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M. Borghi, M. Milani, R. Paoluzzi

Paper 133-04 1

INFLUENCE OF NOTCH SHAPE AND NUMBER OF NOTCHES ON THE METERING

CHARACTERISTICS OF HYDRAULIC SPOOL VALVES

1)

M. Borghi,

1)

M. Milani,

2)

R. Paoluzzi

1)

Department of Mechanical and Civil Engineering – University of Modena & Reggio Emilia

Via Vignolese, 905/b – 41100 Modena – Italy

2)

IMAMOTER – CNR

Via Canal Bianco 28 - 44044 Cassana (Ferrara) - Italy

Keywords Hydraulic components, spool valves, metering edge, metering notches, experimental

characterization, discharge coefficient, axial flow-forces

ABSTRACT

The paper presents theoretical and experimental results of studies on the influence of shape and number of spool

notches on the discharge characteristics (discharge coefficient, velocity coefficient and flow angle) of a

hydraulic distributor metering edge.

The flow-rate vs. pressure drop and the steady state axial flow-force vs. pressure drop diagrams are determined

for spools with different configurations of multiple notched metering edges. Various combinations of the shape

and number of the notches modulating the metering area of the passage between supply and drain ports were

investigated and correlated with flow-rate, axial flow-force and pressure drop, in order to get estimates of

discharge coefficient and flow angle.

The procedure is applied to all the data collected during the experimental activity, and shows the behaviour of

the flow characteristics in both the fully turbulent and transitional region of motion. The influence of notch

shape and number on the metering edge flow characteristics is evaluated as well.

1. Introduction

The modulation of fluid power, in modern electro-

hydraulic systems for both industrial and mobile

application, is obtained by proper metering

characteristics of the notches in the metering edge

of the valve spool.

The design of this type of components and the

simulation of their steady state and dynamic

behaviour, is made difficult by the peculiarities of

flow conditions as a function of metering area and

by the complex dependencies on fluid flow. In the

early 60s’ pioneering studies carried out at M.I.T.

focused the attention of researchers on the

fundamental role played by the variability of

discharge coefficient and efflux angle on the

behaviour of a metering edge. This was found to

affect significantly the variation of power

modulation function performed. The effect of

metering edge geometry and of spool-seat coupling

were highlighted by Blackburn et al. (1960) and in

Merritt (1967). Viersma (1980) describes the role

played by stationary and dynamic efflux

characteristics in hydraulic servo-systems, where

the transition from laminar to fully turbulent flow is

handled by a tailored function describing the

variation of the discharge coefficient. The

experimental study presented by Johnston et al.

(1991) details on one hand the effect of the shape

on the metering characteristics of poppet and disk

valves, on the other highlights, in particular for

fully turbulent flows, how the design parameters

could influence these characteristics, mainly in

terms of flow forces acting on the moving element.

The detailed experimental investigation carried out

by Lugowsky (1993), shows that axial flow forces,

and consequently discharge characteristics, are

conditioned not only by opening of the metering

edge, but also by geometries of spool and seat and

by other factors, like the Coanda effect.

The ever increasing demand for numerical tools for

sizing and simulation of fluid power components,

forced many researchers to address the topic of

characterization of static and dynamic flows of

incompressible fluid through geometries

representing the metering edge of hydraulic valves.

Several papers by the Authors were devoted to

Computational Fluid Dynamics (CFD) simulation

and to experimental verification of typical

discharge coefficient values found in industrial

components. They reported results on sharp edged

geometries (Borghi et al., 1996, 1997 and 1998),

conic profile (Borghi et al., 1999) and on typical

‘compensated’ profile for the minimization of axial

flow forces (Borghi et al., 2000). Work by Ellmann

and Piche (1996) and Wu et al. (2002), was

concerned with developing a semi-empirical

expression describing the variability of the

discharge coefficients in fixed-geometry orifices.

Wu et al. (2003) extended this work by considering

the fluid flow at very small openings. All these

approaches can be powerful tools for the modelling

of this kind of simple components but,

M. Borghi, M. Milani, R. Paoluzzi

Paper 133-04 2

unfortunately, on one hand they are not directly

applicable to complex shapes and, on the other

hand, they need the knowledge of the saturated

flow characteristics in order to give information in

the whole operational envelope of the variable

metering orifice. Other interesting results providing

a better insight into the problem have been obtained

by Elgamil (2001), Gromala et al. (2002) and Del

Vescovo and Lippolis (2002).

The results presented in this paper are concerned

with steady-state analysis of metering edges with

notches. The aim of the paper is to show what

effect the shape and number of notches may have

on the discharge characteristics of orifices under

different operating conditions.

The first part of the paper presents the experimental

set-up used. Eight different spools with different

notches have been tested over a wide range of input

hydraulic power. The shape of the grooves was

derived from typical industrial designs and

metering edges had two, three or four grooves.

Experimental data have been used to estimate the

characteristic of a given metering edge and,

consequently, the discharge coefficient and efflux

angle variations as a function of flow conditions.

The investigation drives to the definition of both

saturation values for discharge characteristics and

variation of coefficients as a function of shape and

number of considered notches.

