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The use of an oscillating vane gauge to determine gas composition of binary gas mixtures
The use of an oscillating vane gauge to determine gas composition of binary gas mixtures received 29 September 197& in revised form 21 November 1978
F H Dyksterhuis, Department of Physics, University of Queensland, St Lucia, Brisbane, 4067, Australia
A description is presented of a continuously driven oscillating vane gauge, which can be used to determine the composition of binary gas mixtures. Consideration is given to operation not only in the molecular and viscous pressure range but also in the intermediate 10-100 mtorr range.
Introduction
Light source I
A variety of gauges exists to measure pressures in a range of 10-500 mtorr. Many of these gauges show some sort of dependence on the molecular properties of the gas and in principle these could be used to differentiate between gases, once the relevant properties and pressure are known. The most suitable of these gauges appears to be the viscosity type manometer: their operating principles are relatively simple and their dependence on the nature of the gas is rather strong. This paper will describe such a gauge. It will be shown how it can be employed to determine the gas composition of binary gas mixtures in a pressure range of 50-500 mtorr. The gauge enables direct measurement of the composition of a mixture and avoids problems of differential wall adsorption--such problems may be serious if an attempt is made to deduce the composition from initial partial pressure measurements. Besides being relatively cheap the gauge also overcomes the difficulties encountered in mass-spectrometry, such as the necessity of using a continuous flow between gas sample and gauge head, because of the low operating pressure of the mass-spectrometer, and the intrinsically cumbersome calibration for different gases.
I
I
I
I
I
I I
I
~
Lens
Razor blade i support ~
Aluminium 11/vane
I I ii II
1
'i
I1 II
E
/', I
Driving plates
I t
I i
0 The vane gauge
unit
Vane gauges have already been used by others t to measure pressures in the range 10-2-10 - 1 mtorr, the nature of the gas being known. In the present work, which was done at pressures in the range 10-500 mtorr, the pressure was measured independently and the vane gauge was used to determine the composition. The vane, which oscillates as a pendulum (Figure 1) is made of a rectangular piece of aluminium foil with a thickness of 0.05 mm (0.002 in.) and has a length of 10 cm and a width, L, of 5 cm. It is suspended from two razor blades by a frame of tungsten wire, measuring 2 × 5 cm, the total length 1 of the vane and frame, thus, being 12 cm. The bottom plate of the vane (5 × 1 cm) is separated from two driving plates by a distance d of approximately 2-3 ram, with which it forms a capacitor. The amplitude of oscillation is maintained by a feedback control system in which the amplitude of the vane is firstly sensed by a photocell and subsequently maintained at a predetermined value by application of an appropriate voltage
Vacuum/volume 29/numbers 4/5.
,
Photocell Figure 1. Schematic diagram of the vane.
between vane and driving plates. The required voltages will be shown to be related to the damping torque acting on the vane and since this torque will depend on the pressure and the nature of the gas surroundings, they can be used ultimately to find gas composition, if the pressure is known. The relationship between driving voltage and damping coefficient a is found from the equation of motion of the vane:
I~ + =0 + MghO = ~,
(1)
where I is the moment of inertia of the vane, M its mass and h the radius to the centre of mass of the vane. The driving torque is obtained on inspection of the driving mechanism. If a
0042-207X/79/0601-0149502.00/0
© Pergamon Press Ltd/Printed in Great Britain
\
- -
149
F H Dyksterhuis: The use of an oscillating vane gauge to determine gas composition of binary gas mixtures voltage V is applied between the vane and a bottom plate the horizontal component of the force is:
LeoV2 d '
F=½
where L is the width of the vane. A voltage pulse V could thus be used to supply an instantaneous torque and set the vane into motion, whereupon then the damping coefficient could be ascertained by observing the rate of decay of the amplitude. A more elegant method is to maintain the oscillation at a fixed constant amplitude and to relate the voltages required in order to do this to the damping coefficient. In the present system this latter technique was opted for but instead of supplying a suitable voltage pulse at each cycle, in fact two voltages are supplied. The pair of driving plates have potentials -4- v sin oJt, respectively, all potentials being measured relative to earth, v being constant and oJ always being equal to the resonant angular frequency of the oscillation; the potential Vo of the vane is maintained by a feedback loop at a value sufficient to ensure a constant amplitude of oscillation. The driving torque is then:
ILeo 4 - - d - vV° sin cot.
I
J
I
I
I. . . . . . .
Figure 2. Block diagram of feedback system: A l-A~ : amplifier stages, R: full wave rectifier, S: sample-and-hold, B: band pass filter, L: low pass filter, C: comparator.
(3)
If this is substituted in equation (l), the latter may be solved for 0 to furnish a relationship between 0, a and the supplied voltages. At resonance the amplitude of the vane is then found to be:
4eoLIVov Oo = dot(co2 _ ct2]412)1/2 •
T
(2)
(4)
Since oJ2 >~, a2/412 in the present case, this may be approximated
a d c reference voltage, which is taken from a stabilized supply and can be set to predetermine the value of 0o. The error-signal is then smoothed by a filter with a time constant of 1005 (which is responsible for a readout delay) and further amplified to give Vo. This voltage, which is measured with a digital meter, depends on a (cf. equation (5)) and can be used in association with independent pressure measurements to determine the gas composition. A detailed description of the electronic feedback system will be published elsewhere 2.
to :
K Vov
0o - - - ,
(5)
where K is a constant.
