The flow of gases in narrow channels


Material Information

The flow of gases in narrow channels
Series Title:
Physical Description:
46 p. : ill. ; 27 cm.
Rasmussen, R. E. H ( Rasmus Ebbe Hansen ), 1901-
United States -- National Advisory Committee for Aeronautics
Place of Publication:
Washington, D.C
Publication Date:


Subjects / Keywords:
Aerodynamics   ( lcsh )
Laminar boundary layer   ( lcsh )
federal government publication   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )


Measurements were made of the flow of gases through various narrow channels a few microns wide at average pressures from 0.00003 to 40 cm. Hg. The flow rate, defined as the product of pressure and volume rate of flow at unit pressure difference, first decreased linearly with decrease in mean pressure in the channel, in agreement with laminar-flow theory, reached a minimum when the mean path length was approximately equal to the channel width, and then increased to a constant value. The product of flow rate and square root of molecular number was approximately the same function of mean path length for all gases for a given channel.
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by R.E.H. Rasmussen.
General Note:
"Report date August 1951."
General Note:
"Translation of "Über die strömung von gasen in engen kanälen." Annalen der Physik, Band 29, Heft 8, August 1937."

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University of Florida
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Full Text

252-? r7 1 / 17




By R. E. H. Rasmussen


This report deals with the measurements of the air flow T per
second per unit of pressure difference through various channels at average
pressures of from 0.00003 to 40 cm Hg. Hydrogen, oxygen, carbon dioxide,
argon, and air were utilized.

The flow channels consisted of:

(1) Narrow annular slits between optically plane glass plates in
Christiansen prismatic devices

(2) A rectangular slit between ground and soot-blackened glass

(3) A cylindrical slit between coaxial cylindrical surfaces of

(4) A porous plate (filter plate) of sintered glass

It was demonstrated that the flow rate T at high pressure increases
linearly with the mean pressure in the channel in agreement with the
laminar-flow theory. The width of the annular slits, of from about 3 to
10P, was measured according to Christiansen's data by means of Herschel
interference; the optically obtained slit width was about 0.2p larger
than that obtained from the flow data.

At decreasing pressure, T assumes a minimum, if the mean path
length X is approximately equal to the slit width a; the minimum
value Tmin is approximately equal to the value obtained by Knudsen's
molecule flow formula. Hence, Xmin : a, Tmin L TKn; this holds for all
channels with well-defined slit width and for all gases.

Uber die Stro'mung von Gasen in engen Kanalen." Annalen der Physik,
Band 29, Heft 8, August 1937, pp. 665-697.

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At further decreasing pressure, T increases again and ultimately
assumes a constant value To, when the mean path length has become sub-
stantially greater than the length of the channel. The most accurately
determined test values of To are tolerably agreeable with the values
obtained from Clausing's formulas by an only approximately correct

It was shown that the quantity TV/M = f is approximately the

same function of the mean path length for all gases for a particular
channel, hence, independent of the gas. This rule may be of practical
significance for determining the flow resistance of a channel for
different gases within a random pressure zone.

The effect of the divergence from the cosine law on the molecule
flow, identified by Knauer and Stern, was investigated. It was found
that it amounts to only a few percent of the total flow.

The decrease of T from the value To to Tmin is a consequence
of the collisions of the molecules.


The principal subject of the present report is the quantitative
study of the flow of various gases in well-defined annular slits formed
in Christiansen's prismatic apparatuses. In the course of the investi-
gation, it was further found necessary to study the flow in different
channels also; a part of these measurements is included in the

The best known earlier investigations by Graham, Kundt, Warburg,
and Christiansen are path-breaking in this phase of research. Christiansen
investigated the flow in a narrow annular slit between the plane surfaces
of two rectangular glass prisms, the spacing of which was measured by an
interferometer. It was found that the flow velocity is governed by the
internal friction of the gases, when this spacing is great in comparison
to the mean free path length. For very narrow slits, Graham's law is
applicable: The rate of diffusion of a gas is inversely proportional to
the square root of its density.

In 1909, Martin Knudsen made a theoretical and experimental study
of the flow in long cylindrical capillary tubes and gave a complete

IM. Knudsen, Ann. d. Phys. 28, p. 75, 1909-
M. Knudsen, Ann. d. Phys. 35, p. 389, 1911.

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explanation of this flow for the entire pressure range of from p = 1 atm.
to p = 0. At high pressure in the laminar-flow region where the mean
path length X is small compared to the tube radius, Poiseuille's law
with slip correction is applicable. For the lowest pressures in the
molecule region of flow where >\ r, Knudsen's formula

T M- 4 p- F 7- j r3

is applicable.

To = the gas volume measured by the product of volume and pressure
flowing through the tube in unit time at unit pressure difference (Gaede
used the letter G). M = molecule number of the gas; R = gas constant;
0 = absolute temperature; r = radius; L = length of tube. In the
boundary zone, where X and r are of equal order of magnitude, T
passes through a minimum so that T is about 5 percent less than T .
min O

Gaede2 measured the flow of hydrogen in a slit (about 4. in width)
and found a very low minimum so that To (at the lowest pressure) was
about 112 percent greater than Tmini obtained at pressure p = 2.3 cm Hg.
Since the width of the slit was not measured accurately, the absolute
values of the flow could not be compared with the theoretical values.

Smoluchowski5 solved the molecule flow problem for infinitely long
cylindrical channels of constant cross section on the assumption of the
cosine law. This result can be written


where J signifies a definite integral

J = JJfdS cos cp ds = J dS r da

dS = an element of the cross section S, ds = an element of the cross-
section periphery, cp = the angle between the normal of the element ds
and the distance r from dS to ds, do = the angle at which ds is
seen from dS.

2W. Gaede, Ann. d. Phys. 41, p. 289, 1913
3M. v. Smoluchowski, Ann. d. Phys. 33, p. 1559, 1910.


Clausing4 calculated the molecule flow in special short channels,
using the nonuniform molecule flow for predicting the absorption periods
of gas molecules.


