Measurements on different valve seat configurations

Citation for published version (APA):

Hoogsteen, R. (1988). Measurements on different valve seat configurations. (TU Eindhoven. Vakgr. Transportfysica : rapport; Vol. R-956-S). Technische Universiteit Eindhoven.

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

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MEASUREMENTS ON DIFFERENT VAtVE SEAT

roNF

ICURAT IONS

RON HOOCSTEEN

December 1988

Begeleiders: Ir. P.T. Smulders

Ing.

J. Diepens

R 956 S

WIND ENERGY GROUP

Technical University Eindhoven

Faculty of Physics

Laboratory of Fluid Dynamics

and

Heat Transfer

P.O.

Box

513

5600 MB

Eindhoven. the Netherlands

Consultancy Services

Wind Energy

Developing Countries

p.O.

box8S 3800 ab amersfoort holland

SUJIInary List of symbols 1. Introduction 2. Theory 2.1 Disc valve 2.2 Ring valve

2.3 Valve ahead of the seat 3. Experimental design

3.1 Construction 3.2 Scaling

3.3 Measuring methods -4. Results

~.1 Resistance of the valve

-4.2 Pressure drop

-4.3 Disc valve compared with ring valve -4.-4 Approximations for Cf anf Cp

5. Theory compared with experiments

5.1 The closed valve

5.2 Convective model. Lindners model

5.3

The axial flow model

6. Discussion 1. Conclusion 8. Literature Graphs Appendices 1.1 2.1 2.14 2.20 3.1 3.3 3.-4 5.1 5.2 504 6.1 1.1 8.1

Summary

This traineeship consisted of two parts:

1. Designing and building a suitable construction of a valve/seat test rig of a piston pump. On this construction the static forces on the piston valve had to be measured. (static

=

piston valve velocity is zero) 2. Measuring the force on the piston valve. the pressure drop over the valve/seat configuration and the flow. All these quantities as a function of the valve seat distance (the lifting height) for the various valve/seat configurations.

Scaled models were made of five different valve/seat configurations as they are used in the CWO 49. 81. 108. 161 and 265 pumps. For the several valve/seat configurations a dimensionless force (CF) and a dimensionless pressure drop (Cp) were computed.

Agreement wi th the theoretical model of Lindner and the theory as it is

presented by S. Orbons was looked upon. Furthermore. an approximation of the Cp and CF functions was computed. For all the configurations. except the CWO 265 ring valve. a general function is valid:

Cp

=

1.6 (hlRv)-1.365 4.10-2

<

hlRv

<

2.10-1

CF

=

6.6 (hlRv)-1.236 4.10-2

<

hlRv

<

2.10-1 Agreement with these functions is within 15%.

g gravitational acceleration

~ dynamic viscosity

v kinematic viscosity

p density

Pa pressure ahead of the valve/seat configuration

Pb pressure behind the valve/seat configuration h valve/seat distance (lifting height)

Vo :velocity of the water in the approach duct

Vg velocity of the water in the valve/seat gap

Rv radius of the disc valve

Rs outer radius of the seat inlet

Rd radius of the duct

Rs1 inner radius of the seat inlet (with ring valve)

Rs2 outer radius of the seat inlet (with ring valve)

R,. inner radius of the ring valve Rv outer radius of the ring valve

Avs area of the valve/seat gap

cI> flow of water

Fv force acting on the valve

m/s2 kg/ms m2/s kg/m3

Pa

Pa

m m/s m/s N

Dimensionless symbols Cf = 1 i j 2 -Fv 2pVoAy Cp =

£r~-2PVo Jl =

£r~-

_2pV g Cc = coefficient of contraction Re = YaRd_v

1 Introduction

The Consultancy Services Wind Energy Developing Countries (CWO). an organization funded by the Ministry of Development cooperation. aims to help governments. institutes and private parties in the Third World in

their efforts to use wind energy and to promote the interest for wind energy in developing countries. CWO emphasises the use of windmills coupled to single acting piston (water)pumps.

One of the demands made upon these pumps is that they can be produced locally. Therefore the design must be kept as simple as possible.

Development of new prototypes is mainly done in the Nether lands. The development includes both theoretical and practical work.

Part of the practical work is this report of measurements of the static force acting on the piston valve. performed on valves of different type and size. Detailed measurements could lead to a better theoretical

understanding of the forces within the pump. A better design of the pump should always be based on this theoretical knowledge.

2.1

2 Theory

2.1 Disc valve

In this chapter a number of models for the flow-resistance of a disc shaPed valve will be described. The valve is part of a valve/seat

configuration:

fig.D1; valve/seat configuration

The models are valid for different areas of the lifting height h. Generally. the force acting on the valve is dependent on the decsease in pressure over the valve/seat configuration.

R

v

FV=(Pa-Pb)As+J (P(r)-pb)2Trdr

Rs

in which Per) is the pressure over the valve/seat overlap.

D1

Four different areas for the lifting height h can be distinghuished. Hence four different lIOdels will be developed.

1. The closed valve

fig.D2; the closed valve

2. The radial flow model

fig.D3; small lifting heights for the radial flow model

The resistance of the smallest passage is representative for the total resistance of the valve. In this area it is the valve/seat gap

(AV8=2JrRvh) .

This model can be subdivided into two other models namely:

2a. viscous model 2b. convection model

2.3

3. The model according to Lindner

fig.D4; larger lifting heights for Lindners model

Usually, this model is used for all lifting heights. Here it forms a sort of transitional model between the radial flow model and the axial flow model.

The average flow velocity in the valve/seat gap is still representative for the valve resistance.

4. The axial flow model

fIg.D5; even larger lIfting heIghts

for the axIal flow model

In this model the flow through the valve/duct gap

(Ad-A

v ) Is

With this strict division in four areas of lifting heights it should be

noted that the model for a specific area still has a certain influence in the adjacent area. The bounderies are not rigid.

The models, mentioned above. will now be worked out.

1. The closed valve

The top view of the seat is as follows

o

o

o

o

O

o

0

000

00

fig.D6: top view of a valve seat

For the sake of simplicity. the six holes in the valve seat. forming the inlet, are replaced by a ring of the same total area as that of the six holes combined.

fig.D7: simplified valve seat

In first order approximation p(r ) in (Dt) is supposed to be

proportional.

2.5

lim. condo P=P. for r=R.

This results in:

P(r)=Pb+(P.-Pb)[~:~Rv]

Substitution of (02) in (01) yields F_{v=T P.-Pb}( _{) [R}2_v+ 3R~Rv-R~-2R~]_{~ 3(R.-Rv)} 2. The radial flow model

D2

D3

This model will be used if hlRv«l holds. In this situation the flow

through the valve/seat gap plays an imPOrtant role for the valve

resistance. The components of the gap velocity

V

g are defined as follows:

I

yJ

u.

L-- - - : : ~

~

_{fig.DB; definition of the gap velocity}

~

cOmPOnents

In deriVing the model. starting points are the equations of continuity

and Navier-Stokes for incompressible flow. These will be applied to the

flow in the valve/seat gap. The equations will be given in cylindrical

coordinates. Note that the velocities are rotational synmetric (no 8-dependence) .

