O

o

o

CO

LU

CO

Di

O

Q

Q

O

N84-2127 6

DEPARTMENT OF CHEMICAL SCIENCES

SCHOOL OF SCIENCES AND HEALTH PROFESSIONS

OLD DOMINION UNIVERSITY

NORFOLK, VIRGINIA

ANALYSIS OF FLOW DYNAMICS THROUGH

SMALL DIAMETER GAS SAMPLING SYSTEMS

By

Kenneth G. Brown, Principal Investigator

Final Report

For the period July 1, 1983 to January 1, 1984

Prepared for the

National Aeronautics and Space Administration

Langley Research Center

Hampton, Virgini a 23665

Under

Master Agreement NAS-1-17099

Task Authorization No. 22

Dr. George M. Wood, Jr., Technical Monitor

General Research Instrumentation Branch

February 1984
DEPARTMENT OF CHEMICAL SCIENCES

SCHOOL OF SCIENCES AND HEALTH PROFESSIONS

OLD DOMINION UNIVERSITY

NORFOLK, VIRGINIA

ANALYSIS OF FLOW DYNAMICS THROUGH

SMALL DIAMETER GAS SAMPLING SYSTEMS

By

Kenneth G. Brown, Principal Investigator

Final Report

For the period July 1, 1983 to January 1, 1984

Prepared for the

National Aeronautics and Space Administration

Langley Research Center

Hampton, Virginia 23665

Under

Master Agreement NAS-1-17099

Task Authorization No. 22

Dr. George M. Wood, Jr., Technical Monitor

General Research Instrumentation Branch

Submitted by the

Old Dominion University Research Foundation

P. 0. Box 6369

Norfolk, Virginia 23508

February 1984
ANALYSIS OF FLOW DYNAMICS THROUGH

SMALL DIAMETER GAS SAMPLING SYSTEMS

By

Kenneth G. Brown*

INTRODUCTION

The removal of gas material through a capillary opening in a surface is

analyzed. The gas, from which the sample is removed, is moving past the

surface at supersonic velocities. A variety of possible conditions of temperature, pressure and composition are discussed in an effort to emulate

conditions that (1) might be found at the surface of a vehicle traversing

the altitude range 100-50 km, or (2) possibly exist at the surface of a

model in the stream of a high enthalpy wind tunnel. We will be concerned

with determining (1) the throughput of the capillary for conditions of different lengths and different L/a (length/radius) ratios, (2) the total

throughput when the surface in question contains many hundreds of these capillaries, (3) the effect of the capillaries upon the composition of the

analyzed gas, (4) the effect of the capillary or capillaries .upon the gas

stream itself, and (5) the implications of the calculations upon the possible implementation of this type of device as an inlet for a mass spectrometer to be developed for analyzing the upper atmosphere.

MODEL PARAMETERS

As this capillary or assembly of capillaries is to be mounted at the

surface of a reentry vehicle, it is necessary that the surface conditions as

a function of altitude can be obtained either from experiment or calculation. The two most important parameters in determining the flow

'Associate Professor, Department of Chemical Sciences, Old Dominion

University, Norfolk, Virginia 23508
characteristics are the temperature and pressure at the surface. The pressure at the surface of a reentry vehicle can be determined using a program

i

developed by Shinn, Moss and Simmonds [l] assuming a shape for the model and

a suitable flight path angle. These calculations have been performed by

Eide [2] at altitudes of 55 km, 64km, 72 km and 84 km. The shape of the

model was a combination of a sphere and cone to produce a blunt-nosed

reentry vehicle. One point at the surface was selected for all of the

calculations in this report. The dimensionless coordinates of this point

are (.013, .017) (see ref. 1), a point slightly behind the point that

represents the merger of the sphere and cone used to generate the shape of

the model. The calculated wall values at this point for the above altitudes

are given in table I.

Table I. Surface values for the parameters as a function of altitude.

Altitude 55 km 75 km 73 km 84 km

Pressure (atm) .0686 .0339 .0151 .00405

Temperature (K) 1480 1620 1590 1560

density (g/cm) 1.53 10'5 6.67 10~6 2.87 10"6 7.92 10~7

viscosity (poise) 5.59 1Q-4 5.93 lO'1* 5.94 lO'1* 5.9 lO-^SV

CALCULATION OF THE MEAN FREE PATH

To help determine whether the flow will be molecular, viscous, or transition, the mean free path for the various gases under consideration must be

determined at each altitude. To estimate the mean free path the following

equation developed by Sutherland [3] will be used:

X = 8.589 (n/P) (T/M) cm (1)
where = mean free path

n = viscosity in poise

P = pressure in mm

T = temperature in K

M = molecular weight

The results of these calculations are shown in table II for 0 atoms, 02,

1^ and CCfe. The largest mean free path (MFP) is for 0 atoms at 84 km

Table II. The mean free path of the various species. All values are calcu

lated using equation (1). All values divided by 10'1* cm.

