Two methods for dynamic measurement of the valve height in a piston pump

Two methods for dynamic measurement of the valve height in

a piston pump

Citation for published version (APA):

Ceelen, van der, B. J. (1988). Two methods for dynamic measurement of the valve height in a piston pump. (TU Eindhoven. Vakgr. Transportfysica : rapport; Vol. R-939-S). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1988

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Two methods for dynamic measurement of the valve height in a piston pump

B.J. van der CEELEN

Report of the traineeship. accomplished .between December 1987 and July 1988

September 1988

coaches: Ir. J.W. Cleijne Ir. P.T. Smulders

WIND ENERGY GROUP

Technical UniversIty Eindhoven Faculty of Physics

R 939 S

Laboratory of Fluid Dynamics and Heat Transfer P.O • .Box 513

5600

Me

Eindhoven. the Netherlands

ConsuHancy Services

Wind Energy

Developing Countries

p.O. box 85 3800 ab amersfoort holland

SUMMARY

During this traineeshlp two methods for dynamic measurement of the valve height in a waterpump have been examined.

In the first method named the optical method a displacement of the object to be measured results in a different part of a light beam being

transmitted. Experiments proved that this method is capable of dynamic valve height measurement. The appearance of air bubbles in the water however renders the method unusable.

The second method called the M.R.S. method uses three magnetoresistive sensors and a permanent ring magnet built in in the valve. A transducer containing three magnetoresistive sensors was built and afterwards calibrated for the valves height in a range of ten millimeters and for its tilting.

The results of this calibration were that the transducer is linearly proportional with the valve height and that the valves tl1tinft does not affect the valve height calculated from the transducers output. During this calibration the transducers output drifting responsible for the error in the calculated valve height was measured. This error proved to be less then 0.2 millimeter.

TABLE OF roNTENTS page SUMMARY •••••••••••••••••••••••••.•••••••• 1 TABLE OF CONTENTS .•••.••••.•••.••••..•••• 2 LIST OF SYMBOLS .••••••••••••••••••••••••• 3 CHAPTER 1 Introduction ... 4 CHAPTER 2 Theory ... 7

CHAPTER 3 Optical method ... 9

3. 1 Theory ... 9 3.2 Experimental setup ... 12 Experiments 12 Results 12 CHAPTER 4 MRS method ... 15 4.1 Theory ... 15 4.2 Experimental setup ... 18 Experiments 18 Results 18 CHAPTER 5 Conclusions ... 25 REFERENCES. . . . 27 APPENDIX .••••••••.•••..••••••..•••..••.. 28

L 1ST OF SYMBOLS symbol h h' m

e

and IP P Po H Vout v R a 'J x r a .6.a I description valve 11ft height valve lift height calculated from transducers output height of the three screws in the

calibration device vector perpendicular to the valves plane spherical coordinates of m resistivity isotropic resistivity maximum change in resistivi ty magnetic field output voltage

output voltages of the three sensors in the transducer

average output voltage valve radius units m m m ohm ohm ohm A/m

v

v

v

m valve angle in sourcesensor -plane

angle of light beam angle of micrometer x-coordinate of intersection x-coordinate of intersection x-coordinate of valve edge

x-coordinate light source width of light source dimensionless intensity m m m m m

CHAPTER 1 INTRODUCTION

One of the activities of CWO (Consultancy Services Wind Energy Developing Countries) is deSigning single acting piston waterpumps for use in and by developing countries. These have to be as reliable and simply designed as possible. Each design is a compromise between energy losses and big pump rod forces. Optimal design therefore requires exact calculation of pump rod forces. This calculation cannot be done without understanding valve motion and valve closure time.

In the past some methods for measuring valve closure and opening time have been used .

One method uses the pressure peak in the water that occurs when the valve closes. This method however only gives information about valve opening and closure times. See [DOE 80] and

[lOO

88].

Another method is visual determination of valve closure time. This method is a stroboscopic measurement of valve height. Measurement of valve

behavior during a whole pump cycle costs much time. See [LOO 88J. A third method uses hall effect proximity switches to determine valve opening and closure times. See [HIL 83].

The results of these three methods do not agree with the mathematical model of valve motion.

[JAN

87].

Therefore a new method for dynamic valve motion measurement was to be developed.

The method to be used had to fulfil the conditions and criteria stated below.