-eq. of continuity; v.v=O

aw

1

a

-a_z + -_r i=<a_r ru)=O uh w=O(R) -eq~ of Navier-Stokes; p(!.V!)=PS-vp+~2!

The eqs. of Navier-Stokes will be worked out in the r- and z-direction.

r-direction:

au

au

...!£E.

[1

a

au

u

a

2

u]

uar + w_az = _fA'~~r + v _?r'~a

I"'=-a

_r)--i-;:;:=2a_{r'" z} DS

[~:] [~:]

z-direction:

00

Since the terms of (00) are an order h/Rv smaller than the

corresponding terms of (DS) they can be neglected. This, combined wi th the assumption h/Rv«1, leads to significant simplification of the eqs. (04),

(DS) and (00):

aw

1a

-a

_z

+:i::<a

_{r r} ru) = 0

au

au

~

B

2 u

uar

+ w_Bz

= -

pBr + v_BZ2

~=

pg

The two submodels 2a and 2b are grounded on these simplified eqs.

D1

DB D9

2.7

2a. The viscous model

This model will be used i f Re«Rylh holds. The eqs. can be simplified

even more: 8w+.!£..J )=0

'liZ

ra?ru ~_ 82 u 8r -

Ttaz2

~=

pg

boundery condo u=O for z=O

z=h P=Pb for r=Ry P=P. for r=R.

From (011) follows with u=O for z=O. h

u

=

~~(z2-hz)

The total water flow through the valve/seat configuration is h

<P

= 2'nJudz o Substituting (013) in (014) yields ... - 2:!:. ~ h3 't' - 6T} 8r

And of course ~VoAd. hence with (015) follows

~ = _ &nVoAd = K1!

8r ~T r

Integrating (016) gives

boundery condo P=Pb for r=Ry

P=P. for r=R. This gives and 010 011 012 013 014 015 016 017

P.-Pb= Kiln (::] From (012) follows:

018

019 Equation 012 represents the (constant) force of gravity and the upwards force of the water which have to be substracted from the actual

measurements to obtain the force the flow is exering on the valve. Filling in 018 in 01 results in:

Fy

= ~K1

(Ay-A.)

=

3;~3Ad (Ay-A.)

On defining the force coefficient Cf as follows

we have. combining (021) and (022)

Cf

=

~.!l

(1 _ A.)

'l"h Va Ay

And defining the pressure drop coefficient as follows:

P. - Pb = Cp ~pV~ we have: D20 021 D22 D23

2.9

2b. The convection model

This model will be used i f Re)}Rv/h holds. The eqs. now become: Ow 10

11_uZ + ::i::{ ru)_rur = 0 024

au

au

.1.£E.

D25

uor + w_oz = pBr

~=~

~

Furthermore, the vorticity of the flow is supposed to equal zero: !.X!.=O. Hence

au

Ow

OZ

=

or

boundery condo u=O for r=R.

027

P=P. for r=R. P=Pb for r=Rv

Substi tuting (027) in (025) and integrating we have ~p(U2+~)+p= K2

Actually, K2 is Bernoulli's constant following from:

~pV~ + Pg = P. + ~pV~

in which Pg is the pressure in the gap. When we neglect ~pV~ in this equation we obtain 028.

We can rewrite this with (014) which here becomes ~= 2rrhu = VoAd

Hence, with w«u, (028) now can be written as

Filling in the boundery conditions in (029) we obtain

and

028

029

roo

The force acting on the valve can be found by substituting (030) and (031) in (01)

Fv

=

~~~AvA~(l+l~)

Ava Av

The force coefficient then becomes

Cf

=[~~J2(1+l~)

The pressure drop coefficient becomes:

3. The model according to Lindner

s

This model can be derived from the same eq . which go for the

convection model. However. Lindner gives for the pressure decrease an expression with

V

g instead of

Va.

032

D33

~pV2

P.-Pb = ~JL D34

Jl

where Jl is an emperical coefficient dependent on the ratio of the areas of

the valve/seat gap and the seat inlet: a=Ava/Aain. For Jl and a a relation

was found by Lindner

Jl = 0.00-0 . 2 Jl

=

0.-ta-

0.36 a<0.3 0.3<a<1.0 D35 D36 The relation was found from measurements performed on valves of different

2.11

f

-\

'"

...

-

...

....

r-

r-~

-, I 0 U III \2

,.

-fig. D9; Lindners curve

In the area for which Lindners model is used here. the valve/seat gap

is large enough to assume

P.

to be uniform. Hence

P(r)= Pll

With this assumption (01) simply becomes

With (034) the force acting on the valve becomes

Fv

=

~pX:Av

J.L

=

l£~~Av

[Ad ]2

J.L

A

v •

The force coefficient then will be

1[A ]2

Cf

=

i?

A:.

The pressure drop coefficient is the same as the force coefficient.

037

D38

4. The axial flow model

For lifting heights large enough the ratio Avs/(Ad-A v). becomes larger than unity. In other words: the valve/seat gap is no longer the smallest passage. The flow in the valve/seat gap now becomes more axial. Now

(Ad-Av) is representative for the valve resistance.

---,

~/c'"

~~

fig.DI0: axial flow for large lifting heights

The flow through (ArAv) will be a contracted flow. For the contracted flow a contraction coefficient is defined(see App. A).

M

A d-v= Cc:Ad-v

M s

where A d-v is the actual area for the flow. In the following eq. the quantities with an asterisk are related to the contracted flow.

It is assumed that the vorticity anywhere in the flow is zero. The D40

2.13

eq. of continuity then follows(see fig.DIO):

Bernoulli's eq. gives:

041

Combining (041) and (042) Vi

*

can be eliminated:

P'.-Pb

=

~Pv~[[

..

~:_J2-I]

=

~pV~ (["~:_v]2

-1]

043

fig.DII; the momentum eq. is applied to the dashed line

The momentum eq. gives(see fig. 11):

2

-p·.Ad

+

P'bAd -

pAdV~

+

pA*d-vV~

=

-Fv

Combined with (041) the force acting on the valve becomes:

F

v

=

(p·.-p·b)Ad

+

PAdV~[I~:_v]

On filling in (043) we obtain:

1 2

(Ad

]2

F

v

₌

_{2PVolCcAd_v}

_-1

_Ad

The force coefficient then will be:

~

)

2

Cf -

Ad

-1

Ad

-

CcAd-v

Av

The pressure drop coefficient becomes:

045

2.2 Ring valve

The theory thus far is intended for disc shaped valves. However.

measurements were performed on another type of valve. too; the ring valve. The valve/seat configuration is slightly altered.

~_ _R_v-4t / "

fig.Rl; ring valve/seat configuration

As with the disc valve. the models for the ring valve. too are valid for certain areas of lifting height h. The general expression for the force acting on the valve. however changes with the valve/seat configuration:

R.l

R

v

FV=(Pa-Pb)T(R~2-R~1)+J

(P(rll-pa)2rrdr+J (P(rl2-Pa)2rrdr Rl

R

r

R.2

where P(rli is the pressure over the valve/seat overlap. Three areas of lifting height are distinguished. Hence. three models will be developed.

2.15

1. The closed valve

fig.R2: the closed valve

2. The model according to Lindner

fig.R3: all kinds of lifting heights for Lindners model

Opposed to the use of Lindners model with disc shaped valves

(transition model). Lindners model with ring valves is used in a more general manner.

3. The axial flow model .e2:I::=:::Ee:a3 ./" / ' / ' / ' / '

/

./" , / '

fig.R4; very large lifting heights for the axial flow model

In the model the flow through the valve/duct gap + the inside area of the valve

(Ad-A

v ) is representative for the resistance of the valve.

The models. mentioned above. will now be worked out.

1. The closed valve

The top view of the ring valve seat is as follows

fig.RS; top view of the ring valve seat

For the sake of simplicity the four holes in the seat are replaced by a ring of the same total area as that of the four holes combined.

2.t7

fig.R6; simplified ring valve seat

In first order approximation Plr)l and Plr)2 in (Rt) are supposed to be proportional to r. boundery condo Pl Rr) 1 = Pb PlR51)1

=

Pa i=1,2 R3 R2

The derivation is analogous to that of (D2). Hence:

Plr)l

=

Pb +

(pa-Pb)fR:~~RrJ

r-Ry

Plr)2

=

Pb + (Pa-Pb) R -R₅₂

y

On

substituting (R2) and (R3) in (Rt). we obtain the force acting on the

valve.

2. The model according to Lindner

The expression which Lindner gives for the pressure decrease over the valve/seat configuration (034) is valid for the ring valve. too. This is because of the general empirical coefficient JJ.. The dependence on a is.

- 0

Sa-

0 • 2

J.L - •

J.L

=

0.4<%-0.36

a<O.3 0.3<a<1.0

The derivation of the force acting on the valve is analogous to that of (038). Hence:

Fv =

~pV~ ~ ~:J2Av

The force coefficient then will be

1 [A

]2

Cf =

j?

LA:.

The pressure drop coefficient is the same as the force coefficcient.

The difference with the disc shaped valve is to be found in the

R5

R6

expression for Avs in the area ratio a. For the disc valve the expression

for a is as follows:

~

\\\\\\\\\\\\\i

I I I , I I I I ~

I

~

