Cryogenic Flow in Corrugated Pipes 2nd CASA Day: April 23, 2009 Patricio Rosen

Outline Motivation Pipe Flow Preliminaries Fluid Flow Equations CFD Navier Stokes CFD k-e Turbulence Model Conclusions Further work

Outline Motivation Pipe Flow Preliminaries Fluid Flow Equations CFD Navier Stokes CFD k-e Turbulence Model Conclusions Further work

Motivation Corrugated Pipes/Hoses Portable Flexible Several Application Areas LNG Transport Development of DTSE (Dual Tank Stirling Engine) Goal Describe and Predict Flow Behavior in Corrugated Hoses in an Efficient Way

Outline Motivation Pipe Flow Preliminaries Fluid Flow Equations CFD Navier Stokes CFD k-e Turbulence Model Conclusions Further work

Preliminaries P ¡ = : ¢ f ½ v 4 R Darcy Weisbach Equation Straight Pipe Poiseuille Flow Non-straight Pipes? Roughness Actual Shape L R P 1 2 v z P 1 ¡ 2 = : ¢ L f ½ v z 4 R v z = 2 µ 1 ¡ r R ¶ R e : = 4 ½ v z ¹ f = 6 4 R e

Moody Diagram Experimental Results Moody Diagram Fully Turbulent Colebrook Equation

LNG Composite Hose Moody Prediction Taking corrugation as roughnes Measurements Water LNG f = . 4 5 f = . 5 8 Moody Diagram is a poor indicator for the friction factor Find a better Alternative (CFD) f = . 1 3

Outline Motivation Pipe Flow Preliminaries Fluid Flow Equations CFD Navier Stokes CFD k-e Turbulence Model Conclusions Further work

Set up saves lots of computation time Fluid Flow Equations Assumptions Incompressible Flow Steady Flow Gravity negligible Isothermal Flow One-phase Cylindrical and Periodic Hose Fixed Wall No Swirl Cylindrical Symmetry Navier-Stokes v ¢ r = ¡ 1 ½ @ P + º h 2 i z Periodicity v ( R z ) ; = r L P + k Set up saves lots of computation time

Analytic Expression for DFF From continuity Rewrite z-momentum Using Divergence Theorem For Poiseuille Flow v ¢ r z = ( ) ¡ r ¢ ( v z ) = ¡ 1 ½ P e + º : ¢ P L : = i n ¡ o u t 1 j µ Z z d S ¹ @ v ¶ Pressure “Friction“ Skin “Friction“ f = 6 4 ¹ D ½ v R e : ¢ P L = 8 ¹ v R 2 :

Outline Motivation Pipe Flow Preliminaries Fluid Flow Equations CFD Navier Stokes CFD k-e Turbulence Model Conclusions Further work

CFD Navier Stokes f 8 Setting > = < : + f Discretization @ P r = ~ z + f Discretization Velocity Pressure

Navier Stokes Re=2.13 Re=213 Re=2713 Re=676

Validation Several Corrugations (Re=213.5)

Forces at Wall Re=213 Re=2713 Pressure and Viscous Forces scale with Re in Laminar Regime and Skin Friction Dominates

Same Friction Factor as for Straight Pipes in Laminar Regime

Outline Motivation Pipe Flow Preliminaries Fluid Flow Equations CFD Navier Stokes CFD k-e Turbulence Model Conclusions Further work

CFD k-e Turbulence Model Re=177 Re=843 Re=1.737e4 Re=3860

Several Periods Re=177 Re=847 Re=3860

At High Reynolds Numbers the Pressure Forces become Dominant

Friction Factor One Period Several Periods

Outline Motivation Pipe Flow Preliminaries Fluid Flow Equations CFD Navier Stokes CFD k-e Turbulence Model Conclusions Further work

Conclusions Correct Prediction of the Friction Factor Problem Solved? (One phase, adiabatic flow) Problem Solved? Sensibility of Results (needs validation) Cryogenic Liquids not yet manageable Expensive computation time for dynamic flow computations 4 hours Computation (NS Example)

Outline Motivation Pipe Flow Preliminaries Fluid Flow Equations CFD Navier Stokes CFD k-e Turbulence Model Conclusions Further work

Towards a 1D Model v ( r ; ) = + ^ d z ( R v ) = ¡ 1 ½ P Z ^ r ; + ` º : = 1 ¡ ( ) Z r ; d A ¡ i n ¡ o u t P ( z ) = 1 j ¡ Z d A d z ( R 2 v ) = ¡ 1 ½ P Z ^ r ; + ` º · @ ¸

Thanks for your attention!

RANS

k-e Model Summary

FEM for Navier Stokes

Navier Stokes Weak Form