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A SD-TDR- b3- 7542fPart lil

MOLECULAR FLOW AND THE EFFUSION PROCESS IN THEMEASURDIENT #01 VAPOR PRESSURES

Robert D. FreemanOklahoma Statc University

TECHNiICAL DOCUMIENTARY REPORT ASD--TDR-63-754, PL. IIINovember, 1967

This document has been approved for public releaseand sale; its distribution is unlimited.

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Air Force System-is CommandWright-Patterson Air Force Base, Ohio

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........... .. .......................

I• • I01 --... ... Dk~jk,%V

MOLECUL-AR FLOW AND THE EFFJSION PROCESS IN THEMEkSURIMENT OkE VAPOR PRESSU"S

Robert D. Freeman

This document has been approved for public releaseand sale; its distribution is unlimfted.

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FOREWORD

This report was prepared by the Research Foundation and theDepartment of Chemistry, Vkidhouma State University, Stillwater, Oklahoma,under USAF Contract AF 33(657)-8767, This contract was initiated underProject No. 7360, "The Chetnistry and Physics of Materials", Task No.736004, "Special Problems in Katerials Physics". The work was adminis-tered under the direction of the Air Force Materials Laboratory, Researchand Technology Division, with Mr. Paul W. Dimiduk, MAYT, as projectengineer. This report was submitted in August 1967 for publication.

This report is an account of the research accomplished betweenI September 1964 and 31 Niay 1967.

Thý author wislIcs to acknowledge the many contributions of thegraduate sutdents ai.d research associates who have been associated withthis research program, and who should be credited with co-authorship ofthe various scctiorb. They are E. A. Elphingstone (section V), R. E.Gebelt (sections IV and V), J, G. Edwards (section II), and Ruth C. Erbar(section III). The exccllent craftsmanship of the machinists and instrumentmakers in our deparcmental Machine and Instrument Shop has been invaluableznd is gratefully acknowledged.

This technical documentary report has been reviewed and is approved.

Chief

Thermo and Chemical Physics BranchMaterials Physics DivisionAir Force Materials Laboratory

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iItAi ABSTRACT

Our extensions to conical orificts of Clausing's analysis of angular

distribution of molecules effusing from cylindrical orifices has resulted

in numerical values for transmission probabilities and recoil-force correction

factors which are tabulated. With these results, it is demonstratcd that

the optimum orifice geometry for (1) recoil force measurements is a diverging

conical orifice with semi-apex angle of 30', (2) delivery of maximum fraction

of effusing molecules onto (or into) a target (aperture) is a long cylindrical

orifice.

Modifications to the angular distribution apparatus are described and

experimental results given for four orifices and two gases over the pressure

range 5 to 900 dyn/cm2. The most interestng aspect oi the results, one

apparently not previously noticed in angular distribution results, is the

presence of maxima and minima in plots of A vs. 0, where A is the (experimental

value - theoretical value) of Pe, the fraction of effusing molecules which

flow per steradian at angle 0 from the orifice axis. Vhese maxima and minima

have been correlated with the relative contribution from the orifice wall

to the total flux at angle 0.

Addltolual Lesuits for experimental transmission probabilitics of orifices

determined by the Multicell technique are generally in agreement with theo-

retical values within 2 to 5%.

The Riker technique for simultaneous determination of vapor pressure by

rate of effusion and by recoil force measurements has been refined to the point

that recoil force data are as repro'-cible as rate of effusion measurements.

Several sources of spurious recoil force have beer identified and eli.inated.

A new furnace and a ;aodificd automatic control system for the microbalance

are described.

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TABLE OF CONTENTS

Section

I. INTR DIITION . . . . . . . . . . . . . . . . . . . . . . . . . I

II. THEMRE•TICAL ANALYSIS OF MOLECULAR FLOW THROUGHCONICAL ORIFICES ............ ........................ 1

Ill. MEASUR1EMNT OF ANGULAR DISTRIBUTION OF MOLECULARFLOW THROUGH CONICAL ORIFICES ..... .............. .. 19

A. Experimental Apparatus ........ ................. ... 19

B. Experimental Results ............ .................. 24

C. Discussion ................ ....................... 33

IV. THE MULTICELL TECHNIQUE FOR EXPERIMENTAL DETERMINATIONOF TRANSMISSION PROBABILITIES FOR MOLECULAR FLOWTHROUGH CONICAL ORIFICES ............ ................. 57

V. THE MIKER TECHNIQUE .................... 67

A. Modifications to Apparatus ............ .............. 67

B. Experimental Results ........ .................. ... 71

VI. CALORIMETRIC STUDIES OF VAPORIZATION PROCESSES ... ....... 7I

REFERENCES ......................... ............................ 76

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ILIST OF ILLUSTRATIONS

Figure Page

1. Conical OrLifices and Parameters: A, DivergingConfiguration, T positive; Bi, Converging Con-figuration, T negative .................. ...................... 3

2. The Transmiseion Probability W as a functionof log (L/ro) at various values of T ..... .................. 7

3. The Recoil Force Correction f as a functionof log (L/ro) at various values of T ...... .................

4. Collimating effect of various orifice geometrieson effusing molecules ............. ...................... ... 4

5. CollmatLing effect of various orifice geometrieson effusing molecules ............. ...................... .. 15

6. Optimization of orifice geometry for recoil force ..... ......... 17

7. Diagram of the Experimental Apparatus ....... ............... 21

8. Experimental angular distribution data(Normalized beam intensity) for Nitrogenthrou h orifice 1 (T = 25.650 and

(1/,, •.....)......................... 2)

9. Ex erimertal [1(e)] and Normalized TheoreticalLQnCoso] Angular Distribution Data for T - 0.0O',L/rm , 2.44 ................... ........................... .. 27

10. Experimental [I(G)] and Normalized TheoreticalLQnCose] Angular Distribution Data for T - 30',L/rm - 2.C, with Nitrogen ........... .................... .. 28

Normalized Experimental LI(C)] and Thuoretical [QnCos0]Angular Distribution Data for Nitrous Oxide throughOrifice 2 (T - 8.5000, L/rm 1 10.08) ..... ............... 29

12. Normalized Experimental [I(e)] and Theoretical EQnCOSO]An ular Distribution Data for Nitrogen through Orifice5 (T -58.93°, L/rm - 11.01) .................. 30

vI

Figure Page

15. Norm'alized Experimental [1(0) ] and Theoretical [QnCOsC ]Angular DistIi bmt iou Dat a for Nitrous Oxide thuoughOrifice 3 (T - -58.95', L/rit . .i.O.) . .......... . 31

), . Normalized ExperitIMntal [1(0) ] and Ti•eoretical [QnCos6]Angular Distribution Data for NiLrogen through Orifice1 (T - 25.050, L/rm , 4.010) ......... ....... 52

25. f(A) plots for 120 aerieE ..................... 38

10. 6 plots for 120 aeries . . . . . . . . . . . . . . . . . . . . . . . 39

17. f(L) plots for 230 series . . . . . . . . . . . . . . . . . . .. . 4o

18. f(L) plots for 320 SerIes .................. . 41

19. f(W) plots for 350 series ..................... 42

20. f(6) plots for 420 series . . . . . . . . . . . . . . . . . . . . . 43

21. 62 plot (cf. equation 2u) for Run 125. Comparewith Run ll25 in Figule, 5 ............... .................... 44

22. The Critical Angles and the Angular Ranges fora Conical Orifice ................. ........................ 45

25. A plot for K. C. Wang's angular distribution data 27

for orifice with Tf - 0.00 and L/ro - o.954 .... ............... 418

2h. A plot for K, C. Wang's angular distribution data2n

for orifice with T 0.0' and L/ro = 2.59 .... ............ 49

- ... J . I,,. .. .. OiL ...... 52

26. Normalized projected area of inside wall of Orifice 4 ......... ... 53

2'[. Niralied projected area of inside wall of Orifice 2.................54

28. Normalized projected area of inside wall of Orifice 3 ... ........ 55

29. Weight loss through orifices with various (L/D) ratiosvs. theoretical transmission probability ............ .. 62

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LIST OF TABLES

Tabble Page

1. Transmissioii rrobsbllitlcs and Recoil ForceCorecttouns for Conical Orifices ............. ................. 4

2. Orifices, Gases, and Pressurvs Used inAngular l)istribution Studied. .......... ................... ... 26

3. Cotapari.so ol Expcrimental and TheoreticalTiansmisiion PlobabilitiCs: Run '7 .......... ................ 61

l. Experimental TranEmission Probabilities forEight Cylindrical Orifices (Set ) ..................... 63

5. Experimental Transmission Probabilities forEight Cylindrical Orifices (Set II) .............. ............... 64

6. Experimental Transmission Probabilities forEight Conical Orifices (Set III) ......... ................. ... 65

7. Miker Data for Vaporization of Tin ....... ............... ..... 72

8. Identification of Major Comunercial Componentsof Apparatus ................. ........................... ... 76

vii

LIST OF S(Y!MOLSDe fined]

Symbol o n Page Definition or Explanation

AL 51. projection at angle e of arca of orifice exit.

A 5] projection at angle 8 of area of orifice entrance.

51 projection at angle 6 of orifice wall area visiblefrom molecular beam detector.

AE 51 projection at angle e of the area A (1 0; DL ); seep. 51.

A(1)0;D L) 51 area of overlap cf circles rTr 2 , and Trr1 pro-0jected at angle 6 onto the plane of 1717'

•~--o

a. 58 cross sectrona) area of ith orifice.

a 16 integration limit.n 2

a 11 cross sectional area of orifice entrance, •r

o 0

b 60 constant in equation for straight line.

b 16 integration limit,n

D 58 diameter of orifice.

dx differential of x; other differentials are listedunder the function differentiated,

exp designates experimental value.

F 18 force exerted by molecules impinging on targetsubtending angle y.

F 18 force exerted by all molecules impinging on target,o from an ideal orifice,

F(LI/D,T) 59 function of L/D and T.

f 2 recoil force correction factor.

4 recoil force correction factor for convergingconical orifice.

f 18 angular recoil force correction factor.

dfr 10 incremental recoil force correction factor,

f(A) 36 the fractional difference (Px-P )/_ .

gi 58 total mass effusing from cell in time T.

do 58 incremental effusing mass.

298 66 standard enthalpy change.

I 58 represents value of an integral.

33 measured ion current at angle e, ++2 33 relative molecular beam intensity, 1 /.+

6 e wi index.

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0. 58 mass effusing per u it area or ifice in t tine "'rI

159 mass effusing per unit area of M i Irice in Lime C I -

Ce 1St ed f uron I ensi t- squares I itc thirough Lcx p utIie topoi lnt s6

J .59 nlmass cllusintu per- unit a112a 01 ideal o)if ice int iic ".

K. 58 exp1rijental transmission probabi i t\.

K 59 experimenta 1 transmission probability for ideal0orifice.

Kp 60 experimental transmission probability, fromi experimental points.

K1 60 experimental transmission probability, fromleast-squaces line through experimental points.

K 60 experimeýntal transmiission probability, average.

L 2 length of orifice along its axis.

D 58 molecular weight.

rn 58 the number of cells in a mu Iticcll experitment

"1 60 constant ill equation for straight ine.

Ny 12 number of molecules which effuse per second into

a cone with semi-apex angle y

N 12 number of molecules which enter the orif ice through00

o ~aao per second.

NL 12 number of molecules whi ch effuseL from the orificeexit per second.

dN 19 number of molecules per second which pass throughthe incremental solid angle dL.

dN (L) 33 number of molecuILcs which effuse per second into-iic elid aI I e at:.

dNe (L) 33 number of me lc cu l s wihich effuse pIer second intothe solid angle I 2"' Sia3 A.

N, (L) 34 numbCr of molecules whichi effuse per second intothte finite, but small, solid angle 2.

p 58 pressure in effusioln CelI.

P 72 presoure iii e inus ion cell de ermined by effusionn1t aisureVnli, t .

P 72 plessu1e iD effusio( ceCI d et Crmi njd by r'ec0iforce measureme.Lnt.

35 probab i liiy density funt ion for effusing mo lec les.

35 P0 , cxperimenita 1.

i y,

I

p 35 P , thcolettical,

P 35 F6, experimenta I, at J =L1o6

Q 33 aL)gula1 dist ribut ion funct ion; Q =1 for idealor if ice

Qn 12 7Q .

12 Q with n1=

1. 58 gas constant

r 2 radius of int rance to orifice.

r 3 radius of exit from orifice.

r 4 radius of smaller end of orifice.m

T 2 angle between orifice axis and orifice wall,measured in a plane which contains the orificeaxis .

T)(0 10 Clausing's angular distribution function.

th designates theoretical value.

dt 58 differential of time.

W 2 transmission probability (theoretical).

W 12 angular transmission probability.

dW 12 incremental transmission probability.

y 12 angle.

A 36 the difference (P2-P t)

AH298 66 standard enthalpy change.

6,5 36 functions of P x and I1' 26 3 angle, especially the angle between a molecular

trajectory and the orifice axis.

o 58 temperture, absolute (section IV only).

6* 37 6*=arctan (r +rL)/.L; see Figu/e 1,

kao 33 molecular flux incident on orifice entrance.

1 33 index.

TT 3.14159

"" 58 interval of time.

oi 16 total recoil force for effusion from ideal orifice.

do 16 incremental recoil force.

V(x) I0 normalized molecular flux on orifice walls.

2 d 34 solid angle subtended at orifice by molecular beamdetector aperture.

subscript; refers to a solid angle.

& 12 incremental solid angle.

x

Si-CTION I

INTRODUCTION

The Knudsen effusion techniquei, 2 is widely used, in varied guises,

to obtain vapor and/or dissociation pressure data, especially at high

temperatures. In actual laboratory operation the conditions under which

effusion occurs are rarely, if ever, the ideal conditions assumed in

the derivation of the simple Knudsen equation. This report summarizes

work on several approaches designed to clarify the understanding of the

effusion process under non-ideal conditions. More detailed introductory

paragraphs are included in each of the following sections.

SECTION II

THEORETICAL ANALYSIS OF MOLECULAR FLOWTHROUGH CONICAL ORIFICES

Vapor pressures and the composition of vapors at high temperatures

are oftpn determined by effusion techniques 3, e.g., Knudsen effusion,

torsion-Knudsen-effusion-recoil, and target-collection methods; furthermore,

mass spectrometry of high temperature vapors is often facilitated by the

use of an effusion cell as the vapor source 3 . Derivation of thermodynam-

ically significant data from the results of effusion experiments requires

knowledge of the relation between the measured quantities and the equilib-

rium pressure in the cell; this relation must take into account, among

other factors, the non-ideality 4 of the geometry of physically-realizable

effusion orifices.