Feedback control system With the driving voltages is associated a feedback system (Figure 2) which has a dual function. Firstly it will supply a sinusoidal signal o, which is of constant amplitude and at the exact resonance frequency of the vane (the frequency is a weak function of the damping coefficient a, for which reason an external oscillator cannot be used). Secondly it will provide the high voltage Vin such a way, that the amplitude 0o is maintained at a constant value independent of the damping. The oscillations are sensed by a photocell, a beam of light focussed on the cell being intercepted by the folded edge of the moving vane. The photocell output is fed into a one stage amplifier and then taken into a limiter to produce a rail-to-rail square wave of the same frequency. The signal then goes to a band-pass filter tuned at the vane frequency of -~1.6 Hz. The resulting sinewave is now of correct frequency and constant amplitude (approximately 3 V peak to peak) and is fed back to one of the driving plates directly and after inversion to the other driving plate. To obtain the high voltage V, the photocell output is firstly fullwave rectified. The resultant is then taken into a sample-andhold stage, where the peak value of each full-wave rectified cycle is sampled, where after the signal is then compared with
Theory of operation The calculation of the damping coefficient a will now be considered. The damping mechanism will depend on the pressure and, therefore, the low pressure case and high pressure case will be dealt with separately. Let us fi~'st regard the vane gauge in a gas at low pressure, such that the mean free path ,Xis much greater than the relevant physical dimensions of the apparatus. Distinction will have to be made between the top section of the vane and the folded bottom plate. Let the total damping coefficient a be the sum of the individual ones, i.e. ct, = ~ t +ct2
(6)
where at and a2 denote the damping coefficients due to the top and bottom sections, respectively. The force per unit area exerted on the vane sections can be calculated from the rate of change of momentum of the molecules hitting and those leaving the vane 3 and the damping coefficients at and a2 are subsequently calculated from the torque acting on the top and bottom sections of the vane:
¢q=½nm~L(13-s3)=klP~
M
(7)
,
(8)
and
ct2 = 4Al2p
= k2
F H
Dyksterhuis: The
use of an oscillating vane gauge to determine gas composition of binary gas mixtures
where kt and k2 are constants. The solution for the low pressure case has, thus, become:
(k, +
Vo = (og + *tz)
= ~"---~v
k, ,]o °Px/--T IM = K ' p x / M"
(9)
It is worth noting that, though equation (9) assumes perfect molecular accommodation, even when specular reflection occurs, the general form of equation (9) is still preserved, apart from a numerical factor. As can be seen, the feedback voltage Vo depends on the molecular mass of the gas and can be used to distinguish between different gases, once the pressure is known. It may also serve to determine the partial pressures of a binary mixture. For a mixture with partial densities nt and n2-where the subscripts designate the two different species, it follows that:
Vo = K t ( p t x / M ,
+ p2v/M2),
? = - - ) , Pl Pt + P2 the above then yields (11)
from which the value of ~, may be determined if p and Vo are known quantities. At high pressures, where the mean free path is short compared to the dimensions of the apparatus, the damping coefficient will be determined by the viscosity coefficient ~7 of the gas, which is defined as the tangential force acting per unit area per unit of velocity gradient in the gas*. The force acting on the bottom plate of the vane will be (disregarding edge effects): U
F = A,7 ~ ,
(12)
if u is the tangential velocity of the plate and d the distance to the wall. It then follows that the torque acting on the vane due to this force will be:
12A t = --~ r/O.
L/) "t" = ~ - r / ~ (/3 _ S3)
(13)
(16)
and, therefore, (/3 __ S3) 0g = ~ ~ q = k t ' D
q.
(k I' + k2)'0 o Vo ----r/ = K2r/.
Again the feedback voltage I/0 will be able to provide information about the nature of the gas. It is noted that Vo is in fact no longer pressure dependent, but a simple function of gas viscosity only. If a mixture of gases is used, the voltage Vo may serve to ascertain the relative percentages. The feedback voltage Vo can then be written as:
Vo = k 2 ( @ , in which (,7) represents the viscosity of the mixture. A~ expression for the viscosity of mixtures has been derived ~1~ Sutherland 6, which is based on the transfer of momentdm through the gas from a moving plate to a fixed wall. The momentum transfer will depend on the momentum which is transferred in each molecular collision and on the rate at which these occur. The latter factor he calculated in accordance with first principles of the kinetic theory of gases, but the first factor he was not able to derive rigorously. Instead he suggested a plausible expression which showed agreement with experimental results obtained by Graham some 50 years earlier. Later this expression was modified by Wilke 7 to make it a more readily applicable one, which describes the viscosity of gas mixtures to an accuracy of 2 ~ :
(,7) =
~'
+
q -----k ' 2 q .
r/2
(19)
1 + ~b2t~,/1 - y '
where
Al 2 Ct2 = ~
(18)
Ku
1 + 4,,2(1 - r)/r The damping coefficient due to the bottom plate is, thus,
(17)
The final solution is then:
(10)
so that if ), indicates the percentage of the first species (i.e.