~~~

~

fig.R7: area of valve/seat gap with a disc shaped valve

21rRvh 2hRv

a

=

~---n2-

=

7ii2---n2-D

T(R

s2-R.l) (R.2-R.l)

2.19

~~

I

~

I

I

I : :

I

I I I t I I

I

&C~~~

fig.RS; area of valve/seat gap with a ring shaped valve

3. The axial flow model

For lifting heigths large enough the ratio

Avs/(Ad-Av)

becomes larger than unity. In other words: the valve/seat gap is no longer the smaller passage. The flow in the valve/seat gap now becomes more axial. Now

Ad-Av

is representative for the valve resistance. The derivation of the force acting on the valve is analogous to that of (016). Hence:

1 2

r

Ad

]2

Fv

=

2PVOLCc(Ad-Av) - 1 Ad

The force coefficient then becomes:

Ad

r

Ad

)2

Cf

=

Av LCc(Ad-Av) - 1

The pressure drop then becomes:

R7

2.3 Valve ahead of the seat

The theory as i t is presented thus far can only be applied to the

various valve/seat configurations when the flow first passes the seats. In real pumps, however, i t is also possible that the flow first passes the valve. This occurs when the piston is moving up from its lowest position and the valve is not yet fully closed. The water above the piston-valve then flows back. In our construction we only had a flow in one direction. We could imitate the two flow directions by placing the valves ahead of or behind the seats.

It therefore is necessary to develop a theory for this back-flow. But the derivation of the formulas is very much analogue to the derivation of the corresponding formulas for the other flow direction as it is given thus far. Hence, we confine ourselves giving only the resulting formulas (which will, as you can see, quite resemble the other formulas). For a derivation I'd like to refer to the report of S. Orbons (nr. R ~99 S). Disc valve, model nr. 1.. F_v

₌

( ) [R2

3R~R.-R~-2R~]

T

P.-Pb

• -

3(R.-Rv)

2a: The expressions for Cp and Cf remain the same

2b: Cf

₌

_~h2AvA~ A2

C -

d

p - :t?~ii2

[1

+

In[~:]]

~: The expressions for Cf and Cp remain the same.

Ring valve. model nr. 1:

F_v

₌

(

_P.-Pb

_{) [RT} 2_{.2-.1 -}R2 3R~R.1-R~1-2R~_3(R.1-R 3R:2Rv-2R~-R~2]

r ) - 3(Rv-R.2)

3.1

3 Experimental design

3.1 Construction

For a proper visualization of the construction on which the

measurements were performed. see fig. El (drawing by Jan Diepens). Except for the top part of the pumprod and the bolts which hold the upper and

lower part of the duct together. everything. valves and seats included. is made of brass. This material is both easy to work with and will not rust.

The construction had to be made such. that it would be easy to change swiftly from one set of valve/seat to another. drawings of the five

different valve/seat configurations are in AppendiX C. therefore. the duct consisted of two parts. held together by eight bolts. Between those two parts the various seats could be fitted. To prevent the pwnprod from causing unnecessary friction by going through the seat. a small pipe was made to fit between the worm of the pumprod and the seat. Because the holes in the center of the valves were made smaller than the holes in the center of the seats (M12 resp. M16) the valves could be fixed to the pumprod. clamPed between a nut and the small pipe (see fig. E2).

(})---

-

@t o L @t -111---""1 jTAtIifUlPW $I-IE SWIjIOP5!lF51-« I~'1rT"FR-- _ Willi! mU ..a~--+- 1'lJlR'" _B!lUT'''''';0 ••.!!!I!..!!li-"-~

1

~-

--. -

f

~

3.2

valve

I:-...~...

fig.£2; detail of the valve/seat configuration

At the top of the construction, the upper, thin part of the pumprod goes through the top plate of the duct. Of course, water leaks from the hole in it. This leakage could be reduced, but the hole had to be large enough, not causing friction on the pumprod. This would affect the measurements. The part upon the top plate of the duct was made to carry off the leakage water. The dynamometer was stationed between two parts of the pumprod of which the upper part has screwthread on it. The valve-seat distance (i.e. the lifting height) was adjustable by screwing the nuts up or down. The accuracy was that large that the 11miting factor was the accuracy of the marking gauge we used (i.e. 0.05 DB.). Only for very small lifting heights (tenths of DB.) the effect the strain has on the pumprod is of the same order of magnitude as the lifting height itself (in fact, this causes the vibrations of the construction; see chapter 6)

The change of length can be computed with the elasticity aodulus E. (for brass

E

is 10.1010 _{Pa. for steel 20.10}10 _{Pal Taking the force on the} pumprod to be approximately 600 N in this area. the change of length w11l be 0.1 DID.

3.2 Scaling

Because we had to perform measurements on five different valve/seat configurations. used in pumps of various dimensions. the original

valve/seat configurations had to be scaled to the test rig we used (i.e.

