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A vibrating vane gauge for pressure measurement
A vibrating received
6 July
G Jones’
vane
gauge
for pressure
measurement
1977
and G T Roberts.
Department
of Scieflce,
This paper describes a vibrating vane cantilever vibrating at a frequency in of gauge sensitivity on the proximity be the basis of a pressure measuring
The Polytechnic
of Wales,
Pontypridd,
Mid
Gfamorgan,
wales
pressure gauge, comprising of a circular vane fixed to a the range 30-70 Hz. The results show a marked dependence of the vane enclosure, and indicate that the gauge could instrument.
1. Introduction
This paper extends the theoretical work of Roberts’ and describesa seriesof experimentsusedto verify the theoretical predictions. The gaugedescribedis of the vibrating cantilever type with a circular vane fixed to the free end and oscillating with a frequency within the range 30-70 Hz. Gauge characteristics have been evaluated for the pressurerange lo-‘-10-l torr for a variety of different gasesand vapours. In addition to the commonly quoted advantagesof an oscillating vane gauge of (a) absolute reading (providing the gas composition is known), (b) beingwithout hot filamentsor electricaldischarges and (c) beingmadeentirely of bakeablematerials,the cantilever type gauge described is (d) unaffected by orientation, (e) unaffectedby minor movementor vibration of the support and (f) may be calibrated (with the aid of an adequate control system) to give a direct reading. The factors limiting the sensitivity and accuracy of the systemare governed mainly by the choice of materials, design and mode of construction of cantilever, cantilever support and vane. Therefore, it is firmly believedthat further work into theseaspectsof gaugeconstruction would extend the capabilitiesof the gauge. The theoretical modeldevelopedby Anderson’ and Roberts’ for a compound pendulumtype gaugeis extended for a cantilever type gaugefor the caseof moleculesfully accommodated on the vane surfaces.Roberts’ also predicted an increasein damping due to the variation in pressurebetweenthe leading and trailing surfacesof the vane when only a small gap exists betweenvane edgesand vane enclosure.The theoretical model of this prediction is developedfurther and the phenomenonis alsoverified by the experimentalresults. 2. Theoretical
model
For the purposeof the theoreticalmodel,the vane andcantilever are assumedto be asshown in Figure 1. The cantilever extends from the support A, to the centre of the vane B, is of length L, massper unit length I+, Young’s modulus E, and moment of area I, and the vane is a light inflexible disc of radius R,, fixed to the cantilever such that when the systemoscillatesthe plane of the vane is normal to its displacement.The inertia of the systemis assumedto be due to two factors, namely the massof cantilever, and a lumped massML at B. As in the caseof a
dx
I----
Figure1. Theoreticalmodelof vaneand cantilever. pendulumtype vane, the damping force due to collisionswith gasmoleculesis proportional to velocity. For easeof treatment the damping forces due to the cantilever material and support are also assumedto be proportional to velocity, therefore the dampingforce at any point of vane or cantilever may be written in the form t-j. Lagrange’sequation may be applied, which for damped oscillations where Y is the generalizedcoordinate, takes the form (seeBickley3)
(1) where T and V are the kinetic and potential energiesof the systemrespectivelyand K is the dissipationfunction. From the assumptionsalready detailed the valuesof V, T and K may be shown to be v=-
3EIY2 2 L3 I
2T=
L,uj2 dx + ML?’
Vacuum/volumeZb/number1.
Pergamon
Press/Printed
in Great
(2)
s0 L-R,,
*Head of Physicsat Mountain Ash Comprehensive School, Mid Glamorgan,Wales.
l
2F?-
2K= f 0
Britain
L+R,
L+R,
r, jJ2 dx +
r2j2 s L-R,
dx +
rJj2 s
0
dx,
I 13
G Jones and G T Roberts: A vibrating vane gauge for pressure measurement where r, and r2 are the damping functions dependent on pressure, due to cantilever and vane respectively, and r3 is the inherent damping within the system independent of pressure. Equation (I) may be solved with the aid of equation (2) provided the variation of y with x is known. In practice the system is only lightly damped (w* > p*), hence the normal mode, undamped solution for y is assumed to apply. This solution may by written in the form y(x, 1) = c sin(of
+ $)5(x),
(3)
where t(x) = sinhclx - sin as -
sinh olL+ sin olL cash aL+ cos aL
1
(cash ax - cos ax)
PW2 a=El and = -(cash P
(4)
cash a Lcos a L+ 1 a Lsin a L-
fJ = y
sinh a Lcos aL).
