research article dynamic modeling and analysis of a high...

Research ArticleDynamic Modeling and Analysis of a High Pressure Regulator

Muhammad Ramzan and Adnan Maqsood

Research Centre for Modeling & Simulation, National University of Sciences and Technology, Islamabad, Pakistan

Correspondence should be addressed to Adnan Maqsood; [email protected]

Received 30 June 2016; Revised 11 October 2016; Accepted 12 October 2016

Academic Editor: Tarek Ahmed-Ali

Copyright © 2016 M. Ramzan and A. Maqsood.This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in anymedium, provided the originalwork is properly cited.

Pressure regulator is a common device used to regulate the working pressure for plants and machines. In aerospace propulsionsystems, it is used to pressurize liquid propellant rocket tanks at specified pressure for obtaining the required propellant massflow rate. In this paper, a generalized model is developed to perform dynamic analysis of a pressure regulator so that constantpressure at outlet can be attained. A nonlinear mathematical model of pressure regulator is developed that consists of dynamicequation of pressure, temperature, equation of mass flow rate, and moving shaft inside regulator. The system of nonlinear andcoupled differential equations is numerically simulated and computation of pressure and temperature is carried out for requiredconditions and given design parameters. Valve opening and mass flow rate are also found as a function of given inlet pressure andtime. In the end, an analytical solution based on constant mass flow rate assumption is compared with nonlinear formulation. Theresults demonstrate a high degree of confidence in the nonlinear modeling framework proposed in this paper.The proposedmodelsolves a real problem of liquid rocket propulsion system. For the real system under consideration, inlet pressure of regulator isdecreased linearly from 150 bar to 60 bar and outlet pressure of nearly 15 bar is required from pressure regulator for the completeoperating time of 19 s.

1. Introduction

Pressure regulator is normally a dynamic open valve thattakes a high varying inlet pressure and converts it to anearly constant and lower desired outlet pressure. It consistsof set screw, working spring, main shaft, valve seat, inletoutlet chamber, sensing orifice, pressure chamber, rollingdiaphragm plate, and return spring as shown in Figure 1.Constant downstream pressure is obtained by variable valveopening area. This variable valve opening area is generateddue to opposite force balancing between spring loading andpressure acting on diaphragm plate in pressure chamber.Rolling diaphragm moves upwards as downstream pressureincreases and moves backward as downstream pressuredecreases for maintaining constant outlet pressure.

Vujic and Radojkovic [1] studied the procedure of form-ing nonlinear dynamic model of gas pressure regulator. Themodel showed the self-exciting oscillations of systems withcertain amplitude and frequency without the presence ofoutside disturbance. The results reported the effect of eachdesign parameter for self-exciting oscillations and suggested

methods to correct them. In the end, it was proposed thata linear model is sufficient for evaluation of stability andtransient response if flow through valve is laminar, dryfriction is negligible, and the motion of valve and diaphragmis not constrained. These sets of assumptions are signifi-cantly contradictory to actual operations. Specifically, mixingtriggers a highly turbulent phenomenon inside pressureregulators and the situation further worsens under highpressure environments. Similarly, Sunil et al. [2] developedthe linear mathematical model of pressure regulator forcryogenic application pressure regulator. The model wasvalidated with the experimental test of cryogenic pressureregulator, developed by Liquid Propulsion Systems Centre(LPSC) of Indian Space Research Organization. The effectof cryogenic temperature was measured in the regulatedpressure taking into account spring load variation, designchanges, and fluid property. It is interesting to note thatthe pressure differential equation in regulator flow volumewas considered without taking the valve clearance factor andassuming the constant temperature in all regulator volumesin both studies.

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016, Article ID 1307181, 8 pageshttp://dx.doi.org/10.1155/2016/1307181

2 Mathematical Problems in Engineering

(a)

Outlet

Spool or main shaft

Control spring

Return spring

Inlet Sensing orifice

Damping chamber

Diaphragm plate

(b)

Figure 1: Cross-sectional view of pressure regulator and its com-ponents: three-dimensional view (a) and two-dimensional crosssection (b).

