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MATHEMATICAL MODELLING OF PRESSURE
REGULATOR FOR CRYOGENIC APPLICATIONS
Sunil S.Liquid Propulsion Systems Centre,
ISRO, Valiamala,Thiruvananthapuram, Kerala, India
Ullekh PandeyVikram Sarabhai Space Centre,
ISRO, Thiruvananthapuram,Kerala, India
Jeevanlal B. S.Liquid Propulsion Systems Centre,
ISRO, Valiamala,Thiruvananthapuram, Kerala, India
M. RadhakrishnanLiquid Propulsion Systems Centre,
ISRO, Valiamala,Thiruvananthapuram, Kerala, India
C. AmarasekaranLiquid Propulsion Systems Centre,
ISRO, Valiamala,Thiruvananthapuram, Kerala, India
ABSTRACT
Pressure regulator is a dynamicpneumatic device used for maintaining
a constant pressure at the control
volume irrespective of varying input
pressure in both terrestrial and space
applications. In cryogenic propulsion
systems, the design of regulator
becomes complex due to variation in
temperature from ambient to 80K. The
stability of regulated pressure
becomes a function of variation of
modulus of rigidity of loading
elements, dimensional variation ofelements, thermo-physical properties
of the gas etc. A non-linear dynamic
mathematical model of cryogenic
pressure regulator is important for the
study and investigating the
performance of regulators in the
specified working conditions in
respect of the initial transients,
temperature effects, variation in input
and output conditions and alsoinherent self exciting oscillations. The
proposed work is to develop a non -
linear dynamic mathematical model of
cryogenic pressure regulator. The
result of the model is then validated
with experimental test results of a
newly developed cryogenic pressure
regulator developed by Liquid
Propulsion Systems Centre (LPSC) of
Indian Space Research Organisation
(ISRO).
INTRODUCTIONLiquid propellant powered rocket
engines form an important stage for
any launch vehicles due its
controllability and high specific
impulse. Amongst these, cryogenic
(LOX and LH2) fueled stage has the
highest value of specific impulse. The
liquid propellant storage tanks are
Proceedings of the 37th
International & 4th
National Conference on Fluid Mechanics and Fluid Power
FMFP2010
December 16-18, 2010, IIT Madras, Chennai, India
FMFP2010 223 .
Proceedings of the 37th National & 4th International Conference on Fluid Mechanics and Fluid Power
December 16-18, 2010, IIT Madras, Chennai, India.
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pressurised during the engine
combustion time in order to maintain
the NPSH for the propellant pumps. In
cryogenic propellant stages, the high
pressure gas for the pressurisation
system is stored in a gas bottle
immersed in propellant tanks with thesame fluid temperature. The regulator
is isolated with a start valve which
opens before starting of engine. The
high pressure gas is regulated either in
single stage or two stages depending
on the regulated pressure precision
requirement. Figure 1 shows
schematic of a typical system.
Fig. 1. Schematic of the pressurisation
system
The basic working principles of gas
regulators can be found in Brasilow,
1989 and Glen et al., 1970. Krigman,
1989 discuss the selection and design
criteria for different types of pressure
regulators and valves. However there
is a dearth of published literature
addressing the modeling of dynamic
gas systems such as pressure
regulators, valves etc. owing to
proprietary concerns. Delenne et al.,
1999 and Delenne, 2000 reported that
operating instabilities can cause
metering perturbations and affect the
operability of shut off and relief
valves. ALLAN Simulation (Jeandel
et al., 1993) developed a general code
for simulating and modeling dynamic
systems. Rami et al., 2007 used this
code to simulate a pressure regulating
station of a natural gas network. Theyfound that the operating conditions
and installation requirements affect
the stability. Naci et al., 2008
developed a comprehensive dynamic
model of a pressure regulator and
linearized it using Taylors series
expansion and carried out sensitivity
analysis and optimization of important
design parameters.
A regulator designed for cryogenic
fluids differs from ordinary regulator
for; material compatibility, dynamicsealing and sensing with bellows and
temperature compensation for
reference loading. None of the known
publications modeled pressure
regulators for low temperature
applications. The present study adopts
the modeling procedure proposed by
Naci et al., 2008 for modeling a
pressure regulator for cryogenic
applications. The inlet pressure
reduces with time with temperature
decreases from ambient to the
propellant temperature during flow.
The model gives the effect of
cryogenic temperature in the regulated
pressure taking into account of
reference load variation, dimensional
changes and also the fluid property
changes.