In order to create the same boundery conditions as there are in the original pumps the Reynold numbers had to be the same. Hence:

Re or

=

Remodel [VaRd]

=

[VaRd]

V or V model

Rd . or Va.mo Rd.mo

=

V

a . or

The Reynold numbers mentioned in the tables (App. B) correspond to the flow which was measured with the valve at very large lifting height.

3.4

3.3 Measuring methods

Three different variables were measured as a function of the lifting height. Namely the force acting on the valve. the pressure drop over the valve/seat configuration and the flow through the valve/seat gap.

1. The force measurements

The force acting on the valve was measured by a dYnamometer (E8M 6202) at the top of the construction. The E8M 6202 was connected to a bridge amplifier (CA 3(0). The output from the amplifier was measured by a voltage gauge. For the output to have any meaning the E8M 6202 was

calibrated first. This was carried out with the E8M 6202 already huH t in into the construction. thus taking into account small deviations due to material stresses and static friction. The calibration was performed with a number of weights. Each of them having a weight of 100 N. The results can be found in graph 1. Because graph 1 is a straight line. the relation force-voltage can also be expressed by the following formula:

F=a.V+b

with the parameters a and b yet to be determined. They can be calculated with the least of squares method. It results in:

F=289.0.V+l+4.8

Taking into account the accuracy of the dYJWDOllleter and the amplifier and

2. The pressure measurements

Initially it was intended to measure the pressure drop with a pressure difference gauge. Because the only thing we are interested in is the

pressure drop over the valve/seat configuration. not the absolute value of the pressure ahead of or behind the valve/seat configuration. that would be sufficient. But the output given by our pressure diference gauge turned out to be dependent on the absolute value of the pressure. Therefore. we had to turn to another method of measuring. We now had to use two pressure gauges. read them out and substract the two values (of course. taking into account the pressure drop caused by the distance between the two

connections on the duct (pgAh). see fig. E1). This introduces another error. but there was no alternative. After one day of measuring. however. one of the pressure gauges broke down. This was due to vibrations which occured when the valve was ahead of the seat, in the viscous area. We then had only one solut'ion left: we had to measure with one pressure gauge connected to both measuring points. We could read out the two values seperately because the leads connecting the pressure gauge to the measuring points could be shut off by taps. Advantage of this way of measuring was that we didn't need to make corrections for the height difference of the measuring points, nor for the offset of the amplifier.

3.6

Another difficulty in measuring the pressure difference was the distance from the seat to the connections for the pressure gauges. At first this was 110 mm. One valve however. had a thickness of 46.5 mm. Considering we had to make measurements up to a lifting height of 30 mm.

it is obvious that this valve would certainly affect the flow around the connecting points. The velocity there will increase and hence the pressure decrease. The pressure drop then measured will not be totally due to the valve/seat obstruction. The only solution was to make other connections. further away from the seat. Now the distance connection-seat is 330 mm. This is large enough to assume that the flow there will not be affected by

the valve/seat configuration. On the other hand it is still far enough away from the water inlet at the top of the duct to consider the pressure distribution uniform. The pressure gauge was attached to a multi-channel amplifier. The amplifying factor was adjusted such that the readout would

. be in bars (in fact. of course. these are volts). Hence no calibration was necessary.

3. The flow measurements

In order to measure the flow of water passing the valve/seat gap we had a prograDIDable flow gauge at our disposal (IItI 6531). It was positioned at the end of the duct. After progranning it well. the indication on the flow gauge was a percentage of a programmmed maximum value (in our case 300

l/min.). Another option was a readout in amperes. For practical reasons we took the second option. Therefore a calibration was required. Only to make sure that the output of current was linear with the percentage readout. This was indeed so. as graph 2 shows. In the same way as we did with the

force-voltage relation we now can calculate a current-percentage of flow relation:

<I> = a.1 + b

Because the flow is nothing less than the mean velocity times the cross sectional area of the duct it follows directly that:

Vo

=

1/<41.163

With I in milli-Amperes and Vo in metres per second. The flow gauge itself was accurate up to IX of the full scale. so the limiting factor were the readout mistakes. Hence. Vo was accurate up to 0.05 m/s. One difficulty we had to conquer; the flow gauge works with magnetic fields. Since the

magnetic permeability of water differs from that of air. the output is sensitive to the presence of air bubbles in the flow let alone a flow not big enough to fill the whole duct. But we had to measure the flow even at very small lifting heights. that is. very small flows. Therefore. behind the flow gauge we made a sort of goose neck in order to keep the flow gauge filled with water. Since the danger of air bubbles in the flow

appears only after the flow has passed the (very small) valve/seat gap.

the top of the goose neck had to be at a higher level than the valve/seat

configuration. Also behind the flow gauge we put a tap to control and

3.B

This was done in order to make measurements at different flows (see fig.

_ _ l -. i ":

~-D'

n

~

-.

-~' , f

I

'

} I I

~\..

I

i

~

£3).

fig. £3; with a goose neck the flow measurements are accurate even at zero flow

4 Results

4.1 Resistance of the valve

For general purposes we don't deal with the actual force on the valve but with a dimensionless force Cf, defined by:

Fv

=

Cf

*

~pV2Av

Because the measurements were performed on scaled models of the pumps the logical thing to do is to rescale the height as measured to its true

values. Another possibility, however, is to make the height independent of the dimensions of the model. The second option has certain advantages. First: in rescaling measurements for the smaller pumps (CWO 49, 81, 108) the true height would be smaller than the height we measured. For very small lifting heights problems arise because a flow would be measured at a lifting height, too small for water to pass through (because of its

viscosity). Second: in making the height dimensionless the results for the different pumps are easier to compare. Therefore the graphs show Cf and Cp

as a function of hlRv •

Before discussing the results it should

be

noted that for all the valve/seat configurations there is an area of lifting height. O<h<l 1IIIl.,

4.2

This was due to vibrations of the construction which occured in that specific area.

First of all we investigated the influence of Reynolds number on the

coefficient of resistance,

Cf.

Therefore we measured

Cf

for the CWO 49

pump at two different flows. As graph no. 3 shows, the two lines for either flow are quite the same. Only for very small lifting height the difference is up to 30%. For the practical area of interest (i.e. 2<h<10 mm.) the difference is always less than lOX. Hence for all the other Pumps we could do with measurements at just one flow.

In the even Roman numbered graphs the results for the valve resistance are presented. From the graphs we can conclude that for large lifting

heights

Cf

is inclined to go to a constant value. Variations in the

lifting height have no effect on the

Cf

value. For the two smallest Pumps,

however, we couldn't reach large enough lifting heights to observe this

constant value. Still we expect the

Cf.

analogue to the other pumps, to go

to a constant value. For very small lifting heights (i.e. h<2 mm.)

Cf

is

growing very rapidly (in fact.

Cf=

m for a closed valve).

It is also remarkable to notice that it doesn't make a difference for

Cf

whether the flow first passes the valve or the seat. For the CWO 49 and

81 the configuration with the valve behind the seat has slightly lower

Cf

values. For the CWO 108 and 161 it is just the other way around. The only

lifting height is always higher when the valve is behind the seat.