Shahani et al. [3] studied the dynamic equation ofpressure and temperature with the assumption of adiabaticprocess happening in high pressure regulator.Themodel wassimulated using numerical tools and verified experimentally.It was found that as outlet volume increases, the stability ofoutlet pressure increases and as spring load increases, outletpressure increases proportionally.The control spring stiffnessand diaphragm area were two sensitive parameters identifiedthat effect downstream pressure of regulator. It was suggestedthat, for the better control of outlet pressure of regulator,these two parameters should be designed and manufacturedcarefully.

Zafer and Luecke [4] investigated the stability character-istics and showed the cause of vibration and possible designmodification for eliminating the unstable vibrating model. Acomprehensive linear dynamic model of self-regulating highpressure gas regulator was developed for stability analysis.Root locus technique was used to study the effect of variationof various design parameters on system dynamics. It wasconcluded that damping coefficient, the diaphragm area, andupper and lower volumes are most important design param-eters that affect the stability. An improvement in stability wasreported with the decrease in flow path between regulatorbody and lower pressure chamber. Similarly, Delenne andMode [5] performed experimental and numerical simulationto identify the relative influence of several parameters onemergence and amplitude of oscillations. However, no dis-cussion on regulator design and model parameters is made.Experimental work performed in various studies cannot beextended as they give results subject to specific geometricshape.

Rami et al. [6] developedmathematicalmodel, performedexperimental analysis, measured performance, and foundoperating conditions that increase stability. The numericaland experimental results were in good agreement withrespect to each other. It was concluded that oscillations

in downstream pressure increase for small volumes andhigher upstream pressure. The study showed that openingof valve and driving pressure have less effect on oscillationof downstream pressure. Moreover, the length of sensinglines from downstream chamber to damping chambers ofregulator has only little influences on the oscillation ofdownstream pressure.

With the advancement of Computational FluidDynamics(CFD), several studies are directed towards complex flowgeneration and visualization inside pressure regulator. Ship-man et al. [7] carried steady and unsteady simulation forthe analysis of gas pressure regulator for rocket feed system.Comparison of the numerical results and experimental datafrom NASA Stennis Space Center was found to be satisfac-tory. Ortwig andHubner [8] worked onmechatronic pressurecontroller for Compressed Natural Gas (CNG) engines. CFDsimulation for analysis of shock and vortices formation nearthe valve opening was performed and potential for theimprovement in geometry with the help of CFD results wasdiscussed. It was concluded that flow forces are dependenton pressure differential. Moreover, steady state flow forcesare almost independent of valve opening. Saha [9] performedthe numerical simulation of a pressure regulated valve to findout the characteristics of passive control circuit. Commercialsoftware was used to analyze the flow forces on differentinterfaces of the moving shaft of regulator. A special UserDefined Function (UDF) for varying inlet pressure andreducing the valve opening by delta amount for transientanalysis of pressure regulator was written. Flow convergencewas checked at each time step using UDF and calculationwas allowed to move to next time step only after meetingthe required criteria for convergence. Similarly, Du andGao [10] carried numerical study of complex turbulent flowthrough valves in a steam turbine system. Their main aimwas to understand the flow behavior through the complexconfiguration operating under different conditions, so as toderive the necessary information of optimizing the valvedesign. However, dynamic prediction cannot be performedusing CFD simulations as they are very expensive and notpractical in real-time environment. The model presented inthis paper tries to fill this void.

From operations perspective, Yanping et al. [11] discussedthe dynamic model of pressure reducing valve. The dynamicprocesses of valve for pressurizing, startup, and differentoperating conditions by numerical simulations and experi-mental investigation were discussed. It was concluded thatincreasing the area of damping hole or decreasing the volumeof damping cavity can not only reduce redressing time ofthe pressure regulating valve, but also reduce overshoot.Moreover, increasing the stiffness of the main spring canreduce the redressing time of the pressure regulating valve.From review of relevant studies, it is clear that a complexphenomenon occurs inside pressure regulator that includescompressibility, choking, turbulence, fluid structure interac-tion, flow expansion with recirculation, and flow separation.