Vo = g x p [ y n / M t + (1 - y)x/M2]
found experimentally to be of order of the distance from vane to the wall (for the present apparatus). Assuming this distance to be constant for all elements dA of the vane surface, one can proceed to calculate the torque acting on the vane through integration. This will yield:
(14)
It will not be possible to calculate the torque acting on the top section of the vane in terms of the dimensions of the vane, because of the velocity component of the gas normal with respect to the plate and the complicated gas flow patterns resulting from this. Following an investigation by von Ubish s, in which he considered drag forces exerted on oscillating ribbons, one may conclude that the force acting on the top of the vane per unit area will still be characterized by the same variables and may be written as:
[1 + ,/'7 d,ldmdm 0 " " ] 2 q~12 =
2x/2(ml/ma),/z
(20)
and
[1 + J ~ 2 / ~ , ( m l / , n 2 ) 1 1 " ] 2 ~b21 = 2n/2(1 + m 2 / m t ) 1/2
(21)
It is, thus, seen that from the effective viscosity (~) of the mixture the composition parameter y can be determined.
Treatment of the intermediate pressure regime
dF u d--A = q ~ ,
(15)
where D is a characteristic distance determined by the dimension of the vane and the vacuum container. This distance is
Satisfactory theoretical treatments exist in the limits of high and low pressure (equations (9) and (18)), but this is not the case for the intermediate range, although some approximate expressions have been suggested s. This will necessitate an 151
F H Dyksterhuis: The use of an oscillating vane gauge to determine gas composition of binary gas mixtures entirely different, empirical approach. From equations (9) and (18) it follows that, if
then f ( x ) = K i x at low pressures and f ( x ) = K 2 at high pressures, where the constants KI and K2 are independent of the nature of the gas. The experimental results to be reported, suggest that at intermediate pressures, when the mean free path is comparable to the dimensions of the apparatus the function f ( x ) is still independent of the nature of the gas. If one plots Vo/'q vs x = pv'M/~I, one obtains a 'universal curve' which depends only on the geometrical constants of the apparatus and the applied voltage Vo. Thus, a connection can be established experimentally in the intermediate pressure regime where a theoretical treatment is mathematically too involved. The empirical connection between the low and high pressure regions which can be expressed as a polynomial (see Figure 4) not only represents a universal curve for different gases, but for gas mixtures as well. From the uniqueness of this curve it will be possible to ascertain the composition of a binary gas mixture. If the polynomial fit for any known gas is found as V0 = . f ( x ) = a + b x + c x 2 + dx 3, r/
(23)
then for a mixture this curve may be used to calculate an effective mass ( M ) and viscosity (-q), where
Vo
)
=f(p~./.~ \
)'
(24)
given the values of p and Vo. Bearing in mind that @/) is a function of the mixing ratio (equation (19)) and may be written as v(y) and that M is likewise a function of y, which can be calculated from equation (I 1) as
= [vUM,
+ ( i - ~)xl'M2] ~
and can be written as ( M )
Vo
.(,):it
(25)
= M ( y ) one finds
[p~/M(?)'~
).
(26)
By letting
F(,) = V°
Mixing ratios have been determined for a number of argoncarbonylsulphide (Ar-OCS) mixtures using the vane gauge. The vane gauge itself is located approximately in the middle of a bell jar with a 30 cm base. Total pressures were measured with a newly developed gauge 9, where the mechanical pressureinduced deflection of a bellows is measured by means of a sensitive capacitance displacement bridge I°. This bellows gauge has a sensitivity of better than 0.25 mtorr at all pressures and has been calibrated against a McLeod gauge using argon as test gas. The vane gauge indications of mixing ratio were compared against the ratio as determined from pressure data. For the particular pair of gases used in this test, wall absorption is not a problem and partial pressure measurements may, therefore, be used for the determination of the composition. The bellows gauge has the advantage of being able to register condensable gases, which are often troublesome when McLeod gauges are used. Furthermore, the presence of mercury from McLeod gauges could be eliminated, which was essential for correct vane gauge readings, the vapour pressure of mercury being enough to cause substantial vane gauge errors. Both gases argon and carbonylsulphide were purchased commercially, and had a purety of 99.9 and 97.5 ~o, respectively, as stated by the manufacturer. The viscosity values for Ar and OCS were deduced from tabulations published by Hirschfelder ~1 as 194.1 and 122.1 p.P (10-TNs/m2), respectively, at a tempe~'ature of 20°C. Figure 3 shows a plot of voltage vs pressure for several A r - O C S mixtures. It can be seen that even at high pressures the voltage is still not entirely pressure independent. For this reason equation (18) cannot be applied without introducing errors. Instead a universal curve has been constructed based on the data of pure Ar and OCS, as shown in Figure 4 and the gas composition of the mixtures has been determined by solving equation (27) for y. This was done numerically by trial and error. The values obtained were at pressures as high as practical, in order to reduce errors to a minimum. There is good agreement between percentages as deduced from the universal curve and the ones based on pressure data, as is illustrated in Table 1. On average partial pressures can be measured to + 3 % of the total pressure. This is a typical value for most combinations of gases, although sometimes larger errors may be expected for unsuitable gases. The accuracy will depend strongly on the viscosities of the pure components, as well as their molecular masses. In the intermediate to high pressure range it is generally advantageous to work with gases, which have a large viscosity difference, but such that their values of x / M / ~ are not too different.
