Effect of Quadratic Fluid DampingEffect of Quadratic Fluid Dampingin Twoin Two--Phase Liquefied Natural GasPhase Liquefied Natural Gas

Andrew KimmelAndrew KimmelUniversity of Nevada, RenoUniversity of Nevada, Reno

Department of PhysicsDepartment of PhysicsReno, Nevada, USAReno, Nevada, USA

AIChE Spring National Meeting 2007AIChE Spring National Meeting 20077th Topical Conference on Natural Gas 7th Topical Conference on Natural Gas

UtilizationUtilization

Houston,Texas, USA April 22Houston,Texas, USA April 22--27, 200727, 2007

IntroductionIntroduction

1.1. TwoTwo--phase fluid flows are often subject to phase fluid flows are often subject to pressure induced oscillations. The volume pressure induced oscillations. The volume of the vapor bubbles increases or decreases of the vapor bubbles increases or decreases differently if the pressure fluctuations are differently if the pressure fluctuations are compressing or expanding.compressing or expanding.

2.2. Consequently, compressing pressure Consequently, compressing pressure fluctuations in a twofluctuations in a two--phase pipe flow cause phase pipe flow cause less displacement in the direction of the less displacement in the direction of the pipe flow than expanding pressure pipe flow than expanding pressure fluctuations. fluctuations.

Axial displacement Axial displacement ∆∆xx==∆∆z(tz(t) is smaller for compression ) is smaller for compression than for expansionthan for expansion

Asymmetric Displacement for Compression and ExpansionAsymmetric Displacement for Compression and Expansion

Introduction (continued)Introduction (continued)

rVg −=∆

Volumetric Ratio:Volumetric Ratio:lg

g

VVVr+

= r=0 for pure liquidr=0 for pure liquidr=1 for pure vaporr=1 for pure vapor

Using the Using the Ideal Gas Law:Ideal Gas Law: gg

g

VVV

ppp

∆+=

∆+gg

g

VVVpp∆+

∆−=∆

Large Large ∆∆p (increasing p (increasing pressures) the gas condenses:pressures) the gas condenses:

+∞=∆ gVSmaller Smaller ∆∆p (p+p (p+∆∆p 0) p 0) mixture becomes all gas:mixture becomes all gas:

For a control volume where For a control volume where VVgg+V+Vll=1=1

Introduction (continued)Introduction (continued)

)1( rVm l −== ρρ

(for a control volume where V(for a control volume where Vgg+V+Vll=1)=1)

gVz ∆≈

The density of the gaseous natural gas is negligibly The density of the gaseous natural gas is negligibly small compared to the density small compared to the density ρρ of the liquefied natural of the liquefied natural gas. Therefore the oscillating mass m depends only on gas. Therefore the oscillating mass m depends only on

the volume of the liquid the volume of the liquid VVll

For pipe flows with constant cross sectional area For pipe flows with constant cross sectional area the axial displacement the axial displacement z z of the fluid is linearly of the fluid is linearly

dependent to the volumetric change dependent to the volumetric change

Introduction (continued)Introduction (continued)

The liquidThe liquid--vapor system generates an vapor system generates an asymmetric spring force:asymmetric spring force:

)()(tzr

tzpF+

=

This asymmetric spring force causes a bending This asymmetric spring force causes a bending of the resonant characteristic towards low of the resonant characteristic towards low

frequenciesfrequencies

Asymmetric Response for Free and Forced OscillationsAsymmetric Response for Free and Forced Oscillations

Dotted line indicates free oscillationsDotted line indicates free oscillationsSolid line indicates forced oscillationsSolid line indicates forced oscillations

Amplitude Response SpectrumAmplitude Response Spectrum

Spectrum for several values of exciting amplitude GSpectrum for several values of exciting amplitude G

1.1. Displacement of the vapor bubbles and Displacement of the vapor bubbles and surrounding liquid are subject to quadratic fluid surrounding liquid are subject to quadratic fluid dampingdamping

2.2. Quadratic fluid damping is due to the viscosity Quadratic fluid damping is due to the viscosity of the LNGof the LNG

3.3. The effect of the quadratic damping depends on The effect of the quadratic damping depends on the ratio of liquid to vapor; the ratio of pressure the ratio of liquid to vapor; the ratio of pressure fluctuations; exciting frequency of the pressure fluctuations; exciting frequency of the pressure fluctuationsfluctuations