There is dearth of published literature which brieflydiscusses the dynamic simulation of high differential pressureregulator. In this study, a generalized nonlinearmathematicalmodel is developed that computes regulator outlet pressure by

Mathematical Problems in Engineering 3

incorporating at the same time the dynamic equation of valveclearance, differential equation of temperature in regulatorvolume, and equation of mass flow rate by finding pressuregradients in the inlet volume and outlet volume as a functionof time.Model is simulated for two cases. For the first case, theinlet pressure of reservoir is increased from 60 bar to 150 barat the rate of 4.74 bar/s. Results of pressure, temperature,massflow rate, and valve opening are computed. The model is ofutilitarian nature and is specifically simulated for particularapplication in rocket tank feed system for propulsion ofrocket in second case. Specifically, the reservoir pressure isreduced from 150 bar to 60 bar at the rate of 4.74 bar/s andmodel is simulated for 19 s for controlling outlet pressure near15 bar. Mass flow rate and valve opening are required as afunction of time formaintaining 15-bar pressure at outlet portwith respect to varying high inlet pressure. Initial parametersof regulator are set in away that it gives initial valve opening of2.3mm. With the known value of constant mass flow rate forvarying high inlet pressure and keeping constant temperatureat inlet and outlet, an analytical solution is also proposed fordynamic equation of pressurewith varying high inlet pressureas a function of time.

2. Geometric Description

Apressure regulator normally consists of tightening screw onits top to provide the load force on working spring. This loadforce controls the outlet pressure. Other main parts are innermoving shaftwith valve whose basic function is to restrict theflow and provide a function of pressure drop.

Overall flow volume of pressure regulator is dividedinto four regions and for each flow volume (Figure 2), wedevelopedmathematical model for pressure and temperatureas a function of time. First-order forward differencing schemeis used for the discretization of dynamic equation of pressureand temperature. Equation of moving inner shaft is reducedto two sets of first order differential equations with initialconditions and computed numerically using Runge-Kuttamethod. Mass flow rate, valve opening, and outlet pressureare computed as a function of time and inlet pressure. Ananalytical solution is found for finding controlled outletpressure as a function of time with some assumptions.

Here 𝑉in is the inlet volume, 𝑉1 is intermediate volume,𝑉out or 𝑉reg is outlet volume, and 𝑉dam is the dampingvolume in cubic meter. 𝑃 (bar) and 𝑇 (Kelvin) are pressureand temperature in corresponding volumes. ��in is the inletmass flow rate (kg/s) from high pressurized compressed aircylinder to inlet volume and othermass flow rates correspondto flow from one volume to another volume (Figure 3).

3. Mathematical Modeling

For the development of dynamic equation of pressure weuse the assumption of adiabatic and reversible process insideregulator ducts, so we can use the isentropic relation betweenpressure (𝑃), density, and specific heat ratio (𝛾)

𝑃( 𝑉𝑀)𝛾 = const. (1)

m1

ms

P1T1V1PdamTdamVdam

mexit

PregTregVout

mdammin

P0T0Vin

Figure 2: Flow volumes of pressure regulator and correspondingquantities.

It is pertinent to note that the entire volume is divided intofour subvolumes,𝑉in,𝑉1,𝑉out, or𝑉reg and𝑉dam.𝑉in here is theinlet volume, 𝑉1 is intermediate volume, 𝑉out or 𝑉reg is outletvolume, and 𝑉dam is damping volume. Differentiating withrespect to time and after simplification, dynamic equation ofpressure is found and is given in

𝑑𝑃𝑑𝑡 = (𝑅𝛾𝑉 ) ∗ (𝑇in��in − 𝑇out��out + 𝑇𝑠��𝑠) − 𝛾𝐴𝑃𝑉 ��, (2)

where 𝐴 is the valve clearance area, 𝑥 is the valve clearancelength in mm, 𝑅 is the ideal gas constant, 𝑇in is the inlet tem-perature, 𝑇out is the outlet temperature, 𝑇𝑠 is the temperatureof damping orifice, and𝑚𝑠 is the mass flow through dampingorifice.

