane. It can be shown that the radius of this projected area, Rp, is R/(1-t/z). The 

variable φomax is the angle along φ where the radii of the aperture entrance and the projected area circles intersect to 

form a 90 angle, as shown in Figure 10. It can be shown that: 

  



 





     t r

z omax r, z arcsin (36)

Upstream Reservoir

Downstream Volume

r 

-φ

z 

α

t 

R 

roII

-φ

γ

γ

φoII

R 

roII

roII

z·tan(α)

t·tan(α)

Figure 9.—Development of the reservoir’s contribution to region III of the 

downstream volume. The shaded reservoir region contributes to the 

downstream flux and density.
NASA/TM—2011-216970 14 

In region IIIa, the point of intersection for the radial lines used to define φomax lies within the radius of the 

aperture, R. This creates some complexity for determining the limits of integration for Equations (27) and (30). For 

φφoII in region IIIa, it can be shown that the distance ro to the edge of the shaded region is roII as given in 

Equation (34). So, the limits of integration are from roII to R between φoII and π. However, between φomax and φoII, 

integration occurs between roII and roIII, as shown in Figure 10. Using Figure 9 and the same process as that used to 

determine Equation (34), it can be shown that roIII is given in non-dimensional form by: 

 

   

z

t 1

sin z

t r cos 1 zt r

r r, z,

2

2

oIII



  





 





       

  (37)

In region IIIb the point of intersection for the radial lines used to define φomax lies outside the aperture entrance. 

This simplifies the the limits of integration for Equations (27) and (30). As shown in Figure 10, the distance ro to the 

edge of the shaded region is still roII of Equation (34). And the limits of integration are from roII to R between φoII

and π. 

The border between regions IIIa and IIIb is determined by placing the point of intersection for the radial lines 

used to define φomax at the aperture radius, R, and solving for the downstream radius as a function of axial location. 

So, region IIIa is defined by the following radial and axial border conditions: 





















 





 







    1 z

t 1

t

z r t

z 2

and z  t (38)

Defining the border conditions for region IIIb requires defining the border between regions IIIb and IV. For 

Figure 6, it can easily be shown that this border is given by r = (2·z/t – R). So, region IIIb is defined by the following 

radial and axial border conditions: 





 



    





















 



 





   1

t

z 1 r 2 zt 1

t

z 2

and z  t (39)

φomax 

Aperture Entrance

Projected Area 

roII

Rp φoII

b. Region IIIb

φomax 

Aperture Entrance

Projected Area 

roII

roIII

Rp

φoII

a. Region IIIa 

Figure 10.—Top view of the reservoir’s contribution to region III of the downstream volume. 
NASA/TM—2011-216970 15 

4. Reservoir Contribution: Downstream Volume—Regions IV 

As shown in Figure 6, there is no direct line of sight from the reservoir to region IV of the downstream volume. 

As a result, the particle flux and density contributions from the reservoir will be 0 for this region, which is defined 

by the following radial and axial border conditions: 





 



   1

t

z r 2 and z  t (40) 

5. Reservoir Contribution: Upstream Volume 

Narasimha solved for the particle flux and density surrounding a thin circular aperture (i.e., so thin that the 

thickness can be neglected) (Ref. 13). He found that there was symmetry about the aperture such that the upstream 

density could be determined with downstream density. For a spherical coordinate system, the reservoir particle 

density is given in non-dimensional form by: 

n r,     1 n r, o,up o (41) 

where no,up is the reservoir’s contribution to the upstream particle density. Transforming coordinate systems from 

spherical to cylindrical, it can be shown that the reservoir’s contribution to the upstream particle density is given in 

non-dimensional form by: 

n r, z 1 n r, z  o,up   o (42)

Here, no is given by Equation (30). By analogy, the reservoir’s contribution to the upstream particle flux is given in 

non-dimensional form by: 

r, z r, z  o,up  o (43) 

Here, Γo is given by Equation (27). Later, it will be shown that Equations (27) and (30) decrease as z and r increase 

in magnitude. So at locations far upstream of the aperture, Equation (42) approaches no (or unity in non-dimensional 

form), as would be expected because this is the reservoir’s density. However, Equation (43) approaches zero at these 

locations because the net axial flux should be zero at these locations due to the isotropic velocity assumption. 

6. Wall Contribution: Upstream, Interior, and Downstream Volumes—Region I 

As shown in Figure 7, there is a direct line of sight from any location to the entire aperture wall surface as long 

as the following radial border condition is met: 

r  1 (44) 

This is true for any volume (upstream, interior, or downstream), so there is no axial border condition. As a result, the 

limits of integration for Equations (28) and (31) are constants (i.e., independent of r and z) and equal to 0 to π for φ

and 0 to t ̄ for z ̄ w. 

7. Wall Contribution: Downstream Volume—Region II 

Figure 7 shows region II of the downstream volume. It can easily be shown that region II is defined by the 

following radial and axial border conditions: 





 



    1

t

z 1 r 2 and z  t (45)
NASA/TM—2011-216970 16 

Figure 11 shows the geometry of region II in greater detail. Integration for this case is slightly more 

complicated in that the integrals of Equations (28) and (31) have to be split into the summation of two integrals. It 

can be inferred from the figure that the first integral’s limits of integration will be between φwt and φwII, for zw from 

zwII to t. The second integral’s limits of integration will be between φwII and π for zw from 0 to t. 