MODELING OF PRESSURE
REGULATORThe modeling approach applied
corresponds to the usual method ofbreaking down the systems to a set of
sub systems reduced to their essential
behaviour, making assumptions,
approximations and mixing empirical
and analytical approaches. The main
subsystems at the lowest level are
fluid domain, mechanical elements of
regulator and flow through the
GAS
LOX
PROPELLANT TANK
REGULATORSTART VALVE
PRIMARY SECONDARY
REGULATOR
RELIEFVALVE
BOTTLE
SELECTED FOR
MODELING
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compartments. In order to develop
model, three control volumes are
defined and are used in dynamic
analysis viz. the low pressure
chamber, the reference and damping
chamber (refer Fig. 2.). Each volume
is characterised by pressure, volumeand density as function of time. For
analysis the volumetric flow through
the system is tracked with these
control volumes.
Fig. 2. Schematic representation of
primary pressure regulator
Poppet Area and Travel
Requirement
The amount of poppet area and travelis computed based on the mass flow
through the regulator, pressure
available at the inlet chamber and
outlet pressure required. The flow
through the seat can be either sonic or
subsonic depending on the ratio of
outlet pressure to inlet pressure. The
standard equation for mass flow rate
m through a valve orifice area As(Baline, 1979) is,
cr
in
Lk
k
in
Lk
in
Linsd
cr
in
Linsd
PP
Pif
P
P
P
P
T
PCAC
PPPif
TPCAC
m )1(1
2
1
1
(1)
and
1
2
11
1
1
2
;)1(
2;
1
2
kk
cr
kk
kP
kR
kC
kR
kC
(2)
The poppet opening xa depends on theconfiguration of seat and poppet. For a
flat seat and poppet configuration
which selected for the present model,
the poppet opening is given by;
s
s
a d
Ax
(3)
Static PerformanceFor any regulator there are two major
performance characteristics to be
analysed; lockup pressure variation
and flow pressure variation. These
variations are caused by one or
combination of
- Regulator inlet pressure variation
- Thermal effects
- Reference load change due to the
regulator poppet travel- Spring and bellow hysterisis
- Friction of sliding parts
Among the above, the effects of the
last two are neither quantifiable nor
repeatable. Hence it is better to
approximate a minimum value and
validate the same with the
experimental results. The variation of
the outlet pressure due to other factors
can be estimated as presented below
Force Balance Equation
The free body diagram of the regulatoris shown in Fig. 2. The force balance
equation in steady state condition is
given as
0)()(
)()()(
apbpbUBLin
abRDaababarr
xKFAPP
APPxKFxKF(4)
and AUB = As - ApbRearranging Eq. (4) to get
pbabrtotabrtot
UBLinaRDpbatottot
KKKKandFFFwith
APPAPPFxKF
0)()( (5)
At steady state condition (i.e. no flow
condition) PD = PL and PR = Patm.
Equation (5) is used to estimate thelockup pressure and Eq. (1) to (3) is
used to calculate poppet opening as a
function of regulator inlet pressure.
Dynamic PerformanceThe regulator is a feed back system
device and its overall performance is
derived from the combination of
equation of motion of moving parts
+
Reference
chamber
Breathing
hole
Damping
chamber
Low pr.
chamber
High pr.
chamber
(Pin, in)
Damping
hole
(PL, VL, L)
(PD, VD, D)
(PR, VR, R)
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and continuity equation for flow
through the regulator. The modeling
procedure is as described.
Generic Equations of Gas DynamicsIn a gas pressure regulator, the flow
direction in process lines is likely to
change at any time. Hence modelswith fluid mechanics problems
relating to stationary or linearised
approaches in the vicinity of a
working point are not valid. Present
model is based on the physical
behaviour of compressible fluid flow.
Equations used for modeling are one
dimensional flow of compressible,
viscous, Newtonian fluid, the principle
of conservation of mass and equation
for flow through orifices.
Equation of StateThe relationship between pressure
variations and changes in temperature
and density needs to be set through an
equation of state. In the usual form:
ZRTP
(6)
The ideal gas equation (with
approximation Z=1) cannot be used inthe present case because of the large
pressures and low temperatures. Peng -
Robinson equation (Peng et al., 1976)
is most widely used for Cryogenicfluids and has been adopted here. The
equation in pressure explicit form is ;
222)( bbVV
a
bV
RTP
mmm
where22 )]1)(26992.054226.137464.0(1[
rT
withc
rT
TT and ` is the acentric
factor expressed by;
1log10
c
vp
P
P
(where `Pvp is the
saturated vapor pressure of the gas atT = 0.7Tc)Coefficients `a and `b are functionsof the critical properties;
2
2245724.0
c
c
P
TRa ,
c
c
P
TRb 0778.0
The Peng-Robinson cubic expression
in `Zis represented as;
0)()23()1( 32223 BBABZBBAZBZ (7)
where2RT
PaA
and
RT
bPB
Equations Governing Pressure
ChangeAssuming the process is adiabatic and
reversible; the second law of
thermodynamics provides arelationship pressure and density of
fluid.