In the first place it had to be capable of detecting displacements in the range of zero to ten millimeters. Secondly the frequency range had to be wide enough to cope with valve closure times of five milliseconds. Furthermore a noncontacting type of measurment was desirable because by such a type a newly designed valve could easily and quickly be tested. Lastly, the method had to be

waterresistant. Most methods of displacement measurement and approximation detection as found in the literature have been

considered. The results of comparison with the criteria are listed below in table 1.1. f' R N C R " R A- D R A 0 0 A A E 0 I E N N N N T S 0 f Q G T G E I I f U E A E R S T I E C T I C N

"

T 1 A- 0 U C I I 0 N N l method y 0 E N G M T A T l I M. E S capacitive II

-

+ resistive II

-

-

+ induction \I

-

+ interferometric II + +

-

+ II fotonic sensor II + + +/- + + optical II .+ + + + magnetoresistive It +

-

+ + acoustic II + + + + + eddl current II +/- ₊ +/- + +

Table 1.1 several methods and the criteria

+ indicates yes - indicates no

The additional difficulties for the fotonic sensor. a device that emits and simultaneously receives the reflected light, consIst of the fact that it uses the valve itself as a mirror that should not rotate ( what it does ).

The acoustic method has not been chosen because of obvious difficulties in interpretation of the output signals. The eddy current method uses sensors that are very large relative to the range of measurement. and therefore would disturb the flow pattern in the pump. Concludingly. the magnetoresistive sensor (MRS) method and the optical method seemed to be the most suited.

In chapter 2 some general theory about valve position is given. The optical method is discussed in chapter 3, where theory concerning this method as well as experiments and results are given.

Chapter 4 deals with the MRS method.

CHAPTER 2. THEORY.

In a single acting piston pump like the CWO 67 S pump a valve has three degrees of freedom. (See fig. 2.1). The first one is the valve height

relative to the upper piston disc. h. If we take the z-axis along this motion. the valve can furthermore to some extent rotate about the x- and y-axis (6 degrees maximum for both). The valve can also rotate about the z-axis. but due to the circular symmetry of the valve this degree of freedom can be neglected. The shift of the valves centre relative to the z-axis has neither been taken into account.

I'I

!

I

rUMt> rod

i '

IJO 1,,( S~"P

cj

1

· ·

UG\lve.

h

r--

----I. - - IApptr pi~\-on di,<:

i

/

Y

r~!;:

....

,:t:

The position of the valve in this model is therefore described by the valve height h and the orientation of the vector ~. if we define ~ as a vector perpendicular to the plane of the valve.

1

x

fig. 2.2 Valve height and orientation

i!;'

Y'

fig. 2.3 Orientation described by two angles

The orientation of the valve can be described by the two parameters e and

~, as shown in figure 2.3. In order to describe the valve position. h,e

CHAPTER 3 OPTICAL METHOD

3.1 Theory

Measurement of the valves height and orientation can be done by maki~

the valve partially stop a light beam. The light source however should be parallel. and thin compared to its width (about 1 and 10 rom respectively) so that the beam lies in a plane. The valves height and tilting results in a part of the light beam ,being stopped. The unstopped part of the light is transmi t ted to a light sensor.

-1

fig. 3.1 LIght source and sensor used to measure valve height and

rotation

See figure 3.1 . The light source is placed at x

=

a and has a width

alo~ the x-axis of Aa. The x-coordinate of intersection of the light

rays 1

and

2

and

the line trough the valve are XiI and xi2' These two coordinates are given ~

XiI

=

h + (a-Aal2 ltanp tan/3 - tana

X 12 = h + (a+Aa/2 ) tan/3 tan/3 - tana

The X - coordinate of the valves edge X is given by

r

X

=

R cos a

r

(3.1 )

,

A

The ratio of the widths of the transmitted and original beams is I A and I is given by ,.. I

=

_{x i2 - xr} x i2 - xi l

Substitution of (3.1) and (3.2) in (3.3) gives

'"

I

=

h ~ (a+~a/2 ) _ Rcosa(tanP - tana)

~a*tanP ~a ~a*tanP

" "

so that I

=

I(h.a) if a, ~a, R and ~ are constants.

'" A

I is a linear function of h and the sensitivity of 1 for h is

'"

81

=

_1~-=

8h ~a*tanP

'"

The sensitivity of I for a Is

"

81

=

8a

R *(sinatanP + cosa)

~a*tanP

For small values of a (3.6) is

(3.3)

(3.4)

(3.5)

(3.6)

(3.7)

In order to get high sensitivity, the source should have a width that Is as small as possible and

P

should be small too.

Since three observables are to be measured. at least three source

sensor-pairs have to be used. When they are placed as shown in figure 3.2 h.

a

and ~ can be calculated from the sensor outputs.

fig. 3.2 Three sensor-pairs used to measure valve position

3.2 Experiments

The main experiment concerning the optical method was a measurement of a fotodiodes output voltage as function of valve height.

For this experiment the experimental setup #1 as shown in figure 3.3 was used. J fui-od;oeic 2 ~ppl!1

.3

rolA IT

lme

+e ,.