The limits zwII, φwII, and φwt must be defined as functions of r and z. The limit zwII is also a function of φ, which 

can be inferred from Figure 11, so the double integrals of Equations (28) and (31) must be integrated with respect to 

zw first. The limit zwII is determined as was done for the reservoir. Note that the triangles at z = t and z in Figure 11 

share a common angle, γ. The law of cosines is applied to the angle φ for the triangle at z to determine tan(β). The 

law of cosines is applied again to both triangles at t and z for the angle γ and the cos(γ) terms of each equation are 

equated to determine zwII. It can be shown that zwII is given in non-dimensional for by: 

     

r 1

r t t 2 z 2 r t z cos φ z r, z,φ 2

2

wII 

          (46)

To solve for φwII, the equation above is set equal to 0 and solved for φ. The result in non-dimensional form is given 

by: 

 





























 



   

  

 

t

z 2 r 1

t

z r 1 2

r, z arccos

2

wII (47)

To solve for φwt, the equation for zwII above is set equal to t and solved for φ. The result in non-dimensional form is 

given by: 

  



 



  r

1

φwt r, z arccos (48)

Upstream Reservoir

Downstream Volume

r 

z β

t 

R 

φ

γ

R 

zwII

(t - zwII)·tan(β)

γ

R 

(z - zwII)·tan(β)

φwII

φwt

Figure 11.—Development of the aperture wall’s contribution to 

region II of the downstream volume. Only the shaded wall 

surface contributes to the downstream flux and density. Only 

half of the shaded region is shown. 
NASA/TM—2011-216970 17 

8. Wall Contribution: Downstream Volume—Region III 

Figure 7 shows region III of the downstream volume. It can easily be shown that region III is defined by the 

following radial and axial border conditions: 





 



   1

t

z r 2 and z  t (49)

Figure 12 shows the geometry of region III in greater detail. Integration of Equations (28) and (31) for this case is less 

complicated than region II. It can be inferred from the figure that the integral’s limits of integration will be between φwt

and π for zw from zwII to t. Note in Figure 12 that the equations for φwt and zwII are the same as those for region II. 

9. Wall Contribution: Upstream Volume—Region II 

Figure 7 shows region II of the upstream reservoir. It can easily be shown that region II is defined by the 

following radial and axial border conditions: 





 



    

t

z 1 r 1 2 and z  0 (50) 

Figure 13 shows the geometry of region II in greater detail. As with the downstream case, integration for this case is 

slightly more complicated because the integrals of Equations (28) and (31) have to be split into the summation of 

two integrals. It can be inferred from the figure that the first integral’s limits of integration will be between φwt and 

φwIIn, where integration along zw will be between 0 and zwIIn. The second integral’s limits of integration will be 

between φwIIn and π, where integration along zw will be between 0 and t. 

Figure 12.—Development of the aperture wall’s contribution to 

region III of the downstream volume. Only the shaded wall 

surface contributes to the downstream flux and density. Only 

half of the shaded region is shown. 

R 

Upstream Reservoir

Downstream Volume

r 

z β

t 

R 

φ

γ

zwII

(t - zwII)·tan(β)

γ

R 

(z - zwII)·tan(β)

φwt
NASA/TM—2011-216970 18 

The limits zwIIn, φwIIn, and φwt must be defined as functions of r and z. The limit zwIIn is also a function of φ, 

which can be inferred from Figure 13, so the double integrals of Equations (28) and (31) must be integrated with 

respect to zw first. The limit zwIIn is determined as was done for the reservoir. Note that the triangles at z = 0 and -z in 

Figure 13 share a common angle, γ. The law of cosines is applied to the angle φ for the triangle at z to determine 

tan(β). The law of cosines is applied again to both triangles at t and z for the angle γ and the cos(γ) terms of each 

equation are equated to determine zwIIn. It can be shown that zwIIn is given in non-dimensional for by: 

   

2 wIIn

1 r

1 r cos z r, z, 2 z



       (51)

To solve for φwIIn, the equation above is set equal to t and solved for φ. The result in non-dimensional form is given 

by: 

 

























 

  

 

t

z 2 r

t

z r 1 2

r, z arccos

2

wIIn (52)

The equation for φwt is given in Equation (48). 

10. Wall Contribution: Upstream Volume—Region III 

Figure 7 shows region III of the upstream volume. It can easily be shown that region III is defined by the 

following radial and axial border conditions: 





 



   

t

z r 1 2 and z  0 (53) 

Figure 14 shows the geometry of region III in greater detail. Integration of Equations (28) and (31) for this case is 

less complicated than region II. It can be inferred from the figure that the integral’s limits of integration will be 

between φwt and π, where integration along zw will be between 0 and zwII. Note in Figure 14 that the equations for 

φwt and zwIIn are the same as those for region II. 

γ

Upstream Reservoir

Downstream Volume

-z 

t 

zwIIn

r 

β

R 

φ

γ

R 

zwIIn·tan(β)

R 

(zwIIn - z)·tan(β)

φwIIn

φwt

Figure 13.—Development of the aperture wall’s contribution to region II of the 

upstream volume. Only the shaded wall surface contributes to the downstream 

flux and density. Only half of the shaded region is shown. 
NASA/TM—2011-216970 19 

Figure 14.—Development of the aperture wall’s contribution 

to region III of the upstream volume. Only the shaded wall 

surface contributes to the downstream flux and density. 

Only half of the shaded region is shown. 

C. Equations for Flux and Particle Density Surrounding a Cylindrical Aperture 

The limits of integration defined in the previous section can be substituted into Equations (27) and (28) for 

particle flux and Equations (30) and (31) for particle density. Because the surrounding aperture had to be divided 

into separate volumes and regions as illustrated in Figures 6 and 7, the radial and axial border conditions defined in 

the previous sections are used to determine the appropriate particle flux or density equation. The results are listed in 

Tables 1 to 5. The particle flux in the axial direction is given by Equation (26), with the reservoir’s contribution 

equations listed in Table 1 and the wall’s contribution equations listed in Table 2. The particle density is given by 

Equation (29), with the reservoir’s contribution equations listed in Table 3 and the wall’s contribution equations 

listed in Table 4. Table 5 lists the variable limits of integration equations used in Tables 1 to 4. 

