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Stability of gas pressure regulators
Naci Zafer
a,*
, Greg R. Luecke
b,1
a
Department of Mechanical Engineering, Eskisehir Osmangazi University, Makine Muhendisligi Bolumu,
26480 Bati Meselik, Eskisehir, Turkey
b
Department of Mechanical Engineering, Iowa State University, Ames, IA 50010, USA
Received 1 May 2005; received in revised form 1 June 2006; accepted 2 November 2006
Available online 3 January 2007
Abstract
Gas pressure regulators are widely used in both commercial and residential applications to control the operational pres-
sure of the gas. One common problem in these systems is the tendency for the regulating apparatus to vibrate in an unsta-
ble manner during operation. These vibrations tend to cause an auditory hum in the unit, which may cause fatigue damage
and failure if left unchecked. This work investigates the stability characteristics of a specific type of hardware and shows
the cause of the vibration and possible design modifications that eliminate the unstable vibration modes. A dynamic model
of a typical pressure regulator is developed, and a linearized model is then used to investigate the sensitivity of the most
important governing parameters. The values of the design parameters are optimized using root locus techniques, and the
design trade-offs are discussed.
2006 Elsevier Inc. All rights reserved.
Keywords: Dynamics; Modeling; Stability; Vibrations; Pressure regulator
1. Introduction
Gas regulators are devices that maintain constant output pressure regardless of the variations in the input
pressure or the output flow. They range from simple, single-stage [1,2] to more complex, multi-stage [3,4], but
the principle of operation [5] is the same in all. High pressure gas flows through an orifice in the valve and the
pressure energy in the gas is converted to heat and flow at the lower, regulated, pressure. The orifice faces a
movable disk that regulates the amount of gas flow. A flexible diaphragm is attached to the disk by means of a
mechanical linkage. The diaphragm covers a chamber such that one side of the diaphragm is exposed to atmo-
spheric pressure and the other is exposed to the regulated pressure. When the regulated pressure is too high,
the diaphragm and linkage move the disk to close the orifice. When the regulated pressure is too low, the disk
is moved to open the orifice and allow more gas pressure and flow into the regulator. On the opposite side of
the diaphragm, an upper chamber houses a wire coil spring and a calibration screw. The screw compresses the
0307-904X/$ - see front matter 2006 Elsevier Inc. All rights reserved.
doi:10.1016/j.apm.2006.11.003
*
Corresponding author. Tel.: +90 222 239 3750x3387; fax: +90 222 239 3613/229 0535.
E-mail addresses: nzafer@ogu.edu.tr (N. Zafer), grluecke@iastate.edu (G.R. Luecke).
1
Tel.: +1 515 294 5916; fax: +1 515 294 3261.
Applied Mathematical Modelling 32 (2008) 61–82
www.elsevier.com/locate/apm
spring, which changes the steady state force on the diaphragm, allowing for the adjustment of the regulated
pressure set point. If the regulated gas pressure rises above the safe operational pressure, an internal relief
valve is opened to vent the excess gas through the upper chamber and into the atmosphere to prevent the dan-
ger of high pressure gas at the regulator outlet.
Little information is published regarding these devices, due to concerns over proprietary information. One
reported study concerning high-pressure regulators is done by Kakulka et al. [6]. The regulator studied was a
piston pressure-sensing unit that had a conical poppet valve that regulates the gas flow. The dynamic effects of
restrictive orifices and the upstream and downstream volumes were addressed in the modeling and analysis.
However, the source of the oscillations in the downstream exiting area, as well as the damping and the friction
effects in the physical system, were left out. Several researchers have, on the other hand, addressed the
unwanted oscillations and noise. Waxman et al. [7] eliminates the noisy oscillations with the implementation
of a dead-band achieved by two micro switches. The design includes a stepping motor activated with the signal
from a differential pressure transducer. Baumann [8] proposes a much less expensive solution, the use of a sta-
tic pressure reducing plate with multiple holes. Ng [9] compares the effectiveness and cost of several methods
that reduce or minimize the noise. Ng [10] addresses pressure regulators for liquids and names cavitations, the
damage caused by continuous formation and collapse of microscopic bubbles, to be the cause of hydrody-
namic noise. Cavitations produce noise, vibration, and even cause significant damage. Ng states that the
use of quiet valves, or an orifice with multiple holes are not the solutions to this problem, since they are expen-
sive and the small passages are most likely to be plugged by solid particles in the flow. Dyck [11] states that a
larger restrictive orifice improves flow performance, but a small one makes the system more stable and is less
sensitive to downstream pressure fluctuations. Liptak [12] gives an equation for the offset in the regulated pres-
sure with changing flow and shows that any decrease in this offset pressure decreases the stability of the reg-
ulator, resulting in a noisy regulator with oscillatory pressure cycling. To stabilize the system he suggests using
larger downstream pipe, a more restrictive flow from orifice to the lower chamber, straight lengths of pipe
upstream and downstream. He also points out that maintaining gas flow at less than sonic velocities and elim-
inating changes in flow directions would reduce the noise. It is obvious that these changes are very restrictive
from the design and installation perspective, and are not guaranteed to stabilize the system.
In this study, we develop a comprehensive dynamical model for a gas pressure regulator from first principles
in order to gain a better understanding of its behavior. We first model an existing regulator and use empirical
data as necessary to identify parameter values for the model. Using a linearized version of this model, we inves-
tigate the effects of parameter variations using classical root-locus techniques. Our motivation is to design a tool
that allows for the identification of the most influential system parameters on the stability of the system, and to
allow an assessment of any effects that changes in these parameters have on stabilization of the regulator.
A schematic diagram of a typical gas pressure regulator (American Meter Gas Regulator, Model 1800) is
shown in Fig. 1. High pressure gas flows through an inlet orifice that is opened or closed by a disk and linkage
Fig. 1. Operational diagram of a typical gas pressure regulator.