The flow measurement was patterned after Knudsen's method: When
a flow passes in T seconds from a tank of volume V' and pressure p'
through a channel to another tank of volume V" and pressure P", and
the flow.may be re.--rded as steady in each time element dt, so that

-d(V'p') = d(V"p") = T(p' p")dt

where T is constant for every definite mean pressure and for small
value of the pressure difference, then

T = 1 V 1 n -p (1)
T V + V p2 p2

Subscripts 1 and 2 refer to the measurement before and after the flow.

The description of the apparatus falls, obviously, into two parts:
the set of pressure gages, and the flow channels. The pressure-gage
assembly is represented in figure 1. The volume V' consists of the
flask K', the two McLeod gages 1, and lk, the pressure gage M', and
the necessary connections of glass tubes. The flow channel indicated
at K is discussed later.

The volume V" is completely symmetrical with V' and just about
as great. The pressure gage M4 is mounted for direct reading of the
pressure difference p' p". Two small flasks Ph contain phosphorus
pentoxide as drying agents. The appartus is connected with a mercury
diffusion pump through the cock HE. The tube L leads to the gas
generator or gas tank. With the small pipette k and the manometer M,
appropriate quantities of gas can be measured and fed into the apparatus.

Pressure measurements in the 3 to 40 cm Hg range are made with the
gages M' and M", and the pressure difference p' p" is read on
gage M4 with a cathetometer. In the 0.1 to 3 cm Hg pressure range,

P. Clausing: Over den Verblijftijd van Moleculen ..., Amsterdam,

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the McLeod gages l, and 2y, whose pressure tubes had a cross-section
area of about 0.4 cm2, were used. The lowest pressures were recorded
with the McLeod gages lk and 2k. The cross section of the pressure
tubes was about 0.015 cm2 and the manometer volume about 250 cm3. The
pressure readings of the gages were then compared direct several times,
and the equation pl'V' + p111"V" = p2'V' + p2'V" checked at each flow
measurement. It ensures a very effective check of the pressure measure-
ments. The uncertainty of these measurements is usually less than 1/2
percent, but naturally it is much greater for the lowest pressures.

The volume of the apparatus was defined by Mariotte's
volumes of flasks K' and K" being measured by weighing
Illustrative of such a measurement are the following three
determinations of the volumes V ', Vo", (V +V") :

V V" + V "

cm3 cm3 cm3
1298.5 1368.0 2669.7

1301.0 1366.1 2670.2

1298.0 1370.6 2669.3

1299.2 1368.2 2669.7


law, the
with water.

Since the equation V0' + V = (V' + V")o is to be fulfilled, the
values V0' = 1300 cm3, V. = 1369 cm3, (V + V")o = 2669 cm3 were
involved. Subscript o indicates that the values apply when the pressure
in the apparatus is zero. Owing to motion of the mercury in the gages,
the volume varies with the pressure, hence V = V + ap and analogously
for Vp". The inside diameter of the manometer tubes was around 1.5 cm,
hence, the area was about 1.8 cm2 and a = 0.9 if p is given in cm Hg.

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The formula (1) is applicable when assuming constant volumes.
Gaede- has shown how the variation of the volume should be allowed for
in the calculation, and that it yields a correction factor

V (V "
by which the quantity o o must be multiplied. He further called
V + VO

attention to the fact that the small volume v between the flow channel
and the stop cock in the feed line yields a correction factor
p p "
v v 1 1
ky = 1- with which the ratio -1 n1 must be multiplied.
V' V" P P
Gaede's corrections were taken into consideration in all measurements.
Every alteration of the apparatus, every exchange of the flow channels
and so forth, was followed by a determination of the volumes, and the
insertion of the correct values in the formula. Care was taken to
keep v from getting greater than necessary, as a rule, only a few cm3
(determined by weighing out with water or mercury).

All tests were run at room temperature. The temperature tK of the
flow channel was read on a thermometer mounted as near as possible to the
channel, while temperature tM was read on a second thermometer mounted
on a level with the receptacles of the McLeod gages. Because of the
difference in height and the natural temperature distribution in the
room, tK was usually several tenths of a degree higher than tM. This
difference was sufficient to prevent a distillation of the mercury from
the manometers up to the flow channel. All pressure measurements were,
in addition, corrected for this temperature difference. Lastly, all test
data were reduced to the same temperature, 200 C, by means of Knudsen's
and Sutherland's formulas for the temperature relationship between mole-
cule flow and internal friction. All of the temperature corrections were
very small, usually less than 1/2 percent.

For one part of the measurements, a pair of gas traps (locks),
cooled with ice or liquid air, were blown into the feed lines AK and
BK (fig. 1) to prevent the mercury vapor and any other condensable
impurities from passing into the flow channel. These measurements were
corrected for the apparent volume increase caused by the cooling. They
amounted to about 20 cm at the most.

W. Gaede, Ann. d. Phys. 41, p. 289, 1913.

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All measurements had to be corrected for the flow resistance of the
intake line. This resistance was unusually high for the prism apparatus I
because of unsuitable design. In this case, the correction was so deter-
mined that, while widening the slit from 1 to 2 mm, the resistance was
lowered to a disappearingly small value. After that, T was measured as
usual; for these measurements, the resistance of the inlet line alone
determines the measured values, which are indicated by TR. The uncor-
rected value of the flow is indicated by TR+K and the corrected value
for the inlet pipe resistance by TK. Therefore, for each pressure,
according to the definition of T

TR(P P")R = (P' -P")K TRK( pl)

together with

p' p" = (p' p")R + (p' p")K

the partial pressure decrements through the inlet lines and the channel
carry the subscripts R and K. From these equations follow

Ty. = TR+K T (2)

All measurements were corrected in the same manner.

For prism apparatus I and for the porous filter plate, the corrections
at the lowest pressures amounted to more than 50 percent. For prism
apparatus II and the other channels, the corrections were less than
10 percent.