Tihe effect of o .:ifice ge y',';h rate. Of -oleCUlZI effusion

throngh the orifice has bccu analyzed by several investigaLors5- 0 , whose.-r'sults have usually been given as nmoerical values for the transmissionI

probability, i.e., the probability that. a molecule which enters the orifice

through one end will exit from the opposite end. The effect of orifice

geometry upon the angular distribution of molecules effusing from an

orifice has received less attention but an analysist m for cylindrical

orifices has been made and the results have been used to obtain the

"recoil-force correction factors" applicable to torsion-Knudsen-effusion-

recoil measurements 3 .

In Part I12 of this report and in a conLribution to "Condensation and

Evaporation of Solids"'13 (subseqoLntly cited as CES) we hive presented

an anialysis of the flow of rarefied gabes through and from conical orifices.

Equations were derived for calculation of the incident molecular flun:

density along the orifice walls, the transmission probability of the orifice,•

the angular uistribution of molecules leaving the orifice, and various

other functions. In Part I, but not in CES, the results of extensive com-

putatioas of the various functions were reported; for convenience, one

table and two Figures from Part I are repeated herein, and the results

discussed briefly. Subsequently, these results are compared with other

theoretical results; comparison with experimental results is deferred

to Section III C and IV of this report.

In the last portion of this Section we shall exploit the angular

distribution functions for conical orifices, which have not previously

been available, to consider the determination of an optimum orifice geometry

(within the class of right circular cones) for particular experimental

configurations.

A. RESULTS OF THE NUMERICAL CALCULATIONS

in Figure 1 the various critical orifice parameters are defined by

illustration. Values of the transmission probability E, and the recoil-

force correction factor f are given in Table 1, and in Figures 2 and 3,

respectively, for conical orifices with O(L_/ro)f1O and with -90!CT3+O9,

i.e., for both converging (_1 negative) &nd diverging (T positive) conical

orifices. Transmission probabilities are given only for the diverging

2

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rL FLOW

Figure 2. Conical Orifices and ParamcLcrs: A, DivcrgIngConfiguration,, T positive; BConvc-rgivtg Conliguration,,

T negative.

TABLE I

TRANSMISSION PROBABILITIES AND RECOIL FORCE CORRECTIONS

FOR CONICAL ORIFICES

T (L/r ) a w f (f*/r 2)am _ _ __ _ _m

0 0,1 0,952399 0.9683220.2 0.909215 0.937308

0.4 0.82408 0.87847

0.6 0.77115 0.82471

0,8 0.71778 0.77620

1.0 0,67198 0,732692,0 0.5142 0.57254.0 0,3566 0.4024

6.0 0.2754 0,31258.0 0.2253 0.2564

10.0 0.1909 0.2177

100 0.1 0.967347 0.986104 0,9803300.2 0,93835 0.97200 0.961270,4 0.88938 0.94451 0.92584

0.6 0.84990 0.91904 0.89445

0.8 0.8176 0.8960 0.86701.0 0.7908 0.8756 0.8432

.20 0.7058 0.8035 0.7621

4. 0, C 0.730- 01.6899

6.0 0.6051 0.7107 0.65898.0 0,5895 0.6970 0.6425

10.0 0.5802 0,6892 0.6328

200 0.1 0.978646 0,999546 0.988540

0.2 0.96027 0.99816 0.97764

0.4 0.93057 0.99407 0.95804

0.6 0.90793 0,98941 0.94150

0.8 0.89034 0.98492 0.92770

1.0 0.87642 0.98087 0.91625

2.0 0.8370 0.9682 0.8813

4.0 0.8108 0.9633 0.8564

6.0 0.8022 0.9660 0.8480

8.0 0.7984 0.9699 0.8442

10.0 0.7963 0.9736 0.C421

14• • n m • •

T (_L/r_) a _(_ f/r_ 2)a

Im30 0,1 0.986915 1.008954 0.9680,2 0.976141 1 .016307 0.9881980.4 0.95973 1 .02782 0.978480.6 0.94812 1 .03652 0.970790.8 0.9396 1.0434 0.96488.0 0.93338 1 .04919 0.96004

12.0 0.9177 1.0687 0.94734.0 0.9095 1.0908 0.94016.0 0.9073 1.1044 0.93818.0 0.97065 1..1140 0.9373

10.0 0.9060 1.01212 0.9369

40 0.1 0.992680 1.014589 0.9970830.2 0.987008 1.026941 0.9944940.4 0.97902 1.04683 0.99034.06 0.97389 1.06220 0.98732.0 0.97044 1 .07444 0.985131 0 ( 0.96806 1 0844 6 0.98352

2.0 0.96288 1 11632 0.97974440 0.9607 1.1462 0997806.0 0.990381 .1618 0.97768.0 0.9601 1,1720 0.9775i0.0 0.9599 1,1793 0,9774

45° 0.1 0. 994775 1 .016077 0. 9981040. 2 0. 990881 1 .029628 0. 9961.o80.4 0.985662 1 .051287 0.9939490.6 0.98249 1.06782 0.992210.8 0.98047 1 .08086 0.99101i .( 0' f'-;912 I .0914 3 (j. , e o 12.0 0.9764 1.1242 0.98834.0 0.9754 1.1536 0.98756.0 0.9752 1.1664 0.98748.0 0.9751 1.i778 0.9873

10.0 0.9750 1.184 4 0.9873

50° 0.1 0.990420 1 .016729 0.9988310.2 0.993877 1 .030084 0.9978550.4 0.99060 1 .05267 0.996420.6 0.98863 1 .0t918 0.995490.8 0.9d772 1 .(08202 0.9q4t w1.0 0.98701 1 .0i2 31 0.99/4472.0 0.98570 1.12355 0.993634.0 0.9,353 1I .1504 0.99336.0 0.98',2 1 . 1636 0,99338.0 0.9851 1 .1717 0.9932

10.0 0.9,b l 1. , 74 0. 9 . 2

T1 (L/rm)a m ,(U/r 2)

600 0,1 0.996591 1.015712 0.99913440.2 0.997715 1.028324 0.999374

0.4 0.996761 10.47307 0.9990260.6 0.99L304 1.060889 0.998831

0.8 0.9960u 1.07108 0.998731.0 0.99592 1.07904 0 .9866

2.0 0.99571 1.10207 0.998544.0 0.9957 1.1205 0.99856.0 0.9957 1.1289 0.99858.0 0.9956 1.1338 0.9985

10.0 0.9956 1.1372 0.9985

700 0.1 0.999631 1.011961 0.999936

0.2 0.999450 1.020864 0.9998950.4 0.999296 1.033229 0.999853

0.6 0.99924 1 .C'4141 0.99983

0.8 0.99921 1,04724 0.99982

1.0 0.99920 1.05160 0,99982

2.0 0.9992 1.0b35 0.99984.0 0.9992 1.0721 0.9998

800 0.1 0.999966 1.006048 0.999999

0.2 0.999957 1.009755 0.9999970.4 0.999951 1.014072 0.9999960.6 0.999Y5 1.01051 1.000000.8 0.99995 1.01809 1.00000

1.0 0.99995 1.01919 1.00000

2.0 1.00O 1.0219 1.0000

4.0 1.0000 1.0236 1.0000

a. r = radius of smaller end of orifice. (L/r m2) i8 the appropriate

recoil force correction for Convergin orifices when (L/!) for the

orifice is calculated with r rather than ro (see Figure 1 and

references 12 and 11).

d

1.0 -- 60°

40

0.9- 3Q0

250w

220

0.8 200

0.7 150

-1,0 OX) 1.0Log (L/ro)

Figuie 2. The Transmission Piobability W as a fun'ti-,n oflog (L/ro) at various values of 1'. 7i

U

1.20450

5Q0

7

115 /����600

/// 7300

HO /7 700

250

1.05

� 220

[00

200

0.95

0.90

� 50 100-1.0 0 1.0

Log (L/r0)

I igure �. The Rec�i1 Force Crrecti �n a� a func tio1 �g (i/�>) at various vaJ.u�s oL T.

8

configuration because, for a given o0. i flCe geomt ry, W is thc iwme fox gas

flow in either directions: 14. The loguritLULic scale on ihc abscissa ofFigures 2 and 3 is ,sed only to siwplify picscntaticn of the results.

Table ) and Figure 2 reveal no unexpected va):, atiOi of W; it decreases

with increasing / and increases ,s LI Increases, as would be predicted

intuitively. However, Figure 3 reveals an unexpected variation of f with

T: for T 5 L2'° f is ,e~ate, than unitv. Hence, the recoil force generated

by molecular effusion irow a conical orifice with T > 220 is greater than

that from the corresponding 4 ideal orifice. This result arises because

(W) the average angle at which molecules effuse from conical orifices is

smaller than for the idt:al oritice, and (2) as T increases above 200, the

transmissloni probability rapidly approaches unity; the momental component

along the coUical olifice axis is therefore greater than that along the

ideal orifice axis and the recoil force is greater. This "focusing"

effect of coi:ical orifices is a uiaximupm at -- %°, the precis-- 'alue depending

Oi (!jrn/':o).

NuraeriVel values for various other quantities, e.g., the incident

molecular flux on the erifice wall and the angular distribution of effusing

molecules, have been calculated but are not presented here; an extensive

tabulation of these functions is available elsewhere 4.

B. COWPARISON WITH OII UR THEORETICAL RESULTS

It Was shown il, Part I and in CE'S that, where comparison was possible,

the rCsults presentcd were in agreement with those of other workers. In

particular, our r t..i.a are in excellent agreement with the transmission

probabilities of Iczbowski, et a].ý' for conical orifices, ar'd witli the

closely-bounded values for the transmission probabilities o,- cylindrick.l

orifices obtained by DeMarcus and Hlopper"a. In view of the different

approaheL, the independent verification et derivations, and the :onsider-

ably different numerical techniques used ii solving the integral eqaations,

by four groupsf 5 , P0, 1 , 14 there can be little doubt of the validity of

the numerical values for the transmi.sion probability for cylindrical and

conical orifices; these values apply rigorously only to the assumed model,

of course.

We know of no o.h. r nlys. s ol. zL11iiar distribution to- conical

orilices, and for cylindrical orifi-,,s only of ClaIsiUg's analysis-".

Clausing assumed that Lhe normalized incident density (or molecular flux)

on the walls of a cylindrical orifice could be expressed as a linear

function of the distance, along the orifice axis, from the entrance.

The accuracy of this approximation has been discussed by Edwardslo whe

has also shown that, with this spproximnation, Clausing 's distribution

functionas ~i T(O) may be derived in detail directly irom our more general

distribution junctions which apply to conical and cylindrical orifices;

it has also been possible to integrate in closed form the resulting

angular distribution equation foi __yjlindrical orifices.

One further argument may be advanced for the validit.y of our angular

distribution results. In solving the various equations 3 the first quantity

obtained is 4(x) the normalized incident density on the orifice walls;

these values of '(x) are then used to obtain W4 in a direct way. The same

values of ,(x) are used in the more Involved computation of k' by numerical

integration of the angular distribution functions. The two values of W

are in good agreement with each other and, as already noted, with tbC

values of Iczkowski, et al.6 which would seem to indicate that both ý(x)

and the angular distribution functions are correct.

It is difficult to compare our results with those of Davis, et al. 1X;

their paper lacks detail in both derivatt•.. and results. Our impiiession

is that their derivation is not rigorous. In any case the. re-.ulto shown

in their Figure 7 for the transmission probability of "ceonical nozzles"

appear to be higher than those of Table I by several percent for small

angles T and lower for large angles T.

Sparroo and Jonssorn•3 have also analyzed the mass flow through conical

orifices and have used the results to formulate analysis of energy transfer

between gas Oad orifice walls. If, in their equation (12) aad the ordinate

of their Figure 5, 12 is made zero, the resulting term is equivalrint to

our transmission probability W. The results in their Figure 5 appear to

be in good agreement with values of L% in 'fable 1, and !(/i') of their

figure (2) appear to agree with out •(x) 2. Sparrow and JOhkcSon obtain

a number of useful but seldom used relations (theii equations 6-11), VOse

of which we have a.so generated'. 3'-.

10

II

Richley and colleagues hav. iucently reported a sceies of analyses

of jolcul ar fow Hthrough fyl indri -a1 tubes "a C -converging and divergingjiob d4.tubes- aud slotsb, s and zbrHoadh cylindri cal iubys .iL &: contributIon

f from surfice diffusionwcV The investigation by Cook and kiehley of

angular distribution from cylindrical oriliecs is based otn Clausing 'S

analysis 1 0 which has already been ,orrr.latcd with the present results.

in their analysis 9b of flow through conve,:gieng anf diverging tubes and

slots, Riebhhy and Iceynolds obtain, by 'tterative solution of the appro-

priate Fredholm Integral equation, values for the normalized incident

density 4(x) on the orifice wall ( their f"lux ratio D-/a') and use these

values to obtain -lie flux distribut.ion evcer the plane of At! exit end of

the orifice; finally, thI exit plane flax distribution is integrated over

the exit area of the orifice to obtain the traunmiasuien probability E

"their Pt), The minor discrepancies which exist between their results

and oLajs appear to arise from (i) their use of an iterative solution for

r:.,/n. raLlier tItan cur 3,more di rec-t soltý eon 3, 14 for 4(x), and (2) their

introduction of an addiLional numerical integration (i.e., to evaluate

the exit plane flux distribution) in dhe camaputational sequence leading

to W, rather than calculation of U4 ditrectly from the 4(x) in one step1 3

C. OI'TIMIZATTON OF ORIFICE GEOMLETRY

An orifice geometry will be considered optimnem if, for any given rare~~~~~•.-1 -- .......... tIF r s eC t t )

of effuslon, the quantity. ben ".asnred is L,,it. ze wh respect to the

orifice paranetersT and (24"ro) ; the effusion rate tay. always be adjusted,

if necessary, by varying the orifice area a( while maintait•ing a fixed

geometry, ie.,, fixed values of T- and (L/re). This criterion for Optiin.

zation is directly applicable when any one of the following typical con-

dftion"s exist: (a) a very small. amount of sample is available; (b) intro-

duction of new sample into the apparatus requlres ýa comparatively long

down-time (as in mass spectrometry); (c) the materil2 itde-r study vaporlizes

incongruently and vaporization chiaracuerisi.c1s change with the compositrion

of the condensed phase; and (d) a low evap'oration coefficient i'posesC the

need to minimize the total flux from the cell, thereby minimizing the dis-

-lacereant of the actual pressure from the equilibrium value. Under each of

[I

these conditions it is clearly desirable to maximize the measured quantity

while simultaneously minimizing the total rate of effusion of sample from

the cell.

We shall now determine Lhad orifike gComleltry which maximizes each of

the following: (1) the near-axial flux density of the effusing molecular

stream; (2) the r~coil force generated on the effusion cell; and (5) the

force exerted on a targetU suspended in the effusing gas stream.

1. •timit.on of Moleculas Beam Intensity on an4d ZCr the Orifice

Axi s. In Part I1 and CFS S3 we obtainee cxprcz:hons for what we shall

here call the irremcntal transmission probability LWO, _.e., the

probability that a molecule which enters one end of the orifice will

exit from the opposite end into the incremental solid angle & located

at angle 0 from the orifice axis. With the solid angle ..w expressed in

spherical coordinates and the assumption of circular s}nmetry in the

distribution around the orifice axis, the equations may be wiitten as

dW0 = 2Q. Sine CosO dO; (a)

the n (n = 1,2,)) are complicated functions of the orifice parameters

L, (L/2o), and, depending on the range in which G lies, also of e.