2. Nomenclature

a, b, c, d, a

1

, b

1

Polynomial coefficient

Dp, ∆p

Pressure drop

A Area Q Flow rate

F Force Re Reynolds number

C

D

Discharge Coefficient R Hydraulic Resistance

C

V

Velocity Coefficient

ρ

Fluid density

*

V

C

Flow Coefficient

ν

Fluid kinematic viscosity

D

H

Hydraulic diameter

θ

Jet Angle

S Wetted Perimeter

Subscript

AX Axial component T Turbulent

EXP Experimental TH Theoretical

Superscript

SAT Saturation

3. Experimental Analysis

The hydraulic components for industrial application

show a wide variety of notch types. The notches are

designed to control hydraulic power supplied to

actuators, by varying the area of the metering edge.

The shape of these notches, as well as their number

and angular position, affect the dynamic behaviour

of the fluid, as they determine the input/output jet

angles and discharge coefficient. The angular

position of the notches determines the symmetry of

the flow through the orifice.

Among the large number of possible edge

configurations, only the three notch shapes shown

in Figure 1 have been considered for the

experimental investigation. The first notch, Figure

1-TYPE A, has a rectangular shape ended by a

semicircle, the second, Figure 1-TYPE B, is

obtained by connecting three semicircles with very

short rectangles (not visible in Figure 1), while the

third notch, Figure 1-TYPE C, has a triangular

section, in both axial and radial direction.

As shown in Figure 1, the three different types of

notches have the same axial length (5.25 mm),

radial depth (2.20 mm), and overlap when the spool

is centered (2.25 mm). All the prototypes were built

with a high control on dimensional tolerances,

maintaining the radial clearance between spool and

sleeve less than 5 µm, and the error in metering

edge opening less than 0.1 mm. This choice of

clearance and positional error permits use of

standard, commercial spools in test rig, and

focusing of the research on the influence of the area

gradient on the orifice flow characteristics.

In order to study the influence of notch number and

their angular positions, three different

configurations were designed and tested: two

notches at 180°, three notches at 120° and four

notches at 90° angular spacing.

The experiments were carried out using the test

bench presented in Figure 2. The circuit schematic

is shown in Figure 2-a. A variable displacement

pump (2) is connected to a volumetric flow-meter

measuring the flow-rate supplied by the pump (3,

range 0-150 l/min, class 0.2 absolute) to a variable

orifice (6) representing the metering edge, and to a

second volumetric flow-meter (10, range 0-150

l/min, class 0.2 absolute). The supply pressure is set

by a relief valve (5), and the pressure drop across

the variable orifice is measured using two piezo-

electric pressure transducers (4 and 5, range 0-600

bar, class 0.2 absolute). A detail of the test bench is

shown in Figure 2-b. A fine pitch worm screw is

used to adjust the axial position of the notched edge

under test and also to measure the axial force

M. Borghi, M. Milani, R. Paoluzzi

Paper 133-04 3

applied to the spool which is marked (8), (range 0-

150 N, class 0.1 absolute). The equipment which

supports the piezo-electric micrometer (range 0-15

mm, class 0.1 absolute) is marked (9).

NOTCH TYPE A NOTCH TYPE B NOTCH TYPE C

Figure 1: The notch shapes designed to perform the efflux characteristics experimental characterisation.

(a

(b

Figure 2: The test bench used for the experimental activity. (a Circuit sketch. (b A detail of the mechanical

equipments designed to vary and to measure the metering opening, and to measure the axial force acting on the

spool.

The tests were performed in order to determine the

steady-state flow-pressure characteristics of each of

the eight notched edges considered. The opening of

the metering edge were in the range 1 to 4 mm, in

order to reduce the influence of the notch

boundaries on the metering edge discharge

characteristics.

Table 1 shows the openings used in the steady state

characterization, for each one of the cases

considered. Some of the data reported in Table 1

can be reviewed referring to Figure 3, which shows

both the axial location of some of the experimental

openings along the non dimensional spool travel,

and the correspondent non dimensional minimum

value of the geometric area 1 . The spool

displacement was non-dimensioned against

maximum notch length and orifice area against its

maximum value. The total efflux area was

calculated, for each metering edge, as the sum of

the areas geometrically defined by each notch at

given spool axial travel.

As shown in Figure 3, the steady state

characteristics were collected for the metering

1

This is the minimum value of the cross sectional

area of the flow passage across the metering edge,

evaluated at each axial spool position.

M. Borghi, M. Milani, R. Paoluzzi

Paper 133-04 4

edges designed with the Notches Type A in the

following positions:

• an opening representative of the initial area

gradient (in the range 1 to 1.4 mm);

• a second opening close to the area

discontinuity (geometrically located 2.25 mm

far from the notch opening);

• a third opening placed in the region with

constant metering area.

The Notch TYPE B introduces two discontinuities

into the metering area gradient, and the

experimental activity was carried out in order to

catch the metering edges behaviour in two extreme

conditions:

1. close to the discontinuity;

2. in positions where discontinuity effect can be

neglected.

The area function corresponding to the introduction

of Notches TYPE C is quadratic with the notch

opening. As shown in Figure 3, the measurement

points chosen for this edge configuration were set

far from the notches boundaries, in the range

between 1.60 and 3.60 mm.

