ArticlePDF Available

Abstract and Figures

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 characterization 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 characterization of through-flow behavior of pneumatic valves of various geometries.
Content may be subject to copyright.
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 SZENTEand 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·(κ1)·pdown
pup 2
κ
pdown
pup κ+1
κ
if pdown
pup >pdown
pup crit (subsonic flow), (2a)
Cm=2·κ
R·(κ1)·2
κ+11
κ1
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
κ1
=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 ·AAjet.(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
AAjet. 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
κ1·R·Tup ·1pdown
pup κ1
κif pdown
pup >pdown
pup crit
,(6a)
v2
jet =2·κ
κ1·R·Tup ·1pdown
pup κ1
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=
1pdown
pup
2·κ
κ1·pdown
pup 1
κ
·1pdown
pup κ1
κ
if pdown
pup >pdown
pup crit
,(8a)
Cq=
1pdown
pup
2·κ
κ1·pdown
pup 1
κ
crit ·1pdown
pup κ1
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
-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 8asthis 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 =CqCqcrit·K1+Cqcrit +K2
if pdown
pup >pdown
pup crit
,(9a)
Cqcorr =CqCqcrit·K3+pdown
pup ·K1K3
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
References
[1] MACK, J., ABS-TCS-VDC Where Will the Technology Lead Us?, Sale international, 1996.
[2] SZ˝
OCS,K.–K˝
OFALUSI,P.–NÉMETH,S.,Fékrendszerek, Maróti-Godai Könyvkiadó Kft.,
1997.
[3] SZENTE,V.–VAD,J.–LÓRÁNT,G.–FRIES, A., Computational and Experimental In-
vestigation on Dynamics of Electric Braking Systems, Proc. 7th Scandinavian International
Conference on Fluid Power, May 2001, Linköping, Sweden, 1(2001), pp. 263–275.
[4] PERRY, J. A., Critical Flow Through Sharp-Edged Orifices, Trans. ASME,71 1949.
[5] BIDEAUX,E.–SCAVARDA, S., A Pneumatic Library for AMESim, Proc. ASME’98 Confer-
ence, November 1998, Anaheim, California.
[6] BUSEMANN, A., Hodographenmethode der Gasdynamik, Zeitschriftfür angewandte Math. und
Mech.,17 No. 2 (1937).
[7] OSWATITSCH,K.,Grundlagen der Gasdynamik, Springer-Verlag, 1976.
[8] BROWER,W.B.–EISLER,E.–FILKORN,E.J.–GONENC,J.–PLATI,C.–STAGNITTI,J.,
On the Compressible Flow Through an Orifice, Trans. ASME,115 (1993).
[9] GRACE,H.P.–LAPPLE, C. E., Discharge Coefficients for Small-Diameter Orifices and Flow
Nozzles, Trans. ASME,73 (1951), pp. 639–647.
[10] JOBSON, D . A., On the Flow of a Compressible Fluid Through Orifices, Proc.IME,169(1955),
pp. 767–779.
[11] TSAI,D.H.–CASSIDY, E. C., Dynamic Behavior of Simple Pneumatic Pressure Reducer, J.
Basic Eng.,83, (1961).
[12] SZENTE,V.–VAD, J., Computational and Experimental Investigation on Solenoid Valve Dy-
namics, Proc. IEEE/ASME International Conference on Advanced Intelligent Mechatronics,
July 2001, Como, Italy, 1(2001), pp. 618–623.
[13] LAJOS,T.,Az áramlástan alapjai,M
˝
uegyetemi Kiadó, 2000.
[14] ISO 6358:1989, Pneumatic Fluid Power. Components Using Compressible Fluids. Determina-
tion of Flow Rate Characteristics.
[15] SZENTE,V.–VAD, J., Computational and Experimental Investigation on the Flow Characteris-
tics of Small-Scale Pneumatic Solenoid Valves, 2nd International Conference on Heat Transfer,
Fluid Mechanics and Thermodynamics, June 2003, Victoria Falls, Zambia.
[16] FLUENT documentation, v6.1.18, 2002, http://www.fluent.com
[17] MCCLOY,D.–MARTIN,H.R.,Control of Fluid Power: Analysis and Design, Chichester,
Ellis Horwood, 1980.
... The radial leakages are simulated by internal by-pass (R). The mass flow rate through the valves is calculated via Equation (3) [25]. ...