The Flow Channels

The prism apparatus I is shown in figure 2. The top picture shows
the prisms (in part in vertical section) mounted in a clamping device,
the arrangement of which is readily apparent. The middle figure shows
the prisms from above; one, Pi, is seen in horizontal section. The
bottom picture shows the hypotenusal faces of the two prisms. In the
face of the one prism P1 three concentric channels Kl, K2, and K3
have been ground, each one about 1 mm wide and 1 mm deep. Three conical

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holes H1, H2, and H3, into which the glass tubes 1, 2, and 3 were
ground, and sealed with apiezon grease, serve as leads to the channels.
Two of these tubes are inlets, the third was closed and merely served
as a stopper.

For the series of measurements, recorded in figure 10, channel K3
was connected with volume V' and K, with V". The flow was along
K3-) K2. The spacing of the prisms was gaged by placing a strip of
tin foil 5 mm wide between the faces around the outermost channel K .

Correct tightening of the wing nuts F makes the ring lie flat
between the faces, and become parallel. The apparatus was sealed with
shellac dissolved in pure alcohol, which was made to run down in the
groove between the prisms. This seal was perfectly satisfactory after
the alcohol evaporated. The radii of the two annular slits were

B1 = 1.1952 cm

R2 = 1.8897 cm

in the wide annular slit between K2 and K3 and

Rl = 0.8994 cm

R2 = 1.0919 cm

in the narrow slit between K1 and K2.

The high inlet resistance of prism apparatus I and the subsequently
large corrections made it desirable to make a number of measurements with
another prism apparatus of more adequate inlet design (fig. 3). This
apparatus had only one annular slit between the radii i1 = 0.8123 cm
and R2 = 1.5984 cm. The inlet lines were no less than 5 mm in diameter
at any point.

The circular channel in the hypotenuse surface of the prism P1
had a trapezoidal cross section of about 2 mm in depth; it was 4 mm wide
at the base and 8 mm on top. The clamping device had an unusual feature.

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The tension was elastic; the wing nuts F did not press directly on the
beam L, but on a pair of helical springs S which transmitted the pres-
sure on L. The stiffness of the springs was measured and the compression
read from scales etched in the blocks C. Thus, the pressure on the
prisms was known and it was possible to obtain equal pressure in both

The distance of the prism surfaces was determined by Christiansen's
method by measuring the angle between the Herschel interference fringes.

When monochromatic light of wave length X is so reflected that
the angle of incidence i is very near to the boundary angle of the total
reflection, the reflecting power of the surface will be very high, and
the light is reflected several times (fig. 4). The effect of the inter-
ference of the reflected beams J- Jr is such that the reflected
x 1 2
intensity is 0, when cos b = 2m, (a = distance of the surfaces,

m = a whole number). Bearing in mind that cos b = 1 sin2i

and that in the case of total reflection m = 0, cos b = 0, sin io = f.t

one obtains the equation

2 2no x x M. 2'
sin i sin21 =2 2 (3)

for defining the directions (im) of zero reflected intensity. In

i + 3 = p, n sin a = nj sin p (4)

As the angles between the dark fringes in the reflected light are
small, it is permissible to put sin2i sin2 = Bin2i di .
Differentiation of the equations (4) leaves n cos co da = -nl cos 0o di,
which, when introduced in equation (3) and resolved, gives

a o /no cos D m2 2 K 1
2 V2n cos a cos i Vdm -_ m ) (5)

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where dal, ml is the angle between the mth and the mith dark line.
If specifically m ml = 1, it is seen that the quantity x = 1'

where nsa indicates the angle between two adjacent fringes, is to be
constant. Thus, the angles Ai act as the digits 3, 5, 7 In
transmitted light, the complementary image, light stripes on dark back-
ground, are -isible, but the spacing of the stripes is unchanged (fig. 5).

The fringe system is observed in a telescope set to infinity, the
angular distances are measured on a micrometer. As a rule, the first
three to six fringes were used. Quantity x was defined as average
value or else computed by a different equalization of the individual
values, after which a is calculated by formula (5). Since the quantity
K(x o is constant for a certain color, it can be found when the refraction

conditions and the prismatic angle are measured.

The distances were measured at several points on the prism surfaces.
An illustrative example of such a measurement is given in figure 6.
Decisive for the molecule flow is the average value of the second power
of the distance, and for the laminar flow, the average value of the third
power of the distance. In the cited example

.a 2|2 2 = 9.21 : 0.020

The fluctuations of the room temperature had no measurable effect on
the prism spacing; but a increases a little with the pressure in the
apparatus so that a = ao(l + cp). In the foregoing example c = 1.5 X 10
(p in cm Hg.) Since p < 40 cm Hg, it is seen that a becomes, at the
most, 0.6 percent higher than ao.

Since the annular slits of the prism apparatuses were formed between
fine optical surfaces on which, according to Knauer's and Stern's
investigations, at grazing incidence, a part of the molecules is mirror
reflected, it was of great interest to measure the molecule flow in slits
between rough surfaces for which the cosine law is rigorously applicable.
For this purpose, the apparatus represented in figure 7 was resorted to.
The slit was formed between the ground surfaces of two thick round pieces
of plate glass P1 and P2 which were mounted in the clamping device as
indicated. The lower surface of the top plate carried the three channels
K1, K2, K and the three conical openings with the glass tubes 1, 2, and
3. Tin foil with rectangular sectors 1, 2, 3, 4 was placed between the
glass plates, thus forming two rectangular slits, one between K2 and
K(3, the other between K1 and K3. The width a of the slit is governed

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by the thickness of the tin foil. The dimensions L the length of the
slit in flow direction and b were measured with the cathetometer.
Only the slit between K and K3 was used for flow measurements; its
dimensions were L = 0.97 cm and b = 1.32 cm. Owing to the mat surface
the slit width a could not be measured optically, but was determined on
the basis of the laminar-flow data. A measurement of the thickness
of the tin foil produced no appropriate determination of a because the
large surfaces were not level enough. The grainy irregularities of the
surface were of the order of magnitude of 0.01 mm. To increase this
roughness still more, they were blackened in some tests with soot from
a wide flame of burning turpentine oil. The apparatus was sealed with
Picein, as indicated in figure 7.

