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01-82-0427
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' . ' . .. -i '? ~ . ---~ (, ' ) ,. C" ·: . ~ ~."7 ~-
BOE I NG~~ 1sEtI~~~ - ··--LABORATOR1 ES
An Experimental Investigation of Mass Flow Through ShQJ:J,
Circular Tu-bes in the Transition Flow Regime II
A.K. Sreekanth
April 1965
' ' i I I
' ' .. _i ' _._.,,
Dl-82-0427
liOl:.INC sc.;u.:NIIFIC RESEARCH LABORATORllS
FLIGHT SCIE.NC1:.S LABORATORY REPORT NO. 95
A?, EXPI-..RIMENTAL INVESTIGATION OF MASS FLOW
THROUGH ShORT CIRCULAR 1'UliES IN THE TRANSlllON FLOW RE.GIME
by
A. K. Sreekanth
April 1965
'
..
•
AN EXPERIMENTAL INVESTIGATION OF MASS FLOW
THROUGH SHORT CIRCULAR TUBES IN THE TRANSITION FLOW REGIME
A. K. Sreekanth
Boeing Scientific Resea~ch Laboratories, Seattle, Wa1hington
ABSTRACT
The mass flow of nitrogen gas through short circular
tube, was measured in the tran1ition flow regime with the length
to-diameter ratio, varied between 0.005 and 1 and with pre11ure
ratio• across the tubes varied between 1 and 20. The range of
Knud1en numbers (ratio of upstream mean free path to the diameter
of the tube) covered was between O.l to 1.7. Compari1on1 are made
of the measur~ments with existing theorie1, and a semi-empirical
equation is developed to correlate the measured data at all of
the pressure ratios, Knudsen numbers, and length-to-diameter
ratios covered in the present study •
.... ------... - .. "' .....
TABLE OF CONTENTS
I Introduction • • • • • • • • • • • • • • • • • 1
II Experimental Arrangement and Measurements • • • • • • • • • • • • • • • • • 2
III Accuracy • • • • • • • • • • • • • • • • • • • 5
IV Results and Discussion. • • • • • • • • • • • 6
V Conclusions • • • • • • • • • • • • • • • • • 33
References. • • • • • • • • • • • • • • • • • 35
Acknowledgement • • • • • • • • • • • • • • • 36
Notation • • • • • • • • • • • • • • • • • • • 37
Tables •• • • • • • • • • • • • • • • • • • • 39
1 I
I I
I
BLANK PAGE
I. INTRODUCTION
The theoretical and experiruental investigation of the flow
of a raref h:d gas throu~h tubes h3d i ta ori~in more than fl f ty years ago
when Knudsen studied the flow of pure gas•is in lon~ circular tubes. He
~erived a theoretical expression for the mass flow under free molecular
flow conditions and from an experimental study deduced a semi-empirical
formula for the transition from molecular to Poiseuille flow. A few years
later, Clausin~ theoretically investi~ated flow through short tubes under
free molecular conditions. Almost all of the experimental work done since
nn the flow of rarefied ~••es thrnt,f9h tubes has been c-f'nfi ncd t ~ ·;c ·.:y ! c-..
pressure ratios across the tube, althouih considerable amount of work baa
been done recently at high presRurc ratios for convcr~ent-<livergent no~zles
l 2 using nitrogen and argon. Hass flow studies at very high pressure ratio•
3 4 have also been done through short tubes using cesium gas. Liepmann
investigated the transition flow through a circular orifice at extremely
hi~h pressure ratio• to check the near frer. molecular flow theory b~ing
developed at that time.
moderate pressure ratios.
5 Liepmann's work was extended recently to low and
The present work is an extenaion of the orifice
{low 1tudy, in which the thickness of the orifice 11 gradually increa1ed
to result in a abort tube.
) -
II. EXPERIMENTAL ARRANGEM£N1' AND MEASURE?-U::NTS
Apparatus
All the experiment s were conducted in the Boeing Scientific
Research Laboratories' continuous flow, low density gaa dynamics facility.
It conai1ts of a large stainless steel vacuum tank, 44" in diameter and
108" long, one end of which is connected to a large vapor .,.~n4Jter pump
(E'cizJards 18 B 4) which has a rated unbaffled pumping speed of 2800 liters
-3 per second at 10 torr. An orifice plate or a short tube separate• thi1
tank into upstream and down1tream chambers (Fig. 1). Nitrogen gas at room
0 temperature (22.5 C) is admitted into the upstre.am chamber through a ma11-
flow tran1ducer and a throttle valve. Downstream pressure is controlled
by the vacuum pump and an adjustable leak. The pressures in the two
chambers are measured and monitored by two sets of thermistor ~auges
calibrated against a precision mercury McLeod gauge.
Ma11 Flow Transducers
C0111Dercially available Hastings-Raydist mass-flow transducers,
with ali~ht modifications for increased accuracy, were used to measure
the mass flow. Two sets of transd ucers were useJ. one having a ran~e of
0 - 20 standard cc/min. and the other O - 100 standard cc/mira. The output
of the transducers were read on a precision (Laboratory Standard) millivolt
recorder, and the mass-flow meters were calibrated both before and after
the experiments against a precision primary standard Porter low-flow
calibrator.
Short Tubes
holes of l '' diameter drilled in 1811 diameter flat aluminum plates
•
l,
THaOTTLf N,---,...-9", VALVE
COLD TRAP
U"dio.
VALVE
- 3 -
...LD• 100• T 1-L, Varied fl'Offl 0.005• to 1.0"
--..... i------68"
TO - DlffUSK')N
,UMP
Figure 1. Schematic diagram of the experimental appa~atus.
,, - 4 -
of variou1 thickne11e1 were used as short tube,. Four geometries were
teated. All had the same diameter of one inch, but the lengths were
0.233, 0.493, 0.756, and 0.995 inche1. The orifice plate con1i1ted of
a circular hole, again one inch ln diameter drilled in a 0.005" thick
copper 1heet epoxied to a bra•• ring of 12" in1ide diameter. The 1hort
tube and orifice plate, were fitted with "O" ring 1eal1 into the dividing
wall in the tank. The size, of the vacuum tank and the orifice and short
tube plate, were 1uch that for all practical purpo1e1 one could consider
them a1 being in an infinite plane wall separating two larae chamber,.
Using nitrogen at room temperature (22.5°c) the mass flow throu~h
tubes and orifice was mea1ured for variou1 upstream and downstream pre11ure
condition,. For each of the tubes the maa1 flow• were measured at five
Knud1en number values, 0.124; 0.247; 0.494; 0.986; and 1.69. (The Knudaen
nmber in this case is defined as the ratio of upstream mean free path to
the diameter of the tube or orifice.) For each Knudsen number th~ ran~e
of pres1ure ratios covered varied from 1 to 20. The up1.tream pres1ures
were kept con1tant at 15.9, 7.96, 3.98, 1.99, and 1.16 micron• Hg. pre11ure1,
corresponding to the above Knudsen number value,, and only the down1tream
pre11ure wa1 changed to obtain the various pre1sure ratio conditions. Some
measurements were also taken at an upstream pressure of 23.8 microns Hg,
corresponding to a Knudsen number of 0.083 for small pressure ratios across
the tube or orifice.
•
- 5 -
111. ACCURACY
The pressure readings were accurate tot 1%, corresponding to
the ahaolute accuracy of the McLeod gau~e used for calibrat i ng the
thermistor gauges. ln the pres1ure range of the present experiment,, the
output of the thermiator v1. pre11ure i1 linear. Since the same pair of
thermistor• was used to mea1ure both up1tream and down1tream pre11ure1,
the ratio of millivolt output, should then repre1ent the pre11ure ratio•
and the uncertainty therein was within t ~%. For flow rate• greater than
3.0 • tandard cc/min. the flow readings were accurate within t 1~%. For
lower flow rates, it is probable that the error might be as hiRh •• t 3%
of !ts absolute value. Repeatability of the experiment• wa1 within t 1% •
I
- 6 -
IV. RESULTS AND DISCUSSION
The plots of the flow rate per unit cross-sectional area of the
tube vs. pressure ratios at various Knudsen number ran~es for all of the
tube eeometries tested are shown in FigureR 2 to 6. Referring to these
figures, it can be observed that at a Knudsen number of 0.12 the flow
rate is still increasin~ at pressure ratios as high as 20. 4 Liepmann
has pointed out that for the case of the flow of a continuun, gas through
a circular aperture, the pressure ratio required to choke the flow is one
or two orders of ma~nitude ~reater than for a convergent-divergent (Laval)
nozzle. The same argument can be extended to the case of short tube• in
wnich, due to small length-to-diameter ratios, the flow through them ls
still like an orifice-type flow. If one assumes that the flow ls still
behaving as a continuum-type flow at Kn• 0.12, the present experiments
substantiate the fact that pressure ratios required to choke the flow in
orifices and short tubes are very much higher than for a Laval nozzle.
For thl:! calculation of the Knudsen number, the mean free path
was determined by the followln~ relation
• .lli_ /RT, 5P, 2w
(1)
The value of the viscosity,~, used was 1760 • 4 • 10- 7 poise, correspondin~
to a temperature of 22.5°c.
Figure~ 7 to 11 show plots of nondimcnsionallzed mass flow vs.
pressure ratios at various Knudsen numbers. The nondimenslonalized quanti-
ties ~/mfm were obtained by dividin~ the measured mass flow rate by the
theoretical free molecule flow rate, which in tum ls obtained by multi
plyln~ the theoretical orifice free molecule efflux rate by the appropriate
•
- 7 -
Ciausing factor,
m • fm (2)
where A• cross sectional &iea of the tube; k • Clausin~ factor. It is
assumed that the temperature of the gas is the same on either side of the
tube or orifice and that the reflection of molecules from tube walls is
c"'11pletely diffuse. The Clausin~ factors were obtained by interpolatin~
0 between the values given in Demarcus' report . For tube ~eometries used
in t~e present experiments, the Clausing factors used are listed in
Table I.
TABLE I
Values of Clausin~ Factor for lube Geometrics
Used in the Experiment
Length/Diameter Clausing Factor, k
0.9946 0.5515
0. 7562 0.5797
0.4926 0.6752
0 .2325 0.8125
o.oos 0.9952
Cross plots of m/mfm vs. Kn at different pressure ratios are
presented in Fi~s. 12 to 16. Tables 2 throur,h)l list the mass flow data.
• 'It can be seen from these plots that at low Knudsen numbers the deviation
·I
- 8 -
from the free molecular flow values is ~reater at lower pressure ratios
and gradually decreases aa pressure ratio is increased. At high Knudsen
numbers the deviation becomes almost constant at all pressure ratios
for a given Knudsen number. It was also observed from measurements that
the m/mfm value decreases slightly as the length of the tube is increased
for the same diameter at all Knudsen numbers and pressure r~cios. Graphs
illustrating this effect at pressure ratios of 17, 10, 5, 3, and 2 are
shown in Figures 17 to 21 . 1'he scatter in the m/mfm at high Knudsen numbers
for pressure ratios cl~se to one in Figures 7 to 11 is probably due to the
fact that theoretical mfm were calculated by the pressure difference
(P1 - P2) across the tube for a given pressure ratio; since we are subtract
in~ two quantities of equal magnitude, a very ali,ht error in the pressure
ratio readings will be magnified in the pressure difference. (The present
results of m/mfm for the orifice differ by 2 - 3% from those reported
5 earlier. This difference is attributed to the alight uncertainty involved
in the measurement of the pressures. In the earlier work the thermistor
gauges were calibrated ag3inst a conmercially available McLeod gauge. The
absolute accuracy of this ~auge was found to be not as good as that of a
primary standards McLeod ~au~e a~ainst which the thermistor gau~es were
calibrated in the present runs and mass flow measurements repeated.)