The integral of dWe over 0 < I nr/2 is just the transmission probability

Y (designated W0 in Part I and CES to distinguish a calculation from

angular distribution considerations).

We now define the quantity Wy,

w • 0 0 2N% Sine Cose dO, (2)

which may be called the angular transmission probability, i.e., it is

the ratio of the number Ny of molecules which effuse per second into

a ccne which is coaxial with the orifice and has semi-apex angle h',

to the number No of molecules which enter the orifice through ro per

second.

The quantity to be maximized is the ratio of N- to the total

number NL of molecules which effuse from the orifice, i.e., NyiN1 a

and from equation (2),

w1/w - (2/W) 0 Q SinG Coso d. (5)

If we restrict Y to Range I1:' 4 (i.e., to 0 : Y < JTJ), %n beccnues 9a

12

and is independent of 0; equation (5) may be now integrated to obtain

W-Y/W = (Q2/w) Siw• 1 ; o 0 ITI. IQuite obviously, the fraction of effusing molecules which flow into

the cone defined by the orifice axis L. h aitile y can be varied by

varying y; the pertinent problem is to maximize that fraction for a

give" y determined by a particular experimental apparatus. Considering

then that Y is fixed, we re-write equation (4) as

Wy/(W Sin- Y) = (Q3/W), (o0 ! Y IT 1), (5)

and note that the quantity (QQ2 W) now to be maximized is a function

solely of orifice geometry.

A plot, prepared from data tabulated elsewhere1 4 , of (QVW) vs. T

for various (L_/_ro) is given in Figure 4. It is readily apparent that

a maximum exists at T Q 0' for each (.L/ro) and that the

maximum becomes sharper as (L/to) increases. We now note that under

the restriction 0 ! Y ý 1.T1, when T = 0', can have only the value

0°; hence, the points in Figure 4 for T = 0* apply only to the beam

intensity alS the orifice axis where Y = 0%

To determine whether W-Y/(W Sin2 •Y) is also a maximum at T = 0'

when )' has typical experimental values of 5' and 10', values of _WY

for cylindrical orifices (T = 0°) have been calculated. in effect by

integration of equation (2), but in fact from a closed-form solution1 4' 15.

The results, plotted as WT/(L_ Sin y), are in Figure 5 in which the

curves tor T - -i0' and for T Ž +10' duplicate those of Figure 4

since in these ranges (N • ITI).

It is evident in Figure 5 that for y = 5', _ Sin2 Y) is

larger for T - 00 than for any T Ž 50; similarly, for y = 10'

(_Y/W Sin2 Y) is larger for T = 0' than for any T - 100. It appears

to be true in general (although we have not made extensive computations'0confirm , "a s- e HT /_ N -- A -Itfy Y, Lj-2 A

L~lUJ LfIO VL a O U'lIu a 6 VCLt Y, ý= /. ýLL T j

is larger for T 00 than is (QVW) for any T Ž_! y.

For a short ori.fice (L_/ro < 2.0C and a given y, the two quantities

(_WE)T=- and (_W,/W Sin2 Y)T0 are tie same within 1i; for longer

orifices the difference is somewhat greater.

13

\U

i i I I I I I "* I *-******1

L/r'm ifIO0

10

4.0

6.0Q,W

3.0--4.0

i/- 0.2

-80 -60 -40 -20 0 20 40 60 80

T, deg

Figure 4. Collimating effect of various orifice geometries oneffusing molecules.

14|

U

'I I ! I I

VALUE Al T0"FOR y 5 0 '

50-- VALUE AT T=0 050-100° 10° FOR 1"&-00°

L-/rm:

.50-

100-40

80-

w

W Sin0T L/rm-

10060-80--

30 -8.0

60--

40-

,40-- 6

2 0 I4 o

-80 -60 -40 -20 0 20 40 60 80

"l)deg

Figure 5. Collimating effect of various orificu geometriesor effusing molecules.

15

Thcrc rcmai-ns the queLiuit vE ihe relative value of 04y/W Sin- y)

for T - 00 and for some T < y, e.g., for T - 0' and T 50 with Y - 10'.

For these cases (Wy/_ Sinp- y) is larger for the conical orifice

(T > O) and increases slowly with increasing T.

2. O•timization of the Effusive Recoil Force

In CES3 (equation 19) it was demoiustrated that the "incremental

recoil-force correction factor" dfo (designated dFU in CES), (i.e.,

the recoil force !• gerierat, A on a cell by effusion of molecules into

the incremental solid angle d at angle e from the orifice axis,

divided by the total recoil force 0i for the corresponding 4 ideal

orifice: dfr = do/oi) is related to dWo by

dfo = (3/2) CosG dWo

or, with equation (I),

df 0 = 3Qn SinG Cos 2 0 do. (6)

The (vttaWl,) xecoil force cr~reotion -factor";-9 is then given :by='n/2 3 bn

f = Y6=0 dfo = n~l f ar 3Qn Sine Cos 2 6 do; (7)

The integration limits (anbn) are discussed in CES and reference 14.

The factor f is, of course, the quantity tabulated in Table 1.

Optimum orifice geometry in this situation requires maximization

of recoil force for any given rate of effusion, which is equivalent

to maximizing the ratio (f/v). In Figure 6 this ratio is plotted vs.

_T for 'Vtiuus (".! l It is evident from Figures 2 and 3 that a maximum

must occur in a plot of (4/W) vs. T. However, it is rather surprising

that this maximum occurs at, or very near, T - +30' for a very wide

range of !1ro. It is quite clear from Figure 6 that the optimum

orifice for recoil force measurements would have a semi-apex angle T

of +30' (i.e., diverging) and would be as long as practicable (within

the range of 1/1o covered in Figure 6).

3. •£utuization of the Force Exerted on a Target in an Effusing

boLecular Beam: We BsbaL restrict our consideration to cases in

which the target in circular and coaxial with the orifice, and in

which the molecules striking the target either all condense or all

16

12 { " I I q i i I I |

L/rm 100- O

60/

1 20 -

20116

f

1.0

1 08-

9''f I

-60 -40 -20 0 20 40 60 80

T, de g

Figure 6. Optimization of orifice geometry for recoil force

Figure

I

rcvvaporize. The angle subtended at the •rifice by the target is

designatcd 2)'; ) then has the same meaning as in equation 2.

If all molecules condense on tle target, the ratio of the force

exerLed on the target to the force FO which would be exerted by

all wolecule. effusing from the corresponding4 ideal orifice, is given

by equation ('I) with integration over 0 < 0 • y:

S= FY/FO fy 3Qn SinQ Cos 2 0 do. (8)

It Y is restricted to 0 Y y • jT so that only (1 (- constant) is

required in the integration, equation (8) becomes

fy - Q1(1 - Cos-y).

We wish to maximize this force at a given flow rate and for a given Y;

therefore, we write

f" .

The quantity to be maximized is Q2H, as in subsection 1 above, and

the arguments given there apply.

If the molecules revaporize from the target, there will be

exerted on the targct an additional force which, with all extra-

orifice parameters fixed, will depend on the molecular flux onto the

target 3 . Maximization of this flux at a given total flux from the

orifice was the subject of aubsectlon 1 above; again, *•W is the

quantity to be maximized.

SECTION III

MEASLUREI.ý.NT OF ANGULARDISTRIBUTION OF MOLECULAR FLOW

THROUGH CONICAL ORIFICES

Of the various quantities which arc derived in the theoretical anal-

ysis 12. 1314 (see Section II) and which are amenable tv experimental study,

the most critical is the angular distribution of molecules effusing froman orifice, i.e., the variation with 0 of the number dN• of molecules which

pass per second from the orifice into the incremental solid angle dX located

at angle e from the orifice axis. Thie theoretical analysis predicts that

Sis proportional to Rn cos 0; _Q is the complicated function of orifice

parameters which arises from the non-ideality of the orifice (for the ideal

orifice, _Q, is always unity). Measurement of dN or an equivalent quantity

would provide experimental data which could be compared directly with the-

oretical values for 2n cos 0.

A. E)O?.RIMENTAL APPARATUS

To accomplish these measurements the apparatus described briefly in

this section (and in detail in Part 1Ii1) has been constructed. It is

designed to allow a study of the effusicn of a permanent gas (e.g., N2 ,

lie, CO,2 ) at any suitable pressure from any orifice with a geometry which

can be machined into a small circular plate. Permanent gases are used

as effusants So that the apparatus cani loe operated at room temperature;

concern that the reservoir-orifice systm may not be isothermal is there-

by minimized. For this advantage the ability to study the effusion process

as a function of temperature is sacrificed.

The angular distribution of effusinig molecules is determined by a

molecular beam method 2 2 incorporating a modulated beam technique2 -3 . The

reservoir from which the molecules effoac can be rotated on an axis which

4 -

passes through, and is parallel to, the outer face of the orifice under

study. Two stationary collimating orifices and the effusion orifice under

study define a molecular beam, the beam is modulated by a mechantcal chopper,

and its intenbity is determined by a neutral-beam detector.

The apparatus, a diagrammatic horizontal cross-section of which is

shoun in Figure 7(, consists of five principal components: (i) The Main

Vacum Chamber, which can be maintained at a pressure very luw wiLh respect

to the y-ressure in the gas reservoir, and wunich contains the rotating

effusing cell; (2) The Gas Reservoir, a large chamber from whiclh gas

flows to the effusion cell and in which the pressure can be kept constant;

(3) The Buffer Chamber, a small indCpe.ndenUtly-pumped chambLr which is

separated from the main chamber by a plate containing the first collimating

orifice, and which contains a chopper capable of interrupting the beam

about one hundred times per second; (4) The Detector Chamher, an independ-

ently-pumped volume which is separated from the buffer chamber by a plate

containing the second collimating orifice, and which contains an election-

impact molecular beam detector; (5) The Detector Electronics, which con-

sists of a power supply for the beam ionizer and a system to amplify,

measure, and record the ion current from the ion collector.

Each of these components was described in detail in Part I121; we shall

describe here only significant modifications.

1. Modulation of Mocular Beam: As experimentel techniques were

refined, it became apparent that the frequency of the mechanical

beam chopper (Figure 5, Part II) was not stable; the cause was

found to he an overloaded driving motor. When the motor was re-

placed by one with higher torque, a new motor mounting and chopper

housing (items 6 an,. 32, Figure 5, Part II) was fabricated from

stainless steel (rathel than brass). The new drive mechanism per-

formed satisfactorily at times, but at others exhibited erratic

chopping frequency. This difficulty arose from binding in the

bearings which support the choppel" shaft; the binding in turn

was caused by a slight misalignment of the new motor mount/chopper

housing when it was welded into the vacuum system.

At this point an electronically-driven tuning fork chopper

20

!

I

/-'8 |1'

17

16 L--JL----J Lm--- 7 ,-tJLw A/ /I1 13 1 r'04

2

16o~0 3 6 9 12 .. 1

SCALE -INCHES ____-

II

1. The Main Vacuum Chamber2. Ihc Gas Reservoir5. The Pluftcr ChamberL The DetecLor Chamber5. The Rotating Effusion cc1lU. Thi Beam Collimating Orifices

. ht: Beani Chopper8. The Bcam Ionizer9. Connectors for Tygon lubulation Whichl Carries Gas

from the Rcservoir to the Rotating Cell1'. Pumping ,1],'s Which Accommodate Liquid NitLrogen Traps11. Class Windows12. Rotary Vacuum Seal Through Wbich the Cell is Rotated15. Brass Bellows211. Clobe ValveI5' Copper pe , iameter1(t. Port for Attaching 5-Liter Stairtles Steel lank

17, Valve for Introducing Eifusant, GasJ,3. Connections to the Lqt.uibar Prtssure Mctei

Figu-re 7. Diagram of thc Lxpcrimental Appaiatus-

g2[

(type Wo) was obtained. atorianl frequency of the chopper is 320 ttz;

the chopping vanes attached to the tines are 10 mm high and have a

maximum aperture of 8-10 mm. An electrenic signal synchronized with

the mechanical oscillations of the tuning fork Is an inherent feature

of the driving circuit, and is readily available for use as reference

signal to the lock-in amplifier.

The new chopper was mounted in the BUFFER CHAMBER (hIgure 7),

buL directly o.n the flange which is welded onto the MAIN VACUUM CHAMBER

and which mates with the BUFFER CILAMBER flange. A new BUFFER CHAMBER

equipped with aluminum-foil-scaled flanges 21 was fabricated from

stainless steel.

No particular problems have been encountered wit00 the tunivg

fork chopper; it was used in obtai.ning essentially - the data

reported in Section B.

c2, Detector ChOsber: The Dctector Chamber descr'ibed in Part II

opercated sat.istactorily, except that the uliath•.e piessure was rel-

atively high ( L1&C torr). Conbequently, the life of the cathode

was seriously shortened and the emission current available from the

cathode was low. A new chamber was fabricated from stainless steel;

aluminum foil flanges"'' were used and all cla,;tomer 0-rings were

eliminated; a teore efficient liquid nitrogen trap ("Cryosorb") was

inserted betw.,een the 2" diffusion pump and the chamber.

"Aftrar a s!'art baec-ut at. I002--c' C, the Ce:w DatoCotor Cham1e-r

caes 5x1&" torr azd operates at 5xI0-7 torr vrith a bears, from

a 0.1 tc.irr source, entering the chamber.

The basic beam detector design and electronics have undergon:e

no significant change. In the improved vacuum system the beam

det!ct-.or hLas performed quite 6,atisfactorily; cathode life is iemark-

ebly ]engthened (ne qtiantitative data, as yet) and the emvission

current Is increased by a factor of live.

3. Valve,- The sensitivity of the bena detector cathode to oxygen

made it desirable to keep the detec(tor chamber evactuated coettlruouSl.y.

To achieve isolation of the Detcc; (.and But icr) Chamber whO le the

Main Chamber was opened, c.g., to change rcifices, a 0sii3ng, r (ri nmg

0')

seaIed valve wab instaliled in. the Main Chamber in such a way that

closu•e of the valve seaýled tl,h first collimating orifice from the

Main Cham-ber. The design of the valve was adapted from that of

Shecffieldut, It has worked satisfactorily.

)1. Alli•mont of Orifices, The apparatus was designed and constructed

to petro.it di.rect, visual alignment of the beam defining orifices,

if the Firad•ay ilon collector is removed from the beam detector

assembly. Aligrnent of the effusion orifice and the various colli-

matLing orifices is rather easilly accomplished by viewing with a low-

powered telescope along the beam axis. However, with only this check

on alignment it is possibhi foi the plane of a given orifice to be

tilted appreciably froln the i.,d perpendicularity to the beam

axis and the tilt. bc uuidot-e0table through the telescope.

