Oe4625 Lecture 1.

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Oe4625 Lecture 1.

  1. 1. OE4625 Dredge Pumps and Slurry Transport Vaclav Matousek October 13, 2004 1 Dredge Pumps and Slurry Transport Vermelding onderdeel organisatie
  2. 2. OE4625 Dredge Pumps and Slurry Transport Vaclav Matousek October 13, 2004 2 Dredge Pumps and Slurry Transport Vermelding onderdeel organisatie
  3. 3. OE4625 Dredge Pumps and Slurry Transport October 13, 2004 3 Dredge Pumps and Slurry Transport Vermelding onderdeel organisatie
  4. 4. Keywords: Transport (Horizontal, Vertical, Inclined) Pipe (Length, Diameter) Pump (Type, Size) Goals: Design a pipe of appropriate size Design pumps of appropriate size; Determine a number of required pumpsOctober 13, 2004 4Dredge Pumps and Slurry TransportVermelding onderdeel organisatie
  5. 5. Objectives: Prediction of energy dissipation in PIPE Prediction of energy production in PUMP Prediction models: Pressure drop vs. mean velocity in PIPE Pressure gain vs. mean velocity in PUMPOctober 13, 2004 5Dredge Pumps and Slurry TransportVermelding onderdeel organisatie
  6. 6. Part I. Principles of Mixture Flow in Pipelines1. Basic Principles of Flow in a Pipe2. Soil-Water Mixture and Its Phases3. Flow of Mixture in a Pipeline4. Modeling of Stratified Mixture Flows5. Modeling of Non-Stratified Mixture Flows6. Special Flow Conditions in Dredging Pipelines October 13, 2004 6Dredge Pumps and Slurry TransportVermelding onderdeel organisatie
  7. 7. Part II. Operational Principles of Pump-Pipeline Systems Transporting Mixtures7. Pump and Pipeline Characteristics8. Operation Limits of a Pump-Pipeline System9. Production of Solids in a Pump-Pipeline System10. Systems with Pumps in SeriesOctober 13, 2004 7 Dredge Pumps and Slurry Transport Vermelding onderdeel organisatie
  8. 8. 1. BASIC PRINCIPLES OF FLOW IN PIPE CONSERVATION OF MASS CONSERVATION OF MOMENTUM CONSERVATION OF ENERGY October 13, 2004 8 Dredge Pumps and Slurry Transport
  9. 9. Conservation of MassContinuity equation for a control volume (CV): d ( mass ) = ∑ (qoutlet − qinlet ) [kg/s] dtq [kg/s] ... Total mass flow rate through all boundaries of the CVOctober 13, 2004 9Dredge Pumps and Slurry Transport
  10. 10. Conservation of MassContinuity equation in general form: ∂ρ + ∇. ( ρV ) = 0 ∂tFor incompressible (ρ = const.) liquid and steady flow (ə/ət = 0) the equation is given in its simplest form ∂ vx ∂ v y ∂ vz + + =0 ∂x ∂y ∂zOctober 13, 2004 10Dredge Pumps and Slurry Transport
  11. 11. Conservation of MassThe physical explanation of the equation is that the mass flow rates qm = ρVA [kg/s] for steady flow at the inlets and outlets of the control volume are equal.Expressed in terms of the mean values of quantities at the inlet and outlet of the control volume, given by a pipeline length section, the equation is qm = ρVA = const. [kg/s]thus (ρVA)inlet = (ρVA)outletOctober 13, 2004 11Dredge Pumps and Slurry Transport
  12. 12. Conservation of Mass in 1D-flowFor a circular pipeline of two different diameters D1 and D2 V1D12 = V2D22 [m3/s] V1 V2 D1 D2October 13, 2004 12Dredge Pumps and Slurry Transport