The steady-state characteristic curves (namely the

pressure drop vs. flow-rate curve and the axial

flow-force vs. pressure drop curve) were collected

for a large number of points in a range limited by

the available hydraulic power (about 100 kW). The

flow-rate was varied from 0 to 120 l/min and the

relief valve setting from 0 to 300 bar. The steady-

state characterization was performed, for each

metering edge and for each metering opening, with

a step by step increase in the flow-rate. Thanks to

the resistance offered by the volumetric flow-meter

(3), 2-4 bar of backpressure, depending on the

measured flow-rate, no cavitation phenomena were

detected along the discharge line, downstream the

metering section.

For each operating condition, the metering edge

characteristic data (the triple Q – ∆p – F) was

determined using the averaged values obtained

during experiments. The data were collected by the

acquisition system at 250 Hz over a time interval of

0.2 s. All the data collected during the experiments

are summarized in Figures 4 to 6.

Notch Type Configuration X

OP,A

X

OP,B

X

OP,C

2 at 180° 1.05 mm 2.30 mm 3.20 mm

3 at 120° 1.40 mm 2.40 mm 3.40 mm

TYPE A

4 at 90° 1.00 mm 2.15 mm 3.20 mm

2 at 180° 2.10 mm 3.00 mm 4.10 mm

3 at 120° 1.45 mm 2.45 mm 3.45 mm

TYPE B

4 at 90° 1.50 mm 2.50 mm 3.50 mm

2 at 180° 1.60 mm 2.60 mm 3.60 mm

TYPE C

4 at 90° 1.60 mm 2.60 mm 3.60 mm

Table 1: Metering Opening Tested for the Efflux Characteristics Determination.

4. Saturated efflux characteristics of the

notched edges

The steady-state characteristics of a metering edge

as such are not a key-factor for design purposes.

The main reason is that the typical operating

condition of a component is dynamic, or at least

non-stationary. To make the steady-state

investigation effective for design purposes, it is

necessary to extract information and general

guidelines not directly available from the

experimental evidence, such as the discharge (C

D

)

and the flow (C

V

cos θ) coefficients variation with

the metering edge geometry and the fluid flow

characteristics.

Many authors (Merrit, Blackburn et al., Borghi et

al., Vaughan et al., Lugowsky, Johnston et al.),

show that a good design of hydraulic valves is

based on:

• fluid flow coefficients characterization for

all active ports;

• effect of geometry on the above mentioned

coefficients;

• influence of the variation in the operating

conditions on the points above.

In this perspective, the stationary characterization is

a straightforward way to acquire all the information

above, indirectly showing how a specific design

choice might affect the flow conditions through a

specific valve port. The estimated characteristic

curve, in all the cases investigated, is in fairly good

agreement with theoretical predictions. Flow rate

M. Borghi, M. Milani, R. Paoluzzi

Paper 133-04 5

vs. pressure curves are more or less quadratic with a

vertex in the origin. Axial force versus pressure

curve are interpolated fairly well by a line through

the origin.

In this work, the experimental data have been

interpolated using a third order polynomial in Q-∆p,

and a first order for F-∆p.

A least square fitting procedure showed that the

pressure drop can be expressed by a function of

flow rate:

32

EXP EXP EXP EXP

∆paQbQcQ=⋅ +⋅ +⋅

(1)

and the axial force is a function of pressure drop

according to:

EXP EXP

Fd∆p=⋅ (2)

0.00

0.25

0.50

0.75

1.00

0.00 0.25 0.50 0.75 1.00

x/x

MAX

A/A

MAX

A

1.05 mm

B

2.3 mm

C

3.2 mm

2 Notches

Type A

at 180°

0.00

0.25

0.50

0.75

1.00

0.00 0.25 0.50 0.75 1.00

x/x

MAX

A/A

MAX

A

1.45 mm

B

2.45 mm

C

3.45 mm

3 Notches

Type B

at 120°

0.00

0.25

0.50

0.75

1.00

0.00 0.25 0.50 0.75 1.00

x/x

MAX

A/A

MAX

A

1.60 mm

B

2.60 mm

C

3.60 mm

4 Notches

Type C

at 90°

Figure 3: Actual Metering Opening Positions vs. Metering Edges Total Geometric Efflux Area.

M. Borghi, M. Milani, R. Paoluzzi

Paper 133-04 6

0

100

200

300

0 30 60 90 120

Q [l/min]

Dp

[bar]

2 Notches

Type C

at 180°

B

C

A

0

30

60

90

0 100 200 300

Dp [bar]

F

[N]

B

C

A

2 Notches

Type C

at 180°

0

100

200

300

0 30 60 90 120

Q - l/min

Dp

[bar]

4 Notches

Type C

at 90°

B

C

A

0

50

100

150

0 100 200 300

Dp [bar]

F [N]

B

C

A

4 Notches

Type C

at 90°

Figure 4: Characteristic Curves determined for the Metering Edges having 2 and 4 Notches Type C.