4.2 Pressure drop

In the same way we made the force on the valve dimensionless we make the pressure drop dimensionless:

.. C

*

21pV2

ap

=

p

The pressure drop was measured over the whole valve/seat configuration. Hence. Cp can be interpreted as the the resistance of the valve/seat

configuration. So. with

Cf.

we can write down an expression for the resistance of the seat only:

(Pa - Pb)Ad

=

Fvatve

+

F•••

t

P. - Db

F

v +

F.

'''="2~_2PVO

-

_-

""="2-

_2PVoAd

Cp

=

C.

+

Cf Av

Ad

in which

C.

is the resistance coefficient for the seat. It is easy to see that. comparing the corresponding functions for

C

p and

Cf.

the share of

resistance of the seat in the total resistance is decreasing with larger lifting heights. Thus for smaller lifting heights the seat becomes more important for the total resistance. Even better: for very small lifting height the combination determines the total resistance.

4.4

C

p -

Cf~~

=

C

s = f(h)

In which f(h) is representative for the resistance of the seat only. The results are in the odd Roman numbered graphs.

4.3 disc valve compared with ring valve

Looking at the curves for the different pumps we see that the general features are the same. But. in taking a closer look. we see distinct differences. In the practical important area (4.10-2< hlRv <2.10-1) the

Cf

and Cp values of the ring valve are always lower than the corresponding

values of the disc valves. The difference is up to 30% (for hlRv=6.10-2) and slowly diminishing with growing lifting height as at hlRv=2.10-1 the

values match. For very large lifting heights the

Cf

and

C

p value is still

20% lower than that of most pumps with disc valve. Exception is the CWO 81 which has for very large lifting heights a

Cf

value much lower than all the other pumps. (for large lifting heights it does make a difference what the configuration is. So only compare curves of the same valve/seat

As we already said. the difference between the configurations for the

Cf and Cp functions is very small. We can therefore compute a function

which is valid for both cases. We do this for the practical imPOrtant lifting heights between 2 and 10 Dm. (4.10-2< hIRv <2.10-1) by taking

averages of the functions computed in App. B. The agreement with the

average functions is within 10%. Except for the CWO 49 for which the Cf

value for the valve ahead of the seat is up to three times as big as the

Cf value for the valve behind the seat.

CWO 49:

Cf

=

3.769 (h/Ryr 1 .538 (only valid for h>4 mm.)

CWO 81:

Cf

=

4.098 (h/Ry)-1.434

CWO 108:

Cf

=

8.090 (h/Ry)-1.115

CWO 161:

Cf

=

7.490 (h/Ryr 1 . 158

CWO 265:

Cf

=

1.115 (h/Ry)-1.723

The two smallest pumps have a

Cf

function that is steeper then those of

the CWO 108 and 161. Concerning this one might try to make two different

functions describing the Cf behaviour for the

om

49 and 81 on the one

4.5

confine ourselves to make just one function for all the (disc valve) pumps. The result is:

Cf

=

6.6

(hlR

_v)-1.236

The agreement with this function is within 15%. The CWO 49 only for h>4mm. We now do the same with the Cp functions.

CWO 49:

C

p

=

3.345

(hlR

v)-1.671

CWO 81:

C

p

=

6.435

(hlR

v)-1.462

CWO 108:

C

p

=

10.730

(hlR

v)-1.166

CWO 161:

C

p

=

9.890

(hlR

v)-1.161

CWO 265:

C

p

=

1.795

(hlR

v)-1.651

We see the same global differences between the smallest pumps and the other two as with the

Cf

functions. We again can make one function that is valid for all (disc valve) pumps:

5 Theory compared with experiments

In this chapter we will check the different models for Cf and Cp as

described in the chapter Theory. For the disc valve three models were derived. completed with Lindners model. For the ring valve only two models were derived. Just for the closed valve and the constant value for very

large lifting heights; for all lifting heights in between Lindners model was used.

5.1 The closed valve

Only when the valve is ahead of the seat we can check this model. For when the valve is behind the seat we had to screw the valve against the seat (as tight as possible) in order to prevent leakage. Hence. the force then measured is not totally due to the watercolumn. The ring valve still had a small leakge at zero lifting height because of its construction. The value of the force given in the table was measured at zero lifting height and a flow less than IX of the full flow. The models for the configuration with the valve ahead of the seat show very good. to bad agreement.

5.2

theory(N) expS(N) diff. (X)

CWO 49 ₅₀₁ 605 21

81 509 480 6

108 468 517 10

161 605

54i

11

265 310 37 88

table 1: forces on the closed valves

5.2 The viscous model

The viscous model could not properly be checked because of lack of measurements. When the valve was ahead of the seat vibrations occured in the viscous area. When the valve was behind the seat there wasn't a flow in the viscous area. We only had a watercolumn of 10 metres at our

disposal (this is 1.105 NIM2) , apparently not enough to squeeze the water

through the very small valve/seat gap. From the scarce results we can only conclude that the order of magnitude is well computed by the viscous

model.

5.3 Convective model. Lindners model

because it turned out that the configuration had no influence on the Cf

and Cp values. as in Lindners model is. too. However. the outcome of

Lindners model is much higher than the actual results: up to four times too big. eight times for the ring valve. Only for very small and very large lifting heights the values given by Lindners model match with the experiments.

The convective model does make a difference betweenthe two configurations. The best agreement. however. is obtained with the convective model. valve ahead of the seat. The convective model. valve behind the seat gives results which are too low. The convective model. valve ahead of the seat should only be used for small lifting heights

(i.e. 2.10-2< hlRv <5.10-2) . The agreement here is within 10%. For the

area of lifting heights from hlRv

=

5.10-2 to where Cp and Cf become

constant there is really not a good model. For a good view on the area in which the convective model can be used see graphs I to X.

5.4

5.4. The axial flow model

This model gives the Cp and Cf values in the area of lifting heights

where they have become constant. The model doesn't make a difference

between the two configurations. A good comparison with the Cp values is

not possible because for large lifting heigths the pressure drop becomes very small (0.02.105 N1M2) and taking into account the instability of the

pressure drop readout (0.01.105 _NIM2_{). we must conclude that the}

measurements in this area are not accurate enough for good comparison with

the model. Fortunately in this area of lifting heights the values for Cp

and Cf do not differ much. So, in first order approximation we can take Cf

and Cp the same.

theory ahead diff(X) behind diff(X)

CWO 49 24.0 28.9 20 35.8 49

81 18.9 11.6 39 16.6 12

108 45.4 31.3 18 41.2 9

161 44.8 42.4 5 41.8 1

265 10.3 9.5 8 29.1 183

table 2; Cf values for very large lifting heights

well resembled by the model. However. the exact values do not always show good agreement. From the large difference with the CWO 265 we must

conclude that the axial model is not fit to compute the Cf value for very large lifting heights of the ring valve. when the valve is behind the seat. In graphs I to X the values for the axial model are given.

6.1

6 Discussion

In the chapter concerning the results of the valve it was pointed out that in a certain area of lifting height vibrations occurred. which would make measuring impossible.