Therefore, to ins11 parallelism. o. th; planes of all beam-

defining orifices while si. it! naeously establishing all orifices to

he coaxial, a second alig0ment -rocedure was adopted. A low-powered

telescope was fitted with a Gaussian eyepiece and the telescope

operated as an autocollimator: A lighr Source 1,n rhe side of the

eyepiece iluminmated a cress-haixr tlhe image of whitch was then pro-

jected through the telescope onto an optJcally flat (both sides)

front-surfaced mirror atteched fir•nly to the oiftce plate being

aligned; the oriiice plate was adjusted until the reflected image

of the cross-hair coincided i.n thC Cyepiece with the image of the

act-al ,'cto.ts-. hair; coincidence, o! t!he two imazgcs requires3 that the

mirror (ro-ifice plate) bc perpendicular to the liglhit (molecular)

beam.

I

-" a)

I

B. ELIER.IMENTAL RESULTS

Given the apparatus described abo.'e and the desired experimental

measitrement, i.e4, the atngular distribution of the effusing beam, experimental

procedure was rather straightforward. After a given orifice was inserted

into the Rotating Effusion Cell and aligned, the entire system was evacuated,

e.g., oveinight. At the beginning of a run the Gas Reservoir was ibolated

from the vacuum pumps (by closing valve i);. Figure 7) and filled with gas

(helium, nitrogen, nitrous oxide) to the desired pressure as measured by

a variable capacitance sensor (Equibar 120). Gases were obtained from high-

pressure cylinders; pressure in the reservoir was controlled WiL1I a variable-

leak valve (Type 9101-M) in the line between the usual cylinder regulator and

the reservoir. After steady state flow was established throughout the gas

flow system, the pressure in the reservoir remained surprisingly constant

(I to 2); a precision pressure regulator originally planned for insertion

in the gas line between the cylinder regulator and the variable-leak valve

was not required. Purity of the helium and nitrogen used was >99.5% and

of the N•O >98.0%.

While steady gas flew was being established, the electronic circuitry

was energized, adjustment of the lock-in amplifier checked, and, in partic-

ular, the filament current of the beam detector was adjusted to provide an

electron emission current of J.ClrA. The isolating valve (section III.A.5)

was then opened; the molecular beam could then pass to the detector and

measurements were begun.

With the gas piessure in the reservoir constant, beam intensities in

arbitrary units (i.e., the output from the lock-in amplifier) were recorded

with the Effusion Cell rotated to orientations varying by angular Increments

of 5' between 0' and + 9 0' and also between 0' and -90'. Typical concordance

between data for +e and for -0 is illustrated in Figure 8.

Angular distribution data have been obtained for five orifices with

thee .. t various Pressures; .... C.. . . .. 2 givcUs Ui d L of ti % lus

parameters.

In Figures 9-14 the results of various measurements are plotted as I

.ys. 0; _I8 is the relative molecular beam intensity normalized to 1.0 at

24

R U N I e'_'RUN 126]

ORIFICE NUMBER 1NIT ROGEN AT 266 dyn/crn2

00 0 0 E)O - 900

"9 CURVE SKETCHEDI90 THROUGH POINTS

80

70

I, 60-t40

.50

40-

30-

20 -

10

0 10 20 30 40 50 60 70 80 90

G, deg

Figure 8. Experimental angular distribution data (Normalized

beam intensity) for Nitrogen through orifice 1 (T - 25.65'

and (L/rrm) 4 4.010).

TABLE 2

ORIFICES, GASES, AND PRESSURESUSED IN

ANGULAR DISTRIBIUTION STUDIES

PRESSURE 0 oOrifice ,ASN,11ber T,deg L/rm r,,, Cal le NA NS0

= -2 -0

25.65 4.01o o.o46o 190., 70.0, 670., 200., 195., 69.0,45.0, 20.2 72.7, 4i.o, 45.0, 22.0,27.5, 20.2, 6.97.0

2 8.50 10.08 0.04.55 190., 71.0, 66o., 210., 207., 70.0,41.5, P1.0 70.0, 22,0, 41.5, 20.0,

7.0 6.23 -58.93 11.01 0.0292 230., 74.0, 700., 180., 690., 200.,

41.5 66.0, 14.o 69.o, 24.04 27.99 1.987 0.0951 -- 126., 12o.,

91.0, 75.0,4o.o, 24.0

5 0.0 2.2 39 0.0782 45.5, 53.0 70.0, 46.o,

25.5, 22.0,ti. '9

1.0 o He, 60.7d cm 2

-20.9 \ -/ He, 44.0d cm-,o . { 1(N ,) 11o d "-

N,93.3d cm

" 1. Q Cos f-L/r x 2.42

0.7 QNCOS L/r = 2.00

0 ?

0.

L~L 1 . l LI I L I0 10 20 30 40 50 60 70 80 90

8, Degrees

Figure 9. Experimental [1(e)] and Normalized Theoretical

[QnCose] Angular Distribution Data for T 0.00, L/rm = 2.h4.

27i

"o- .073 Torr

1.0 (8) a - .024 Torr

-A - .126 Torr

0.9 • QN Cos8 -

0.89

0,7

0.6-

C 0.5

0u 0.4

zuj 0.3-

020

0., -

01

01,I - 1 tI..

0 10 20 30 40 50 60 70 80 so8, DEGREES

Figure 10. Experimental [1(e)] and Nonnalized Theoretical

[QnCose] Angular Distribution Data for T = 30•, L/rm = 2.0,

with Nitrogen.

28

U

I I I I I

S- 8 27 dyn/cmn

2- 26 66 dyn/cm2100 1 ( G) 3- 55 33 dyn/cm -4- 93.46 dyn/crn 2

90 5- 2760 dyn/cm2

II OnCosoS80 -

F-4 \\ \

0 .70-

0 60 -\

0'50 \

S \ \\ 3

740 -\ 4

<' It

•'•-- "-•i

II0- 3C) -

10--

Figure Ii. Normalized Experimental [I(0)] and Theoretical

[QnCosO] Angular Distribution Data for Nitrous Oxide through

Orifice 2 (T - 8.5000. L/rm = ]O. O8).

29

T ------- I-

I ]- 18 67 dyn/cmr100 2- 8800 dyn/cm2

He) 3- 240.0 dyn/cmn2

9 , 4- 933.0 dyn/cm2

v \ \\\

80 Q~s

0 70 \\2 \ \

• o _ C/o

Li 50A A\\

N 50- 34 0 C

z \ \

S\\30 0\,

.20-

0 10 20 30 40 50 60 70 80 900, deg

Figure 12. Normalized Experimental [I(e)] and Theoretical

[QnCose] Angular Distribution Data for Nitrogen through

Orifice 3 (T - -58.930, L/rm - 11.01).

30

I1- 32 00 dynicrn 2

2- 92 00 dyn/cmL

100 N\3- 266 0 dyn/cm 24- 9200 dyn/cm

2

\ ,\.90 X,,\QnCoOS

\,,,.80\ ' \,o 70 \ \\",

\\\ \ \\\\\ \ \,

.60 ¶

a)5.6 \ \\

C.)

Ld 3.4 N3,0- \ \

N\ \j\A\

cr \ \",.\

0 30 2 \ ,\\.

~~A\ \ ,\\

.20- \ X,

\~\

0 10 20 30 40 50 60 70 00 90

0,deg

Figure 13. Normalizet. Experimental [I(e)] and Thlorctical

[QnCOe] Angular Distribution Data for Nitrous Oxide through

Orifice 3 (T = -56.95°, L/rm = lI.CI).

51

1 - 9 33 dyn/crn 2

100 2- 26 93 dyn/crn 2

3- 36 6G dyn/cr2

1(0) 4- 54 66 dyn/cm290C - 5- 96 93dyn/cm2

/ 6- 26G 6 dyn/cm2

I-- 7- 8933 dyn/cn 2

so - ~ 7-/-

Q3 70 ,

<:0D 6O

t-O5 0ncos\

C70

0 50

140 "'

O& ,

7

o) 30

0 I1 20 30 40 50 60 70 80 90 i

202E),deg

Figurc 14. Norn-alized Experimental [I(e)] an, Theoretical

[QCos8j Angular Distribution Data for Nitrogen through

orifice 1 (Tf = 25.C5*, L/rm .4cl)

0=00, i.e., I -M /_+o, with-j0 representing the measured ion current at o.

In Figures 9 and 10 the experimental points are shown, and the lines

are drawn through theoretical values. In Figuires 11-14, the dashed lines

represent smoothed curves drawn through the experimental points for both

positive and negative angles; the scatter of points about the smoothed curve

in Figure 8 is typical, i.e., not the minimal. The solid line in each

figure is drawn through theoretical values which are discussed in the follow-

ing section.

C. DISCUSSION

1. Remarks on T : Implicitly in Part 112 and CES1'3 and explicitly

in Reference 14 the theoretical angular distribution is expressed as

dNo(L) - 2qto Qn Sine CosO do; (9)

dN0(L) is the number of molecules which effuse per second into the

solid angle al1 Sine do, po is the molecular flux incident on the

entrance of the orifice, and Qn (n - 1,2,5) is the complicated function

of T, Lro, and 0 which describes deviations from the "ideal" cosine

law distribution. We now define

QV Q Qn/n (v V n - 1 2, 3), (i0)

insert ro explicitly (ro and L are normalized to ro - I in the theo-

retical analysisl2" 4 ), and write in terms of a generalized incre-

mental solid angle dx:

dNo(L) - 1onro2 Q, Cosa (2n Sine dO) (n:)dNo(L) = •olnr 0

2 Q, CosO dw.

The transmission probability W may be expressed as (number of mo!eculee

effus,,• from orifice)/( numbe enterin, g nrifce), r with eqa, n (1))

W (14/onrc 2 ) f PoTro-2 Q% Cos du

-" Cos coe (b.

- �/�Q. Coso (2v Sine do). (c)

Equation (12c) may be compared with equation (1), if equation (10) is

noted.

5.5

We assume that the aperture of the mokLcula." beam detector

subtends Pt the effusion e-ifLce 6 solid angle (f2d) sufficiently small

that Q CUSE1 may be considered constant over d, For two anglets G

and 8' we may new write

N•(L) ' n y"ro2 Qv' Coso' ;d (a)

and (1.)

N-f(L) , o % coseo Cjd. vCb)

The ratic of equation (15b) to (13a) is, if 0' is ta!:en saqt

Equation (i14) gives the th'eojetical value of the ratto o2 (number

enteriilg dptec or at O) to (number e;itering rt e - C').

"Ihe basic assumption concerning the operatLon of the bezia

detector is that the mjasured ion current I_+ is proportional to the

number of neutral melecule- entering the detector, or

- 'e (15)

We now define the symbol * to mean 'is (theoretically) predicted to

be equal to", an6 combine equations (]4) and (15) to obtoin

QV ~Cosa (16)

For any orifice, at 0 = 0' equation (20 becomes

la= 1,+ = 1 t (Q)o • 1 - 1; (r7)

hence, both the experimental and the theoretical, resul•s are aelf-

normalizinr, to unity at 8 - 0°. Equation (16) Ia the baails for the

form of the graphs in subsect on B.

As indicated in equation (1y), the plotting method suggested by

equation (16) and used in Figures 8-11', forces sgreement between

experimental and theoretical results at 0 = 0'. Furthermore, the

nature cf the experiment essentially forces agreement at 8 = 90*.

Therefore, whatever the actual nature of the discrepancy between

.5'4

experimental and theoretical results, in plots ot 1I anfL_- Co°o/QQv)o

3 E- 0 the apparent discepancy near 0'O0 is small and any real

discrepancies are forced to appear in midranger of 0 and are therefore

overemphasized. To circumveut this difficulty Phipps and Adamsfl'3 have

introduced, and Wahlbeck and Waug'"' have also used, a probability

density junction Po which we now conaider.

The experimental probability density function px is defined' "a by

PX = . +'1'!/-2 1 + (2, Sine do). (].8)

Obviously, I f normalized ion currents (cf. equation 15) are substituted

for T P6 ia unchanged:

/ I C./ I (2nSio dO) (19)

We alec note that st.uce I_ 3.0-90

l/e'~I (2, Sine dO) (20)

and

xX

It is apparent from equation (21) that 9 plot of Px vs. e will

differ from a plot of I .s. X enl% y by the factor pX. For a given setpx

of LiO, 91 data fo is obviously fixed, but there i no requiremenL on

the constancy of PX from run to run; hence, Px is not self-normalized

and can reflect di:--repanciee between experiment6l and theoretical

data at e - 0° as well as at other valuez of L.

The interpretation of--9 as a probabilicy density funct-jon fol.iows

immediately from its definition (equation (18) or (19)): P" dw Is

the probabqlity that an effusiE molecule will travspeicc. duo at 6,

or stated differently, _g is the fraction (of efu.ýsirt molecules)

wich Hlow, per &teradian, at e. With this_ interpretation of _P-nd With ...... '- • . ... . .•aand.. wtl%,i aantl an l3Ye5Si [I)t fo•- tile cureapording theoretical

quantity, t may bc obtalned:

t= dN,) (L) ,. !L1°r°•2 Qý, Coso d

dw¾oo' W

or,

tQP (Q Cos)/W (22)

351

I

The quanitity Pt also is not self-normalized, but may be converted

to a self --nor•nalized form by multiplying by ,/(0 •" the result ia

then to be compared with equation (16):

W Pt/(Qv)o - (QCoso)/(Qv)o t Io (1

FroM equetions (23) and (21) one ob:ains

x (Q)o/W. (24)

From the above obteorvaLions we draw the following conclusions:

(i) A plot of P vs. e revcals no information not alreadyx

pro:ided by a ploL of .9 vs. 0 and by the value of P0

(cEf. eqatlo 21)

(2) If one. attempts to nornalize V- via equation (25) and compare

the i-e.lilta with ]-G (e.g., 81 1(I - P'XW/(Q,)o), one obtains

nothing new, since

= ~V(.o ~~i-PW/(, o (25)

and the bracketed port:ion of the equation is a constant

for a given run. If, en the other hand, one attempts

to compare I directly with _•(t )O (equation 2)) by

defining

0/ [Q6 -QV)o],

one finds that

=[I(Q )/WPt] - 1, (26)

and, by virtue of equations (17) and (22), 8,2 is inhcrently

nornalized to zero at 0 - 0'. Furthermore, a plot of 82 '_.

i is, except fur a shift in the zero point of the ordinatex

caused by non-identity of P0 and (,)o/W (cf. equation 24),

ideiica" to the more useful fovin now to be descrj!bed.

(5) The moat informative scheme for comparing eXperTimental and

theorecical results to that obtained by defining, and plot-

ting Is. C, the quantitiee A and f(A)

x t

f(A) - A/P ( - (-

A plot of f(A) vs. 0 provides a direct, non-normalized

indication of the discrepancy betweenL experimental and

theoretical results, expressed as a fraction of the theo-

retica value. The only restriction on the value of f(A)t xis that -oo Z 0 and Pso (i.e., I~o) is adjusted to zero

by subtracting background ion current; therefore, A 0.