The measurements indicated that the magnitude T / = f0) for
a given channel is solely a function of the mean path length and independent
of the gas. For a more accurate check of this rule, flow channels
were used which first remained geometrically constant during the time
interval of the test series with different gases. Two channels, both
satisfying the cited demand, were used for this purpose: a cylindrical
slit between two coaxial cylindrical surfaces and a porous body, a filter
plate of sintered glass. The cylindrical slit was obtained by fitting
an accurately milled brass plug (compare fig. 8) in an accurately
drilled hole of a solid brass block. Figure 8 represents a cut through
the axis of the apparatus. The cylindrical surfaces A and B fit the
holes exactly, while the diameter of the surface C is about 0.03 mm
less. The tubes RI and R2 serve as inlets; so the flow in the
cylindrical slit is parallel with the axis. The slit was L = 0.487 cm
in length, the diameter of the surface C was 1.9997 0.0001 cm. The
diameter of the hole itself was not measured, but that of the employed
drill was 2.0019 cm. The apparatus was sealed by means of stiff stop-
cock grease applied to the groove R. The channels h were drilled to
prevent an almost closed dead-air space from forming below the part B.

The porous filter plate was fused in a glass tube of the form
depicted in figure 9. The diameter of the plate was about 1.9 cm, its
thickness about 0.2 cm. The size of the pores was not,known from the
start. An upper limit is obtained, according to Weber0 by recording the
pressure required to force air bubbles through the filter when it is wet
and coated with a layer of water. At around 13 cm Hg, the air started
to penetrate at a certain point, but an increase of the pressure to about
17 cm already forced air through at many places. This proved that the
filter was homogeneous to some extent. On computing the surface tension
of the water at 73 dynes/cm and assuming complete wetting, it is found

S. Weber, Teknisk Tidsskrift 1917, Nr. 37.

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that the narrowest spot of the widest pore of the filter has a diameter
of about 17i when the pore is regarded as circular.


Flow measurements were made with hydrogen, oxygen, carbon dioxide,
and argon.

Hydrogen was produced in the Kipp apparatus from zinc and dilute
hydrochloric or sulfuric acid. It was purified and dried in a solution
of potassium hydroxide, a saturated alkaline solution of potassium
permanganate, concentrated sulphuric acid, and lastly, in phosphorous

Oxygen was taken from a commercial tank and then dried in sulfuric
acid and phosphorous anhydride.

Carbon monoxide was manufactured in Kipp's apparatus from marble and
dilute hydrocholoric acid, washed in distilled water and then dried like
the other gases. An assay with absorption in potassium solution indicated
99.9 percent purity.

Argon was obtained from the factory guaranteed 99.5 percent pure;
it was used without being treated.

The Measurements with the Annular Slits

The principal test data are correlated in table 1. The lines 2 to
4 give the slit dimensions with the optically defined values of the slit
width a. Line 6 gives the minimum values of TO (M = molecule
number of gas). Line 7 gives the values of T Vi at the lowest pressures
at which tolerably reliable measurements could be made. The other lines
are discussed below. The accuracy of the data is characterized by the
digit number, which is such that the uncertainty is one or several units of t
the last digit.

The several measurements are represented in figures 1Q, 11, 12,
and 13. The quantity TV\S is plotted against the logl0 og .
X = mean path length measured at 1 cm Hg, 20 in units 1P. The reason

for this choice of representation is found in the following paragraphs.

NACA TM 1301 13

At high pressures in the laminar-flow region the internal friction
of the gas governs the flow. Assuming that the speed of the gas along
the surface is zero

T 6? P

where i = coefficient of internal friction, Z = natural log.

Bearing in mind the slip, one obtains the boundary condition

uot = TI with uo = velocity along the surface, t the coefficient of ex-
ternal friction, and d1u the velocity gradient. With 5 = 1 = slip

T n(a3 + 6a2) p = Alp + BI

where Al and Bl are constants inside of a pressure area in which
tp is to be regarded as constant.
By kinetic theory, n = cpfl, with p = gas density, n = average
value of thermal velocities, and c = numerical factor.

The definition of X is that given in Landolt-BEornstein's tables7

1 (0076 cm Hg) = 1 o cm (7)
1 8 0.49 1,013,250


=2 273'R. R = 83.15 x 106 erg = gas constant
o 8 M degree mol

TLandolt-Bdrnstein, Tables 1, 1923, p. 119.

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The employed numerical values were:

Gas X2 (0076 cm Hg) IX (2001 cm Hg.) '1200 C



x 10-8
x 10-8
x 10-8
x 10-8




10- 4

C = Sutherland's Constant

With these numerical values

n200 = 0.717 x 10" x X1 x



for all gases.

Inserting (7a) in (6) and bearing in mind that 1 cm Hg
per cm2 one obtains

T !M = 0.971 x 109 x -


= 13,296 dynes

P. + 5.83 x 109 -2 -U = A + B

if p is measured in cm Hg and \1 in p.. From it, it is seen that
T / at high pressure becomes the same function of the mean path
length for all gases as a result of the definition of X.

From figure 14, where the measurements with the annular slit in
prism apparatus I are reproduced, it then also follows that T TiiM
at high pressure is linearly dependent on p/\ and that the dependence
is the same for all gases, which simply implies that the measurements are
in relative agreement with the employed values n20o

Starting from the concepts of the molecule flow, it must be assumed
that the rule T A = f(,\ must hold also at the lowest pressures, where

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the internal friction has either only little or else no significance at
all for the flow.

And so figure 10 actually indicates that the dependence of T SRM
on p/X is nearly the same for all gases, not only at the high pressures,

but also in the entire pressure range explored. The discrepancies lie
at the limit of instrumental accuracy.

The foregoing indicates that the minimum of T lies at the pressure
where the mean path length Xm is equal to the slot width a.

The height of the minimum can also be approximated quite simply in
view of the validity of the relation Tmin 2 TKn, TKn being calculated
from Knudsen's general formula

FRe!)w 3 21cLt dl (9)
TKn = 2 LW S S2

which, applied to a circular slit, gives

TKn = a (10)


The values TKn iT computed from equation (10) are given on line 8
of table 1.