k
P
Constant (8)
By considering the time differentials
of Eq. (8) together with definition of
density, it can be shown that;
m
m
V
V
P
P
k
1 (9)
and Qm
(10)
Vm (11)
Since the density for a fixed
operational flow rate is constant ,
volumetric flow rate will be used
rather than more conventional mass
flow rate. Equation (9) provides basis
for analysis and modeling of the
regulator. It describes the relationship
between pressure, volume and mass
flow rate for a chamber. The
governing equations for different
chambers are
a) Low Pressure ChamberEven though the pressure at the outlet
of low pressure chamber tends to be
steady, the pressure inside the low
pressure chamber changes with time.
The pressure inside damping chamber
and low pressure chamber fluctuates
as the regulator moves towards
equilibrium. These fluctuations
compress the gas inside and cause
density change. This density change is
small when compared with the density
change when gas flows from highpressure to low pressure cham ber. The
change in density can be accounted
with an expansion ratio.
Since the low pressure chamber is
rigid, the volume change, 0
LV . The
mass balance for low pressure
chamber is given by equating mass
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flowing in and out of the chamber.
DoutinL mmmm (12)
Substituting Eq. (10) in the above
equation,
DoutinL QQWQQ (13)
and Lin
W
is used to account for thechange of density of inlet gas to that
of the outlet gas. It is also assumed
that L=out=D, because thedifferences in pressure are small
compared to the difference in pressure
between these and inlet pressure.
Combining these equations and
substituting in Eq. (9) gives the outlet
pressure as a function of the flow
crossing the control boundary.
)( DoutinL
L
LL QQWQV
P
kP
(14)
b)Damping ChamberThe change in volume of damping
chamber is related to movement of the
actuator by
abaD AxV
(15)
The positive motion of actuator causes
the damping chamber volume to
decrease, hence the sign is negative.
The overall governing equation for
damping chamber is
)( abaDD
DDD AxQ
VPkP
(16)
Minus sign of QD denotes theconvention that the positive flow
direction ofQD is out of the cavity.
c) Reference Chamber
UsingabaR AxV
, the overall governing
equation for reference chamber is;
)( abaRR
RRR AxQ
V
PkP
(17)
Equations Governing FlowThe dynamic response characteristic
of regulator is controlled by the gas
entering and exiting the regulator inlet
and outlet, breathing hole of reference
chamber etc.
a) High Pressure ChamberThe volumetric flow rate through the
regulator is proportional to the area of
poppet opening or the poppet travel.
ainin xCQ (18)
Cin is obtained from the slope of thecurve plotted between volumetric flow
rate Qin and poppet travel xa as shownin Fig. 3.
b)Reference ChamberThe flow in and out of the reference
chamber can be neglected since the
breathing hole is designed with
enough area so as to avoid pressure
build up inside the reference chamber.
0RQ (19)
Flow rate Vs Poppet lift
0.0E+00
5.0E-04
1.0E-03
1.5E-03
2.0E-03
2.5E-03
3.0E-03
3.5E-03
4.0E-03
0.0E+00 2.0E-05 4.0E-05 6.0E-05 8.0E-05 1.0E-04 1.2E-04 1.4E-04 1.6E-04 1.8E-04 2.0E-04 2.2E-04
Poppet lift (xa) in m
Flow
rate(Qin
)in
m3/s
Qin = 18.99xa
Fig. 3. Poppet travel (xa) requirementwith variation of regulator inlet flow
rate (Qin)
c) Low Pressure ChamberThe flow going out of the regulator is
proportional to the outlet area
available and pressure difference .
Using the square root relationship
(Baline et al., 1979);
)(2
atmout
out
dorout PPCAQ
(20)
At any instant the outlet pressure will
dependent on losses occurring within
the regulator from the low pressure
chamber. This basically depends on
the configuration and geometry of the
passage. The common pressure drops
within the regulator are;Load droop (Pld) is the decrease inregulated pressure caused by a
decrease in reference load as metering
valve opens from its closed position to
full flow condition.
