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CRANFIELD
INSTITUTE OF TECHNOLOGY
VORTEX RATE SENSOR
by
December, 1970
CRANFIELD INSTITUTE OF TECHNOLOGY DEPARTMENT OF PRODUCTION ENGINEERING
VORTEX RATE SENSOR by
-0. Brychta
S U M M A R Y
An expression for the gain of the Vortex Rate Sensor is developed and compared with the actual results. Some problems of practical uses of the Vortex Rate Sensor are discussed.
C O N T E N T S
Page No. SUMMARY
1. INTRODUCTION 1 2. VORTEX CHAMBER GAIN 1
3. GAIN OF THE SENSING SYSTEM 2 3.1 The Sensitivity of a Cylindrical Sensing System 3
3.2 Two Tubes Sensing System 4 4. GAIN OF VORTEX RATE SENSOR 5 5. USE OF VORTEX RATE SENSOR 7
6. CONCLUSION 8 7. REFERENCES 8
1. INTRODUCTION
The vortex rate sensor is a fluidic device proposed for the sensing of the small speeds of rotation. The sensor (Fig. 1) consists basically of two coaxial discs separated by a cylindrical coupling ring, one outlet sink and a suitable pickoff element. The fluid flows through the coupling element created by a grill (a large number of the radial flow passages) and discharges at one sink tube. The radial flow between the two coaxial discs is modified by the viscous shear and by a vortex created by the input rate (w) of the unit about an axis parallel to its axis of symmetry. The radial flow entraining at the radius of the coupling into the vortex chamber is inclined by the tangential velocity of the unit rotation and the inclination (a) of the resulting velocity vector of the fluid at the radius pickoff location results from the given
circulation. The change of the velocity vector inclination from the radius of the coupling ring to the radius of the sink hole can be defined as the gain of the vortex chamber (G ) .
w
The resulting inclination of the fluid velocity vector (a) is sensed by the
differential sensing system located downstream of the sink hole. The differential sensing system converts the velocity vector inclinéition into the pressure
difference (AP) measured on the outputs of the sensing system. The conversion factor of the sensing system can be called the pickoff gain (G ) .
The gain of the vortex rate sensor (G) results from the gain of the vortex rate chamber and the gain of the sensing system.
G = G G
Ü) a
G = ^ . f (1) Aüj Aa
2 . VORTEX CHAMBER GAIN
The ideal angular movement of the fluid inside the vortex chamber, providing that the strength of the vortex is not diminished by the shear stress on the walls, is determined by Eqn. (2) expressing the conservation of the circulation:
^2 ~ TpQ = konst.
V = ^PO-'^PO = konst. (2)
where r is the circulation of the fluid on the coupling ring wall, Pp^ on the radius of the pickoff location.
The circulation in Eqn. (2) is expressed as the product of the tangential velocity (v) and the proper radius on which the tangential velocity is acting. Using Eqn. (2), the theoretical or ideal gain of the vortex chamber, that means the final
angle (Aa) of the inclination of the streamlines on the radius of pickoff for the low input rates (1-500 deg/s) can be expressed by Eqn. (3):
2
-where (U ) is the radial sink (maximal) velocity in the sink hole, (Au) is the s
input rate of the vortex chamber, (R) is the radius of the coupling ring, (r ) is the radius of the pickoff location.
The value of the actual vortex chamber gain is decreased by the viscous motion in the vortex rate chamber. The factor decreasing the gain is called the viscous efficiency (E) and can be defined as the ratio of the actual circulation (r.) of fluid generated by the input rate (w), divided by the ideal circulation
(r.) of the fluid in the vortex chamber.
E=-^ (4)
i
The actual circulation and the viscous efficiency has been calculated by
T. Sarpkaya in ref. 1. In Ref. 1 it has been shown that the viscous efficiency (E) is a function of the term:
E = f( / ^ } (5)
^ vR
where (Q) is the total flow rate through the sensor, (b) is the spacing of the bottom and top of the chamber, (R) is the radius of the coupling ring, (v) is
the kinematic viscosity of fluid. The function (E) can be found in Fig. 2 where it can be seen that the viscous efficiency is a step function of its argument in the range of zero up to the value of four. To get a reasonable viscous efficiency, the argument of the function should be larger than the value of four.