'I

Ink:.

n1;()efer t.Vlfh

"a

~C'r

blode

S ft'sij./{Je Jense , if p,.~sel1f )

6

fibre

'1

kA$t!.r

fig. 3.3 Experimental setup #1

The light source was a 0.3 mW He-Ne laser.

The output voltage of the sensor. a fotodiode. was measured with a digital multimeter. and had a range of 0 to 150 mVolts. The output

voltage as a function of the razorblades displacement (from 0 to 5 mm in steps of 0.25 mm) was measured for three different angles 1 (1

=

25. 27

and 29 degrees). This was done for both a round (2 mm diameter glass) and a rectangular (5 x 1 mm perspex ) fibre. Since the light coming out of each of the two fibres was diffuse. sensor and source had to be put close together in order to acchieve an output voltage high enough for the output voltage in case of no illumination to be neglected. As a result only angles 1 (see fig. 3.3) in the range of 25 to 30 degrees could be measured since other angles would not have left enough room for the micrometer with razorblade to be fitted in between source and sensor.

Using the round fibre the output characteristic. that is the output

voltage as function of h. was obtained for the three angles 1. See figure 3."1 •

fig. 3."1 The fotodiodes output voltage as function of h for three angles 1

All other experiments had a qualitative character. The effect of air bubbles in th~ water as well as fibre shape and beam collimation were tested.

The effect of the light source being not parallel on the output linearity was measured using a positive lense (see figure 3.3 ).

In order to measure the effect of air bubbles on the transmission of the light from the fibre to the sensor, and thus on the sensor output

voltage, a fotodiode was built in in a perspex tube (see figure 3.5 ). By doing so the fotodiode was made waterreslstant.

1 fofoaiode

2 f,.uppllj

J

mUI+ltne.f~,.

4

hypoderNH~

Z>/Jri"[je

$

fibre

I>

'a..sf!.r

This sensor and one end of a round fibre were put under water. and the output voltage was measured. Then. using a hypodermic syringe, air bubbles were injected in the water between source and sensor. and while doing so, the output voltage was measured again.

A positive lense used to collimate the outcoming beam slightly improved linearity. Injecting air bubbles between source and sensor made the output voltage drop to as little as 10% of the original output voltage.

CHAPTER -4 MRS METHOD

-4.1 Theory

Another method of measuring h.

e

and ~ Is using three magnetoresistive sensors and a valve with a built-in permanent ring magnet. Similar to the method of occultation. the output voltage of the sensors are dependent of the distance from magnet to sensor. and orientation of the magnet

relative to the sensor. The sensors are made of permalloy. a ferromagnetic alloy. Such an alloy has a magnetic field-dependent resistivity. When a magnetic field is applied to the alloy. It will be

magnetized. This magnetization induces an anisotropic resistivity: if the magnetization makes an angle

e

with the direction of current through the alloy. the resistivity is given by

where

(-4.1)

is the isotropic resistivity

is the change in resistivity resulting from a 900

rotation of the magnetization from the direction of current

I

Lsupo!Yf __ 1

-$tgl'!&l ,+Wl)

T~ KMZlO m,gnetoresinive sensor I tKene development

tor detecting m~netit·fteld variations.

When the applied magnetic field H is normal to the direction of current and

Ho

(comprising the demagnetizing and anisotropic fields) makes an angle 0 with the direction of current, then sinO

=

HlHo.

Substituting

P

=

Po

+ APm.x[1-H21H~

]

P

=

Po

for

H <

Ho

for

H

~

Ho

(4.2)

In order to linearize P as function of H. gold stripes are used to rotate the current direction through 450

(see fig. 4.2). Equation (4.2) then becomes

1/2

P

=

Po

+

Apmax

+

APmaxH

[1-H21H~

]

2

Ho

which is linear with H for small values of

HlHo.

(See fig. 4.3) .

.... =

..

,

..•.

, ,

,

-GoIdlitripwontllo....-...v _ _ _ tho ~

_ion Chtouth 45". TIIilIi _ _ tho ~~.

fllold _ .... ltlict 01 tho MAS

fia.

4.2 Cold stripes used to rotate the current dIrection trough 45 degrees

In 0 t>arbet-polt MRS. resiltMty _ _ I~ witll llIH<lHol. The lit _ _ tilt _ 1 _

d\lnge A.IJ/Apf'MX ift tHistillhy at • fUftCtfon of

H/H" lAp· P - p" - AP ... m

fig. 4.3 Resistivity as function of the magnetic field

Since the resistivity of the sensor depends on temperature. the sensitivity of the sensor is temperature dependent too. The specifications indicate that the temperature coefficient of the sensitivity is -0.4 %/K.

