Cryogenic Flow in Corrugated Pipes 2 nd CASA

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

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

Outline • Motivation • Pipe Flow Preliminaries • Fluid Flow Equations • CFD Navier

Outline • Motivation • Pipe Flow Preliminaries • Fluid Flow Equations • CFD Navier Stokes • CFD k-e Turbulence Model • Conclusions • Further work Mathematics Department 1

Outline • Motivation • Pipe Flow Preliminaries • Fluid Flow Equations • CFD Navier

Outline • Motivation • Pipe Flow Preliminaries • Fluid Flow Equations • CFD Navier Stokes • CFD k-e Turbulence Model • Conclusions • Further work Mathematics Department 2

Motivation • Corrugated Pipes/Hoses • Portable • Flexible • Several Application Areas • LNG

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 Mathematics Department 3

Outline • Motivation • Pipe Flow Preliminaries • Fluid Flow Equations • CFD Navier

Outline • Motivation • Pipe Flow Preliminaries • Fluid Flow Equations • CFD Navier Stokes • CFD k-e Turbulence Model • Conclusions • Further work Mathematics Department 4

Preliminaries • Darcy Weisbach Equation L ½v 2 z P 1 ¡ P 2

Preliminaries • Darcy Weisbach Equation L ½v 2 z P 1 ¡ P 2 = : ¢ PL = f 4 R • Straight Pipe • Poiseuille Flow µ vz = 2 vz 64 f = Re 1¡ R L r 2 ¶ R 2 Re : = • Non-straight Pipes? • Roughness • Actual Shape Mathematics Department P 1 vz 5 4 R½vz ¹ P 2

Moody Diagram • Experimental Results • Moody Diagram • Fully Turbulent • Colebrook Equation

Moody Diagram • Experimental Results • Moody Diagram • Fully Turbulent • Colebrook Equation Mathematics Department 6

LNG Composite Hose • Moody Prediction • Taking corrugation as roughnes f= 0. 045

LNG Composite Hose • Moody Prediction • Taking corrugation as roughnes f= 0. 045 • Measurements • Water • LNG f= 0. 058 f= 0. 13 • Moody Diagram is a poor indicator for the friction factor • Find a better Alternative (CFD) Mathematics Department 7

Outline • Motivation • Pipe Flow Preliminaries • Fluid Flow Equations • CFD Navier

Outline • Motivation • Pipe Flow Preliminaries • Fluid Flow Equations • CFD Navier Stokes • CFD k-e Turbulence Model • Conclusions • Further work Mathematics Department 8

Fluid Flow Equations Navier-Stokes ¢ r v= 0 h i 1 @P v +

Fluid Flow Equations Navier-Stokes ¢ r v= 0 h i 1 @P v + º r 2 vr ¡ r v ¢r vr = ¡ ½ @r r 2 1 @P = + º r 2 vz vz ¡ ½ @z Periodicity v (R(z); z) = 0 v (r; 0) = v (r; L ) P (r; 0) = P (r; L ) + k. L Mathematics Department • Assumptions • Incompressible Flow • Steady Flow • Gravity negligible • Isothermal Flow • One-phase • Cylindrical and Periodic Hose • Fixed Wall • No Swirl • Cylindrical Symmetry Set up saves lots of computation time 9

Analytic Expression for DFF • From continuity v ¢r vz = r ¢(vz v

Analytic Expression for DFF • From continuity v ¢r vz = r ¢(vz v ) ¡ vz r ¢v = r ¢(vz v ) • Rewrite z-momentum 1 r ¢(v vz ) = ¡ r ¢(P ez ) + º r ¢(r vz ): ½ • Using Divergence Theorem µZ ¶ Z 1 @vz ¢ PL : = P i n ¡ P ou t = P n z d. S ¡ ¹ d. S : @ n j¡ i n j ¡ ¡ Pressure “Friction“ • For Poiseuille Flow 8¹ v. L ¢ PL = : R 2 Mathematics Department f = 10 Skin “Friction“ 64¹ 64 = : D ½v Re

Outline • Motivation • Pipe Flow Preliminaries • Fluid Flow Equations • CFD Navier

