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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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