0

100

200

300

0 30 60 90 120

Q - l/min

Dp

[bar]

2 Notches

Type A

at 180°

A

B

C

0

40

80

120

0 100 200 300

Dp [bar]

F [N]

A

B

C

2 Notches

Type A

at 180°

0

100

200

300

0 30 60 90 120

Q - l/min

Dp

[bar]

3 Notches

Type A

at 120°

A

B

C

0

40

80

120

0 100 200 300

Dp [bar]

F [N]

A

B

C

3 Notches

Type A

at 120°

0

100

200

300

0 30 60 90 120

Q - l/min

Dp

[bar]

4 Notches

Type A

at 90°

A

B

C

0

40

80

120

0 100 200 300

Dp [bar]

F [N]

A

B

C

4 Notches

Type A

at 90°

Figure 5: Characteristic Curves determined for the Metering Edges having 2, 3 and 4 Notches Type A.

M. Borghi, M. Milani, R. Paoluzzi

Paper 133-04 7

0

100

200

300

0 30 60 90 120

Q - l/min

Dp

[bar]

2 Notches

Type B

at 180°

A B

C

0

50

100

150

0 100 200 300

Dp [bar]

F [N]

A

B

C

2 Notches

Type B

at 180°

0

100

200

300

0 30 60 90 120

Q - l/min

Dp

[bar]

3 Notches

Type B

at 120°

A

B

C

0

50

100

150

0 100 200 300

Dp [bar]

F [N]

A

B

C

3 Notches

Type B

at 120°

0

100

200

300

0 30 60 90 120

Q - l/min

Dp

[bar]

4 Notches

Type B

at 90°

B

C

A

0

50

100

150

0 100 200 300

Dp [bar]

F [N]

B

C

4 Notches

Type B

at 90°

A

Figure 6: Characteristic Curves determined for the Metering Edges having 2, 3 and 4 Notches Type B.

Tables from 2 to 4 show the values of the

polynomial coefficients computed for all the

experimental conditions examined. For the sake of

simplicity, the regression coefficient R was not

reported, but it is worth mentioning that its value is

always higher than 0.9997. The fact that the Q-∆p

experimental characteristic curve is better described

by third order than second order polynomials is at

least partly due to measurement errors intrinsic in

the instrumentation used. However, other

considerations apply. The flow through a restriction

can be classified, according to the amount of the

flow rate, as laminar, for low Reynolds numbers,

transitional and turbulent at higher Reynolds

numbers. The linear relationship between Q and ∆p

used for laminar flows, turns into a second order

curve as the flow rate enters the turbulent range. It

implies that the pressure drop could be described by

the following linear combination:

2

TH1EXP1EXP

∆p=aQ +bQ⋅⋅ (3)

where linear and quadratic contributes are linearly

combined to describe the overall characteristic of

the restriction. In this paper, the coefficients of the

linear combination are computed using a least

square approximation, in order to give the best

possible approximation of the curve for each notch

type and opening. Tables from 5 to 7 show the

values of

()

11

a,b coefficients thus determined. The

regression values in this case are always higher than

0.97. Equation (3) can be rearranged as:

TH 1

1

2

EXP

EXP

∆pb

=a +

QQ

(4)

The terms at the second member of Equation (4)

equal the square of the characteristic resistance of a

turbulent orifice for an incompressible fluid, and

include the correction factor fitting the actual data

behaviour to the ideal parabolic correlation. The

resistance is usually expressed as:

ρ⋅⋅

=

2AC

1

R

GEOD

T

(5)

where C

D

is the discharge coefficient, ρ is the fluid

density (850 kg/m

3

) and A

GEO

is the reference cross

sectional area of the orifice.

M. Borghi, M. Milani, R. Paoluzzi

Paper 133-04 8

TYPE A Opening

a

[bar/(l/min)

3

]

b

[bar/(l/min)

2

]

c

[bar/(l/min)]

d

[N/bar]

A -1.88E-03 3.13E-01 -4.75E-01 0.1557

B 1.41E-04 4.38E-02 5.26E-01 0.3170

2 at 180°

C 7.71E-05 3.65E-02 3.79E-01 0.3669

A 1.80E-04 3.81E-02 5.26E-01 0.3108

B 6.04E-05 1.81E-02 2.77E-01 0.4609

3 at 120°

C 3.38E-05 1.52E-02 1.41E-01 0.4944

A 5.19E-04 2.84E-02 7.37E-01 0.2031

B 2.50E-05 1.30E-02 1.03E-01 0.4310

4 at 90°

C 1.60E-05 8.83E-03 9.97E-02 0.5121

Table 2: Polynomial Coefficients for Experimental Data Interpolation – NOTCHES TYPE A

TYPE B Opening

a

[bar/(l/min)

3

]

b

[bar/(l/min)

2

]

c

[bar/(l/min)]

d

[N/bar]

A 1.67E-04 3.58E-02 5.32E-01 0.3239

B 7.45E-05 1.89E-02 2.64E-01 0.5138

2 at 180°

C 3.58E-05 1.59E-02 1.60E-01 0.6715

A 1.01E-04 3.96E-02 4.64E-01 0.4535

B 5.68E-05 1.21E-02 2.88E-01 0.4940

3 at 120°

C 2.26E-05 9.30E-03 1.34E-01 0.6622

A 2.33E-04 1.17E-02 5.67E-01 0.3830

B 3.40E-05 7.81E-03 2.01E-01 0.4849

4 at 90°

C 1.31E-05 6.43E-03 8.12E-02 0.6949

Table 3: Polynomial Coefficients for Experimental Data Interpolation – NOTCHES TYPE B