For the creeping flow (flow in which the inertia terms can be
neglected in comparison to viscous terms in the Navier-Stokes equa ~ion) of
7 a continuum gas through a circular orifice, Roscoe has shown that the
pressure drop is related to volumetric flow, Q, by the relation
t:.P • 24 Q!!. 03
(3)
where~ is the viscosity of the gas and D the diameter of the orifice.
- 9 -
A similar expression relating the pressure drop to flow rate has been
8 obtained by Weissberg for short tubes
== ~ [1 + !2- L] 3 3n D
D
(4)
where L • length of the tub~. Both the equations (3) and (4) show that
for a fixed Land D the pressure drop is linearly proportional to the
volumetric flow. For the case of free molecule flow, the relation
between pressure drop and flow rate is
== 0.245 k • Kn (5)
where Kn• Knudsen number based on diameter and k the Clausing factor,
Q, lJ • and Kn values being evaluated at upstream conditions. (For the
derivation of the above equation the relationship between viscosity and
the mean free path as given by Equation (1) was used.)
From the measurements it was found that in the tran~ition flow,
for pressure ratios less than about 1.2, the flow rate was directly pro
portional to the pressure drop but the proportionality constant was a
function of Knudsen number and tube geometry. This is shown in Figure 22.
For the determination of the quantity Q/~P, the measured flow rates as
a function of pressure ratios were plotted and a smooth curve was drawn
through the experimental points. From this curve a cross plot of Q vs ~p
was made and the quantity Q/~P was determined. Table 32 lists the values
thus obtained for various tube geometries and Knudsen numbers. A semi
empirical equation which was developed (see below) to correlate with the
mass flow measurements was also foJnd to fit fairly well with the measured
flow rate at extremely low pressure ratios, as shown in Figure 23. The
empirical equation is valid or.ly in the transition or near free molecule
flow regimes.
- tu -
9 Willis has theoretically analyzed the mass flow through a
circular orifice at very hir-h pressure ratios in the near free molecule
flow re~ime. His final calculations are restricted to one iteration in an
inte~ral iteration scheme. Comparison of this result with the orifice data
at a pressure ratio of 17 is shown in Fi~ure 24. The theory underestimates
the mass flow slightly for Reynolds number, Re • ( DPl ) less than one. '/RT 1 lJl
As pointed out by Willis there is no reason to expect the first iterate
solution to ar,ree with data for Re> 1 since further iterations are then
necessary. Although comparison is made of the experimental data at a press
ure ratio of 17 with the theory which assumes infinite or very large
pressure ratio across the orifice, the m/mfm values for high Knudsen numbers
have almost become independent of pressure ratios from 5 to 17 (Figure 7)
and it is therefore reasonable to assume that the same values hold ~ood for
lar~er pressure ratios.
Milligan2 in his work on nozzle characteristics in the transition
regime did some experiments on the flow throu~h a short tube of L/D • 0.312.
His plot of c0 vs. D/2~ show very little dependence of c0 on the pressure
ratios 2 to 20 over the entire transition flow regime. (C0 • discharge
coefficient, defined as the ratio of actual mass flow to one-dimensional
isentropic mass flow for choked conditions
C • D
m meas ti,
isen •
2 y+l
m meas (6)
where y is the ratio of specific heats, R • the gas constant, A• cross
sectional area of the tube. The subscripts l refer t~ upstream conditions.)
- 11 -
Fi~ure 25 is a plot of the present results for a short tube of
L/D • 0.233. The abscissa and ordinate h~ve the same meanin~ as described
above. The plot bhows a very si~niflcant dependence on the pressure ratios
in the transition re~im£, an observation contrary to those of Milligan.
The Poiseuille formula for the mass flow for lon~ circular tubes
with slip boundary conditions is
m [ ,rl)
4 p 6LP + 0.519 o
3 6LP]
• 128 uRl C
(7)
Pl+ p2 where~• 2 and 6P • r 1- P2• u • viscosity of the ~as. c • avera~e
molecular velocity, L • length C'f the tube.
The al ,ve equation is similar to that 1tiven in Present's booklO
except that it is modified by the use of Equation (1) for the relation
between viscosity and mean free path.
From the measurem~nts it was found that for a ~iven Knudsen
number. based on upstream conditions and tube diameter, the decrease
in mass flow as the tube len~th was increased was proportional to L(l+D/L).
at a 11 of the pressure ratios. It was a 1 so observed that these measured
values could be well correlated with Equation (7) when the equation was
multiplied by a factor (L/L+D). The modified Poiseuille equation is
• m • [-~4__ p /\P + 0.519 o3 t,p] _l __
l2t3 uRT - L( l+D/L) C
(8)
This semi-empirical equation wJ 11 predict the mass flow fairly well for all
pressure ratios, for the ranr.e of Knudsen numbers and L/D ratios covered
.. - 12 -
in the present experiment when the viscosity and temperature are evaluated
at upstream conditions.
The comparison between the measured values and the Equation (8)
are shown in Figures 26 to 30. The \;·~dinate represents the volumetric
flow per unit cross sectional area of the tube or orifice times L(l+D/L).
The solid line represents Equation (8) converted into proper units plotted
in the graphs. It could be seen that the a~reement i1 very good at high
Knudsen numbers, and as the Knudsen number is decreased the semi-empirical
theory overestimates the mass flow by 5% at hi~h pressure ratios. Although
the experiments were done at a fixed diameter of 1• · for all the tubes and
orifice, the semi-empirical relation developed will be valid for other
diameter••• well since Equation (8) can be converted with a little algebra
to
m/unit area. (9)
Thus for a ~iven Knudsen number, L/D, and pressure ratio, the ratio of
mass flow per unit cross-sectional 3rea is independent of the diameter
of the tube. The ratio of mass flow as ~iven by the Equation (9) to the
theoretical free molecule flow value is
l 2k( l+L/D)
(10)
(k • Clausing factor)
40
t o'-' 0 N
""O C C 3 C, ... ... 0 ._.
0 -0 ........ -C
rs 2 i)
E V
----·e -\,i V
> 10
r.J. -C
°' ~ 0 ~
0
I-
~ 0 5
- i 3 -
10
I
1-,-r · D · ~
l - r . < o - - j
P1
• 3 98 • 10 J Torr
1 99 • 10 J Torr
• I 16 • 10 J Torr
15
Pressure Ratio, Pi/P2 -
20
Kn= ,\1 /D
0 494 0986 1690
25
Figure 2(a). Fiow rat e at vari ous i'>ress urc ratios for an orifice
' I ;v 0 N
""O C C
s ._.
~ " -C
N
E u -., E -~
-~ ·> Ji 0
°' )
..2 ~
20,
15
10 ---- -
5
5
P1 Kn= \ 110 P, • D • P, \ r l
• 15 9 • 10 3 Torr 0 .124
b =ooos
10 15
Pressure Ro tic,, P1 /P2
7 90 • 10 3
Torr 0 .247
20 25
Figure 2(b ). Flow rat e at various pres sure ratios for an orifice
(~ = 0 . ()05) . l)
f u
0 0 N
,:, C: 0 t:: 0 ~
0 ,()
" -0 N
E u -C: ·e -u u
·> Ji 0 ex J
.Q ~
,o
30
20
10 --
0 0 5
- 14 -
lL p • ::, ••
I C
·: r ( l - ~ ~ 1 • , . - ! • ' ; ,,
0
•
10 15
P1
399 • 10-~Torr
1 9Q • 10 1 Tor , ! i 7 x · 0 3 Torr
Pressure Ratio, Pi/P2 -
Kn= -'i/D
0 ,93
0987 1 ()00
I - 7
25
Figur e 3(a). Fl 0"' r at , · a t vari o us pre s sure ratios fot ~ short tub~
havi n g a 1e n ~ r ;, t o J ~a md c r rat~,, of 0. 2 33.
t - , I u
0 16 ~ 0 N I ,:,
C:
l 0 .. P1 Kn= A1 /D .. P, • D • P_, 0 12 15 9 • 10 3 Torr ~ '' 1r-1
• 0.12, 0
0 7.98 • 10 3 Torr 0 .2•6 -0
" b = o JJ,~
I -0
'"" 8 I E -~ u -C
E -u u
J ---------·> ~--0
ex
J 0 .Q
~ 0 5 10 15 :,0 25
Pressure Ratio, P1/P2 -Figure 3(b). Flow rat e at various pressure ratios for a short tube
having a llmgth to diameter ratio of 0.233.
L
t ;u 0 N
"i o 3.0 .. .. 0 ~
~ " -0 2.0 N
E ~ ., E -u 10 ~
·>· .. 0 ~
• 00 0 ~
- 15 -
P, - · D • P1
,\ . ~I~ l
P1
• 3 98 • 10-3 '!'orr o I 99 • 10 3 Torr
• 116 • 10-3 Torr
Kn= 'A.i/O
0493 0986 1690 - . - __ _ .....
. 1
I I
____ J I
l , I ___.o---~, ~~~-~----- :
I ' I
5 !O 15 2C, )O
Pressure Ratio. P1/P2
Figure 4(a). Fl o· . .- rat e: a t v a rious ~re ssure r a tio s fo ~ a s hort tube
t 20
;u ~
1 ._ 15
J ~ -0
hrtv i ng :J le :1 g t i, t o cii ame t e r r a ti o o f 0,49 3 .
----- -----------P,
• !~.9 • 10·3 Ton o 7.96 • 10-3 Ton
I !
Kn•'A.1/D
0.124 0.247
N 101--- -#--- -~-----<f---E ~
·I -u ~ 5ll--,D,J~~~~:::e:::::::~=~:-::~_:_:_:_:~_:_j__c_:_:_:_:_:_:_:_:~_.._---~
10 15 20 25 30
Pressure Ratio, Pi/P2
Fi~ure 4 (h). Fl m, rat t' at va ri o u s p r ess ur e ratios for a short tu1J e
i1 av i11~ a lcn ~ t h to d i ame t e r ratio of 0,493.