/Px/M(~)~
v/(,'---~-ft ~
)
(27)
the value for y may be obtained by solving F(y) = 0 numerically. All that is required is a knowledge of the total pressurep and the vane gauge feedback voltage Vo, the molecular mass and viscosity of the pure components. The above treatment has not taken into account the small damping force, which will arise from friction in the suspension system of the vane. To overcome this a small feedback voltage V, (typically approximately 0.3 V) is required, which can be added on to the Vo described above. If the measured control voltage is V then the Vo in the above treatment will be equal to ( V - - V,).
152
Results
Discussion
The vane gauge will be able to measure gas composition to an accuracy of 3% for suitable binary gas mixtures. The overall accuracy is limited by the fact that equation (19) is only an approximate expression, giving viscosity values to an accuracy of 2% v. In addition to this are the limitations posed by the feedback control system, which introduces a further l ~o error in the determination of the feedback voltage Vo. The gauge is most effectively used above approximately 50 retort, where the gas viscosity is the dominant parameter. Below this pressure the masses of the gas components will become more and m o r e important. In the pressure region of say 1-50 mtorr an assessment of the gas composition will strongly depend on both
F H Dyksterhuis: The use of an oscillating vane gauge to determine gas composition of binary gas mixtures
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I 300
1 400
I .500
Pressure - rnTorr Figure 3. Plot of voltage vs pressure for AJ'-OCS mixtures.
oa
n
i %.
>o - Ar
[] -OCS
04
I 5
i
t
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Figure 4. Plot ofpv'M]~ vs (V-- I"=)1",7for pure Ar and OCS and the universal curve based on these data.
Table 1. Vane gauger•suits compared against compositions based on pressure data 7oAr based on pressure data
7oAr based on universal curve
82.1 75.5 61.5 54.3 41.0 33.7
82.9 73.3 61.7 55.9 44.7 33.6
444444-
2.9 3.1 3.1 3.1 3.4 3.4
viscosity and mass. For example, for the gases formaldehyde and helium ( C H 2 0 and He), which are of interest in current investigations in microwave spectroscopy (Dyksterhuis and HeckenberglZ), the feedback voltage as a function of pressure is as shown in Figure 5. At very low pressure (p < 1 mtorr) the damping force will depend mainly on molecular mass. Thus formaldehyde will require a higher feedback voltage than helium. At high pressures, however, the viscosity will become dominant and, hence, helium would require the higher feedback voltage. The two curves in Figure 5, therefore, intersect and at that pressure the two gases will become indistinguishable. This illustration indicates that it will generally be desirable to operate the vane gauge near the viscous pressure region or near the molecular region. Which is preferred will depend on the 153
F H Dyksterhuis: The use of an oscillating vane gauge to determine gas composition of binary gas mixtures
°
i
,.=
Pressure
Figure 5. Representation of the dependence of the feedback voltage on pressure for He and CH20. molecular properties. For a C H 2 0 - H e mixture it appears advantageous to operate the gauge in the molecular pressure region, rather than near the viscous region, because of the large difference in mass of the two components. Since in the viscous gas range the damping force arises mainly from the folded edge of the vane, one could, thus, decrease the viscous drag force acting on the vane, by increasing the distance between the edge and the driving plates (equation (14)). This would shift the operating pressure to slightly lower values. In the present arrangement, however, it is not possible to attain an operating pressure in the molecular region (p < 1 mtorr), without substantial modification of the feedback system. Nevertheless, gauges could be designed with an optimum performance at any pressure, with a range of about two decades by a suitable choice of dimensional parameters.
Conclusion The vane gauge can be a very useful tool to differentiate between gases and can be used successfully to determine gas composition to a reasonable accuracy. It can be used over a wide range of pressures, this range depending on the actual design of the gauge. It can be made a very useful instrument in that it may overcome difficulties associated with wall absorption, since the gauge provides a continuous readout, thus, being able to monitor changes in gas-composition due to absorption instantly.
154
Acknowledgement This work was undertaken in the microwave research laboratory of the University of Queensland. The author wishes to express his thanks to Professor Parsons and Dr N R Heckeaberg for their stimulating discussions and Dr D W Hainsworth for his contribution to the instrumentation of the apparatus.
References i W E Austin, Vacuum, 19, 1969, 319. 2 D W Hainsworth and F H Dyksterhuis, Rev scient Instrum, to be published. 3 G Carter, Vacuum, 22, 1972, 225. + S Dushman, Scientific Foundations of Vacuum Technique. 2nd edition, Wiley, New York (1962). s H yon Ubish, Vacuum, 14, 1963, 89. s W Sutherland, Phil Mag, 40, 1895, 421. C R Wilke, J chem Phys, 18, 1950, 517. " H W Drawin, Zanyew Phys, 14, 1962, 369. 9 F H Dyksterhuis, M.Sc Thesis, University of Queensland, to be submitted. 1OF D Stacey, J M W Rynn, E C Little and C Croskell, J S c i lnstrum (J Phys E) 2, 1969, 945. 1 j O Hirschfelder, C F Curtiss and R B Bird, Molecular Theory of Gases and Liquids. Wiley, New York (1954). 12 F H Dyksterhuis and N R Heckenberg, J Mol Spectrosc, to be published.