The integrals remaining in Equations (27), (28), (30), and (31) could only be solved in two cases, which 

included the flux equations for region I of the downstream and interior volumes and for the upstream volume 

(see Table 1). The remaining integrals in Tables 1 to 4 must be solved numerically. 

It can be shown that the particle density equation for the reservoir’s contribution in region I produces a 

singularity at r = z = 0. This is easily overcome by noting that the reservoir’s contribution to particle density is 1/2 at 

the aperture entrance (i.e., z = 0), and is independent of radial location (Ref. 11). 

The particle transmission probability, P, through the cylindrical aperture (i.e., the fraction of particles incident 

upon the aperture inlet that eventually pass through) can also be determined (Ref. 7). The probability is given by 

integrating the total particle flux radially at any axial location and dividing by the particle flux at the aperture inlet: 

 

 

  

 

  



 

    

    

    



0

2

0

R

0

o

2

0 0 2 r, z r dr

r dr d

r, z r dr d

P z (54) 

Within the cylindrical aperture (i.e., 0  z  t), the upper limit of integration along r changes from ∞ to R (or 1 in 

non-dimensional form). Because the transmission probability can be calculated at any axial location and still yield 

the same result, this equation also serves as a useful tool to verify the equations in Tables 1, 2, and 5. And because 

particle density equations utilize the same equations except for the cos(δ) terms, it can indirectly verify the equations 

in Tables 3 and 4. 

γ

Upstream Reservoir

Downstream Volume

-z 

t 

zwIIn

r 

β

R 

φ

γ

R 

zwIIn·tan(β)

R 

(zwIIn - z)·tan(β)

φwt
TABLE 1.—EQUATIONS FOR THE RESERVOIR’S CONTRIBUTION TO THE PARTICLE FLUX IN THE AXIAL DIRECTION FOR THE DIFFERENT REGIONS OF FIGURE 6 

Region Equation Border conditionals 

Downstream and Interior volumes

I    



 

 

2 2 22

22

o

r41rz

1rz 1 1

2 z,r 1 r 

II 

 

  

                       

               

 

 

                 

                 









 

  







 





















  













 

  







 



 

  









  











d

zsinr

cosr

arctan

zsinr

cosr

rzzsinr

rz

zsinr

cosr1

arctan

zsinr

cosr

rzcosr21zsinr

rzcosr

z

d

zsinr

cosr

arctan

zsinr

cosr

rzzsinr

rz

zsinr

cosr,z,rr arctan

zsinr

cosr

rzcosr,z,rr2,z,rrzsinr

rzcos,z,rrr

z

z,r

z,r

2 2 2 3

2 2 2 2

222 2 2

22

2 2 2 3

2 2 2 2

22 2 2 2

22

2

z,r

0

2 2 2 3

2 2 2 2

222 2 2

22

2 2 2

oII

32 2 2 2

22

oII 2 oII 2 2 2

22

oII

2

o

oII

oII

z

t

r1  and tz 

IIIa 

 

  

                       

  

                         

 

 

                 

  

                         









 

  







 





















  









 

  







 



 

  









  













d

zsinr

cosr,z,rr arctan

zsinr

cosr

rzcosr,z,rr2,z,rrzsinr

rzcos,z,rrr

zsinr

cosr1

arctan

zsinr

cosr

rzcosr21zsinr

rzcosr

z

d

zsinr

cosr,z,rr arctan

zsinr

cosr

rzcosr,z,rr2,z,rrzsinr

rzcos,z,rrr

zsinr

cosr,z,rr arctan

zsinr

cosr

rzcosr,z,rr2,z,rrzsinr

rzcos,z,rrr

z

z,r

z,r

2 2 2

oIII

32 2 2 2

22

oIII 2 oIII 2 2 2

22

oIII

2 2 2 3

2 2 2 2

22 2 2 2

22

2

z,r

z,r

2 2 2

oIII

32 2 2 2

22

oIII 2 oIII 2 2 2

22

oIII

2 2 2

oII

32 2 2 2

22

oII 2 oII 2 2 2

22

oII

2

o

oII

oII

maxo





 





 1 t

z 1 z

t r z

t

2

and tz 

IIIb  

 

                 

  

                         









 

  







 





















  



 

d

zsinr

cosr,z,rr arctan

zsinr

cosr

rzcosr,z,rr2,z,rrzsinr

rzcos,z,rrr

zsinr

cosr1

arctan

zsinr

cosr

rzcosr21zsinr

rzcosr

z

z,r

z,r

2 2 2

oIII

32 2 2 2

22

oIII 2 oIII 2 2 2

22

oIII

2 2 2 3

2 2 2 2

22 2 2 2

22

2

o

oII



 





 





 1 z

t

2r1 t

z 1 z

t

2

and tz 

IV   0z,r o  



  1 z

t

2r and tz 

Upstream volume

-    



 

 

2 2 22

22

o

r41rz

1rz 1 1

2 1z,r 0z 

NASA/TM—2011-216970 20
TABLE 2.—EQUATIONS FOR THE APERTURE WALL’S CONTRIBUTION TO THE PARTICLE FLUX IN THE AXIAL DIRECTION FOR THE DIFFERENT REGIONS OF FIGURE 7 