The one difficulty presented by the use of f(A) is thatnx tat E > --70' P0 and P may be rather small (e.g. 0.02 of

the value at o = 00); the difference A may be small com-

pared to experimenta( egror in ob0aini2g 1., but b( A) canbe quite large (e.g., 0.20 - O.LO) because PG is small.

In other words, the fractional errors are magnified as Pt

approaches zero. for this reason, one might prefer to

plot t va, 0; both types of plots are illustrated subse-

quenL nly.

2. Exiicrimenta] Results: Figures 15-20 present various portions ofx tou- results in the form: (P-/Po) - I vs. O, i.e. f(A) vs. e, except

for Figure 16 which is a replot of Figure 15 in the form (P -) vs.

f, i.e., A vs. e- Comparison of Figures 15 and lb illustrate the

point made in the preceding suction: the large excursions of f(A)

at 0 > 70' do not realistically reflect the dEactepancy between P x

and P0. At 0 < 60' plots of A vs. ( and of f(A) y.-. 8 are very simildr

except for scaling.

Figure 21 is a replot in the form [(1 0 (Q )o/Wpt) - 1] i. e,iLe. 6 .. e, (cf. equation 26) of Run 125 in Figure 15, and is

in(luded to illustrate two points: (1) a plot of 62 vs. e is nor-

malized to zeto at 0 = 0', and (2) except for point (1) and scaling,the shape oL the curve is the saame as that of f(A) va. e in Figure 15.

B~efore we examirte- the V'riou's curves in dec1 it =y be useful

to note again the geowetry cf the corilcal orifice and the two important

angles T and U*; thuse aie illubtrated in Figure 22 for the diverg-

in, orifice (T > 0), For G 1 e : I, tke detector may receive molecules

, froed 1lU of tihe ciiv.lar e:ttrance (a the oifLice and frow all

elements of the onificL wall; this coý iespuud~li• to Renge 1 and n - 1

/

06 I,

! 20 SERIES

( L/ro) = 401

0 4

N lROI "OE N

III

02-

p.-- I,/;.I,I,

I,7

I / ',t"- _ •

-04 - R N i,

0 12 66

A 127 8933•0

- -- --- I ----------L

5 I . 5

0 10 20 30 40 50 60 70 80 go

E), degFo

02ue]. f()pot o 2 eis

0a

4

120 SERIES030 (L/ro) --- 4 D6

8 NITROGEN

020 0.5

010- -04

0p0.

7 - ~

-010- O02

PRESSURE,

"0 20- RUN drycm2 01o 121 9.33

O 123 36660 12 I5 9693L 127 893 30

- I0 10 20 30 40 50 60 70 80 900, deg

Figure 16. A plots for 120 series.

I ~ I

230 SERIES

04 -o (L/ro) z 10.08

NITROUS OXIDE

02 ' 7'

i I -,

-02

'd PRESSURE,RUN dyn/cm2 \•

o o 231 8 27 I- 0 232 2666

-06 o o 233 55.33 ."" " 234 9346

v 235 276.00S I I I I 1 I. .~~ j

0 10 20 30 40 50 60 70 80 900, degi

Figure 17. f(A) plots for 23C series.

4o

320 SERIES0.6- (L/rm) = I 101

NI TROGEN

04 4OC

0

! i)

0~0

0.- 00

-021•

PRE55URE.-04 -RUN dyn/cm2 \'

o 321 1867 3o 322 88 Ou

A53~ 24000C' 324 93330 \2

-C I I I0 10 20 30 40 50 60 70 80 90

Odeg

Figure. 18. f(L) plots !or 5ýC series.

24 1

"0 G

o C i i I =- r

4 330 SLRIES-(L/rr) 1 10NITROUS OXIDE

0-1-

0~

-0Th

3PRESSURE,

-04- RUN dyn/crn 2 ,

o 331 3200 0o

[] 332 9200 Ln f

6 333 26660 It

) 334 92000 F- cD

0 10 20 30 40 50 60 70 80 90O, leg

Figure 19. f(L) plots for 33C series.

I I I I I I I I I I I II I I2

420 SE.FILS011- (L /ro) = 1,"99

02

0~0

-02A

CC

V L(0

*CF

PRES SURE,RUIN d yn /cm1

2

0 42 1 3200-06- -1 23 97 33

on .26 168 00

0 ic, 20 '30 40 -50 60 70 80 900, deg

Figuc 2.f(L) ploLth for .Xseiic3.

i / -

04 /

02-

O2

"--. RUN 125

0

-02

0 i0 20 30 40 50 60 70 80 900, Oeg

Figure 21. &: plct (cf. equation :'C,) for Run 125. Comparewith Run 125 in Figure 15.

4414

DEl [EC ORA PERT UR Elý

GASFLOW Lz-

j E): T

II ;

/x-

E) ?I J ~ * *// / // // JI

2 1, /"

"I,' //! / "

Figur /-I T,, .,_/ A/ a

8 C i Orifice//\

, \

/ G•_T \

Figure ,;2. The Criti cal Angles and thc Angular Ranges for

a Conical Orifice

(cf. equation 9 and 10). For T <. a < e0, the detector may receive

molecules directly from elements of the orifice wall over the entire

length of the orifice and frow par of the entrance to the orifice,

but a portion of the wall, and of the entrance, are shielded from

the detector by the outer rim of the orifice; this case corresponds

to Range II and n - 2. Finally, for o0 g 0 o T/2 the detector

receives molecules 2oly from 2_ of the orifice wall; molecules which

traverse the orifice entrance cannot proceed directly to the detector;

this is Range II113, and n - 3. For the converging orifice (T < 0),

the above statoments are also valid (i) if T is replaced by ITI, and

(2) if the description of Range I is modified to read: ,,receive

molecules directly only from - por~tLon of the ocifice entrance".

The significance of T and 0* with regard to angular distribution

m iay be summarized: For 0 < -, II elements of the orifice wall

(_T > 0), o: no elements (• < 0), contribute to the flux at the detector

aperture. As .jfp•rLeta in the range ITs 0 C * a decrtai

portion of both the orifice entrance and the orifice wall contribute

to the detected flux. For 0 > e*, the orifice entrance makes no

contribution, all molecules reach the detector from a portion of the

orifice walls, and this portion decreases to 0 as e -4 •/2.

We shall now examine the features of the curves in Figures 15-20,

beginning with Figures 20 and 15. The agreement between experimental

and theoretical values for P0 for the 420 series (Figure 20) is very

good except in the range 40' • 8 • 600 where there is a maximum inX tI

each curve. Similarly, in the 120 series the discrepancy t, P9 - P0

for the lower pressure runs (121. and 12 )) is zero within 2 to 4%,

i.e., very nearly within experimental error, for.q <,T; however,

there is again a maximum in the curves at @ i- 40'. At higher pressures

(Runs 125 and 127) f(A) is -0.05 to -0.10 at 0 <T, the maximum in

each curve is high' e and ib ishifted to larger 9. We now note the

correlation, in Figures 15, 16, and 20, of the maxima with the angles

T and 0*, and that the magnitude of both the maxima and the discrep-

ancy f(6) increases with increasing pressure and with increasing

46 1

Support L foT .is last gencial i zat •ln wati soight in Wang's data

for t11e effusion of cesium chloride !hiough 1 cylindiical (T ',. (. )

copper sld nickel orifices; data required for the t vs. I plots in

Figures 2" and 2'4 have been calculatd• by us fr +/1 + del~~~ .... /, u0fou ;G] dr-ta

tabulated by Wang-7. Maxima in the various curves at 10-15* and min nima

at (00-70' are immediately obvious (but note that the L scale is expanded

by a factor of five compared with Figures 15-21). Thc discrepancy A

does indeed appear to dec-ecase with increasing pressure, but this asscrtion

must be qualificd: Run 75 in Figure 2ji i- "out of order" for no apparent

reason; in Figure 25 Runs 21 and 2u were made with one experimental

configuration, Runs 2i', 28, and 55, with another; within each group

A decreases with increasing pressure; the reason for the discrepancy

between the groups is not apparent. The shift of the maximum to higher

0 with increasing pressure is qujite evident tn Run 91, Figure 214, hut is

not clearly exhibited in Figure 23, perhaps because data were not taken

at sufficiently high pressures

It might appear that Wang's data, then,provide support for, if not

confirnration of, our generalization about the variatiou of A with (vL/r)

and with pressure. However, we were surprised that for low pressures

the maxima in Figures 25 and 2b occur in the same angular range (10-151)

despite the difference in (L_'r) for the two orifices. Upon investigation,

we find that in Wang's apparatus7? the angle from the orifice to the annulus

which surrounds the baffle plate in the front oven is in the range 10 to

15'. It is therefore not clear whether the maxima in the curves from

Wang's data arise from gas ilow-orifice phenomena or from the baffle-

ennulus acting as a (reatively) concentrated source of molecules.

An obviouti question, then, is whether the maxima in Figures 15, 36

and 20 arise from a cell effect rather than an orifice effect. The

cylindrical iuterior ot our simulated Knudsen Cell is 1.00" in diameter

and 1.10" high; the bottom of the cell is completely open to the flow

nf g- f'nm I-t 14,i The int-lor ,.1 of the cell is macro-

scopically smooth. Hence, the only discontinuity in the emitting surface"seen" by the detector through the effusion orifice is at the "bottom

corner" of the cell; the angle front the orifice to the "boLtom corner"

47

l~14 I I T I

VNt, NG'S COPPER ORIWICEI2- (L/io) 0934

CESiUM CILORIDL

!0

08

•.7

00

04

--T- 0 00 PRESSURE,RUN dyin/c 2 -

V 21 1 30S27 27 0 '1

o 28 1330 *CD

0 26 2670o 35 3330

IOL I ,

0 10 20 30 40 50 60 70 80 90

Figure 2j. A plot for K. C. Wang's angular distribution data'-,

for otifice with T = o.0° and L/ro = 0.954.

48

141 -r- i r r-- i 1

WANG'S NICKEL. ORIFICE

12I\ ( L/ro) : 2 59

59 CESIUM CHLORiDE

00

i

7\

Oc',

O04 1l•i/ • / "

02'"I~ /

g° /0

02-

- 0

0

z HPRESSURE,

O RUN dyn/cm 2

-08V 75 2 70o 67 400

-10 C3 59 54 7r-

0 7!1 1 ,O0n I1I 409 30

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

I0 20 30 '10 50 60 70 80 90

l'ig ic g I .C A plot fto K. C, Wag'Igis angul ai distlil ution data7

fo ] or i ice With T 1 0.Q0 a;id ./ ) - ,

is -25°. For 0 greater than "--•Q molecules can travel directly irtxu

-he cell wall through the orifice (it 0* > 25') to the detector; for 0

less than •25° the detector cannot "see" the cell wall and receives no

molecules directly from it. Wc now examine Figures 29 and 2D for evidence

ci a cell effect an find that in Figure 16 V is so near 25' that one

cannot hope to distinguish cell effects from orifice effects; in Figure

20 the eituatiori is similar. However, the data for Figures 15 and 19

were obtained for a converging orifice with T - -58.9' ; in this case the

orifice presents no obstruction to molecules traveling along a 2 • 250

trajectory from the region of the surface discontinuity to the detector.

Figures 17 and 18 provide no evidence (i.e.,, no maxima no minima, no

breaks in thle curves) for a cell effect at or near 0 - 25', but there

is a definite change (more pronounced at higher pressures) in the curvature

of the various curves in the narrow range ITI a * 0* . We therefore

conclude that the deviations from theoretical behavior, exhibited by the

curves in the various figures, are orifice effects.

We return our attention to Fiires 16 and 20 and consider the two

mechanisms frequently cited as causes of deviation from theoretical

behavior: specular reflection from the ovifice wall and surface diffusion

along the orifice wall 9 . One would expect any enhancement of intensity

by specular reflection to uccur at angles 0 ý_T. Enhancement at angles

O > T requires specular reflection through angles greater than 2T, and

the maxima of Figures 16 and 20 would require 2L~f~etkg• enhancement

by specular reflection through angles > 50* with no concurrent depletion

at smaller angles (jr. especially Figure 20). Available experimental

dataeS on angular distribution of molecules reflected from surfaces do

not support these rather stringent requirements.

If surface diffusion occurs along the orifice wall, the concentration

of molecules resident within any incremental area on the wall is expected

to be greater than if surface diffusion did not occur; consequently, the

flux from the orifice wall to the detector Is expected to be greater.

Therefore, one might expect any discrepancy between theoretical and

experimental data for angular distribuLions, which results from surface

diffusion, to be most prominent in the range(s) of 0 in which the flux

50

from the orifice wall rc6a makt.5 thu MaXilifll tCl8LivL cou1Libuton Lo the

total flux into the detector.

The area cA the projection of the orifice wall onto a pla'nc per-

pendicular to the orifice-detector line defined by o may be calculated

from the following equations (sec Figure 1 tor Symbols):

AL - TrL Cos 0

A0 "rol, Cos C (28)

AE A(DIo; t 1 ) Cos 0

[A(Do; DL) is tihe area of overlap of the circlC T1 -1 , with the pro-

jection at anglc 0 onto the plane of that circle of the circle J~rL;

derivation of this quantity has been described in detaill32, 3•4 .4

1i f ~ Aw - A L

I< 0 <i 0l, Aw = AL - AE

if e a Iii nd T > 0. 0, A, - Al, -A

If 0 a•i'- aid T < 0.0, Aw = 0.0

The resulls of these c ltions for the eril-ces ior whi --2.

distribution dat.a were obtained are shown iln Figures 25-26 wherein,

for convenience in plotting, the areas have been nornalized to AL( Oo.O)-n.O.

In the ranges 0 > 0•* and 0 : T, the variation of A,, with 0 is given by

Cos - lowever, in the range T o - ,Aw exhibits a rather different

behavior which produces a miniiinim (or virtual mainituim) int Aw at 0 -

and a ,naxinli.m (or virtua! .ii4xi,,u,,) at "

As 0 increases over tl.c range 'i 0 0 4z for a diveigi ng (T > O.O)

orifice, the entrance of the oriflice is eclipsed by thle outer rim of thle

exit, and therefore the fLtLoIsl contr-ibuti on from the wall to the

total flux to the detector increasos to uniLy at 0 C,4 1 - In the range

o g 0t) over wlichi the fractional coil. cibutlon from t he wall is uni ty,

the projected wall a "seen' by the d2tector is a maximum at 0 U"

Therefore, if there were an increase ii, the flux Irom the oritice wall

above that predicted by our extension of Clausing's Model, one should

expect to see maxitua in Z vs. 0 plots t' or near e =

This is precisely what Is observed in Figures 15, it.), JY, and 20, and

in lieu of any acceptable Llternative. we tentatively conclude that the

51

U

00 -- " -- , , , I90o

80

70o-

LLJ 5Q -

10 h.30-

20- ej

I0 __

0 . I j de g. •_-•

0 10 20 Y) 40 50 60 70 80 90E), d:eg

Figure 25. Normalized projected area of inside wall of Orifice 1.