(5)
From equations (3) and (4) it may be shown that for any function F(x) X2 F(x)j2 dx = Y2~(F, X,, X2), s Xl
m [(u + v)~ + (u + v)u]exp(Bu2) s0
(6)
i
Pb = !!$
m[(u - v)’ + (u - v)rr]exp(Bu2) du, s0 where A = (2m3/nkT) ‘I* , B = m/2kT, and n, and nb are the molecular density on the respective sides of the vane. Therefore, with reference to Figure I and equation (5) +(r2, L - R,, L + R,) becomes &r2,
L-R,,
L+&)=&
which is a function independent of both x and t. Equation (2) may now be rewritten in the form 2T = [-(p,
du
L+R,,
X2 1 x, $(F, x,,X,) =52o dx,J s R2(x)
where
where /3,, is a constant representing the pressure independent inherent damping. Numerical values of &.I, 0, L) and ML are determined from known values of L, R,, p, E, I and w with the aid of equations (4) and (5). Therefore a theoretical determination of r2 will enable experimental results to be compared with theory. The damping function r2 may be determined by following the method described by Roberts’. Hpwever, a more rigorous determination is currently presented of the additional damping due to the pressure variations between the leading and trailing surfaces of the vane, for the case of a circular vane oscillating in a cylindrical enclosure. Roberts’ showed that the pressures in front of and behind an element of the vane travelling with a velocity V (normal to the vane surface) is given by
0, L) + MJ p2
(3)
and 2K=[$(r,,O,L-Ru)+6(r2,L-R”,L+RJ+
_ r2t2(x) dx s L R, 2(P/ - Ph) L+Ru CR: = vt2(L) s L-R, - (L- x)~]“~~~(x) dx.
Expressions for 11~and nb can be derived for conditions where the mean free path of gas molecules is much greater than vane and enclosure dimensions, by considering the molecular flow of gas past the vane. Figure 2 represents the section of vane and
+ 4(rJ, 0, L + RJ P2 = 4’(r, L,R,)p’
*
2Re P
3
A
and hence on substituting in equation (I ) it may be shown that [Q,
0, L) + M,]Y
+ t$‘(r, L, R”)p + 5
I---
Y = 0.
G-ti -.-.-----.-
This equation when solved for the case of light damping yields Y = Y. exp( -flt)sin(yt
2R,-+
20
;
+ Y),
where 1 1
Figure
Of the three terms within 4’(r, L, R,), r3 is independent of gas parameters, and for the type of gauge investigated it may be shown that +(rl, 0, L-R,) is less than 0.5% of Q(r2, L-R,, L + R,), therefore by neglecting the rl term, equation (7) becomes
2p = 4(r2r L - Ry, L + RJ + 28 O’ N4 0, L) + ML 14
(8)
2.
C
Section of vane and enclosure at vane centre.
enclosure at the centre of the vane, the vane being displaced by a distance Y. Let the centre of the vane be moving with a velocity V, therefore in time dr the molecular density of ABGH increased by approximately no Vdt/a, where n, is the molecular density of the gas at equilibrium. Consequently, a molecular flux is set up between the respective sides of the vane which may be represented by the expression
G Jones
and
G T Roberts:
A vibrating
vane
gauge
for pressure
measurement
+(n, - n,+R~.f(a, R,, R,), where f is a function procedure described
and
of a, R, and R, only. By following by Roberts’, it can be shown that
thus
the 1 =2
hence
from
~-4
cos
dS ss
0; dS,
cos2 2
+ s
S2
r2
0,
dS2
.
I
Values of/(a, R,, R,) have been determined numerically with the aid of the substitution detailed by Kennard4 and are shown in Table I. Note that f is independent of a/R,.
