Flow in Corrugated Pipes

Transcription

1 Flow in Corrugated Pipes CASA Day April 7, 2010 Patricio Rosen Promotor: R.M.M. Mattheij Supervisors: J.H.M. ten Thije Boonkkamp J.A.M. Dam

2 Outline Motivation Fluid Flow equations Method of slow variations Friction factor estimates Validation (Sinusoidal pipes) Further work

3 Outline Motivation Fluid Flow equations Method of slow variations Friction factor estimates Validation (Sinusoidal pipes) Further work

4 Based mostly on: Direct Measurements Models (lumped) Generate Huge Libraries Network Model P1 D P =f L D Ū 0 ρū P2 Expensive experiments for models of new components (different geometry) Need for automation L

5 LNG composite hose Moody Prediction Taking corrugation as roughnes f=0.045 Measurements Water f=0.058 LNG f=0.13 Moody Diagram is a poor indicator for the friction factor CFD computations can handle the problem. But they are time consuming when: performed within a network solver, and for optimization.

6 Outline Motivation Fluid Flow equations Method of slow variations Friction factor estimates Validation (Sinusoidal pipes) Further work

7 Governing Equations R Γ in X = 0 Γ R(X) Ω Γ out X X = L ( UU X +VU R =ν U XX +U RR + 1 ) R U R 1 ρ P X, UV X +VV R =ν (V XX +V RR + 1R V R 1R ) 2V 1 ρ P R, Axisymmetric Pipe Navier-Stokes Axisymmetric Flow No-slip, flow rate U X +V R + 1 R V =0, U(X,(X))=V(X,(X))=0, 0 X L Q= Γ in UdS

8 Darcy Friction Factor From continuity, V U = (UV) Rewrite axial momentum (UV)= 1 ρ (Pe X)+ν ( U), Using divergence theorem (periodicity) P = 1 Pn X ds Γ in Γ }{{} P P µ U Γ in Γ n ds, }{{} P S Pressure Friction Skin Friction Need to estimate these integrals

9 Outline Motivation Fluid Flow equations Method of slow variations Friction factor estimates Validation (Sinusoidal pipes) Further work

10 Dimensionless Navier-Stokes a D R X u = Ū U 0, v = V Ū 0, x = X D, r = R D, p = P ρū2 0. Re:= Ū0D ν. L ɛ:= a L Re(u u x +v u r )=u x x +u r r + 1 r u r Rep x, Re(u v x +v v r )=v x x +v r r + 1 r v r 1 r 2 v Rep r, u x +v r + 1 r v =0.

11 Method of slow variations Slowly varying wall ( ɛ ) Dh D X = R(X), ɛ: a L, ɛ 1 Proper scaling Fixes the geometry Transfer geometric characteristic into a coefficient in the equation Allows asymptotic expansion

12 Slowly varying variables Scaling x=ɛx,r=r, u=u,ɛv=v,ɛ 1 p=p. Fix domain Independent of We can consider L L a D 1 r x Solve for leading term ɛ=0 a D

13 Scaled equations ɛre(uu x +vu r )=ɛ 2 u xx +u rr + 1 r u r Rep x, ɛ 3 Re(uv x +vv r )=ɛ 4 v xx +ɛ (v 2 rr + 1 r v r 1 ) r 2v r Rep r, u x +v r + v r =0. ɛre 1 u 0 rr+ 1 r u 0r Rep 0 x=0, Rep 0 r=0, u 0 x+v 0 r+ v 0 r =0. u 0 (x,r)= Leading Order Solution 1 2h(x) 4 ( h(x) 2 r 2), v 0 (x,r)= h (x)r ( h(x) 2 2h(x) 5 r 2), p 0 (x,r)= 2 Re x 0 1 h(ξ) 4dξ.

14 Outline Motivation Fluid Flow equations Method of slow variations Friction factor estimates Validation (Sinusoidal pipes) Further work

15 Direct estimation of friction factor Asymptotic solution in original variables U(X,R)=2Ū0 Friction factor f = 64 Re R(0) 2 ( ) 1 R2, R(X) 2 R(X) 2 V(X,R)= R (X) R(X) RU(R,X), P(X,R)= 16ρŪ2 0 R(0) 3 L Re R(0) 4 1 L 0 R(X) }{{ 4dX, } CF1 X 0 1 R(ξ) 4dξ. Only requires 1D integration. times faster 103 than CFD

16 Sinusoidal pipe R a 1 X R(X)=1+ a 2 h(x)= 1 2 +a 4 L ( 1+sin ( 1+sin ( )) 2π L X π, 2 ( πx a π 2 )).

17 Direct estimate CF1 D=2, a=2 Accurate when a L 1 Independent of L f 10 1 L=3 L=5 L=10 L=50 64/Re CF1 Idea: Leading term + Integral formula Re

18 Estimation with integral formula P P := 1 Γ in Γ Pn X ds= 16ρŪ2 0 R(0) Re [ R(L) 2 L 0 1 L R(X) 4dX 0 ] 1 R(X) 2dX. P S := µ Γ in Γ Pressure Friction U n ds=8µū0 L Skin Friction 0 [ 1 ) 2 ] R(X) 2 1+( R (X) dx. f = 64 Re L R(0) 2 R (X) 2 L 0 R(X) + R(0) 2 2 R(X) }{{ 4dX } CF2 Friction Factor. Basically same cost as CF1

19 Integral estimate CF2 D=2, a=2 Still works for a L 1 Now captures changes in L Applicability? f 10 1 L=3 L=5 L=10 L=50 64/Re Accuracy? Re

20 Outline Motivation Fluid Flow equations Method of slow variations Friction factor estimates Validation (Sinusoidal pipes) Further work

21 Validation R a 1 X L 0 a 2 0 L 40 0 Re 10 3 Comparison with CFD Err:= f f f, f :with CFD f :estimate CF2

22 Applicability Err 10% Err 20%

23 Err 30% Err 40%

24 Error Isosurfaces

25 Contours Re L a=0.2 Err 8%

26 Contours Accurate if a L Re 1 Max error Err 30% Re a= L

27 Outline Motivation Fluid Flow equations Method of slow variations Friction factor estimates Validation (Sinusoidal pipes) Further work

28 Further work The method for computing the friction factor is very efficient, 10^3 times faster than CFD, which allows explore shape optimization Attempt higher accuracy with higher order approximations Further effects: extend methodology for handling pipe flow with heat load

29 THANK YOU FOR YOUR ATTENTION