TYPE C Opening

a

[bar/(l/min)

3

]

b

[bar/(l/min)

2

]

c

[bar/(l/min)]

d

[N/bar]

A -1.17E-02 2.41E+00 -4.79E+00 0.1167

B -7.01E-03 5.93E-01 -1.52E+00 0.3008

2 at 180°

C -1.21E-03 1.80E-01 -5.13E-02 0.4454

A -1.44E-02 1.07E+00 -2.99E+00 0.2197

B -1.30E-04 1.17E-01 1.98E-01 0.5296

4 at 90°

C 4.29E-05 3.50E-02 2.21E-01 0.8328

Table 4: Polynomial Coefficients for Experimental Data Interpolation – NOTCHES TYPE C

Fitting the polynomials it is possible to derive the

variation of the characteristic resistance as a

function of the flow rate for each and every

metering edge as:

()

12

1

TEXP 1

EXP

b

RQ =a+

Q

⎡⎤

⎢⎥

⎣⎦

(6)

At the same time, the effect of the flow rate on the

discharge coefficients for each metering edge is

derived combining equation 5 and 6

()

()

12

DEXP

1 1 EXP GEO

ρ 1

CQ =

2a+bQ A

⎡⎤

⋅

⎢⎥

⋅

⎢⎥

⎣⎦

(7)

Recalling equation 4, it can be noted that the

correction term becomes smaller and smaller as the

flow rate increases, allowing the definition of a

limit value for the characteristic resistance of a

given geometry as:

EXP

12

SAT

TH

T1

2

Q

EXP

∆p

Rlim a

Q

→∞

⎛⎞

==

⎜⎟

⎝⎠

(8)

This definition is obviously valid in the operating

range where the flow can be considered as being

fully turbulent. It can be used to compute the limit

value of the discharge coefficient for a given

geometry, once the flow conditions meet this

turbulence criteria, i.e. Reynolds number above

transitional values, corresponding to values of the

discharge coefficient approximately constant. Using

the definition given in equation 7, this limit is given

by:

EXP

12

SAT

DD

Q

GEO 1

1 ρ

C=limC=

A2a

→∞

⎡

⎤

⋅

⎢

⎥

⋅

⎣

⎦

(9)

The computation of the saturated discharge

coefficient in all the geometries investigated, lead

to the results presented in Table 8. From a general

point of view, it is noted that the saturated

discharge coefficient in all the cases examined is

bounded between a minimum value of 0.48 and a

maximum of 0.74. For notches TYPE A and B it

presents a minimum value for the three notches

M. Borghi, M. Milani, R. Paoluzzi

Paper 133-04 9

configuration, while it decreases as the total number

of notches increases for the notches TYPE C.

Notches of TYPE A show a behaviour more or less

independent from their number, but strongly

influenced by their axial position. Starting from

values close to 0.65/0.67, for the smaller openings,

SAT

D

C

drops to 0.54/0.56 at intermediate openings

and rises back to 0.62/0.64 at large openings. This

is a clear effect of the discontinuity in the area

function at intermediate openings, with subsequent

discontinuity in the area gradient. The flow

conditions are significantly affected, and the

discharge coefficients decrease more than 18%. It is

also worth noting that the discharge coefficient at

the minimum opening of the metering edge, where

the area gradient is positive, has a slightly higher

value than at large openings, where the area

gradient is zero.

Notches of TYPE B have a more complex area

function. In this case the discharge coefficient

depends on both opening and notch number.

Comparing configurations having the same number

of notches, the highest value of the discharge

coefficient is always reached at maximum opening.

The lowest value is reached at intermediate

openings in the case of three and four notches, at

minimum opening for the two-notches one. Clear

tendencies are not present in this case, possibly due

to the complexity of the area gradient function, a

piecewise linear function with two discontinuities.

As a consequence, the discharge coefficient for two

notches is an increasing function of opening, from

0.62 to 0.69.

The case of three notches has a minimum value of

discharge coefficient of 0.54 at intermediate

openings and is 0.65 at small and large openings.

The four notches configuration has a maximum

value of the discharge coefficient at large openings

and minimum value at intermediate openings, but

shows values significantly lower (10% to 14%)

than those reached in other configurations. This

kind of behaviour could be the result of errors in

measurements or data handling (at present not

discovered by the authors), however, the absence of

a systematic error only suggests that this

configuration should be better investigated by

means of detailed numerical simulations of the flow

field which are beyond the scope of this work.

Notches of TYPE C have the highest values of the

discharge coefficient at small openings. Two

notches configuration always performs better than

that with four notches. Discharge coefficients

values decrease as the opening increases. It is worth

noting that the value changes significantly during

the spool travel: in the two notches configuration it

drops from 0.74 to 0.57 (-23%), and in the four

notches configuration from 0.70 to 0.53 (-24.3%).

