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PERIODICA POLYTECHNICA SER. MECH. ENG. VOL. 47, NO.2, PP. 131–142 (2003)

A SEMI-EMPIRICAL MODEL FOR CHARACTERISATION OF

FLOW COEFFICIENT FORPNEUMATIC SOLENOID VALVES

Viktor SZENTE∗and János VAD∗∗

Department of Fluid Mechanics

Budapest University of Technology and Economics

H–1111 Budapest, Bertalan Lajos u. 4 – 6., Hungary

∗Phone: (+36-1)-463-3187, Fax: (+36-1)-463-3464

∗∗Phone: (+36-1)-463-2464, Fax: (+36-1)-463-3464

e-mail: szente@simba.ara.bme.hu, vad@simba.ara.bme.hu

Received: July 1, 2003; Revised: July 15, 2003

Abstract

A semi-empirical model has been elaborated for analyzing and predicting the ﬂow characteristics of

small electro-pneumatic (EP) valves within a wide range of pressure ratio. As a basis for character-

ization of ﬂow coefﬁcient, an analytical model has been established for a simpliﬁed geometry. This

model has been corrected corresponding to more complex valve geometries, utilizing the results of

axisymmetric quasi-3D (Q3D) computations using the Computational Fluid Dynamics (CFD) code

FLUENT.By this means, a semi-empirical modelling methodology has been elaborated for charac-

terization of through-ﬂow behavior of pneumatic valves of various geometries.

Keywords: pneumatic valve, ﬂow coefﬁcient, computational ﬂuid dynamics.

1. Introduction

Electro-pneumatic (EP) valves are widely applied in several areas of industry. The

knowledgeonthedynamic ﬂowcharacteristics ofsuch valvesisespecially important

inthe caseswhenthey areintegratedin fast-response, controlledﬂuidpowersystems

as control devices. A typical application of this kind is the realization of control

functions in intelligent EP braking systems of motor vehicles [1], [2]. The dynamic

ﬂow characteristics of EP valves inﬂuence the successful operation of the entire

ﬂuid power equipment.

As illustrated in SZENTE et al. [3], a simpliﬁed 1D simulation tool can be

effectively used in design, research and development regarding controlled ﬂuid

power systems. The EP valve models integrated in such systems must represent

reliably the transmission characteristics of the EP valve, without time-consuming

and practically unnecessary resolution of 3D ﬂow details. A key factor in such 1D

models is the ﬂow coefﬁcient Cq, representing the contraction of the isentropic or

sonic gas jet in the oriﬁce cross-section.

In the last decades, several attempts have been made to describe the pressure

ratio and valve geometry dependent ﬂow coefﬁcient of pneumatic oriﬁces. The

classic measurements by PERRY [4] provide truly empirical data – summarized in

132

V. SZENTE and J. VAD

the ‘Perry polynomial’ – for sharp-edged circular oriﬁces over the entire pressure

ratio range. Since the through-ﬂow geometry is considerably different in the case

of pneumatic solenoid valves, the applicability of Perry data is doubtful in this

area. Contrarily, the lack of knowledge compels the researcher even recently to use

the Perry polynomial in pneumatic simulation in certain cases [5]. BUSEMANN

[6] presents an analytical discharge coefﬁcient model for a two-dimensional planar

slit using the tangent-gas approximation, whereas OSWATITSCH outlines a more

comprehensive model [7] for a Borda-type oriﬁce. Both models, however, analyze

the subcritical regime only. BROWER et al. [8] suggest a new analytical model

based on Busemann over the entire pressure ratio range, but for axisymmetric sharp-

edged circular oriﬁces. There are several other sources with measurement data, e.g.

GRACE and LAPPLE [9], JOBSON [10], or TSAI and CASSIDY [11]. They usually

suggest some constant Cqvalues, and are mainly concerned about sharp-edged

oriﬁces or poppet valves, both ofwhich are quite different from the geometry used

in EP valves. The above suggest the necessity to establish a widely applicable

model for prediction of ﬂow coefﬁcient for characteristic EP valve geometries.

This paper presents a semi-empirical model, supplying reliable information

on the ﬂow transmission characteristics of the EP valve. It is based on a simpliﬁed

analytical model using the law of linear ﬂuid momentum. The data of the analytical

model have been corrected with use of CFD results supplied by the ﬁnite volume

CFD code FLUENT and thus, a semi-empirical model has been established. The

application of the model is demonstrated in a case study EP valve. The semi-

empirical model has been extended to a number of different EP geometries, serving

as a knowledgebase for future developments.

2. The EP Valve of Case Study

The valve under investigation is applied in fast-response pneumatic systems as con-

trol valve providing e.g. pressure signal for relay valves. Such miniature valves

must provide rapid, pulsed ﬂuid transmission between enclosures of relative pres-

sures in the orders of magnitude of 10 bar and 0 bar within a time period in the

order of magnitude of 0.01 s. It is of critical importance to elaborate a reliable ﬂuid

dynamical model for the valve to be applied in design of the ﬂuid power hardware

and its control.