Region Equation Border conditionals 

I          

   

w

t

0

32 2 2 2 2 w

2 2 w w w w zd

r41rzz

1rzzzz zba2z,r 







 



  1 r 

Downstream and Interior volumes 

II 

             

 

       

 

 

 

 

           

     

 













 







 



 

 









 

 













 

 







 



 

 









    











d

cosr21r

z

arctan

cosr21r

b

cosr21rz

a

cosr21r

tz

arctan

cosr21r

b

cosr21rtz

tba

cosr1 1

d

,z,rzz

cosr21r

arctan

cosr21r

b

cosr21r,z,rzz

,z,rzba

cosr21r

tz

arctan

cosr21r

b

cosr21rtz

tba

cosr1 1 z,r

z,r

2 2 22

2 2 2 2

z,r

z,r

2

wII

2 2 2 wII

wII

2 2 2 2

w

wII

wt

wII



  1 z

t

2r1 and tz 

III              

 

       

 

 

 













 

 







 



 

 









 





 

d

,z,rzz

cosr21r

arctan

cosr21r

b

cosr21r,z,rzz

,z,rzba

cosr21r

tz

arctan

cosr21r

b

cosr21rtz

tba

cosr1 1 z,r

z,r

2

wII

2 2 2 wII

wII

2 2 2 2

w

wt



  1 z

t

2r and tz 

Upstream volume 

II 

     

 

       

 

 

     

 

 

           

     

 













 







 



 

 









   













 







 



 

  





 



    











d

cosr21r

z

arctan

cosr21r

b

cosr21rz

a

cosr21r

tz

arctan

cosr21r

b

cosr21rtz

tba

cosr1 1

d

cosr21r

z

arctan

cosr21r

b

cosr21rz

a

cosr21r

,z,rzz arctan

cosr21r

b

cosr21r,z,rzz

,z,rzba

cosr1 1 z,r

z,r

2 2 22

2 2 2 2

z,r

z,r

2 2 22

2

wIIn

2 2 2 wIIn

wIIn

w

wIIn

wt

wIIn



 

z

t

21r1 and 0z 

III      

 

       

 

 

     

 













 







 



 

  





 

 

   



 

d

cosr21r

z

arctan

cosr21r

b

cosr21rz

a

cosr21r

,z,rzz arctan

cosr21r

b

cosr21r,z,rzz

,z,rzba

cosr1 1 z,r

z,r

2 2 22

2

wIIn

2 2 2 wIIn

wIIn

w

wt



 

z

t

21r and 0z 

NASA/TM—2011-216970 21
TABLE 3.—EQUATIONS FOR THE RESERVOIR’S CONTRIBUTION TO THE PARTICLE DENSITY FOR THE DIFFERENT REGIONS OF FIGURE 6 

Region Equation Border conditionals 

Downstream and Interior volumes

I 

  1

2 z,rno  1 r  and 0z 

            



 







 







 d zsinr rz rzcosr21zsinr rzcosr

2

z

z,rn

0

2 2 2

22

22 2 2 2

22

o 1 r  and 0z 

II 

    

           

 

 

          



 







 









 





 

 



 









d

zsinr

rz

rzcosr21zsinr

rzcosr

2

z

d

zsinr

rz

rzcosr,z,rr2,z,rrzsinr

rzcos,z,rrr

2

z

z,rn

z,r 2 2 2

22

22 2 2 2

22

z,r

0 2 2 2

22

22

oII 2 oII 2 2 2

22

oII o

oII

oII

z

t

r1  and tz 

IIIa 

 

  

         

  

           

 

 

     

  

           





  



















  





 





















d

rzcosr,z,rr2,z,rrzsinr

rzcos,z,rrr

rzcosr21zsinr

rzcosr

2

z

d

rzcosr,z,rr2,z,rrzsinr

rzcos,z,rrr

rzcosr,z,rr2,z,rrzsinr

rzcos,z,rrr

2

z

z,rn

z,r

22

oIII 2 oIII 2 2 2

22

oIII

22 2 2 2

22

z,r

z,r

22

oIII 2 oIII 2 2 2

22

oIII

22

oII 2 oII 2 2 2

22

oII

o

oII

oII

maxo 





 





 1 t

z 1 z

t r z

t

2

and tz 

IIIb  

 

     

  

           





  

















 

d

rzcosr,z,rr2,z,rrzsinr

rzcos,z,rrr

rzcosr21zsinr

rzcosr

2

z

z,rn

z,r

22

oIII 2 oIII 2 2 2

22

oIII

22 2 2 2

22

o

oII



 





 





 1 z

t

2r1 t

z 1 z

t

2

and tz 

IV   0z,rno  



  1 z

t

2r and tz 

Upstream volume

-             



 







 







 d zsinr rz rzcosr21zsinr rzcosr

2

z

1z,rn

0

2 2 2

22

22 2 2 2

22

o 0z 

NASA/TM—2011-216970 22
TABLE 4.—EQUATIONS FOR THE APERTURE WALL’S CONTRIBUTION TO THE PARTICLE DENSITY FOR THE DIFFERENT REGIONS OF FIGURE 7 

Region Equation Border conditionals 

I                           







 







 



 







 d cosr21rz cosr21rbzzba cosr21rtz cosr21rbtzzba cosr21r 1cosr

2

1

z,rn

0

22

2

2 2

2

2 w 1 r 

Downstream and Interior volumes 

II 

     

       

   

         

     

 

 

 

 

       

   

     

   









 







 



 













  



 