Outline • Motivation • Pipe Flow Preliminaries • Fluid Flow Equations • CFD Navier Stokes • CFD k-e Turbulence Model • Conclusions • Further work Mathematics Department 11

CFD Navier Stokes Setting P = : P~ + f z z 8 >

CFD Navier Stokes Setting P = : P~ + f z z 8 > < > : @P @r = @P @z = @P~ @r @P~ @z + fz fz fz Discretization Velocity Pressure Mathematics Department 12

Navier Stokes Re=2. 13 Re=213 Re=676 Re=2713 Mathematics Department 13

Navier Stokes Re=2. 13 Re=213 Re=676 Re=2713 Mathematics Department 13

Validation Several Corrugations (Re=213. 5) Mathematics Department 14

Validation Several Corrugations (Re=213. 5) Mathematics Department 14

Forces at Wall Re=213 Re=2713 Pressure and Viscous Forces scale with Re in Laminar

Forces at Wall Re=213 Re=2713 Pressure and Viscous Forces scale with Re in Laminar Regime and Skin Friction Dominates Mathematics Department 15

Friction Factor Same Friction Factor as for Straight Pipes in Laminar Regime Mathematics Department

Friction Factor Same Friction Factor as for Straight Pipes in Laminar Regime Mathematics Department 16

Outline • Motivation • Pipe Flow Preliminaries • Fluid Flow Equations • CFD Navier

Outline • Motivation • Pipe Flow Preliminaries • Fluid Flow Equations • CFD Navier Stokes • CFD k-e Turbulence Model • Conclusions • Further work Mathematics Department 17

CFD k-e Turbulence Model Re=177 Re=843 Re=1. 737 e 4 Re=3860 Mathematics Department 18

CFD k-e Turbulence Model Re=177 Re=843 Re=1. 737 e 4 Re=3860 Mathematics Department 18

Several Periods Re=177 Mathematics Department Re=847 19 Re=3860

Several Periods Re=177 Mathematics Department Re=847 19 Re=3860

Re=177 Re=3860 At High Reynolds Numbers the Pressure Forces become Dominant Mathematics Department 20

Re=177 Re=3860 At High Reynolds Numbers the Pressure Forces become Dominant Mathematics Department 20

Friction Factor One Period Mathematics Department Several Periods 21

Friction Factor One Period Mathematics Department Several Periods 21

Outline • Motivation • Pipe Flow Preliminaries • Fluid Flow Equations • CFD Navier

Outline • Motivation • Pipe Flow Preliminaries • Fluid Flow Equations • CFD Navier Stokes • CFD k-e Turbulence Model • Conclusions • Further work Mathematics Department 22

Conclusions • Correct Prediction of the Friction Factor (One phase, adiabatic flow) Problem Solved?

Conclusions • Correct Prediction of the Friction Factor (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) Mathematics Department 23

Outline • Motivation • Pipe Flow Preliminaries • Fluid Flow Equations • CFD Navier

Outline • Motivation • Pipe Flow Preliminaries • Fluid Flow Equations • CFD Navier Stokes • CFD k-e Turbulence Model • Conclusions • Further work Mathematics Department 24

Towards a 1 D Model vz (r; z) = vz (z) + v^z (r;

Towards a 1 D Model vz (r; z) = vz (z) + v^z (r; z) Z ¡ in 1 v (r; z)d. A ¡ (z) ¡ ( z ) z Z 1 P(z) = P d. A j¡ (z)j ¡ ( z) vz : = ¡ out d 2 (R vz ) = 0 dz Z R ( z) d 2 2 1 d 2 2 = v^z (r; z)dr + P (R(z); z)R`(z)R(z)+ (R vz ) ¡ (R P ) ¡ 2 dz ½dz dz 0 ½ · 2º Mathematics Department ¸ @vz @v (R(z); z) ¡ R(z) z (R(z); z) R(z) @r @z 25

Thanks for your attention!

Thanks for your attention!

RANS Mathematics Department 27

RANS Mathematics Department 27

k-e Model Summary Mathematics Department 28

k-e Model Summary Mathematics Department 28

FEM for Navier Stokes Mathematics Department 29

FEM for Navier Stokes Mathematics Department 29

Navier Stokes Weak Form Mathematics Department 30

Navier Stokes Weak Form Mathematics Department 30