4.2 Experiments

A very detailed calibration of the valve height h and the two angles

e

and ~ had to be carried out. For this purpose a calibration-device was built. see fig 4.4 .

I

j' 14

J

HRS~/11

< , } J I C ) ' : I i

I~

fig. 4.4 The calibration device

The transducer in this calibration device is also shown in figure 4.5 . The transducer consists of a solid plastic covering with three KMZ 10 B sensors casted in.

8~_

1 2. Lxdvc ! e/k/tllllte.1 Q""pllfl~" and StApp'IJ

J

~r~w$

l.f

baS(. S" (}al(lf!

'J

c.t

;dc

b

frM~alA"r

1

"'tAl "'me fers

S K-t

"",.ile,..

The transducer had to warm up after having been switched on. When warmed up, the valve height and angle 9 were set using the three screws (see Appendix A.8). These screws were all given a certain height using a vernier caliper. The three output voltages were measured with digital mul timeters.

The valve height was increased in steps of one millimeter from 3 to 10 millimeters. At each valve height the output voltages were measured for

e

=

0, ~ = O.The angle ~ was set using the graduation. Furthermore at each height (except at 10 millimeters) for

e

= 2.5 degrees the output voltages were measured at values of ., of O. 50, 120. 180. 2-40 and 300 degrees. At a valve height of 8 mm and at five different angles

e ( o.

1. 2. 3. 4 and 5 degrees) the output voltage of each sensor was measured for the six values of ., mentioned above.

The output voltages that were actually measured were the minimum and maximum output voltage each sensor could have at a certain height and orientation. Since the assumed rotational symmetry of the magnets field proved not to hold, the output voltages were affected by rotating the valve about the z-axis. It turned out that by doing this each sensor output had a minimum and maximum value. The shift of the centre of the valve from the z-axis also affected the output voltages in this manner. As a result at each valve height h and orientation. a minimum and maximum of the output voltages could be achieved by rotating and shifting the valve.

An example of the results is given in figure 4.7 .

t

o

mAX. mi n.

fig. 4.7 Minimum and maximum output voltage of sensor 1 as function of ~ for h

=

8 DID.

e

=

5 degrees

Notice that at each angle ~ a minimum and maximum output voltage occ rs. Furthermore. if we look at ~

=

60 degrees and at ~

=

240 degrees we see that the output voltage of this sensor at h

=

8 mm.

e

=

5 degrees can vary between 1.28 and 1.54 volts.

These two voltages are plotted against the valve height for e = 0 and 2.5 degrees for each of the sensors (see figures A.I to A.6 Appendix).

An example of this fi iven in figure 4.8 .

r.~~'I~~~V.~/'~~~~--~~~~~~~~--~~ '.~ -," (2 ',I " c ~ r--~--- . . . ---;---.----~__r___ 3,? l,;,:J 1;0 ,,., 11'1" ~,6 11 0 '_0~."" ...-.. h m mM.

ig. 4.8 Minimum and maximum output voltage of sensor 1 as function of h for

e

=

0 degrees

When the average of each sensors maximum and minimum is taken it follows that the relation between this average and the valve height is

h = 1.26

*

v -

2.34 o for e = 0 o for

e

=

2.5 (4.4)

were v is the average output voltage in volts h is the valve 'height in millimeters

From the calibration at 8 mm for different angles of

e

followed that a sensors maximum and minimum output voltage varied with. as a cosine (see figure

4.1

above)

and

that the angle

e

only affected the amplitude of this cosine. The average taken of a sensors absolute maximum

and

minimum is therefore hardly affected bye. This is confirmed by equation 4.4 where the relation for

e

=

2.5 degrees only slightly differs from the relation for

e

=

0 dAvrAAR_

Since ambient temperature was not constant, the temperature of the transducers sensors wasn't either. Due to the sensors' temperature dependent sensitivity a thermal drift in the outputs occured. This

thermal drift was measured by recording two sensors' outputs during 75 hours. During this experiment the valve was set to 0 mm and taped to the base. The output of the x-t writer is shown in figure 4.9 •

'4.tif\<'1Ib

'''',s

. "'0·"·

. :~ ," ., _c

~

r---~L---~---~_;---_1

/"

i

~ 10 i ~o i

fig. 4.9 Output voltage as function of time at h

=

0,9

=

0

The thermal drift in the sensors' output appeared to be about 20 mVolts.

When the supply of the transducer was switched on, the sensors heated up and finally reached a thermal balance with the environment. The output voltage of each sensor decreased when heating up. This proces was

recorded with the x-t writer.

r - - - ---.---~-,.-'I'"'!_~_r;__t_,._"""!'"_:T~~~'!1'I'i

\::.l~

__

u u . u u u u m u u m u m u _ u u _ u __ . .