62 N. Zafer, G.R. Luecke / Applied Mathematical Modelling 32 (2008) 61–82
attached to a diaphragm. The diaphragm moves in response to the balance between pressure inside the reg-
ulator body and the adjustment spring force. As the regulated pressure increases, the disk closes to restrict
the incoming gas. When the regulated pressure is too low, the disk opens to allow more gas into the body cav-
ity. The stability of the system depends on the amount of damping in the system, and much of the damping
comes from flow restrictions within the regulator. In order to develop our model, we define three control vol-
umes that are used in the dynamic analysis, identified in Fig. 2: the body chamber, the upper chamber and the
lower chamber. Each control volume is characterized by pressure, volume, and the density as a function of
time. For the purpose of this analysis, these control volumes are used to track the mass flow through the
system.
2. Gas dynamics governing equations
Modeling of operation of the gas pressure regulator is based on the physical behavior of compressible fluid
flow. The modeling in this work uses the fundamental principles of ideal compressible flow, the principle of
conservation of mass, and well-known expressions for flow through orifices [13,14]. For the development of
the pressure regulator model, we assume that the operating fluid is a perfect gas. Kinetic theory is then used
to express the state of a particular control volume according to the ideal gas equation:
PV ¼mRT ;ð1Þ
where Pis the pressure, Vis volume, mis mass, Ris a gas constant, Tis temperature. Assuming the process is
adiabatic and reversible, the second law of thermodynamics provides a relationship between the pressure and
the density of the fluid:
P
qk¼Constant:ð2Þ
Fig. 2. Pressure regulator schematic.
N. Zafer, G.R. Luecke / Applied Mathematical Modelling 32 (2008) 61–82 63
By considering the time differentials of Eqs. (1) and (2) together with the definition of density, one can easily
show that
1
k
_
P
Pþ
_
V
V¼
_
m
m;ð3Þ
where kis the specific heat ratio, and
_
m¼qQ;m¼qV:ð4Þ
Because the density for a fixed operational flow rate is constant, volumetric flow rate will be used, rather than
the more conventional mass flow rate. Eq. (3) provides a basis for analysis and modeling of the pressure reg-
ulator and describes the relationship between pressure, volume, and the mass flow for a particular control
volume.
2.1. Lower chamber
Applying Eqs. (3) and (4) to the lower chamber, we get
1
kL
_
PL
PLþ
_
VL
VL¼QL
VL
:
Note that the minus sign indicates that our convention of the direction of positive flow, Q
L
, is out of the cav-
ity. The motion of the diaphragm is related to change in volume by
_
VL¼_
xdAd;
where _
xdand A
d
are the velocity and the surface area of the diaphragm. Because of the sign convention chosen
for the diaphragm motion, a positive change in the diaphragm position causes the lower chamber volume to
decrease. Although A
d
has a nonlinear relationship with x
d
, it is assumed constant for linear simulations. This
assumption can be made because the operational inlet flow rates are, in general, small, less than 0.01 m
3
s
1
,
and the diaphragm travel also remains relatively small for these flow rates. For the nonlinear simulations, the
more accurate empirical relationship shown in Fig. 3 is used.
The overall equation governing the pressure–flow relationship in the lower chamber is then
_
PL¼kL
PL
VLðQLþ_
xdAdÞ:ð5Þ
Linearizing this equation using a Taylor series expansion and neglecting the higher order terms, we have:
e_
PL¼kL
PL0
VL0 ðe
QLþe_
xdAdÞ;ð6Þ
where the notation ‘‘’’ is used to express an incremental change of the related quantity.
Fig. 3. Empirical data for diaphragm travel vs. diaphragm area.
64 N. Zafer, G.R. Luecke / Applied Mathematical Modelling 32 (2008) 61–82
2.2. Upper chamber
A similar analysis for the upper chamber yields the differential equation governing the change in the pres-
sure of the upper chamber:
1
kU
_
PU
PUþ
_
VU
VU¼QU
VU
;
where _
VU¼_
xdAd. The overall equation governing the pressure–flow in the upper chamber and the linearized
form are then
_
PU¼kU
PU
VUðQU_
xdAdÞ;ð7Þ
e_
PU¼kU
PU0
VU0 ðe
QUe_
xdAdÞ:ð8Þ
2.3. Body chamber
Although the flow out of the regulator is not steady, the gas pressure in the connected lower chamber and
body chamber cavities fluctuates as the regulator moves to equilibrium. For a compressible gas, these fluctu-
ations also compress the gas and change the density of the fluid. However, the changes in density in these
chambers is small compared with the change density as the fluid moves from the high-pressure inlet to the
lower pressure, regulated, body pressure. We can account for this change in density with an expansion ratio.
Solving Eq. (3) for the body chamber using the outlet pressure gives:
1
kout
_
Pout
Pout ¼
_
mbody
mbody
:
Note that there is no change in volume for the body chamber, so that _
Vbody ¼0 and the mass balance for the
body chamber in Fig. 1 is obtained by summing the mass flow rates in and out of this chamber.
_
mbody ¼_
min _
mout þ_
mL:
Substituting _
m¼qQin this equation for each of the control volumes, it follows that
Qbody ¼jQin Qout þQL;
where Q
in
is the inlet flow-rate, and the expansion ratio
j¼qin
qbody
is used to account for the change in density of the inlet gas to that of the outlet gas. Again, note that q
body
=
q
out
=q
L
is assumed because the differences in pressure are small compared to the difference in pressure
between these and the inlet pressure. Combining these equations gives the outlet pressure as a function of
the flow crossing the control boundary:
1
kout
_
Pout
Pout ¼jQin Qout þQL
Vbody
;
_
Pout ¼kout
Pout
Vbody ðjQin Qout þQLÞ:ð9Þ
Using a Taylor series expansion, we also obtain the linear, incremental, equation:
e_
Pout ¼kout
Pout0
Vbody0 ðje
Qin e
Qout þe
QLÞ:ð10Þ
N. Zafer, G.R. Luecke / Applied Mathematical Modelling 32 (2008) 61–82 65
2.4. Flow governing equations
Fluid enters and exits the gas regulator through three flow holes, the inlet valve, the outlet orifice, and
through the relief valve on the top of the upper chamber, each of these flow components contributes to the
dynamic response characteristics of the overall regulator.