At the lowest pressures, T yiM becomes constant independent of the
pressure, as is plainly seen in figure 13 (hydrogen in prism apparatus II),
where the measurements have been carried out to about p = 3 x 10-5 cm Hg.
Throughout the entire pressure zone in which the mean path length was
relatively large compared to the slit width R2 R1 (marked on all
graphs), T is practically constant (T ). In these measurements, the
inlet lines were cooled by liquid air, so that the flowing hydrogen was
not contaminated by mercury vapor. Another effect of the cooling was
that the McLeod measurements were more reliable and this made it possible
to extend the measurements so far into the low pressure zone. In the
other test series, the boundary values T, were less accurately determined.

Ann. d. Phys. 28, p. 76, 1909.

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A calculation of the free molecule flow in slits of the form employed
here has never been attempted, to the writer's knowledge. An attempt at
an accurate treatment of this problem resulted in difficult calculations
and will not be mentioned. Professor N. Bohr pointed out that an
approximate solution could be obtained by appropriate use of Clausing's
formula for the molecule flow in a rectangular slot. This formula,
which is valid for b >> L >> a, can be written as

T VM R]R(1 a2b (11)
Cl 2 t 2 ait/ L

For example, putting L = R2 R1, b = i(R2 + R1) gives for the flow
in the circular slot

2, R 2i)a^Rp + RA
TC1 M = 1) R + 1)-2-) (12)

and this expression must become exact when (R2 + RI) RH R1 >> a.

The values of line TC1\/M in table 1 are computed from formula (12);
the ratio R2 + R1 is also given in the table for the various circular
R2 R1
slits. It is seen that the computed values T CL\/ are greater every-
where than the observed T VFM. This is due, in part, to the fact that
the measurements were not extended to sufficiently low pressures in
all test series, and in part to the fact that the mercury vapor was
frozen in only one of the aforementioned series and therefore had a
retarding effect on the flow in the other series; lastly, the condition
R2 + R1 >>R2 1 is not fulfilled and it is easily seen that the very
noncompliance with this condition must result in a divergence in the

direction of computed value >observed value. The ratio R2 + R1
R2 B1
is greatest for the small circular slit, namely, 10.4, and here also the
agreement between observation (1.6) and calculation (1.72) is fairly

It was found that all test data can be represented in close
approximation by

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T =A + B + C log a (13)
---T -+ D
2 R1

The values of the constants A, B, C, and D, which are dependent on the
slit, but not on the nature of the gas, were determined graphically and
entered in lines 9 to 12 of table 1. The curves of figures 10 to 14
are computed from equation (13) with these constants.

At high pressures, the last term (C) in equation (13) is dis-
appearingly small and by comparison with equation (8) it is seen that

A = 0.971 x 109- -. From this equation, one of the quantities A
or a can be computed when the other is known. Computing a from the
A values gives the values indicated with at in table 1; they are
all smaller than those defined optically. The difference Aa = asopt astr
is in all cases about 0.15 0.24. No satisfactory explanation of this
discrepancy could be found. It was possible to attribute it to
the inertia forces which were not allowed for in the calculation of
equations (6) and (8); but the calculation indicated that they were only
about 1/10b of the friction forces and hence were altogether insignificant.
On computing the coefficient Al of formula (6) for each gas and then
a str by means of the value T1200, almost the same value is obtained for
all three gases H2, 02, and C02, which, at about 9.0O, is in good agree-
ment with the value 9.02P obtained in this case from equation (8). This
implies that the employed gases were sufficiently pure. To be completely
sure in this respect, a series of measurements with atmospheric air were
carried out. The air was dried in sulphuric acid and phosphorous
anhydride. The inlet tubes were cooled with ice in order to reduce the
mercury-vapor pressure in the slit to a low value. The measurements
were carried out in such a way that for each value of the pressure p
(average values) several (2, 3, or 4) independent measurements of the
quantity T were made at decreasing values of the pressure difference.
The value of T for the same p never indicated a systematic course,
which confirms that the inertia forces are disappearingly small; hence,
the use of the average value of T for each p. The measurements were
limited to the high-pressure range. The results are represented in the
following table:

NACA TM 1301

Atmospheric Air in Prism Apparatus II; aopt = 8.15p

p T C' (T C') (T C') A
obs obs ber o-b







In order to be able to use all the measurements for determining
the slope coefficient Al, the curvature of the T, p curves must be
taken into account to some extent. For this reason, the C term of
formula (13) was computed and subtracted from T obs. The values of the

D = 1.5, 0.67 = 0.12; X = 4.7-
l/M 29

were utilized. The correction term is indicated with C' in the table.
The corrected term (T C')obs is to be linearly dependent on p.
Smoothing resulted in T C' = 0.0287p + 0.1550; the values computed
from it along with the differences Ao-b are reproduced in the table.
_____7 -4
Inserting Al = 0.0287 and 20 = 1.820 x 10" in equation (6) gives
astr =7.984.

An optical measurement of the slit width accompanied the flow
measurements. The results were aopt = 8.171 measured with green
mercury light and aopt = 8.134 measured with yellow mercury light;
average value aopt = 8.15 0.02i. The difference aopt astr= 0.17P.
This example proves without a doubt that the nonagreement of the two methods;
is real, and that it cannot be explained by insufficient purity of gas
and subsequent uncertainty of the value of n. The optical test method
is not gone into any further. Unless stated otherwise, the optically
determined values of the slit widths are used in this report.

Returning to the discussion of formula (13), it is noted that, by
comparing the expressions (12) and (13) for X = m

NACA TM 1301

B = b RT a R2 + ) bC
R R c
2 1

where c and b are pure numbers. From this, it is seen that the
ratio B = b should be constant, which (compare table 1) is not
C c
altogether true. The discrepancies are largely due to the previously
cited insufficient fulfillment of the condition R2 + R1 >R2 R1,

As stated before

nmin "a a,

min Kn

3 V2 V M 1 2


is valid for the minimum.