~.

- i n _.2

I

1

, .& ' i

20 :to i

I .

fig. 4.10 The sensors' warming up proces.

The stability of the sensor output before and after amplification was measured using an x-t writer. The amplified and unamplified signal of sensor 1 were measured during sixteen hours, as shown in figure 1.11 .

Vt .;JfIltP ',//((1 .) i.. ... -) : ! ' , "I~ 1 " " ,

fig. 1.11 Sensor output stability test

From this experiment followed the fact that the three channel amplIfier and 12 Volt supply were stable within

IX.

Finally, the measuring method and its calibration were tested. This was done by setting the three screws to a certain height and comparing the valve height calculated from the sensor outputs with the valve height calculated from the three screw-heights.

o The valve heights at which this was done were 0.0 mm. with

e

=

0 ,

o 0 0

5.15 mm. with

e

=

5.1 , 5.53 mm. with

e

=

1.2 • 6.00 mm. with

e

=

0 and

o 0

At all these combinations of heights and angles

e

this test was carried out for some different angles ~.

The results of the testing of the measurment method are represented below in figure 4.12 1m

i

h';~

mm / /. I

I

I

ls-I

r _ • .. ,..:....#J ,..1 _ _ - _ • i4t.1 "'1.1 ....

fig. 4.12 The real valve height h and the calculated valve height h'.

From this the conclusion can be drawn that the transducer is linear over a range of 10 mm with an error less then 5%.

CHAPTER 5. roNCLUS IONS.

5.1. Optical method.

Experiments proved that a transducer with reasonable linearity and range of measurement can be constructed even with a light bundle being not parallel, being only 4 mm wide and without focussing transmitted light on the fotodiode. Therefore the conclusion that a linear transducer with a range of over 20 mm can be constructed seems to be justified.

There are however two disadvantages to the method.

The first disadvantage consists of the effect of air bubbles in the water on the transmission of the light. Each air bubble in the water scatters incoming light as a negative lense.

The second disadvantage is caused by the fact that in a waterpump there is not much room for three transducers of this type to be built in. When these advantages can be eliminated, the very wide frequency respons allows dynamic valve height measurement.

5.2. MRS method.

Experiments proved that using the MRS transucer valve height can be staticly measured with an accuracy better then 5% full scale in a range of ten millimeters.

The measured valve height is neither affected by the tilting of the valve nor by the shIfting of the valves centre from the z axis.

Since the thermal drifting is about 20 mY. the error in h' can be calculated from (4.4) and is about 0.14 mm. The results of the method

test (see figure 4.12) however shows that the maximum error in h' is about 5% full scale. This error only occured at a valve height less then 3 mm that does not lie within the range in which the transducer is

calibrated (3-10 mm). Within the range the transducer was calibrated. the error in the measured valve height never exceeds 0.14 mm. This error can be explained with the thermal drifting of the transducers output.

The experiments also showed two minor disadvantages of the transducer. In the first place there is the long time the transducer needs to warm up when switched on (about 45 minutes).

In the second place the transducer drifts due to thermal effects. what causes the error in the valve height.

Since the frequency range of the sensors in the transducer is wide

REFERENCES

1. [OOE 80]

2. [LOO 88]

3. [HIL 83]

4. [JAN

87]

van der Does. P.

Metingen van klepsluittijden van de Tunesiepomp Internal report R ~2~ D. Eindhoven University of Technology. January 1980.

van Loon. E.M.L.

Metingen aan het klepgedrag van zuigerpompen Internal report

R

897

S.

Eindhoven University of Technology, January 1988.

Hilbers. M.

Drie deelmetingen aan een pompopstelling voor de • Tanzania pomp'

Internal report R 641 S, Eindhoven University of Technology. December 1983.

Janssen. W.

Berekening van de kleppenbeweging in zuigerpompen Internal report R 860 S. Eindhoven University of Technology, June 1987.

APPENDIX

A.l Optical method measurement results

h V V V

In mm. in mVolts in mVolts in mVolts .o.h=O.Ol mm. .o.va::O.5mV .o.v.O.5mV .o.v=O.5m'f

'J

=

25° '1

=

27° 'J

=

29° 0.00 3 97 41 0.25 5 103 52 0.50 8 108 62 0.75 12 114 71 1.00 16 119 80 1.25 22 123 88 1.50 31 127 96 1.75 39 131 103 2.00 48 137 107 2.25 57 140 114 2.50 67 142 123 2.75 76 1-43 131 3.00 84 144 138 3.25 91

·

.