Quadratic DampingQuadratic Damping

Quadratic Damping (continued)Quadratic Damping (continued)

4.4. Pressure changes act as a force in the axial Pressure changes act as a force in the axial direction of the pipedirection of the pipe

5.5. The compressible gaseous volume and the mass The compressible gaseous volume and the mass of the liquid volume form an oscillating systemof the liquid volume form an oscillating system

6.6. Friction forces generated are opposed to the Friction forces generated are opposed to the velocity vectorvelocity vector

7.7. Friction forces are proportional to the square Friction forces are proportional to the square of the displacement velocity vector times the of the displacement velocity vector times the damping factor, Ddamping factor, D

8.8. The exciting force depends on the exciting The exciting force depends on the exciting amplitude and the square of the exciting amplitude and the square of the exciting frequencyfrequency

Absolute Amplitude Response with Quadratic DampingAbsolute Amplitude Response with Quadratic Damping

Absolute amplitude for several values of the damping parameter CAbsolute amplitude for several values of the damping parameter C, , with the exciting frequency set to 0.1with the exciting frequency set to 0.1

Steady State Asymmetric Oscillations with Quadratic DampingSteady State Asymmetric Oscillations with Quadratic Damping

Dotted line indicates free oscillationsDotted line indicates free oscillationsSolid line indicates forced oscillationsSolid line indicates forced oscillations

Steady State Asymmetric Oscillation Spectrum with Steady State Asymmetric Oscillation Spectrum with Quadratic DampingQuadratic Damping

Spectrum of the steady state oscillations for different values oSpectrum of the steady state oscillations for different values of the f the exciting amplitude parameter with the damping parameter set to 0exciting amplitude parameter with the damping parameter set to 0.45.45

ConclusionsConclusions

1.1. Oscillations with quadratic damping Oscillations with quadratic damping demonstrate a complicated patterndemonstrate a complicated pattern

2.2. The natural frequency has a minimum value if The natural frequency has a minimum value if the volumes for gas and liquid are equalthe volumes for gas and liquid are equal

3.3. Resonant amplitude occur at the critical Resonant amplitude occur at the critical frequency which is below the natural frequencyfrequency which is below the natural frequency

4.4. Exciting frequencies above the natural Exciting frequencies above the natural frequency are less critical than exciting frequency are less critical than exciting frequencies below the natural frequencyfrequencies below the natural frequency

Conclusions (continued)Conclusions (continued)

5.5. Damping effect and energy dissipation depends Damping effect and energy dissipation depends on the liquid to vapor volume ratioon the liquid to vapor volume ratio

6.6. For a small damping effect the oscillations are For a small damping effect the oscillations are more critical and have to be closely monitored more critical and have to be closely monitored for plant safety and reliabilityfor plant safety and reliability

7.7. The damping effect and energy dissipation The damping effect and energy dissipation causes the twocauses the two--phase LNG to heat up and create phase LNG to heat up and create more vapormore vapor

8.8. Oscillation patterns change with a change in the Oscillation patterns change with a change in the vapor to liquid ratiovapor to liquid ratio

ReferencesReferences1.1. Henry, R.E. "Pressure Wave propagation in twoHenry, R.E. "Pressure Wave propagation in two--phase phase

mixtures" Heat transfer Minneapolis, Chemical mixtures" Heat transfer Minneapolis, Chemical Engineering Progress Symposium Series No.102, 1970, Engineering Progress Symposium Series No.102, 1970, Volume 66, American Institute of Chemical Engineers, Volume 66, American Institute of Chemical Engineers, New York.New York.

2.2. Kimmel, A. "Pressure induced nonKimmel, A. "Pressure induced non--linear oscillations in linear oscillations in twotwo--phase LNG pipe flow", phase LNG pipe flow", AIChEAIChE Spring National Spring National Meeting 2006, Orlando, FL, USA, April 23Meeting 2006, Orlando, FL, USA, April 23--27, 2006 27, 2006

3.3. Kimmel, A. Kimmel, A. ““Effect of quadratic fluid damping in twoEffect of quadratic fluid damping in two--phase liquefied natural gasphase liquefied natural gas””, , AIChEAIChE Spring National Spring National Meeting 2007, Houston, TX, USA, April 22Meeting 2007, Houston, TX, USA, April 22--27, 200727, 2007

Andrew KimmelAndrew Kimmel