,,, t 24
.u 0 N
~ 20 0
t: 0 ~ 1.6
~ " -0
12 N
E u ·-C
!_ 08 u ~
> Ji 04 0
°' J 0
a: 00 5
- 16 -
- ----- -·----~ - ~
Y P1 ·3
P, • D • P, • 3 99 • 10 Torr -', n l 0200 • 10 3 Torr • 1.16 • 10·
3 Torr
b= o 756 I j
10 15 20 25
Pressure Ratio, Pi/P2
Kn= A1/D 0493 0 .984 1.690
30
Figure S(a). Flow rate at various pressure ratios for a short tube
having a l ength t o diameter ratio of 0.756.
t 12.0
;;u 0 N
~ 0 .. 0
10.0
~ 80
~ " 0
6 .0 N
E u -C
~ 40 ~ ~
·> of 2.0 -0
°' J
y P1 • D • P, .,\, n l
b = 0 7S6
P, • 15.9 • I0 3 Torr o 7.98 • 10 3 Torr
I
r I
Kn= Ai/0 0.124
0 .247
0 a: o ........... __._ ....................... ...._ ........ __..__._ ....................... ...._...__~-------...... -----------
0 5 10 15 20 25 30
Pressure Ratio, P1/P2
Figure S(b). Flow 1·ate at various pressure ratios for a short tube
having a length to diameter ratio of 0.756.
-~·-------·---------------- . --- ----
t
~
0 I-
~ 1.6
" 0 r... 1.2 E u -C
! 0.8 u
.!:!.
·> o., Ji 0
°' ) 0
u: 5 10
- 1 7 -
u P • D • P.
• 1-r7-P1
• 3 98 • 10 3
Torr 0 199 • 1ff3 Torr
• 116 xlff 3 Torr
15 20 25 30
Pressure P.atio, Pi/P2 -
Kn= -\ 1/0
049A 0986 lo90
35
Figure 6(a). Flow rat e at various pres s ure ratios for a short tube
having a length to diameter ratio of 0.995.
t ;u o 12.0---~---~- r---N
~ C 1
~ 10.0 - ------=-=~:i::;;...;.~-+-~;;;;;;;~-----...... -r---r---1 ~ L.J 0 -o 8.0
" --0
e 6.o
~ C .E ~ , .o $ ·>
•>' 2.0 0 °' • 0 ~ 5 10
P, •0 •P, ,\~n -l l D = 0 995
15 20
P, • 15.9 x 10 3 Torr o 7.97 x 10·3 Torr
I
Kn: \1/D 0.12,
0.247
--7 - - -- -- -----
25 30 35
Pressure Ratio, P1/P2 -
Figure b(b). Flow rate at various pressure ratios for a short tube
hav1ng a len ~th to diameter ratio of 0.995.
2.1
1.9
1.7
t .! 1.5
~ ·E
1.3
1.1
- 18 -
r l
P1 • D • P7
• 15.9 x 10 3 Torr o 7.96 x 10-3 Torr
• 3.98 x 10 3 Torr A 199 x 10·3 Torr 0 u~ X 10-3 Torr
A, f .L l
~ =0005
Kn=A1/D
0.124 0.247 0.494 0.986 1.690
5 10 15 20 25
Pressure Ratio, Pi/P2
Figure 7. Nondimensiona liz e d mass flow vs pressure ratio at various
L Knuds e n numb er s for an orifi ce (~ = 0.005).
2.1 --- - ---.- ·- --
1.9 - - -
1.7
t j 1.5
·E -·E
1.3
1.1
-r - --- ---- ~ ----- ·, I I
l P1
• 159 X 10 l Torr
P, • D • p~ 0 7.98 x 10 3 Torr ,\ , -ff\ • 3 99 x 10·3 Torr
6 199 X IQ J Torr
~ =om ~ (J 117 11 10 3
Torr
Kn= 1'1 /D 0.124
02'6 0 .493 0.987
1.690
09......_. ........ _.,_,___.,_ __ ....l......J..- ___ ,_ ,_,_......__.__.__.__._ ___ ....._....._ ....... _._ ...... .-.._.._~
0 5 10 15 20 25 30
Pressure Ratio, Pi/P2 -
Figure 8. Nondimens iona liz ed mass flow vs pr~ssure ratio at various
L Knudsen numbers f or a s hort tube, 0
• 0.233.
l ,.
I
•
- 19 -
21
19 P, l"n= .\ 1·D
. 15 9 • 10 3 Torr 0124
17 ' I
~ . C . ~ ,, 7 Q6 • 10 J Torr 0247
• 398•101
Torr 0 4Ql
t E
19~ • 10 3 Torr 0 986
116 • 10 3 Torr 1690
·e 15 --..;;. ·E
1.3
11
5 10 15 20 25 30 35
Pressure Ratio, Pi/P2 -
Figure 9. Nondimensionalized mass flow vs pressure ratio at var~ous
Knudsen numbers for a short tube,~ .. 0.493.
2.1 ....----- i --1 1.9
1.7
t E -e 1.5
~
1.3
~ I
t I • 11 "»~ * d a
~ 0 0
09 0 5 10
P, .. \ ,
- r - -- --,
P, • 15 9 x 10·3 Torr
o 7.98 x 10 3 Tor~
• 3.99 • I0'3 Torr
n 2.00 • 10 3 Torr . j
•' l.16 x 10 Torr
i
I
Kn= .\1/D 1
0124 0 .247
0 .493
0.984 !690 ,
---- i
I I I
0 ' ol • -
Orr:, lo-0 Q
' ' ' ' ' ' I I j j I ' I 15 20 25 30 35
Pressure Ratio, Pi/P2 -
" Figure 10. NondimensionalizeJ mass f101,,; vs pressure ratio at various
Knudsen numbers for a short tube, ~ '"' ll. 756.
,,
2.1
1.9
1.7
t J
·E 1.~
:e--1.3
1.1
-~ ---- -
-... - \ --
:_\ :\_
"II
~
... ~ --...
... -.. ·- ~ . ... - - -•- -...
I I I I I I I I
5
- 20 -
I ~ -- I I
P, Kn= ",ID L,.J • 15.9 x 10-3 Torr 0.12 ..
~-- -- - ~ - ~- o-~ o 7.91 x 10-3 Torr 0.2,1 -~~l • 3.98 x 10-3 Torr 0 .494
~•0995 A 1.99 X 10·3 Torr 0986
-- - -- a 1.16 x 10-3 Torr 1.690 -
---
- - - -- . - - - -
I I I I I I I I I I I I I I I I I I I I I I
10 15 20 25 30 35
Pressure Ratio, P1/P2
Figure 11. Nondimensiona lized mass flow v s pre ssure ratio at various
Knudsen numbe r s for a short tub e ,~• 0.995.
- 2 1 -
2.0
18
1.6
t 1 , ...
·E -·E 12
10 -
08 001 010 10
L. P • D • p
I t-1 l
l - '"'" 5 o -
----
P1/P2
20 0 30 a 50
• 100 ----i 6 170
frN Molecule Flow limit
-::\
100
Fi~ure 12 . Nondimensionalizt•ii ma ss f lo.,_· \' S Kn ud st->n num1'cr at various
L pressure ratios for an orifice, D • 0.005.
- 22 -
20
l P1/P2 18 • 20
p • D • P_, -~ fl 0 30
0 50 16 b =om~ • 100
t 6 200
l I~ ·E -·E
12 frN Molecule r•.,w lim,t
10
O BL---l-~-L......1.....1.--...1..J..1. __ .....1._...1....-1.......1.._._1...1.. ...... __ _.__..._...._ ...... ......._......,.
001 010 1.0 100
Kn= 'At/D-
Figure lJ. Nondimensionalized ma ss flow vs Knudsen number at various
pressure r a tios for a short tube, ~ = 0.233.
t
20
18
16
12
10
Pi/P, 2.0
0 30 • 5.0
-1 I
7 = ;~~ l
----~
FrN Molecule
_____ ~mil I
08'--~~.............._~___._ ............... ...........,_.........__,____.___.~J 001 010 1.0 10.0
Fi gure 14. ~ondimensionalized mass flow vs Knudsen number at various
L pressure ratios for a short tube, D • 0.493.
-
20....----- --- - - -,--
18
1.6 -
t ! 1., - -
·E -·E 12
1.0 -
- 2 J -
- - --- - ~ -------- - --,
u P • D • P
' .(7.
P1/P2 20
C 3 0 Cl 50
• 100 6 30 0
- ~
I I
Fre1t Molecule ' Flow l1m1I
~ · 0.8 L--------1.---l.-..1...J..J...J.. __ ....__..,___......__.1....1, .......... ___ ....,_____..._.._ .......... --LJ
0.01 0 10 I 0 10 0
Figur e 15. Nondimensionaliz e d mass flow vs Knuds e n numb e r at vari r us
L pressur e rati o s for a s hort tube, D = 0.75b,
2.0
1.8 - -
16
e u --e -·E 1 2 -
10
p • D • pl
P1/P2
• 2 0 (, 3 0 ,\,nl n 5.0
• 100 6 30.0 .... J
frff Molecule Flow l1m1t
Fi~ure 16. Nondimension a lizc d :1:as s flow vs KnuJs e n numbt! ." a t various
pre ssur e rati os f o r a short t Jbe, !:. = U.99 5 . D
1.7 ..... ~
._
l.5 .....
t .....
! 1.3 ·E ......_ ·E
-.....
-~
1.1 ..... .... ....
0 .9 0
-
-
-
I I I I I
0 .2
- 24 -
7-\ P, /P, = 17.0 l
' I
Kn=0.12• -
0 .2,1 u
o.•9• A .. n.986 - -• l .6YU
I I I I I I I I I I
o.• 0 .6 0.8 1.0
L/D ....
Figure 17. Nondimensionalized mass flow vs length-to-diameter ratio 0f
tubes at var i ous Knudsen numbers at a pressure ratio of 17.
1.7
1.5
t ./ 1.3
~
11
~
.
0.9 0
• -
0.2
1
(:; s 10.0)
Kn s 0.12• _ -I 0.2,1
\ - - ....
o.•9• • 0.986 • -
I - -1.690 D
I o.• 0.6 0.8 1.0
L/D-
Figure 18. Nondimensionali ze d mas s flow vs l ength-to-diameter ratio of
tube s at • ari 0us Knudsen numbers at a pressure ratio of 10.
1.7
1.5
t J l J
·E 1-
1.1
0.9 0
T -I I -I . ..
I
--
I I
I 0.2
- , -- --, - .
·1 I
B I
Kn= 0.12,
0
--I I I
0.2,1! I I I .., ... o.,9, i
• -0.9AA I
... ,AoO I -
~
I
I I I o., 0.6 08 1.0
L/D-
Figure 19. Nondime ns ionalize <l .nass flow vs l ength-to-diameter ratio cf
tube s a t va ri c us Knud sen numb e rs at a pressure r&t~o of S.
1.7 ~
1.5
t J
I l
1- 1.3
1.1
0.9 0
I I
-
0
.....
-A
0.2
l
~ Kns0.12, -
I
I -
0.2,7 I ... -
o.,9, -0.984.'! -
... ,"90 ~
A ~
o., 0.6 0.8 1.0
L/D~
Fi~urc 20. Nondime ns i onali zed mas s fl ow v ~ len~th-to-diameter ratio of
tubes at vari ous Knuds en numbers at a pressure ratio of 3.