F H Dyksterhuis, Department of Physics, University of Queensland, St Lucia, Brisbane, 4067, Australia
A description is presented of a continuously driven oscillating vane gauge, which can be used to determine the composition of binary gas mixtures. Consideration is given to operation not only in the molecular and viscous pressure range but also in the intermediate 10-100 mtorr range.
Introduction
Light source I
A variety of gauges exists to measure pressures in a range of 10-500 mtorr. Many of these gauges show some sort of dependence on the molecular properties of the gas and in principle these could be used to differentiate between gases, once the relevant properties and pressure are known. The most suitable of these gauges appears to be the viscosity type manometer: their operating principles are relatively simple and their dependence on the nature of the gas is rather strong. This paper will describe such a gauge. It will be shown how it can be employed to determine the gas composition of binary gas mixtures in a pressure range of 50-500 mtorr. The gauge enables direct measurement of the composition of a mixture and avoids problems of differential wall adsorption--such problems may be serious if an attempt is made to deduce the composition from initial partial pressure measurements. Besides being relatively cheap the gauge also overcomes the difficulties encountered in mass-spectrometry, such as the necessity of using a continuous flow between gas sample and gauge head, because of the low operating pressure of the mass-spectrometer, and the intrinsically cumbersome calibration for different gases.
I
I
I
I
I
I I
I
~
Lens
Razor blade i support ~
Aluminium 11/vane
I I ii II
1
'i
I1 II
E
/', I
Driving plates
I t
I i
0 The vane gauge
unit
Vane gauges have already been used by others t to measure pressures in the range 10-2-10 - 1 mtorr, the nature of the gas being known. In the present work, which was done at pressures in the range 10-500 mtorr, the pressure was measured independently and the vane gauge was used to determine the composition. The vane, which oscillates as a pendulum (Figure 1) is made of a rectangular piece of aluminium foil with a thickness of 0.05 mm (0.002 in.) and has a length of 10 cm and a width, L, of 5 cm. It is suspended from two razor blades by a frame of tungsten wire, measuring 2 × 5 cm, the total length 1 of the vane and frame, thus, being 12 cm. The bottom plate of the vane (5 × 1 cm) is separated from two driving plates by a distance d of approximately 2-3 ram, with which it forms a capacitor. The amplitude of oscillation is maintained by a feedback control system in which the amplitude of the vane is firstly sensed by a photocell and subsequently maintained at a predetermined value by application of an appropriate voltage
Vacuum/volume 29/numbers 4/5.
,
Photocell Figure 1. Schematic diagram of the vane.
between vane and driving plates. The required voltages will be shown to be related to the damping torque acting on the vane and since this torque will depend on the pressure and the nature of the gas surroundings, they can be used ultimately to find gas composition, if the pressure is known. The relationship between driving voltage and damping coefficient a is found from the equation of motion of the vane:
I~ + =0 + MghO = ~,
(1)
where I is the moment of inertia of the vane, M its mass and h the radius to the centre of mass of the vane. The driving torque is obtained on inspection of the driving mechanism. If a
0042-207X/79/0601-0149502.00/0
© Pergamon Press Ltd/Printed in Great Britain
\
- -
149
F H Dyksterhuis: The use of an oscillating vane gauge to determine gas composition of binary gas mixtures voltage V is applied between the vane and a bottom plate the horizontal component of the force is:
LeoV2 d '
F=½
where L is the width of the vane. A voltage pulse V could thus be used to supply an instantaneous torque and set the vane into motion, whereupon then the damping coefficient could be ascertained by observing the rate of decay of the amplitude. A more elegant method is to maintain the oscillation at a fixed constant amplitude and to relate the voltages required in order to do this to the damping coefficient. In the present system this latter technique was opted for but instead of supplying a suitable voltage pulse at each cycle, in fact two voltages are supplied. The pair of driving plates have potentials -4- v sin oJt, respectively, all potentials being measured relative to earth, v being constant and oJ always being equal to the resonant angular frequency of the oscillation; the potential Vo of the vane is maintained by a feedback loop at a value sufficient to ensure a constant amplitude of oscillation. The driving torque is then:
ILeo 4 - - d - vV° sin cot.
I
J
I
I
I. . . . . . .
Figure 2. Block diagram of feedback system: A l-A~ : amplifier stages, R: full wave rectifier, S: sample-and-hold, B: band pass filter, L: low pass filter, C: comparator.
(3)
If this is substituted in equation (l), the latter may be solved for 0 to furnish a relationship between 0, a and the supplied voltages. At resonance the amplitude of the vane is then found to be:
4eoLIVov Oo = dot(co2 _ ct2]412)1/2 •
T
(2)
(4)
Since oJ2 >~, a2/412 in the present case, this may be approximated
a d c reference voltage, which is taken from a stabilized supply and can be set to predetermine the value of 0o. The error-signal is then smoothed by a filter with a time constant of 1005 (which is responsible for a readout delay) and further amplified to give Vo. This voltage, which is measured with a digital meter, depends on a (cf. equation (5)) and can be used in association with independent pressure measurements to determine the gas composition. A detailed description of the electronic feedback system will be published elsewhere 2.
to :
K Vov
0o - - - ,
(5)
where K is a constant.