Knudsen's general formula


0 dl

is obtained by computing the tangential motion quantity B1

given off

on the channel wall per unit time per unit area, and by putting the total

Lmotion quantity
motion quantity

Blo dl equal to the driving force (p' p")S.

A premise of this calculation is that B1 can be proportional to
the average value of the flow velocities over the channel cross section.
In consequence, the calculation can be valid only for channels with
somewhat homogeneous cross sections, for example, slits of constant width,
circular-cylindrical pipes, and so forth, as already noted by v. Smoluchowski
and Clausing. Furthermore, these researchers have indicated that an exact
solution on the assumption of the cosine law for X = a* produces different
and greater values of the quantity T than Knudsen's calculation. Then
it is readily apparent (for instance, on examining the Smoluchowski-
Clausing calculation, or in the calculation of the pure effusion in a
slit; compare on the next page) that a very large part of the free

TKn = V M w'

NACA TM 1301

molecule flow in the slit is due to molecules which fly approximately
parallel to the slit walls and therefore cover great distances between
two collisions against the walls; this holds for infinitely small

If the pressure has such a value that a (small) number of collisions
occur (in the zone L > X > a), these far-reaching molecules will con-
tribute much less by reason of the mutual collisions. Against it is a
contribution of molecules which have participated on mutual collisions
but which at the beginning (that is, when X is still great compared
to a) is very much smaller than the decrease in flow caused by the
collisions. The total effect of the collisions is a decrease in flow
in the zone X = m to X 2 a.

In the minimum zone X a, the assumptions of Knudsen's calculation,
particularly the assumptions of constant flow velocity over the channel
cross section, appear to be fulfilled to some extent, hence Tmin TKn.

The discussion of the circular-slit measurements is concluded with
a few remarks about the pure effusion and the effect of the mirror reflec-

The effusion, that is, the contribution to the flow due to molecules
which pass the slit without impinging on the wall, is easily computed.
The result is discussed because it is a fine example of the importance
of the cosine law.

The flow in the annular slit is assumed to be from the inside toward
the outside:

I. Assumption: The molecules are emitted from the cylindrical
surface 2iR1 X a according to the cosine law, so that

TEfI F = 2 R9 a2J

NACA TM 1301

where J is the defined integral

0J=R x 1

= ( arc sin
( 1

+ 2-


II. Assuming, by way of contrast, that the molecules proceed
from the plane circular areas 2R1 2 according to the cosine law

TEfI M = 2n R a 22 R12
R22 R12

The results are, like the assumptions,
the numerical significance, the ratio



not identical. To indicate


2 R2
2 1

was computed for several cases

R2 R ( JEff
R2 1 I ^T

Prism apparatus II .
Prism apparatus I, wide slit .
Prism apparatus I, narrow slit .





NACA TM 1301

It is apparent that the ratio E-S- -41 when _2 -1, which is
also apparent from the expression for J, and for the rest is immediately

It is difficult to decide which of the assumptions is preferable
for the annular slits, or whether a third might not be better still; for
this reason, the effusion cannot be indicated with a high degree of

The foregoing calculations are applicable only in the absence of
mutual collisions of molecules; the effusion decreases considerably with
increasing pressure. An exact calculation of it is not of interest in
view of the uncertainty of the first calculation and the small value
of the effusion; hence, the decision to put the effusion at pressure p
1 1 X A -

equal TO TEf e

t nnl-: y l

The values of the effusion by assumption I, shown in table 1 below
T EfI VM amount to a few percent of the flow.

The individual measurements T v/i
the effusion are plotted in figure 13.
the uncorrected values, which were also
in formula (13).

with and without correction for
The other graphs contain only
used for defining the constants

Knauer and Stern have proved divergences from the cosine law for
grazing incidence toward polished surfaces. Since precisely the molecules,
moving grazingly toward the walls, contribute much to the flow, it is
readily apparent that they produce a great increase in the flow when
they are mirror reflected. The writer believed that up to a certain
time interval, this reflection had great importance for the marked
increase of flow in the zone from T to T But a more accurate
min o
calculation, indicated in the following, showed that with the small values
of the mirror reflection, cited by Knauer and Stern, its effect is, at
the most, a few percent of the free molecule flow. This is in good
agreement with the fact that the ordinary calculation of the molecule
flow (Clausing's formula) explains the observed great values T com-

NACA TM 1301

By calculations, not repeated here, it is found that when Tr
denotes the increase in flow due to the mirror reflection, for small values
of the reflecting power

Tr/M < {Mr (E + 2Fa)da (14)9

where r, = the reflecting power when a is the grazing angle and
am = the upper limit of the grazing angle for which a > 0

E = R 1a,

F = h[t/R + arc cos > ( 2 12 2R2
2 I2 2R 1 2

For the reflecting power of a polished surface (speculum metal)
compared to hydrogen, Knauer and Stern have given the following values0

a = 1 x 10-3 1.5 x 10-3 2 x 10-3 2.25 x 10-3 Radian
obs = 0.05 0.0 0.015 0.0075
raber = 0.050 0.033 0.016 0.0075

These results are closely approximated by the expression
r = 0.084 34a, as seen from the line rber- Introducing this expression
for r along with the constants RI, R2, and a for the prismatic
apparatus II in formula (14) gives Tr\{WM< 0.075 for hydrogen. Reflec-
tion measurements on other gases have never been published to the writer's
knowledge. But an explanation of the reflections by wave mechanics indi-
cates that the reflection of a given surface diminishes when the number
of molecules of the reflected gas increases because the associated wave

The calculation is to be found in: Om Luftarters Stromning in snaevre
Kanaler. Copenhagen 1936, pp. 63-69.
10Ztschr. f. Phys. 53, 1929, p. 779.

NACA TM 1301

length I h is inversely proportional to M and the condition for
mirror reflection is Asin a < 2, where A is the mean height of the
irregularities of the surface. Therefore, if r is assumed to be

r = I/(0.084 3a) (15)

for other gases.

The values Tr /M in table 1 were computed from equations (14) and
(15); they are, as will be recalled, the upper limits.