... where A is the actual orifice area, C q is Perry's coefficient from his correlation [26], C m is the flow coefficient as also described by Szente et al. [25], P up and T up are the upstream pressure and temperature. The mechanical losses (ML) are modelled as a fixed frictional loss (one constant to be estimated). ...
... where A is the actual orifice area, Cq is Perry's coefficient from his correlation [26], Cm is the flow coefficient as also described by Szente et al. [25], Pup and Tup are the upstream pressure and temperature. ...
Article
Full-text available
This paper presents a virtual model of a scroll compressor developed on the one-dimensional analysis software Simcenter Amesim®. The model is semi-empirical: it needs some physical details of the modelled machine (e.g., the cubic capacity), but, on the other hand, it does not require the geometrical features of the spirals, so it needs experimental data to calibrate it. The model also requires rotational speed and the outlet temperature as boundary conditions. The model predicts the power consumption and the mass flow rate and considers leakages and mechanical losses. After the model presentation, this paper describes the test bench and the obtained data used to calibrate and validate the model. At last, the calibration process is described, and the results are discussed. The calculated values fit the experimental data also in extrapolation, despite the model is simple and performs calculations within 7 s. Due to these characteristics, the model is suitable for being used in a larger model as a sub-component.
... The measured values of the discharge coefficient have been compared with theoretic predictions of the contraction coefficient ( [14,15]) as well as with the published values of the discharge coefficient determined on both experimental [9] and numerical [15] basis. A comparison of results is shown in Fig. 3. ...
... The measured values of the discharge coefficient have been compared with theoretic predictions of the contraction coefficient ( [14,15]) as well as with the published values of the discharge coefficient determined on both experimental [9] and numerical [15] basis. A comparison of results is shown in Fig. 3. ...
... The figure relates the discharge/contraction coefficient to the outlet-to-inlet pressure ratio. As can be seen in the figure, differences between the values of C D published in the relevant literature are significant, which is primarily bound to different assumptions compressed air installation pressure regulating and flow metering line concerning the analysed geometry: side-wall opening (authors' results), orifice ( [9,14]), a complex geometry [15]. Despite of the discrepancy between the presented sources, it may be observed as a general rule that both coefficients decrease significantly (by 10e40%) with the increasing pressure ratio. ...
Article
The paper discusses chosen issues concerning damaged gas pipelines. Attention is paid to modelling the steady-state flow of natural gas in distribution pipelines, and the most commonly applied models of isothermal and adiabatic flow are evaluated for both the ideal and the real gas properties. A method of accounting for a leakage by means of a reference flow equation with a discharge coefficient is presented, and the dependency of the discharge coefficient on pressure is demonstrated both with literature data and the authors' experimental results. A relevant computational study of a pipeline failure is presented for a high- and a medium pressure pipeline. The importance of an appropriate choice of the flow model (isothermal or adiabatic flow of real or ideal gas) is demonstrated by the results of the study. It is shown that accounting for the variability of the discharge coefficient is required if medium pressure pipelines are analysed. However, it is eventually shown that the impact of the discharge coefficient on the predicted outflow rate is of lesser importance than that of the applied flow model.
... It indicates that the flow coefficient is affected by system pressure. Some research [24,45] also proves it. ...
Article
Full-text available
The air leakage in sintering machines affects the technological and economic indexes of the sintering process. It is of great significance to monitor and estimate the key areas. Mathematical models of sintering air leakage through holes in the steady-state process are given based on the fluid mechanics to predict the flow rate and effect on the key area. It was found that the hole model is the application of constant orifice outflow in the computation of sintering air leakage. The counter-flow bed model is suitable for predicting the flow rate through a complete break in sintering wind boxes. Furthermore, This paper proposes a new hole–bed generalized model to cover all the possible hole diameters for further high-precision application. The model connects the leakage hole diameter with the sintering process for the first time and establishes their coupling relationship. The pressure state in the sintering system depends on the ratio of the leakage hole area to the sintering bed area. The proposed fast estimation models are a step forward in developing more precise and powerful calculation tools to foresee the effects and consequences of sintering air leakage. It has a good prospect for reducing and replacing complex manual measurement and bringing some insight into the state of the art that could be improved in the future.
... The Borda-type orifice was employed to describe the dispersion flow through a leak hole, as shown in Fig. 3. The momentum conservation in the control volume of a Borda-type orifice is characterized by (Szente and Vad, 2003) ...