Because the pressure drop is very large from the inlet pressure through the orifice, the sonic, or critical, flow
into the regulator is proportional to the throat area [16], or the plunger travel. This provides a linear relation-
ship between the flow into the regulator and the effective flow area between the orifice and the disk. This effec-
tive area is dependent on the annular distance between the face of the disk and the orifice and the specific
geometry of both the disk and orifice, but is more or less constant in a specific valve.
Qin ¼Cinxp;ð11Þ
where the constant C
in
is obtained from empirical data shown in Fig. 4. This equation is already linear, and the
incremental representation is:
e
Qin ¼Cin~
xp:ð12Þ
Flow in or out of the upper chamber occurs through a relief cap with a small vent hole. Generally, the upper
chamber relief cap ventilation hole regulates flow during smaller adjustments of the diaphragm, and a spring-
loaded relief plate prevents pressure build-up during large diaphragm motions or in the event of a rupture of
the diaphragm. Using the assumption of a small orifice area when compared to the upper chamber cross sec-
tional area, the flow through the ventilation hole in the upper chamber is expressed with the well-known
square root relationship
QU¼CUt ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
PUPatm
p;ð13Þ
where C
Ut
is the nonlinear flow coefficient. With P
atm
assumed constant, this equation is linearized using
Taylor series expansion to yield
e
QU¼CUe
PU;ð14Þ
with CU¼CUt
2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
PU0
Patm
p.
Note that for values of P
U
close to the atmospheric pressure, or e
PUclose to zero, the linearized flow coef-
ficient C
U
gets large and leads to large flows in response to small pressure changes to maintain the equilibrium
conditions. Theoretically, this square-root relationship leads to an infinite slope of the pressure–flow curve,
and this is borne out by the experimental data for flow through the upper chamber orifice at various pressure
differences, shown in Fig. 5. However, as the pressure difference gets very small, this theoretical square-root
Fig. 4. Empirical data defining flow into the regulator as a function of the plunger travel.
66 N. Zafer, G.R. Luecke / Applied Mathematical Modelling 32 (2008) 61–82
relationship breaks down and leads to a linear relationship with a very high gain. For the modeling in this
work, the empirical data depicted in Fig. 5 is used to get the flow coefficient for the nonlinear square-root
model, C
Ut
. For the linear model, C
U
is the slope of the curve in Fig. 5, and with low flow rates, C
U
will
be very large.
Flow out of the regulator is modeled using the assumption that the outlet orifice area is variable, which
affects the gas pressure in the regulator body. This approach assumes that the flow demand to the regulator
is not separated from the body by additional dynamics from subsequent piping, and that the changes in flow
demand can be modeled as a variable orifice area at the regulator outlet. Using this approach, the flow out of
the orifice is:
Qout ¼ACdffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Pout Patm
p;ð15Þ
where ‘‘A’’ represents the variable area or demand from downstream. Linearization of this flow relationship
about an equilibrium state has a slightly different result, because the pressure difference between the outlet and
the atmosphere never goes to zero. The outlet flow is a function of two variables, the pressure drop,
(P
out
P
atm
), and the flow area, A. This linearization leads to
Qout ¼Qout0 þoACdffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Pout Patm
p
oAPout¼Pout0
A¼A0
ðAA0ÞþoACdffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Pout Patm
p
oPout Pout¼Pout0
A¼A0
ðPout Pout0Þ
or
e
Qout ¼C10 e
AþC20 e
Pout;ð16Þ
where C10 ¼Qout0
A0,C20 ¼A2
0C2
d
2Qout0. While we continue to use empirical data to find typical value for the discharge
coefficient, C
d
, in Eq. (15), for a given equilibrium condition the linear model uses the constant coefficients in
Eq. (16).
The flow into and out of the lower chamber is a complex function of the flow of gas through the regulator
and the shape of the flow cavity. The classical square-root relationship shown in Eq. (15) does not do a good
job of describing the pressure–flow relationships found for regulators experimentally. Fig. 6 shows a typical
relationship for the lower chamber and body chamber pressures at various steady state flow conditions. This
data indicates that the lower chamber pressure is lower than the outlet pressure in the body of the regulator.
This is known as the ‘‘boost effect’’ and is caused by the venturi effect of the dynamic flow through the valve
body. In practice, this boost effect is carefully designed into the regulator as a means of obtaining constant
regulation pressure over a wide range of flows, but using a model such as that in Eq. (15) means that there
should always be flow from the lower chamber into the body. Since the lower chamber is a fixed volume at
steady state flow, this clearly cannot happen. Multiple flow paths, the dynamics of the fluid, and the geometry
of internal obstructions make it difficult to develop an effective analytic model, but one approach is to imagine
Fig. 5. Empirical data for the upper chamber defining pressure as a function of the flow.
N. Zafer, G.R. Luecke / Applied Mathematical Modelling 32 (2008) 61–82 67
a reversed pitot tube as shown in Fig. 7, where the tube is oriented to face downstream to allow measurement
of the venturi effect of the moving fluid. The static pressure head (the effective lower chamber pressure), mea-
sured at P
L
is found using Bernoulli’s equation:
P
L¼PLþ0:5qoutv2
out ¼PLþqout
2A2
m
Q2
out;ð17Þ
where A
m
is the outlet orifice effective area, and P
L
is the measured pressure inside the lower chamber (which is
also the pressure inside the tube in Fig. 7).
At steady state, the pressure difference, P
LPout, should be zero, since there is no flow in or out of the
closed lower chamber, and we can use the experimental test data in Fig. 6 to compute the effective coefficient
on the last term in Eq. (17):
P
L¼Pout )Pout PL¼P
LPL¼qout
2A2
m
Q2
out:
The pressure difference from Fig. 6,P
out
P
L
, should be proportional to the square of the flow, and from
Fig. 8a, the empirical data for lower flow rates shows that the effective venturi coefficient is:
qout
2A2
m¼5:6106:
Natural gas is mostly methane, and the density at atmospheric pressures is approximately 0.7 kg/m
3
. Using
this density leads to an effective outlet area of A
m
= 2.5 ·10
4
m
3
.