Reconsidering equation 7, the discharge coefficient

as a function of flow rate can be rearranged as:

()

12 -12

1

DEXP

GEO 1 1 EXP

-1 2

SAT

1

D

1EXP

b

1 ρ

CQ = 1+

A2a aQ

b

=C 1+

aQ

⎡

⎤⎡ ⎤

⋅⋅

⎢

⎥⎢ ⎥

⋅⋅

⎣

⎦⎣ ⎦

⎡⎤

⋅

⎢⎥

⋅

⎣⎦

(10)

TYPE A Opening

a

1

[bar/(l/min)

2

]

b

1

[bar/(l/min)]

A 0.2325 0.0064

B 0.0612 0.0019

2 at 180°

C 0.0473 0.0018

A 0.0520 0.01

B 0.0265 0.0025

3 at 120°

C 0.0200 0.0050

A 0.0718 0.0018

B 0.0167 0.0013

4 at 90°

C 0.0115 0.0013

Table 5: II Order Characteristic Polynomial

Coefficients – NOTCHES TYPE A

TYPE B Opening

a

1

[bar/(l/min)

2

]

b

1

[bar/(l/min)]

A 0.0525 0.1417

B 0.0289 0.0015

2 at 180°

C 0.0210 0.0020

A 0.0510 0.2200

B 0.0200 0.0980

3 at 120°

C 0.0126 0.0400

A 0.0375 0.0350

B 0.0136 0.0150

4 at 90°

C 0.0086 0.0030

Table 6: II Order Characteristic Polynomial

Coefficients – NOTCHES TYPE B

TYPE C Opening a

1

[bar/(l/min)

2

] b

1

[bar/(l/min)]

A 2.1000 0.3000

B 0.3500 0.1500

2 at 180°

C 0.1400 0.1000

A 0.5970 0.5470

B 0.1100 0.2800

4 at 90°

C 0.0400 0.1340

Table 7:

II Order Characteristic Polynomial Coefficients – NOTCHES TYPE C

M. Borghi, M. Milani, R. Paoluzzi

Paper 133-04 10

The comparison between data on different metering

edges arrangements (experimental or derived

quantities) can be made using either:

1.

comparison of flow characteristic as a function

of spool axial travel;

2.

comparison of flow characteristic as a function

of actual cross sectional area of the equivalent

orifice.

the result is that the curves must be rescaled

according to a non-linear transformation which

needs the definition of the area function of each and

every configuration investigated. This function

varies with the spool position and shows

discontinuities. The only workaround to overcome

these difficulties, is the introduction of a parameter

able to describe the flow conditions and the

geometry. The Reynolds number of a metering

configuration can be defined as:

H EXP GEO

EXP

D(Q A )

4Q

Re =

νν

⋅

⋅

=

⋅ S

(11)

where ν is the kinematic viscosity (32 cSt), D

H

is

the hydraulic diameter of the metering edge, while

S is the wetted perimeter of the reference cross

sectional minimum area of the orifice. It is not

worthy to underline that both the hydraulic

diameter and the wet perimeter vary with the

metering edge opening, introducing the effect of the

notched edge geometry on the Reynolds’ number

definition. In this way, it is possible to consider

equation 10 in the new form:

()

-1 2

SAT

1

DD

1

b

41

CRe= C 1+

aRe

⎡⎤

⋅⋅⋅

⎢⎥

⋅

⎣⎦

S

ν

(12)

At the same time, the axial flow force as a function

of the pressure drop across the metering edge can

be expressed using the traditional Von Mises

approximation (Merritt, Blackburn):

()

AX D GEO V

F=2C ReA C cos ∆p

ϑ

⋅⋅⋅⋅⋅ (13)

Equation 13 considers as positive all forces acting

towards a closure of the valve, and drives to a direct

definition of the flow coefficient:

*

VV

CCcos

ϑ

=⋅ (14)

defined as the product of velocity coefficient and

cosine of the jet flow angle. Combining equation 2

with equations 13 and 14, the variation of the efflux

coefficient with flow conditions in the generic

metering edge is given by:

()

*

EXP EXP

DGEOV

EXP TH

FF

=2 C Re A C d

∆p ∆p

≅⋅⋅⋅= (15)

hence:

()

()

*

V

DGEO

12

1

1

SAT

DGEO

d

CRe

2C Re A

b

41

d1+

aRe

C2A

==

⋅⋅

⎡⎤

⋅⋅⋅

⎢⎥

⋅

⎣⎦

=

⋅⋅

S

ν

(16)

The flow coefficient is therefore a function of

geometry and flow conditions, but it is also directly

affected by the saturated value of the discharge

coefficient. Equation 16 shows that the flow

coefficient has a limit of +∞ as Reynolds number

tends to zero, indicating that the approach proposed

here cannot be applied to the analysis of flow at

very low Reynolds numbers. Conversely, Equation

16 holds for the actual metering edges operating

conditions, where flow can be described mainly

with reference to the fully turbulent (or, at least,

high transitional) conditions. In the wide range of

variation of the Reynolds number where Equations

16 holds, the flow coefficient has a saturated value

defined by:

*,SAT *

VV

SAT

Re

DGEO

d

C=limC

C2A

→∞

=

⋅⋅

(17)

Recalling the flow coefficient definition (Equation

14), the variation of the jet angle with the Reynolds

number of the incompressible flow through a given

geometry is described by the function:

()

()

*

V

V

CRe

Re = arccos

C

⎡

⎤

⎢

⎥

⎢

⎥

⎣

⎦

ϑ

(18)

Which presents a saturated value expressed as:

SAT

SAT

VD GEO

d

= arccos

CC 2A

⎛⎞

⎜⎟

⋅⋅⋅

⎝⎠

ϑ

(19)

Table 9 summarizes the values of the saturated jet

angle computed for all the geometries investigated.