(6)
The actual vortex chamber gain (G ) , assuming the viscous efficiency, is given
by Eqn. (7): '^
2
G = E — V (7)
^P0"s
Assuming for example: R = 50 mm; rp^ = 1 mm; U = 250 m/s; E = 0,68, the
resulting vortex chamber gain will be: G = 6,8 ms. The value of the chamber gain determines that, for the input rate of 1 rad/s, the relative angle of the streamline to the radial direction on the radius of the pickoff location will be 6,8 m rad = 0 , 3 9 degrees.
3. GAIN OF THE SENSING SYSTEM
where (AP) is the output pressure difference, (Aa) is the vector angle of the sink velocity, (Kp_) is the sensitivity of the pickoff system. As a matter of fact, the sensing system used in the vortex rate sensors is a miniaturized "Yawmeter" system used in the wind tunnels.
3.1 The Sensitivity of a Cylindrical Sensing System
The cylindrical sensing system (Fig. 3) creates a tube divided inside by a dividing wall placed in the tube in the midsection, and two pressure pickoff holes are
drilled near the midsection. The sensing tube is placed in the perpendicular direction with respect to the sink hole center line, while the pickoff holes with respect to the sink hole center line are located at an angle (a). The distance of the pickoff holes from the sink hole center line is equal and creates the radius distance (fp-.) on which is sensed the angle of the sink velocity
vector (U ) . s
Knowing the statical pressure distribution with respect to the position (a) on the cylinder, it is possible to calculate the maximum sensitivity of the pickoff, The statical pressure distribution on the cylinder is related by Eqn. (9):
P-P a 2P"s
2
4 sin a (9)
where (P) is the statical pressure on the cylinder, (P ) is the pressure of the ambient atmosphere. Differentiating Eqn. (9) by (da), the expression for the sensitivity of one pickoff is related by:
P-P
2^"s
K^- = 2 = - 8 sinacosa (10)
The maximum pickoff sensitivity results from differentiating eqn. (10) by (da) and equalizing the result with zero:
2 P-P
2P"g 2 2 = - 8(cos a - sin a) = 0
sina - ± y
-x-a = ± 45° (11) The point of the maximum sensitivity of the pickoff position is given by a= 45 .
The maximum sensitivity of the cylinder pickoff system using two pickoff holes is given by Eqn. (12):
, . 0 , ^ 0
K^ =2,8 sin 45 cos 45 = 8 (a in rad) MAX
_ 4
-or:
K^ = 0,14 (a in degrees) (12) MAX
The diagram in Fig. 4 (Ref. 2) shows the variation of sensitivity of a cylindrical "Yawmeter" with Mach number and position of the pressure hole. It can be seen that the position of the pickoff corresponds to the introduced calculation, while the sensitivity is about 70% of the calculated value:
K^ = 0,1 (a in degrees). The disadvantage of the cylindrical sensing system used in the vortex rate sensor is the low flow rate through very small pickoff holes giving low power output of the sensing system.
3.2 Two Tubes Sensing System
The two tubes vortex rate sensing system can be seen in Fig. 5. Two tubes of the outside diameter smaller than the sink hole radius are positioned in the rectan-gular position to the sink hole. The centers of both sensing tubes are on the datum line of the sink hole diameter, while the inlet section of each tube is faced at a certain angle with respect to the center line of the sink hole.
Assuming that the pressure in the tube is given by the component of the sink velocity (u ) , the sensitivity of one pickoff tube can be expressed by Eqn. (13):
"(i
P-P
1'
K^ = — ^ , — = 2 sinacosa (13) 2^"s
The maximum sensitivity of one pickoff tube, according to the position of the inlet face (a) of the hole, is given by differentiation of Eqn. (13) by (da):
•fP^s . 2 2 — = 2(sin a-cos a) = 0 da
sma
^/F
a = ± 45° (14)
Using the result of Eqn. (14) in the expression (13), the sensitivity of the two pickoff tubes system results:
K^ = 2.2 sin 45°cos45° MAX K^_ = 2 (a in rad.) MAX K^ = 0,035 (a in degrees) (15) MAX
The results obtained from the tube yawmeters introduced in Refs. 2 and 3 are quite different. In Fig. 6 (Ref. 3) it can be seen that the sensitivity is 0.05 (a in degrees) using the angle a = 35°, while the theoretical value of the gain
calculated from Eqn. (13) for a = 35° is only 0,0328, about 35% less than the actual value.