)(
)(
sab
pbabr
ldAA
KKKP
(21)
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Drop due to sudden expansion (Pld)of gas from high pressure chamber
low pressure chamber.
outout
ex
UP
28
12
(22)
Frictional pressure drop (Pfr): in the
flow passage between low pressurechamber and the regulator outlet.
out
out
outoutfr
d
UflP
2
(23)
Drop due to turbulence (Pfr) creatednear the seating area.
Loutth
t
UUKRP
2)1(
22 (24)
Total pressure drop (P) within theregulator is given by
tfrexld PPPPP (25)
The pressure at the outlet of the
regulator during flow isPPP Lout (26)
d)Damping ChamberThe flow between low pressure
chamber and damping chamber is
through the damping orifice. Using
Poiseuilles equation (Baline et al.,
1979);
d
d
LDD
l
dand
PPQ
128
4
(27)
Mechanical Governing EquationsThe dynamic response of the system is
also contributed by the mechanical
moving elements. From the free body
diagram the various forces acting on
the each element can be found. Figure
4 gives the simplified free body
diagram for the system.
Fig. 4. Free body diagram of regulator
The equivalent mass Me of the systemis given by;
333pb
paprab
e
MMM
MMM (28)
The simple dynamic analysis of the
moving parts is given by the equation.
0
FxcxM aae (29)F is given by the Eq. (5). The forceof laminar damping is given by
(Dragoljub et al., 2001);
4
2128
d
abd
aa
d
Aland
xxc
(30)
Dynamic System ResponseThe mechanical and fluid equations
generated in the previous sections are
used to simulate the performance of
regulator controlling equations.
Combining Eq. (14), (16), (17) and
(29) together with Eq. (18) to (20),
(26), (27) and (30) to get the final
controlling equations of the system.
LDaba
D
DDD PPAxV
PkP (31)
)( abaR
RRR Ax
V
PkP
(32)
)(2
atmoutoutdor
LDain
L
LLL
PPCA
PPxWC
V
PkP
(33)
!
0)()(
)(
UBLinabRD
pbtotatotaae
APPAPP
FFxKxxM (34)
Effects of Cryogenic Temperature
in Regulator PerformanceThe regulator at the start of
functioning, the body as well as the
gas temperature will be equal to
ambient temperature. When the flow
is initiated the low temperature gas
cools the regulator body partsincluding the spring and bellow
materials. For modeling and analysis
the following effects are to be studied,
Rate of CoolingSince the construction of regulator is
very complex, a thermo structu ral
analysis of regulator to find out the
rate of cooling and reaching
+
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equilibrium is very difficult. The
result of physical tests conducted on a
prototype is used as the input for
mathematical modeling. Figure 5
shows the temperature of regulator
body Vs time of flow. For
simplification it is assumed that thetemperature of body will be same as
the regulator internal elements. The
curve-fitted equation of temperature as
a function of time is given by
!
3007846.10053.0106
102107104
236
49512615
tttx
txtxtxT(35)
Regulator body temperature Vs Time of flow
0
50
100
150
200
250
300
0 100 200 300 400 500 600 700
Time of flow in seconds
Regulatorbody
temperatureinK
Fig. 5. Experimental results showing
variation of regulator body
temperature with time of flow
Variation of Spring and Bellow
StiffnessAs the temperature of spring and
bellow decreases, the stiffnessincreases due to increase in modulus
of rigidity of springs. A special
Nickel-Iron-Chromium alloy is used
as spring material and Titanium
stabilized austenitic stainless steel is
used for bellow construction. The
variation of modulus of elasticity Erwith temperature for spring material is
obtained from the published results
(UNS N09902).
!
1162
34
1021027892994.2031454.0xTxTTTEr (36)
The change in spring constant Kr isobtained from the standard equation
for the spring (Joseph et al., 1989);
)1(28 3
4
r
r
rr
wr
E
nD
dK
(37)
Similarly the variation of modulus of
elasticity with temperature for bellow
material is given by (Unknown, 1977).1172 10210713380/ xTxTEE pbab (38)
The stiffness of U-shaped and straight
wall configuration bellow is given by(Glen et al., 1970);
YLh
qEtRKorK
cb
bbmpbab
139.4
3
3
(39)
Yis a factor depending on the value o f
hq2 and is taken as 0.5. Refer Fig. 6
for the details of bellow configuration.
Fig. 6. Details of bellow with U-
shaped and straight wall configuration
Change in DimensionsThe cooling causes change in
dimensional clearance between sliding
parts and also changes referencelengths of spring room. The
immediate effect of former is change
in coulomb friction. Since the change
is neither quantifiable nor repeatable,
the effect is added by a minimum
force against movement of actuator
after reaching the lowest temperature.
This will be verified with
experimental results.