Development of dynamic equation of temperature iscarried out from first law of thermodynamics. Consideringthe density change is occurring in regulator duct, we canwritesum of all energy added to ducts of pressure regulator by theenergy equation as

𝑑𝑑𝑡𝐸 + 𝑑𝑑𝑡𝑊 = 𝑑𝑑𝑡𝑄 + 𝐻in − 𝐻out, (3)

where 𝐸 represent total internal energy, 𝑊 is the rate ofchange of work, and 𝑄 is heat transfer inside and outsideof pressure regulator. Here we are assuming that no heattransfer is taking place. 𝐻in and 𝐻out are the inlet and outletenthalpy of air mass in regulator ducts. After simplification of


A5

A11A1

A2 A3

A4A6

F0 + x ∗ Kcspring

Fval + x ∗ Kcval

Figure 3: Force balance on various area of moving shaft.

energy equation, following dynamic equation of temperatureis found:

𝑑𝑑𝑡𝑇 = ( 𝑅𝑇out𝑃out𝑉out)2

∗ [(𝛾 𝑇in𝑇out − 1) ��in − (𝛾 − 1) ��out]− (𝛾 − 1) 𝑃𝑉out 𝐴��.

(4)

For the mass flow from one flow volume to other, followingequation is used:

if 𝑃out/𝑃in < 0.528

�� = 𝐶𝐴( 2𝛾 + 1)1/(𝛾−1)√ 2𝛾𝛾 + 1 1𝑅𝑇in𝑃in; (5)

if 𝑃out/𝑃in > 0.528

�� = 𝐶𝐴√ 2𝛾𝛾 − 1 1𝑅𝑇in𝑃in√(𝑃out𝑃in )

2/𝛾 − (𝑃out𝑃in )(𝛾+1)/𝛾, (6)

where𝐶 is the discharge coefficient and its value can be takenas 0.8, 0.9, or 1. A is the corresponding cross section area formass flow from one volume to another. Equation of movingshaft is developed from Newton formulation based on itsupward and downward motion and is given by

�� = (𝐴11𝑃in − 𝐴1𝑃in − 𝐴2𝑃reg − 𝐴3𝑃reg + 𝐴4𝑃reg + 𝐴5𝑃reg + 𝐴6𝑃reg − 𝐹𝑐sign (��) − 𝐾𝑐spring𝑥 − 𝐹0 − 𝐹valve − 𝐾𝑐valve𝑥)𝑀 ,𝑥 (0) = 2.3mm,�� (0) = 0,

(7)

where 𝐾𝑐spring and 𝐾𝑐valve are the control springs and returnspring rates.𝑀 is the mass of moving shaft and 𝑃in is the inletpressure. 𝑃reg is the regulated pressure or outlet pressure. 𝐹0is the control spring preload or initial force set on spring.𝐹valve is the return spring precompression. 𝐹𝑐 is dry frictionbetween shaft seal and regulator. It is also called lubricatedor coulomb friction. Also here sign is a MATLAB� signumfunction which gives the moving shaft opposite direction byobserving its velocity.

For all four volumes, derivatives of pressure and temper-ature are replaced by their time differencing for computationof pressure and temperature. Here the discretized equationof pressure and temperature for outlet volume based on first-order forward differencing scheme is shown in (8) and (9),respectively:

𝑃reg (𝑡 + Δ𝑡) = [( 𝑅𝛾𝑉reg)∗ (𝑇in (𝑡) ��in (𝑡) − 𝑇out (𝑡) ��out (𝑡) + 𝑇 (𝑡)𝑠 ��𝑠 (𝑡))− 𝛾𝐴 𝑠𝑃 (𝑡)𝑉reg 𝑥 (𝑡)] ∗ Δ𝑡 + 𝑃reg (𝑡) ,

(8)

𝑇reg (𝑡 + Δ𝑡) = [( 𝑅𝑇out2 (𝑡)𝑃out (𝑡) 𝑉out)∗ [(𝛾 𝑇in (𝑡)𝑇out (𝑡) − 1) ��in (𝑡) − (𝛾 − 1) ��out (𝑡)]− (𝛾 − 1) 𝑃 (𝑡)𝑉out 𝐴exit

𝑥 (𝑡)] ∗ Δ𝑡 + 𝑇reg (𝑡) .(9)

4. Simulation Results and Discussion

Above nonlinear and coupled differential equations of eachflow volumes are simulated numerically by finite differencemethod. Second-order differential equation of moving shaftis converted to two sets of first-order differential equationsand solved numerically by MATLAB built-in routine ode45based on Runge-Kutta method.

