Cryogenic Flow in Corrugated Pipes 2 nd CASA
- Slides: 31
Cryogenic Flow in Corrugated Pipes 2 nd 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 Mathematics Department 1
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 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 Stokes • CFD k-e Turbulence Model • Conclusions • Further work Mathematics Department 4
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 Mathematics Department 6
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 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 + 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 ) ¡ 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 Stokes • CFD k-e Turbulence Model • Conclusions • Further work Mathematics Department 11
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
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 Regime and Skin Friction Dominates Mathematics Department 15
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 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
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
Friction Factor One Period Mathematics Department Several Periods 21
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? • 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 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; 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!
RANS Mathematics Department 27
k-e Model Summary Mathematics Department 28
FEM for Navier Stokes Mathematics Department 29
Navier Stokes Weak Form Mathematics Department 30
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