The cause of these vibrations is in fact the finite value of stiffness of the pumprod. Of course when a force is exerted on the rod its length will change. In the case that the valve is ahead of the seat the pumprod will exert a force on the closed valve pointed upwards. The force of the watercolumn on the valve (and hence on the rod) is pointed downwards. When

the valve is now lifted from the seat. just enough to have flow. this flow exerts an extra force on the valve. causing the pumprod to become longer and hence reclosing the valve again. Then the extra force of the flow drops out. The pumprod pulls the valve open again causing the water to flow which exerts again the extra force etc .• etc .•

When the valve is behind the seat the vibrations cannot occur because the force of the pumprod on the valve as well as the force of the flow are pointed downwards. In other words: when the pumprod pushes the valve open. the flow tends to make the valve/seat gap even larger.

The conclusion from this DI1st be that the force on the valve has a maximum in the viscous area before its drops off to a constant value for

Deviations from the theory can be explained from the several

conditions that are supposed in order to derive the formulas. Such as there are: laminair flow. uniform pressure distribution before and behind the valve. zero vorticity of the flow. neglection of viscous sublayers (except of course in the viscous model). linear pressure drop over the overlap of valve and seat. We also worked with a simplified seat. The theory as it is now is not yet fully developed. The seperate models are only valid in a certain area. links between them by transition models would certainly improve the agreement with experiment.

1.1

1 Conclusion

From the experiments done on the different types of valves. several conclusions can be drawn. First of all it is clear that no appropiate model is available for the ring valve. In contrast to the disc valves for which we have a set of models wi th which the Cf and Cp values can be

computed. See graph I to X.

Also is shown that the ring valve has much lower Cf and Cp values than

the disc valves. When there are no construction difficulties the ring valve should always be used.

The models (except Lindners model) give different values for the Cp and

Cf dependent on the valve/seat configuration. It turned out that the

difference in the two configurations is negligable. Only for very large lifting heigths the configuration with the valve behind the seat has a higher Cf and Cp value.

All measurements were performed with only one flow. From a few

measurements it was concluded that Cf and Cp were not flow-dependent.

However. it is reconmended to look upon this Reynolds dependency Dlch closer to get a better view on it.

The programs that were written to compute approximations for Cp and Cf

and to compute the values of these coefficients for the different models are presented in App. C. Although all the programs do work it is usefull

Literature

1. A.J. Ward-Smith. Internal fluid flow. 1980

2. S. Orbons. De stationaire weerstandskracht van een klep in een

zuigerPOmp; Stageverslag

TU

Eindhoven vakgroep Transportfysica afd.

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'j::-=::=-:4-

OCRA:1

~

I':d8

:~-~~

••

::~:

-: : . 1

I..

i I ~ \ i .,)( valve behind the seat

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I

2 I 4. 5 6 78910J I l !

valve behind the seat convection model (valve ahead of the seat) Lindners model approxima tion 3

,

~ -~ )( --, ~-~-===~t~;:L.

:~~=F~'--

:

~ - -

-..

.:: =1-: -- .

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1 I 1 , I

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"t"l'

I'

I

.

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.1

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-f--- -1--- -1--- -1---

--

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convection model (valve ahead of the seat)

Lindners model

.-

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I

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i I

I .

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i

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j

1

I

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--~----:-~t-~

-

'~-~l=

-

~

=r

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108:

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i

1

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- - -.

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~-~==

:

valve ahead of the seat

-1=-~-+-~:-

' T : ' -

·--,,:-:x

valve behind the seat

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~

::.-:-.:.

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-I

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1

I

-I

j .

I,

1 t-l---+--~ f--I--+'-++-il-+----+-.:.~ I -, . I 1: I()i. , I

- -_.-- - - -valve behind the seat

- -j-!----\1---t--'r-'+---l-+-t-1-H1'_ _-_.::.7;·_·-'ft---

_+_..._--1+_.-++-+-++_--'_-:' )(

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1-- - i=f -'-~r--

3 4 4 5 6 7 I , j L !~~ i _i _.-. , '_.._-,-- .-__~ 1 _ _ ---+---- : • ~-- - _.! ----~

valve behind the seat convection model (valve ahead of the seat) Lindners model approximation -~-~-~ - f - - - f I ----1 ~- . t· T 5 6 7 I 4 3 -~-- -- 1--2 5

6

7 8910' 3 -1 -1-1-1-1 t --r 1-1Q ; ~.) 8 7 --- - ---- - ---

Appendix A

At Reynolds numbers above 4.103

• which is roughly correct for the flow

between the valve and the duct. a flow. passing an orifice. has the

following general features. As the flow passes through an orifice plate it separates away from the surface of the orifice to form a discrete jet. which contracts to a minimum cross-sectional area Ave some way downstream of the plate. The station at which this occurs is known as the vena

contracta. The coefficient of contraction.

C

e • is defined as follows:

where Ao is the cross-sectional area of the orifice.

v~//rl~//

_ _ _ _---.;;...~ I Al

/ /

/

f

2 fig. Al flow .3 through an orifice

/

-+JI

for Re>4.I03

We will now make a JDOdel of the flow through the orifice in order to derive an expression for the coefficient of contraction Ce • We assume

A2

1. the flow is incompressible. hence p is constant

2. between stations 1 and 3 (see fig. AI) the flow in the jet is inviscid 3. the contribution to the momentum balance due to shear stresses at the wall and due to the seperated flow region at staion 3 is negligible 4. the pressure p is constant over the cross section at station 3 The energy eq. then becomes:

P1+~pV~=P2j+~pV~j

=P3j+~pV~j A2

A3

where V3j is the mean velocity in the jet at station 3. (which is. of course. not the same as the mean velocity at staion 3)

The continuity principle applied to the through-flow yields:

PV1=APV2J~PV3J=PV4=ptd

where A is the ratio of the cross-sectional area of the orifice (Ao ) and

the cross-sectional area of the approach duct (Ad). , is the total water flow.

The momentum balance between station 3 and 4 is:

A4

AS Combining (A2).(A3) and (A4) results in:

,~

=

(~]2

Defining the pressure drop coefficient. K. for the orifice plate by the relation:

A6 on substituting (A6) in (AS). we have:

A7 Ward-Smith (1971) used measurements of the pressure drop coefficient K

to deduce the variation of Cc with A for orifices with square edges. He

found the following relation:

O · 8 r - - - , - - - r - - r -...- - r----..----r-...-...,---,

••

0-50

02 0-4 06 o-e to

) A

fig. A2. dependence of Cc on the area ratio A

81

Appendix 8

table 1. CWO 49; valve behind the seat; Re

=

2.3.104

H/Rv 0 co 2.21.10-3 co 4.50.10-3 co 5.31.10-3 3.44.104 9.55.10-3 2.62.104 1.29.10-2 5.81.103 1.59.10-2 4.23.103 1.12.10-2 2.90.103 1.83.10-2 2.23.103 2.13.10-2 1.85.103 2.39.10-2 1.11.103 2.84.10-2 8.11.102 3.01.10-2 6.81.102 3.46.10-2 5.21.102 3.85.10-2 3.85.102 4.30.10-2 2.93.102 4.88.10-2 2.45.102 5.23.10-2 1.96.102 5.59.10-2 1.12.102 6.28.10-2 1.50.102