 

















d

cosr21rz

cosr21rbzzba

cosr21rtz

cosr21rbtzzba

cosr21r

1cosr

2

1

d

cosr21r,z,rzz

cosr21rb,z,rzzzba

cosr21rtz

cosr21rbtzzba

cosr21r

1cosr

2

1

z,rn

z,r

22

2

2 2

2

2

z,r

z,r

2 2

wII

2

2

wII

2

2

2 w

wII

wII

wt 



  1 z

t

2r1

and tz 

III                                     







 



 





 



 





 

d

cosr21r,z,rzz

cosr21rb,z,rzzzba

cosr21rtz

cosr21rbtzzba

cosr21r

1cosr

2

1

z,rn

z,r 2 2 wII

2

wII

2 2

2

2 w

wt



  1 z

t

2r

and tz 

Upstream volume 

II 

                                 

 

 

       

   

     

   









 







 



 













 









  



 

















d

cosr21rz

cosr21rbzzba

cosr21rtz

cosr21rbtzzba

cosr21r

1cosr

2

1

d

cosr21rz

cosr21rbzzba

cosr21r,z,rzz

cosr21rb,z,rzzzba

cosr21r

1cosr

2

1

z,rn

z,r

22

2

2 2

2

2

z,r

z,r

22

2

2 2

wIIn

2

wIIn

2 w

wIIn

wIIn

wt 



 

z

t

21r1

and 0z 

III                                     







 









  



 







 

d

cosr21rtz

cosr21rbtzzba

cosr21r,z,rzz

cosr21rb,z,rzzzba

cosr21r

1cosr

2

1

z,rn

z,r 2 2

2

2 2

wIIn

2

wIIn

2 w

wt



 

z

t

21r

and 0z 

NASA/TM—2011-216970 23
NASA/TM—2011-216970 24 

TABLE 5.—VARIABLE LIMITS OF INTEGRATION USED IN TABLES 1 TO 4 

Table use Type Limit equation 

Tables 1 and 3 

Radial 

 

   

z

t 1

sin φ z

t r cos φ 1 zt r

r r, z, φ

2

2

oII



 













      



 

   

z

t 1

sin φ z

t r cos φ 1 z

t r

r r, z,φ

2

2

oIII



 



 





      



Angular 

 



































 







   







 





   





 





  

 

z

t 1 zt r 2

z

t r

z

t 1 1

r, z arccos

2 2

oII

  



 





   t r

z φomax r, z π arcsin

Tables 2 and 4 

Axial 

     

r 1

r t t 2 z 2 r t z cos φ z r, z,φ 2

2

wII 

         

   

wIIn 2 1 r

1 r cos z r, z, 2 z 

      

Angular 

 





























 



   

  

 

t

z 2 r 1

t

z r 1 2

r, z arccos

2

wII

 

























 

  

 

t

z 2 r

t

z r 1 2

r, z arccos

2

wIIn

  



 



  r

1

φwt r, z arccos
NASA/TM—2011-216970 25 

V. Far-field Equations Using Clausing’s Angular Distribution Equation 

A commonly accepted far-field expression will be presented here for later comparisons with the more rigorously 

developed equations above. Clausing was the first to develop a far-field expression for the angular distribution of 

particles downstream of a cylindrical aperture (Ref. 3). His equation assumes that the aperture is a point source and that 

the flux of particles from the walls of the aperture is linear as a function of axial thickness. The angular distribution of 

particles downstream of a cylindrical aperture, T(α), in the far-field is given by (Refs. 4 and 5): 

 

         

 

 

 











































 



  

    





 



  































   

     























































     













     



t

2 if α arctan

2

t tan α

1 2 κ

3 π

4 κ

and

t

2 if α arctan

2

t tan α

2

t tan α 1 1

1 2 κ 3 π

4

2

t tan α 1

2

t tan α

2

t tan α 1 κ arcsin π2 1

T α

2

3

2

2

(55)

where: 

t 4

4 2

t 4 t

κ

2

2





   (56)

Figure 15 shows the geometry for a single aperture that is treated as a point source. It can easily be shown that: 

  2 2 L  r  z  t (57)

and: 

  

  2 2

r z t

z t

cos cos

 

     (58)

Equations (57) and (58), along with T(α), can be substituted into Equations (5) and (9) to determine the downstream 

particle flux and density from a cylindrical aperture in the far-field. Note that all variables are independent of dAo

because a point source is assumed, so that integral reduces to π·R2

. The resulting equation for the far-field flux, Γfar, 

is given in non-dimensional form by: 

   

 

   

2 2 2

2

2 2

far

r z t

z t

r z t

z t

r, z T arccos

 

 









































 

   (59)

while the far-field density, nfar, is given in non-dimensional by: 

       2

3

2 2 2 2

far

r z t

z t

r z t

z t T arccos

4

1

n r, z

 

 









































 

   (60)
NASA/TM—2011-216970 26 

The transmission probability of particles through a cylindrical aperture can also be determined by integrating 

Clausing’s angular distribution function over all azimuthal and polar angles and dividing by that of a diffuse 

distribution. It can be shown that this probability, Pfar, is given by (Ref. 5):

   

 

            

      

        

 

 

  









2 T cos sin d

cos sin d d

T cos sin d d

P

2

2 0

0

2

0

2

0

2

0

far (61) 

VI. Results 

The following sections will present the results of the particle flux and density equations developed in this study. 

First, an assessment of the linear wall flux distribution assumption will be made. Then, the axial flux and density 

equations from this study will be applied to a cylindrical aperture and compared to those of Clausing’s far-field 

expression to accomplish the following: 1) the commonly accepted far-field expression will be used to verify the 

rigorously developed equations in the far-field; and 2) the rigorously developed equations will be used to determine 

the efficacy of the far-field expression in the near-field. A typical ion thruster aperture thickness-to-radius ratio of 

1.25 will be used throughout. Finally, the axial flux and density equations will be verified using three different 

methods. 