Feedback control system With the driving voltages is associated a feedback system (Figure 2) which has a dual function. Firstly it will supply a sinusoidal signal o, which is of constant amplitude and at the exact resonance frequency of the vane (the frequency is a weak function of the damping coefficient a, for which reason an external oscillator cannot be used). Secondly it will provide the high voltage Vin such a way, that the amplitude 0o is maintained at a constant value independent of the damping. The oscillations are sensed by a photocell, a beam of light focussed on the cell being intercepted by the folded edge of the moving vane. The photocell output is fed into a one stage amplifier and then taken into a limiter to produce a rail-to-rail square wave of the same frequency. The signal then goes to a band-pass filter tuned at the vane frequency of -~1.6 Hz. The resulting sinewave is now of correct frequency and constant amplitude (approximately 3 V peak to peak) and is fed back to one of the driving plates directly and after inversion to the other driving plate. To obtain the high voltage V, the photocell output is firstly fullwave rectified. The resultant is then taken into a sample-andhold stage, where the peak value of each full-wave rectified cycle is sampled, where after the signal is then compared with
Theory of operation The calculation of the damping coefficient a will now be considered. The damping mechanism will depend on the pressure and, therefore, the low pressure case and high pressure case will be dealt with separately. Let us fi~'st regard the vane gauge in a gas at low pressure, such that the mean free path ,Xis much greater than the relevant physical dimensions of the apparatus. Distinction will have to be made between the top section of the vane and the folded bottom plate. Let the total damping coefficient a be the sum of the individual ones, i.e. ct, = ~ t +ct2
(6)
where at and a2 denote the damping coefficients due to the top and bottom sections, respectively. The force per unit area exerted on the vane sections can be calculated from the rate of change of momentum of the molecules hitting and those leaving the vane 3 and the damping coefficients at and a2 are subsequently calculated from the torque acting on the top and bottom sections of the vane:
¢q=½nm~L(13-s3)=klP~
M
(7)
,
(8)
and
ct2 = 4Al2p
= k2
F H
Dyksterhuis: The
use of an oscillating vane gauge to determine gas composition of binary gas mixtures
where kt and k2 are constants. The solution for the low pressure case has, thus, become:
(k, +
Vo = (og + *tz)
= ~"---~v
k, ,]o °Px/--T IM = K ' p x / M"
(9)
It is worth noting that, though equation (9) assumes perfect molecular accommodation, even when specular reflection occurs, the general form of equation (9) is still preserved, apart from a numerical factor. As can be seen, the feedback voltage Vo depends on the molecular mass of the gas and can be used to distinguish between different gases, once the pressure is known. It may also serve to determine the partial pressures of a binary mixture. For a mixture with partial densities nt and n2-where the subscripts designate the two different species, it follows that:
Vo = K t ( p t x / M ,
+ p2v/M2),
? = - - ) , Pl Pt + P2 the above then yields (11)
from which the value of ~, may be determined if p and Vo are known quantities. At high pressures, where the mean free path is short compared to the dimensions of the apparatus, the damping coefficient will be determined by the viscosity coefficient ~7 of the gas, which is defined as the tangential force acting per unit area per unit of velocity gradient in the gas*. The force acting on the bottom plate of the vane will be (disregarding edge effects): U
F = A,7 ~ ,
(12)
if u is the tangential velocity of the plate and d the distance to the wall. It then follows that the torque acting on the vane due to this force will be:
12A t = --~ r/O.
L/) "t" = ~ - r / ~ (/3 _ S3)
(13)
(16)
and, therefore, (/3 __ S3) 0g = ~ ~ q = k t ' D
q.
(k I' + k2)'0 o Vo ----r/ = K2r/.
Again the feedback voltage I/0 will be able to provide information about the nature of the gas. It is noted that Vo is in fact no longer pressure dependent, but a simple function of gas viscosity only. If a mixture of gases is used, the voltage Vo may serve to ascertain the relative percentages. The feedback voltage Vo can then be written as:
Vo = k 2 ( @ , in which (,7) represents the viscosity of the mixture. A~ expression for the viscosity of mixtures has been derived ~1~ Sutherland 6, which is based on the transfer of momentdm through the gas from a moving plate to a fixed wall. The momentum transfer will depend on the momentum which is transferred in each molecular collision and on the rate at which these occur. The latter factor he calculated in accordance with first principles of the kinetic theory of gases, but the first factor he was not able to derive rigorously. Instead he suggested a plausible expression which showed agreement with experimental results obtained by Graham some 50 years earlier. Later this expression was modified by Wilke 7 to make it a more readily applicable one, which describes the viscosity of gas mixtures to an accuracy of 2 ~ :
(,7) =
~'
+
q -----k ' 2 q .
r/2
(19)
1 + ~b2t~,/1 - y '
where
Al 2 Ct2 = ~
(18)
Ku
1 + 4,,2(1 - r)/r The damping coefficient due to the bottom plate is, thus,
(17)
The final solution is then:
(10)
so that if ), indicates the percentage of the first species (i.e.
