Density distribution of a molecular flux from a short cylindrical tube

DENSrry DISTRIBillION OF A MOLECULAR FLUX FROM A SHORrl' CYLINDRICAL TUBE

FEBRUARY 1967

by

rR ~~HSO E HOGI:SWCOl OBH

J. H. de Leeuw VLlEGTUIGa&U':;:~UNDE

and E. O. Gadamer

Bi~UOIHEfK

,

_,

_.

DENSITY DISTRIBUTION OF A MOLECULAR FLUX FROM A SHORT CYLINDRICAL TUBE by

J. H. de Leeuw*" and

, E.O. Gadam,er**

r' '

* Institute for Aerospace Studies

** Unit for Applied Mathematics, McMaster University, Hamilton, Canada

t " ,

, J

J

J

ACKNOWLEDGEMENTS

The authors are grateful to Professor G. N. Patterson, Director of the Institute for Aerospace Studies, for his encouragement and continued interest in the reported research. Thanks go to Professor

D.J.

Kenworthy and

Mr.

K.K. Ghosh, both at McMaster University for their assistance in the

,

preparation of the Fortran IV program.

, , "

SUMMARY

The radial density distribution of the molecular flux emerg-ing from a short cylindrical tube, which conneets two low-density gas cham-bers, is calculated both in the exit plane of the tube and just downstream of that plane. The calculations are based on the kinetic theory of gases. It is assumed that the upstream density is low enough so that free-molecule conditions prevail at the tube inlet, and that the distribution function in the upstream chamber is a Maxwellian with zero average velocity. The mole-cules emerge into a near-vacuum which is maintained in the downst~eam chamber. It is assumed, furthermore, that all reflections of molecules from the tube wall are of the diffuse type, that is, the momentum and thermal accommodation of the molecules leaving the wall are complete. This wall may in general be at a temperature different, from the upstream gas temperature , but in the present calculations both these temperatures are assumed to be the same.

Computations have been made for tube length-to-diameter ratios 0.25, 0.50, 0.75, and 1.00, using the IBM 7040 electronic digital computer facility at MeMaster University.

The study was undertaken primarily to provide theoretical re-sults complementing an experimental investigation, carried out at the Institute for Aerospace Studies, of the density distribution in a low-density air jet emerging from a cylindrical tube of length-to-diameter ratio 1.00. In these experiments, the gaseous fluorescence induced by a thin electron beam pro-jected across the gas jet was utilized. We have found satisfactory agree-ment between our experiagree-ment al and calculated results .

.

- - - - -

-. ./ l . 2.

3.

4

.

5

.

6

.

TABLE OF CONTENTS

INTRODUCTION

THE FIVE CALCULATED CONTRIBUTIONS TO THE DOWNSTREAM DENSITY

MOLECULES FLYING STRAIGHT THROUGH THE TUBE

MOLECULES

REFLECTED FROM THE TUBE WALL

THE FORTRAN IV PROGRAM AND THE COMPUTED RESULTS

CLAUSING'S FUNCTION

REFERENCES

iv 1

6

8

12

18

43

50

,

1. INTRODUCTION

We have recently carried out an experimental investigation of the radial density distribution of the molecular air flux emerging from a short cylindrical tube (Ref. 1). For the purpose of these experiments we have developed what may be called an "optical electron beam probe" . This is an instrument for the measurement of the local density in a rarefied gas flow using the electron beam generated in an electron gun. The method makes use of the possibility of exciting a fairly intense and well confined gaseous fluorescence by projecting a thin but powerful electron beam into the rare-fied gas. The fluorescence coincides with the beam and therefore has the form of a th in luminescent line traversing the space of the flow field. We measured the fluorescent light output from a small selected volume, approximately 1 mm3 in size, traversed by the thin beam whose diameter was approximately 1 mmo This output depends on, and therefore measures, the gas density at the "point" surrounded by the small volume. The method can be used in the study of three-dimensional flow fields by controlling the position of the electron beam in the field (this takes care of two dimensions), and by selecting the light coming from a small section of the thin beam (this takes care of the third dimension). Specially designed traversing mechanisms make the three-dimensional probing feasible. The light emanating from a small section of the electron beam selected at a time, the section being defined by suitabl~ optical stops, is focussed by a lens onto the cathode of a

photomultiplier tube. The tube generates a photocurrent which is proportional to the incident light intensity. The photocurrent, therefore, becomes the measure of the gas density at the selected beam station and is directly cali-brated against that density. Figure 1 shows one of the experimental

arrange-ments described in Ref. 1. .

A modified arrangement, also described in Ref. 1, is shown in Fig. 2. This arrangement features two low-density gas chambers connected by a tube of 1.00 cm length and 1.00 cm diameter. It was used for the mea-surement of the radial density distribution. The electron beam penetrates the air jet just downstream of the tube exit plane. The distance from the exit plane to the center of the beam was 0.75 mm, which is as close as is experimentally feasible. The measured results are shown in Fig.

3.

In a similar set of experiments the distance was increased to 1.50 mm; Fig.

4

shows the measured results.

All experimental results reported in Ref. 1 show an initially gentle but then more rapid decrease of the density from the centre of the exit plane to its edge. This is what one would expect. The task of the present contribution is a theoretical calculation of that densi ty distribu-tion. In the following sections we develop the required distribution formula on the basis of the kinetic theory of gases. The underlying assumptions are that the upstream density is low enough so that a Knudsen gas exists inside the tube connecting the two chambers, and that molecular reflections from the tube wall are completely diffuse. The distribution is then found as a superposition of five independent contributions (Section 2). Numerical calculations using an IBM 7040 computer follow the derivation of the dis-tri but ion formula. We have found generally good agreement between the theo-retical and experimental results. This indicates that the assumptionsmade are realistic .

FIG. 1 Electron Beam induced fluorescence in a low-density air flow. Left: Test section of the UTIAS low-density wind tunnel with a Mach-4 nozzle installed. The lower end of the electron gun is seen at the top, a metal cup collecting the beam current at the bottom. A model is

supported by a horizontal bar and can be traversed across the jet.

Center: Electron beam induced fluorescence. The model is removed. The higher air density within the jet is clearly indicated by a greater brightness. (The white spot on top has no significance.)

Right: The model is moved into the flow. Rarefaction behind the model is indicated by the reduced beam brightness. The variations in the width of the fluorescent line are partlv real but primarily due to overexposure.

2

j t~ad

gtass

window

needl.e va/ve

air

dry~r air Intake

upstream

chQmber

~'ectron

gun

sock.~t

Pyrex

!}I.ass

t

e

to

diffusion

pUYnp

eLectron

exit

nozzte

downstret:lm

chomher

to dlffus/on pUYnp

+-r--- -cm

3

1. 5 y 10-6 mg 1.0 .5 I I

o

.4-

x 10- 2

rA

1 -.3_

~

.2-LL

. - .1-0 2 3

4

5 6 turns of calibrated sp 1. 0 cm indle

FIG.

3

Distribution of the air density measured across low-density jet emerging from a cylindrical tube of 1.00 cm length and 1.00 cm diamete

_6,

Dis]ance of center of beam from exit plane of

tube 0.75 mmo Upstream density

3.48

x 10- mg/cm. Downstream density measured in center of

jet 0.905 mg/cm

3 .

Measured downstream-to-upstream density ratio in center of jet 0.260. Measured ambient downstream-to-upstream density ratio approximation 0.1. (The theoretical downstream-to-upstream density ratio in center of jet for 0,1 ambient ratio is 0.277, see Section

5.)

Upstream Knudsen number 2.3.

t~ad

gtass

window

needte

va/ve

air

dryer

air /nioke

upstream

chQmber ~/ectron

gun

S 0 k~t

Pyrex !jtoss

t

e

to

dl'ffusion

pump

electron

e.xit

/ r-"-""-'::_Ylc.>z z I~

downstream

chamh~r to dtffus/on pUYYrp

+" 1. 5 x 10-6 mg cm

3

1.0 . 5 I

o

.4-

x 10- 2

rA

1 - --- _{- -1} -.3_

~

.

. 2. - .1-0 2 3

4

5 6 turns of calibrated sp 1. 0 cm indle

FIG.

3

Distribution of the air density measured across low-density jet emerging from a cylindrical tube of 1.00 cm length and 1.00 cm diamete

_6.

Dis~ance of center of beam from exit plane of tube 0.75 mmo Upstream density

3

.4

8

x 10- mg/cm. Downstream density measured in center of jet 0.905 mg/cm3 . Measured downstream-to-upstream density ratio in center of jet 0.260.

Measured ambient downstream-to-upstream density ratio approximation 0.1. (The theoretical downstream-to-upstream density ratio in center of jet for 0.1 ambient ratio is 0.277, see Section 5.) Upstream Knudsen number 2.3.