It will be seen that whenever the reflection contributes to the flow,
the rule T 9[W = f,) cannot be rigorously fulfilled; it can hold only
for the quantity (T Tr) M.

The expression (14) was obtained on the assumption that no mutual
collisions occur. Maxwell computed the effect of the mirror reflection
on the slip at higher pressures and found that

R E) 2- f (16)11

where f is the fraction of the incident molecule sent out diffused from
the surface; the part 1 f is mirror-reflected. Knauer's and Stern's
values for the reflection are 1 f = 38 x 10-8 f being practically
equal to unity.

With these values, the formulas (6) and (16) give

BV n= ja2t fR = jr \ 'A_ (17)
1 R2 2
1 1

1L. B. Loeb: The Kinetic Theory of Gases. 2nd edition, p. 288.

NACA TM 1301

which was to be the contribution of the slip alone. The quantity
B1 ,I is shown in table 1; it is, in all cases, smaller than the constant
term B in equation (13).

Retaining the factor 2 f in equation (17) and equating B1, vV

to the term B in equation (13) gives the values for f which are
shown in table 1 and are much smaller than unity, as well as being
contradictory to the afore-mentioned small values of reflection. Since
it is not likely that the reflecting power of the surface at higher
pressure would be so much greater, it would seem that Maxwell's value
of the slip coefficient in the manner attempted here should not be

We will now discuss measurements with the other gases.

The measurements with the rectangular slit between ground glass
plates (fig. 7) were made for the purpose of ascertaining the flow in
a channel in which no mirror reflection occurs. Three test series with
hydrogen were run off:

I. Flow between the pure ground plates
II. Flow between the same plates blackened with soot
III. Slit as in II, but the air traps of the inlet tubes were
cooled with liquid air

The principal data of these three test series are correlated in
table 2; the individual results for series II and III are represented
in figures 16 and 17.

The slit width a could only be determined by means of the laminar
flow data. At high pressures

(a + 2t)3b a3b a2tbp
12LD 12LD p 2LT

or T = Alp + B1, Al and B1 being constants within a certain pressure

Figure 17 indicates the linear relationship between T and p at
high pressure. It is seen that the slope Al can be fairly accurately
defined. The value of a,, computed from the equation A1 = 2Lb
is the average value a3; but the surface irregularities are of the
same order of magnitude as a (about 0.01 mm).

NACA TM 1301

The minimum lies at the pressure where X = a. Knudsen's formula,
applied to the rectangular slit, gives

4f 2|R' a2b
TKn = 3 x M L~

The values TKn /i computed from it are given in table 2. The agreement
between Tmin and TKn is less good than for the flow in the circular
slits; Tmin is about 20 percent greater than TKn. The reason for
this is, in part, that the slit width is not well-defined because of
the irregularities and, in part, that the flow for the same reason
differs from the flow between smooth surfaces.

To compute the flow To at the lowest pressures the writer attempted
to use Clausing's formula

T E 9 (l+ L) a2b
SCl 2 v M 2 a L

the formula holds for b L > a, which is not entirely complied with
because b is only 1.4 times L. The calculated values TCi are about
25 percent greater than the observed values To; this discrepancy is
due to the nonagreement of the geometric assumptions.

The most significant result of these measurements is that they show
that the decrease of T in the zone from X = oo to X 2 a is not
produced by the mirror reflection, but rather by the collisions of the
molecules among one another.

The measurements with the cylindrical slit (fig. 8) and with the
gas filter (fig. 9) were made for the purpose of checking the rule
T VM = i(x) in absolutely constant channels. Hydrogen and carbon
dioxide were used.

The results for the cylindrical slit are represented in figure 18.
At high pressures, the measurements with the two gases are in complete
agreement, as indicated by the fact that when the slope Al in the
equation T = Alp + B1 is graphically defined and a is computed from
the equation A, = A 6-, a = l16.1l for CO2 and a = 15.9p for H2,
if the previous values T20 are utilized. The slit width a being
very small compared to the radius R and the length L, the cylindrical
slit can be regarded as a rectangular slit of width 2nr in the calcu-
lation, and this produces the preceding formula for Al.

NACA TM 1301 27

The position of the minimum is, as for the other slits, determined
by the fact that min 2:a.

The application of Knudsen's formula yields

3 L
which, on inserting the numerical values, gives TKn 5N = 5.50, in
close agreement with the value Tmin VT (compare fig. 18).

In the range X 220p to X 1000i, there is a small discrepancy
between T 'WF for the two gases so that T P/d" is always greater

than T VM(CO for the same value of --. The difference is about
4 percent, at the most. A plausible explanation is that the true effec-
tive path length Xe in the range = co to X 2 a is not the same
as the path length computed from the equation K = This is apparent

from the remark that X1 and hence X is determined by the internal
friction in such a manner that only the collisions, which occasion great
direction changes, are regarded as cut-offs of free path-lengths. But
in a slit-in molecule flow the collisions initiating only small direction
changes are also significant because (as stated previously) a great part
of the transport in a slit is due to molecules, which fly approximately
parallel to the slit walls, and this contribution to the flow is greatly
diminished even by small divergences. Therefore, the effective path
length Xe must be less than K for the same value of the pressure in
the low-pressure zone.

Knauer and Stern, Mais and Rabi, and others have demonstrated that
the same holds true for the effective path length which is determined
by scattering of molecular rays. Mais12 adduces that the effective
center distances for collisions between potassium atoms and hydrogen
or carbon dioxide molecules, etc., are

a12 = 7.2 and 13.3 A for deviations < 4.5'
"12 = 5.7 and 9.6 A for deviations < 1.50
12 = 5.0 and 5.9 A for great deviations

The effective mean path length Xe 1 decreases therefore much more
in C02 than in H2.

12W. H. Mais, Phys. Rev. 45, p. 773, 1934.

NACA TM 1301

On assuming that T \M= f(Le the length A B, cut off on a

horizontal line in figure 18 between the M(TV, t) curves, is equal

to the log10 of the ratio of the effective path lengths. Hence, it
follows that XeH2 is everywhere greater than Xeco2 for the same
value of -. The ratio eE2 is around 2, when logl0 -L = -2.5.
X1 XeCO2 X1
This appears to be in good agreement with Mais's results.