Upstream Reservoir

Downstream Volume

r

L

z - t 

Cylindrical 

Aperture 

α

n

δ

Figure 15.—Geometry for determining downstream 

axial flux and density in the far-field. 
NASA/TM—2011-216970 27 

A. Assessment of Linear Wall Flux Assumption and Transmission Probabilities 

As was shown earlier in Figures 3 and 4, the linear axial wall flux distribution assumption of Equation (14) 

accurately models the true distribution over a range of aperture thickness-to-radius ratios. However, it would be 

useful to determine a figure of merit for setting the maximum thickness-to-radius ratio that can utilize this 

assumption. One figure of merit is the particle transmission probability. Figure 16 shows the error between the true 

transmission probabilities (i.e., those determined numerically per Reference 7 with an error of 0.25 percent) and 

those determined with Equation (54), which assumes a linear distribution assumption. The figure shows that 

transmission probability errors with a linear wall flux distribution assumption are 4 percent for thickness-to-radius 

ratios 16. However, in state-of-the-art ion thruster perforated grids, the thickness-to-radius ratio is within 1.25, so 

the transmission probability error would be 0.1 percent. 

Verifying the linear distribution assumption as the cause for this error is essential for assessing its impact on axial 

flux and density calculations. To better understand the cause of the transmission probability errors, Figure 17 plots the 

transmission probability of Equation (54) as a function of axial location for a thickness-to-radius ratio of 32. This 

thickness-to-radius ratio was selected because these errors increase at larger ratios. The transmission probabilities 

should be constant throughout the axial direction in Figure 17. However, the figure shows that the probability is not 

constant within the aperture. So, the transmission probability can be in error by as much as 11 percent for this 

thickness-to-radius ratio. It can be shown that this disparity generally increases for larger thickness-to-radius ratios. 

Also shown in Figure 17 is the transmission probability when the linear wall flux distribution assumption is replaced 

with a 5th order polynomial curve fit equation. The polynomial curve fit equation produces a nearly constant 

transmission probability as a function of axial location whose average value is nearly identical to the true transmission 

probability. This is significant, because the flux difference between the true and linear distributions never exceeded 

1 percent of Γo, and yet the impact on transmission probability was a nearly 7 percent error. 

‐7

‐6

‐5

‐4

‐3

‐2

‐1

0

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 4 8 12 16 20 24 28 32

Error,

%

Transmission Probability, P

Thickness‐to‐RadiusRatio, t ̄

True Probability

Error

Figure 16.—True transmission probability as a function of aperture thickness-toradius ratio and the error between the true probability and that assuming a 

linear wall flux distribution. 
NASA/TM—2011-216970 28 

0.066

0.068

0.07

0.072

0.074

0.076

0 4 8 12 16 20 24 28 32

Transmission Probability, P

Non‐dimensionalAxial Location,z ̄

Linear Fit

Polynomial Fit

Figure 17.—Transmission probability with linear and polynomial curve fits to the 

actual wall flux distribution for a thickness-to-radius ratio of 32. 

Because the transmission probability is the integral of the axial flux per Equation (54), the errors plotted in 

Figure 16 would roughly represent the errors expected for flux, and therefore density, calculations. This assertion 

was checked for a thickness-to-radius ratio of 32. While the maximum transmission probability error was about 

7 percent, the maximum flux error was within 8 percent. So, if a maximum acceptable flux and density error is 

4 percent, a linear wall flux distribution assumption can be used for thickness-to-radius ratios 16 according to 

Figure 16. 

B. Particle Flux and Density Surrounding a Cylindrical Aperture 

1. Near-field Results 

The axial flux surrounding a cylindrical aperture with a thickness-to-radius ratio of 1.25 is shown in Figure 18. 

The plot was determined with Equation (26) and the equations from Tables 1, 2, and 5. As expected, the axial flux 

decreases rapidly further from the aperture in the upstream, downstream, and radial directions. This is because the 

flux tends to be equal in all directions upstream of the aperture and the flow expands rapidly downstream of the 

aperture. It is only within the cylindrical aperture that the aperture and its walls align the flux of the particle flow in 

the axial direction. It is interesting to note that there is symmetry in the axial direction about z = t/2. 

Figure 19 shows the contributions of the reservoir and aperture walls to the surrounding axial flux. As expected, 

the non-dimensional axial flux from the reservoir is 1 (or the reservoir flux, Γo) at the aperture inlet plane and this 

decreases rapidly as the flow expands in all downstream directions. The wall’s contribution produces a net negative 

(i.e., upstream) axial flux near the aperture inlet because of upstream particle reflections from the aperture walls. 

The particle density surrounding a cylindrical aperture with a thickness-to-radius ratio of 1.25 is shown in 

Figure 20. The plot was determined with Equation (29) and the equations from Tables 3 to 5. Far upstream from the 

aperture inlet, the non-dimensional particle density is constant and equal to 1 (or the reservoir density, no). However, 

closer to the aperture inlet, the figure shows a depleted density zone. This depletion zone is caused by particles 

escaping through the aperture. The figure also shows that the density drops rapidly from the upstream to 

downstream regions and through the aperture. Figure 21 shows only the particle density downstream of the aperture. 

The particle density drops rapidly as particles expand into vacuum, dropping to less than 10 percent of the upstream 

density within 1 aperture radius of the downstream surface. 
NASA/TM—2011-216970 29 

Aperture Wall 

Aperture Wall 

r

z

Flow 

Direction 



Aperture Wall 

Aperture Wall 

r

z

Flow 

Direction 

Flow 

Direction 

r

z

o w

a. Reservoir’s contribution. b. Wall’s contribution.