Vo = g x p [ y n / M t + (1 - y)x/M2]
found experimentally to be of order of the distance from vane to the wall (for the present apparatus). Assuming this distance to be constant for all elements dA of the vane surface, one can proceed to calculate the torque acting on the vane through integration. This will yield:
(14)
It will not be possible to calculate the torque acting on the top section of the vane in terms of the dimensions of the vane, because of the velocity component of the gas normal with respect to the plate and the complicated gas flow patterns resulting from this. Following an investigation by von Ubish s, in which he considered drag forces exerted on oscillating ribbons, one may conclude that the force acting on the top of the vane per unit area will still be characterized by the same variables and may be written as:
[1 + ,/'7 d,ldmdm 0 " " ] 2 q~12 =
2x/2(ml/ma),/z
(20)
and
[1 + J ~ 2 / ~ , ( m l / , n 2 ) 1 1 " ] 2 ~b21 = 2n/2(1 + m 2 / m t ) 1/2
(21)
It is, thus, seen that from the effective viscosity (~) of the mixture the composition parameter y can be determined.
Treatment of the intermediate pressure regime
dF u d--A = q ~ ,
(15)
where D is a characteristic distance determined by the dimension of the vane and the vacuum container. This distance is
Satisfactory theoretical treatments exist in the limits of high and low pressure (equations (9) and (18)), but this is not the case for the intermediate range, although some approximate expressions have been suggested s. This will necessitate an 151
F H Dyksterhuis: The use of an oscillating vane gauge to determine gas composition of binary gas mixtures entirely different, empirical approach. From equations (9) and (18) it follows that, if
then f ( x ) = K i x at low pressures and f ( x ) = K 2 at high pressures, where the constants KI and K2 are independent of the nature of the gas. The experimental results to be reported, suggest that at intermediate pressures, when the mean free path is comparable to the dimensions of the apparatus the function f ( x ) is still independent of the nature of the gas. If one plots Vo/'q vs x = pv'M/~I, one obtains a 'universal curve' which depends only on the geometrical constants of the apparatus and the applied voltage Vo. Thus, a connection can be established experimentally in the intermediate pressure regime where a theoretical treatment is mathematically too involved. The empirical connection between the low and high pressure regions which can be expressed as a polynomial (see Figure 4) not only represents a universal curve for different gases, but for gas mixtures as well. From the uniqueness of this curve it will be possible to ascertain the composition of a binary gas mixture. If the polynomial fit for any known gas is found as V0 = . f ( x ) = a + b x + c x 2 + dx 3, r/
(23)
then for a mixture this curve may be used to calculate an effective mass ( M ) and viscosity (-q), where
Vo
=f(p~./
)'
(24)
given the values of p and Vo. Bearing in mind that @/) is a function of the mixing ratio (equation (19)) and may be written as v(y) and that M is likewise a function of y, which can be calculated from equation (I 1) as
= [vUM,
+ ( i - ~)xl'M2] ~
and can be written as ( M )
Vo
.(,):it
(25)
= M ( y ) one finds
[p~/M(?)'~
).
(26)
By letting
F(,) = V°
Mixing ratios have been determined for a number of argoncarbonylsulphide (Ar-OCS) mixtures using the vane gauge. The vane gauge itself is located approximately in the middle of a bell jar with a 30 cm base. Total pressures were measured with a newly developed gauge 9, where the mechanical pressureinduced deflection of a bellows is measured by means of a sensitive capacitance displacement bridge I°. This bellows gauge has a sensitivity of better than 0.25 mtorr at all pressures and has been calibrated against a McLeod gauge using argon as test gas. The vane gauge indications of mixing ratio were compared against the ratio as determined from pressure data. For the particular pair of gases used in this test, wall absorption is not a problem and partial pressure measurements may, therefore, be used for the determination of the composition. The bellows gauge has the advantage of being able to register condensable gases, which are often troublesome when McLeod gauges are used. Furthermore, the presence of mercury from McLeod gauges could be eliminated, which was essential for correct vane gauge readings, the vapour pressure of mercury being enough to cause substantial vane gauge errors. Both gases argon and carbonylsulphide were purchased commercially, and had a purety of 99.9 and 97.5 ~o, respectively, as stated by the manufacturer. The viscosity values for Ar and OCS were deduced from tabulations published by Hirschfelder ~1 as 194.1 and 122.1 p.P (10-TNs/m2), respectively, at a tempe~'ature of 20°C. Figure 3 shows a plot of voltage vs pressure for several A r - O C S mixtures. It can be seen that even at high pressures the voltage is still not entirely pressure independent. For this reason equation (18) cannot be applied without introducing errors. Instead a universal curve has been constructed based on the data of pure Ar and OCS, as shown in Figure 4 and the gas composition of the mixtures has been determined by solving equation (27) for y. This was done numerically by trial and error. The values obtained were at pressures as high as practical, in order to reduce errors to a minimum. There is good agreement between percentages as deduced from the universal curve and the ones based on pressure data, as is illustrated in Table 1. On average partial pressures can be measured to + 3 % of the total pressure. This is a typical value for most combinations of gases, although sometimes larger errors may be expected for unsuitable gases. The accuracy will depend strongly on the viscosities of the pure components, as well as their molecular masses. In the intermediate to high pressure range it is generally advantageous to work with gases, which have a large viscosity difference, but such that their values of x / M / ~ are not too different.
/Px/M(~)~
v/(,'---~-ft ~
)
(27)
the value for y may be obtained by solving F(y) = 0 numerically. All that is required is a knowledge of the total pressurep and the vane gauge feedback voltage Vo, the molecular mass and viscosity of the pure components. The above treatment has not taken into account the small damping force, which will arise from friction in the suspension system of the vane. To overcome this a small feedback voltage V, (typically approximately 0.3 V) is required, which can be added on to the Vo described above. If the measured control voltage is V then the Vo in the above treatment will be equal to ( V - - V,).