, 9
1 7
t 1 5
E 0 74
·E ·E
1 3 0 49 ~
0 980
11 H u t
1690 r,
Q9 L-________ _. ________ ~_..._._------
o r, : 0 4 0 6 0 8 l 0
L 10 - -
1 ' • t J'. \ ' - \.. 1 t l 1 -
) ' . I .
r-: ',,:
• 1 '
- :>. 7 -
f,.. Molecule flow Theo,y I
'b ~ .10 ~ - -- -----+--...... C--,l~~:::.,__+---------1 --&- Continuum Flow
l,mitt • • l/D•0.005 (., 0233 6 0.493 0 0.756 • 0.995
Kn='A.t/D
Figure 22. The quantity 7 as a function of Knudsen number for !lpl) p 1
small pressure ratios, p ~ 1.2. 2
10
-..... ,o + ---=:... 0.1- --
l!t"I
::t 0
I (00245 +0.255Kn)
l/0 • . oos
0/!;
~ ~ :!~~ I C .756 • .995
.0IL-~....J..~.1...l ul lL.1.1..J1 j1...-.__,j-._l ..i.i-i~.._.11u.1.i..ll'-----'--li.....i.l...il...il...il,.....I I 0 .01 0.1 1.0 10
Kn=~i/0 -
Figure 23. Comparison of the measured data with the semi-empirical
Pl ~quation (8) at small pressure ratios p ~ 1.2.
2
- 28 -
1.5
f 1.3 - ----- ,.,~-
J ~ ·E ~~ £ ,~
1.1 ~ - --------..-----t-Ar""----::--
1.0 10.0
D1 Pi Reynolds Number _. {Rf, /J
1
100
Figure 24. Comparison he tween theory and experiment
o f th e nonui me ns i on r1li zed mas~ flows vs Reynolds
nurnh1.:r f or thL· ca ~c. ot" a thin orifi ce .
- .~Y -
lL P, • 2r • P7
u
l .11.0
A, -ff\ P,/P,=2 i,=0233
5 10 20
·E --1 ·E II a u
--~•=r---- ----- ----.---.---- ----.--------r--frH Molecule Flow Limits
I o.1 L-_.....1.._.J_...L,....L....,L..1.,.,L,'-,i.__......__...._...._ .............. ~,o o
0.1 1.0 .
r/]\.1-
Figure ?5. Disc harge coefficient, c0
v s inverse of
Km idsc n number (hascd 0 1~ radiu s ) for a tube o f
L () 2 ·1 " • t . d . . i5 = . , J at vc'.lr:ous pn: ~s~ire ra 10 con 1t1ons.
0
t 4 0 ,
u 0 N
-0 C O 3 0 ... J 0 -0 r,
c , o E ~ C
E ---u u 10 -a ,~ +
----~
- w -
P1 = 116 • 10·3 Torr .Kn =A 1/D = 169
u L/0 = 0995 P • D . P 0 0 756
'17 I\ 0 493 n 0 233 • 0005
Eq 18 )
·> () .._._ _____ ...__...___......__...__._._._._._._._._...__ ....... ~..._...._..._.._........, J 10 15 20 25 30
Pressure Ratio, P1/P2
Fi gu rl · ~ ti , Cll!:1p..iris o 11 o t tl1l · :rw ,1surl' d flow rate for va ri o us tub e geometrit.."s
F i .:11 rv
;it .i l, n 11ds t·11 numh t· r , , f l . h CJ with tlH.' scmi-L"mpirical e quati o n (8).
6 0
t 5 0
OU 0 N
-c, 40 C 0 ... 0
I-
~ 3 0 "'--0
E u
---- 2 () C
E ----u ::'
- 10 a l~ + _, >
I\
n
I - -- - - --
0 4':'J
0.2'33 • 0005
Eq 8 ~
---_J o ....... ..__ ______ ~ ____ ...__ ___ ....__ ___ ___._ ____ .__ ___ ,
(' 5 10 15 20 25 .30 3.'i
Pressure Rati,,, P1/P1 -
' -- I • Cor.ipar i son of Uw measured flow rate for various tube geome tries
at a Knudsen number of 0.986 with the semi-empirical equation (8).
---------·--------
- JI -
12 - 1
t I
OU 0
10 N .,, C 0 .. .J .. 8 P,=398x10 Torr , Kn:A i/0=0493 0 .,_ L
0 • - = 0995 ,0 u D
"' 0 0 756
- 6 P • D • Pl A 0 493 0 \ .f7i- I E 0 0 233 u • 0 .005 ........ C
Eq 8 ·e 4
---u ~
2 +----0 /-' + _, 0 ·> 0 5 10 15 20 25 30 35
Pressure Ratio, PiiP2-
Figure 28. Comparis on of the measu r e d flow r a t e f or various tube geome tri es
t 25
OU 0 N
20 .,, C 0 .. .. 0 .,_
0 15 ,0 i",
-0
E 10 u ........
C ·e -\,/ ~
5
01_, + _, 0 ·>
at a Knuds e n numbe r of O. 494 with the semi-empirical equation ( 6) •
..
0 5 10
. r
•
P1=797 • 10·3 Torr . Kn= .\1 /D= 0 247
u P, • D • P1
'.ff
15 20
Pressure Ratio, P1/P2 -
• .l = 0995 D .
o 0756
A O 493
a 0.233
• 0.005 Eq.8
- - +- - --- - - -I
I
25 30 35
Fi ~urc 29. Compari s on of th ~ meas ure d flow rate for various tube geometries
at a Knuds en numb e r of 0 . 247 with the semi-empir i c6l equation (8).
60
0 t E C) !!J • I'. r.1:1 . /\ <, ... :. . : ~ u u ..__ co 40 -0 EN ...__
P1=159x10 J u~ Torr Kn = ,\ 1/D= 0 124 u C
0 p · D • f • L/D =0995 - ... 0 756 ......J ... 20 , r _J _ --7 0 - 0 0 ...... . f- /\ 0493
+ 0 :7 0233 -0 • 0005
......J ,-...., Eq (8 ) >
(1 () 5 10 15 20 25 JO
Pressure Ratio, P1/P2
i . i 1.:1 1 r, · l( I . ,, r var i uu s tul ,( • ~l< 11: ,t · tril' ;.;
- 3J -
V. CONCLUSIONS
The mass flow throur,h short tubes has been measured for the
ran~e of pressure ratios from l to 20 in th~ transition flow re~ime. The
Knudsen numbers based on diameter varied from 0.1 to 1.7. For the case
of creeping flow, the pressure drop is linearly proportional to volumetric
flow as predicted bv both continuum and free molecule flow theories (for
fixed Kn). The slope of 0 vs. 6P was a function of Knudsen number and
lcn~th-to-diamcter ratio of tubes, and the results show a ,·ery smooth
transition from the constant continuum value to a linear function of Knudsen
number in Cree mol~cule flow. At hi~h pressure ratios, for Reynolds number
around one, the measured values are slightly hi~her than Willis' first
iterate solution, for the case of an orifice. The mass flows were hi~hly
dependent on the pressure ratios for a ~iven tube ~eometry and pressure
ratios ~rcater than 30 are required before it becomes constant for a ~iven
upstream condition. For a ~iven upstream pressure, the decrease in the mass
flow as the tube lcn~th, L, is increased (for fixed diameter, D, of the tube)
was found to be proportional to L(l+D/L). A semi-empirical equation was
developed to predict the mass flow throu~h short tubes, and this equation
is identical to Poiseuille equation with slip boundary conditions multiplied
by a factor which is a function of tube ~eometry alone. This equation was
found to be valid at all of the pressure ratios and Knudsen numbers covered
in the present experiments. At hir,h Knudsen numbers, the experiments
showed that the measurements were approachin~ asymptotically the theoretical
values predicted by Clausin~ for short tubes.
BLANK PAGE
- 35 -
REFERENCES
1. F. O. Smetana, A.S.M.E. Paper 63-WA-94 (1963).
2. M. W. Milligan, AIAA Journal 2, 6, 1088 (1964).
3. H. Cook and E. A. Richley, NASA TN D-2480 (1964).
4. H. Liepmann, JouT. of Fluid Mech. 10, 1, 65 (1961).
5. A. K. Sreekanth, Proc. of Fourth International Symp. or, Rarefied Gas
Dynamics (1964) to be published by Academic Press; also Boeing
Document Dl-82-0322 (1963).
6. W. C. Demarcus, AEC Report K-1302, Part III (1956).
7. R. Roscoe, Phil. Mag. (Ser. 7) 40, 338 (1949).
8. H. L. Weissberg, Phys. of Fluids 1, 9, 1033 (1962).
9. D. R. Willis, Jour. of Fluid Mechanics ~. 1, 21 (1965).
10. R. D. Present, Kinetic Theory of Gases, McGraw-Hill (1958).
- 36 -
ACKNOWLEDGEMENT
The author would like to express his thanks to Dr. H. Harrison
and R. Goldstei,1 for reviewin~ the ~anuscript, and to Mr. R. Bowey for his
help in conducting the experiments.
l
A
C
u
k
Kn
L
• m meas
p
4P
Q
R
R e
~fm
-----------------
- )7 -
NOTATION
cross-sectional area of tube or orif ice
average molecular speed
diamet e r of the tube or orifice
Clausing factor
Knudsen number,= l
l)
length of tube or orifice
theoretical free molecule mass flow 0 oz A)
measured mass flow
upstream pressure
downstream pressure
pressure drop across the tube or orifice, (P1
- P2)
volumetric flow at pressure P1 and temperature T1
gas constant
Reynolds num~er, •
0 upstream temperature~ 22.5 C
theoretical free molecule volumetric flow
- 38 -
v meas - measured volumetric flow
y - ratio of specific heats
\
1 - upstream mean free path; ~(ems)• P
1(microns) 4.992 ---
ii 1 - 0 -7 viscosity of nitrogen at 22.5 C • 17(~.37 x 10 poise
(Subscript 1 refers to upstream conditions.)
- 39 -
TABLI•, 2
pl l S. 91 - '\
T 22.s"c: ~ l 0. 11 1.1 X IO T n r· r; cm~; L 0.0127 cm; I ~1
D 2.S4 Kn O. l z !~; L o.oor;. ..:: cm; o [)
p • ( I • 760 To rr & 20°C) m 2 c-r 1 tnlrl al m"" ~
p / p • • -1 V Th0ory \" "' l ' 2 X 10 Torr frn m0a s f rn
1 • 1 26 I 4 • 1 '\ 8.102 17.92 2. l 'i8
1 • 081 14.72 s.sso l 2. 2 l 2. 199
I 2. SO 1 • 27'\ 68.28 96. l !~ 1 • '~07
16.61 0.9S8 69.72 98.00 I . 407
9.09 1 • 7 so 66.04 94.98 l • 4 '\ 8
6.9s 2.289 61.S2 9).26 1. 468
s.821 2.712 6 I • 4 7 91.80 1 • 491
4.740 1.1S6 '18.Sl 89.69 1. 'il2
1.6s8 4. ·,49 '11.92 86.09 1 • '196
2.790 ·,.702 47.62 79.92 1 • 678
2.780 r;.721 1~7.r;2 79.92 1.682
2.190 7.264 40. 'JS 71.90 1 • 782
I. 800 8.818 12.98 62.06 l. 882
l. S08 IO. r; SO 2,;.00 49.99 1 • 999
I . 121 12.010 18.09 18.09 2.10s
l. 211 11. 1 20 1).02 28.09 2. 1 ,;7
l • 1 S 2 l'J.81 9.791 21.46 2. 190
,, I
7 . C) i 'i '(
p ,' p I 2
I • OH i,
I • I 2 1!