Simulation results have been derived for two cases. Infirst case, we increase the inlet pressure of reservoir from60 bar to 150 bar at the rate of 4.74 bar/s and note theoutlet pressure of regulator. In second case, pressure in


2 4 6 8 10 12 14 16 18 200Time (sec)

−0.5

0

0.5

1

1.5

2

2.5

3 V

alve

clea

ranc

e (m

m)

Figure 4: Valve opening as a function of time for increasing inletpressure.

the reservoir or compressed air cylinder is reduced from150 bar to 60 bar at the rate of 4.74 bar/s and overall regulatorperformance is checked corresponding to outlet pressure. Foreach case, outlet pressure, mass flow rate, temperature, andvalve opening are computed.

4.1. Increasing Inlet Pressure from 60 bar to 150 bar. Load orcontrol spring compression on pressure regulator is initiallyset in such a way that initial valve clearance of regulator is2.5mm. When the inlet valve of regulator is opened, highpressure fluids begin to flow from reservoir to different vol-ume of regulator. Pressure in each volume of regulator beginsto increase. The pressure in outlet volume is continuouslychecked in damping chamber through damping orifice. Asthe pressure in the outlet volume of regulator reaches above15 bar, pressure creates a force on the diaphragm plate againstthe load force of spring which was set for a 15-bar pressureat the outlet. As a result, the main shaft of regulator movesupward and valve clearance reaches its minimum openingposition. After that, the moving shaft of regulator oscillatesup and down such that pressure of nearly 15 bar is maintainedin outlet volume of regulator as shown in Figure 4.The outletpressure approaches to 15 bar around 0.8 s and valve openingreduces to its minimum level. Steady state oscillations ofabout 0.25mm are observed afterwards. The magnitude ofoscillation is about 0.1mm which is negligible.

Initial pressure at outlet was 1 bar, near about 0.8 seconds,and outlet pressure reaches 15 bar and then a minor oscilla-tion in outlet pressure is seen for whole simulation time inFigure 5.

Initial valve clearancewas 2.5mmdue towhich highmassflow rate is seen initially form volume 𝑉1 to volume 𝑉out. Asthe pressure in volume 𝑉out reaches 15 bar, the valve openingreduces and a sudden decrement inmass flow rate can be seen

60 70 80 90 100 110 120 130 140 15050Inlet pressure (bar)

2

4

6

8

10

12

14

16

18

20

Out

let p

ress

ure (

bar)

Figure 5: Outlet pressure (N/m2) variation against inlet pressure(N/m2) and over entire simulation time.

2 4 6 8 10 12 14 16 18 200Time (sec)

−0.5

0

0.5

1

1.5

2M

ass fl

ow ra

te (k

g/s)

Figure 6: Mass flow rate at valve opening as a function of time.

in Figure 6. Subsequently, mass flow rate value oscillates nearabout 0.24 kg/s for maintaining nearly 15 bar at outlet.

Initial temperature of volume 𝑉out was 290K (standardatmosphere temperature).With the increase in pressure from1 to 15 bar, temperature also increases. Due to compressibilityeffect, temperature does not decrease sharply but it decreasessmoothly as shown in Figure 7.

4.2. Decreasing Inlet Pressure from 150 bar to 60 bar. Thiscase corresponds to simulation of real problem of liquidpropulsion system. Initial pressure and temperature in each


2 4 6 8 10 12 14 16 18 200Time (sec)

200250300350400450500550600

Tem

pera

ture

(K)

Figure 7: Temperature in outlet volume as a function of time.

2 4 6 8 10 12 14 16 18 200Time (sec)

0

0.5

1

1.5

2

2.5

3

Valv

e ope

ning

(mm

)

Figure 8: Valve opening as a function of time.

flow volume are taken as 1 bar and 290K. Time increment(Δ𝑡) is taken as 0.001 s. Valve opening (Figure 8) is 100%(2.3mm) nearly for 0.7 s. Afterward outlet pressure reachesnearly 15 bar and valve opening reaches its minimum level.Valve clearance value oscillates with magnitude of 0.02mmand increases as well with the decrement of inlet pressure tomeet the mass conservation.