(8)
CR,2- ( L - x)‘] “2<2(x)
ds
cos
SI
0, cos
0; dS,
r:
+
S2 1 cos2 7
s
0,
d&
r2
investigation
.(9)
Function f‘(a, R,, R,) is determined by evaluating the molecular flux through the annulus formed between vane and enclosure. The flux is assumed to be due to two molecular fluxes resulting from diffuse reflection from the cylindrical sections of the enclosure of the area S, and diffuse reflections from the flat ends of the enclosure of area Sz. With reference to Figure 3, the total flux through an element dS of the annulus from the volume on one side of the vane, where the molecular density is n, therefore becomes
[S
ri
[S St
3. Experimental
;dS
0, cos
-I
The vibrating vane gauge investigated is shown in Figure 4. It consists of a circular aluminium vane A, supported by a steel cantilever B, rigidly clamped to the solid brass enclosure C. The displacement and hence amplitude of the vane is determined electronically from the capacitance between the vane and a suitably shaped electrode D. For each vane used the error in recorded values of displacement and amplitude was considered to be less than 2%. The vane is set in oscillation with an amplitude of approximately I mm, with a driving coil (not shown), the driving current being at the resonant frequency of the vane. Values of damping coefficient ,9 were determined for various pressures, gases and vane dimensions with the aid of the following procedure. The gauge is initially evacuated to below lob5 torr for 20 min and the appropriate gas introduced at a predetermined pressure. The vane is then set in oscillation with the driving coil and as the driving current is switched off the amplitude decay is recorded. Experiments were carried out on three vanes of diameters 18.0, 16.4 and 6.5 mm in an enclosure of diameter 20.0 mm and length IO mm, with helium, air, argon, carbon dioxide and Freon-12, in the pressure range lO-4-lO-2 torr. Variation of 8 with pressure, molecular weight and gauge dimensions were investigated and compared with the theoretical investigation described.
(i) Variation /?,, between of weeks, Figure 5, Freon-12 are linear
of /3 with pressure. On account of the variation in different gauges and with time intervals of the order the dependence of p on pressure is represented in where values of (8-p,,) are displayed for air and for the three vanes used. The variations in (8-j&) with pressure to within the experimental accuracy of
D
B
J Figure
3. Molecular
Table
1. Values
of/(o,
flux through
annulus
Figure
S.
R,, R,) for various
values
of
4. Simplified
view
of gauge.
RJR.
RJR,
0.95
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
f(a, R,,. R.)
0.337
0.732
I .759
3.256
5.560
9.382
16.41
31.61
75.00
0.1 309.3 15
G Jones and G T Roberts:A vibrating vanegaugefor pressuremeasurement of j with gauge dimensions. The dependenceof fl on gaugedimensionsand mechanicalpropertiesof the cantilever are demonstratedin Table 2 and Fig. 5. The increasein gauge sensitivity due to the pressurevariations betweenthe leading and trailing edgesof the vane is clearly shown.For example,the theoretical value of p/P[(RT/M)‘l’] for a vane of dimensions similar to vane 1, but vibrating in an infinite enclosureis 2.19, hencethe effect describedhasincreasedgaugesensitivity by a factor 4.1. (iii) Variation
2
4
6
s
vane
2
Vane
I
vane
2
vane
3
Vane
3
fOrI
’
Figure5. Experimentalvaluesof 8 - 8, for air 0 and Freon-120.