6.65.10-2 _1.36.102 7.42.10-2 1.31.102 8.58.10-2 _1.18.102 9.94.10-2 _9.83.101 1.23.10-1 6.89.101 1.58.10-1 5.33.101 3.08.10-1 3.03.101 5.59.10-1 2.55.101

B3

table 2. CWO 49; valve behind the seat; Re

=

1.1.104

HlRv Ap(.105 Pal F(N) Cr 0 1.11 1609 110 110 2.15.10-3 _{1.11 1102 110 110} 4.30.10-3 1.11 607.9 110 110 5.37.10-3 1.08 546.8 3.68.105 4.96.105 9.67.10-3 0.96 453.8 3.40.104 4.95.104 1.18.10-2 0.83 386.2 1.41.104 2.07.104 1.50.10-2 _0.70 323.0 5.40.103 8.03.103 1.72.10-2 0.63 286.0 4.20.103 6.26.103 2.04.10-2 0.48 217.5 2.60.103 3.93.103 2.47.10-2 0.29 131.7 1.34.103 2.01.103 2.90.10-2 0.20 90.0 8.61.102 1.30.103 3.22.10-2 0.15 11.2 6.63.102 9.50.102 3.76.10-2 _0.11 55.1 5.03.102 6.83.102 4.19.10-2 0.09 44.7 4.01.102 5.49.102 4.40.10-2 0.08 38.7 3.48.102 4.83.102 4.94.10-2 0.06 31.9 2.91.102 3.60.102 6.17.10-2 0.05 28.0 2.45.102 2.98.1~ 8.06.10-2 _{0.03 18.5 1.60.10}2 _1.16.102 9.17 .10-2 _0.01 13.4 1.15.102 5.80.101 1.24.10-1 ₀ 10.4 8.85.101 0 2.13.10-1 _{0 5.9 5.02.10}1 ₀ 6.28.10-1 0 4.2 3.58.101 0

table 3. CWO 49; valve ahead of the seat; Re

=

2.0.104 HlRv Ap(.105 Pal F(N) Cr 0 1.03 605.0 CIO CIO 3.22.10-3 1.03 469.3 CIO CIO vibrations 2.58.10-2 1.07 623.4 1.14.104 1.33.104 3.01.10-2 0.90 512.2 4.27.103 5.11.103 3.44.10-2 0.81 424.6 1. 74.103 2.26.103 3.97.10-2 0.59 333.4 1.19.103 1.43.103 4.51.10-2 0.48 273.2 8.56.102 1.02.103 5.37.10-2 0.34 197.3 5.34.102 6.27.102 6.12.10-2 0.26 146.6 3.70.102 4.47.102 6.55.10-2 0.21 122.2 3.00.102 3.51.102 8.16.10-2 0.16 83.7 1.96.102 2.55.102 9.45.10-2 _{0.12 64.6 1.50.10}2 _1.89.102 1.08.10-1 0.10 51.5 1.18.102 1.55.102 1.25.10-1 _0.08 42.6 9.65.101 1.23.102 1.45.10-1 0.06 33.4 7.48.101 9.15.101 1.65.10-1 _{0.05 27.7 6.18.10}1 _7.59.101 1.97.10-1 0.04 22.3 4.95.101 6.03.101 2.24.10-1 _{0.04 19.1 4.22.10}1 _6.03.101 2.82.10-1 _{0.04 15.2 3.40.10}1 _6.08.101 3.39.10-1 _0.02 12.5 2.89.101 3.14.101

B5

table 4. CWO 81; valve behind the seat; Re

=

2.1.104

H/Rv Ap(.105 Pal F(N) 0 1.01 1481 CD CD 1. 11.10-3 _1.01 760.4 CD CD 2.22.10-3 _{1.02 554.8 2.95.10}9 _3.46.109 8.88.10-3 _0.95 457.1 2.47.104 3.27.104 1.22.10-2 1.01 468.4 6.92.103 9.53.103 1.44.10-2 0.92 421.9 3.98.103 5.54.103 2.22.10-2 _{0.74 332.8 1.49.10}3 _2.11.103 2.77.10-2 _{0.54 242.6 8.16.10}2 _1.16.103 3.55.10-2 0.42 184.7 5.35.102 7.77.102 4.22.10-2 0.30 132.0 3.50.102 5.07.102 5.22.10-2 0.22 96.8 2.41.102 3.50.102 6.33.10-2 0.16 70.9 1. 70.102 2.44.102 9.21.10-2 _{0.11 50.4 1.17.10}2 1.63.102 1.04.10-1 0.09 40.5 9.32.101 1.32.102 1.30.10-1 0.06 27.4 6.18.101 8.62.101 1.58.10-1 0.05 20.6 4.61.101 7.14.101 1.83.10-1 0.03 17.3 3.85.101 4.26.101 2.14.10-1 0.02 14.9 3.33.101 2.85.101 2.84.10-1 _{0.01 12.5 2.78.10}1 _1.42.101 5.23.10-1 _0.01 9.2 2.03.101 1.41.101 8.51.10-1 _0.01 7.5 1.66.101 1.41.101

table 5. CWO 81; valve ahead of the seat; Re

=

2.1.104 H/Rv Ap(.105 Pal F(N) Cr 0 1.03 479.8 GO GO vibrations 8.88.10-3 1.09 565.3 1.20.104 1.48.104 1.55.10-2 0.88 434.2 3.48.103 4.50.103 2.33.10-2 0.72 357.0 1.58.103 2.04.103 3.11.10-2 0.55 258.9 8.84.102 1.20.103 3.88.10-2 _0.38 176.4 5.08.102 6.99.102 4.66.10-2 _0.29 140.4 3.82.102 5.04.102 6.22.10-2 _0.22 96.2 2.46.102 3.59.102 7.66.10-2 0.16 73.3 1.82.102 2.53.102 9.77.10-2 0.12 52.4 1.26.102 1.85.102 1.22.10-1 0.09 39.0 9.26.101 1.36.102 1.49.10-1 0.07 29.5 6.91.101 1.05.103 1.76.10-1 _0.06 23.8 5.55.101 8.90.101 2.12.10-1 _0.05 19.4 4.48.101 7.37.101 2.24.10-1 _0.05 18.2 4.20.101 7.37.101 2.55.10-1 _0.04 15.5 3.57.101 5.88.101 3.25.10-1 _0.04 11.6 2.66.101 5.85.101 3.91.10-1 _{0.02 9.5 2.17 .10}1 _2.91.101 6.97.10-1 0.01 5.1 1.16.101 1.45.101