-t.TI

.

-1. 5 x 10- 6 mg I 1.0 ; .5 ) O! r-cm 3

.4-

x 10-2

JA

A .3_

.

_.

_.

.

.

.

.

.

.2_

L

~

~ .1-0 2 3 4 5 turns 0 calibrated sp - 1. 0 cm

..

FIG.

4

Distribution of the air density measured across low-density jet emerging from a cylindrical

tube of 1.00 cm length and 1.00 cm diamete

6.

Distance of center of beam from exit plate of

tube 1.50 mmo Upstream density

3.48

x 10- mg/cm

3.

Downstream density measured in center of jet 0.905 mg/cm

3 .

Measured downstream-to-upstream density ratio in center of jet 0.260.

f ndle

-,

2. THE FIVE·CALCULATED CONTRIBUTIONS TO THE DOWNSTREAM DENSITY

The radial distribution of the gas density in a low-density jet emerging from a tube of any geometry that connects two gas chambers

naturally depends on that geometry. For a circular cylindrical tube the distribution must possess radial symmetry. In th~t case the only relevant parameter is the tube length-to-diameter ratio, L. We have made computations

for L

=

0.25, 0.50, 0.75, and 1.00. The distribution is determined by cal-culating the density as a function of a radius variabIe whose origin lies in the center of the cifcular cross-sectional plane. The domain of the variabIe terminates at a distance·from the center equal to the radius of

the tube.

The assumption of free-molecule flow implies that collisions

,between molecules inside the tube are so rare that their effect can be ~neglected. It is possible, then, to assume that a molecule emerging from . the tube comes ei ther directl'y from the inlet area or directly from the tube

wall but never from any point of collision within the tube. The density at

a point in the exit plane, or at a point just downstream of the exit plane, is therefore found by adding a number of contributions each of which is cal-culated independently. From Fig.

5

we see that there are in fact five

rele-vant contributions, and that the de'nsity must therefore be constructed as a superposi tion of these five contributions. Their physic'al nature is as ,follows. , Inlet UPSTREAM plane

I

~

I

I

TUBE Shoulder Typical Point Cross~sectional plane in which density is calculated

-·THE FIVE CALCULATED CONTRIBUTIONS TO THE DOWNSTREAM DENSITY

First contribution: molecules coming directly from the upstream chamber with no reflections at the tube wall.

Second contribution: molecules coming from the upstream chamber and re-flected once or several times at the tube wall.

Third contribution: molecules coming from points downstream of the plane in which the density is calculated.

Forth contribution: molecules coming from the downstream chamber, reflected once or several times from the tube wall, and flying back into the downstream chamber.

Fifth contribution: molecules coming from the downstream shoulder.

Contributions one and two depend on the upstream chamber density; contributions three, four and five depend on the downstream chamber density. In typical experiments the latter density is much smaller than the former. If, on the other hand, both these densities are made equal, there will be no flow. In that case the calculated density distribution must be expected to be uniform across the entire cross-sectional plane because it must at all points be equal to the common density of the two chambers. This requirement affords a convenient check on all distribution calculations.

If the downstream chamber holds a perfect vacuum, contributions three, four and five are zero. If the plane in which the density distribution is calculated coincides with the tube exit plane, while the downstream density is different from zero, then only contribution five is zero.

Though five contributions must be calculated, there are just two types of formulas to be developed. The first is for molecules flying

straight through the tube without striking the tube wall; this type applies

for contribution one and, as will be shown, for contribution five. The

second is for molecules experiencing wall reflections; this type applies

for contributions two and four. Contribution three does not require any elaborate calculation; it is merely one-half the uniform downstream chamber density outside the jet (ambient downstream chamber density); it enters the calculation as a free parameter.

In the following two sections, Sections

3

and

4,

we develop the two principal formulas. They can be applied directly to contributions one and two. To obtain contribution four, the second formula is used with the integration along the tube wall performed in reverse direction, that is, from the exit plane to the inlet plane. Contribution five is taken as one~ half the downstream density minus the density of the molecules flying

straight through an imaginary tube whose inlet plane is the tube exit plane and whose exit plane is the cross-sectional plane in which the density dis-tribution is calculated.

The Fortran IV program used and the computed results are presented in Section

5.

In Section

6

we have computed Clausing's function ETA which controls the effusion of molecules from the tube wall under diffuse reflection conditions, as a function of a tube wal I length variabIe. The table of ETA values is used as an input for the computation of contributions two and four.

In both principal formulas there occur complete elliptic

integrals. They form part of the integrands of our density integrals. The

complete elliptic integrals are called from a subroutine permanently stored

in the computer. One of the integrals has a singularity at the tube exit

plane. However, the density integrals affected by the singularity converge

nevertheless. The resulting numerical difficulties can be overcome by

tak-ing the computation in the immediate neighborhood of the singularity out of

the computer and replacing it by a computation based on an analytical

approxi-mation.

3.

MOLECULES FLYING STRAIGHT THROUGH THE TUBE

We begin the calculation of the density distribution in the exit plane of a cireular cylindrical tube, and in planes just downstream of the exit plane, by determining the contribution made by molecules that fly

directly from the inlet to the exit without striking the tube wall, and by

first restricting the calculation to points lying in the exit plane. The

extension of the result to points just downstream of the exit plane will be

very simple.

First we develop an expression for the infinitesimal

contri-bution to the density at a typical point of the exit plane made by molecules

coming directly from a small area element of the inlet plane (Fig.

6).

This

expression is then integrated over the entire inlet plane to give the local

contribution made by all molecules flying straight toward the selected point

of the exit plane.

'\ \ )" \

/

l

,

~\

\ \ ~

S

~

\

_()~--

!

~

_I-

_-

_{-- -}

_-

-, / I / FIG.

6

One ean assume that the number density of the molecules in the inlet plane is equal to the number density n throughout the upstream

chamber. Considering a point P in the inlet plane, there is at P a small

fráction of that density consisting of molecules which have a specified speed C

and are flying :in a specified direction ep wi th respect to a norm al staniing al the

inlet plane. That fractional number density is

n f(C) dw

where f(C~ is the upstream veloeity distribution function,

dw =

p2

sin cp dep de dC

is the volume element in a three-dimensional velocity spaee,€*-pressed in spherical coordinate·s. The flux of molecules from the area element dA con-taining the point P of the inlet plane, into the element of solid angle dQ containing the point p'(s,e,~) of the exit plane, becomes

n f(C~ dw. C dA cos~ or

n f(C) C2 sin~d~dedC . C dA cos~ Since sin<Pd~de dQ and C2 dQdC

=

dw, the expression becomes

n f(C) C3dQ dC.dÄ cos~

The factor dA cos~ represents what is sometimes referred to as Lambert's law, or the eosine law, and amounts to introducing an effective partiele emitting area element which varies as the eosine of theangle ~ between the direction of motion (direction of s-) ·and the normal on the geometrie area element. At the distance s from the emitting area element the surface element bounding the element of solid angle dO has the area

dA'

=

s2 dO .

Therefore the flux through unit area···a-t P' coming from dA at P becomes

or

n f(C) dw . C dA cosp dA'

n f(C)

c

3 dC dA cosp s2

We may drop the factors dA and dC and then say.that n f(C) c3 cosp

s2

is the flux of molecules of speed C and direction ~ that comes from unit area at a given point P of the inlet plane and passes through unit area at the point P' of the exit plane.

• • • ~ •. ;;!, j ~I

The contribution of this flux to the density at P' is obtained by dividing the last expression.by the speed C. Finally, integration over all speeds yields the density contribution nstraight(s,~)

=

ns(s,~) at P' by all molecules coming straight from unit area at P. Thus

00

ns(s,~)

= n

c~sp

J

f(C) C2 dC

s 0

We' now specify f(C) and evaluate the integrale Th~ gas in the upstream chamber i·s assumed to be at rest and in temperature equilibrium with the chamber walls. Its velocity distribution is therefore Maxwellian, that is

3

where

f3

2C2

Joo

f(C) C2

o

Thus f(C)

= (

~)

3/2

1

1$

With this result the contribution to the density at the typical point P' in

the exit plane made by molecules emitted by unit area at

P

in the inlet plane

becomes C

In order to obtain the density contribution at

P'

made by ALL

molecules flying straight through the tube one must now integrate this

expression over the entire inlet plane. We thus calculate

nstraight ns =

J

ns ( s ,cp) dA

=

inlet plane

J

inlet plane ncoscp dA

47T

s2

To express ns as a function of a radius variable in the exit plane, the

following substitutions are made which follow from the tube geometry and

the notation introduced in Fig.

7.

FIG.