52I

100 I I

90-

80

7J 0

GO

LLJ bo

ODD

30 Ar-

20-

I0

0 1 ____.___._____. __,.___.__._____._.0 10 2.0 30 40 50 60 70 80 90

0, deg

t Figur-e '(6. Normalized projected arca ol inside wall of Oiit•'t,

I

1 0 0 - ---- -

70

60 -

750

,40- °

00 o

,20

0 °0 10 20 30 40 50 60 70 80 90

e,deg •

Figure 2•(. Normalized projected area of inside wall of Orifice 2.

54+

,I

I00 - -T !

90

UOC

60

C a CO30 Goo

20

30

20

0 10 20 30 140 50 60 70 80 90C), deg

Figure 28. Normalized projcctcd area of inside wall of Orifice .

!I

,855i

I

increased flux from the orifice wall results from surface diffusion.

We have not yet made a more detailed analysib of a model including Surface

diffuaLoa; Ruth and lhrth 2 E0 have reported the corresponding tinalysis for

cylindrical orifices.

Other features of the f(L) vs. 0 plots, e.g. variations with pressure

and nature of the gas, are being analyzed and will be reported in the opeu

literature.

56

fU

SECTION IV

TIE MUITICELL TECHNIQUE FOR EXPERIMENTALDETERMINATION OF TRANSMISSION PROBABILITIES FOR

MOLECULAR FL OW TIIROUClH CONICAL ORIFICES

The Knudsen effusion techniquei' 3 has been widely used to obtain

vapor pressure data in the range 1C-- 1Q" atm. The technique has several

variants 3 , but basically the experimentally measured variable is the

isothermal rate of flew of vapor through an orifice, e.g., I nan in diameter

and 0. 5 mm long. The equation which relates the pressure of the gas to

the flux through the orifice contains a factot which is often called the

Clausing factor, or more properly, the transmission probability, of the

orifice, i.e., the probability that a molecule which has entered one end

of the orifice will exit from the opposite end. Numerical values for the

transmission probability are available from theoretical analysess-B 12-- 14

of molecular flow of gas through orifices with various geometries. Although

these theoretical transmission probabilities are widely2- 3 used to correct

the observed rate of flow for the effect of finite o-riflce icigtih, reliable

experimental data with which to compare the theoretical transmission prob-

abilities are scarce, 3 1 ,33.

The Multicell technique, which we describe below, provides a simple

method for obtaining experimental values for transmission probabilities

and also offers a means for critically testing various assumptions which

have been made9430 concerning molecular flow within a Knudsen cell.

Application of the technique to the determination of experimental transmission

probabilities is discussed in this section.

A. THE MULTICELL APPROACH

In a Multicell effusion experiment a number (e.g., 4 to ]0) of ef-

fusion cells2, 3 which are as nearly identical as possible except in the

57

4|

variable under study are subjected simultancously to a given environment

for a given time. Comparison of results (e.g., weight lost from each cell)

for the Pvveral cells in a giv• n experiment permits isolation of the

effect of the variable of interest since all other variables are constant,

or vary simultaneously and identically, for all cells.

In the applicaLion of this approach to the determination of experimental

transmuission probabilities, Knudsen cells are fabricated to be essentially

identical except that each orifice is different from the others; each cell

is loaded with an equal amount of a given sample (e.g., cedmium or mercury),

and then heated simult.neously in vacuum for a givcn time - and at an

appropriate Lemperatur2 e which are identical for all cells. The weight

lost from each cell during the experiment is determined by weighing before

and after the heating period. The weight di lost from the ith cell in

time dt is given by'3 7

dg, ý aiKI[P(M/2TTRO) 2dt] (30)

in which P is the pressure in the cell at temperature 0, M is the molecular

weight of the effusing molecule-, Ki is the experimental transmission

probability and ai the cross-sectional area for the ith orifice. For

the experimental conditions described above, P, 0, and M are, at each

instant, the same for all m cells; therefore, the integral of the bracketed

portion of equation (30) over the interval 0 to T is the same for each cell.

Hence, if .gi is the total weight lost in time T, we write

.rt=o,t=T dgi -gi = aK I

with I representing the integral which is constant for the several cells

in a given run, but which varies from run to run. Since the various 8i

and gi are directly measurable quantities, it is convenient to define

-- i/Ai and then to write

Ji = K- (31)

Clausing's theory predicts5- P 13 that, for conical (-!i/2 < T < rr/2)

and cylindrical (I .- ) orifices, the variation in a given run of the values

Ji for the various cells depends only on the ratio if length L to diameter

D for the orifices and on T the semi-apex ongle of the cone of which the

58

p

orifice is a truncated section. ieiz-fore, if the values if are plotted

axainst some suitable function of (IQ,1) and T, designated by (I.'PDi), andextrapolated to o0, T) one 1ay write; fI111 Eqti.atin (51)ý Jo I 1 i hAch

the subscript designates the condition L ,n = ". h'tL (L,' = O defines5-tý, 13

tiL ,,ideal,, orifice for which tie transmission probtlilltt, X. is unity

1vgardless of the value of T (whichl in fact it mesninglcss when LiD = 0);

therefore,

Hence, the intercoPt at FN 0 ) is just the wcight loss per e'it area Ji

to be ex1pected, under the conditions of the run, frot, an idevl orifice.

With this cxperiment.,] value for i, the experimental transmiss 4 on probability

for each of tle cells is calculated from Equations (5i) and (52):

K, = ai/ao. (33)

Tie i11ert suitable choice for (i•, ),T) appears to be 14, tile theoretical

tra:,smission probability S-u, 13 calculated from the orifice dimensions. This

choice results in a conveniently bounded plot with abscissa values ranging

from zero to one; ':.ie e important, if the theoretical transmission probabilities

predicted by Clausing's theory5 and its extensions(3 '0 4 , 13 are correct, a I-lot

of Ji versus F(LL/D,T)j!. -= WT should give a straight line which may be precisely

extrapolated to ji = 1 (i.e., to L/D a ) to obtain the ir-ercept 30. The

slope of the straight line resulting frov, such a plot should also equal Jo.

B. EXPERIMENTAL RESULTS AND DISCUSSION

The results reported here were obtained with a typical Multicell

configuration: various sets of eight nickel-plated, steel Knudsen cells,

machined to be "identical" except ior the lengths of the cylindrical orifices

and/or the semi-apex angles T of the conical orifices, were placed into

closely-fitting, symntcLrically-arranged cavities in an aluminum (or copper)

block which, during a run, sat inside a s,:ainless steel vacuum chamber.

During the heating period, the entire vactuum chamber and several inches of

connecting stainless steel vacuum line were inside a circulating-hot-air,

constant-temperature oven. The cells were charged with equal amounts of

high-purity cadmium metal.

59

The dimcnzions of the orifices and the detailed results of the various

runs are available elsewhere 3 0 ,40; the results of a typical run are presented

in Tablc 5 and in Figure 29 in which the lenst-squares straight line (Jic -

mW1 4- b) for the dat.a points ie also given.

For each run, the expeiimental transmission probabilities were calculated

i1 two ways: (2) the :,rdinate value of each experimental point (i.e., each

i was divided by j0 (cf. equation 33) to obtain the values Kip; (2) the

coriespondin ordinate value (Jic) on the least-squares line was divided by

Jo to obtain the values Ki C. In Table 5, the transmission probabilities cal-

culated in the two ways from the data of Run 7 are compared with the theoreticalA

transmission probabilities.

For each orifice the transmission probability calculated from the results

of several runs will be more reliable than a value obtained from any single,

arbitrarily-chosen run. "Average" transmission probabilities Ki for the

various sets of orifices, and the transmissioa probabilities Ki calculazed Ifrom the individual runs are given in Tables 4-6. Two values are given for

each orifice: the upper of each pair of numbers is the transmission probability

calculated from the experime ntal poin', (2f. Kip, Table .5), and the lower

number is the value calculated from the least-squares line for the points of

a given run (cf. Ki , Table 3).

As shown in the last culumn of Tables 4- 6 , the discrepancies between the

experimental values for transmission probabilities (foi cadmium vapor passing

through nickel-plated, cylindrical orifices) and the theoretical values lie

in the range ±-% (with two notable exceptions in Table 6, for which we have

no explanation), if the experitental points are considered. If th3 results

from the least-squares lines are used, the discrepancies are appreciably

smaller. Similar agreement for a very limited number of "short" orifices

has been obtained by McKinley and Vance 3 3 and by Carlson, Gilles and Thorn37.

In 6cicral, the experimental values K- are less than the theoretical

lL, Lnd thlere is a sligit trend for the discrepancy to become larger as

increases. These observations are exactly opposite to what would be expected

if surface diffusion8'38 through the orifice contributed significantly to the

total efflux.

The agreement between theoretical predictions and the experimental

results reported here (especially Table 4) is appreciably better than the

60 o ,I

i

TABLE 3

COMPARl',ON OF EXTERIMENIAL AND THEORETICAL!'l-f-'MISSION PROBABILITIES: RUN 7

Orifice, 300(0K-14] 'oo(ycw).L/D Wi gng .' y W

0.195 0.83Y 42.36 0.843 +0.7 0.831 -0.8

0.290 0.77W 38.16 0.759 -2.3 -.767 -1.3

0.414 0.711 35.62 0.709 -0.3 0.69) -1.8

o.048 o.695 32.94 0.656 -6.o 0.682 -1.8

0.5521 o. 664 33.83 o.673 +1.4 0. 649 -2.2

0.522 0.663 31.91 0.635 o.649 -?..2

0.667 0.609 28.85 0. 574 -6.1 0.592 -2.9

0.8oo 0.567 28.55 o.568 +0.2 0.549 -3.4

61

, .

I

500- J0 50 26~

450-

I

400

S, -mg

350

300- , -

060 0.70 080 1 or)

W1

Figure 29. Weight loss through orifices with various (L/D)ratios vs. theoretical transri.ssion probability.

62

II

TABLE I

EXPERII.INTAL TRANSMISSIONI PROBABILiTIESFOR) FICPT CA!,DP ' ORIFICES (SET I)

Orifice, K I_(_o -W)

L/D W Run 4 Run 5 Run 6 Run 7 K W

0.195 0.835 0.85519 o.B,512 0.8H50 0.814:29 0. N46 +1. I0 . 629(b 0,8392 0.-3 78 0.8H305 .83) -0.4

0.290 0.7K( --- 0.7207 0.75h6 0.7"75 0.745 -4.]--- 0.7791 0.7772 0.707 0.'7(5 -0.5

C. 4.34 0.711 0.69)2a 0.7395 0.6888 C 7(k 0.700 -0.40 .6-) 97 5 b 0.7±41 0.71'16 0. 0.706 -0.7

0. 448 0.695 0. 6 651)a o. 0. 991 0-(555 0."670.610 b 0.698 0.C60)o 0, 6)24 0.690 -0.

0. 521 o. 664 O. 0- 610 .o74 o 5 0. 6905 0. U' _ 2 . 689 +3.6o.6477b O, 6672 0.6643 o.)61,,2 0.657 -1.1

0.522 0.663 0.66J3a 0.6572 0.6828 0.635o 0.659 -0.6o. 616j) 0.6665 0,6636 0!.485 0.656 -1. 1

o.66, 0. 609 u o 0 57Oc,,,a ( ý, ,1 C).R -5 k. ,3.,0. 5903) o. 6130 0.6 o096 o. 5921 o. 6oi -1.3

0.800 0.567 0. 5,•9 0. 53 56 0.'5701 G. 568 ]i O.557 ].8o. 5 4 6 53 b . 5716 0. 5679 0. 5485 0.559 -1.4

mU = slope 38.15 35.57 -30.76 52.39b = intercept

at (Wi 0 O) -1.71 +D. 35 +0,.20 -2.1.5Jo= intercept

at (Wi = 1) 36.hL 311.95 8o.9, 50.26

a. The upper number of each pair is the value obtained from the experimentalpoints.

b. The lower number of each pair is the value obtaincd frem the leastsquares line.

u3I

I

TABLE 5

EXPERIMLENTAL TRANSMISSION PROBABILITIESFOR EIGIIT CYLINDRICAL ORIFICES (SET II)

Orifice, K IOo( K-W)

L/R W Run 2 Run 3 Run 4 Run 6 K W

0.4462 0.819 0.0F7, 0.854 0.809 o.8i4 0.838 +2.-o.848b 0.811 0.7'JF 0.802 0.815 -0.5

o.6614 0.754 0.755a 0.752 0.754 0.751 0.748 -0.80.761b 0.742 0-724 0.731 0.740 -1.9

i.016 o0669 o.647a o.649 0.591 0.599 o.622 -7.00.680b 0.652 0.629 o.638 0.650 -2.8

1.291 0.616 0.614a 0.594 0.541 0.577 0.582 -5.5O.603b 0.598 0.570 0.595 0.592 -3.9

1.627 0.565 0.590a 0.555 0.497 0.520 0.556 -4.80.575b 0.541 0.510 0.523 0.537 -4.6

1.946 0.521 0.556a 0.484 0.4C2 0.486 o.497 -4.60.535b 0.49T 0.464 0.477 0.493 -5.4

2.263 0.465 0.508a 0.456 0.419 0.404 0.447 -7.6O.501b 0.460 0.425 0.437 0.455 -6.2

2.575 0.455 0.464a 0.448 0.414 0.446 o.443 -2.6o.469b 0.428 0.390 0.405 0.423 -7.0

m w slope 46.67 66.15 84.28 75.14b - intercept

at (Wj - 0) +1.24 -3.10 -9.07 -6.12Jo- intercept

at (Wi - 1) 47.91 63.05 75.21 67.02

a. The upper number of each petir is the value obtained from the experimentalpoints.

b. The lower number of each pair is the value obtained from the leastasqarea line.

64

TABLE U

EXITERIMENTAL TRANSY I S SION IPIOBAilITILSFOR EIGHT CONICAL ORilFICES (SElT III)

Orifice, K - ___(_-W)

L/R T, dog - W

.-05O 222.0 O.88) 0.87'(a C, -` 5 0 0.8)140. 68( b 0. 86 5 0. 886 -

3.889 26.8 0.884d 0.944 2a O.860 0.901 +1.80.886b 0.88 8 0.885 <0.1

1.005 18.y 0.8665 0,832a 0.956 0... +2.00. 8u6b 0.862 0.864 -0.5

0.421(, 0 0.8252 0. 870a u. 8 57 0.8614 +),.f0 82yb 0.825 0.82P +0.1

J4.051 15-2 0.7375 0.620a 0.5,81 O.oO0 -28.70.742b 0.756 0.759 +0.2

2.1835 9.65 0.6893 0.87ha 0.829 0.853 +25.6o.695b o.686 o.692 +0.3

3.857 9.15 0.6 187 C. 568a 0.595 0.582 -6.0o.625b 0.617 O.62i +0.3

2.072 0 0.5059 0.505a o.493 0.502 -0.80.515b O.5J 4 0. 510 C +0.8

m - slope 57.9-" 35.88b - intercept at (Wi-O) +0.68 -0.17JO= intercept at (Wi-1) 58.65 35.71

a. The upper nurnber o" cach pair is the value n Cpoints.

b. The lower number of each pair is the value obtained from the leastsquares line.