Fig. 6. Empirical pressure and flow relationships.
Fig. 7. Reversed pitot tube.
68 N. Zafer, G.R. Luecke / Applied Mathematical Modelling 32 (2008) 61–82
In our model, we used a maximum 3/4 in. (0.019 m) diameter outlet orifice (hardware outlet diameter) and
for this the actual area is 2.8502 ·10
4
m
3
. Although the flow path for the actual body chamber is more com-
plex than any standard nozzle, it is well understood that flow though an orifice generates a vena contracta such
that the effective flow area is smaller than the actual hole size. Comparing the flow area from the test data with
the actual hardware hole size indicates that we need to use a coefficient for the area contraction of 0.877. This
value for the flow coefficient due to the effect of vena contracta corresponds well to typical published values
between 0.73 and 0.97, depending on the shape of the opening [17].
For our linear model, Eq. (17) becomes
e
P
L¼e
PLþe
Qout
KLð18Þ
with KL¼A2
m
qoutQout0 , representing the boost factor. Fig. 8b shows the experimental pressure difference along with
a linear least-fit approximation of the constant, K
L
leading to 2.3 ·10
5
.
Also shown in Fig. 8b is a cubic least-square regression for the effective pressure that is used in the full non-
linear model. Note that the linear approximation compares favorably with the nonlinear experimental data up
to about 0.007 m
3
s
1
. In simulation, this model of the boost effect resulted in a satisfactory response for both
the linear model and the nonlinear model at both low and moderate flow rates (as illustrated later in Section
3). At very high flow rates, the venturi effect begins to fall off, and for the nonlinear model a cubic regression is
used to develop a more accurate representation of the pressure–flow relationship. This polynomial has been
implemented as the function for P
L, the fictitious equivalent pressure of the lower chamber:
P
L¼PLþfðQoutÞ:ð19Þ
The resulting model output for steady state conditions is shown for both the linear model and the nonlinear
models in Fig. 8b. Using this effective pressure in the lower chamber, the flow between the body and the lower
chamber is then
QL¼CdL ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
P
LPout
p:ð20Þ
In order to determine the flow coefficient, a test was performed by removing the diaphragm and pumping air
from the lower chamber and out through the valve body. This data is shown in Fig. 9a for flow from the lower
chamber to the valve body chamber for a particular valve, and this substantiates the use of Eq. (19) in the
model. The data was taken by removing the diaphragm and just flowing air from the lower chamber out of
the body, with no venturi effects. The discharge coefficient for Eq. (20) was found by plotting (P
L
P
out
)
vs. Q2
L, as shown in Fig. 9b, and finding the square root of the slope. The value of the nonlinear discharge
coefficient used in simulations is C
dL
= 5.5 ·10
4
.
Fig. 8. Pressure difference between lower chamber and body chamber for various flow rates.
N. Zafer, G.R. Luecke / Applied Mathematical Modelling 32 (2008) 61–82 69
We can linearize Eq. (21) as
QL¼QL0 þoCdL ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
P
LPout
p
oP
LP
L¼P
L0
Pout¼Pout0
ðP
LP
L0ÞþoACdffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
P
LPout
p
oPout P
L¼P
L0
Pout¼Pout0
ðPout Pout0Þþh:o:t;
QL0 þ1
2CdL ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
P
L0 Pout0
pðP
LP
L0ÞðPout Pout0 Þ
:
Defining CL¼1
2CdL ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
P
L0Pout0
pand using e
PL¼e
PLþe
Qout
KL, we develop a linear flow model for lower chamber:
e
QL¼CLðe
P
Le
PoutÞ¼CLe
PLþe
Qout
KLe
Pout
!
:ð21Þ
The linearized discharge coefficient, C
L
, is slope of the line in Fig. 9a at the particular flow rate of interest.
2.5. Mechanical system governing equations
The mechanical parts of the system also contribute to the dynamic response of the system. The gas pressure
regulator is represented with a simplified model as shown in Fig. 10. Free body diagrams are given in Fig. 11.
A simple dynamic analysis of the free body diagrams leads to
Fig. 10. Pressure regulator.
Fig. 9. Pressure–flow relationship between the lower chamber and the body chamber.
70 N. Zafer, G.R. Luecke / Applied Mathematical Modelling 32 (2008) 61–82
M€
xdþb_
xdþKxdþAdðPLPUÞ¼PinAp=L;ð22Þ
Me
€
xdþbe_
xdþK~
xdþAdðe
PLe
PUÞ¼0;ð23Þ
where the equivalent system mass is a combination of the mass of each part
M¼JL
R2
2þmp
L2þmdþms
3:
Here we have used the traditional analysis for the effective mass of a spring, based on the concept of
conservation of total energy in the spring [15], even though this is likely a negligible component of the total
inertia. Because we make an assumption that the mechanical linkage shown in Fig. 10 is rigid, the inertia and
damping are reflected by the square of the motion ratios, where L=R
2
/R
1
. Note that the diaphragm and the
plunger displacements are related by x
d
=Lx
p
and that the effect of any flow forces on the plunger has been
neglected.
3. Dynamic system response
The mechanical and fluid equations developed in the previous sections are used to simulate the operation of
the gas regulator. Combining the incremental equations (6), (8), (10) and (23) together with Eqs. (12), (14),
(16) and (21) to express the system as a set of dependent differential equations at a steady state operating point,
we obtain the four governing equations for the system;
e_
PL¼kL
PL0
VL0 CLe
PLCLC10
KLe
AþCL1C20
KL
e
Pout þe_
xdAd
;
e_
PU¼kU
PU0
VU0ðCUe
PUe_
xdAdÞ;
e_
Pout ¼kout
Pout0
Vbody0
j
Cin
L
~
xdþCLC10
KLC10
e
AþCLC20
KLC20 CL
e
Pout þCLe
PL
;
Me
€
xdþbe_
xdþK~
xdþAdðe
PLe
PUÞ¼0:
These equations are also illustrated by the block diagram shown in Fig. 12. The numbers in parenthesis in the
figure correspond to equation numbers in the text.