7

coscp =

g.

s2=~2+

T2 . T2 À 2 + À' 2 + 2ÀÀ' COS?/I so that

s ' ,

'I'

s2

=

t

2 + À2 + À,2 +

2

ÀÀ'co~.

Also dA

=

ÀdÀ d7fJ. With these substitutions

the integral becomes

We non-dimensionalize as follows.

t

t

À À' n R R' s L

=

_{d 2r} =

r;

_r Ns The integral n then reads N L

fR

_aRf"[4L

2 d'!/!.. s 27T 2 + R,2 +

2RR'COS~J3/2

o

0 +R

=

~

J1R

aR [[4L

2 d'!!!. + R2 + R,2 +

2RR'COS~J3/2

0

This double integral can be separated by reducing the integration over ~ to ~

a ~omplete elliptic integrale To this end one substitutes cos~

=

1 - 2

rumF

_2.

Thus N L

Jl

i

7T

d'!/!.. s

=

~

R dR [4L2 _{+ R2 + R,2 + 2RR' (1-2}

_{. !l!.}

_{)J3/ 2}

o

0 Slon 2 . 2L

1

{/2

d '!/!..'

Jo

R

aR

o

[4L

2

7T _{(R+R,)2 _ 4RR' . 2}

_~'

_J3/2 + Slon

4RR'

_{.. 2?'I' J3/}2 Slon 'r where k

2

=---~

4RR'

s 4L2 + (R+R,)2

The last form of the integral over ~' can be evaluated in closed terms. One obtains

where

J

7T/:_2 __

~~d~'!/!..~'

__

~~

[1 k2 sin2 ?,;, J3/ 2

o

s 'r k2 Slon . 2?ft 'r' d'r' ?ft s , k

f

1 s

is a complete elliptic integrale The density contribution then becomes

,

.

N s 2L'

Jl

R dE E(k_{s )} ==

7T .

0 [-4-L 2-+-(-R-+-R -, )-2=-J 3---'/"-2-(-1-_

k-~-)

.

or

.We see that this result is a function only of the dimensionless radius variable of the exit plane ~ R', and of the parameter L. It is thus the re-quired expression for the contribution made by straight flying molecules to the dimensionless cross-sectional density distribution in the exit plan~.

In order to extend the validity of OUT result to cross-sectional planes just downstream of the exit plane, it is only ncessary to replace the tube length-to-diameter ratio, L, by the dimensionless di stance

fro~ the inlet plane to the cross-sectional plane, that is, by the distance between the two planes divided by the diameter of the tube. This is possible because the straight flying molecules "do not 'know" the tube walL The new configuration is equivalent to that of a tube of slightly larger length. The cross-sectional planes of experimental interest are those in which a thin electron beam could lie which is projected a.cross the low-denSi ty jet emerging from the ttibe. The new dimensionless length is therefore denoted by' Lel' (The experiments discussed in Section

2

were carried out With an L == 1.000 tube and the .electron beam lying in the cross-sectional planes at , Lel == 1.075 and Lel == 1.150, respectivelyo)

4.

MOLECULES REFLEC,TED FROM THE' TUBE WALL

We now proceed with the calculation of the density distribu-tion in the exit plane by determining the contribution made by molecules that have experienced any number of collisions with the tube wall. The procedure is similar to that followed in 'Section

3.

We

first seek an ex-pression for the infinitesimal density contribution made at a typical point in the exit plane by molecules cOItling from a small area element of the wall. This expression is then integrated over the entire wall area to give the total contribntion of reflected molecules at theselected point of the exit pl~e. .

.The calculation is somewhat more complicated, however, because one may not assume that there exists a uniform emission of molecules along the tube wall as was the ·case across the inlet plane in Section

3.

We there-fore introduce a function Tj(-x,.e)·which specifies the local flux of molecules emitted by unit area of the tube wall. x is a tube length variable such that at the inlet x == 0 and at the exit x ==.e. This function which is known as Clausing's function must be determined separately. We defer the task of determining Clausing's functiGn .until aftèr the development of the general expression for the radial density distribution in the exit.plane

and

assume for the purpose of that derivation that Tj(x,.e) is a well behaved known function.

12

\ / .

In order to arrive at an expression for the contribution to the density at a point in the exit plane made by the molecules coming from a small area element of the wall, one makes use of an important property of diffuse reflection. This is the fact that the flux of molecules diffusely emitted from an element of the wall area obeys the cosine law. The area

element can therefore be considered equivalent to a small orifice in a wall of zero thickness behind which there rests a virtual Maxwellian gas. The virtual gas density depends on the tube length variable x; we denote it by n*(x). The virtual gas leaks through the orifice with the emerging flux obeying the cosine law. We may thus use an expression identical in form to that derived earlier for the inlet plane (Section

3),

namely,

n _wa ll(x,s,~)

=

n _w (*,s,~)

=

n*(x) cos ~ 47T s2

We must now calculate n*(x) in terms of the Clausing function ~(x,t). Since ~(x,t) is defined as the flux of molecules from unit wall area, and since an element of the wall area can be replaced by a virtual orifice emitting molecules from a Maxwellian gas, we may write (c.f. Section 3)

~

(x, t) =

J

n* ( x) f ( C) dw C cos

~

w

n*(x)

1:

3

r(c) dC 0

rdeJ

7r/~intp

costp dtp .

The two angular integrations give 27T and 1/2, respectively. Thus

We tntroduce the mean random speed of ~he fictitious molecules behind the wall

c

=

V

R ₂ T _7T

where T is the wall temperature ~nd R is the gas constant of the real gas flowing through the tube. Hence

~(x,t)

C or n*(x)

=

4 11

(x,t)

C

With this result the contribution to the density at a point P' of the exit plane made bymolecules emitted from unit area at a point p(x) of the wall

becomes

(

)

n(x,t)

cosp

n x,s,~ = 2

w 7T C s

In order to obtain the density contribution at P' in the exit plane made by

all molecules emitted from the tube wall one must now integràte this ex-pression over the entire wall area. We thus calculate

n w

J

wall area wall area TJ(x,t) coscp 7Të s2 àA.

We wish to express nw as a function of the radius variable in

the exit plane and therefore make the following substitutions which follow

from the tube geometry and the notation introduced in Fig.

8.

FIG.

8

r + ÀI cos7j;

S

(.e- x)2 + T2

(t-

x)2 + r 2 + ÀI2 + 2rÀI cos7j;

d.A r d7j; dx

With these substitutions the integral becomes

n

w 7T

r _

J

tTJ(x,t) dx [7T

(~

+ À; cos 7j;) d7j;

C 0 0

[Ct-x)

+ r +ÀI2 + 2rÀI COS7j;]3j2 As previously, we non-dimensionalize:

t

d 2r ' RI ÀI r N w ~

, ETA (X,L)

n

T)[x(X), t(L)]

TJ in TJin is the constant flux of molecules into the tube through unit area of

the inlet plane. lJin can be expressed in terms of the upstream number density by the relation

TJin n

ë

=

'4

The dimensionless Clausing function may thus be written as

ETA(X,L)

=

The integral then reads

4

1)[x(X), .t(L)]

nC

Since ETA(X,L) has not yet been specified and will in gener al be a numerically defined function, the integration with respect to ~ should be performed first if possible. As in the case of the straight flying molecules, the integral

over ~ can in fact be evaluated if one expresses it in terms of complete elliptic integrals. We thus substit ute

and write cos

~

=

1 - 2

sin2~

2

J

2rr (1 + R' coSlj; )

d~

o

[4(L_X)2 + 1 + R,2 + 2R'

cos~]372

1.

r

4(L_X)2 + 1 + R,2 + 2R' COf?'l/J d'l/J 20 [4(L_X)2 + 1 + R,2 + 2R'

cos~172

j

dik

=

~o

[4(L_X)2 + 1 + R' 2 + 2R'

cos~]172

Now 2

J

1r/2 d?j;'

[

4

(L-X) 2 + (R' +1) 2 ] 1/2 0 \ ,;='1===1

4

rn

Rt-, ===s=i=-n;)'2 ='IjJ'

V

4(L_X)2 +(R' +1)2 2 K(k_w) where

is a complete elliptic integral whose argument is

k

w

Similarly 1

J2

7T d1J;

~

1 2

=

~

[4(L_X)2 + 1 + R,2 + 2R' _ 4R'Sin2

.

+

p/2

27T d JI!. =0

1

[4(L_X)2 + 1 +

R~2

+ 2R' - 4R' sin2

-%-

]3/

2

/ '

) 2]3/2

171/'2 -

-

dW'

3/

·

2 R +1 0 [1 _ 4 R' sin2

yJ'

'J 4(L_X)2+(R'+1)2

-

' [4(L_X)2 + 2 where d'ljl'

is another complete elliptic integral (c.f. Section

3

·

).