Figs. 1aand 1bshow the simpliﬁed scheme of the valve (SZENTE and VAD

[12]). Thevalvebody is equipped withﬂexible seal andcontactsurfaces. In absence

of solenoid excitation, the valve body is kept at its closed end-position by the return

spring. The solenoid is energized by DC voltage. The frame and the jacket assist in

development of a magnetic circuit. The resultant magnetic force displaces the valve

body against the return spring. As a consequence, a ﬂow cross-section develops

through the oriﬁce. The original valve has a valve seat with angle of α=8

◦(see

the explanation for αin Chapter 4).

As a ﬁrst step of modelling the ﬂow transmission characteristics of the EP

SEMI-EMPIRICAL MODEL

133

valve, an analytical model has been elaborated.

inlet port

outlet port

orifice

valve body

return spring

solenoid

jacket

frame

(a)

(b)

Fig. 1. (a) Scheme of the EP valve, (b) Detailed view of the valve

3. Analytical Model

Theanalyticalmodel of thevalveuses theﬂuid momentumlawappliedonto aBorda-

type oriﬁce. The scheme of the oriﬁce is shown in Fig. 2. The Borda-type oriﬁce

is a circular, sharp-edged, straight, short pipe section immersed in the enclosure

serving as the source of incoming ﬂow. For incompressible ﬂuids, the analytical

model for a Borda-type oriﬁce represents a ﬂow coefﬁcient of 0.5 [13], justiﬁed by

measurements. The novelty of the present model compared to the one proposed

by Oswatitsch is that it provides ﬂow coefﬁcient data over the entire pressure ratio

range, thus taking compressibility effects into account as appropriate.

The upstream surfaces of control volume presented in Fig. 2extend sufﬁ-

ciently far from the oriﬁce to guarantee no through-ﬂow and undisturbed static

pressure pup. The control volume excludes the Borda-type oriﬁce. Downstream of

the oriﬁce, the control volume ends at the vena contracta (i.e. the limit of applica-

bility of the isentropic law).

The following assumptions have been taken for the present model:

• The ﬂow is stationary through the oriﬁce,

• The effects of force ﬁelds are negligible,

• The ﬂow is isentropic (inviscid ﬂow with no heat transfer) upstream of and

inside of the vena contracta (narrowest cross-section of pneumatic jet). This

means that even for sonic ﬂow (throttled expansion), the shock losses appear

only downstream of the vena contracta.

134

V. SZENTE and J. VAD

The mass ﬂow rate qmthrough the oriﬁce is a function of upstream absolute pressure

pup, upstream temperature Tup, oriﬁce cross-section A, ﬂow coefﬁcient Cqand mass

ﬂow parameter Cm([5], [14]):

qm=A·Cq·Cm·pup

Tup ,(1)

where

Cm=

2·κ

R·(κ−1)·pdown

pup 2

κ

−pdown

pup κ+1

κ

if pdown

pup >pdown

pup crit (subsonic ﬂow), (2a)

Cm=2·κ

R·(κ−1)·2

κ+11

κ−1

if pdown

pup ≤pdown

pup crit (transonic ﬂow), (2b)

pup pdown

A

Ajet

Fig. 2. Scheme of the Borda-type oriﬁce

and the critical pressure ratio is

pdown

pup crit =2

κ+1κ

κ−1

=0.528 if κ=1.4.(3)

SEMI-EMPIRICAL MODEL

135

According to [5] and [14], all the parameters except the ﬂow coefﬁcient Cqcan be

assessed using explicit functions or measurements, therefore, to build ananalytical

or semi-empirical model, the function of Cqhas to be determined.

By applying the ﬂuid momentum law to the control volume, the following

equation can be obtained, assuming that the process is isentropic in the control

volume, and, according to the energy equation, the total enthalpy remains constant

in the control volume:

−ρ·v2

jet ·Ajet =pjet ·Ajet −pup ·A+pdown ·A−Ajet.(4)

In this equation, it has been assumed that the static pressure valid in the down-

stream ﬂow ﬁeld inﬂuences the development of the jet on the annular cross-section

A−Ajet. Therefore, it is supposed that even in case of throttled expansion, the

oriﬁce is suitably short to avoid its blockage by shocked ﬂow from the downstream

ﬂow ﬁeld.