For the nonlinear model, four independent equations govern the dynamics of the system. These equations
are obtained by combining Eqs. (5), (7), (9) and (22) together with Eqs. (11), (13), (15) and (20);
Fig. 11. Free body diagrams.
N. Zafer, G.R. Luecke / Applied Mathematical Modelling 32 (2008) 61–82 71
_
PL¼kL
PL
VL
_
xdAdCdL ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
P
LPout
p
;
_
PU¼kU
PU
VU
_
xdAdþCUt ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
PUPatm
p
;
_
Pout ¼kout
Pout
Vbody
j
Cin
LxdACdffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Pout Patm
pþCdL ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
P
LPout
p
;
M€
xdþb_
xdþKxdþAdðPLPUÞ¼PinAp=LþFc:
Here, P
Lis the equivalent pressure of the lower chamber described by Eq. (17) for the pitot, or by Eq. (19) for
the cubic fit approaches. Lower and upper chamber volumes may be approximated by V
L
=V
L0
A
d
x
d
and
V
U
=V
U0
+A
d
x
d
, where initial diaphragm position is xd0 ¼L
Cin Qin0 ¼L
Cin
Qout0
qNL and A
d
is a function of x
d
de-
scribed in Fig. 3. Because x
d0
50, a force to initially calibrate the regulator is required. This is done by adding
the term F
c
=Kx
d0
+A
d
(P
L0
P
U0
)P
in
A
p
/Linto the mechanical system equation.
The simulation is used to verify the basic operational characteristics of the system, including the time
response and the stability of the regulator. The full nonlinear model is used to validate the overall operation
of the regulator, including the transient response to large and small changes in outlet flow rates and steady
state pressure and flow conditions. Our main objective is to show that the simulations operate in a reasonable
way in response to normal inputs, and in a manner consistent with observed behavior of the physical gas reg-
ulator. Fig. 13 shows the simulation results for the steady state outlet pressure as a function of the outlet flow
rate. The three modeling approaches are compared with the empirical data, and it is clear that there is a dif-
ference in the steady state response using the linear and nonlinear models, particularly at higher flow rates.
Fig. 13 also shows the input values used to test the models: a small flow demand of 0.001 m
3
s
1
and larger
demands of 0.0065, 0.0071, 0.0092 and 0.0098 m
3
s
1
, along with the outlet orifice areas used to generate these
flows. First, the linear model will be compared to the nonlinear simulations to show that the linear model is
valid for small amplitude response about an equilibrium point. Next, using the linear model, we will apply the
powerful root locus techniques to investigate the effects of changes in various parameters on the system
response and stability.
Fig. 14 shows the time response of both the linear and the nonlinear models to a small step input in flow
demand, corresponding to case I in Fig. 13. The initial outlet flow rate was set to Q
out0
= 3.9329 ·10
4
m
3
s
1
and the step change for the outlet orifice area was taken as e
A¼1:5355 105m2. Note that the sudden
change in the outlet valve orifice area causes the pressure to drop, and then come back to the steady state.
The sudden change first causes a drop in the regulated pressure, which is then restored as the plunger moves
to a new steady state location. The small steady state errors are caused by the differences in the model assump-
Fig. 12. System block diagram.
72 N. Zafer, G.R. Luecke / Applied Mathematical Modelling 32 (2008) 61–82
tions, and while there are differences in the amplitude of the oscillations, the frequencies and settling times
match well.
Operational data for typical gas regulators show that the hardware has a tendency to exhibit dynamically
unstable behavior under certain operating conditions. This instability causes the regulator to vibrate, or hum,
although the gross operation of the regulator is not affected. Indeed, one major problem with this dynamic
instability is that the causal observer may become alarmed by the noise, requiring replacement of the unstable
regulator. One common factor of instability is the coupling of the upper chamber with a large volume dis-
charge tube for venting purposes. While changes in many other factors, including temperature, flow, and
atmospheric pressure, affect the unstable response, empirical evidence indicates that it is possible to tune this
discharge volume to induce the unstable behavior regardless of other factors.
In order to establish the conditions for the regulator to hum, we studied the time response of the regulator
with an upper chamber volume about four times larger than the nominal value of the actual hardware,
V
U0
= 0.0025 m
3
, and the time response is shown in Fig. 15. This condition caused instability in both the non-
linear and the linear model. The time response of both the linear and the nonlinear models at these large upper
chamber initial volumes predict the frequency of oscillation at about <133 Hz. Fig. 15b also shows that the
frequency is the same for both the linear and nonlinear models, although there is a phase difference between
them. This phase difference is caused by a very small difference in frequency between the nonlinear models and
the linear model, which adds up over many cycles. The small lag gets larger if the initial displacement from the
equilibrium is made larger [18].
Fig. 16 shows the time response of the regulator models for large and small inputs at the intermediate flow
rates of cases II and III in Fig. 13. In the center plot, there are two step changes in the flow demand, a large
change at time = 0 corresponding to an initial outlet flow area of A
0
= 1.6903 ·10
5
m
2
and changing to
Fig. 13. Outlet pressure and flow relationships: (I) A= 3.2258 ·10
5
m
2
; (II) A= 2.7493 ·10
4
m
2
; (III) A= 3.013 ·10
4
m
2
; (IV)
A= 3.93 ·10
4
m
2
; (V) A= 4.2344 ·10
4
m
2
.
Fig. 14. Time response to step change in outlet area, A= 3.2258 ·10
5
m
2
.