The denominator can be simplified.

=

[4(L_X)2 + (R'+1)2]3/2 [1 -

.

4~'

2J 4(L-X) +(R' +1)

Thus

Finally the density distribution becomes N w = 2 1 7T .

J

L ETA(X,L)

{~

1 1 -

~

[4(L_X)2 -1 + R,2] I2 } dX

o

=

~ LETA(X,L~

_2K(kw) _ .2[4(L_X)2 + R,2_1]

E(~w)

"

~

koc

27T

01 .

'l

V

4 (L-X)2 + (R'+l)2

V

4 (L-X)2+(R' +l)2[4(L-X)2+(R'

or

N (R' ,L)

w ETA(X,L) {K(kW)

4(L-X)2 + (R'+1)2

where k;

=

4R' This result is a function only of the

dimensionless radius variable R' of the exit plane, and of the parameter L.

It is thus the required expression for the contribution, made by all mole-cules reflected from the tube wall, to the dimensionless cross-sectional density distribution in the exit plane of the tube.

We extend the validity of our result to cross-sectional planes

just downstream of the exit plane, as we have done earlier in the case of the

straight flying molecules. In principle it is again only necessary to replace the tube length-to-diameter ratio, L, by the dimensionless distance from the inlet plane to the cross-sectional plane, Lel' that is, by the distance between the two planes divided by the diameter of the tube. Actually, how-ever, the replacement is necessary only in the integrand but not in the

specification of the upper limit of the integral which may remain just L.

The reason for this is that between Land Lel there is no tube wall, and

consequently no emission of wall-reflected molecules can occur. The emission

controlling function ETA is therefore zero between the two stations, and so is the entire integrand because i t is proportional to ETA •.

5 • THE FORTRAN IV PROGRAM ANTI THE COMRJTED RESULTS

We have prepared a Fortran IV program for the formulas derived in Sections

3

and 4, and have used the program to compute numerical results with the IBM 7040 computer at McMaster University. Interspersed in the pro-gram are a number of comments to make its various sections more readable. The following additional remarks should be made. The program has been executed three times, that is, for an integer variable M assuming the values 1, 2,

and

3.

M controls the position of the cross-sectional pl"ane in which the density distribution is calculated. If M

=

1, the plane coincides with the tube exit plane; if M = 2, it is 0.75 mm downstream of the exit plane;

and if M

=

3,

it is 1.50 mm downstream of the exit plane (c.f. the last paragraph of Section

3).

(We could have introduced a DO statement covering

all three M-cycles, but since each M-cycle requires about 45 minutes computer time this would not have been practical. We have therefore staggered the computations by making M an input and by requesting machine time for only

one value of M at a time.)

For reasons required by the Fortran language, the notation of the program is of course not identical to the notation used in previous

sections. But the symbols used in the program are sufficiently suggestive to make the necessary translations easy. For example, the complete elliptic integral E(ks ) is denoted in the program by COMPE(SK), and the complete elliptic integral K(k_w) is denoted by COMPK(WK), etc. RIN, REX, and REL

are respectively the dimensionless radius variables in the inlet plane of the tube, in the exit plane, and in the cross-sectional plane just

down-stream of the exit plane where the density distribution is calculated.

The symbol ARG is used throughout for integrands. The plane of the variable REL is referred to as the plane of the electron beam.

There are four major DO cycles, one for each of the four tube length-to-dimaeter ratios chosen. These are L = 0.25, 0.50, 0.75, and 1.00. Their index is denoted by I. Within each major DO cycle there are 51 minor DO cycles, whose index is denoted by

J,

according to 50 intervals along the independent variable REL. And within each minor DO cycle there are either 51 DO cycles, minor to the minor DO cycles, according to 50 intervals along the integration variables RIN or REX, or there are 101 such DO cycles according to 100 intervals along the dimensionless tube length integration variable X. Their index is always denoted by K. Finally, there are, where applicable, six additional DO cycles, whose

index is denoted by N, one for each of the six downstream-to-upstream density ratios chosen. These are 0,0.05, 0.10, 0.15, 0.20, and l.O.

(The ratio 1.0 is included for checking purposes, c.f. Section 2.) The density ratios enter the calculations of contributions three, four, and five.

A separate section in the program is given to each of the five contributions discussed in Section 2. The five sections are preceded by a general section containing a dimension statement and statements for time monitoring, input reading, and general definitions applicable to all subse-quent sections. The five sections are followed by a short section for the computation of the total density distribution, and then by two final sec-tions, one for punching and one for printing of the results. Only the major DO cycle (pertaining to the tube length parameter, index I) runs through all sections. All minor DO cycles are closed within their respec-tive sections. All integrations are carried out by using Simpsonls rule. A special effort has been made to program the punching of the results in publishable table form.

From the result tables we see that the density invariably has a maximum at the center of the cross-sectional plane, at RI = REL = 0.00, and that it tapers off to a minimum reached at the terminal point of the radius variable, at RI = REL = 1.00. This is in agreement with our experi-mental evidence. Figure

9

shows a comparison between experimental and calculated results.

The last column of each result table exhibits, with few exceptions, the value 1.0000. Where variations occur they are negligibly smalle The value 1.0000 is of course precisely the dimensionless density one must expect under the conditions of the last column, because that column represents the case when the downstream-to-upstream density ratio is one. This is a confirmation that our theory is consistent within its assumptions.

No results appear in the last lines of those tables where the cross-sectional plane coincides with the tube exit plane. In that case Lel =L (or TLELE = TL) and simultaneously RI = REL = 1.00. That is the only case when the integrand of the formula derived in Section

4

becomes infinite.. The reason for the occurrence of the infinity is that the argument ~.T (or WK) of

t_{he complete elliptic integral K(kw) (or COMPK(WK) ) is one and th at} K(~)= 00

when ~ = 1. We have simply programmed this point out.

The following table is to serve as a guide for finding the various sections of the program and the tabulated and plotted results.

r-General section Page 21

Contribution one 22

two 23

Program three 24

four 25

five 26

Total density distribution 27

Punch results 28

Print results

~ 29

Distance of

Tube length electron beam L

from exit plane

/, 10.0 rnrn

o

rnrn 0.25 30 0.50 31 0.75 32 1.00 33 Result tables 0.75 mm 0.25 34 0.50 35 0.75 36 1.00 37 1.50 mrn 0.25 38 0.50 39 0.75 40 1.00 41 Cornparison with 0.75 mm 1.00 42 experiment 1.50 mrn 1.00 42 (Fig. 9) 20

C D~NSITY DISTRIBUTION ALONG THE ELECTRON BEAM IN THE TUBE EXIT C PLANE, AND IN TWO PLANES SOMEWHAT DOWNSTREAM OF THE EXIT PLANE C OCTOBER 1964

DIMENSION ETA(4,101), ARG1(Sl), ONE(Sl), ARG2(101), TWO(Sl), 1 THREE(51,6), ARG4(101), FOUR(51,6), ARGS(51), FIVE(51,6), 2 TOTAL(51,6)

C NEXT TWO LINES FOR TIMING PURPOSES ONLY WRITE (6tl)

1 FORMAT (lH) READ 2, M 2 FORMAT (11)

C (M CONTROLS THE DISTANCE OF THE ELECTRON BEAM FROM THE TUBE EXIT C PLANE AND ASSUMES THE VALUES 1, 2 AND 3)

READ 3, ((ETA( I,K), 1=1,4), K=1t1011

3 FORMAT (29X,F6.4,9X,F6.4,9X,F6.4,9X,F6.4)

C DEFINE THE DIMENSIONLESS DISTANCE EL FROM THE TUBE EXIT PLANE C TO THE ELECTRON BEAM

EL = 0.075*FLOAT(M-1)

C DO CYCLES, EACH AT A FIXED VALUE OF THE DIMENSIONLESS TOTAL C TUBE LENGTH TL

DO 200 1=1,4

IF (M.EQ.1.AND.I.EQ.2) CALL TIMNOW TL = 0.25*FLOAT(I)

C DEFINE THE DIMENSIONLESS DISTANCE FROM INLET TO ELECTRON BEAM TLELE = TL + EL

C

CONTRIBUTION ONE. MOLECULES ENTERING INLET AND FLYING STRAIGHT

C

TO EXIT

C

DO CYCLE"S. EACH AT FIXED STATION ALONG THE ELECTRON BEAM

DO 30 J=I.51

C

DEFINE THE DIMENSIONLESS RADIUS VARIABLE ALONG ELECTRON BEAM. REL

REL

=

(1.0/50.0)*FLOAT(J-l)