87

table 6. CWO 108: valve behind the seat; Re

=

1.9.104

H/Rv Ap(.105 Pal F(N) CF 0 1.04 1603 GO GO 5.04.10-3 1.04 691.0 GO GO 5.04.10-3 1.04 598.6 6.84.105 9.19.105 1.11.10-2 0.96 476.5 1.31.104 2.05.104 1. 71.10-2 _0.86 409.1 3.14.103 5.11.103 2.52.10-2 0.62 279.1 1.03.103 1.76.103 3.93.10-2 0.29 130.8 3.59.102 6.15.102 5.44.10-2 0.19 93.6 2.42.102 3.79.102 6.75.10-2 0.14 76.6 1.93.102 2.72.102 7.96.10-2 0.11 60.2 1.48.102 2.09.102 1.07.10-1 0.08 42.9 1.03.102 1.48.102 1.33.10-1 _0.06 32.5 1.69.101 1.10.102 1.59.10-1 0.05 26.5 6.24.101 9.10.101 1.88.10-1 0.04 24.1 5.68.101 1.28.101 2.40.10-1 0.03 22.3 5.26.101 5.46.101 3.86.10-1 0.02 21.2 4.97.101 3.63.101 6.80.10-1 0.02 17.6 4.12.101 3.62.101

table 7. CWO 108; valve ahead of the seat; Re

=

1.8.104 HlRv Ap(.105 fa) F(N) Cr 0 1.04 517.3 5.73.105 8.90.105 8.06.10-3 _0.96 572.4 6.06.103 7.86.103 1.31.10-2 0.76 452.0 2.60.103 3.38.103 1. 71.10-2 0.61 364.1 1.36.103 1.76.103 2.52.10-2 _0.39 235.7 6.78.102 8.68.102 2.82.10-2 _0.34 204.1 5.63.102 7.25.102 3.43.10-2 _0.24 145.7 3.76.102 4.79.102 4.13.10-2 0.19 111. 7 2.79.102 3.66.102 5.54.10-2 0.13 71.8 1. 73.102 2.42.102 6.75.10-2 0.09 51.6 1.21.102 1.64.102 9: 17.10-2 0.07 36.1 9.33.101 1.40.102 1.09.10-1 _0.06 30.4 7.89.101 1.20.102 1.34.10-1 0.05 26.2 6.81.101 1.00.102 1.57.10-1 0.05 23.5 6.07.101 9.97.101 1.79.10-1 0.04 21.5 5.53.101 7.96.101 2.18.10-1 0.03 19.1 4.90.101 5.96.101 2.51.10-1 0.03 17.3 4.44.101 5.95.101 3.38.10-1 _0.02 14.6 3.73.101 3.95.101 4.17 .10-1 0.02 13.1 3.34.101 3.94.101 7.02.10-1 0.02 14.6 3.73.101 3.95.101

89

table 8 CWO 161; valve behind the seat; Re

=

1.1.104

HlRv Ap.(.105 Pa) F(N) Cr 0 1.10 1606.1 CD CD 4.04.10-3 1.11 134.5 3.23.109 3.11.109 9.08.10-3 _{1.04 599.5 3.54.10}4 _4.14.104 1.12.10-2 0.94 525.6 5.25.103 1.24.103 2.02.10-2 _{0.11 396.3 2.23.10}3 _3.09.103 2.83.10-2 _{0.44 243.1 9.28.10}2 _1.29.103 3.53.10-2 _{0.31 111.3 5.11.10}2 _8.06.102 3.94.10-2 _0.22 126.6 3.81.102 5.18.102 5.35.10-2 0.18 95.4 2.82.102 4.10.102 6.16.10-2 0.13 13.0 2.10.102 2.89.102 1.31.10-2 0.09 61.4 1.15.102 1.98.102 8.98.10-2 _{0.01 49.5 1.39.10}2 _1.52.102 1.12.10-1 0.06 38.4 1.01.102 1.29.102 1.26.10-1 0.05 33.1 9.11.101 1.01.102 1.52.10-1 _{0.04 21.4 1.51.10}1 _8.51.101 1.19.10-1 0.04 23.2 6.41.101 8.51.101 2.35.10-1 _{0.03 19.1 5.44.10}1 _6.39.101 6.69.10-1 _{0.01 14.9 4.18.10}1 _2.12.101

table 9 CWO 161; valve ahead of the seat; Re

=

1.6.104 HlRv Ap(.105 Pal F(N) Cr 0 1.10 541.2 GO vibrations 1.21. 10-2 1.03 641.0 5.30.103 6.56.103 1.92.10-2 0.61 367.4 1.60.103 2.05.103 2.02.10-2 _0.57 340.0 1.44.103 1.86.103 2.72.10-2 0.35 209.2 7.59.102 9.80.102 3.23.10-2 0.25 146.0 5.00.102 6.61.102 4.54.10-2 0.14 80.2 2.59.102 3.49.102 6.16.10-2 0.09 50.7 1.60.102 2.19.102 7.37.10-2 0.07 39.1 1.22.102 1.69.102 9.59.10-2 _0.05 29.2 9.08.101 1.20.102 1.28.10-1 0.Q04 21.8 6.75.101 9.58.101 1.64.10-1 0.03 17.3 5.35.101 7.17 .101 2.11.10-1 0.03 13.7 4.25.101 7.17.101 2.51.10-1 0.02 12.2 3.79.101 4.78.101 6.45.10-1 0.02 13.7 4.24.101 4.78.101

Btl

table 10 CWO 265; valve behind the seat; Re

=

1.2.104

H/Ry _Ap(.105 _{Pal F(N)}

Cr 0 1.05 2026.3 2.23.107 5.10.106 5.29.10-3 1.05 1031.0 1.42.106 3.11.106 1.06.10-2 1.02 452.9 1.84.105 2.05.105 1.48.10-2 0.69 268.2 5.88.103 1041.103 2.22.10-2 0.35 122.2 1.42.103 2.00.103 2.96.10-2 0.11 62.6 6.55.102 8.18.102 3.59.10-2 0.10 35.8 3.60.102 4.96.102 4.23.10-2 0.01 26.8 2.65.102 3.42.102 4.65.10-2 0.05 20.9 2.05.102 2042.102 1.12.10-2 0.02 11.9 1.16.102 9.60.101 1.22.10-1 0.01 6.0 5.82.101 4.19.101 1.10.10-1 0 6.0 5.82.101 0 1.12.10-1 0 3.0 2.91.101 0

table 11 CWO 265; valve ahead of the seat; Re

=

2.1.104 HlRv p(.105 Pal F(N) Cr 0 0.95 37.2 3.00.105 3.84.106 3.17.10-3 0.96 308.1 1.20.106 1.85.106 4.23.10-3 0.99 372.8 9.86.105 1.29.106 vibrations 1.48.10-2 _0.87 438.3 3.19.103 3.13.103 1.90.10-2 0.59 302.2 1.42.103 1.37.103 2.33.10-2 0.42 209.5 8.16.102 8.07.102 2.96.10-2 0.31 147.8 5.20.102 5.39.102 3.49.10-2 0.24 114.7 3.84.102 3.97.102 4.02.10-2 0.20 85.8 2.75.102 3.17.102 4.65.10-2 _{0.16 67.6 2.12.10}2 2.47.102 5.39.10-2 _0.12 54.8 1.69.102 1.83.102 6.34.10-2 0.10 43.8 1.33.102 1.50.102 7.40.10-2 _0.09 34.9 1.05.102 1.34.102 9.51.10-2 0.06 22.9 6.81.101 8.81.101 1.35.10-1 0.04 11.6 3.39.101 5.76.101 1.84.10-1 0.02 6.9 2.01.101 2.87.101 2.75.10-1 0.02 4.2 1.21.101 2.86.101 5.48.10-1 0.01 3.3 9.5.100 1.42.101

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