In Equation 19 a velocity coefficient C

V

of 0.98

was used, as an average of the values found in

literature.

Table 9 show that for TYPE A notches the

saturated jet angle has values very close to the

theoretical value found by Von Mises (69°) for an

inviscid, incompressible fluid flowing from a high

pressure confined volume to an unconfined volume

at low pressure.

The configuration with four notches has jet angle

close to 75°, with variations as a function of spool

position lower than 0.4°. Configurations with two

and three notches have values, respectively close to

69° and 70°, with larger variations with spool

position, but in any case lower than 1.5°.

This means that for TYPE A notches it is possible

to define a unique, constant value of the saturated

efflux angle, independent of the axial spool travel.

The behaviour shown by TYPE B notches is

completely different. The two notches configuration

shows values of

SAT

ϑ

significantly decreasing as the

spool travel increases, going from 69.5° at

minimum opening to 63° at maximum opening.

The three notches configuration shows the lowest

SAT

ϑ

value (approximately 61°) at minimum

opening and values close to 70° at intermediate and

large openings. The four notches configuration has

a maximum value of

SAT

ϑ

for intermediate

M. Borghi, M. Milani, R. Paoluzzi

Paper 133-04 11

openings;

SAT

ϑ

values start from 69.5° at small

openings, increase to 74.5° at intermediate positions

and slightly decreases to 72.5° at large openings.

These results confirm that TYPE B notches have a

much more complex behaviour than all the other

types considered, and these anomalies can be

properly understood only by a deeper investigation

of the flow field inside the metering section of the

valve.

TYPE C notches seem to force the saturated flow in

a direction mainly determined by the radial depth of

the notch, having a nominal angle of 25°, with a

final nozzle effect constraining the efflux angle.

In both configurations tested (two and four

notches),

SAT

ϑ

shows a similar behaviour, varying

from 37.3° at small openings to 38.5 at large

openings. At intermediate openings it reaches its

minimum value, respectively 33.5° and 34.5°.

This configuration show no dependency on the

notches number; it is therefore possible to describe

all of the configuration with a single function

SAT

ϑ

of the spool position.

TYPE A Opening

SAT

D

C

TYPE B Opening

SAT

D

C

TYPE C Opening

SAT

D

C

A 0.67 A 0.62 A 0.74

B 0.55 B 0.64 B 0.69

2 at 180°

C 0.62

2 at 180°

C 0.69

2 at 180°

C 0.57

A 0.67 A 0.65

B 0.56 B 0.54

3 at 120°

C 0.64

3 at 120°

C 0.65

A 0.65 A 0.55 A 0.70

B 0.54 B 0.48 B 0.61

4 at 90°

C 0.63

4 at 90°

C 0.59

4 at 90°

C 0.53

Table 8: Saturated Discharge Coefficient defined adopting the Equation (9)

TYPE A Opening

SAT

ϑ

(°)

TYPE B Opening

SAT

ϑ

(°)

TYPE C Opening

SAT

ϑ

(°)

A 69.4 A 69.6 A 37.4

B 68.4 B 65.8 B 33.3

2 at 180°

C 68

2 at 180°

C 62.8

2 at 180°

C 38.5

A 70.6 A 61.3

B 69.4 B 70.8

3 at 120°

C 70.8

3 at 120°

C 69.6

A 75.2 A 69.6 A 37.2

B 74.8 B 74.6 B 34.4

4 at 90°

C 75.0

4 at 90°

C 72.4

4 at 90°

C 38.6

Table 9: Saturated Jet Angle defined adopting Equation (19) (C

V

= 0.98)

5. Analysing the main features of notched

edges flow characteristics.

As shown by equations 12 and 18, the experimental

data set collect the stationary characterization of all

the metering edges considered in this work. Their

use makes possible the description of discharge

coefficient and jet angle. In figures from 7 to 14 the

values determined for C

D

and θ adopting Equations

(9), (12), (18) and (19) are plotted as a function of

the square root of Reynolds number, as defined in

equation 11.

The curves give a synthetic view of the

characteristics analysed in the previous paragraphs,

and allow a fairly good identification of the

transition Reynolds number, as the limit value

above which a flow can be considered as fully

turbulent with a sufficiently high confidence level.

Figures from 7 to 9 show that TYPE A notches, for

all notch numbers, have a transition Reynolds

number notably low (500÷900). The transition from

the laminar to the transitional and turbulent efflux

conditions is clearly defined, and the saturated

values are stable. Beyond the transition value,

discharge coefficients approach the values given by

equation 9, and it is confirmed that intermediate

openings (close to the area gradient discontinuity

point) show values significantly lower than small

and large openings. It is also worth noting that

discharge coefficients for minimum and maximum

opening tend to get closer and closer as the number

of notches increases. At the same time the jet angle

at Reynolds number beyond the transition,

approaches the value given by equation 19. This

value is practically independent from the valve

opening for each configuration and slightly

increasing with the number of notches.