65

Ly'pi'-.a a Slecnt~ft between IlicasUted vapol pre'Ssu S~i reported by two laho,:rat cries,

IIOth Of Which USC Supp~losedly t he saiiie Knudsen effusion tcchnique, and is

b~Ctter: Lhnu the reprvuducibiIi ty in vapor pressure mtuasurcfments often Obtai ned

within a given laborat-ory. It wou~'d tlicrcfcrc appear thaLtLhe source of

thlese discrcpancics in vapor preSSUrL incasu,:cments may lie in phetnomena other

thtan flow of gas through an orifice.

Concurrently With our work, Macur, Edwards, slnd Wahlteck 4 .1 have attempted

to circu"-,-nt this probleiu Ot poor reproducibility in Knudsen effusion measure-

ment~s by using a Multicell technique in Whiichl thc em1phasis iS onl the aimul-

tancous determination Of several vapor pressures at the Sanme temp-erature; the

average deviuajo: in the pressures .calculated from the Several Cells in a

given run range from 1. to .3for indium~l and from I to 81 or gallium. The

agreemý-ntA from run to run (i.e., With chan~ge in temperature), as reflected

in the calculated Va1Lues Of AW,0,o of vaporization, is equally impressive;

the uncertainty quoted is ±0.10 Kcal/mole.

SECTION V

THLE MIlKER TECIINIQ11E

The Miker techinique for deicroil nation Of vIIpor pIeeSSures and Of molecular

weights of vapors at hig;h temperatures was described in detail !n Part Il" .

In this technique a vacuum microbalance is used tO obtain bothI the rate

of effusion from a suspended Knudsen cell and the recoil force which is

excrted oil the cell as a result of effusie..

The results reported in Part Il' dewonstiatecd the general validity

of the techn~que, and also demonbt rated the needI for vax ions re ii netuni s

if tile techniqiic was to provide a significant imp~rovemen~lt over tYp~ical

Torsion-Knudsen effusiun-ReA I data. Tile various refinemenlt-ls will bi:

rep or ted.

A. MODiFICATiONS TO APPARATUS

The early results 21 with the Miker appAratus indicated two major de-I

ficiencies: (1) When powcr to the furnace Was terminated in the courbc

oi meesuring the recoil force, the furnace cooled so slowly that rn u~n-q

desirably large correction for effusion during cooling was required.

(2) Manual control of the vacuum microbalance was satisfactory for rate

of effusion mcasurements, but not for recoil force measurements; aot~omatic

balance control appears to be a requirement for valid recoil force measure-

ments. Most of the modifications to be eescribed were made in thle course

of overcoming these two aeri4ciencies.

1. Furnace and Power Supply: Rapid cooling by thuý furnace appcared

to be Moat easily achieved through thie use of elements with very low

heat-capacity and with high surface area. Tunigsten muesh furnace

elements were considered butL rejected in favor of ýVaf)Iite LBape for

the following reasons: (1) Graphite tape retains its flexibility

C1lilpIeri It l ICtS Wk.''10 11t , ct C an I . a' ! it LiJ I SO 5hih tha vaL po r rooat ionl

1i 0i1 t ii' 1t))piii L,2 cI nmt'L apS d a I)rob1 1iii ; (3) Furnaice falbricil trol

insji cti Ae ',It I S r L''3d Iy aWCOU11p IiSlIe ill our shllpj) (4) The 1rt'siS-

. i vrty' of Crpti is so h i~ltha olne may easi Iv falriteate o furnace

X!i lj opemrateIs ~Ii frt( ,1y traom ai SCR- :nm en]I led , I bOAA, I I0k', bIil~ lýIine;

pore1 ;It at Lilavlae-:ar level is noel' more1 e~ý.i ii N andlt:!d anid

inlt roduLced ni aU theI) VflCIl111lll Le5tIII t hat[I i ,, it' IILIOwe r ti Dag, - veryI)

high cutrent levelts required by metaOl io heat'In', elements.

111e fu~rnace h~as a eyliIIdr.ica I ea~nfigerat ioui and eoonsiatab of six 13-cm

eiigthris of I .27-en- wide. p r'apluto tape lie Id at. t~i opaiaj butt n by maeLIi rued

graphite virings vsith pro pi13t iti set ews . ileLIocr r ica y , the three lengtihs

of t,1ape wh i C formf on1l10 le I ud'IA are7 illar I as aeLii'. othel

th t tit' two set,,, ot t.hreet ;1ie ti. h Ti tiV t't'i.'S (16'~e ('d IC rl i 2-

raly h the bot-tom gpitering. 3'hie res5LrtaCC' Of tilt' furnAIce is

-I X6 Ami aIt W C, t th no ia rat( i~oi sli e I ding, za current,1 oi -5OA rilsA

1) rod uces a tecmpera tture of 13000 C in1 . CC1 a c ll Supen~d*e inl t 1e C011Le

aof the: MI urn cs. V/eIower to Lt lie. I urnsý ie j ii L ertii fI P t-e-d , tie s kisilOndo 0

cell clool S inl lal at aaI ý rdt a of - 2 5-30 dIeg C/sec.

As, inticl tate a hove,, power toa the furnaýce li contr 1 ,' led withi a

single. I10A iliulcltroldeca ertireW gýate of Which is driven

by a ''Silicontrol'', oulse: gene ratilq uIi It

Tilt' other urnits tl the( contrlY0 100p are a tIlogStel'irienluli tietr-

moe oupl,, wh bi.cih resaponds to a ia oge s ir, furnace teicipera :. re, a stab In'

varxable re.ll'rene,' vantage, an elect roilic nlull detc~tac whlich resporids

to ainy error voA!Ltage between_ tlhe tblerliacaupl1e and the! reference voltage,

and a Coat cal 1e wh:ci' accepts t-he ariLpirt of the null. dc ect:rrr, provides

proport ionl)l, iie~rivat ive , 0110 integral cant rot actions, aInd riuppfles orn

opi apr ja e dn current to d~rive tlhe SiIi cantra(l uniit . Thle thermocouple

is used only for cont rol ; t~L~iU~ rtoircs arecr:sie with ail 6ut oca tic prhotoa-

elect~ric 'HIiOLOVulit ic'' yoree'

2. Automatic Control of Mi croha lance: Several changes hove been madc

in thle aut omiatic control system, aIl though Liu: basic cont rol loop

(Figure 20 in reference 2 1) relam inIS hef. same . The magneItl i C ampli L f icr

Which Suppli ed current to tlhe compensat ing coilIs has bee~n replaced

with a we 11- egolat~ed dc power Sul-ply and all emlitter- follower circuit

whi-in is driven by thle output of the CoIntrOller. This substitution

eliminated a significant ripple voltage from thL coils.

The "Photopot" light beam posit ion bensor 21has been replaced

with one of liter manufacture, which is wore stable and Which Canl

he readily operated inl high vacuum. The ne(w 'PhOtOFoL" Wasi. mounted

inside the balance chamber about 6.3 CM frOM theIL mirror affixed to thle

center of tile ha lance beam. The light beam originates , outside the

vacuIum system, in a 50W projector lamp, is defined by a 0.010-cm sliL,

reflected through a lens and window into the balance. chamber and ont-o

the balance mirror, and focused onto thle "Phiotopot_'. The output of

the projector lamp is monitored and controlled by a modification oi

the circuit described by RosenthaI4 Power for the 'Phot~opot" bridge

Circuit is now supplied by mercury batteries.

The weight change whiich can bv compensated by change in Current

throu~gh the ha lance coils (from 0 te clO00M) is 28mg. To avoid1 ~ opening the vacuum systeml to re-tare the ha lance after each 25-30mg

weight loss, a taring mechanism operable from outside the vacuum

system Was installed - A bel lows- sea led slinft d rivcn ocutS ide thei

vacuum chamber by a micrometer screw, carries four _,20-mg ring Weights

formed from Alumel wire 0.020 cm in diameter. As thle mictometer screw

is rotated, thle shaft is translated downward (upward) and the four

ring weights are sequentially pIaLcC on (removed from) a cross-arm

at tach~ed to the balance beam.

3. Miscellaneous Modifications: In the Miker technique effusing

vapor is directed downward, directly toward th-e window/prism thlrough

whlichi opt icalI pyromecter measurement s arc modeý. A shutte~r prot cets

the window during 'Lbe Interval bet~wcen pyrometerl reading; evL-n so,

the window acquires a significant de,.posit rathler rapidly. To a vL)id

e,e ning the entire vacuum syst em toe the otmosphecre whil c the wfndowI/

69

prism is rcmotvcd, cle,.ed, and Icplaccd, a very thin; bellows-

sealcd gate, v•oivy uas installed in thtic window mount and between the

window/prisr, ittself -nd tl',e furnace cl.'sbcr. 'The valve is a mod fi-

cat ion of the des igi reported b N ftSI te , and has perlormed

sat isfactori ly.

The effect of building erid equipment vibrattions on the micro-

bholance has been minimized by mounting the vacuum system on a steel-

reinforced concrete block (2x2x3 ft. high) which was cast in place

on the basement flooa; mechanical vacuum pumps in the vicinity have

been mounted on vibration-absorbing pods, etc.

For increased pumping capacity, there has been installed in the

vacuum system a larger diffusion pump (PMC-720) and a liquid nitrogen

trap (Cryo-sorb).

4. L, usioius Recoil Force s in measurements of rate of effusion and

of recoil force made immediate]y after tile modificaLions described

above were effected, the measured recoil force was much larger (e.g.,

by a factor of 1.5-2.0) than exp:ected from the rate of effusion

measuremcr,. s A ma jor source of the spurious portion of the recoil

force has been identified as a, interaction between the fields

su.:urrounding the furnuce power !eads and the magnet attached to the

belamer beam. This interaction has been minimized, and all but elim-

inated, by arranging the furnace power leads and the furnace elements

themuelves in a configuration symmetrical with respect to generation

of spurious iagneti c fields In, the balance chamber.

A second ma jar source of spurious recoil force was inadequate

shielding of the "Photopot" from stray light in general, and specifi-

cally from light from the furnace. Str-ay light from the furnace

caused a shift in the null point of the "Photopot"; when the furnace

power was terminated to measure the recoil force, the stray light

from the furnace was also terminated. The nail point of the "Photopot"

shifted bak to tLie "furnace cold" position, but since this shift

coincided with a recoil force measurement, the shift was incorporated

into the apparent recoil force. Shielding of the "Phot.opot" was

relatively simple, once the difficulty had been identified.

70

The residual spurious forces which remain appear to arise from

pressure-dependent phenomena, e.g., thermomolecular flow forces4.

However, these resldual spurious forces now constitute only -1- of

a typical expected recoil force, compared with --50 to 100% before

the various corrective actions were taken.

B. EXPERIMENTAL RESULTS

We have only recently eliminated (or minimized) the spurious recoil

forces discussed above. The experimental data we report were obtained

with the modified furnace and automatic control system, but before the

spurious forces were identified and eliminated, and are presented to

indicate the precision which has been obtained in recoil force measurements

with the microbalance.

The data in Table 7 for the vaporization of tin were obtained with a

graphite Miker cell2l the conical orifice of which is described by T -

28.80, (L/ro) 4.98, W = 0.908, and f = 1.088.

The results of Table 7 are typical illustrations of the reproducibility

obtainable with the Miker system. It should be noted in particular that

the reproducibility in measurements of recoil force is at least as good

as the reproducibility ip. measurements of rate of effusion.

We expect to observe similar reproducibility and good agreement

between effusion and recoil measurements in subsequent vapor pressure

deter-ainations.

71

TABLE 7

HMiRM DATA FOR VAPORIZATIONOF TIN

Rate of Recoil Recoil Preusure1 -. din/cm2

Temp, effusion, Current, Maoaui

1466 1.061 0.306 l00.4 11.2 21.7

1470 1.078 0.310 101.7 11.4 22.0

1473 1.138 0.298 97.7 12.• 21.1

1473 i.o67 0.302 99.1 11.4 2i.4

1486 1.156 0.328 107.6 12.3 23.2

1489 1.070 0.327 107.3 11.4 23.2

SRecoil mass is that mama which, under the acceleration of gravity,counterbalances the recoil force; balance calibration = 0.328 mg/mA.

t "at of eff.,1e.AF.. ,gi~..o., P from rccoal

The "accepted" value45 is 7.8 dcy/cm2 at 14800K.

72

U

I

SECTION VI

CALORIMETRIC STUDIES OF

VAPORIZATION PROCESSES

Determination of equilibrium vapor pressures and/or decomposition

pressures, with subsequent calculation of enthalpy of vaporization and/or

decomposition and of enthalpy of formation of the gaseous species evolved,

occupies a key role - one is tempted to say the keystone role - in high

temperature chemistry (IITC). From the various measurements of (supposedly)

equilibrium pressures in various laboratories have arisen numerous dis-

crepancies: between Knudsen eifusion data and mass spectrometric data;

between Knudsen effusion data and Langmuir vaporization data; and even

between two sets of Knudsen effusion data, both of which were obtained by

reputable workers using, supposedly, the same technique. Attempts to

resolve these discrepancies have resulted in, among other things, the

formation at the 196.6 Gordon Research Conference on HTC, of a committee

to study establishment of vapor pressure standards above 500°K and below

10-3 atmosphere.

In addition to refinement and standardization of effusion techniques,

an endeavor to which the previous sections of this report is devoted,

new approaches are needed. A more or less obvious one, except for

experimental difficulties encountered at high temperatures, is direct

calorimetric measurement of enthalpy of vaporization and/or decomposition.

Sunner and MorawetZ46 (at the University of Lund, Sweden) 1.,;e studied

intensively the problem of calorimetric measurement of heats of vaporization

of various hydrocarbons and other organic compounds at 25'C. They have

successfully measured heats of vaporization with a precision and accuracy

better than 0.1 kcal/mole for materials with vapor pressures as low as 10-4

torr at 250C. They have also measured differences in heat of vaporization

as a function of the effusion orifice geometry.

I.)

N

In conjunction with work supported by another contract we became

interested two to three years ago in the possibility of direct calorimetric

measurement of enthalpies of decompositi'Ln, e.g. A(,) - B(s) + C(g). Our

initial approach to the direct measurement of enthalpy changes was essentially

to copy the furnace and calorimeter configuration of Speros and Woodhouse's

'ýquantitative differential thermal analysis" (QDTA) system47 and to make

various changes in the electronic control circuity and the output sigial.