  13. 13. Conservation of MomentumNewton’s second law of motion: d ( momentum) = ∑ Fexternal dtThe external forces are- body forces due to external fields (gravity, magnetism, electric potential) which act upon the entire mass of the matter within the control volume,- surface forces due to stresses on the surface of the control volume which are transmitted across the control surface.October 13, 2004 13Dredge Pumps and Slurry Transport
  14. 14. Conservation of Momentum In an infinitesimal control volume filled with a substance of density the force balance between inertial force, on one side, and pressure force, body force, friction force, on the other side, is given by a differential linear momentum equation in vector form DV ∂ρ = Dt ∂ t ( ) ρV + ρV .∇V = −∇P − ρ g∇h −∇.T October 13, 2004 14 Dredge Pumps and Slurry Transport
  15. 15. Conservation of MomentumClaude-Louis George Navier Stokes October 13, 2004 15 Dredge Pumps and Slurry Transport
  16. 16. Conservation of Momentum in 1D-flow In a straight piece of pipe of the differential distance dx (1D-flow), quantities in the equation are averaged over the pipeline cross section:  ∂V ∂V ∂ h  ∂ P τo ρ +V + g  + + 4 = 0  ∂t ∂ x ∂ x ∂ x D October 13, 2004 16 Dredge Pumps and Slurry Transport
  17. 17. Conservation of Momentum in 1D-flowFor additional conditions :- incompressible liquid,- steady and uniform flow in a horizontal straight pipe dP dP 4τ o − A = τ oO , i.e. − = dx dx D for a pipe of a circular cross section and internal diameter D. October 13, 2004 17 Dredge Pumps and Slurry Transport
  18. 18. Conservation of Momentum in 1D-flow P − P2 4τ oFor a straight horizontal circular pipe 1 = L D P1 P2 t0 V D t0 L October 13, 2004 18 Dredge Pumps and Slurry Transport
  19. 19. Liquid Friction in 2D Pipe FlowThe force-balance equation generalized for 2D-flow dP 2 gives the shear stress distribution in a cylinder: − =τ dx r October 13, 2004 19 Dredge Pumps and Slurry Transport
  20. 20. Liquid Friction in 2D Pipe FlowNewton’s law of liquid viscosity F  dVx (valid for laminar flow): τ = = µf −  A  dy  October 13, 2004 20 Dredge Pumps and Slurry Transport
  21. 21. Liquid Friction in 2D Laminar Flow in PipeThe generalized force-balance equation for dP 2 the 2D-flow in a cylinder − =τ + dx r  dv x  Newton’s law of liquid viscosity (valid for laminar flow) τ = µf −   dr  = Velocity distribution in laminar flow in a pipe dv x dP r = dr dx 2 µ f October 13, 2004 21 Dredge Pumps and Slurry Transport
  22. 22. Liquid Friction in 1D Laminar Flow in Pipe D/2The integration of the velocity 1 8gradient equation gives a Vf = ∫∫ vxdA = 2 ∫ v rdr xvalue for mean velocity in pipe AA D 0 2 D  dP  and thus a relationship between Vf =   pressure drop and mean velocity 32 µ f  dx  which is the required pressure-drop dP 32 µ f V f model for laminar flow in pipe = 2 dx D October 13, 2004 22 Dredge Pumps and Slurry Transport
  23. 23. Liquid Friction in 1D Laminar Flow in PipeA comparison of the pressure-drop dP 32 µ f V f model for laminar flow in pipe = 2 + dx Dwith the general force balance dP 4τ o (driving force = resistance force) = for pipe flow = dx Dgives the equation for the shear 8Vf stress at the pipe wall in laminar τo = µ f flow D October 13, 2004 23 Dredge Pumps and Slurry Transport
  24. 24. Liquid Friction in 1D Turbulent Flow in Pipe The wall shear stress for turbulent flow cannot be determined directly from the force balance and Newtons law of viscosity (it does not hold for turbulent flow). + Instead, it is formulated by using dimensional analysis. A function τ0 = fn(ρf, Vf, µf, D, k) is assumed. The analysis provides the following relationship between dimensionless groups τo  k = fn  Re,  1  D ρ f Vf2 2 October 13, 2004 24 Dredge Pumps and Slurry Transport