C

DO CYCLES, EACH AT A FIXED STATION IN THE INLET PLANE

DO 20 K=1.51

C

DEFINE THE DIMENSIONLESS INLET RADIUS VARIABLE RIN (INTEGRATION

C

VARIABLE FOR THIS CONTRIBUTION)

RIN

=

(1.0/50.0)*FLOAT(K-1)

C

USE EXISTING SUBROUTINE TO COMPUTE COMPLETE ELLIPTIC INTEGRAL

C

COMPE(SK) FOR STRAIGHT FLYING MOLECULES

SK

s

SQRT(4.0*RIN*REL/(4.0*TLELE**2+(RIN+REL)**2»

COMPE(SK)

=

E(SK,

3.1415926/2.0)

C

COMPUTE INTEGRAND FOR STRAIGHT FLYING MOLECULES

20 ARG1(K)

=

2.0*TLELE*RIN*COMPE(SK)/(3.1415926*SQRT(4.0*TLELE**2

1

+(RIN+REL)**Z)*(4.0*TLELE**2+(R1N-REL)**2»

C

INTEGRATE USING SIMPSON'S RULE

EVEN

=

0.0

DO 21 K=2,50,2

21 EVEN

=

EVEN + ARGl(Kl

000 = 0.0

DO 22 K=3,49,2

22 000

=

000 + ARGl(K)

30 ONE(J)

=

(1.0/3.0)*(1.0/50.0)

1

*(ARG1(1)+4.0*EVEN+2.0*ODD+ARG1(51»

C

THIS COMPLETES THE COMPUTATION OF CONTRIBUTION ONE

.

-C CONTRIBUTION TWO, MOLECULES ENTERING INLET AND REFLECTED FROM WALL C DO CYCLES, EACH AT A FIXED STATION ALONG THE ELECTRON BEAM

DO 50 J=1,51

REL = (1.0/50.0)*FLOAT(J-l)

C DO CYCLES, EACH AT A FIXED STATION ALONG THE TUBE WALL DO 40 K=1,101

C DEFINE THE DIMENSIONLESS TUBE LENGTH VARIABLE X (INTEGRATION C VARIABLE FOR THIS CONTRIBUTION).

C NOTE -- SINCE ETA IS ZERO FROM X=TL TO X=TLELE,

C WE NEED TO INTEGRATE ONLY FROM ZERO TO TL, NOT TO TLELE. C HENCE THE INTEGRAND IS COMPUTED FOR 101 STATIONS

C CORRESPONDING TO 100 INTERVALS BETWEEN x=o AND X=TL X = TL*(1.0/100.0)*FLOAT(K-l)

C USE EXISTING SUBROUTINE TO COMPUTE COMPLETE ELLIPTIC INTEGRALS C COMPE(WK) AND COMPK(WK) FOR MOLECULES REFLECTED AT THE WALL

WK = SQRT(4.0*REL/(4.0*(TLELE-XI**2+(REL+l.01**2» COMPK(WK) = F(WK, 3.1415926/2.0)

C (COMPE WILL BE NEEDED TOO, BUT IT HAS ALREADY BEEN DEFINED) C COMPUTE INTEGRAND FOR REFLECTED MOLECULES

40 ARG2(K)

=

(ETA(I,K)!(3.1415926*SQRT(4.0*(TLELE-X)**2+(REL+1.0) 1 **2»)*(COMPK(WK)-(4.0*(TLELE-X)**2+REL**2-1.0)*COMPE(WK) 2 !(4.0*(TLELE-X)**2+(REL-l.O)**2»

C INTEGRATE USING SIMPSON'S RULE EVEN

=

0.0

DO 41 K=2,100,2

41 EVEN = EVEN + ARG2(K) ODD

=

0.0

DO 42 K=3,99,2 42 ODD

=

ODD + ARG2(K)

50 TWO(J)

=

(1.0/3.0)*(1.0/100.0)*TL

1 *(ARG2(1)+4.0*EVEN+2.0*ODD+ARG2(101»

C

CONTRIBUTION THREE, MOLECULES DOWNSTREAM OF ELECTRON BEAM

C

DO CYCLES, EACH AT A FIXED STATION ALONG THE ELECTRON BEAM

DO 60 J=1,51

DO 59 N=1,5

RATIO

=

0.05*FLOATCN-l)

59 THREE(J,N) = RATIO*O.5

60 THREECJ,6) = 0.5

C CONTRIBUTION FOUR, MOLECULES ENTERING EXIT AND REFLECTED FROM WALL C DO CYCLES, EACH AT A FIXED STATION ALONG THE ELECTRON BEAM

DO 80 J=l,Sl

REL = (1.ü/SO.0)*FLOAT(J-1)

C DO CYCLES, EACH AT A FIXED STATION ALONG THE TUBE WALL DO 70 K=l,lOl

X = TL*(1.0/100.0)*FLOAT(K-1)

WK = SQRT(4.0*REL/(4.0*(TLELE-X)**2+(REL+1.0)**2» KK = 102 - K

C COMPUTE INTEGRAND FOR CONTRIHUTION FOUR

70 ARG4(K) = (ETA(I,KK)/(3.141S926*SQRT(4.0*(TLELE-XI**2+(REL+1.0) 1 **2) )1*(COMPK(WK)-(4.0*(TLELE-XI**2+REL**2-1.01*COMPE(WKI 2 /(4.0*(TLELE-XI**2+(REL-l.01**21)

C INTEGRATE USING SIMPSON'S RULE EVEN = 0.0

DO 71 K=2,100,2

71 EVEN = EVEN + ARG4(KI ODD x 0.0

DO 72 K=3,99,2 72 ODD = ODD + ARG4(KI

SIMPS = (1.0/3.01*(1.0/100.01*TL 1 *(ARG4(1)+4.0*EVEN+2.0*ODD+ARG4(10111 DO 79 N=l,S RATIO

=

O.OS*FLOAT(N-l) 79 FOUR(J,NI = RATIO*SIMPS 80 FOUR(J,6) = SIMPS

C THIS COMPLETES THE COMPUTATION OF CONTRIBUTION FOUR

C CONTRIBUTION FIVE, MOLECULES BETWEEN DOWNSTREAM SHOULDER AND

C ELECTRON BEAM

C NOTE -- M=l REPRESENTS THE HYPOTHETICAL CASE WHEN THE ELECTRON C BEAM LIES IN THE EXIT PLANE. IN THIS CASE CONTRIBUTION FIVE

C IS ZERO IF (M.GT.ll GO TO 89 DO 88 J=1,51 DO 88 N=1.6 88 FIVE(J,Nl = 0.0 GO TO 109

C DO CYCLES. EACH AT A FIXED STATION ALONG THE ELECTRON BEAM 89 DO 100 J=1,51

REL = (1.0/50.0l*FLOAT(J-ll

C DO CYCLES, EACH AT A FIXED STATION IN THE EXIT PLANE. THIS PLANE

C PLAYS NOW THE ROLE OF INLET PLANE

DO 90 K=1,51

REX = (1.0/50.0l*FLOAT(K-ll TLELE

=

EL

SK

=

SQRT(4.0*REX*REL/(4.0*TLELE**2+(REX+RELl**2ll C COMPUTE INTEGRAND FOR CONTRIBUTION FIVE

90 ARG5(Kl

=

2.0*TLELE*REX*COMPE(SKl/(3.1415926*SQRT(4.0*TLELE**2

1 +(REX+RELl**2l*(4.0*TLELE**2+(REX-RELl**2ll

C INTEGRATE USING SIMPSONtS RULE

EVEN

=

0.0 DO 91 K=2,50,2

91 EVEN = EVEN + ARG5(Kl ODD

=

0.0

DO 92 K=3,49,2 92 ODD = ODD + ARG5(Kl

SIMPS = 0.5 - «1.0/3.0l*(1.0/50.0l 1 *(ARG5(1l+4.0*EVEN+2.0*ODD+ARG5(51lll DO 99 N=1,5 RATIO

=

0.05*FLOAT(N-ll 99 FIVE(J,Nl

=

RATIO*SIMPS 100 FIVE(J,6l

=

SIMPS

C THIS COMPLETES THE COMPUTATION OF CONTRIBUTION FIVE

C COMPUTE THE TOTAL DIMENSIONLESS DENSITY ALONG THE ELECTRON BEAM C FOR DOWNSTREAM-TO-UPSTREAM DENSITY RATlOS 0, 0.05, 0.10, C 0.15, 0.20 AND 1.00