143 3.50 96 _{• •} 145 3.75 102 _{• • • •} 4.00 113

·

.

4.25 121 4.50 128 4.75 135

·

.

_{• •} 5.00 141 _{• • • •} 5.25 145 _{• •}

· .

5.50

..

·

.

·

.

A.2. M.R.S. calibration results hi h2

h3

h

e

_'P

V1

V

2

V3

mm mm mm mm degr degr Volt. 6h .. O.1 Mm. 6V .. O.02 V 10.00 10.00 10.00 10.00 0 0 1.68 1.54 1.65 1.81 1.66 1. 73 9.00 9.00 9.00 9.00 0 0 1.54 1.43 1.56 1.67 1.55 1.65 8.13 9.44 9.44 9.00 2.5 0 1.52 1.41 1.61 1.64 1.52 1.68

· .

• •

·

.

• •

·

.

60 1.50 1.45 1.58 1.61 1.56 1.66

·

.

·

.

• • • • 120 1.52 1.47 1.54 1.64 1.58 1.62

·

.

_·

_.

_·

_.

180 1.57 1.45 1.52 1.70 1.56 1.60

·

.

_·

.

·

.

·

.

240 1.61 1.41 1.55 1.73 1.53 1.64

·

.

·

.

• • • • 300 1.57 1.40 1.59 1. 70 1.50 1.67 hi _mm h2

h3

h

e

tp

V1

V

2

V3

mm mm mm degr degr Yolt. 6h=D.l Mm. 6Y=D.02 V 8.00 8.00 8.00 8.00 0 0 1.39 1.29 1.46 1.52 1.41 1.55 7.13 8.44 8.44 8.00 2.5 0 1.39 1.29 1.50 1.51 1.40 1.58

· .

·

. ·

.

• • • • 60 1.37 1.32 1.48 1.48 1.44 1.57

·

.

• • • • • •

·

.

120 1.38 1.35 1.44 1.50 1.45 1.53

·

.

·

.

• • • • • • 180 1.42 1.33 1.43 1.54 1.44 1.50 • • • •

·

.

• • • • 240 1.44 1.29 1.44 1.56 1.40 1.52

·

.

• • • •

·

.

• • 300 1.42 1.27 1.46 1.54 1.37 1.56

h1 h2 h3 h

e

.,

V1 V2 V3 mm mm mm mm degr degr Volt. Ah_O.l Mm. AY=O.02 V 7.00 7.00 7.00 7.00 0 0 1.22 1.16 1.30 1.34 1.27 1.4.0 6.13 7.44 7.44 7.00 2.5 0 1.20 1.13 1.35 1.31 1.23 lA3

·

.

• • • •

·

.

·

.

60 1.18 1.16 1.32 1.27 1.27 lAl

·

.

• •

·

.

• • 120 1.19 1.18 1.28 1.30 1.28 1.37

· . ·

.

• •

·

.

• • 180 1.23 1.16 1.26 1.34 1.27 1.34

·

.

·

.

• • • • 240 1.25 1.13 1.28 1.36 1.23 1.36 • • • •

·

.

• • 300 1.24 1.11 1.31 1.35 1.20 1.40 h1 h2 h3 h

e

If> Vi V2 V3 mm mm mm mm degr degr Vo It. Ah=O .1 mm. AY=O.02 V 6.00 6.00 6.00 6.00 0 0 1.10 1.04 1.20 1.21 1.15 1.29 5.13 6.44 6.44 6.00 2.5 0 1.08 1.02 1.23 1.18 1.11 1.31 • •

·

.

• • 60 1.05 1.04 1.21 1.14 1.15 1.30 • •

·

.

• •

·

.

• • 120 1.06 1.06 1.18 1.16 1.15 1.26 • • • • • • • • • • 180 1.09 1.04 1.16 1.20 1.14 1.23

·

.

• • • • • • 240 1.11 LOI 1.17 1.21 1.11 1.25

·

.

• • • • • •

· .

300 1.10 0.99 1.19 1.20 1.08 1.28

h1 h2 h3 h

e

'I' V1 V2 V3 mm mm mm mm deg,. deg" Volt. ll.h =0. t am. AV=O.02 V 5.00 5.00 5.00 5.00 0 0 0.95 0.91 1.06 1.05 1.01 1.15 4.13 5.44 5.44 5.00 2.5 0 0.93 0.89 1.09 1.02 0.97 1.17 • • • • • • • •

·

.

60 0.91 0.91 1.08 0.99 1.00 1.16

·

.

·

.

• • • • 120 0.92 0.92 1.05 1.01 1.01 1.13 • •

·

.

• • 180 0.94 0.91 1.04 1.04 1.00 1.10

·

.

· .