I • 200
I • 27 l>
I • 1, '\ 7
I • 0 0 C)
1 • o!,o
I • 08'!
I • I 1, IJ
I • I 1! IJ
1 • 220
I • '\ 2 I
I 7. 0 ', 0
1 2 .H0
I '2 .H7
b. 2 7
7. 09
'LI 87
' • q ,, '2
I • (, -, 2
I • ', -; 0
I O - \ .I, ·r ,q· r; I
I' 2
- '\ x IO r, , r· , .
7. "iG2
7. T\<l
6.(, 2 <)
G.c '\ 1J
r; • r; '\(,
7. 88 ,,
7 • ll 1! q
7. '\ ' \I)
6.9 'i 4
6.<) 'l 'J
(,. ·i20
(1 • 0 2 '2
'i • i 1-+ 0
0 • 1J (> 7
0.(>21
C,.(118
0.881
I • I 2 2
2. I ·S6
2. ·'-'9 6
' \. '2 ' \()
,, . 8' )
- 40 -
TABLE '\
22. r;0c; x
1 o.627'i cm; L
x, D
O. 2 1J 7; L n 0.00'i.
( c c · I III i II a t 7 (, 0 T n r r· & 2 0 ° C ) • v· V f . ·1·1 Ill l('()f'Y llll'il~
!J. O 9 S
<>. 184
8.026
I 1 • 280
v. '\ ·~o
I • 4 27
2. 8 7 '\
4.668
'" • (, 6 8
6.691
9.016
I I • 270
., !" • () ~
., '-' • 2 I
'\4. 2 I
., 2 • () C)
'\ I • l 8
·n . 8 7
29.71
27.0i
2 -; • '" (,
22.0·~
18.00 I 4 . (> ·i
6.808
12.670
l7.1'i
O.GGO
2.22
IJ. 68
7.s4
7.s1
10.s7
I 'L 97
l 7. I 7
44.0l
41.66
41.61
'J 2 . 'JO
40.77 41.s ·,
19.,;9
17.01
-~ 'i • I 8
'\ I • '\ 7
26.48
2 2. IO
17.82
0.0127 c m~;
• Ill
lllf'a~
,,. r· m
I • 70 I
I. 676
I • 662
1 • 6 l 4
I.S78
1.s17
I • 997
1. s ss
I • 6 29
l • 6 l 4
l • 6 I 2
l.'378
l • 'i49
l. '324
l. 260
1.276
I. 274
I. 28'i
1 • 107
1. 101
1.112
1 • ·368
1 • 181
l. 424
1.471
I • S08
l.~47
"
- 41 -
TAHLI-. !.s
p '\. 98 - '\ (I
I • 2 "i !.i X IO Ton· ; T 22.r; C; xi cm:--;: L 0.01 ~7 < Ill ; 1 I
xt D 2 • -; '.s Kn 0 • '.s C) '-' ;
L o.oo s . C' Ill ; I) lJ
"2 (c-c ,.111in il t 7(>0 Torr · & •
fl m 20 C ) Ill I• i I :--;
p I p . . 10- ·\ \ " \ '
"' f I . 2 X '> r· r fm T11 : •01 · y OH•,l:--; Ill
4.680 0. 8,0 l !L60 I 7. I 9 I • I 78
'\. '\'\ 2 I • I 9 ,1-' I'\. 00 l'i.S "-i I • I C)6
2. '\60 1. MH> 10.70 I 'LO& I • 2 21
2.000 I • 990 CJ. 281 1 l • -'J 7 I • 2 '\ -;
I • 670 2. "\8'\ 7.448 9. ·w7 I. 2-;o
I. 029 '\. 868 -; • 2 2 l.i o.64o I • 2 2 '"i
1 .O"iO 1.790 8.861 I. 1 1 0 I . 2 "i ' l
1 • 08 'l 1.668 I • '.s -; -; l • C) 20 l • '\ l 9
1 • I l.i4 1. ,, 79 2,117 2.99 I. 279
I • 200 '\ •. , I 7 ., • 0 9 2 ., • <J l) C) I • 29'\
1 • '\60 2. 9 2(> 4. 9 I (> 6,178 1. 297
l '\ • '\ l 0.2<)9 I 7. l 7 19,98 I • 161
1·,.21..i 0. '\O 1 l 7. 1 6 19.98 1 • I 64
17.46 O. 228 17.-;o 20.26 I • 1 i 7
8.64 o.'.i61 16. ,,1 19.1 •-; I • I 66
It. IO 0. '\'i9 1 6. 8CJ 19,68 1 • 16 S
r;, l-'O 0. 7'\ 7 I r;. 11 17,90 1 • 184
6.60 0. 60 '\ 1.-;.71; 18.S8 I • I 80
4.so o.884 14.44 17. l 4 I • 1 87
- 42 -
TABLE 5
- '\ 0
X l p I • 991 X 10 Ton· ; T ?.2.'i C; 2. 'i0'i cm~; L 0.0127 cm; I I
X 1 () 2. 'i4 Kn - 0.986; L 0.00'i. cm;
D ()
p2 •
(cc / min al 760 Tc,rr & 20°c) m mea~
p I p • v - '\ vr Th(' OI'Y 111
I 2 X 10 Tor· r m m<- as fm
·,. o·w 0.6'i8 6.226 7.04 l.110 -2.210 0.902 'i.088 "i.79 1 • l '\ 7
I • 770 1 • 1 26 '.i • o 4 '\ 4.64 1 • 147
1 • ,, 8 1.J l. '\41 1.011 1.4'i l • I 18
1 • 294 l.'i l.~o 2 • l 1 '\ 2.41 1 . 1 40
I • 186 l.680 l. 460 1.65 l.129
l.ll'i I • 787 0.960 l • 1 0 I. I '-' 'i
1 • 076 l.8'i2 o.6'i8 0.71 1.079
1 • 040 1 • 916 0 • '\ 'i 9 0 • ,, 1 I • l 4 1
1 'L ·10 0. I '~8 8. Ll 0 1.J 9. 41 l • 091
17.66 0 • l 1 '\ 8. 7(· 9 9.614 l.096
7.80 0.2'i6 8. 100 8.958 1 • 1 0 'i
l 'L 'i 'i o. 147 8.609 9.455 l.098
'i. l.Jl)'i 0. 16'\ 7.602 8. /-4,29 1.108
1.604 0. S 51 6.717 7.557 l • l 2 'i
'}. 0 27 o.6'i8 6.226 7.014 1.129
l I • 810 0. 168 8.'il2 9.150 1.098
p
~-
- 4 3 -
TAI\Ll•, (,
i I I 6 ..., I O - 1 ·1· 1· • ~ x , , r· r ;
1 4.292 ems; L 0.0127 ems:
D 2 • i 1.J < 111 s ; K r 1
I) o.oo--;.
J> I J> I 2
I • O () 2
I • I 76
I • '\ I O
I • -; (, C)
I • 880
2. v;o 2. CJ8'\
I'\. GG
~Lr;I&
i.&4
'\ • CJ '.i
18. '.io
1 'L (lO
<l.71
(>. '\()
18. 'i0
I'\. 81
'\ • '\ 7
I' 2
I O - '\,,. X Cl IT
I • OCJ i
C .989
o.88H
9. 7 '.i I
0 • () I CJ
o. 4<Jr;
O.'lCJO
0. 08 r;
0. I'\ 7
0. 20(>
0.29i
O. 06 '\
0.086
0. ~ 20
0. 182
O.Obl
0. 08',
O. 14 'i
{< T ' 111i11 ;tl 7&0 T11r-r- .<-. II
20 C) ~ {
f III l'IH•ot · y nir•a :~
. m
1111' i l S
-,,-,f-. -rn ----------
0. ' \ I 8 0 • ' \ 'i I • 09CJ
0.811 0.90 I • I 1 0
I • 28 '.i I • -'-' 2 I • I 06
I. 970 2. I 2 1.07 --;
2. i ' \ 8 2.79 l.OCJ9
'\ • I I 'i ' \. 17 I • 082
'\. GO 1 1. 91 2 I. 086
i. 028 'i. 40 I I. 074
,, • 7 8 r; i. I l.,11 I. 07 r;
,, • l.,16 ' \ _,,. 8 2 r; I • Ob I
4. o l.,18 4. '\6 I • 077
'l.l'\O r;. 49 1.071
'l.02'\ r;. 40 1.07,
l.,i. 86,; r;. 27 1.084
4. --;7,; 4.96 I. 081
r;. 110 r;. ,;4 l • 079
r;.O'\'\ r;. 41 l • 079
'L 8 l r; 4 • l r; l • 087
p I
21.H S x -1 10 T o rr· • T ' I
D - 2.S4 c m; Kn
I • O 1 1
l .0'16
1. 21'1
1 • l 7 2
1. l 17
p2
-1 x 10 T o r-r
21.-;9
22. 'i 9
19.11
20.1'1
21 • 1 r;
- 44 -
TAHLF. 7
,_, -~ r: 0 c "' O · O 9 L -= ,,. ) ; "l - • t:. cm;
>.
0
1 0 • 0 8 2 I~; ~ .: O • O O r; •
0.0127 r. m~ ;
• (r e / min :-. l 760 T o r· r & 20 ° c)
• V V f 111 Th«~ o r y mf: as
m mP a s
rtifm ----------
1 • 21 2 4.214 '\. ,~ 7 6
S .877 16.S6 2.817
21 • 1 8 ~6.24 2.6r;r;
16.12 44.14 2.717
I 1 • 66 ' \ 2. 97 2.826
- 45 -
TABL~ 8
pl l S. 88 - ·1
T 22. '1°C; 0 • 'l 1 4 '\ X 1.0 Torr· ; ~. - ('fn!';; L 0. ';906 <'fllSt 1 ~ 1
I
I) 2. r;!" Kn 0 • I 2 '\ 7 ; L 0 .212i. <. Ill t
I) D
p2 .