55 km

0 8.86 km

02 6.26

N2 6.7

C02 5.34

65 km

19.9

14.1

15.0

12.0

73 km

44.3

31.3

33.5

26.7

84 km

162

11.5

123

98

and is approximately 160 microns while at the lower altitudes the MFP drops

to approximately 5 microns for COz- This variation in MFP would indicate

that at a capillary radius of 10 microns the flow may be molecular at high

altitudes with a X/a = 16, with an increase in the viscous nature of the

flow at lower altitude. It is more likely, however, that the bulk of the

flow will be in the transition region where the flow will be neither molecular nor viscous. In order to determine the relative importance of these

two types of flow the calculations will be performed assuming (1) that the

flow is of either nature, and (2) a treatment of the flow that tries to

simulate the transition (slip) region will be utilized. The results of

these different calculations will then be compared.
VISCOUS FLOW

For the purposes of this report, it will be assumed that the viscous

flow will be given by Poiseuille's equation corrected for length. This

equation is given below:

\ SHnRT L J

where a = radius of the capillary (cm)

L = length of the capillary (cm)

M = Molecular weight

P = pressure in microns

TI = viscosity in poise

R = gas constant

Q = flow in micron-liters/sec.

N = Avogadro number

All viscosities are calculated using the equations and programs developed in

reference 4. The exit side of the capillary will be maintained at a pressure of 10-5 torr, or less, so that P can be ignored. Dividing the above

equation by the pressure yields the conductance, F, which has units of

liters/second. The calculated flow for pure 1^, air, and CO^ in air are

compared in figure 1 as a function of inlet pressure for different L/a ratios. Since 02 and CO^ are minor constituents the viscosity does not

change and the curves for the three cases overlay at the assumed temperature

of 1500 K. (This temperature represents an average for the temperatures

presented in table I. At a constant temperature and pressure there will not
itr

10'

10C

O

O»

10

u

O

10"

X

/

_ / L/a = 20

10 20

P(mm)

30 40 50

Figure 1. Viscous flpw, as a function of pressure, for Nitrogen,

Air, and Air-1% CO^ at three different values of L/a. As

the viscosities for all three are the same there is only

one curve at each L/a. (-1500K, ... 300K)

5
be a capillary induced effect upon the composition of the removed gas for

viscous flow.

MOLECULAR FLOW

If the MFP is greater than the characteristic dimension, in this case

the radius, then the flow will be governed by collisions with the opening

rather than with other molecules. In addition, if the length of the capillary is short enough the flow will remain molecular throughout the capillary. To calculate the flow the following formula developed by Clausing is

used:

l

F = 3.638KA(T/M) (3)

where F is in liters/sec, A is the area of the orifice, and T and M

are the temperature and molecular weight, respectively. K is a constant

which corrects the flow for the length of the capillary. Values of K are

quoted in standard reference books as a function of L/a. In the limit of

large L/a (XLOOO)

K = 8a/3L . (4)

The calculated molecular flow rates for 0 atoms and Na are compared to

the calculated viscous flow rates in figure 2, for a = 100 and 10 as a function of a/L. An immediate observation is that, if the flow is molecular,

the observed flows through the capillary will have to be corrected for

molecular weight. Also, for a very small radius capillary the flow will be

greater in the molecular case than in the viscous case, at the pressures

considered.
10

10

o

o

10

-1

mi i i i i INI 11 i i t

10

L/a

Fiqure 2. A comparison of calculated molecular flow for 0 atoms

(—) and N2(...) with calculated viscous flow (-)

a) a = lOy b) a = IQOy.
COMBINED FLOW

The above calculations represent the two limiting cases for flow. It

is necessary, however, to account for the possibility that the flow will be

in the transition (slip) region. Knudsen [5] observed that as the pressure

increases, the flow through the capillary actually reached a minimum value

in the transition region. One possible model, which attempts to account for

the transition region and combine molecular and viscous flow, was proposed

by Scott and Dullien (6,7). They assumed that the fraction G(a/A) of the

collisions with the walls which resulted in the particles being backscattered, not transmitted through the tube, or which occurred without an intermittent intermolecular collision is given by:

/cose\/2a \ " ~ 4 n2 HO \- 0 /\

G(aA) = - / / sin e cos 8 e \ slne /\ x

 d e d +. (5)

n o o

An approximate solution for the above equation is:

<x(a/A) = e

sinh f Ji-\ (6)

This assumption results in the following equation for the flow:

nL - e

c,-«k-

-

sinh

2a -sinh-1 /2a \

(Ap + g)+ Ce —J (7)

8
where p = the average pressure and A, B and C are defined below.

n a1*

A = , (8)

8nL

4 n a3RT

B = , (9)

3 Mv L

16 a3RT

C = . (10)

3 Mv L

All of the symbols have been defined above except for v which is:

1

/8RT\ j

v = /

(ii)

Equation 7 reduces to the Knudsen formula when X»a and becomes the

Poiseuille equation if a»X.