Figure 19.—The reservoir’s and wall’s contributions to the axial particle flux surrounding a cylindrical aperture 

with a thickness-to-radius ratio of 1.25. 

Figure 18.—Axial particle flux surrounding a cylindrical aperture 

with a thickness-to-radius ratio of 1.25.
NASA/TM—2011-216970 30 

Aperture Wall 

Aperture Wall 

r

z

Flow 

Direction 

n

Figure 21.—Particle density downstream of a cylindrical 

aperture exit plane with a thickness-to-radius ratio of 

1.25. 

r

z

Aperture Exit 

Flow Plane 

Direction 

n

Figure 20.—Particle density surrounding a cylindrical aperture 

with a thickness-to-radius ratio of 1.25.
NASA/TM—2011-216970 31 

2. Far-field Results and Comparisons with Clausing’s Far-field Expression 

The far-field particle flux and density equations (i.e., Eqs. (59) and (60), respectively) that utilize Clausing’s 

angular distribution function are compared to that calculated with Equations (26) and (29), respectively, and the 

equations of Tables 1 to 5 in Figures 22 and 23. Here, the axial flux and density were determined at fixed radial 

distances from the downstream center of the aperture from 2 to 60 radii while the angle was varied (i.e., a spherical 

coordinate system was used) to determine angular distributions. The figures illustrate that at increasing distances from 

the aperture, the far-field angular distribution becomes equal to that of the Equations (26) and (29). The figures also 

show that the worst-case error in the far-field equation occurs along the axial centerline of the aperture. 

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 10 20 30 40 50 60 70 80 90

NormalizedDownstreamAxial Flux, Γ ̄

Angle, degrees

2 ‐ Far‐field 2

10 ‐ Far‐field 10

60 ‐ Far‐field 60

Non‐dimensional Radial Distance:

Figure 22.—Normalized axial flux angular distributions for a variety of radial 

distances in a spherical coordinate system. The aperture thickness-toradius is 1.25. Solid lines were determined from Equation (59) and dashed 

lines from Equation (26) and Tables 1, 2, and 5. 

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 10 20 30 40 50 60 70 80 90

NormalizedDownstreamDensity, n ̄

Angle, degrees

2 ‐ Far‐field 2

10 ‐ Far‐field 10

60 ‐ Far‐field 60

Non‐dimensionalRadial Distance:

Figure 23.—Normalized density angular distributions for a variety of radial 

distances in a spherical coordinate system. The aperture thickness-toradius is 1.25. Solid lines were determined from Equation (60) and 

dashed lines from Equation (29) and Tables 3 to 5. 
NASA/TM—2011-216970 32 

Because Equations (26) and (29) along with the equations of Tables 1 to 5 represent a rigorous solution for 

downstream particle flux and density, respectively, the far-field flux and density equations were compared to them 

in Figures 24 and 25 to determine the error as a function of radial distance from the downstream aperture center at 

various angles. The figures show that both far-field equation errors reduce to about 5 percent at about 25 aperture 

radii downstream for the 0° case, but within 9 aperture radii for the remaining angles. These figures demonstrate that 

the far-field equations will produce accurate results at 9 aperture radii downstream of a cylindrical aperture with a 

thickness-to-radius ratio of 1.25. 

0.01

0.1

1

10

100

0 50 100 150 200

Magnitudeof Far‐field Particle Flux Error,

%

Non‐dimensionalized RadialDistance

0 degrees

20 degrees

60 degrees

80 degrees

40 degrees

Angle from Downstream

Aperture Center:

Figure 24.—Far-field flux error as a function of radial distance at various 

angles in a spherical coordinate system. The aperture thickness-to-radius 

is 1.25. Error is equal to 100·|Equations (26) to (59)|/Equation (26). 

0.01

0.1

1

10

100

0 50 100 150 200

Magnitudeof Far‐field Particle Density Error,

%

Non‐dimensionalized RadialDistance

0 degrees

20 degrees

60 degrees

80 degrees

40 degrees

Angle from Downstream

Aperture Center:

Figure 25.—Far-field density error as a function of radial distance at various 

angles in a spherical coordinate system. The aperture thickness-to-radius is 

1.25. Error is equal to 100·|Equations (29) to (60)|/Equation (29). 
NASA/TM—2011-216970 33 

C. Verification of Equations—Clausing’s Far-field Expression and Conservation of Mass 

The equations determined in this study for the axial flux and density surrounding a cylindrical aperture were 

verified using three methods. First, comparisons were made to a commonly accepted expression determined by 

Clausing (Refs. 3 and 4). This was done in the prior section for both the axial flux and density in Figures 22 to 25. 

Results showed that maximum differences were within 5 percent at 25 aperture radii downstream and that these 

differences decreased with increasing distance. So, the equations determined in this study compare favorably to that 

determined by Clausing in the far-field downstream volume. 

A second method was to compare the transmission probabilities of the far-field expression (i.e., Eq. (61)) to that 

of the more rigorous Equation (54). Because both equations were determined assuming a linear wall flux 

distribution, both equations should yield the same transmission probabilities. Comparisons were made over a broad 

range of thickness-to-radius ratios, and Equations (54) and (61) were found produce nearly identical results. 

A third method was to verify conservation of mass throughout the axial direction. For mass to be conserved, the 

total axial particle flow rate must be constant (i.e., independent of axial location, z). The total axial particle flow rate 

is determined by integrating the axial flux radially at a given location. Note that this is merely the transmission 

probability multiplied by the particle flux at the aperture inlet, which is a constant. Therefore, merely demonstrating 

a constant transmission probability along z with Equation (54) would assure that mass was conserved. This was 

done over a range of thickness-to-radius ratios and axial locations, and transmission probabilities were found to be 

constant, so mass was conserved. 