RESULTS AND DISCUSSIONSThe block diagram of the system is
shown in Fig. 7. A code wasdeveloped in MATLAB (version 7.0)
to solve and model the fi nal
controlling equations of the regulator.
The values of different constants were
taken from the prototype made. The
initial pressure Pin is taken as 22.5MPa and initial low pressure chamber
pressure PL is taken as 1.06 MPa. The
h/
h/
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results of the program, poppetmovement Vs time are plotted in Fig. 8and the variation of PL Vs time is
plotted in Fig. 9 along with
experimental results.
Results show a close matching of
experimental and theoreticalpredictions. Small deviations during
initial transients may be due to
coulomb friction between the sliding
parts. Change in slope of the curve
with time can be due the assumption
that the temperatures of the internal
components are same as body
temperature. In practice it may not be
true. The contraction of internal
elements and chambers with
temperature also contribute to change
in regulated pressure. This depends on
the thermal conductivity and thermal
mass of the internal elements.
CONCLUSIONSThe paper presents a methodology and
an accurate mathematical model for
cryogenic pressure regulator. The
result of the model is comparable withexperimental results of a prototype
tested to the same conditions.
The limitations of the present model
being the assumption taken for the
effect of temperature conductivity and
thermal mass and also the friction
factors. This can be exactly studied
after conducting a thermo-structural
analysis of the model.
The accuracy of present mathematical
model can be further increased by
adding these results and exact valuesof friction coefficients.
Fig. 7. Block diagram of the system
Poppet movement Vs time
0
0.0001
0.0002
0.0003
0.0004
0.0005
0.0006
0.0007
0.0008
0.0009
0.001
0 5 10 15 20 25 30
Time in Seconds
Poppetmovementinm
-
+
-
-
+
+
+
+ +
+
- +
-
-+
-
+
-
-+
-
+
++
+
+
--+
-
Variation of PL Vs time
800000
850000
900000
950000
1000000
1050000
1100000
1150000
1200000
-5 5 15 25 35 45 55
Time in seconds
Pressure(PL)inPa
Experimental
Theoretical
Fig. 8. Theoretical prediction of
Poppet movement (xa) with time
Fig. 9. Comparison of theoretical and
experimental results for variation of
PL with time.
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NOMENCLATURE
A area (m2)
c viscous friction coefficientCd discharge coefficientd diameter (m)
Dr mean coil diameter (m)
E modulus of elasticity (Pa)f Darcy friction factorF assembled load (N)k specific heat ratio of gas
K spring constant (N/m)
KR recovery factorl length (m)
Lc length per convolution
m mass flow rate (kg/s)
M mass (kg)n number of turns
P pressure (Pa)Pcr critical pressure ratioQ flow rate (m
3/s)
R gas constant (J/kg/K)
T temperature (K)
Tr reduced state temperatureU velocity (m/s)V volume (m
3)
Vm molar volume
W expansion ratioxa poppet or actuator movement (m)Z compressibility coefficient
dynamic viscosity (Ns/m2)
poissons ratio gas density (kg/m
3)
Subscript
ab actuator bellowap actuator pistonatm ambientc critical point
d damping orificeD damping chamberin inlet
L low pressure chamberor regulator outlet orifice
out outletp poppetpb poppet bellowr reference spring
R reference chamber
s seatt time
th throat
tot totalw reference spring wire
REFERENCES
Baline W. Anderson, 1979, Theanalysis and design of pneumatic
systems, John Wiley & Sons Inc.,
NY, USA, pp. 17-31.
Brasilow R, 1989, The basics of gas
regulators, Weld. Des. Fabric. pp.
61-65.
Delenne B, Mode L, Blaudez M,
1999, Modelling and simulation of
a gas pressure regulator, European
simulation symposium, Erlangen,
Germany.
Delenne B, Mode L, 2000,Modelling and simulation of a
pressure oscillations in a gas
pressure regulator, Proceedings of
ASME 2000 FPST, vol. 7, CongressIMECE, Orlando.
Dragoljub Viji, Slobodan
Radojkovi, 2001, Dynamic model
of gas pressure regulator, Facta
Universitatis, Mechanics,
Automatic Control and Robotics,
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El Goli Rami, Bezian Jean-Jacques,Delenne Bruno, Menu Francois,
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vessel and piping, Vol. 84. pp. 234 -
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Glen W. Howell, Terry M.
Weathers, 1970, Aerospace fluid
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Mischke, 1989, Mechanical
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Krigman A, 1989, Guide to
selecting pressure regulators, In
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Applied Mathematical Modelling,
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A new two constant equation of
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