( / ' at 7(>0 Torr «·. 20 °c) m <· c rn I n llll'ilS
P1 / P2 - ·1 ~ • rh ThP n ry V X 10 Torr fm ntPa !'; f 111
l.l S7 I ' \ • 7 2 S 8. l 87 1 7. l 16 2.09';
l • 1 00 l '-' • '-' '.i s .48 ·1 1 I • 69 2 • I ' \ 2
I. 06 'i I '-' • 9 1-' '\. S80 7. 80 '\ 2. 180
I r; . 00 I • 06 '1 6 • l.i ., 79.29 I • -'.i U r;
19.99 0.791.i s 7 • '"6 79.92 I • '\ 91
l2.2 ·'-' I • 297 r;r; . --;2 78.28 1 • 4 l 0
8. 6 l 1 1 • 8 1.i4 r; 1. 4(> 77.41 1 • /" !"8
7.21 2. 1 <)6 S2 .09 76. i 2 1 • 469
6 • '-' O '-' 2. ·'-' 79 S l .01 7 'i .62 l • '.i8 2
S .22 '\ • 0 l.i 2 '.i8. 89 7 ·'-' • 0 1 1 . r; 14
4.09 ., • 880 l.i ·1 • 7 0 71 • 28 1 • ·160
4. o·, 1.940 4s.47 7 1 • 1 1 l.S64
1 •. , 1 4.80 42.19 68.17 1 • 6 21
2.718 r;. 80 18.18 64.47 1. 680
2.29 6. <)'\ 14.08 ,;9. 'J S 1.741
1 • 8 '\ ~ 8.6S'\ 27. s·, 5 1.06 1.Bs s
1.s1r; 10.1s 21. 06 41.21 l • 9 S8
l • 16 2 l 1 • 66 16.07 12.42 2.017
1.228 l 2. 9'\ l l. 21 21.26 2.071
1 • 1 76 11. '1 0 9.06 18.86 2.081
-1 ' P 1 7.975 x 10 Torr; 1 1
Pl / P2
n ::. 2.54 cm; Kn
p2
-1 x 10 Torr
- 46 -
TABLE 9
22."inc; Xl = 0.626 ems; L
Xl D
0.2464; L n 0.212'i.
(cc/min at 760 Torr & 20°C)
Vfm Theory \'meas -------------I • O IO
1 • 029
1 • 1 O 2
l • 29 'l
1. 6(>1
1 S. S8
20. ";6
11 • 46
8.77
5.608
4.451
1.485
2.671
2.011
1 • 669
7.896
7.7r;o
7.2 '\7
6.881
6.024
4.801
O. '\88
o.696 0.909
l • 1 '\O
1 • 422
1. 791
2.288
2.984
1.966
4.778
0.101
o.8'i7
2.810
4. l 66
6.919
12.09
28.42
28.89
27.72
26.91
26.06
24.95
21.55
21.66
19.00
15.15
12. 1 7
o.4so 1. '320
4.102
6. 6 '\ 2
8.064
10.60
17.71
1S.64
)5.98
1'i.07
14.14
11.71
12.616
11.257
29.)12
26.416
21.971
17.967
0.";906 cm;
• m nH•as
111 f' m
t. 495
1 • r;4o
1 • 51 l
l • '19 2
1 • r;68
l • S '\ 2
1.467
1. 2 54
1. 245
l.265
1.276
1. 294
1. )07
1. 127
1 .15)
1 .190
l. 450
1.476
- 4 7 -
TABLE 10
f> l
U 2 • 'i '~ c 111 ~ ; K 11
2 2 • i ( 1 C ; X l l • ~ 'i '\
X l O 4 en., . !_: D • '' D
cm; L 0. 'i906 cm;
0.2 '\2S.
. p2 (cc l mi11 at 760 Tnr· r &. 20°C)
m
Pl / 1'2 lo -·,., \"f ·1·1 \' x • or· r· 111 tPor· y m<'a~
8.08 0. '~ <)'' 1 '\ • '\ O 1 i. 28 l • 149
10.96 O. 'H> ·'-' 1 '\. 7 9 l ';.79 1.14,;
1 6. l '-' O. 2 '-' 7 l '-'-21 I l>. 21 I • l '\ l)
21 • Sb o. 18 •-; I ·'-' • 4 7 l 6 • -' ' \ l • 1 ' \ r;
(J. 6 1-' o.&oo 12.89 14.87 l • 1 i 4
~ r; • 2 '\ 0 • 7 {) 2 12 .27 14.26 1 . l 6 2
" • 4. ·n 0 • C) 20 l 1 • 6 7 1 '3. 64 1 • 1 69
1. :; /.,i 1 • 1 26 ,0.89 12.Br; l • 1 80
2.7,; 1 • /" '-' 9 9 • (, :; 7 1 1. 6 I I • 20 2
2. l 9 l. 820 8.2'-' I+ 1 o. o .'+ 1 • 21 7
I • 7 r; 2.277 6. r; o4 8.08 1 • 242
1 • l..i86 2.682 4.962 6.29 1. 268
1 • ·rn s 1. 0 -; !" 1.r;4r; 4. _5'\ 1.278
I • 2 0 '-' 'L 11 O 2 .s70 1.29 1. 280
1 • l 61 'L426 2. I 29 2.7 ) 1.282
l • 0 24 1.892 o. '\ r;4 o.47 1 • ·128
1 • 0 i 1 1.781" 0.765 0.96 1.255
I • 1 1 6 1.r;71 0.158 2.00 l • 266
- 48 -
TAI\Lb l l
p I
I • 99 2 x - '\
l 0 Ton· ; T l
0.'i906 <'nt;
I) 2. 'i4 c-m~; K11
'\~ .
(cc / mill at 760 T () T' I ' & 20 °c) m lll!'il:--.
(r . p P.-> - ' \
Th«-nr·y V ,,, f
l .... X l 0 Tor· r· fm 1111' il ~ m
l .0 'i 0 I. 897 0. 'H> 2 0 . 1.i20 I • l 60
I • I 00 I. H 11 0.689 O. 7<> l • IO 1
I • l 7 '\ I • (> <Hi I • I ~O I • 29 I. l , I
l • 2 r; O l • S 9 •~ l.'il(> l • 69 I. l l S
2."l.00 0 • 0 <) 0 7.241 7.977 l. IO I
16. so 0. I 21 7. I 2 r; 7.8,;r; I • IO 2
I 2. 2 l.i, 0. l 61 6.96', 7. <,Hq I • l O 1+
8.64 0.212 (>. 7 O 2 7. '.i "\ ,, 1 • l O 9
6.607 0.'\01 6. 1419 7 • l '..i 1 1 • l O 9
; • 077 0. '\')l 6.09'\ 6.790 I • l 1 :..i
14. 1 4o o.481 r;.7,;4 6. i.i.l'i 1 • l 1 8
'LT\7 o.,;97 'i • '\ L 2 'i.961 i • i 2 2
2. 14 (> 7 0.807 , •• --; 1 2 'i.097 1. 1 ·,o
l.8,2 I • 076 '\. /488 '\. 960 l • 11 S
l • '"69 l • '\'i 6 2. '4 2 2 2.719 I. 1 'H
- 49 -
TABLE 12
I • I 6 i -1
Tl 2 2. 'i ° C ; >. I !". 28 i L O. S90o p X 10 T o rr; cm -; ; cm; I
>. I D 2. j !~ Kn I .687;
I. 0.212'i. C' Ill t i) I)
p • (cc / min 760
I) m 2 cl t T1> r r· &. 20 C) m .. a ~
\' f .
I' p2 - ' \ ThP1>ry
\ ' m I X 10 T o rr rn IIH'ti~ fm
2 I • CJ'\ O.O 'i'\ 4. 21 ·'-' '.i. r; I 6 l • 067
If). 80 O.Ol>C) 4. I 71 '-'. 448 l. 066
I I • i <> O. 1 0 l 4.0i2 '-' •. , 2 2 I • 067
7.7i 0.150 1.86, 4. I I 6 l.06'i \
q. iO 0. I 2 '\ ., • <> 6 8 '-' • 2 i -~ l. 07 2
'". 88 0. 2 '\C) 1.'i26 'l.7(,7 I • 068
!". 8 i 0. 2 1-'0 1. i 2 2 1.748 l • 064
I • 08 1.070 0.127 O. ''\60 1 • 1 O 1
I • l 7 0.996 0.644 0.710 1 • 111 .. I • 17 o.8,o I • 200 I .120 1.100 I
I • (>0 0.728 l. 664 l.. 790 1 • 07 ~
I .8'i1 0.629 2.041 2.21 1.083
:.. • ,80 0.4S2 2.71'1 2.92 1. 07 ~
1.706 0.114 1.241 ).478 1.073
!". 98 0.214 1.S4'1 1.806 l. 074
- 50 -
I ' I
! ' \ • 7 I , -1 10 Torr; T
1 0.210-; <m; L o. ,90() rm;
I) 2.,;4 rm s ; Kn D
0.08 29; ~
r2 .
( (' (' 7()0 & 2 0 ' 'c ) m min at To rr IOPilS
• . I ' I ' - -~ \
Th 0n ry \' 111 f I ' ,) X 10 Torr fm 111P ,\ S rn
I • OT\ 22.9'; 2 • 8 C) ,1-' 7. 6 <>2 2.b 1-'8
I • I '! H 21 • 02 10. 2 '-' 27. 77 1
-' 2.712
l . :! O l I 0 • 7 !-' 1 i • I 2 1().2il 2.i96
I • '\ ' \H 17.72 22.81 ,6. '\'}8 2.472
- S 1 -
TABLF. 14
pl 15.87 -1 Tl 22.s 0 c; X 1 O.Jl!J6 L 1 . 2 S 1 = X 10 Torr: ::. = cm~; -= cm~;
D 2.519 Kn Xl
0. 1 24: L o.4926. = ems; ::: ..:: -:
D j)
p2 .
(cc 1mi11 7o O Torr & 20 °c) m at mPas
pl ' p2 -1 • <' ihfm V X 10 To re · rm Th e or \· m• >as
•
17.87 o.888 4 7. 18 66.09 1 • 19;
12.75 1 . 2 lVi 46.27 6S.26 1 . 4 1 0
9.62 1 • 6 ;o 44.98 6 1-L 1 5 1. 411
7.72 2.os6 4~L 68 61.4 S 1. 4 51
6. 21 2.SS/j 42.10 62.0) 1 . -'.i 71
5.10 2.994 40.74 60.78 1 • 492
2).)5 0.680 48.0 5 66.06 1. 37 !:i
4.467 3.553 '38.97 59.29 1.521 11
3.664 4.~31 16.50 57. 14 1 • 56 5
1.167 5.011 1!~. 1"' 54.99 1 • 601
2.637 6.018 11 • 16 51 . 51 l • 6 54
2.104 7. 541 26. J-'4 .'.4-1.70 1.73 5
l.761 9.012 21.69 )9.41 1. 817
1.453 10.922 15.65 29.8 5 1.907
1.269 12.51 10.63 20.94 1.970
1.184 13.40 7.813 15.33 1.962
1.119 14. 18 5 .345 10.82 2.024
1.180 lJ.45 7.654 15.31 2.000
1.094 14. 51 4.2)8 9.002 2.124
- 52 -
TABLE 15
7.955 10-JTorr; Tl 0
XI 0.6275 L 1 . 2 51 pl = X = 22.5 c: = ems; = cm;
D 2.539 Kn X 1
0.247; L o.4926. = ~m; - o = D
p2 .
(cc / min at 760 Torr r~ 20°c) m m0as p 1 ·1 p 2 (rfm
• _, V rtifm X 10 Torr Th0ory meas
1 . 008 7.892 0. l 994 0.)10 1.55)
1. 029 7.731 0.7088 1.0)0 I. 4) J
1.084 7.339 1.949 2.8)0 1.452
1. 154 6.893 3.361 .5.171 1.538
1.309 6.077 5.94) 8.941 1.504
2.010 3.958 12.65 17.681 1.398
i:: 5. l l 0.')168 24.17 29.38 1.216
18.87 o.4216 2).84 29.18 1.224
lJ.84 0.5748 23.3) 28.82 l. 23!..;.