N. Zafer, G.R. Luecke / Applied Mathematical Modelling 32 (2008) 61–82 73
A= 2.7493 ·10
4
m
2
, and a small change at time = 1.0 corresponding to a change in outlet flow area from the
steady state at A
0
= 2.7493 ·10
4
m
2
and changing to A= 3.013 ·10
4
m
2
thereafter. The flow rate settles to
a steady state of Q
out
= 0.0065 m
3
s
1
during the first second, and to Q
out
= 0.0071 m
3
s
1
by the end of the
simulation. Because the step change for the first second is quite large, only the nonlinear models are used in the
simulation. Once the steady state is reached, the flow and pressure values are used to update the linear model
parameters and the response of the linear and nonlinear models are compared for the small amplitude input,
shown in the zoomed portion on the right of Fig. 16. Thus, both the nonlinear and the linear model are com-
pared after the step at 1 s in the simulation. For small amplitude inputs, the linear model dynamics closely
match the nonlinear model simulations, although there are steady state errors predicted by Fig. 13.
This same set of large and small inputs is shown in Fig. 17, but in this case, with a large upper chamber vol-
ume V
U0
=6·10
3
m
3
. Again, the initial, large step input is only simulated using the nonlinear models, and
the linear model is compared to the nonlinear responses for the small step input at 1.5 s. In this case, the linear
model response still follows the nonlinear dynamics, although for both linear and nonlinear cases we see that
the increase in the upper chamber volume has the effect of slowing the settling time of the regulator.
Fig. 18 shows the time response of the regulator for very high flow rates corresponding to cases IV and V
from Fig. 13. These step changes in the outlet orifice area from A
0
= 1.6903 ·10
5
m
2
to A= 3.93 ·10
4
m
2
in the first 1 s and A= 3.93 ·10
4
m
2
to A= 4.2344 ·10
4
m
2
at 1.5 s correspond to steady state flow rates of
Q
out
= 0.0092 m
3
s
1
and to Q
out
= 0.0098 m
3
s
1
respectively. Again, the steady state values of the nonlinear
Fig. 15. Regulator response with large upper chamber volume, A= 3.2258 ·10
5
m
2
.
Fig. 16. Time response to step changes in outlet area, V
U0
=6·10
4
m
3
(nominal value).
74 N. Zafer, G.R. Luecke / Applied Mathematical Modelling 32 (2008) 61–82
model after the first large step input are used update the linear model parameters. Thus, both the nonlinear
and the linear models are compared in the second part of the simulation, shown in the right of the figure. Note
that during the initial response for the large step input, the system is very under damped, as shown by the
oscillations in the left-hand portion of Fig. 18. For the smaller input at the higher flow rate, however, the
dynamics show considerably more damping, indicating that higher flow rates tend to stabilize the system.
Fig. 19 shows the response for this same set of inputs with the larger upper chamber volume, V
U0
=
6·10
3
m
3
. Here again, it is clear that the higher flow rates tend to stabilize the system response.
These simulations show that the mathematical models of the gas regulator produce reasonable response to
changes in flow demands. The dynamic instability seen in real regulators for small flow rates can be repro-
duced in the simulations by increasing the upper chamber volume, which is also true for regulators in actual
operation. The linear model closely matches the nonlinear model close to equilibrium positions and for small
inputs. We will now use linear analysis methods to identify design parameters that have an effect on the sta-
bility of the system.
4. Linear system analysis
In order to understand the effects of various parameters on the stability of the system, an analysis of the
linear system has been examined in the form of a root locus diagram for several parameters. This method
Fig. 17. Time response to step changes in outlet area, V
U0
=6·10
3
m
3
.
Fig. 18. Time response to step changes in outlet area, V
U0
=6·10
4
m
3
(nominal value).
N. Zafer, G.R. Luecke / Applied Mathematical Modelling 32 (2008) 61–82 75
allows for the determination of the effects on stability from changes in each model parameter separately. The
root locus analysis was implemented for various design parameters that could be realistically modified to
improve the stability of the regulator. These include the damping, b, the upper chamber volume, V
U0
, the
lower chamber volume, V
L0
, the discharge coefficient between the upper chamber and the atmosphere, C
U
,
the discharge coefficient between the lower chamber and the body chamber, C
L
, and the area of the dia-
phragm, A
d
.
For the system with nominal parameter values, the system was stable, although there are roots close to the
right half plane. The roots of the characteristic equation for the transfer function of the block diagram in
Fig. 12 are 7306.4, 27.7 ± 801.9i, 62.7 ± 41.1i when V
U0
=6·10
4
m
3
, and 7306.3, 92.3, 21.9,
1.9 ± 655.7i with V
U0
= 0.0025 m
3
.
We use the root locus for the linear model to investigate the effects of changes for various design parameter
on system stability. The parameters chosen for investigation are those that can be realistically changed in the
design of the regulator to improve stability. The nominal values used in the root locus come from the analytic
development and incorporate measured values where possible, and are shown in Table 1.
4.1. Root locus on the damping coefficient
Increasing the mechanical damping in the system has the effect of stabilizing the system. This damping coef-
ficient is very difficult to measure in the real hardware, and in our simulation, this value has been adjusted
heuristically as a means of adjusting the system models to more closely match the experimental performance
of the hardware. The adjustment of the damping coefficient, however, must be made within realistic limits.
Fig. 20 shows a root locus plot for variable damping. There are two plots shown, one for a small upper cham-
ber volume, V
U0
, and one for a large upper chamber volume. While increasing the damping reduces the ten-
dency toward unstable behavior, excessive increases in damping tends to increase the transient response time
of the system, and lead to undesirable steady state effects such as dead-band.
Fig. 19. Time response to step changes in outlet area, V
U0
=6·10
3
m
3
.