Article
High-pressure gas pipe leakage is a topic of great concern in the risk assessment and safety management of pipeline systems. Traditional ideal-gas models with the assumption of fully ruptured leak loss are valid for a large leak hole in low-pressure gas pipelines (3–14 bar). This paper developed a new model for small leaks in high-pressure gas pipelines, which improves the traditional models in two aspects: the introduction of a mass fraction to characterize the proportion of leaked gas flow to in-pipe gas flow, and the integration of the real-gas thermodynamics into the fundamental governing equations to include the gas compressibility factor in the high-pressure conditions (>14 bar). To examine the validity and practicality of the new real-gas model, comprehensive comparison tests were carried out with the results of the benchmark numerical simulations and the hazard modelling software ALOHA, respectively. In the light of prediction accuracy, the new real-gas model shows its advantage over the traditional ideal-gas model, the improved ideal-gas model, and the semi-empirical correlation proposed by previous research. The virtues of high accuracy and short computational time confirmed the effectiveness of the new real-gas model in the responsive evaluation of high-pressure gas pipe leakage with small leak holes.
... Natural gas is a kind of high-quality, efficient, clean energy and raw material. Since the 1970s, the development of natural gas in the world has been accelerated, and the research on gas exploitation, transportation, and storage has received increasing attention [1][2][3][4][5][6][7]. Nowadays, natural gas pipelines account for almost half of the total length of pipelines in the world. ...
Article
Full-text available
Small leaks in natural gas pipelines are hard to detect, and there are few studies on this problem in the literature. In this paper, a method based on the extended Kalman filter (EKF) is proposed to detect and locate small leaks in natural gas pipelines. First, the method of a characteristic line is used to establish a discrete model of transient pipeline flow. At the same time, according to the basic idea of EKF, a leakage rate is distributed to each segment of the discrete model to obtain a model with virtual multi-point leakage. As such, the virtual leakage rate becomes a component of the state variables in the model. Secondly, system noise and measurement noise are considered, and the optimal hydraulic factors such as leakage rate are estimated using EKF. Finally, by using the idea of an equivalent pipeline, the actual leakage rate is calculated and the location of leakage on the pipeline is assessed. Simulation and experimental results show that this method can consistently predict the leakage rate and location and is sensitive to small leakages in a natural gas pipeline.
... Other authors propose a semi-experimental approach to determine the valve coefficients [8] and [9]: the experimental data are used to define a mathematical model by which to determine the air flows for different operation conditions. Either the method suggested by the ISO standard or the ones proposed by other authors are difficult to be applied especially due to the complex measurement chain needed. ...
Conference Paper
Full-text available
Several strategies, in order to improve actuator's control and to increase the bandwidth, consider the relationship between the valve's driving signal and the air flow rate. Such an approach to the control strategy takes advantage of the evaluation of the valve's characteristic parameter known as sonic conductance. The sonic conductance can be measured following the procedure stated by the standard ISO 6358. Nevertheless the measurement carried out according to this standard is very expensive in terms of time and air consumption. In this paper an alternative method to evaluate the sonic conductance is presented. The method is based on a new practical approach: the sonic conductance is evaluated leaving the valve mounted on the actuator and using only the piston's position transducer. The steady state piston's motion allows to determine the sonic conductance. The new approach allows to get the conductance in a very short time, without the need to use a proper test-bench and pressure transducers. Moreover performing the measurements directly on the pneumatic axis allows to characterize not only the valve but the duct connecting the valve to the actuator's chamber too.
... In recent years, numerous researchers have started conducting CFD visualization analyses to obtain more details about the flows in valves, and CFD simulation is considered as a powerful tool to study the characteristics of these valves. Szente and Vad [7] established a semi-empirical model for predicting the flow characteristics of a small electro-pneumatic valve and used quasi-3D computation to correct the model. Dimitrov [8] investigated the flow-pressure coefficient of a pilot-operated pressure relief valve by theoretical analysis and verified the results through experiments. ...
Article
The transient flow and dynamic characteristics of a PWM-controlled solenoid valve are simulated by CFD to investigate the pressure control performance of the solenoid valve under PWM-controlled conditions. A group of cases with a constant valve opening but different pressure drops are used in the steady-state CFD simulations, which cover both sonic and subsonic flows. The mass flow rates obtained from CFD simulations are compared with theoretical results and show a good agreement, indicating that CFD is capable of handling the complex flows. The dynamic mesh technique is used to simulate the movement of the valve spool, and transient CFD simulations are conducted to analyze the dynamic characteristics. The responses of the control pressure and the mass flow rate to the duty-cycle variation are analyzed for 30%, 50%, and 80% duty cycles. The mean control pressure values are verified by experimental data. The flow force exerted on the spool is recorded and the transient force is compared with the steady force. The pressure wave transmission during the valve opening process is captured. The influence of the valve opening time on the control pressure is also investigated, and results show that the former can affect the mean control pressure.