Another arrangement of the pickoff tube (Ref. 2) used as a yawmeter (shown in Fig. 7) gives the sensitivity about 0,064, while the theoretical value for a = 45° gives only 0,035.
From the introduced values of the actual pickoff tube sensitivity can be considered that some other effect amplifies the gain. This effect can be explained by the action of the parallel component of the sink velocity (u ) creating the suction function in the entrance of the pickoff tube. The suction decreases the pressure in the tube built up by the perpendicular compnent of the sink velocity (Ugsina) and amplifies the overall sensitivity through the high sensitivity of the suction function upon the variation of the parallel component of sink velocity with Aa. The introduced assertion is confirmed by the measurement of the static pressure in the tube, with inclined entrance in the turbulent jet. The pressure inside the tube with the dead end, with entrance inclined at a = 20° to the turbulent jet is about 30% of the value calculated from the perpendicular component of the velocity
1 ^ 2 (-sPu sin a) .
I s
It has been found that the face of the pickoff tube with entrance inclined at 20° gives the maximum sensitivity: K^_ = 0,08 to 0,1 (a in degrees). The gain of the tube pick off system is related by Eqn. (16):
1 2
G = (0,08 to 0,l)4)u (16) a 2^ s
4. GAIN OF VORTEX RATE SENSOR
Using Eqns. (7) and (16), the total gain of the vortex rate sensor is related by Eqn. (17): 2 R „ 1 2 G - G G = E -i:^ K„„ T u Ü) a r„„u TO 2 PO 8 s (17)
where (Kp_) is the factor expressing the sensitivity of the pickoff system. Assuming the turbulent flow in the sink hole having a radius (r ) , and assuming the non-compressible flow, the vortex rate sensor gain is related by:
^ = ^ ^ ^oyï/^ ^^«>
rPO
or:
S.h^
'^'>
where (Pp) is the gauge pressure on the radius (R) in the vortex rate chamber, (p) is the density of fluid, (Q) is the total flow rate through the vortex sensor.
6
-by air is dependent on the square root of pressure in the vortex rate chamber (P ) . To keep the pressure (Pp) as high as possible, the pressure drop across the
coupling ring has to be as small as possible. The design of the coupling ring as grill with the relative large flow passages gives the pressure drop of several mm w.g. only.
From Eqn. (19) can be seen the paradoxical requirements: the maximum gain requires that the total flow through the sensor should be as large as possible, but the outlet sink as small as possible. Providing that the coupling does not restrict the flow, the main restriction of the flow creates the outlet sink area, thus giving the flow rate through sensor.
Assuming that the flow rate (Q) at a certain supply pressure is given by the outlet sink area, the spacing of the vortex chamber (b) has to be related to the radial boundary layer thickness. It has been found (Ref. 1) that the boundary layer
thickness decreases as the inward radial flow velocity along the disk increases (Fig. 9 ) .
The Boundary Layer solution (Ref. 1) gives:
(20)
where 6 is the maximal boundary layer thickness on the radius of the coupling ring.
The Integral-Momentum Method (Ref. 1) gives:
(21)
6^
' o
where 0- is the maximal boundary layer thickness in the distance R/4 from the radius of the coupling ring (R). (u^) is the radial velocity on the coupling ring radius (R). The chamber spacing R/b = 30 to 40 has been found satisfactory for the coupling ring created by the radial vanes with the through flow to solid area ratio 1:1.
The statical characteristic of the vortex rate sensor with the following design parameters can be seen in Fig. 10.
Coupling ring radius: R = 50 mm Coupling ring design: 360 vanes
0,508 thickm 4 mm long Spacing : b = 1,6 mm
Sink hole diameter : r = 2,4 mm Pickoff : two tubes
oa 1 mm. id = 0,6 mm; a = 20 Centre line distance
of pickoff tubes : 2r = 1,8 Supply pressure • P„ = ^8 KN/m
s
Total flow rate : Q = 0,9 dm'^/s
The variation of the total flow rate (Q), pressure difference across the coupling
ring (P1-P2)» ^^^ ^^^ entrance velocity (UQ) with different supply pressure (Pg)
of the above mentioned vortex rate sensor can be seen in Fig. 11.