Table 1
Nominal values of parameters used in the linear simulation
A= 3.2258 ·10
5
m
2
A
0
= 1.6903 ·10
5
m
2
A
d
= 0.0139 m
2
A
m
= 2.5 ·10
4
m
2
A
p
= 1.7814 ·10
5
m
2
b= 5 N s/m C
d
= 0.5495 C
dl
= 5.5 ·10
4
C
in
= 2.649 C
L
= 5.9 ·10
6
C
U
= 4.2 ·10
7
C
Ut
= 3.75 ·10
6
C
10
= 23.2672 C
20
= 1.097 ·10
7
k=k
out
=k
U
=k
L
= 1.31 K= 700 N/m
K
L
= 2.3 ·10
5
L=4 M= 0.1491 kg P
in
= 308.2 kPa
P
out0
=P
L0
= 103.15 kPa P
U0
= 101.35 kPa Q
out0
= 3.9329 ·10
4
m
3
s
1
V
body
= 1.6387 ·10
4
m
3
V
L0
= 3.2823 ·10
4
m
3
V
U0
=6·10
4
m
3
q
out
= 0.7 kg/m
3
j= 2.3061
76 N. Zafer, G.R. Luecke / Applied Mathematical Modelling 32 (2008) 61–82
Changes in the upper chamber volume also affect the stability of the regulator. As this volume decreases, the
system becomes more stable, as shown for the root positions when the damping b= 5 N s/m. At the larger vol-
ume, the system is unstable and at the smaller volume, the system is stable. This corresponds with known design
goals, where the upper chamber volume must be large enough to accommodate the diaphragm diameter and
travel, but is designed to be as small as possible. In addition, the oscillatory ‘‘hum’’ can often be induced by sim-
ply attaching a larger volume pipe to the vent discharge valve, effectively increasing the upper chamber volume.
4.2. Root locus on the diaphragm area
Another parameter that is a candidate for design change is the diaphragm area. The root locus for A
d
,
shown in Fig. 21, shows some interesting trends. While the root locus shows that the diaphragm could be
made very small and result in stable performance of the regulator, this size diaphragm is not large enough
to counteract plunger flow forces or even allow mechanical connections necessary for the physical hardware.
The remaining locus indicates that diaphragm area cannot be chosen smaller than 0.0084 m
2
in order to retain
stability. Larger diaphragm areas, on the other hand, improve the response time of the system, although in the
extreme, these roots also become under damped. However, because changes in diaphragm area usually imply
changes in lower and upper chamber volumes, care must be taken when implementing a larger diaphragm.
4.3. Root locus on the upper chamber initial volume
The physical conditions surrounding the onset of instability include a large volume or pipe attached to the
upper chamber. Often, just attaching a vent hose, common in Europe where the regulators are installed
Fig. 21. Root locus on A
d
;e:A
d
= 0.0139 m
2
(nominal), r:A
d
= 0.0084 m
2
.
Fig. 20. Root locus for the variable bwith two values of V
U0
;h:b= 5 N s/m.
N. Zafer, G.R. Luecke / Applied Mathematical Modelling 32 (2008) 61–82 77
indoors, is enough to excite oscillations, or ‘‘humming’’. This indicates the need for examination of the system
stability characteristics as the upper chamber volume parameter, V
U0
, is increased. Note that the root locus
shown in Fig. 22 is for the variable 1/V
U0
, so that origin, typically shown with an ‘‘x’’, is the root locations
for a large upper chamber volume. In this sense, the locus in Fig. 22 is ‘‘backwards’’, with a larger volume
pushing the root locations away from the zeros and toward the poles. A close-up view of the root locus, also
seen in Fig. 22, shows that increasing V
U0
generally decreases the system stability. The roots associated with
these locations will always tend to cause under damped response, but as the upper chamber assumes values
close to nominal, the response due to these roots settles quickly.
Because there are many parameters involved in the overall response, we examine another root locus for
1/V
U0
in Fig. 23, which shows how the system roots are affected by changes in the upper and lower discharge
Fig. 22. Root locus on 1/V
U0
with changing b;h:V
U0
=6·10
4
m
3
(nominal), :V
U0
=12·10
4
m
3
,r:V
U0
=25·10
4
m
3
.
Fig. 23. Root locus for the variable 1/V
U0
with changing values of C
U
and C
L
;:V
U0
=6·10
4
m
3
(nominal value).
78 N. Zafer, G.R. Luecke / Applied Mathematical Modelling 32 (2008) 61–82
coefficients. These coefficients are mainly a function of the hole sizes of the cap in the upper chamber, a design
parameter that is relatively easy to change. The nominal values of all parameters except V
U0
and C
U
are used
in Fig. 23a, where we see that increasing the upper chamber discharge coefficient, corresponding to increasing
the size of the vent hole, improves stability for small upper chamber volumes but does not substantially affect
stability when these volumes get large. As the lower discharge coefficient is decreased, shown in Fig. 23b and c,
the entire locus tends to move away from the imaginary axis, with an associated improvement in stability. As
pointed out in Fig. 22, reducing the initial size of the upper chamber still does improve the behavior of the
system, but pipe and venting installations in the field will always tend to increase this volume and drive the
system toward instability. Most importantly, the lower values of C
L
move the root locus entirely into the sta-
ble region. This indicates that lowering the value of C
L
, by reducing the associated flow area between the lower
chamber and the valve body by at least a factor of two, will enhance system stability over a wide range of
upper chamber volume values.
4.4. Root locus on the lower chamber initial volume
We also explore the effects of the lower chamber volume on the stability of the regulator. The root locus
analysis on V
L0
is shown in Fig. 24a. We can see that in general, changes in the lower chamber volume have
only a slight effect on the system roots, causing very little change in the damping ratio and settling time. Using
the general trend from Fig. 23, that decreasing the lower chamber discharge coefficient improves stability,
Fig. 24b shows that decreasing C
L
tends to pull the root locus into more stable regions, where ‘‘’’ indicates
the location of the nominal design value of V
L0
. This indicates that reducing the size of the flow path between
the lower chamber and the body chamber tends to increase system stability.
Fig. 24. Root locus for the variable 1/V
L0
with changing values of C
U
and C
L
;:V
L0
= 3.3 ·10
4
m
3
(nominal value).
N. Zafer, G.R. Luecke / Applied Mathematical Modelling 32 (2008) 61–82 79
Fig. 23 also indicates that increasing the upper discharge coefficient, C
U
, may improve stability, Fig. 24c
shows the locus with a the nominal value of C
L
, but with a range of values for C
U
, as the lower chamber vol-
ume, V
L0
, is changed. As the upper chamber discharge coefficient is increased by a factor of ten, the locus
moves toward instability. It is natural to try decreasing the value of C
U
, but this change also pushes the locus
toward instability.