10.JJ 0.7701 22.74 28.28 1.244
7.486 1.063 21. 81 2i.39 1.2_)6
).856 I. J 58 20.88 26.52 1.270 L
4.328 1.838 19.36 25.07 1.295
J.092 2.573 17.03 22.58 1.326
2 .192 3.629 13.69 18.79 1.373
1.698 4.685 10.35 14.68 1.418
L
-1 P
1 -= J.984 x 10 To rr; T
1
p 1 I p 2
26. 58
19.81
16.07
11.98
8.609
6.094
4.Bss
3.=;96
2.186
1 • 746
1.339
1.169
1.108
1.0.50
'l :.· 2 • ") 19 C Ill 5 ;
p 2
-1 x 10 Torr
0.1'.i9')
0.200()
O. 2 :.i 79
0.1126
0. 14 6 28
0.6'i ) 8
0.8206
1 • 1 C 8
1.670
2.282
2.97s
').408
3.596
3.794
- ) J -
TAHLE 16
· 22.'i°C;
. >. 1 Kn = - ,-
D
>. l - 1 • 2 :; 1 c m s ; L --
, , L , 6 o. • 914; 0 = o. • 91 .
( c c mir1 at 760 To rr & 20°C) <· ~· f m Th•' o r,. rn" a s
1 2 • I ' \ 1 '\ • (> 1
1 I • 9 7 11 • . '4 ' \
I 1 • 8 2 11.29
I 1 • , -; 11. 0 2
l 1 • 1 !.i 12.6 !.i
10.54 12.05
1 0. 0 l 11.=;2
9. 1 01 10.61
7.122 e.7 ;
S.186 6. 594
3.192 3.9so
l. 823 2.270
1.229 1. 52
0.601 0.73
. Ill
r1i fm
I • 1 2 2
1 • I 2 2
1 • 1 2 :.i
I • l 27
i • \ 1~
1 • 1 ~ 1
I .l'll
1.16';
1 • 19,;
1. 22~
1.238
1. 24 =;
1.217
1.214
, 111 ... :
- 54 -
TABLE l?
P1 = 1.994 x 10-)Torr; T1 = 22.5°c; ~l = 2.504 cm; L = 1.251 ems;
_ >-1 L
2?.72
20 • .53 14.oo
11.93 9.635 6.790 1.044 1.114 1.156
1.325 1.660
1.986 ).056 4.82)
6.624
D = 2.539 ems; Kn - ,r = 0.98.58; D = o.4926.
0.0?193 0.09?1 0.1424
0.1671 0.2070
0.29:37 1.910 1.791 1.725 1.505 1.201 1.004 0.6525 o.4134 0.3010
(cc/min at ?60 Torr & 20°c) t t fm Theory meas
6.082
6.003 5.860
5.781 5.655 .5.379 0.2658 o.6424 u.8512
1.547 2.509 ).1:J)
4.245 .5.002
.5-)5?
6.613 6.519 6.416 6.256 6.114 .5.822
0.290
0.690 0.920 1.68
2.? .5
).4.58 4.?0 5~484 .5.841
..
..-.... -------------------·----
m meas
"'rm
1.087 1.086 1.09.5 1.082 1.081 1.082
1.091 1.074 1.081 1.086 1.096
1.10? 1.096 1.090 ..
..
.
.
. .
• I I
' I
: I
I I
I
- 55 -
TABLE 18
P1
= 1.163 x l0-3Torr; T1 = 22 • .5°c; ~l = 4.2923 ems; L = 1.251 ems;
~l L D = 2.539 ems; Kn= ,r = 1.690; ~ = o.4926.
(cc/min at 760 Torr & 20°c) t f m Theory tmeas
"' meas itifm
29.52 0.03940 3.555 3.671 1.032 20. 70 0.05618 3.502 3.632 1.037 1.5.84 0.07342 3.448 3 • .584 1.039 9.86 0.1180 3.307 3.448 1.043 7.62 0.1526 3.197 ).J22 1.039
7.23 0.1609 3.171 3.293 1 • 038
5.28 0.2203 2.983 3.127 1.048
3.96 0.2937 2.751 2.860 o.040
2.94 0.39.56 2.428 2.540 1.046
2.30 0.5057 2.080 2.180 1.048
1.777 0.6545 1.609 1.680 1.044
1.074 1.083 0.2.531 0.260 1.027 1.184 0.9823 0 • .5718 0.-58.5 1.023 1.)06 0.890.5 0.8623 0.890 1.032
- --- -----·-- ----
- 56 -
TABLE 19
P1 = 23.71 x l0-3Torr; T1 = 22.5°c; ~l = 0.2105 ems; L = 1.251 cm; ~l L
D = 2.539 cm; Kn= ,r = 0.08); D = o.4926.
(cc/min at 760 Ttrr & 20°c) t t fm Theory meas
"' meas
"'rm
1.079 21.97 5.506 14.02 2.546
1.166 20.)J 10.696 26.46 2.474
- 57 -
TABLE 20
pl = 15.93 :x l0-3Torr• T = 22 • .5°c; A1 = o.3134 ems; L = 1.920 cm; ' 1 Al
D 2.539 L = ems; Kn= ,r = 0.124; ~ = 0.7.56.
p2 • (cc/min at 760 Torr 4 20°c) m meas
Pl/P2 10-3Torr (r t ilifm .x fm Theory meas
19.70 0.8086 41.04 56.48 1.376 5.13 3.10.5 34.81 51.81 1.488 5.62 2.834 35 • .54 .52 • .56 1.479
7.53 2.116 37.49 54.03 1.441 14.oo 1.138 40 .14 55.7.5 1.389 19.80 0.805 41.05 .56.29 1.371 4.48 3.556 33.58 .50 • .50 1.504 3 • .5.5 4.487 31.06 47.94 1.543 2.95 5.400 28.58 4.5 • .55 1.594 2 • .524 6.311 26.11 42.75 1.637 2.541 6.269 26.22 42.56 1.623 2.120 7.514 22.84 38.48 1.685 1.850 8.611 19.86 )4.5.5 1.740 1.580 10.082 15.87 28.67 1.80? 1.450 10.986 13.42 24.62 1.8)4 1.3.50 11.80 11.21 20.94 1.868 1.240 12.8.5 8.367 1.5.94 1.90.5 1.163 13.70 6.060 11.79 1.94.5 1.086 • 14. 67 ).422 6.65 1.943 1.28) 12.42 9 • .537 18.00 1.887 1.197 13.Jl 7.116 13.7.5 1.932 1.12.5 14.16 4.804 9.64 2.007 1.094 -14 • .56 3.71.5 ?.4? 2.011
26.20 0.08 . 41 • .58 .56.7.5 1.36.5 ,, 1.09.5 14 • .5.5 ).?.51 ?.4.5 1.986
. .
- 58 -
TABLE 21
pl = 7.980 X l0-3Torr; ~l = 22 • .5°c1 A1 = 0.626 cm; L = D = 2 • .539 cm; Kn - Al = 0.24?; ~ = o.7.5f .• -,r
p2 (cc/min at ?60 Torr & 20°c) Pl/P2 x l0-3Torr t t fm Theory meas
20.81 0.383 20.62 2.5.13 21.00 0.380 20.63 24.98
7.015 1.138 18.57 23.09 10.150 0.?42 19.64 24.12
5.223 1.528 17.51 22.14 4.230 1.88? 16 • .54 21.13 4.230 1.88? 16.54 21.1.5 3.430 2.327 15.34 19.90 2.806 2.844 13.94 18.33 2.896 2.7.56 14.18 18.58 2.300 ).4?0 12.24 16.38 1.708 4.6?2 8.978 12,48
1.027 1.110 0 • .570 0.870 1.084 7.362 1.6?? 2.49 1.026 1.118 0 • .548 0.760 1.212 6 • .584 3.789 5 • .57.5 1.42? .5 • .592 6.481 9.263
1.763 4 • .526 9.374 13.0.56 2.1.53 3.706 11.60 1.5.829
1.148 6.951 2.793 4.204
20.84 0.383 20.62 2.5.08?
27 • .58 0.289 20.8? 2.5.382
•
1.920 cm;
m meas
ilifm
1.219 1.211 1.243 1.228 1.264 1.277 1.279 1.297 1.31.5 1.310 1.338 1.389 1.526 1.48.5 1.387 1.471 1.429 1.392 1.364 1 • .50 .5 1.21, 1.216
.. ,
..
•
. .
..
.
.
•
. .
'
• t J
pl = 3.99 X
D
Pl/P2
2.964 2.540 1.980 1.723 1.512 1.330 1.190 1.181 1.094 1.093 1.04.5 3.240
3.63.5 4.960 7 .. .584
9.970 21.87 11.94
29.43 22.01
- 59 -
TABLE 22
l0-3Torr• T 0 Al = 1.2.51 crns; L = 22 • .5 C; = 1.920 ems; ' 1
Al Kn L 0.756. = 2.539 cm; = - = o.493: ii =
D
p2 • (cc/min at 760 Torr cl 20°c) m meas
x 10-3Torr v t ilifm fm Theory meas
1.346 7.176 8.333 1.161
1.571 6 • .565 7.768 1.183 2.015 .5.360 6.444 1.202
2.316 4.543 5.511 1.213
2.639 3.667 4.477 1.221
3.000 2.687 3.313 1.233
3.3.53 1.729 2.040 1.180
3.378 1.661 2.02 1.216
3.647 0.931 1.160 1.246
3.6.51 0.920 1.1.50 1.2.50 3.818 o.467 0 • .5.50 1.178 1.231 7.488 8.793 1.174
1.098 7.849 9.18.5 1.170 0.804 8.647 10.04 1.160
0 . .526 9.401 10.7.5 1.14) o.4oo 9.743 11.12 1.141 0.182 10.33 11.71 1.133 0.334 9.922 11.29 1.137 0.136 10.46 11.80 1.128
0.181 10.34 11.66 1.128
----- -- -----------------------
- 60 -
TABLE 2)
pl = 1.99? X 10-)Torr; Tl = 22 • .5°c; ~l = 2 • .50 ems; L =
D = 2.539 cm; Kn ~l
0.984; j = 0.156. -,r -
p2 (cc/min at 760 Torr & 20°C) Pl/P2 x l0-3Torr ~fm Theory t
meas
22.450 0.089 5.178 5 • .538 11.500 0.174 4.948 5.300 9.8?0 0. 202 4.8?2 5.254 5.230 0.382 4.383 4.815 5.160 0.J8? 4.3?0 4.796 3.88 0.515 4.022 4.399
2.93 0.682 3.569 3.941
2.93 0.682 3.569 3.921 2.144 0.931 2.893 3.196
1.53 1.305 1.8?8 2.050 1.300 1.536 1.251 1.360 1.126 1.7?4 0.605 0.63 1.810 1.103 2.426 2.66
22.?5 0.088 .5.181 .5.529 )0.)0 0.066 .5.241 5.584
1.920 cm;
rh meas
"'rm
1.07 1.071 1.078 1.098 1.097 1.094 1.104
1.099 1.105 1.092 1.08? 1.041 1.096 1.067 '
1.06.5
.-
'
. .