Fig. 3. Q3D scheme of the valve

Deﬁne Cqas the ratio of the ﬂow- and the oriﬁce cross-section:

Cq=Ajet

A=pup −pdown

pjet −pdown +ρ·v2

jet

.(5)

136

V. SZENTE and J. VAD

Fig. 4. Mach contour plot at pdown/pup =1:10

The exit velocity of the Borda-type oriﬁce for subsonic (6a) and transonic (6b)

ﬂows:

v2

jet =2·κ

κ−1·R·Tup ·1−pdown

pup κ−1

κif pdown

pup >pdown

pup crit

,(6a)

v2

jet =2·κ

κ−1·R·Tup ·1−pdown

pup κ−1

κ

crit if pdown

pup ≤pdown

pup crit

.(6b)

By assuming the following circumstances in the vena contracta

pjet

pup =pdown

pup if pdown

pup >pdown

pup crit

,(7a)

pjet

pup =pdown

pup crit if pdown

pup ≤pdown

pup crit

,(7b)

andsubstituting Eq.(6a) andEq.(6b) intoEq. (5),the resultis theanalytical function

SEMI-EMPIRICAL MODEL

137

of Cq:

Cq=

1−pdown

pup

2·κ

κ−1·pdown

pup 1

κ

·1−pdown

pup κ−1

κ

if pdown

pup >pdown

pup crit

,(8a)

Cq=

1−pdown

pup

2·κ

κ−1·pdown

pup 1

κ

crit ·1−pdown

pup κ−1

κ

crit +pdown

pup crit −pdown

pup

if pdown

pup ≤pdown

pup crit

.(8b)

The analytical model is presented in Figs.7and 8(‘Cq-analytical’ curves). For the

incompressible case represented by the pressure ratio of unity, the model represents

the ﬂow coefﬁcient value of 0.5 formerly deduced for incompressible cases. It is

conspicuous in the ﬁgures that the model represents the trends of the Perry model

qualitatively; serving as a kind of explanation of the underlying physics.

The next chapter reports the CFD campaign carried out on the case study EP

valve, having geometrical cross-sections different from a Borda-type oriﬁce.

0.6

0.65

0.7

0.75

0.8

0.85

0 0.2 0.4 0.6 0.8 1

Pdown/Pup

Cq

20º

14º

8º

0º

-8º

-14º

-20º

Perry

Fig. 5.Cqvalues for different seat angles (α)

138

V. SZENTE and J. VAD

0

2

4

6

8

10

12

0 0.2 0.4 0.6 0.8 1

Pdown/Pup

Cq differences [%]

Fig. 6. Differences between min. and max. Cqvalues

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0 0.2 0.4 0.6 0.8 1

P

down

/P

up

C

q

Cq-analytical Cq-Perry

Cq-analytical-transformed Cq-8° (CFD)

Fig. 7. Transformation for the subsonic region

4. CFD Studies

Inorderto build up adata basefrom whichthe corrections of theanalyticalmodelcan

be deduced, a number of different 3D models were prepared, based on a previously

validatedComputationalFluid Dynamics (CFD)code FLUENT[15],[16]. Because

of axial symmetry, the 3Dvalve model has been transformed to Q3D axisymmetric

domain. Fig.3showsthe2D schemethat hasbeen usedin CFDsimulation. Because

ofpresentlimitations inthe simulationsoftwareavailable, themovementof thevalve

body has not been incorporated into the model. In order to analyze the inﬂuence

of the geometry on the ﬂow parameters, a number of different Q3D models were

SEMI-EMPIRICAL MODEL

139

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0 0.2 0.4 0.6 0.8 1

P

down

/P

up

C

q

Cq-analy tical Cq-Perry

Cq-analytical-transformed Cq-8° (CFD)

Fig. 8. Transformation for both regions

prepared. The current investigations were concentrated on the inﬂuence of the angle

of the valve seat αat the inlet of the oriﬁce, shown in Fig.3, on the ﬂow coefﬁcient.

The seat angle αis positive if a meridional line of the valve seat cone forms an acute

angle with the section of oriﬁce axis at the output port. Therefore, the Borda-type

oriﬁce in Chapter 3 is considered as a valve seat with α=90◦.

The simulation software computed the mass ﬂow rate for the stationary state.

pup,pdown and Tup have been speciﬁed as boundary conditions. A computed Mach

number contour distribution can be seen in Fig.4as example. Such images can be

used in the future for a detailed analysis of ﬂow ﬁeld within the valve. The value

of Cqhas been deduced from the CFD simulation using Eq. (1).

InFig. 5the results for the different geometries are plotted against the pressure

ratio. The values of the Perry model are also shown for comparison purposes. The

main trends of the computed curves are similar to those for the Perry model as

well as for the analytical model of Chapter 3. Fig. 5suggests the tendency that

the increase of the angle decreases the ﬂow coefﬁcient, as the separation bubble

developing at the inlet edge of the oriﬁce reduces the ﬂow cross-section. The angle

variation has more inﬂuence at higher pressure ratios, as it can be seen in Fig.6

Furthermore, it is apparent that the domain can be separated into two parts: at lower

pressure ratios up to 0.5 the relative difference between the Cqvalues for the largest

and smallest valve seat angles almost remains constant, and then starts increasing

linearly from there (see the two jagged lines in Fig. 6, the dashed horizontal line

shows the region where the difference is constant, while the dotted line shows the

region where the difference increases linearly). This partitioning is the same as it

is with the ﬂow coefﬁcient characteristics: Cqremains constant at lower pressure

ratios, and starts to decrease at about the critical pressure ratio. It suggests that

140

V. SZENTE and J. VAD

the analytical model should be corrected differently in the subsonic and transonic

regions.