The pure molecule flow in a cylindrical slit has been computed by
Clausingl3. Application of his formula gives TCi M = 12.3. The
measured value of T VM at the lowest pressures was about 9; the dif-
ference is due, in part, to the fact that the mercury vapors were not
frozen out and, in part, to the fact that the measurements were not
extended to sufficiently low pressures.

Figure 19 represents the test data with the glass filter as flow
channel. The corrections for the inlet tube resistance were rather great
(about 50 percent at the lowest pressures) since the resistance in the
filter was quite small. When X > about 10, Tf/M is practically
constant. The variation of T VT9 = f M is exactly the same for both
gases throughout the entire pressure zone. According to Weber's invest-
4gations of such filters 14, the flow at low pressures should be regarded
as pure effusion, which agrees with the fact that no minimum exists.
This is apparent by an application of Knudsen's theory of effusion
through an opening in a thin platel5.

Np1 = A(v' v")

A = area of opening, NPI = number of molecules per second through
opening, Vt and v" are collision numbers per sec cm2 at both sides
of the plate, where the pressures are p' and p".

When each pore in the porous body can be regarded as a series of
chambers, between which openings of the magnitude Al, A2 .. An
are presented, the notations of figure 20 give

Ni = A( v' v'1) = A2(v'1 v'2) An(V'n-l v)

130ver den Verblijftijd van Moleculen Amsterdam 1932, formulas
151 and 154, pp. 108-109.
148. Weber, Teknisk Tidsskrift 1917, Nr. 37.
15Ann. D. Phys. 28, p. 999, 1909.

NACA TM 1301

from which follows

Ni Ni Ni
Al A2 An


N =

(v' v")

If the porous
total effusion is

plate consists of a number

(i) of such pores, the

N = 1 Ni (' -


Hence, it follows that N is proportional to v' v", or that T is
constant at low pressures. Knudsen's theory is applicable to pressures
where X is great compared to the diameter of the opening A. At higher
pressures the ratio NP1 increases with p. Applied to equation 18,
v' -v,
it indicates that N must increase also with increasing pressure in
good agreement with the measurements.

Translated by J. Vanier
National Advisory Committee
for Aeronautical

Table I

Flow in the Annular Slit


NACA TM 1301

H2 02 CO2 A '
2.00 2.01 1.92 1.98
4.0 3.9 3.8 3.8 ,



,0.10 0.03 0.02 0.02 ,








R1 cm
R2 cm

aopt astrC
R2 + R1
R2 R1

f V







NACA TM 1301

Table II

Flow of Hydrogen in Rectangular Slits

Size of Slit Observed Calculated

L cm b cm a4 Tmin M To0 M TKnAFM TCiM

I 0.971 1.320 19.2 0.94 1.6 0.80 2.14
II 0.963 1.323 18.1 0.92 1.4 0.75 1.89
III 0.963 1.323 18.2 0.92 1.5 0.76 1.92

NACA TM 1301

Figure 1.- Pressure-gage assembly.

NACA TM 1301



: 0

Figure 2.- Prismatic apparatus I.

NACA TM 1301

Figure 3.- Prismatic apparatus II.

NACA TM 1301 35

Figure 4.- Theory of Herschel's interference fringes.

NACA TM 1301

Figure 5.- Herschel fringes photographed in transmitted (top) and
reflected light.

Figure 6.- The hypotenusal surface of prism Pi of the first prismatic
apparatus showing measured distances in units I .

yi i
[. 7

- I



VI 1111 1

-- -'- 'AFA

' '


0 6
S *

9 9 S
S. S

* *

I *

' *

. q
r r b

Figure 7.- Rectangular slit between ground glass plates.

NACA TM 1301




NACA 'TM 1301

-C R,


Figure 8.- Flow channel with cylindrical slit around C.


Figure 9.- Filter plate.

NACA TM 1301



,a )

4) a(


0 a


NACA TM 1301



Figure 11.- Flow of hydrogen in the narrow circular slit of the prismatic
apparatus I. The flow was measured in both directions K1 --> K2
and K2 K1. The results are completely identical within
measuring accuracy.

NACA IM 1301


Figure 12.- Flow of hydrogen in prismatic apparatus II. aopt. 8.17P.
The graph shows the uncorrected values of T 7FM- (.), as well as
those corrected for inlet tube resistance (O) along with the
measurements of the quantity TR \JM at 100 times smaller
ordinate scale.

NACA TM 1301

Figure 13.- Flow of hydrogen in prismatic apparatus II. aopt. = 8.13 i.
The air locks (or traps) of the inlet tubes are cooled with liquid air.
The effect of the effusion is indicated. The upper point of the two
points linked together is not corrected, the lower one is the value
of T \ corrected for the effusion.

NACA TM 1301






+ oH
lo CO 2


i i i i l i 1

0 1 2 3 4 5 6 7

Figure 14.-

Flow in the wide annular slit at high pressure.

Figure 15.- Cut placed normal to the plane of the annular slit; the axis of
rotation lies in the drawing plane.

NACA 2M 1301

Figure 16.

Figure 17.

Figures 16 and 17.- Flow of hydrogen in the rectangular slit.

O Indicates liquid-cooled air traps
Indicates no cooling

NACA TM 1301



Figure 18.-

Flow of hydrogen and of carbon dioxide in the cylindrical slit;
r = 1.00 cm, L = 0.487 cm, a = Ar = 16.0p.

NACA TM 1301

e*Ha oCOa


* C* o*.
6 g Od cB 'D@*

* 8e 8 *0


A=/O /0o3 /10' O /i0

0 LogP

Figure 19.- Flow of hydrogen and of carbon dioxide through glass filter;
the top row of dots and circles indicates the values corrected for
inlet resistance.

Fiur I v s EIc pe yri =
A h p or 3 A4 ie

Figure 20.- Each pore equals a series of chambers.

NACA-LaineL 8-17-51 1000





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