VII. Conclusions 

The equations for rigorously calculating the particle flux and density surrounding a cylindrical aperture in the 

free molecular flow regime were developed and presented. Previous work did not provide accurate near-field 

determinations of particle flux and density. The fundamental equations for particle flux and density from a reservoir 

and a diffusely reflecting wall were developed and presented. Assumptions included, in part, a Maxwell-Boltzmann 

speed distribution, equal particle and wall temperatures, and a linear flux distribution from the cylindrical aperture 

walls. With this information, the equations for axial flux and density surrounding a cylindrical aperture were 

developed. The cylindrical aperture was divided into three volumes (i.e., upstream reservoir, interior, and 

downstream), and the upstream and downstream volumes were further divided into multiple regions to rigorously 

determine axial flux and density. After the base equations for flux and density were determined, the limits of 

integration were developed for all volumes and regions. The complete set of flux and density equations was 

presented along with an equation for the transmission probability. Finally, a commonly accepted far-field angular 

distribution expression was used to develop far-field flux, density, and transmission probabilities for comparisons. 

An assessment of the linear wall flux distribution assumption showed that a thickness-to-radius ratio of 16 

would produce an axial flux and density error of 4 percent and that this error decreased with decreasing thickness-toradius ratios. The axial flux and density surrounding a cylindrical aperture with a thickness-to-radius ratio of 1.25 

was presented. The axial flux was found to be axially symmetric about 1/2 of the aperture plate thickness. Both axial 

flux and density were found to decrease rapidly downstream as the flow exhausting from the aperture expanded into 

vacuum. 

Finally, the equations determined in this study were verified using three methods. Comparisons to a commonly 

accepted far-field expression showed that maximum differences were within 5 percent at 25 aperture radii 

downstream and that these differences decreased with increasing distance. The transmission probabilities of the farfield expression were also nearly identical to those determined from this study. Finally, conservation of mass was 

confirmed in the axial direction. 

References 

1. Kaufman, H.R., “Technology of Electron-bombardment Thrusters,” Advances in Electronics and Electron 

Physics, vol. 36, Academic Press, New York, 1974, pp. 265–373. 

2. Goebel, D.M. and Katz, I., Fundamentals of Electric Propulsion: Ion and Hall Thrusters, John Wiley and Sons, 

Inc, Hoboken NJ, 2008. 

3. Soulas, G.C., “Modeling Neutral Densities Downstream of a Gridded Ion Thruster,” 46th

AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, AIAA Paper 2010-6699, July 2010. 

4. Clausing, P., “The Formation of Beams in Molecular Streaming,” Z. Physik, vol. 66, pp. 471–476, 1930. 
NASA/TM—2011-216970 34 

5. Dayton, B.B., “Gas Flow Patterns at Entrance and Exit of Cylindrical Tubes,” National Symposium on Vacuum 

Technology Transactions, Pergamon Press, New York, 1956, pp. 5–11. 

6. Reynolds, T.W. and Richley, E.A., “Free-molecule Flow and Surface Diffusion Thruster Slots and Tubes,” 

NASA Technical Report R-255, April 1967. 

7. Richley, E.A. and Reynolds, T.W. “Numerical Solutions of Free-molecule Flow in Converging and Diverging 

Tubes and Slots,” NASA Technical Note D-2330, June 1964. 

8. Robertson, S.J., “A Study of Density Variations in Free Molecular Flow Through Cylindrical Ducts due to 

Accommodation Coefficients,” NASA Technical Memorandum TM X-53516, November 1964. 

9. Bird, G.A., Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Oxford University Press, Oxford, 

1996. 

10. Beijerinck, H.C.W., Stevens, P.J.M., and Verster, N.F., “Monte-Carlo Calculation of Molecular Flow Through a 

Cylindrical Channel,” Physica, vol. 83c, 1976, pp. 209–219. 

11. Gombosi, T.I., Gas Kinetic Theory, Cambridge University Press, Cambridge, 1994, pp. 18–35. 

12. Clausing, P., “The Flow of Highly Rarefied Gases Through Tubes of Arbitrary Length,” Ann. Physik, vol. 5, 

no. 12, 1932, p. 961. 

13. Narasimha, R., “Orifice Flow at High Knudsen Numbers,” Journal of Fluid Mechanics, vol. 10, no. 3, 1961, 

pp. 371–384. 
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Gas Flux and Density Surrounding a Cylindrical Aperture in the Free Molecular Flow 

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13. SUPPLEMENTARY NOTES

14. ABSTRACT

The equations for rigorously calculating the particle flux and density surrounding a cylindrical aperture in the free molecular flow regime are 

developed and presented. The fundamental equations for particle flux and density from a reservoir and a diffusely reflecting surface will 

initially be developed. Assumptions will include a Maxwell-Boltzmann speed distribution, equal particle and wall temperatures, and a linear 

flux distribution along the cylindrical aperture walls. With this information, the equations for axial flux and density surrounding a cylindrical 

aperture will be developed. The cylindrical aperture will be divided into multiple volumes and regions to rigorously determine the 

surrounding axial flux and density, and appropriate limits of integration will be determined. The results of these equations will then be 

evaluated. The linear wall flux distribution assumption will be assessed. The axial flux and density surrounding a cylindrical aperture with a 

thickness-to-radius ratio of 1.25 will be presented. Finally, the equations determined in this study will be verified using multiple methods.

15. SUBJECT TERMS

Ion thruster; Ion engine; Free molecular flow; Density; Flux 

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