5. Fitting the Analytical Model to CFD Data

On the basis of the perception that the analytical model follows trends similar to

those apparent in the computational results, it has been supposed that the curve

representing the analytical model can be transformed to the computed ones with

use of simple transformation functions.

The ﬁrst step of the model correction was to select one case for which the

transformation functions can be tested and veriﬁed. This case was the one where the

value of αwas 8◦asthis was the angle of the original valve seat. Numerous attempts

have been made to ﬁnd the simplest solution for transformation. It has been found

that the regions of subcritical and supercritical pressure ratio indeed have to be

treated separately. Fig. 7shows that the analytical model shares the same tendency

as the Perry model, both of which are quite different from the validated CFD values.

It also shows, however, that the correction of the subsonic range can be quite simple,

as the Cq-analytical-transformed curve ﬁts almost perfectly to the CFD data by

using a simple linear transformation, represented later by Eq. (9a). This suggested

that the correction for the transonic range should be a similar transformation, and

should use the parameters from the subsonic range correction function as well.

As it has been mentioned previously, the Cq-analytical-transformed curve is

based on a simple linear transformation. The curve has been rotated around the C

q

value of the critical pressure ratio, then shifted along the Yaxis by a constant value.

Thistransformation producedthe curvewhichcan beseen in Fig.7. Thisfollowsthe

CFD data quite well in the subsonic range, but breaks away in the supersonic range.

To correct this departure, a pressure-dependent transformation has been applied.

This transformation uses one more constant (K3)in addition of the two (K1,K2)

used in the subsonic correction. The ﬁnal curve, using both transformations for the

appropriate region, can be seen in Fig.8, and the two transformations in Eqs. (9a)–

(9b).

Cqcorr =Cq−Cqcrit·K1+Cqcrit +K2

if pdown

pup >pdown

pup crit

,(9a)

Cqcorr =Cq−Cqcrit·K3+pdown

pup ·K1−K3

Cqcrit +Cqcrit +K2

if pdown

pup ≤pdown

pup crit

.(9b)

SEMI-EMPIRICAL MODEL

141

After specifying the correction functions, the values of the constants Kihave been

determined for the αvalues used in the CFD calculations by using the least-squares

method. After specifying these constants for each αvalue, it became apparent that

the variations of the constants can be deﬁned with simple linear functions of αas

it can be seen below in Eqs. (10a)–(10c). The new Kivalues provided by these

functions are capable to keep the corrected Cqvalues within a ±1% maximum

relative difference compared to the CFD data.

K1=0.0028 ·α+0.4307,(10a)

K2=−0.0015 ·α+0.1433,(10b)

K3=0.0003 ·α+0.1482.(10c)

6. Conclusions

An analytical model has been elaborated for description of ﬂow coefﬁcient for a

Borda-type oriﬁce over the entire pressure ratio range. This new model has been

compared to literature data. It formed the basis for a semi-empirical model de-

scribing the ﬂow coefﬁcient of EP valves with various valve seat angles. The

semi-empirical model has been obtained by simple transformations from the ana-

lytical model. The model parameters have been established by ﬁtting the model to

Q3D CFD data obtained by means of code FLUENT. The semi-empirical model is

capable of predicting the ﬂow coefﬁcient within a ±1% maximum relative differ-

ence compared to the CFD data. The model is to be generalized and experimentally

veriﬁed by future studies.

Acknowledgement

This work has been supported by the Hungarian National Fund for Science and Research

under contract No. OTKA T 038184.

Nomenclature

pabsolute pressure [bar]

qmmass ﬂow rate [kg/s]

vﬂow velocity [m/s]

Aoriﬁce cross-section [m2]

Cmoriﬁce mass ﬂow parameter [√◦K/(m/s

)]

Cqoriﬁce ﬂow coefﬁcient [ – ]

Kconstant used in the correction functions [–]

Rperfect gas constant [J/kg/◦K]

Ttemperature [◦K]

142

V. SZENTE and J. VAD

Greek letters

αangle of valve seat [◦]

κspeciﬁc heat ratio [–]

ρﬂuid density [kg/m3]

Subscripts

up upstream values

down downstream values

jet values in the ﬂuid jet at the vena contracta

crit values at critical pressure ratio

corr transformed values

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