10% for vane 1 and 25% for vane 3. The variations in experimental accuracy are due to the increasedsensitivity of vane 1. of /? with molecular weights. The expecteddependence of /? on z/M was observedas for other gauges,provided the pressurewas sufficiently low. The dependenceon dM is shown in Table 2, where values of fi/P[(RT/M)‘/2] have been tabulated for various gasesand gauges.Calculated values of ,~/P[(RT/II~)“~] were determinedwith the aid of equation (9) and the valuesofjgiven in Table I. (ii) Variation
Table 2. Experimentaland theoretical values of fl/P[(RT/M)‘] (m’ kg-‘)
Helium Air Argon Carbondioxide Freon-l2 Calculatedvalue VaneradiusR, (mm) Vaneresonantfrequency(Hz)
16
Vane 1
Vane2
Vane3
9.43
6.66 5.85 5.12
1.56 1.44
5.46 1.33 8.20 38.1
1.68 I.37 3.25 66.9
9.21 8.46 9.42 8.25 12.14 9.00 36.3
Conclusion
The investigation has clearly shown the dependenceof gauge sensitivity on the proximity of the enclosureas predicted by Roberts’. This result is contrary to the work of Austin and Leeks, who failed to observeany significant increasein sensitivity for a slow-movingpendulumtype vane. However, it is not clear from their paper whether or not the conditions assumed by Roberts’ applied, namely that the conductanceabove and below the vane is negligible compared with the conductance along the side of the vane. The theory describedin this paper overcomesthis problem by considering a circular vane in a cylindrical enclosure.This arrangementallowed detailed comparison of theory and experiment to be undertaken, the only difference between the theoretical model and experimental systembeing a smalldecreasein conductancein the vicinity of the displacementelectrode,and a smallincreasein conductance in the vicinity of the groove containing the cantilever. Further work is currently in progresson a slow-movingvane oscillating in a cylindrical enclosure. The gaugedescribed(vane 1 or vane 2) could be used to measurepressurein the range 1O-‘-1O-3 torr, but accuracy decreases considerablyfor the lighter gasesdue to the relatively high valuesof PO.However, the gaugecould be the basisof an instrumentsinceit is almostunaffectedby orientation and small vibration. Further work is projected on gauge design and choice of materials,the aim being a considerablereduction in /IOand hencean increasein sensitivity.
References
’ G T Roberts,J Phys E, Sci Ins/rum, 3, 1970,806. z J R Anderson,Rev Sci Instrum, 29, 1958,1073. ’ W G BickleyandA Talbot, An Inrroducfion to the Theory oj’ Vibrating Systems. Oxford UniversityPress,Oxford (1961). 4 E H Kennard,Kinetic Theory of Gases. McGraw-Hill, New York (1938). 5W E AustinandJ H Leek, Vacuum, 22, 1972, 33 I.
6 July
G Jones’
vane
gauge
for pressure
measurement
1977
and G T Roberts.
Department
of Scieflce,
This paper describes a vibrating vane cantilever vibrating at a frequency in of gauge sensitivity on the proximity be the basis of a pressure measuring
The Polytechnic
of Wales,
Pontypridd,
Mid
Gfamorgan,
wales
pressure gauge, comprising of a circular vane fixed to a the range 30-70 Hz. The results show a marked dependence of the vane enclosure, and indicate that the gauge could instrument.
1. Introduction
This paper extends the theoretical work of Roberts’ and describesa seriesof experimentsusedto verify the theoretical predictions. The gaugedescribedis of the vibrating cantilever type with a circular vane fixed to the free end and oscillating with a frequency within the range 30-70 Hz. Gauge characteristics have been evaluated for the pressurerange lo-‘-10-l torr for a variety of different gasesand vapours. In addition to the commonly quoted advantagesof an oscillating vane gauge of (a) absolute reading (providing the gas composition is known), (b) beingwithout hot filamentsor electricaldischarges and (c) beingmadeentirely of bakeablematerials,the cantilever type gauge described is (d) unaffected by orientation, (e) unaffectedby minor movementor vibration of the support and (f) may be calibrated (with the aid of an adequate control system) to give a direct reading. The factors limiting the sensitivity and accuracy of the systemare governed mainly by the choice of materials, design and mode of construction of cantilever, cantilever support and vane. Therefore, it is firmly believedthat further work into theseaspectsof gaugeconstruction would extend the capabilitiesof the gauge. The theoretical modeldevelopedby Anderson’ and Roberts’ for a compound pendulumtype gaugeis extended for a cantilever type gaugefor the caseof moleculesfully accommodated on the vane surfaces.Roberts’ also predicted an increasein damping due to the variation in pressurebetweenthe leading and trailing surfacesof the vane when only a small gap exists betweenvane edgesand vane enclosure.The theoretical model of this prediction is developedfurther and the phenomenonis alsoverified by the experimentalresults. 2. Theoretical