109 DO 110 J=1,51 DO 110 N=1,6

C PUNCH RESULTS

ELL ~ 0.75*FLOAT(M-11 PUNCH 150, ELL

150 FORMAT (3X,68HTUBE LENGTH 10.0 MM, DISTANCE OF ELECTRON BEAM FROM 1TUBE EXIT PLANE ,F4.2,2HMM)

PUNCH 151, TL

151 FORMAT (3X,21HTUBE LENGTHIDIAMETER ,F4.~/1 PUNCH 152

152 FORMAT (3X,22HDOWNSTREAM-TO-UPSTREAM) PUNCH 153

153 FORMAT (4X,13HDENSITY RATIO,4X,IHQ,8X,4HO.05,6X,4HO.I0,6X,4HO.15, 1 6X,4HO.20,6X,3H1.0/1

PUNCH 154

154 FORMAT (5X,3HREL,16X,45HD I MEN S l O N LES S DEN S I T I IE SI)

DO 155 J=I,51

REL

=

(1.0/50.0)*FLOAT(J-1)

IF (M.EQ.1.AND.J.EQ.511 GO TO 157 155 PUNCH 156, REL, ITOTAUJ,N), N=1,61 156 FORMAT (5X,F4.2,6X,6FI0.4)

GO TO 188 157 PUNCH 158, REL 158 FORMAT (5X,F4.21

C PRINT RESULTS

188 PRINT 189, ELL, TL

189 FORMAT (lH1,68HTUBE LENGTH 10.0 MM, DISTANCE OF ELECTRON BEAM FROM 1 TUBE EXIT PLANE ,F4.2,26H MM, TUBE LENGTH/DIAMETER ,F4.2/)

PRINT 190

190 FORMAT (1X,36HDOWNSTREAM-TO-UPSTREAM DENSITY RATIO,8X,lHO,31X, 1 4HO.05,29X,3HO.1/) PRINT 191 191 FORMAT (4X,3HREL,7X,3HONE,4X,3HTWO,3X,3(4X,5HTHREE,3X,4HFOUR,3X, 1 4HFIVE,3X,7HDENSITY)/) DO 192 J=1,51 REL = (1.0/50.0)*FLOAT(J-l) IF (M.EQ.1.AND.J.EQ.51) GO TO 194

192 PRINT 193, REL, ONE(j), TWO(j), «THREE(j,N), FOUR(j,N),

r

FIVE(j,Nl, TOTAL(j,Nl 1, N=1,3)

193 FORMAT(4X,F4.2,4X,2F7.4,3(4X,3F7.4,F8.411 GO TO 196

194 PRINT 195, REL, ONE(j), «THREE(j,Nl, FIVE(j,N», N=l,31 195 FORMAT (4X,F4.2,4X,F7.4,7X,3(4X,F7.4,7X,F7.4,8X»

196 PRINT 189, ELL, TL PRINT 197

197 FORMAT (lX,36HDOWNSTREAM-TO-UPSTREAM DENSITY RATIO,7X,4HO.15,29X, 1 4HO.20,29X,3H1.0/)

PRINT 191 DO 198 J=1,51

REL = (1.0/50.0)*FLOAT(j-1)

IF (M.EQ.1.AND.j.EQ.511 GO TO 199

198 PRINT 193, REL, ONE(j), TWO(jl, «THREE(j,NI, FOUR(j,NI, 1 FIVE(j,NI, TOTAL(j,N», N=4,6)

GO TO 200

199 PRINT 195, REL, ONE(jl, «THREE(j,NI, FIVE(j,N) I, N=4,61 200 CONTINUE

STOP END

TUBE LENGTH 10.0 MM, DISTANCE OF ELECTRON BEAM FROM TUBE EXIT PLANE

o.

MM TUBE LENGTHIDIAMETER 0.25 DOWNSTREAM-TO-UPSTREAM DENSITY RATIO 0 REL

o.

0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.56 0.58 0.60 0.62 0.64 0.66 0.68 0.70 0.72 0.74 0.76 0.78 0.80 0.82 0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.00 D 0.3869 0.3869 0.3868 0.3867 0.3865 0.3863 0.3861 0.3858 0.3854 0.3850 0.3846 0.3841 0.3835 0.3829 0.3822 0.3815 0.3807 0.3799 0.3790 0.3780 0.3770 0.3759 0.3748 0.3735 0.3722 0.3708 0.3693 0.3678 0.3661 0.3643 0.3625 0.3605 0.3585 0.3563 0.3541 0.3517 0.3491 0.3465 0.3437 0.3408 0.3377 0.3344 0.3310 0.3274 0.3236 0.3196 0.3153 0.3107 0.3057 0.2999 0.05 MEN S 0.4176 0.4175 0.4175 0.4174 0.4172 0.4170 0.4168 0.4165 0.4lf>1 0.4158 0.4153 0.4149 0.4143 0.4138 0.4131 0.4124 0.4111 0.4109 0.4101 0.4091 0.4082 0.4071 0.4060 0.4048 0.4036 0.4023 0.4009 0.3994 0.3978 0.3961 0.3944 0.3925 0.3906 0.3885 0.3863 0.3841 0.3817 0.3792 0.3765 0.3737 0.3708 0.3677 0.3645 0.3610 0.3574 0.3536 0.3495 0.3452 0.3404 0.3349 30 0.10

o

N LES S 0.4482 0.4482 0.4481 0.4480 0.4479 0.4477 0.447~ 0.4472 0.4469 0.4465 0.4461 0.4457 0.4452 0.4446 0.4440 0.4434 0.4427 0.4419 0.4411 0.4402 0.4393 0.4383 0.4373 0.4362 0.4350 0.4337 0.4324 0.4310 0.4295 0.4279 0.4262 0.4245 0.4226 0.4207 0.4186 0.4165 0.4142 0.4118 0.4093 0.4067 0.4039 0.4010 0.3979 0.3947 0.3912 0.3876 0.3838 0.3797 0.3751 0.3699 0.15 0.20 DEN S I T I E S 0.4789 0.4788 0.4788 0.4787 0.4786 0.4784 0.4782 0.4779 0.4776 0.4773 0.4769 0.4765 0.4760 0.4755 0.4749 0.4743 0.4736 0.4729 0.4722 0.4713 0.4705 0.4695 0.4685 0.4675 0.4664 0.4652 0.4639 0.4626 0.4612 0.4597 0.4581 0.4565 0.4547 0.4529 0.4509 0.4489 0.4468 0.4445 0.4421 0.4396 0.4370 0.4343 0.4314 0.4283 0.4251 0.4216 0.4180 0.4141 0.4099 0.4049 0.5095 0.5095 0.5094 0.5094 0.5092 0.5091 0.5089 0.5086 0.5083 0.5080 0.5077 0.5073 0.5068 0.5063 0.5058 0.5052 0.5046 0.5039 0.5032 0.5024 0.5016 0.5007 0.4998 0.4988 0.4978 0.4966 0.4955 0.4942 0.4929 0.4915 0.4900 0.4884 0.4868 0.4851 0.4832 0.4813 0.4793 0.4772 0.4750 0.4726 0.4701 0.4675 0.4648 0.4619 0.4589 0.4557 0.4523 0.4486 0.4446 0.4399 1.0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.000~ 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0001 1.0002 1.0003 1.0001

TUBE LENGTH 10.0 MM, DISTANCE OF ELECTRON BEAM FROM TUBE EXIT PLANE O. MM TUBE LENGTH/DIAMETER 0.50 DOWNSTREAM-TO-UPSTREAM DENSITY RATIO 0 REL O. 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.56 0.58 0.60 0.62 0.64 0.66 0.68 0.70 0.72 0.74 0.76 0.78 0.80 0.82 0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.00 D 0.3126 0.3126 0.3125 0.3125 0.3123 0.3121 0.3119 0.3117 0.3113 0.3110 0.3106 0.3102 0.3097 0.3092 0.3087 0.3081 0.3074 0.3068 0.3060 0.3053 0.3044 0.3036 0.3026 0.3017 0 .. 3007 0.2996 0.2985 0.2973 0.2961 0.2948 0.2934 0.2920 0.2906 0.2890 0.2874 0.2858 0.2840 0.2822 0.2802 0.2782 0.2761 0.2739 0.2716 0.2691 0.2665 0.2637 0.2607 0.2575 0.2539 0.2495 0.05 MEN S 0.3470 0.3470 0.3469 0.3468 0.3467 0.3465 0.3463 0.3461 0.3458 0.3455 0.3451 0.3447 0.3442 0.3438 0.3432 0.3427 0.3421 0.3414 0.3407 0.3400 0.3392 0.3384 0.3375 0.3366 0.3356 0.3346 0.3336 0.3324 0.3313 0.3301 0.3288 0.3274 0.3260 0.3246 0.3231 0.3215 0.3198 0.3181 0.3162 0.3143 0.3123 0.3102 0.3080 0.3056 0.3032 0.3005 0.2977 0.2946 0.2912 0.2871 31 0.10