Article
We numerically investigated high-pressure hydrogen leakage from transportation facilities, focusing on the steady mass flow rate and pressure distribution in a tube during the leakage. We studied steady leakage from a square opening in a square duct as well as leakage from a ruptured cylindrical tube with unsteady closure of a cutoff valve from fully open. A prediction model for the mass flow rate and pressure distribution inside the tube was proposed; such a model would help prevent physical hazards during an accident. We considered changes in the physical quantities according to the fluid dynamics occurring inside the tube. The flow properties were divided into two phases: (i) the unsteady expansion wave propagating inside a tube filled with hydrogen and (ii) the acceleration of hydrogen due to the reduction in the cross-sectional area between the tube and the leakage opening. To close the prediction model, we introduced contraction coefficient models depending on how the hydrogen leakage occurred. The mass flow rate and pressure drop during the leakage estimated by our prediction model showed good agreement with numerical simulation results when the contraction coefficient model was appropriately chosen. This model is considered highly applicable to the construction condition of pressure sensors, the operating conditions of a valve, and the prediction of mass flow rate during an accident.
Article
Several strategies, in order to improve an actuator's control and to increase the bandwidth, consider the relationship between the valve's driving signal and the air flow rate. Such an approach to the control strategy takes advantage of the evaluation of the valve's characteristic parameter, known as sonic conductance. The sonic conductance can be measured following the procedure stated by the standard ISO 6358. Nevertheless, the measurement carried out according to this standard is very expensive in terms of time and air consumption. In this paper, an alternative method to evaluate the sonic conductance is presented. The method is based on a new practical approach: the sonic conductance is evaluated leaving the valve mounted on the actuator and using only the piston's position transducer. The steady state piston's motion allows us to determine the sonic conductance. The new approach allows us to get the conductance in a very short time, without the need to use a proper test bench and pressure transducers. Moreover, performing the measurements directly on the pneumatic axis allows us to characterize not only the valve but the duct connecting the valve to the actuator's chamber too.
Article
Dynamic modeling of flow process inside a pressure regulating and shut-off valve has been investigated using a computational fluid dynamic approach. The valve is designed to reduce high inlet pressure to a lower level of outlet pressure which remains almost constant. With the change of inlet pressure, the change in position of the spool inside the valve was calculated using a force balance approach. The Navier–Stokes equation along with appropriate turbulent closure has been solved for this purpose in the compressible flow regime using ANSYS-FLUENT software with special functions developed for calculation of flow force. The code could predict the spool movement and the final spool position when the spool position is deviated from equilibrium. The final spool position and time required to reach equilibrium, besides the flow parameters, also depends on the value of friction coefficient between spool and the valve-body. Higher values of friction coefficient between the spool and the vale body is found to be associated with faster stability of the spool.
Conference Paper
Full-text available
Electro-pneumatic (EP) components are frequently used in brake systems of commercial vehicles. The simulation of EP brake systems is of great importance in order to understand their dynamics for developing a control logic being robust but fulfilling the modern functional demands. On the other hand, the simulation aids the design of EP components being able to execute the commands of a precision control. The paper presents a flexible computational simulation tool being applied in industrial research and development related to complex mechatronics in brake systems for commercial vehicles. The Electric Braking System (EBS) case study presented herein comprises an air supply unit, an EBS modulator, piping, a diaphragm brake chamber, and the connected brake mechanism. The simulation environment is AMESim® 3.0. Considering the complexity of the EP components and the related phenomena, special models have been elaborated for the solenoid valves, piping, and the diaphragm brake chamber. The simulation results show good agreement with measurement data. The comparative numerical and experimental study confirmed that the simulation tool can be effectively used in design, research and development of EP brake systems.