Initially pressure at outlet volume is 1 bar and after nearly0.7 s, pressure at outlet reaches average value of 15 bar andcontrol spring load force on diaphragm plate is balanced bypressure force on diaphragm plate in opposite direction.Thusa controlled pressure of nearly 15 bar is achieved as shown inFigure 9.

Initially mass flow rate increases till about 0.7 s andpressure at outlet volume reaches 15 bar, valve clearancedecreases, and suddenly mass flow rate also decreases asshown in Figure 10.

Initial temperature was set to 290K at outlet volume withpressure of 1 bar. Due to sudden increase of pressure in outletchamber, average value of temperature suddenly increases forvery short time and then steady state temperature reaches dueto maintained pressure at outlet as shown in Figure 11.

2 4 6 8 10 12 14 16 18 200Time (sec)

−5

0

5

10

15

20

25

Pres

sure

(bar

)

Figure 9: Outlet pressure for whole simulation time.

2 4 6 8 10 12 14 16 18 200Time (sec)

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

Mas

s flow

rate

(Kg/

sec)

Figure 10: Mass flow rate as a function of time.

0 1 2 3 4 5 6 7 8 9 10−1Time (sec)

280

300

320

340

360

380

400

420

Tem

pera

ture

(k)

Figure 11: Variation of temperature in outlet volume.


Vin

Vreg

Figure 12: Two main flow volumes of pressure regulator used foranalytical estimation.

5. Closed Form Solution of Pressure Equation

For quick analysis of outlet pressurewithout involving the fulldesign parameters, an analytical solution of dynamic equa-tion of pressure is obtainedwith some assumption of constantmass flow rate and different inlet and outlet temperature. Forthe development of analytical solution only two flow volumesare taken. Inlet volume𝑉in is the flow volume before the valveopening and outlet volume or regulated volume 𝑉reg is thevolume after the valve opening as shown in Figure 12.

By assuming average value of mass flow rate and usingmass flow rate equation, valve opening from 150 bar to 60 barinlet pressure is found as function of time in (10) and isshown in Figure 13. Here valve opening for analytical solutionis compared with numerical computed valve opening whenoutlet pressure reaches nearly 15 bar. The difference in resultsis due to use of variablemass flow rate with full design param-eters in numerical computation and only constant averagemass flow rate with different inlet and outlet temperature foranalytical solution.

𝐴 (𝑡)= ��[𝐶din (2/ (𝛾 + 1))1/(𝛾−1)√(2𝛾/ (𝛾 + 1)) (1/𝑅𝑇in)𝑃in (𝑡)] .

(10)

Introduced in dynamic equation of pressure, a lineardifferential equation is obtained which is solved analyticallyand solution is given by

𝑃𝑟 (𝑡) = 𝛾𝑅𝑉𝑟 (√𝑇in𝐴 𝑠 (𝑡) 𝐶1𝑃in (𝑡)) ∗ 𝑒−𝐴𝑒∗√𝑇out∗𝐶1

+ 𝐶 ∗ 𝑒−𝐴𝑒∗√𝑇out∗𝐶1 ,𝐶1 = 2𝛾 + 1

𝛾/(𝛾−1) ∗ √ 2𝛾𝑅 (𝛾 + 1) ,

AnalyticalNumerical

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

Valv

e ope

ning

(mm

)

2 4 6 8 10 12 14 16 18 200Time (sec)

Figure 13: Comparison of valve opening as a function of time.

𝐶 = 𝑃𝑟 (0) − 𝛾𝑅𝑉𝑟 (√𝑇in𝐴 𝑠 (0) 𝐶1𝑃in (0))∗ 𝑒𝐴𝑒∗√𝑇out∗𝐶1 .

(11)

Outlet pressure fromnumerical computation is comparedwith analytical solution as shown in Figure 14. Averageconstant mass flow rate used in valve opening in analyticalsolution does not provide any oscillation in mass flow rateso here outlet pressure smoothly reaches 15 bar in case ofanalytical solution for whole simulation time.This is becausewhen the constant mass flow rate assumption is taken thenvalve opening is only the function of inlet pressure and inlettemperature. In this way we are here ignoring the equationof moving shaft and finding valve opening from the massflow rate equation.The constant mass flow rate in steady statecondition can be used just for quick analysis of some initialdesign parameters.