Equation 7 can be compared to experimental data for the flow of

through a glass capillary using the data of Knudsen [5]. The agreement between experiment and theory is quite good with the theory predicting the experimentally observed minimum at approximately the correct pressure. In

figure 3, data obtained from observing the flow of N2, Oa, and C02

(Lewis and Brown, unpublished results) through a gold leak into a quadrupole

mass spectrometer is compared to that calculated using equation 7. The

agreement between theory and experiment is excellent for the 02/N2 ratio

and reasonable for the C02/N2 ratio. (The precise radius of the pinhole
.27

.26

CM

O

o

.25

.24

I I 1 I I I I I I I

4 5

P(mm)

8 10

Figure 3. Comparison of the flow observed for C0a> N2 and Oa

in air to that calculated in equation 6. Calculated

COa/tla --- calculated 02/N2 -, experimental C02/N20,

experimental 0/N x. Temperature = 300K. The calculated

values assume a radius of 20y and a length of 400y.

10
and the radius of the leak were not known in this experiment.) It is apparent that, as the pressure decreases in front of the gold leak, the flow experiences a change in character as evidenced by a sharp change in slope for

the ratio. This alteration in slope is due entirely to a change in nature

of the flow rather than a change in composition. Any observed ratio will

then have to be corrected using equation 7, or any other formalism that

accounts for transition flow. As equation 7 is fairly successful in accounting for flow, changes, it will be used in the remainder of this report to

discuss the sampling process.

EVALUATION OF THE FLOW

The calculated flow rates for Nitrogen as a function of pressure, for

different valules of L/a, are shown in figure 4. At room temperature the

region of molecular flow occurs over a very small range of pressure (5 mm).

At higher temperatures the range of molecular flow seems to be greater. An

expansion of the data in figure 5, for a/I = .025, better illustrates the

minimum in the flow at a pressure of 5 mm. At the higher temperature,

1500 K, the minimum does not occur until approximately 25 mm and is quite

broad. The high temperature represents the temperatures compiled in table I

for the reentry vehicle. At this temperature molecular flow is seen to

occur over a wider range of temperature with the transition region extending

up to 55 mm of pressure. Comparison to table I would indicate that for the

lowest altitude in that table, the flow would still be, at worst, transitional and not molecular.

FLOW RATES FOR MASS SPECTROMETRY

The above calculations and resultant graphs are all for flow through

11
10

c)

u

<u

CO

10C

a)

P(pim)

Figure 4. Comparison of the calculated flow rates for N2

for equation 6. 300K (-), 1500K (---) a) L/a =

40 b) L/a = 20, c) L/a = 8. In all cases

L = 400y.

12
.25

.2

0

u

.15

.1

10 20 30 40 50

P(mm)

Figure 5. Calculated flow rates for Nz as a function of pressure for I/a

= 40 and L = 400y, 1500K (---) 300K (-).

13
one capillary. The proposed inlet will actually consist of an array of capillaries in a multi-channel plate. The number of capillary openings could

vary from one thousand to well over one million. To calculate the flow

through this array it is assumed, as a first approximation, that the flow

through any one capillary is independent of the flow through the adjacent

capillary. The total flow through the array is, then, simply the flow

through one capillary multiplied by the number of capillaries. These flow

rates as a function of altitude, at different L/a, are shown in figure 6.

None of these flow rates is so high that it cannot be handled by conventional, readily available pumping systems, to avoid the accumulation of sample in the mass spectrometer.

EFFECT ON THE FLOW

Using the methodology developed in reference 4, the distance of sample

removal from the surface was calculated. In all cases of a/L, for the

conditions given in table I, the distance from the surface was no greater

than 1Q-6 cm. This distance is an exceedingly small fraction of the boundary layer thickness which, at the point on the surface where the calculations were performed, is approximately 2 cm. It is quite likely that this

distance lies either within, or close to the Knudsen layer. As a result removal of material through this type of effusive device should have little or

no effect upon the overall stream flow.

SUMMARY

In this report the flow through a capillary opening in a surface has

been analyzed. Programs have been written which will be utilized in the

14
I/a

10"

10

10

50

Al +

I

60

.(km)

70 80

Figure 6. The calculated flow rate (in liters/min) for N2

from equation 6 for an assembly of 1000 capillaries

15
analysis of flow experiments which are currently proceeding in the laboratory. The major finding of this report is that for the altitudes given, and

the calculated conditions, the flow through the device will, most likely be

molecular or transitional. As a result most reactions caused by intermolecular collisions will occur on the exterior side of the device and not within

the device itself. In addition, the material removed from the flow will be

well within the boundary layer. This type of inlet would, then, be quite

useful for studying surface chemistry within the boundary layer. These conclusions are amenable to laboratory testing and are currently being examined.

16
REFERENCES

1. J. L. Shin, J. N. Moss and A. L. Simmonds: AIAA Paper 82-0842, June

1982.

2. D. Eide, personal communication.

3. VI. Sutherland: Phil. Mag. 40, 421 (1895).

4. K. G. Brown: NASA technical report NAS-1-17099-8, October 1983.

5. M. Knudsen: Ann. Phys. 28 75 (1909).

6. D. S. Scott, and F. A. L. Dullien: A. I. Ch. E. Journal 8, 293, (1962)

7. D. S. Scott, and F. A. L. Dullien: A. I. Ch. E. Journal 8, 113, (1962)

17