  25. 25. Liquid Friction in 1D Turbulent Flow in Pipe The dimensionless group Re, Reynolds number, is a ratio of the inertial forces and the viscous forces in the pipeline flow Vf Dρ f inertial . force Re = = µf viscous. force Remark: The Reynolds number determines a threshold between the laminar and the turbulent flows in a pipe. October 13, 2004 25 Dredge Pumps and Slurry Transport
  26. 26. Liquid Friction in 1D Turbulent Flow in Pipe The dimensionless parameter on the left side of the dimensional-analysis equation is called the friction factor. It is the ratio between the wall shear stress and kinetic energy of the liquid in a control volume in a pipeline. Fanning friction factor Darcy-Weisbach friction coefficient τo 8τ o ff = λf = 1 ρ fVf 2 ρ fVf2 2 λf = 4ff October 13, 2004 26 Dredge Pumps and Slurry Transport
  27. 27. Liquid Friction in 1D Flow in Pipe 8τ oA comparison of the Darcy-Weisbach friction coefficient equation λf = ρ f Vf2 +with the linear momentum eq. dP 4τ o (driving force = resistance force) − = for pipe flow dx D = dP λ f ρ f V 2gives the general pressure-drop f equation for the pipe flow − =(Darcy-Weisbach equation, 1850) dx D 2 October 13, 2004 27 Dredge Pumps and Slurry Transport
  28. 28. Liquid Friction in 1D Laminar Flow in Pipe dP λ f ρ f V 2A comparison of the general f pressure-drop equation − = dx D 2 +with the pressure-drop eq. for dP 32 µ f V f laminar flow in pipe − = 2 dx D = 64µ f 64gives the pipe-wall friction law for λf = = laminar flow in pipe ρ f DV f Re October 13, 2004 28 Dredge Pumps and Slurry Transport
  29. 29. Liquid Friction in 1D Turbulent Flow in Pipe In turbulent flows there is no simple expression linking the velocity distribution with the shear stress (and so with the pressure gradient) in the pipe cross section. The dimensional analysis provides the following relationship between dimensionless groups  k λ f = fn  Re,   D October 13, 2004 29 Dredge Pumps and Slurry Transport
  30. 30. Liquid Friction in 1D Turbulent Flow in PipeThere are three different regimes with the different wall friction laws:Hydraulically smooth Transitional Hydraulically rough λf=fn(Re) λf=fn(Re, k/D) λf=fn(k/D) October 13, 2004 30 Dredge Pumps and Slurry Transport
  31. 31. Liquid Friction in 1D Turbulent Flow in Pipe October 13, 2004 31 Hydraulically smooth Transitional Hydraulically rough λf=fn(Re) λf=fn(Re, k/D) λf=fn(k/D)
  32. 32. Liquid Friction: Moody DiagramOctober 13, 2004 32 Hydraulically smooth Transitional Hydraulically rough λf=fn(Re) λf=fn(Re, k/D) λf=fn(k/D)
  33. 33. Liquid Friction: Moody DiagramOctober 13, 2004 33 Hydraulically smooth Transitional Hydraulically rough λf=fn(Re) λf=fn(Re, k/D) λf=fn(k/D)
  34. 34. Conservation of energy P1 V1 P2 V2 λLV 2 + + z1 = + + z2 + , ρg 2g ρg 2g 2gD w h e re λ is th e fric tio n fa c to r, fo r o u r c a s e : 2gD3 λ = v 2R 2 e T h e C o le b ro o k fo rm u la fo r λ is : 1 λ = , R e ≥ 4000,  2 .5 1  2 lo g 1 0   Re λ  w h e re R e is R e y n o ld s n u m b e r VD Re = νOctober 13, 2004 34Dredge Pumps and Slurry Transport

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