6. Conclusion

Mathematical model for flow inside a generic pressureregulator is developed and numerically simulated. Dynamicequation of pressure and temperature is developed based onthe continuity and energy equation to capture the pressureand temperature change in regulator ducts as a functionof time. Equation of moving shaft is developed by Newtonsecond law of motion. First-order forward differencing tech-nique is used to discretize the dynamic equations. Resultsinclude dynamic simulation of pressure, mass flow rate,temperature, and valve opening as a function of time and


AnalyticalNumerical

0

2

4

6

8

10

12

14

16

18

20Pr

essu

re (b

ar)

2 4 6 8 10 12 14 16 18 200Time (sec)

Figure 14: Comparison of outlet pressure with time.

inlet pressure. Two scenarios are simulated. In the first case,inlet pressure is increased from 60 bar to 150 bar and 15 baris maintained at outlet. In second case, inlet pressure isdecreased from 150 bar to 60 bar while maintaining outletpressure of 15 bar. Oscillating mass flow rate with averagevalue of 0.4547 kg/s is found for outlet pressure of 15 bar. Theanalytical solution is presented in the end and satisfactorycomparison is demonstrated with numerical results.

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

References

[1] D. Vujic and S. Radojkovic, “Dynamic model of gas pressureregulator,”Mechanics, Automatic Control & Robotics, vol. 3, no.11, pp. 269–276, 2001.

[2] S. Sunil, U. Pandey, B. S. Jeevanlal, M. Radhakrishnan, and C.Amarasekaran, “Mathematical modeling of pressure regulatorfor cryogenic applications,” in Proceedings of the 37th National& 4th International Conference on Fluid Mechanics and FluidPower, FMFP10-CF-30, Chennai, India, December 2010.

[3] A. R. Shahani, H. Esmaili, A. Aryaei, S. Mohammadi, andM. Najar, “Dynamic simulation of high pressure regulator,”Journal of Computational and Applied Research in MechanicalEngineering, vol. 1, no. 1, pp. 17–28, 2011.

[4] N. Zafer and G. R. Luecke, “Stability of gas pressure regulators,”Applied Mathematical Modelling, vol. 32, no. 1, pp. 61–82, 2008.

[5] B. Delenne and L. Mode, “Modeling and simulation of pressureoscillations in a gas pressure regulator,” Proceedings of AmericanSociety of Mechanical Engineers, vol. 7, 2000.

[6] E. G. Rami, B. Jean-Jacques, G. Pascal, and M. Francois,“Stability study and modeling of a pilot controlled regulator,”Material Sciences and Application, vol. 2011, no. 2, pp. 859–869,2011.

[7] J. Shipman, A. Hosangadi, and V. Ahuja, “Unsteady analysesof valve systems in rocket engine testing environments,” inProceedings of the 40th AIAA/ASME/SAE/ASEE Joint PropulsionConference, Fort Lauderdale, Fla, USA, 2004.

[8] H. Ortwig and D. Hubner, “Investigation and CFD simulationof flow forces inside a mechatronic pressure controller for CNGengines,” in Proceedings of the 12th Pan-American Congress ofApplied Mechanics, Barcelona, Spain, 2012.

[9] B. K. Saha, “Numerical simulation of a pressure regulated valveto find out the characteristics of passive control circuit,” Interna-tional Journal of Mechanical, Aerospace, Industrial, Mechatronicand Manufacturing Engineering, vol. 7, no. 5, pp. 936–939, 2013.

[10] X. Du and S. Gao, “Numerical study of complex turbulent flowthrough valves in a steam turbine system,” International JournalofMaterials,Mechanics andManufacturing, vol. 1, no. 3, pp. 301–305, 2013.

[11] J. Yanping, S. Chibing, L. Lin, and L. Youpeng, “Simulationresearch on the dynamic processes of gas pressure reducingvalve,” in Proceedings of the 46th AIAA/ASME/SAE/ASEE JointPropulsion Conference & Exhibit, Nashville, Tenn, USA, July2010.

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