o

N LES S 0.3814 0.3814 0.3813 0.3812 0.3811 0.3809 0.3807 0.3805 0.3802 0.3799 0.3796 0.3792 0.3788 0.3783 0.3778 0.3773 0.3767 0.3761 0.3754 0.3747 0.3740 0.3732 0.3724 0.3715 0.3706 0.3696 0.3686 0.3676 0.3665 0.3653 0.3641 0.3628 0.3615 0.3601 0.3587 0.3572 0.3556 0.3540 0.3522 0.3504 0.3485 0.3465 0.3444 0.3422 0.3398 0.3373 0.3346 0.3317 0.3285 0.3246 0.15 0.20 DEN S I TIE S 0.4157 0.4157 0.4157 0.4156 0.4155 0.4153 0.4151 0.4149 0.4146 0.4144 0.4140 0.4137 0.4133 0.4128 0.4124 0.4119 0.4113 0.4107 0.4101 0.4095 0.4088 0.4080 0.4073 0.4064 0.4056 0.4047 0.4037 0.4027 0.4017 0.4006 0.3994 0.3982 0.3970 0.3957 0.3943 0.3929 0.3914 0.3898 0.3882 0.3865 0.3847 0.3828 0.3808 0.3787 0.3765 0.3741 0.3716 0.3689 0.3658 0.3621 0.4501 0.4501 0.4500 0.4500 0.4499 0.4497 0.4495 0.4493 0.4491 0.4488 0.4485 0.4482 0.4478 0.4474 0.4469 0.4465 0.4460 0.4454 0.4448 0.4442 0.4435 0.4429 0.4421 0.4413 0.4405 0.4397 0.4388 0,.4378 0.4369 0.4358 0.4348 0.4336 0.4325 0.4312 0.4299 0.4286 0.4272 0.4257 0.4242 0.4226 0.4209 0.4191 0.4173 0.4153 0.4132 0.4110 0.4086 0.4060 0.4032 0.3996 1.0000 1. 0000. 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0001 1.0002 1.0003 0.9997

TUBE LENGTH 10.0 MM, DISTANCE OF ELECTRON BEAM FROM TUBE EXIT PLANE O. MM TUBE LENGTHIDIAMETER 0.75 DOWNSTREAM-TO-UPSTREAM DENSITY RATIO 0 REL

o.

0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.56 0.58 0.60 0.62 0.64 0.66 0.68 0.70 0.72 0.74 0.76 0.78 0.80 0.82 0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.00 D 0.2653 0.2653 0.2653 0.2652 0.2651 0.2649 0.2648 0.2646 0.2643 0.2640 0.2637 0.2634 0.2630 0.2626 0.2622 0.2617 0.2612 0.2607 0.2601 0.2595 0.2589 0.2582 0.2575 0.2567 0.2559 0.2551 0.2542 0.2533 0.2523 0.2513 0.2503 0.2491 0.2480 0.2468 0.2455 0.2442 0.2428 0.2413 0.2398 0.2382 0.2365 0.2347 0.2328 0.2308 0.2287 0.2264 0.2240 0.2214 0.2184 0.2143 0.05 MEN S 0.3021 0.3020 0.3020 0.3019 0.3018 0.3017 0.3015 0.3013 0.3011 0.3008 0.3006 0.3002 0.2999 0.2995 0.2991 0.2987 0.2982 0.2977 0.2971 0.2966 0.2959 0.2953 0.2946 0.2939 0.2931 0.2923 0.2915 0.2906 0.2897 0.2887 0.2877 0.2867 0.2856 0.2844 0.2832 0.2820 0.2806 0.2793 0.2778 0.2763 0.2747 0.2730 0.2712 0.2693 0.2673 0.2651 0.2628 0.2603 0.2575 0.2534 0.10

o

N LES S 32 0.3388 0.3388 0.3387 0.3387 0.3386 0.3384 0.3383 0.3381 0.3379 0.3376 0.3374 0.3371 0.3367 0.3364 0.3360 0.3356 0.3351 0.3346 0.3341 0.3336 0.3330 0.3324 0.3317 0.3311 0.3303 0.3296 0.3288 0.3280 0.3271 0.3262 0.3252 0.3242 0.3232 0.3221 0.3209 0.3198 0.3185 0.3172 0.3158 0.3144 0.3128 0.3112 0.3095 0.3077 0.3058 0.3038 0.3016 0.2993 0.2966 0.2926 0.15 0.20 DEN S I T I E S 0.3755 0.3755 0.3755 0.3754 0.3753 0.3752 0.3750 0.3749 0.3747 0.3744 0.3742 0.3739 0.3736 0.3733 0.3729 0.3725 0.3721 0.3716 0.3711 0.3706 0.3701 0.3695 0.3689 0.3682 0.3675 0.3668 0.3661 0.3653 0.3645 0.3636 0.3627 0.3618 0.3608 0.3598 0.3587 0.3575 0.3564 0.3551 0.3538 0.3525 0.3510 0.3495 0.3479 0.3462 0.3444 0.3425 0.3404 0.3382 0.3357 0.3318 0.4123 0.4122 0.4122 0.4121 0.4121 0.4119 0.4118 0.4116 0.4115 0.4112 0.4110 0.4107 0.4104 0.4101 0.4098 0.4094 0.4090 0.4086 0.4081 0.4076 0.4071 0.4066 0.4060 0.4054 0.4048 0.4041 0.4034 0.4026 0.4019 f).4011 0.4002 0.3993 0.3984 0.3974 0.3964 0.3953 0.3942 0.3931 0.3918 0.3905 0.3892 0.3878 0.3863 0.3847 0.3830 f).3812 0.3792 0.3771 0.3748 0.3709 1.0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0001 1.0002 1.0003 0.9976

TUBE LENGTH 10.0 MM, DISTANCE OF ELECTRON BEAM FROM TUBE EXIT PLANE

o.

MM TUBE LENGTH/DIAMETER 1.00 DOWNSTREAM-TO-UPSTREAM DENSITY RATIO 0 REL,

o.

0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.56 0.58 0.60 0.62 0.64 0.66 0.68 0.70 0.72 0.74 0.76 0.78 0.80 0.82 0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.00 D 0.2321 0.2321 0.2321 0.2320 0.2319 0.2318 0.2317 0.2315 0.2313 0.2311 0.2308 0.2305 0.2302 0.2299 0.2295 0.2292 0.2287 0.2283 0.2278 0.2273 0.2268 0.2262 0.2256 0.2249 0.2243 0.2236 0.2228 0.2220 0.2212 0.2204 0.2195 0.2185 0.2176 0.2165 0.2155 0.2143 0.2131 0.2119 0.2106 0.2092 0.2078 0.2063 0.2047 0.2030 0.2012 0.1992 0.1971 0.1949 0.1923 0.1878 0.05 MEN S 0.2705 0.2705 0.2705 0.2704 0.2703 0.2702 0.2701 0.2699 0.2697 0.2695 0.2693 0.2690 0.2687 0.2684 0.2681 0.2677 0.2673 0.2669 0.2664 0.2659 0.2654 0.2649 0.2643 0.2637 0.2631 0.2624 0.2617 0.2609 0.2602 0.2594 0.2585 0.2576 0.2567 0.2557 0.2547 0.2536 0.2525 0.2513 0.2501 0.2488 0.2474 0.2460 0.2444 0.2428 0.2411 0.2393 0.2373 0.2352 0.2327 0.2281 0.10

o

N LES S 0.3089 0.3089 0.3089 0.3088 0.3087 0.3086 0.3085 0.3084 0.3082 0.3080 0.3077 0.3075 0.3072 0.3069 0.3066 0.3062 0.3059 0.3055 0.3050 0.3046 0.3041 0.3036 0.3030 0.3025 0.3018 0.3012 0.3005 0.2998 0.2991 0.2983 0.2975 0.2967 0.2958 0.2949 0.2939 0.2929 0.2918 0.2907 0.2895 0.2883 0.2870 0.2857 0.2842 0.2827 0.2811 0.2793 0.2774 0.2754 0.2731 0.2684 33 0.15 0.20 DEN S I T I E S 0.3473 0.3473 0.3473 0.3472 0.3472 0.3471 0.3469 0.3468 0.3466 0.3464 0.3462 0.3460 0.3457 0.3454 0.3451 0.3448 0.3444 0.3440 0.3436 0.3432 0.3427 0.3423 0.3417 0.3412 0.3406 0.3400 0.3394 0.3387 0.3381 0.3373 0.3366 0.3358 0.3349 0.3341 0.3331 0.3322 0.3312 0.3301 0.3290 0.3279 0.3266 0.3253 0.3240 0.3225 0.3210 0.3194 0.3176 0.3157 0.3134 0.3087 0.3857 0.3857 0.3857 0.3856 0.3856 0.3855 0.3853 0.3852 0.3851 0.3849 0.3847 0.3844 0.3842 0.3839 0.3836 0.3833 0.3830 0.3826 0.3823 0.3818 0.3814 0.3810 0.3805 0.3800 0.3794 0.3789 0.3783 0.3776 0.3 770 0.3763 0.3756 0.3748 0.3741 0.3732 0.3724 0.3715 0.3705 0.3695 0.3685 0.3674 0.3662 0.3650 0.3637 0.3624 0.3609 0.3594 0.3577 0.3559 0.3538 0.3490 1.0 1.0001 1.0001 1.0001 1.0001 1.0001 1.0001 1.0001 1.0001 1.0001 1.0001 1.0001 1.0001 1.0001 1.0001 1.0001 1.0001 1.0001 1.0001 1.0001 1.0001 1.0001 1.0001 1.0001 1.0001 1.0001 1.0001 1.0001 1.0001 1.0001 1.0001 1.0001 1.0001 1.0001 1.0001 1.0001 1.0001 1.0001 1.0001 1.0001 1.0001 1.0001 1.0000 1.0000 1.0000 1.0000 1.0001 1.0001 1.0002 1.0000 0.9938