·

.

240 0.96 0.88 1.04 1.05 0.98 1.12

·

.

• •

·

.

·

.

300 0.95 0.88 1.07 1.05 0.95 1.15 hi h2 h3 h

e

tp V1 V2 V3 mm mm mm mm deg" deg" Volts Ah=O .1 mm. AV=0.02 V 4.00 4.00 4.00 4.00 0 0 0.81 0.78 0.93 0.90 0.87 1.02 3.14 4.44 4.44 4.00 2.5 0 0.79 0.76 0.96 0.87 0.84 1.03 • •

·

.

·

.

• •

·

.

60 0.77 0.77 0.96 0.84 0.88 1.04 • •

·

.

• • • • • • 120 0.80 0.81 0.93 0.88 0.89 1.01

·

.

• • • • • • • •

ISO

0.82 0.79 0.92 0.91 0.88 0.98 • • • • • •

·

.

240 0.83 0.76 0.93 0.91 0.85 1.00 • • • • • •

·

.

• • 300 0.81 0.76 0.94 0.90 0.82 1.02

hi h2 h3 h

e

'P Vt V2 V3 mm mm mm mm de; .. de; .. Volt. Ah=O.l mm. AV=O.02 V 3.00 3.00 3.00 3.00 0 0 0.64 0.64 0.79 0.73 0.73 0.87 2.13 3.+1 3.+1 3.00 2.5 0 0.65 0.65 0.82 0.72 0.72 0.89 • • •• • • • • 60 0.64 0.65 0.81 0.70 0.74 0.88 • • • •

.

.

,

.

·

.

120 0.64 0.66 0.79 0.72 0.73 0.86 ,

.

_{• • • •}

·

.

180 0.66 0.65 0.79 0.74 0.73 0.85

· . ·

, ,

.

• • 240 0.67 0.63 0.79 0.74 0.71 0.86

·

.

• •

·

.

• • 300 0.66 0.63 0.80 0.73 0.69 0.88 0.00 0.00 0.00 0.00 0 0 0.30 0.32 0.46 0.36 0.40 0.52

hi h2 h3 h

e

.,

Vi V2 V3

mm mm mm mm degr degr

Volt.

Ah=O.1 Iftm. AV:O.02 V

8.00 8.00 8.00 8.00 0 0 1.39 1.28 1.43 1.50 1.40 1.52 7.65 8.17 8.17 8.00 1 0 1.38 1.28 1.45 1.49 1.39 1.54

·

. ·

.

·

.

• • • • 60 1.36 1.29 1.44 1.47 1.41 1.53

· .

·

.

• •

·

.

120 1.36 1.31 1.42 1.48 1.42 1.51 • • • • • •

·

.

180 1.38 1.29 1.41 1.50 1.41 1.49

·

.

·

.

· .

240 1.40 1.28 1.42 1.52 1.40 1.51 ,

.

· .

·

.

_{• •} 300 1.40 1.27 1.44 1.52 1.37 1.53 7.30 8.35 8.35 8.00 2 0 1.38 1.28 1.48 1.50 1.38 1.56 • •

·

.

·

.

• • 60 1.35 1.31 1.46 1.45 1.42 1.55

·

.

·

.

· .

·

.

120 1.36 1.34 1.42 1.47 1.44 1.51 • • • • • •

·

.

180 1.39 1.32 1.41 1.52 1.43 1.48

·

.

• • • • • • 240 1.42 1.28 1.42 1.54 1.39 1.50 • •

·

.

• • • •

·

.

300 1.41 1.25 1.44 1.53 1.35 1.53 6.95 8.52 8.52 8.00 3 0 1.34 1.25 1.47 1.45 1.35 1.55 • • • • • • • •

·

.

60 1.31 1.30 1.45 1.41 1.40 1.53 • • • • • • • • • • 120 1.33 1.32 1.39 1.44 1.42 1.48 • • • • • • • • • • 180 1.37 1.30 1.38 1.49 1.41 1.45 • • • • • • • • 240 1.40 1.25 1.39 1.51 1.36 1.48 • •

·

.

·

.

• • • • 300 1.38 1.22 1.43 1.50 1.31 1.52

h1 h2 h3 h

a

cp V1 V2 V3 mm mm mm mm dell" dell" Vo 1 t • .6h:O.l Mm. .6V:O.02 V 6.60 8.10 8.10 8.00 -t 0 1.33 1.24 1.-t9 1.-t3 1.33 1.56

·

.

• • • • • •

·

.

60 1.29 1.30 lA6 1.31 1.-t1 1.53 • •

·

.

·

.

• • 120 1.32 1.35 1.39 lA2 1.~ 1.-t1 • •

·

.