model
For the purposeof the theoreticalmodel,the vane andcantilever are assumedto be asshown in Figure 1. The cantilever extends from the support A, to the centre of the vane B, is of length L, massper unit length I+, Young’s modulus E, and moment of area I, and the vane is a light inflexible disc of radius R,, fixed to the cantilever such that when the systemoscillatesthe plane of the vane is normal to its displacement.The inertia of the systemis assumedto be due to two factors, namely the massof cantilever, and a lumped massML at B. As in the caseof a
dx
I----
Figure1. Theoreticalmodelof vaneand cantilever. pendulumtype vane, the damping force due to collisionswith gasmoleculesis proportional to velocity. For easeof treatment the damping forces due to the cantilever material and support are also assumedto be proportional to velocity, therefore the dampingforce at any point of vane or cantilever may be written in the form t-j. Lagrange’sequation may be applied, which for damped oscillations where Y is the generalizedcoordinate, takes the form (seeBickley3)
(1) where T and V are the kinetic and potential energiesof the systemrespectivelyand K is the dissipationfunction. From the assumptionsalready detailed the valuesof V, T and K may be shown to be v=-
3EIY2 2 L3 I
2T=
L,uj2 dx + ML?’
Vacuum/volumeZb/number1.
Pergamon
Press/Printed
in Great
(2)
s0 L-R,,
*Head of Physicsat Mountain Ash Comprehensive School, Mid Glamorgan,Wales.
l
2F?-
2K= f 0
Britain
L+R,
L+R,
r, jJ2 dx +
r2j2 s L-R,
dx +
rJj2 s
0
dx,
I 13
G Jones and G T Roberts: A vibrating vane gauge for pressure measurement where r, and r2 are the damping functions dependent on pressure, due to cantilever and vane respectively, and r3 is the inherent damping within the system independent of pressure. Equation (I) may be solved with the aid of equation (2) provided the variation of y with x is known. In practice the system is only lightly damped (w* > p*), hence the normal mode, undamped solution for y is assumed to apply. This solution may by written in the form y(x, 1) = c sin(of
+ $)5(x),
(3)
where t(x) = sinhclx - sin as -
sinh olL+ sin olL cash aL+ cos aL
1
(cash ax - cos ax)
PW2 a=El and = -(cash P
(4)
cash a Lcos a L+ 1 a Lsin a L-
fJ = y
sinh a Lcos aL).
(5)
From equations (3) and (4) it may be shown that for any function F(x) X2 F(x)j2 dx = Y2~(F, X,, X2), s Xl
m [(u + v)~ + (u + v)u]exp(Bu2) s0
(6)
i
Pb = !!$
m[(u - v)’ + (u - v)rr]exp(Bu2) du, s0 where A = (2m3/nkT) ‘I* , B = m/2kT, and n, and nb are the molecular density on the respective sides of the vane. Therefore, with reference to Figure I and equation (5) +(r2, L - R,, L + R,) becomes &r2,
L-R,,
L+&)=&
which is a function independent of both x and t. Equation (2) may now be rewritten in the form 2T = [-(p,
du
L+R,,
X2 1 x, $(F, x,,X,) =52o dx,J s R2(x)
where
where /3,, is a constant representing the pressure independent inherent damping. Numerical values of &.I, 0, L) and ML are determined from known values of L, R,, p, E, I and w with the aid of equations (4) and (5). Therefore a theoretical determination of r2 will enable experimental results to be compared with theory. The damping function r2 may be determined by following the method described by Roberts’. Hpwever, a more rigorous determination is currently presented of the additional damping due to the pressure variations between the leading and trailing surfaces of the vane, for the case of a circular vane oscillating in a cylindrical enclosure. Roberts’ showed that the pressures in front of and behind an element of the vane travelling with a velocity V (normal to the vane surface) is given by
0, L) + MJ p2
(3)
and 2K=[$(r,,O,L-Ru)+6(r2,L-R”,L+RJ+
_ r2t2(x) dx s L R, 2(P/ - Ph) L+Ru CR: = vt2(L) s L-R, - (L- x)~]“~~~(x) dx.