5. USE OF VORTEX RATE SENSOR
The statical characteristics, meaning the value of the gain, the linear range, the accuracy compared with the linear range and the dynamic behaviour of the sensor, determine the use of the vortex rate sensor.
The gain of the vortex rate sensors is in the range G = 2 to 4 mm wg per degree/s measured in the deadhead pickoff output, while the linear range of the sensor reaches the value over 360 degrees per second. The statical instability of the output signal which can be seen in every point of the statical characteristic as a long term change of output pressure difference, as well as the high frequency change of the output signal (output noise) determines the accuracy of the sensor. The long term instability of the output signal varies with the ideal statical characteristic (Fig. 10), so that the recorded value of the output is in the range ± AP. It means that the input rate which corresponds to the output signal is in the tolerances given by Eqn. (22).
± Ao) =
f-
(22)
The accuracy of the sensing can be expressed in percentage of the linear range using the Eqn. (23):
ACCURACY = f^ 100 = • ^ 100{%} (23)
where (fi) is the linear range of the sensor. In the introduced case, the long term variation of the output signal was in the range AP = ± 4 m m w g , G = 2,4 mm wg per degree/s, fi = 360 degree/s the accuracy is 0.46% of the linear range of the sensor. The value of the accuracy means that the input rate is sensed with the accuracy + 0,46.10"^ . 360 = ± 1,6 degree.s".
The small input impedance of the pure fluid proportional or bistable amplifier, as well as the high impedance of the pickoff vortex rate system decreases the output V.R.S. gain, when it is compared with the gain obtained by the dead head V.R.S. output. The relatively small gain of the pure fluid amplifiers as well as their high noise sensitivity reduce the accuracy of the rate sensing.
According to the introduced facts, the use of the vortex rate sensor is limited to the sensing of relatively high rates, with maximum value limited by the linear range.
The rate sensor transfer function (Ref. 4) is given by
AP „ -Ts
o _ Ge
w ~ l+Tjs (24)
where (T^) is sensor lag time constant, (G) is gain of the sensor, (T) is the pure time delay constant given by
8
-where (<R b) is the volume of the sensor vortex chamber, (Q) is the total flow through the sensor. The sensor's lag time constant depends on the pickoff system as well as on the output signal transfer arrangement. Both time constants are in the order of 0,01s.
6. CONCLUSION
The vortex rate sensor is a fluidic device with very high sensitivity and accuracy. In the term of the output-input ratio of the dynamic pressure generated by the output-input tangential velocity, the gain of the device is in the order of 5.10' and the accuracy expressed in the percentage of the linear characteristic is below 0.5%. But over this performance, the use of the air supplied vortex rate sensor is limited for sensing relatively large rates. One way to improve the vortex rate sensing system is to design a hydraulic vortex sensor. According to the V.R.S. gain Eqns. (18) and (19), a significant change of the fluid density magnifies the gain. A rate sensing system having a smaller linear range, but larger gain will help to widen the range of application of this fluidic system. The development of output fluid amplifiers with a high input impedance is another part of the system which has to be solved.
7 . REFERENCES 1. Sarpkaya, T. Goto, J.M. Kirshner, J.M. 2. Pankhurst, R.C. Holder, D.W.
"A Theoretical and Experimental Study of Vortex Rate Gyro"
Advances in Fluidics - The 1967 Fluidics Symposium, New York, ASME, pages 218-232, "Wind Tunnel Techniques",
London, Pitman, 1968. 3. Gorlin, S.M.
Slezinger, I.I.
4. Hayes, W.F. Kwok, C.K.
"Wind Tunnel and Its Instrumentation" Jerusalem, Israel Program for Scientific Translation, 1966, pages 195-206.
"Hybrid Fluidic Heading Reference" ASME Publication j^ 70FLCS-10,
Presented at the Fluidics Conference, Atlanta, Ga., June 22-23, 1970.
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