· . · .

• • 180 1.39 1.32 1.36 1.50 lA2 lA2

·

.

· .

·

.

2-tO 1.~ 1.25 1.39 1.53 1.35 1.-t6 • • • • • • • • 300 1.-tO 1.21 1.-t5 1.51 1.29 1.53 6.25 8.87 8.81 8.00 5 0 1.33 1.23 1.50 1.-t2 1.31 1.56

·

.

• • • • 60 1.28 1.31 1.56 1.35 1.-tO 1.5-t • •

·

. ·

.

·

.

120 1.30 1.36 1.39 1.39 1.~ 1.-t6

·

.

• • • •

·

.

180 1.39 1.33 1.35 1.50 1.-t2 1. -tl • • • •

·

.

• • 2-tO lAS 1.25 1.38 1.5-t 1.3-t 1.-t5

·

.

• •

·

.

• •

·

.

300 lA2 1.20 1.-t5 1.52 1.21 1.53

·

..

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hi h2 h3 h

e

V1 V2 V3 V h'

In mm. deor In volt. In mm.

4h.O.1 mm. Av=O.02 volt 4h , .. 0.14mm.

7.00 3.65 4.80 5.15 5.7 1.08 1.03 1.08 1.06 5.31 , , ,

.

,

.

.

,

·

_, 1.09 1.00 1.09 1.06 5.31

.

,

·

.

,

.

_{• •} 1.08 0.98 1.10 1.05 5.24

·

.

.

, , ,

·

_, 1.11 0.93 1.16 1.07 5.38

·

,

·

, 1.05 0.95 1.19 1.06 5.31 h2 h3 h

e

V2 V3 V h' in mm. degr in volts in mm.

Ah::O.1 mm. Av=O.02 volt Ah • =0.14 mm.

7.00 4.80 4.80 5.53 4.2 1.00 1.06 1.22 1.09 5.53

1.05 1.08 1.14 1.09 5.53

, ,

·

_, 1.11 1.06 1.11 1.09 5.53

h2 h3 In mm. deg,.

e

V2 In volts V3 V tn h' mm. .6. h = 0 . 1 mm. .6. v :0.02 volt .6.h·=0.14 6.00 6.00 6.00 6.00 0.0 1.09 1.08 1.21 1.13 5.82 • •

·

.

..

• • • •

·

.

• • • •

.

.

• • • •

.

.

h1 h2 h3 h m

e

In mm. deg,. .6.h=O .1 mm. 5.13 6.44 6.44 6.00 2.5

.

, _{• •}

·

.

• • • •

·

.

1.14 1.04 1.20 1.13 1.08 1.05 1.24 1.12 1.10 1.06 1.22 1.13 1.05 1.11 1.21 1.12 V1 V2 V3 V 1 n volts .6.,,:0.02 volt 1.13 1.04 1.21 1.13 1.11 1.08 1.19 1.13 1.11 1.06 1.20 1.12 1.16 1.08 1.16 1.13 5.82 5.75 5.82 5.75 h' tn mm. .6.h·:0.14 5.82 5.82 5.75 5.82 mm. mm.

h2 h3

e

V1 V2 V3 V h'

In mm. deg" In volts In mm.

6h=0.1 mm. 6v =O.02 volt At. ' .. 0.14 mm.

0.00 0.00 0.00 0.00 0.0 0.33 0.36 0.52 0.40 0.54 •• ,

.

.

.

· .

, , 0.34 0.34 0.52 0.40 0.54 , , ,

.

0.38 0.34 0.48 0.40 0.54 h2 h3 In mm. deg"

e

V1 V2 V3 V In volts 6h .. 0 . l mm. Av=O.02 volt mm. 10.75 10.50 10.45 10.57 0.0 1.90 1.67 1.73 1.77 10.45

..

.

,

.

,

·

.

1.83 1.65 1.78 1.75 10.30

·

.

1.79 1.73 1.75 1.76 10.37

A.S

Assume the three screws are set at h1, h2 and h~ millimeter. This gives three 3-dimensional points on which the valve lies. The vector equation of the plane in which the valve lies can now be written down.

The coordinates of the three points are

Two independent vectors that lie in the plane are

!1

=

E1-

E2 and

The vector equation of the plane is now

with

A1. A2

arbitrary real numbers

The valve height h is described by the vector Since the vector lies in the plane this gives

[

~

]=

r'

+

h,M

!,+

h2M !2

[

o~

]

In fact this are three equations. The first one gives

The second one gives

so that

A2

=

1/3 From the third one now follows that

A vector perpendicular to the plane. m can be calculated from

!

=

!1 X !2 .

and since the z component of

!.

m% =

I!I

cose e can be calculated from