152
Results
Discussion
The vane gauge will be able to measure gas composition to an accuracy of 3% for suitable binary gas mixtures. The overall accuracy is limited by the fact that equation (19) is only an approximate expression, giving viscosity values to an accuracy of 2% v. In addition to this are the limitations posed by the feedback control system, which introduces a further l ~o error in the determination of the feedback voltage Vo. The gauge is most effectively used above approximately 50 retort, where the gas viscosity is the dominant parameter. Below this pressure the masses of the gas components will become more and m o r e important. In the pressure region of say 1-50 mtorr an assessment of the gas composition will strongly depend on both
F H Dyksterhuis: The use of an oscillating vane gauge to determine gas composition of binary gas mixtures
j
.
~
'
,
O
01"/xr
0
J'
>
8?- 1%
o
~
°°
.//
/, / / J ' L ' f • ~
Oj
~
I . • ~
~
•
12o
e/•~"
°/o
•
~
t BOO
•
~
O
,
-
-
~
33.6 N
i 200
I 300
1 400
I .500
Pressure - rnTorr Figure 3. Plot of voltage vs pressure for AJ'-OCS mixtures.
oa
n
i %.
>o - Ar
[] -OCS
04
I 5
i
t
I
i
I IO
i
I
i
i
Pc"M/'q-arbitrary units
Figure 4. Plot ofpv'M]~ vs (V-- I"=)1",7for pure Ar and OCS and the universal curve based on these data.
Table 1. Vane gauger•suits compared against compositions based on pressure data 7oAr based on pressure data
7oAr based on universal curve
82.1 75.5 61.5 54.3 41.0 33.7
82.9 73.3 61.7 55.9 44.7 33.6
444444-
2.9 3.1 3.1 3.1 3.4 3.4
viscosity and mass. For example, for the gases formaldehyde and helium ( C H 2 0 and He), which are of interest in current investigations in microwave spectroscopy (Dyksterhuis and HeckenberglZ), the feedback voltage as a function of pressure is as shown in Figure 5. At very low pressure (p < 1 mtorr) the damping force will depend mainly on molecular mass. Thus formaldehyde will require a higher feedback voltage than helium. At high pressures, however, the viscosity will become dominant and, hence, helium would require the higher feedback voltage. The two curves in Figure 5, therefore, intersect and at that pressure the two gases will become indistinguishable. This illustration indicates that it will generally be desirable to operate the vane gauge near the viscous pressure region or near the molecular region. Which is preferred will depend on the 153
F H Dyksterhuis: The use of an oscillating vane gauge to determine gas composition of binary gas mixtures
°
i
,.=
Pressure
Figure 5. Representation of the dependence of the feedback voltage on pressure for He and CH20. molecular properties. For a C H 2 0 - H e mixture it appears advantageous to operate the gauge in the molecular pressure region, rather than near the viscous region, because of the large difference in mass of the two components. Since in the viscous gas range the damping force arises mainly from the folded edge of the vane, one could, thus, decrease the viscous drag force acting on the vane, by increasing the distance between the edge and the driving plates (equation (14)). This would shift the operating pressure to slightly lower values. In the present arrangement, however, it is not possible to attain an operating pressure in the molecular region (p < 1 mtorr), without substantial modification of the feedback system. Nevertheless, gauges could be designed with an optimum performance at any pressure, with a range of about two decades by a suitable choice of dimensional parameters.
Conclusion The vane gauge can be a very useful tool to differentiate between gases and can be used successfully to determine gas composition to a reasonable accuracy. It can be used over a wide range of pressures, this range depending on the actual design of the gauge. It can be made a very useful instrument in that it may overcome difficulties associated with wall absorption, since the gauge provides a continuous readout, thus, being able to monitor changes in gas-composition due to absorption instantly.
154
Acknowledgement This work was undertaken in the microwave research laboratory of the University of Queensland. The author wishes to express his thanks to Professor Parsons and Dr N R Heckeaberg for their stimulating discussions and Dr D W Hainsworth for his contribution to the instrumentation of the apparatus.
References i W E Austin, Vacuum, 19, 1969, 319. 2 D W Hainsworth and F H Dyksterhuis, Rev scient Instrum, to be published. 3 G Carter, Vacuum, 22, 1972, 225. + S Dushman, Scientific Foundations of Vacuum Technique. 2nd edition, Wiley, New York (1962). s H yon Ubish, Vacuum, 14, 1963, 89. s W Sutherland, Phil Mag, 40, 1895, 421. C R Wilke, J chem Phys, 18, 1950, 517. " H W Drawin, Zanyew Phys, 14, 1962, 369. 9 F H Dyksterhuis, M.Sc Thesis, University of Queensland, to be submitted. 1OF D Stacey, J M W Rynn, E C Little and C Croskell, J S c i lnstrum (J Phys E) 2, 1969, 945. 1 j O Hirschfelder, C F Curtiss and R B Bird, Molecular Theory of Gases and Liquids. Wiley, New York (1954). 12 F H Dyksterhuis and N R Heckenberg, J Mol Spectrosc, to be published.