Conference Paper
Full-text available
A generally applicable, synthetic simulation model and computational tool has been elaborated for dynamic simulation of solenoid valves (SV) applied as control elements in fast-response pneumatic fluid power systems. The SV of case study has been modeled as a system consisting of coupled magnetodynamic and mechanical subsystems. At the present state of investigation, fluid dynamic effects are not considered in the model. The appropriateness of the model has been verified by experimental data. The simulation model resolves the valve body motion and the solenoid current at a high accuracy. It has been pointed out in the concerted numerical and experimental studies that the valve body performs repetitive flexible collision (bouncing) at its opened end-position. Such initial vibrating motion of valve body may affect favorably the SV fluid transmission characteristics, transferring momentum to the fluid in the orifice cross-section. The investigation reveals that the valve body penetrates to the flexible contact surface at its opened end-position. This results in an actual valve body displacement 50 percent higher than the geometrical displacement (determined from the SV geometry with neglect of penetration). Such modified displacement may result in flow transmission characteristics differing significantly from the SV design condition (considering the geometry with no deformation). The SV flow transmission characteristics will be studied in a SV model supplemented with fluid mechanical submodels
Article
This paper presents an analysis of the dynamic behavior of a simple pneumatic pressure reducer. Both the nonlinear and the linearized problems were studied. Some experimental results also were obtained on a working reducer model to check the validity of the analysis. The agreement between the nonlinear solutions and the experimental results was satisfactory. The nonlinear and the linearized solutions were compared in detail so as to bring out the essential features of the dynamic behavior in both cases. The stability problem was studied also, and a set of stability criteria for the linearized case was formulated in terms of the design and operating parameters of the reducer. In the few sample cases studied, these criteria gave correct qualitative predictions of the stability of the reducer in both the linearized case and the nonlinear case. The flow forces on three typical flowmetering values were determined by experimental measurements (Appendix 2). These results were used in the analytical part of the paper.
Article
Discharge coefficients were determined in a standard 1-in. pipe for thick-plate and knife-edge orifices of 1/32-in. to 3/4-in. diam, and for critical (sonic) flow nozzles of 1/32-in. to 1/4-in. diam. With properly constructed knife-edge orifices the discharge coefficients were established within ±0.5 per cent for all sizes, but, with small orifices of 1/4 to 1-hole-diam thickness, the discharge coefficients were not reproducible. Thick-plate orifices (1-hole-diam thick) were found to be as good or better than nozzles in constancy of discharge coefficients when used under critical-flow conditions.
Article
An investigation was made to discover the flow rates of air through sharp-edged orifices in which the pressure drop was below the critical; the relationship of pressure, temperature, and orifice area; and a comparison between the nozzle and the orifice. Equations are derived for expressing orifice flow in both the critical and subcritical regions.
Article
A new theory for the compressible flow through an orifice is presented which provides a significant advance over the approach currently employed. As the flow approaches the critical regime (local sonic condition), measurements diverge from the theoretical result due to the non-one-dimensionality of the flow. Nevertheless, a straight forward correlation is available, and the measurements for different pipe/orifice geometries all appear to lie in the vicinity of a single, universal curve. As the flow approaches the incompressible condition the correlation factor (the discharge coefficient) becomes unity.
Article
By making certain basic assumptions, the author has determined a theoretical expression for the contraction coefficient, C, appropriate to an orifice when transmitting a compressible fluid, either above or below the critical pressure ratio, provided that the corresponding value for incompressible flow, Ci, be known.
Conference Paper
The present development of the simulation and virtual prototyping actually has lead to the development of specialized software packages to industries using fluid power based components as the truck, railway or car industry. These specialized packages such as AMESim (Advanced Modeling Environment for performing Simulations), consist of a robust solver and multidisciplinary libraries of models, since the solving method is strongly linked to the simulation aim and the modeling step itself. AMESim has already proposed hydraulic and one-dimensional mechanical components and some useful tools for the study of fluid power systems. As strong links exist between hydraulic and pneumatic technology, we have implemented a library of pneumatic components that could be used in connection with the other existing libraries. The models arise directly from our pseudo bond graph knowledge and our experience of measuring existing pneumatic components. The first library (the Basic Elements Library) contains the basic elements: sources, modulated dissipative elements, modulated transformers, capacitive elements, sensors, which allow the simulation of any pneumatic based system. In the second library called the Elementary Component Library, we propose a selection of standard components such as relief valves, distributors, servo-valves and spools. The third and last library is the Advanced Component Library and it consists of models for specialists such as pipes, variable volume chambers, and more complex components. In this paper, we propose an overview of this library of pneumatic conrponents and we show its possibilities by simulating of an existing pneumatic systems.