TUBE LENGTH 10.0 MMt DISTANCE OF ELECTRON BEAM FROM TUBE EXIT PLANE 0.75MM TUBE LENGTHIDIAMETER 0.25 JOWNSTREAM-TO-UPSTREAM DENSITY RATIO 0 REL

o.

0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.56 0.58 0.60 0.62 0.64 0.66 0.68 0.70 0.72 0.74 0.76 0.78 0.80 0.82 0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.00 D 0.3250 ·0.3249 0.3248 0.3247 0.3245 0.3242 0.3238 0.3234 0.3229 0.3223 0.3217 0.3210 0.3202 0.3193 0.3184 0.3174 0.3163 0.3151 0.3139 0.3125 0.3110 0.3095 0.3078 0.3061 0.3042 0.3022 0.3001 0.2978 0.2954 0.2928 0.2901 0.2872 0.2841 0.2808 0.2773 0.2735 0.2695 0.2651 0.2605 0.2554 0.2499 0.2439 0.2374 0.2302 0.2224 0.2139 0.2046 0.1946 0.1839 0.1727 0.1614 0.05 MEN 5 0.3587 0.3587 0.3586 0.3584 0.3582 0.3580 0.3576 0.3572 0.3567 0.3562 0.3556 0.3549 0.3542 0.3534 0.3525 0.3515 0.3505 0.3494 0.3482 0.3469 0.3455 0.3440 0.3424 0.3408 0.3390 0.3371 0.3351 0.3329 0.3306 0.3282 0.3256 0.3229 0.3199 0.3168 0.3134 0.3099 0.3060 0.3019 0.2974 0.2926 0.2874 0.2817 0.2755 0.2687 0.2613 0.2532 0.2444 0.2348 0.2247 0.2141 0.2033 0.10 0.15 0.20

o

N LES 5 DEN 5 I T I E 5 0.3925 0.3924 0.3924 0.3922 0.3920 0.3917 0.3914 0.3910 0.3906 0.3901 0.3895 0.3889 0.3882 0.3874 0.3866 0.3857 0.3847 0.3836 0.3825 0.3812 0.3799 0.3785 0.3771 0.3755 0.3738 0.3720 0.3701 0.3680 0.3659 0.3636 0.3611 0.3585 0.3557 0.3528 0.3496 0.3462 0.3425 0.3386 0.3344 0.3299 0.3249 0.3195 0.3136 0.3072 0.3002 0.2925 0.2841 0.2751 0.2655 0.2554 0.2452 0.4262 0.4262 0.4261 0.4260 0.4258 0.4255 0.4252 0.4249 0.4244 0.4240 0.4234 0.4228 0.4222 0.4214 0.4206 0.4198 0.4189 0.4179 0.4168 0.4156 0.4144 0.4131 0.4117 0.4102 0.4086 0.4069 0.4051 0.4031 0.4011 0.3989 0.3966 0.3941 0.3915 0.3887 0.3857 0.3825 0.3791 0.3754 0.3714 0.3671 0.3624 0.3573 0.3518 0.3457 0.3391 0.3318 0.3239 0.3154 0.3063 0.2968 0.2871 0.4600 0.4599 0.4599 0.4597 0.4596 0.4593 0.4590 0.4587 0.4583 0.4579 0.4573 0.4568 0.4562 0.4555 0.4547 0.4539 0.4530 0.4521 0.4511 0.4500 0.4488 0.4476 0.4463 0.4449 0.4434 0.4418 0.4401 0.4~82 0.4363 0.4343 0.4321 0.4298 0.4273 0.4247 0.4219 0.4188 0.4156 0.4121 0.4084 0.4043 0.3999 0.3951 0.3899 0.3842 0.3779 0.3711 0.3637 0.3556 0.3471 0.3382 0.3291 1.0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

TUBE LENGTH 10.0 MM, DISTANCE OF ELECTRON BEAM FROM TUBE EXIT PLANE 0.75MM TUBE LENGTHIDIAMETER 0.50 JOWNSTREAM-TO-UPSTREAM DENSITY RATIO 0 REL

o.

0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.56 0.58 0.60 0.62 0.64 0.66 0.68 0.70 0.72 0.74 0.76 0.78 0.80 0.82 0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.00 D 0.2636 0.2635 0.2635 0.2633 0.2631 0.2629 0.2626 0.2623 0.2619 0.2614 0.2609 0.2604 0.2598 0.2591 0.2584 0.2576 0.2567 0.2558 0.2548 0.2538 0.2527 0.2515 0.2502 0.2488 0.2474 0.2459 0.2442 0.2425 0.2407 0.2387 0.2367 0.2345 0.2321 0.2296 0.2269 0.2241 0.2210 0.2177 0.2141 0.2102 0.2059 0.2013 0.1962 0.1907 0.1846 0.1779 0.1705 0.1626 0.1541 0.1453 0.1362 0.05 MEN S 0.3004 0.3004 0.3003 0.3002 0.3000 0.2998 0.2995 0.2992 0.2988 0.2984 0.2979 0.2974 0.2968 0.2962 0.2955 0.2947 0.2939 0.2930 0.2921 0.2911 0.2900 0.2889 0.2877 0.2864 0.2850 0.2836 0.2820 0.2804 0.2786 0.2768 0.2748 0.2727 0.2705 0.2681 0.2656 0.2629 0.2599 0.2568 0.2534 0.2497 0.2456 0.2412 0.2364 0.2311 0.2253 0.2190 0.2120 0.2045 0.1964 0.1880 0.1794 35 0.10 0.15 0.20

o

N LES S DEN S I TIE S 0.3372 0.3372 0.3371 0.3370 0.3368 0.3366 0.3364 0.3361 0.3357 0.3353 0.3349 0.3344 0.3338 0.3332 0.3325 0.3318 0.3311 0.3302 0.3293 0.3284 0.3274 0.3263 0.3252 0.3239 0.3226 0.3213 0.3198 0.3183 0.3166 0.3149 0.3130 0.3110 0.3089 0.3067 0.3042 0.3017 0.2989 0.2959 0.2927 0.2892 0.2853 0.2812 0.2766 0.2716 0.2661 0.2601 0.2535 0.2463 0.2387 0.2307 0.2226 0.3740 0.3740 0.3739 0.3738 0.3737 0.3735 0.3732 0.3729 0.3726 0.3722 0.3718 0.3713 0.3708 0.3702 0.3696 0.3689 0.3·682 0.3674 0.3666 0.3657 0.3648 0.3637 0.3627 0.3615 0.3603 0.3590 0.3576 0.3561 0.3546 0.3529 0.3512 0.3493 0.3473 0.3452 0.3429 0.3405 0.3378 0.3350 0.3320 0.3286 0.3250 0.3211 0.3168 0.3121 0.3069 0.3012 0.2949 0.2882 0.2810 0.2735 0.2658 0.4108 0.4108 0.4108 0.4107 0.4105 0.4103 0.4101 0.4098 0.4095 0.4092 0.4088 0.4083 0.4078 0.4073 0.4067 0.4061 0.4054 0.4047 0.4039 0.4030 0.4021 0.4012 0.4001 0.3991 0.3979 0.3967 0.3954 0.3940 0.3925 0.3910 0.3893 0.3876 0.3857 0.3837 0.3816 0.3793 0.3768 0.3741 0.3713 0.3681 0.3648 0.3610 0.3570 0.3525 0.3477 0.3423 0.3364 0.3301 0.3233 0.3162 0.3090 1.0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1. 0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.00ÓO _I 1.00qO 1.0000 1.0000