Expressions for 11~and nb can be derived for conditions where the mean free path of gas molecules is much greater than vane and enclosure dimensions, by considering the molecular flow of gas past the vane. Figure 2 represents the section of vane and
+ 4(rJ, 0, L + RJ P2 = 4’(r, L,R,)p’
*
2Re P
3
A
and hence on substituting in equation (I ) it may be shown that [Q,
0, L) + M,]Y
+ t$‘(r, L, R”)p + 5
I---
Y = 0.
G-ti -.-.-----.-
This equation when solved for the case of light damping yields Y = Y. exp( -flt)sin(yt
2R,-+
20
;
+ Y),
where 1 1
Figure
Of the three terms within 4’(r, L, R,), r3 is independent of gas parameters, and for the type of gauge investigated it may be shown that +(rl, 0, L-R,) is less than 0.5% of Q(r2, L-R,, L + R,), therefore by neglecting the rl term, equation (7) becomes
2p = 4(r2r L - Ry, L + RJ + 28 O’ N4 0, L) + ML 14
(8)
2.
C
Section of vane and enclosure at vane centre.
enclosure at the centre of the vane, the vane being displaced by a distance Y. Let the centre of the vane be moving with a velocity V, therefore in time dr the molecular density of ABGH increased by approximately no Vdt/a, where n, is the molecular density of the gas at equilibrium. Consequently, a molecular flux is set up between the respective sides of the vane which may be represented by the expression
G Jones
and
G T Roberts:
A vibrating
vane
gauge
for pressure
measurement
+(n, - n,+R~.f(a, R,, R,), where f is a function procedure described
and
of a, R, and R, only. By following by Roberts’, it can be shown that
thus
the 1 =2
hence
from
~-4
cos
dS ss
0; dS,
cos2 2
+ s
S2
r2
0,
dS2
.
I
Values of/(a, R,, R,) have been determined numerically with the aid of the substitution detailed by Kennard4 and are shown in Table I. Note that f is independent of a/R,.
(8)
CR,2- ( L - x)‘] “2<2(x)
ds
cos
SI
0, cos
0; dS,
r:
+
S2 1 cos2 7
s
0,
d&
r2
investigation
.(9)
Function f‘(a, R,, R,) is determined by evaluating the molecular flux through the annulus formed between vane and enclosure. The flux is assumed to be due to two molecular fluxes resulting from diffuse reflection from the cylindrical sections of the enclosure of the area S, and diffuse reflections from the flat ends of the enclosure of area Sz. With reference to Figure 3, the total flux through an element dS of the annulus from the volume on one side of the vane, where the molecular density is n, therefore becomes
[S
ri
[S St
3. Experimental
;dS
0, cos
-I
The vibrating vane gauge investigated is shown in Figure 4. It consists of a circular aluminium vane A, supported by a steel cantilever B, rigidly clamped to the solid brass enclosure C. The displacement and hence amplitude of the vane is determined electronically from the capacitance between the vane and a suitably shaped electrode D. For each vane used the error in recorded values of displacement and amplitude was considered to be less than 2%. The vane is set in oscillation with an amplitude of approximately I mm, with a driving coil (not shown), the driving current being at the resonant frequency of the vane. Values of damping coefficient ,9 were determined for various pressures, gases and vane dimensions with the aid of the following procedure. The gauge is initially evacuated to below lob5 torr for 20 min and the appropriate gas introduced at a predetermined pressure. The vane is then set in oscillation with the driving coil and as the driving current is switched off the amplitude decay is recorded. Experiments were carried out on three vanes of diameters 18.0, 16.4 and 6.5 mm in an enclosure of diameter 20.0 mm and length IO mm, with helium, air, argon, carbon dioxide and Freon-12, in the pressure range lO-4-lO-2 torr. Variation of 8 with pressure, molecular weight and gauge dimensions were investigated and compared with the theoretical investigation described.