.-
'
. .
21.98 29.97 17.10 11.40
6.879 3.778 3.684 2.710 2.042 1.667 1.348 1.037 1,. 12.5
1.304 1.195 1.120
0.0529 0.0388 0.068 0.102 0.169 0.3078 0.3156 o.4291 0.5695 0.6976 0.8627 1.121 1.034 0.8918
0.9732 1.038
- 61 -
TABLE 24
• (cc/min at 760 Torr & 20°c) • t
Vfm Theory meas
m meas
"'rm
3.012 3.137 1.041
3.051 3.157 1.035 2.972 3.059 1.029 2.880 2.990 1.038 2.698 2.790 1.034 2.320 2.390 1.030
2.299 2.370 1.030
1.992 2.050 1.029 1.609 1.670 1.038 1.262 1.320 1.046 0.814 0.8.5 1.044 0.114 0.105 0.921
0.3.50 0.330 0.943
0.735 0.760 1.034 0 • .516 0 • .520 1.008
0.339 0.330 0.973
- 62 -
TABLE 2.5
pl 23.82 X l0-3Torr· T 0 A1 = 0.2096 cm; L 1.92 = = 22 • .5 c; = cm; ' 1
2 • .539 Kn Al L - 0.7.56. D = cm; = - = 0.083; n -D
p2 • (cc/min at 760 Torr & 20°c) m meas
Pl/P2 x l0-3Torr v t ilifm fm Theory meas
6.60 3.609 .54. 85 87.53 1 • .596
2.29 10.40 36.42 70.01 1.922
3.02 7.887 43. 23 77.93 1.803
4.07 .5.852 48.77 83.10 1.704
4.67 .5.100 50.81 85.18 1.676
1.685 14. 14 26.27 55.60 2.124
1.180 20.19 9.852 23.30 2.365
1.088 21.89 .5. 238 13.11 2.502
- 63 -
TABLE 26
P1
= 15.9 x 10-3rorr; T1 = 22.5°c; ~l = 0.314 ems; L = 2.526 cm; ~l L
D = 2.54 crn; Kn= D = 0.1236; D = 0.9946.
p2 (cc/min at 760 Torr & 20°c) • m meas
_P_l_/P_2 _____ x_l_0_-3_T_o_r_r ___ t_fm_fheory tmeas "'rm
JO.JO 0.525 37.16 50.22 1.351 22.87 0.695 36.75 49.97 1.360
,. 14.92 1.066 35.85 49.35 1.377 11.39 1.396 35.06 48.70 1.389 8.64 1. 840 33.98 47.72 1.404
7.28 2.184 JJ.15 47.52 1.433 6.20 ~. 565 32.23 1J6.58 1.445
4.49 3.541 29.8~ 44.42 1.487
3.42 4.649 27.19 41.80 1.537
2.55 6.235 23.36 37.49 1.605
1.92 8.281 18.42 Jl.08 1.687
1.33 11.95 9.537 17.54 1.839
1.33 11.95 9.537 17.60 1.845 1,18 13.47 5.864 11.28 1.923
1.13 14.07 4. t.,.23 8.403 1.900
1.09 14.59 3.174 6.161 1.941
- 64 -
TABLF. 27
pl 7.966 l0- 3Torr; Tl 0
Xl 0.6266 L 2.526 = X = 22.~ C; = cm; = cm;
D 2.54 cm; Kn ~l
0.2467; ~ = 0.99',,-6. = = 7r .
p2 • ( ' . " t 760 Torr & 20°c) m
c c , m111 me a5 Pl / P2 l0- 3Torr
{r (r ~fm X fm TIH'orv mea5
4.071 1.957 J !.i. °l? 18.44 1.270
2.858 2.787 l2.'l2 16.39 1.109
1.934 4.119 9.298 12.64 1.359
l. 282 6.214 'J. ?, ~ 6.171 1.4~7
l.186 6.717 1.01'~ 4.409 1.460
1.080 7.176 1 • u'.?h 2.110 l • 4Q!.J
1.0:n 7.726 o.i;ao o.84o 1,448
12.95 0.242 18.67 22.15 1.197 2!,,.. 46 0.126 18.47 22.18 1.201
1?.66 o.4~1 18.}t, 21.87 1. 204
11.l'J 0.716 1 i. ~ 2 21.30 1.216
9.12 o.871 17. 1 4 20.96 1.221
6.94 1.148 l. 6. 48 20.JO l.2Jl
5.08 1.56ti l~.~6 19.30 1.249
4.lJJ l. qi7 1 14. 60 18.41 1. 261
. .
. .
- 65 -
TABLE 28
P1 = 3.98 x l0-3Torr; T1 = 22.5°C; ~l = 1.254; L = 2.526 cm; ~- L
26.02
35.10 12.07 8.744
6.577 5.566 4.684
3.781
3.137 2.496 2.016
1.705 1.463
1.195 1.127 1.0'5? 1.019
D = 2.54 cm; Kn=_;!;.= o.4937; = 0.9946. D D
0.151 0.1134
0.3297 o.4552 0.6051 0.7151 o.8497 1.051 1.269
1.595 1.974
2.334 2.720
3.330 3.531 3.76'1 3.906
(cc / min at 760 Torr & 20°c) t v fm Theory meas
9.250
9.347 8.822 8.520 8.1.57
7.892 7.565 7.075 6.552 5.765 !.i,. 849
3.978 3.045 1.571 1.085 O.'l197 0.1788
10.16 10.296
9.817 9.543 9.193 8.932 8.602 8.116
7.575 6.744
5.703 4.748
3.623 1.900 1.290
0.620
0.230
• rn meas 111 rm
1.098 1.101
1.113 1.120 1.127 1.132 1.137 1.147 1.156 1.170 1.176 1.194 1.190 1.209 1.189 1.19) 1.286
- 66 -
TABLE 29
P1 = 1.993 x 10-JTorr; T1 = 22.5°c; ~l = 2.50~ cm;~= 2.526 cm;
~l L
37.27 26.77
17.91
13.26
9.73 9.87 6.91
5.45 4.42
3.58
2.63.5 1.980 1 .418
1.185
1.067
D = 2.54 cm; Kn - D = 0.986; 0 = 0.9946.
0.054 0.074;
0.1113
0.1503 0.2048
0.2019
0.2884
0.3657 o.4509
0.5567 0.7564 1.006
1. 406
1.682
1.868
(cc/min at 760 Torr & 20°c) v ~ fm Theory meas
4.688
4.637
4."48 4.~54 4.322
4.329 4.120
3.933 3.727
3.472 2.989
2.386
1.419
0.752 0.302
4.958 4.901
4.815
4.719 4.594 4.594
4.399 4. 185
3.960
3.690
3.196
2.550 1.~20
0.810
0.320
• m meas
"'rm
1. 058
1.057 1.0~()
1.059 1.06)
1.061.
1.068
1.064
1.061
1.063
1.069
1.069
1.071
1.077
1.060
- 67 -
TABLE JO
pl 1.163 X 10-3Torr; Tl IJ
~1 = 4.292 ems; L 2.526 cm; = = 22.5 C; =
D 2.54 ems; Kn ~1
1.690; L 0.9946. = = n = D =
p2 • (cc / min at 760 Torr & 20°c) m
meas pl/p2 10-3Torr v (r ibfm X fm Theory me~5
25.69 o.453 2.701 2.74 1.014
37.62 0.0309 2.736 2.79 1.020
15.11 0.077 2.625 2.68 1.021
10.38 0.112 2.540 2.59 1.020
6.97 0.1668 2.408 2.47 1.026
6.84 0.170 2.400 2.46 1.025
5 .02 0.2316 2.251 2.30 1.022
3.94 0.2952 2.097 2.14 1.021
J.69 0.3152 2.049 2.13 1.040
J.08 0.3776 1.898 1.92 1.012
2.15 0.5409 1.504 1.53 1.017
1.604 0.7251 1.058 1.08 1.021
1.280 0.9086 0.6149 0.61 0.992
1.187 0.9798 o.4428 o.45 1.016
1.090 1.067 0.2320 0.2) 0.991
- I
·I I
- 68 -
TABLE 31
Pl= 23.72 x l0-3Torr; T1 = 22.5°C; ~l = 0.2105 cm; L = 2.526 cm; ~l L
D = 2.54 cm; Kn= D = 0.0829; D = 0.9946. _
• (cc/min at 760 Torr~ 20°C)
• • Vfm Theory Vmeas
m meas "'rm
1.030 23.03 1.668 4.107 2.462
1.152 20.59 7.565 17.31 2.288
1.162 20.41 a.coo 17.85 2.232 1.024 23.16 1. 3.54 3.013 2.225
6.720 3.530 48.80 75.86 1.555 8.530 2.781 50.61 76.31 1.508
6.69 3.546 ..-d.75 74.93 1.537 4.03 5.886 43.10 70.49 1.63_3
2.823 8.402 37.03 64.41 1.739 2.09 11.35 29.90 55. 77 1.865
1.630 14.55 22.16 44.15 1.992
1.297 18.29 13.12 28.22 2.1.51
1.068 22.21 3.65 8.20 2.246
- 69 -
TABLE 32
[Note: pl in microns Hg; Q = in cc / sec at Pressure P1 and
Temp T1; ~P in dyn e s / sq cm]
' L - = 0.00 5 D
Q "1 Q
pl Kn ~p ~p D~
23.85 0.082 5 ,143 0.05522 15.91 O .124 6,143 0.06597 7.955 0.247 9,215 0.09896 3.980 o.494 14,444 0.1551 ,. 1.993 0.986 25,722 0.2762
1.163 1.690 42,431 o.4556
L 0.2325 D =
23.71 0.083 4,041 0.04340
15.88 O .124 4,827 0.05184
7.975 0.246 7,209 0.07741
3.985 o.493 11,806 0.1268
1.992 0.987 20,300 0.2180
1.165 1.690 34,956 0.3754
L o.4926 n =
23.71 0.083 J,213 0.03450
15.87 0.124 J,8J4 0.04117
7.955 0.247 5,806 0.06235
3.984 o.493 9,500 0.1020
1.994 0.986 16,483 0.1770 .. 1.163 1.690 27,J.89 0.2920
. '
- 70 -
TABLE 32 (Continued)
L 0.7561 ii = •
Q pl Kn 4P
23.82 0.083 2,655 15.93 0.124 3,188 7.98 0.247 4,803
3.99 o.493 7,877 1.997 0.984 14,106
1.163 1.690 22,493
L - 4 0 - 0.99 6
23.72 2,141
1.5.90 2,748 7.966 4,282
3.980 6,886
1.993 12,.500 1.16) 19,9?9
Q "1
4P D~
0.02852 0.03424
0.05159 0.08461 0.1515 0.2416
0.02299 0.029.52 0.04.598 o .-07394 0.1)42 0.2146
41
t
•
.. .
•
• -