In evaluating our QDTA system , we measured enthalpies of fusion of several

metals with a precision of 0.5-1.5% and an accuracy of 1-5% in the range

150-4500 C. Subsequently we investigated the decompostiion of PbCO3 , ZnCO3,

and NH.4C1; with QDTA we were able to determine with reasonable precision

and accuracy the enthalpy change for these decompositions, including the

change for each of three steps in the decomposition of PbCO:

Our Pxperience with the QDTA concept has led us to make several sig-nificant modifications in the furnace-calorimeter configuration and in

the control-output circuitry; we refer to the new system as "Differential

Scanning Calorimetry" (DSC)4 9 . These modifications are'presently (Summer,

1967) being debugged and tested. We expect that with this new DSC w-

shall be able to measure in the range 30-1000%C, endothermic enthalpy

changes resulting frcA fusion, vaporization, decomposition and phase

transition, and perhaps even exothermic enthalpies of reaction.

It is our further opinion that it is feasible, within the present 4

state-of-the-art, to design and build a DSC system which would provide

direct measurement of enthalpy changes in the range 1000-2000*K, and

perhaps to 2500'K. Such a system would make available to "higher" tem-

perature chemists the capabilities presently available in commercial

equipment4 9 up to 50 0 C and expected in our DSC system up to 1000C.

Another virtue of the DSC technique is thist with the one assump-

tion that the rate of energy input to the calorimeter is proportional to

the rate of the (endothermic) process occurring in the calorimeter, the

data one obtains are directly interpretable as the rate of the process

as a function of temperature and of time. It is then rather straight

forward47 P5 0 to obtain rate constants over a range of temperatures, and

hence the activation energy, from a single 1-to-3-hour run. With the

older QDTA system we have obtained•a energies of activation for the various

74

steps in the decomposition of rbC03 and for the homogeneous decomposition(in solution) of a complex organic azo compound.

Tn Suinary, we aie CUnvinced that direct caloriTretric measurement of

enthalpies of decomposition, vaporization, etc., are presently achievable

to 1000'C, and with a relatively modest development effort could be achieved

at 20000 C. While these techniques are not likely to replace effusion

techniques in the near future, they can provide for high temperature

processes data obtained by other-than-equilibrium techniques; such supple-

mentary and complementary data are sorely needed in many high temperuture

chemical systems.

7!

TABLE 8

IDENTIFICATION OF MAJOR COMMERCIAL COMPONEETS OF APPARATUS

Item Suppli.e•

Amplifier, Leek-in Princeton Applied Research Corp.PAR JB-5 Princeton Junction, N.J.

Controller, C.A.T. Leeds and Northrup CompanyType 10877 4901 Stenton Avenue

Philadelphia 44, Penna.

Lamp, Projector Local Photographic8V, 50W, Type 13113C-04 Supply ShopPhilips (Holland)

Null detector, d.c. Leeds and Northrup CompanyType 9834-2 4901 Stenron Avenue

Philadelphia 44, Penna.

Photopot Giannini Controls Corporation55 N. Vernon AvenuePasadena, California

Power Supply, Sorensen Products30V d.c., QB28-i Raytheon Co.

S. Norwalk, Conn.

Pressure Meter, Trans-Sonics, IncorporatedEquibar Type 120 P.O. Box 328

Lexington 73, Massachusetts

Pump, diffusion, oil, Consolidated Vacuum Corporation4 in., PMC-720 1775 Mt. Read Blvd.

Rochester 3, New York

Pyrometer, optical Pyrometer Instrument Co.Photoelectric, "Fhotomatic" Bergenfield, N.J.

Valve, vacuum, Cranville-Phillips Companyvariable leak 5675 E. Arapahoe AvenueCat. No. 9101-M Boulder, Colorado 80301

Silicontrol Pulse Unit VecTrol Engineering Div.Type VS6332AF Sprague Electric Co.

P.O. Box 1089Stanford, Conn.

Trap, liquid nitrogen Granville-Phillips Co.yu-•t, l, 5673 E. Arapahoe Ave.

Boulder, Colorado 80301

Tuning Fork American Time ProductsType 40 61-20 Woodside Ave.

Woodside, N.Y. 11377

76

RIEIRENCES

1. Mi. Knudsen, Ann..P-s. -2-. 75, 9)) (i)@)) ; "Kinetic Theory of Cases,Methuen, London (,1954).

2. J. L. Margrave, In "Physico-Chumical Measurements at Hitgh TempCistures"

(3. M Bockris, J. L. White, and J. D. MacKenzie, eds.). ButterworthsScientific Publications, London (15,59).

3. Various chapters in "Characterization of High Temperature Vapors"(J. L. Margrave, ed.). John, Wiley and Soils, Inc., New Yoark (1967).

h. By definition, an ideal orifice has zero length; if its radius is thesame as the radius of the smaller end of) an actual (conical) orifice,the orifices are said to correspond.

5. P. Clsusing, Physica 0, 65 (1929); Ann. Physik 12, 961 (1932).

6. W. C. DeMarcus, Technical Report K-1502, parts 1-6, Oak Ridge GaseousDiffusion Plant, Oak Ridge, Tennessee (195T); and W. C. DeMarcus andE. H. Hopper, J. Chem. P:Ls. 2.ý, 1344 (1955).

7. E. W. Balson, J. Phys. CheQ,_ 6, 1151 (1961).

8. R. P. -zkows i, J. L. Margrave, and S. M. Robinson, J. Phys. Chem. a,5 229 (1963).

9. W. L. Winterbottom and J. P. Hirth, J. Chem. PhYs. 7, 784 (1962).

10. P. Clausing, Z. Physik 66, 471 (1930).

.11. R. D. Freeman and A. W. Searcy, j. Chem. Phys. 22, 762, 31' (1954).

12. R. D. Freeman, "Molecular Flow and the Effusion Process in the Measurementof Vapoz Pressures", Technical i.;port ASD-TDR-754, Part I, 1965(AD42314O).

13. R. D. Freeman and J. G. Edwards. in "Condensation and Evanoration ofSJulids" (E. Rutner, P. Goldtinger, and J. Birth, eds.); Proc. Intnl.Symposium, Dayton, Ohio, September, 1962. Gordon and Breach, New York(19614). p. 1-1.

1h. J. C. Edwards, Ph.D. Thesis, Oklahoma State University, Stillwater,

1964 (AD469505).

15. J. G. Edwards and R. D. Freeman, to be publi'hed.

16. A.-endix E of referencc 14.

17. D. H. Davis, L. L. Levenson, and N. Milleron in Rarefied Gas Dynamics,(L. Talbot, Ed.) Academic Press. New York (1961). p. 99.

18. E. M. Sparrow and V. K. Jonsson, AIAA Journal 1, 1081 (1963).

77

Ii

1 A. NASA Technical Notes, Lewis Research Center, ClevelAnd, Ohio

a. E. A. Richley and C. D. Bogart, NASA TN D-2115 (February, 1964).b. E. A Richley and T. W. Reynolds, NASA TN D-2330 (June, 1964).c. 11. Cook and E. A. Richley, NASA TN D-2480 (Septembey, 1964).d. T. W. Reynulds and E. A. Richley, NASA TN D-I564 (October, 1961t).e. T. W. Reynolds and E. A. Richley, NASA TN D-3225 (January, 1966).

20. J. W. Ward, "Use of the Knudsen Effusion Method, A Literature Survey"

Report LA-3006, Los Alamos Scientific Laboratory of the University of

California. (Available from the Office of Technical Services) May, 1964.

21. R. D. Freeman, "Molecular Flow and the Effusion Process in the Measurementof Vapor Pressures," Technical Report ASD-TDR-65-754, Part II (AD612953).

22. N. F. Ramsey, "Molcular Bemus", Oxford University Press (1956).

23. W. L. Fite and R. T. Brackknau, Ph2ysicl Review ]1?, !141 (1958).

24. T. H. Batzer and R. H. McFarland, Rev. Sci. Instr. I, 528 (1965);

T. H. Batzer, private communication.

25. J. C. Sheffield, Rev. Sci. Instr. Lb, 1269 (1965).

26. J. Q. Adams, Ph.D. Thesis, University of Illinois, Urbana (1961).

27. K. C. Wang, Ph.D. Thesis, Illin3is Institute of Technology, Chicago (1966).

28. F. C. Hurlbut, Report No. AS-66-10, College of Engineering, University

of California, Berkeley, August, 1966; paper presented at Rarefied GasDynamics Fifth International Symposium, Oxford, England, July, 1966.

29. V. Ruth and J. P. Hirth, (same as 13). p. 99.

30. J. W. Ward, "A Study of Some of the Parameters Affecting Knudsen Effusion",

Report LA-3509, Los Alamos Scienti fic Laboratory of the University ofCalifornia (Available from CrSTI). Also, Ph.D. Thesis, University of

New Mexico, Albuquerque (1966).

31. P. W. Gilles, Ann.Tl. Rev. Phys. Chem. 1., 355 (196].).

52. K. D. Carlson, Ph. D. Thesis, University of Karsas, Lawrence (1960);Argonne National Laboratory Report ANL-6156 (1960).

35. J. D. McKinley, Jr. and J. E. Vance, 7. ghem.. Py. 22, 1120 (1954).

34. K. Motzfeldt, 1. Phys. Chem. 52, 139 (1955).

35. C. I. Whitman, Ch. e.m. Phya. ?C, 161 (1952).

36. A. J. Boyer and T. R. Meadowcroft, Trans. Met. Soc. AI , 388 (1965).

(0

4

.1 37. K. D. Carlson, 1'. W. Gilles, and R. J. Thorn, J. Che.a. I' , 225

(1963).

38. C. Hall, Rev. Set. lnjtr. 5, 1)1 (1962).

39. A. L. Ball, Jr., MS Thesis, Oklahoma State University, Stillwater (196'3).

40. R. D. Freeman and R. E. Gebelt, unpublished resujlts.

41. C. J. Macur, R. K. Edwards and P. C. Wahlbeck, J. V_.. Che.. In,

2956 0.96-6).-

142. N. C. Mushovae, Technical Paper TP-63-11, Sprague Electric Co., NorthAdams, Mass. Figure 8.

143. L. A. Rosenthal, Rev. Sci. Instr. L01 1529 (1965).

0h1. First three papers in ",Vacuum Micxobalance Technique", vol. 5, (K. It.Bchrndt, ed.) Plenum Press, New York (0966).

45. R. Ilultgren, R. L. Orr, P. D. Anderson, and K. R. Kelley, "SelectedValues of TheunodynaMic Properties of Metals and Alloys". J. Wiley andSons, Inc., New York (l96.).

46. S. Sunner and E. Morawetz, Act,-Chem. Scand. 2-., 13 (i)63); privatecolmlunication, June, 19t,(,

147. D. M. Speros and R. L. Woodhouse, J. PLhys. Chetm. ýJ, 26164 (1963).

48. P. D. Gwinup, Ph.D. thesis, Oklahoma State University, Stillwatet (1967).

49. Pe-kin-Elrcr Cotp., Norwalk, Connecticut, Differential Scanning Cal-orimeter DSC-113.L. S. Watson, et al., Anal. Chem. •, 1233 (59Q4).M. J. 0 Neill, Aak. Chew. 56. L?3. (1964).

50. 1i. J. Borchardt and F. Daniels, J. Am. Chem. Soc., L.), 41 (1957).

7)

UNCL.AS;jIFL. E

3e aiy C as[c to DOCUMENT CONTROL DATA -R&D(S~cuui.1 cI... iftarl_. 1Ito -y r ,eefrt *s,4 tndeorng annotaitan must be *nIorod when ih. overall rop.lr in ci-t.di )

[IOrnIGINA 7IN G ACTI'-IT Cf-t Autor 1-0 RLPcOT EEUR TVQ C ;;Fi:

Oklahoma State University UrNCI.SSI FIEDI,ý,t.of Chemistry -1 GROUP

SIi II wa re r, 0!(la h oma 74074 ___ __ _________

_377-LCT IITL E

"ýi ')occil ar Fl ow and the Effusion Process ill the Mtasuremntuoi Vapor Pressuros'

_F ESc C771PT I VE oIN 0 (7

-po a I rej i at, d in, Jn v O. I*..

11101. Re~port, I Septcmber 1964 to 31 Moy 1967

Frceman, Robert D).

6 METPO FT 0;ATi 7a TOTAL NQ OF PAqFP =79- NO Or RLF#

NovIHILO ., 1967 ________ 79 50 ______

8. CONT11ACT OH -RAN~rNO~ SO ORIGINATOR's REPORT1 NUMOICft(S)

AF --3(657)-8767b. 'AOJCC- NQ 7360 ASD-TDR-63-754, P't:. III

Task: 736004 ________________________

Ob. OTHIC.9 K PORT tdO(S) (Any thonu- ove 9--may be samIned

IS ~ ILASLIT/LMITTIN N11~5 This documlent has been approved for public release

andl sales; itS distribution is unlimited.

I I SLIPPL.EMIEN IAMY NOTES la SPON5OIIING MILITARY ACTIVITY

AF Materials Laboratory (MAYT)Wright-Patt~erson AFB, Chio

3i AUSINAC TOr extensions to conical orifices of Claiising's analysis of angular dis-

LribUt-oln of' mol~eculetu effusing from cylindrical orifices has resulted in numericsvalues for transmission probabilities and recoil- ffirre correc.tion factors which aretLbulated. With these results, it is demonstrated tha-t the optimum orifice geonm-etry for (1) recail force measurements is a diverging conical orifice with semi-apex angle of 30c, (2) delivery of ma,:cimum fraction of cffusing molecules onto (onlinto) a targct (,aperture) is a long cylindrical orifice.

Modifications to the angtliar distribution apparatus are described and experi-

mental results given for four orifices and two gases over the pressure range 5 to900 dyn/cm'2. The mostý incerestitag aspect of the results, one apparently not previ-

oaisly noticed in angular distribution results, is the presence of maxima and minimi

in plots of Li VS. £, where 6 is the (experimental value - theoretical value) oiP?,, the fract 'ion of effusing r-olecules which flow per steradian at angle @ from

Leorifice axis, These rnaxlr.:& and minima have been correlated with the relativecontribution from the orifice well to the total fIlux at angle P.

Additional results for expctillental transmission prol abilities of orifices de-

teruirined by the Multicell technique are generally in agreement with theoretical

values within 2 tQ 5%.The Miker technique for simultaneous determination of vapor pressure by rate -)f

effusion and by recoil force measurements has been refined to the point that recoýA

force data are as reproducible as rate of cffusion measurements. Several sourcesof' Spurious recoil force have been identified and eliminated, A new furnace and amodified- automatic control system for the microbaiance are described.

D D I A44 1473 ____UNCLA SSI FIEDSecurity Classification

UlNC LA SSI F1' I'DScict I Iy (IasItfI catliOn

4 Lt, A Ls 1 Li LINK1 CKLY VORDS - _

Vapor: PrcssuruMo]ccular Flow1 ranstttisj ;on I'roboibilifly for Orif ices;Mit CIo hl a 1a1ce

* Molecular Deatiis

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