(i) Variation /?,, between of weeks, Figure 5, Freon-12 are linear
of /3 with pressure. On account of the variation in different gauges and with time intervals of the order the dependence of p on pressure is represented in where values of (8-p,,) are displayed for air and for the three vanes used. The variations in (8-j&) with pressure to within the experimental accuracy of
D
B
J Figure
3. Molecular
Table
1. Values
of/(o,
flux through
annulus
Figure
S.
R,, R,) for various
values
of
4. Simplified
view
of gauge.
RJR.
RJR,
0.95
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
f(a, R,,. R.)
0.337
0.732
I .759
3.256
5.560
9.382
16.41
31.61
75.00
0.1 309.3 15
G Jones and G T Roberts:A vibrating vanegaugefor pressuremeasurement of j with gauge dimensions. The dependenceof fl on gaugedimensionsand mechanicalpropertiesof the cantilever are demonstratedin Table 2 and Fig. 5. The increasein gauge sensitivity due to the pressurevariations betweenthe leading and trailing edgesof the vane is clearly shown.For example,the theoretical value of p/P[(RT/M)‘l’] for a vane of dimensions similar to vane 1, but vibrating in an infinite enclosureis 2.19, hencethe effect describedhasincreasedgaugesensitivity by a factor 4.1. (iii) Variation
2
4
6
s
vane
2
Vane
I
vane
2
vane
3
Vane
3
fOrI
’
Figure5. Experimentalvaluesof 8 - 8, for air 0 and Freon-120.
10% for vane 1 and 25% for vane 3. The variations in experimental accuracy are due to the increasedsensitivity of vane 1. of /? with molecular weights. The expecteddependence of /? on z/M was observedas for other gauges,provided the pressurewas sufficiently low. The dependenceon dM is shown in Table 2, where values of fi/P[(RT/M)‘/2] have been tabulated for various gasesand gauges.Calculated values of ,~/P[(RT/II~)“~] were determinedwith the aid of equation (9) and the valuesofjgiven in Table I. (ii) Variation
Table 2. Experimentaland theoretical values of fl/P[(RT/M)‘] (m’ kg-‘)
Helium Air Argon Carbondioxide Freon-l2 Calculatedvalue VaneradiusR, (mm) Vaneresonantfrequency(Hz)
16
Vane 1
Vane2
Vane3
9.43
6.66 5.85 5.12
1.56 1.44
5.46 1.33 8.20 38.1
1.68 I.37 3.25 66.9
9.21 8.46 9.42 8.25 12.14 9.00 36.3
Conclusion
The investigation has clearly shown the dependenceof gauge sensitivity on the proximity of the enclosureas predicted by Roberts’. This result is contrary to the work of Austin and Leeks, who failed to observeany significant increasein sensitivity for a slow-movingpendulumtype vane. However, it is not clear from their paper whether or not the conditions assumed by Roberts’ applied, namely that the conductanceabove and below the vane is negligible compared with the conductance along the side of the vane. The theory describedin this paper overcomesthis problem by considering a circular vane in a cylindrical enclosure.This arrangementallowed detailed comparison of theory and experiment to be undertaken, the only difference between the theoretical model and experimental systembeing a smalldecreasein conductancein the vicinity of the displacementelectrode,and a smallincreasein conductance in the vicinity of the groove containing the cantilever. Further work is currently in progresson a slow-movingvane oscillating in a cylindrical enclosure. The gaugedescribed(vane 1 or vane 2) could be used to measurepressurein the range 1O-‘-1O-3 torr, but accuracy decreases considerablyfor the lighter gasesdue to the relatively high valuesof PO.However, the gaugecould be the basisof an instrumentsinceit is almostunaffectedby orientation and small vibration. Further work is projected on gauge design and choice of materials,the aim being a considerablereduction in /IOand hencean increasein sensitivity.
References
’ G T Roberts,J Phys E, Sci Ins/rum, 3, 1970,806. z J R Anderson,Rev Sci Instrum, 29, 1958,1073. ’ W G BickleyandA Talbot, An Inrroducfion to the Theory oj’ Vibrating Systems. Oxford UniversityPress,Oxford (1961). 4 E H Kennard,Kinetic Theory of Gases. McGraw-Hill, New York (1938). 5W E AustinandJ H Leek, Vacuum, 22, 1972, 33 I.