4.5. Root locus on the upper chamber discharge coefficient
The analysis of the lower chamber volume in relation to upper and lower discharge coefficients warrants a
closer look at the stability effects of these two coefficients. Fig. 25 shows the roots of the system as C
U
, the flow
coefficient in the upper chamber, changes for various fixed values of C
L
, the discharge coefficient for flow
between the lower chamber and the valve body control volumes. With C
L
at the nominal value, 5.9 ·10
6
,
it is clear that for both very large and very small values of C
U
, the roots tend toward oscillatory or even unsta-
ble behavior. Again, this compares with the known behavior of the regulator, in that very small vent holes
produce unstable oscillations, as does removing the upper chamber cover completely. Notice that as the value
of C
L
is decreased, which corresponds with decreasing the flow hole between the lower chamber and the body
chamber, both end points of the root locus, and those in between, are pulled into the stable left half plane. This
also indicates that reducing the flow area between the body and the lower chamber will enhance the stability of
the regulator.
We also note that there is an optimum C
U
value, located at the left-most point on the curve. Indeed, this
value of C
U
was a relatively constant value for a large range of C
L
, and corresponds to a upper chamber vent
hole about 3 times larger than the nominal size. This provides a guide for selecting the most robust value of the
upper chamber discharge coefficient for all cases of the lower chamber parameter.
4.6. Dynamic simulation with improved parameters
Our root locus analysis indicates that very simple changes to the flow areas in the two chambers can have a
significantly improve the performance of the regulator. In order to verify this, Fig. 26a shows the time
response of the full, nonlinear simulation with changes in the upper and lower discharge coefficients. In this
case, the input is a small amplitude change in outlet demand, and the response to both small and large upper
chamber volumes is shown. The upper chamber flow diameter has been increased by a factor of 3, based on the
root locus of Fig. 25, and the effective lower chamber flow diameter has been decreased by a factor of ten,
based on the root locus of Fig. 23. Recall that with the nominal values for the flow diameters, an upper cham-
ber volume of 0.0025 m
2
was sufficient to cause instability at low output flow rates using both the linear and
nonlinear models as shown in Fig. 15a. Fig. 26 shows that these two simple changes the upper and lower dis-
charge coefficients are sufficient to stabilize the system response for smaller output flow rates regardless of the
size of the upper chamber volume.
Fig. 25. Root locus on C
U
with changing C
L
;:C
U
= 3.9 ·10
6
.
80 N. Zafer, G.R. Luecke / Applied Mathematical Modelling 32 (2008) 61–82
Larger output flow rates tend to stabilize the system as shown in Figs. 17–19. However, even at the low flow
rates and with the larger upper chamber volumes, the changes in the two flow coefficients stabilize the oper-
ation of the regulator as shown if the response of Fig. 26a. Although both of these design changes
(C
Ut
= 3.375 ·10
5
,C
dl
= 5.5 ·10
6
) are very easy to implement in hardware, care must be taken to allow
for sufficient flow capability through the two chambers and out of the upper chamber vent hole in case the
inlet flow orifice from the high pressure gas is stuck open, either though debris or mechanical failure. As noted
earlier, the upper chamber vent is often a dual stage, spring loaded plate with a small discharge hole specif-
ically for this reason.
5. Results and conclusion
This study establishes methodology to accurately model and analyze the behavior of self-regulating high
pressure gas regulators. The model used has been developed from first principles to couple the mechanical
and fluid system dynamics. A linear version of the model has also been developed to allow the application
of root locus techniques to study the stability of the system with changes in various design parameters.
Estimation of important parameters, such as damping and discharge coefficients, was based on available
steady state empirical data. Both the linear and nonlinear models produce transient and steady state responses
that compared favorably with the expected behavior of the typical gas regulator. Small errors in the steady
state response of the model can be attributed to the simplifying assumptions in the gas dynamics involved with
the venturi effect of the flowing gas. Transient response characteristics of the linear model match the nonlinear
model for small amplitude inputs, although there are significant differences in steady state values for large
amplitude inputs.
One of the main reasons for the development of this model was the fact that these types of regulators tend
to vibrate or hum when the relief port is connected to an extended pipe and the flow rate is kept small. This
phenomenon was simulated as an increase in the upper chamber cavity on the regulator. Simulation results
support the loss of stability with an increase in this volume, indicating that the model provided an accurate
representation of the hardware.
In order to gain insight into the possibility of stabilizing the system through redesign, root locus techniques
were used on the linear model to predict the effects of changes in various design parameters on the system
response and to identify the most influential system parameters. Physical parameters that were shown to have
a significant effect on stability include the damping coefficient, the diaphragm area, and upper and lower
chamber volumes. Added damping, however, is difficult to control when relying on friction, may require add-
ing hardware for accurate control, and increases the system dead-band and response time. Physical limitations
on size restrict the effectiveness of changes in the sizes of the chamber volumes and the diaphragm area.
The effect of the size of the flow paths from the upper and lower chambers was also seen to have a signif-
icant effect on the system stability. Using the root locus approach, we find that reducing the size of the flow
Fig. 26. Time response with changes in the upper and lower flow paths.
N. Zafer, G.R. Luecke / Applied Mathematical Modelling 32 (2008) 61–82 81
path between the regulator body and the lower chamber provides a distinct improvement in stability. In addi-
tion, we show that there is an optimum size for the discharge hole on the upper chamber. We use the root locus
techniques to show that decreasing the effective diameter of the lower chamber flow path by a factor of 10 and
increasing the diameter of the upper chamber flow path by about three times theoretically improves the sta-
bility of the regulator. While there are limits on the allowable sizes for these holes, based on the discharge flow
that must be allowed in the case of catastrophic failure of the regulator, we show that using these changes in
the nonlinear simulation eliminates the vibration and hum and provides satisfactory performance of the
regulator.
Acknowledgement
The authors are grateful to the anonymous referee for the insightful comments and suggestions which
helped greatly improve the quality of this paper.
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