M. Borghi, M. Milani, R. Paoluzzi

Paper 133-04 12

0.00

0.20

0.40

0.60

0.80

0 50 100 150 200Re^

0.5

C

D

Opening A

Opening B

Opening C

30

50

70

90

0 50 100 150 200Re^

0.5

θ

Opening A

Opening B

Opening C

Figure 7: Discharge Coefficient and Jet Angle distributions with the Reynolds’ Number variation – 2 notches

TYPE A metering edge.

0.00

0.20

0.40

0.60

0.80

0 50 100 150 200Re^

0.5

C

D

Opening A

Opening B

Opening C

30

50

70

90

0 50 100 150 200Re^

0.5

θ

Opening A

Opening B

Opening C

Figure 8: Discharge Coefficient and Jet Angle distributions with the Reynolds’ Number variation – 3 notches

TYPE A metering edge.

0.00

0.20

0.40

0.60

0.80

0 50 100 150 200Re^

0.5

C

D

Opening A

Opening B

Opening C

30

50

70

90

0 50 100 150 200Re^

0.5

θ

Opening A

Opening B

Opening C

Figure 9: Discharge Coefficient and Jet Angle distributions with the Reynolds’ Number variation – 4 notches

TYPE A metering edge.

0.00

0.20

0.40

0.60

0.80

0 50 100 150 200Re^

0.5

C

D

Opening A

Opening B

Opening C

30

50

70

90

0 50 100 150 200Re^

0.5

θ

Opening A

Opening B

Opening C

Figure 10: Discharge Coefficient and Jet Angle distributions with the Reynolds’ Number variation – 2 notches

TYPE B metering edge.

M. Borghi, M. Milani, R. Paoluzzi

Paper 133-04 13

0.00

0.20

0.40

0.60

0.80

0 50 100 150 200Re^

0.5

C

D

Opening A

Opening B

Opening C

30

50

70

90

0 50 100 150 200Re^

0.5

θ

Opening A

Opening B

Opening C

Figure 11:

Discharge Coefficient and Jet Angle distributions with the Reynolds’ Number variation – 3 notches

TYPE B metering edge.

0.00

0.20

0.40

0.60

0.80

0 50 100 150 200Re^

0.5

C

D

Opening A

Opening B

Opening C

30

50

70

90

0 50 100 150 200Re^

0.5

θ

Opening A

Opening B

Opening C

Figure 12: Discharge Coefficient and Jet Angle distributions with the Reynolds’ Number variation – 4 notches

TYPE B metering edge.

0.00

0.20

0.40

0.60

0.80

0 50 100 150 200Re^

0.5

C

D

Opening A

Opening B

Opening C

0

20

40

60

0 50 100 150 200Re^

0.5

θ

Opening A

Opening B

Opening C

Figure 13:

Discharge Coefficient and Jet Angle distributions with the Reynolds’ Number variation – 2 notches

TYPE C metering edge.

0.00

0.20

0.40

0.60

0.80

0 50 100 150 200Re^

0.5

C

D

Opening A

Opening B

Opening C

0

20

40

60

0 50 100 150 200Re^

0.5

θ

Opening A

Opening B

Opening C

Figure 14:

Discharge Coefficient and Jet Angle distributions with the Reynolds’ Number variation – 4 notches

TYPE C metering edge.

M. Borghi, M. Milani, R. Paoluzzi

Paper 133-04 14

TYPE B notches give the characteristics plotted in

figures 10 to 12. The observations introduced in the

previous paragraphs on this type of notches apply

also on these curves; however, the graphs seem to

show a transition square root of Reynolds’ number

close to 75÷100, although in some cases it is not

well identified. The differences among the various

jet angles are more significant, when comparing the

same configuration and for different configurations

as well.

TYPE C notches plots are shown in figure 13 and

14. In this case the flow conditions can be

considered saturated only for values of the square

root Reynolds’ number above 100. This limit is

higher the higher is the number of notches. The jet

angle tend to approach its asymptotic value only for

Reynolds numbers particularly high, with the

notable characteristic that, for all configurations,

minimum and maximum opening have asymptotic

values very close to each other and higher than

those typical of intermediate openings.

6. Conclusions

This paper presents a thorough and critical analysis

of stationary flow characteristics in different

configurations of metering edges with timing

notches, which are typically used in industrial

hydraulic valves. The study is developed starting

from experimental data gathered at different valve

openings for eight different configuration of

metering edges. The investigation allow to

highlight trends and overall effects on discharge

coefficient and jet angle of shape, number and

position of timing notches. Among the main results

shown in the paper, the effect of the discontinuities

in area gradient are highlighted, together with the

saturated flow conditions (asymptotic values of

relevant flow parameters), where discharge and

flow coefficients can be considered constant. It has

been shown that the saturated values of the

discharge coefficient are reached at values of the

Reynolds number depending on shape and number

of the notches on the metering edge, giving also

some hints on how these indications can be used to

improve valve design or to increase the

effectiveness of valve metering characteristics.

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