8Flow Through a Vertical Tube Flow through a circular tubeFlow through a vertical tube
9Flow Through a Vertical Tube Hagen-Poiseuille Equation
10OutlineFlow Through a Vertical TubeFlow Through an AnnulusExercises
11Flow Through an Annulus Liquid is flowing upward through an annulus (space between two concentric cylinders)Important quantities:R : radius of outer cylinderκR : radius of inner cylinder
12Flow Through an Annulus Assumptions:Steady-state flowIncompressible fluidOnly Vz component is significantAt the solid-liquid interface, no-slip conditionSignificant gravity effectsVmax is attained at a distance λR from the center of the inner cylinder (not necessarily the center)
22Shell BalancesIdentify all the forces that influence the flow (pressure, gravity, momentum flux) and their directions. Set the positive directions of your axes.Create a shell with a differential thickness across the direction of the flux that will represent the flow system.Identify the areas (cross-sectional and surface areas) and volumes for which the flow occurs.Formulate the shell balance equation and the corresponding differential equation for the momentum flux.
23Shell BalancesIdentify all boundary conditions (solid-liquid, liquid-liquid, liquid-free surface, momentum flux values at boundaries, symmetry for zero flux).Integrate the DE for your momentum flux and determine the values of the constants using the BCs.Insert Newton’s law (momentum flux definition) to get the differential equation for velocity.Integrate the DE for velocity and determine values of constants using the BCs.Characterize the flow using this velocity profile.
24Shell Balances Important Assumptions* The flow is always assumed to be at steady-state.Neglect entrance and exit effects. The flow is always assumed to be fully-developed.The fluid is always assumed to be incompressible.Consider the flow to be unidirectional.*unless otherwise stated
25Design Equations for Laminar and Turbulent Flow in Pipes
26Outline Velocity Profiles in Pipes Pressure Drop and Friction Loss (Laminar Flow)Friction Loss (Turbulent Flow)Frictional Losses in Piping Systems
27Velocity Profiles in Pipes Recall velocity profile in a circular tube:What is the shape of this profile?The maximum occurs at which region?What is the average velocity of the fluid flowing through this pipe?
31Outline Velocity Profiles in Pipes Pressure Drop and Friction Loss (Laminar Flow)Friction Loss (Turbulent Flow)Frictional Losses in Piping Systems
32Recall: Hagen-Poiseuille Equation Describes the pressure drop and flow of fluid (in the laminar regime) across a conduit with length L and diameter D
33Hagen-Poiseuille Equation Pressure drop / Pressure loss (P0 – PL):Pressure lost due to skin friction
34Mechanical energy lost due to friction in pipe (because of what?) Friction LossIn terms of energy lost per unit mass:Mechanical energy lost due to friction in pipe (because of what?)
35Friction FactorDefinition: Drag force per wetted surface unit area (or shear stress at the surface) divided by the product of density times velocity head
36Friction FactorFrictional force/loss head is proportional to the velocity head of the flow and to the ratio of the length to the diameter of the flow stream
37Friction Factor for Laminar Flow Consider the Hagen-Poiseuille equation (describes laminar flow) and the definition of the friction factor:Prove:Valid only for laminar flow
38Outline Velocity Profiles in Pipes Pressure Drop and Friction Loss (Laminar Flow)Friction Loss (Turbulent Flow)Frictional Losses in Piping Systems
39Friction Factor for Turbulent Flow Friction factor is dependent on NRe and the relative roughness of the pipe.The value of fF is determined empirically.
40Friction Factor for Turbulent Flow How to compute/find the value of the friction factor for turbulent flow:Use Moody diagrams.- Friction factor vs. Reynolds number with a series of parametric curves related to the relative roughnessUse correlations that involve the friction factor f.- Blasius equation, Colebrook formula, Churchill equation (Perry 8th Edition)
41Moody Diagrams Important notes: Both fF and NRe are plotted in logarithmic scales. Some Moody diagrams show fD (Darcy friction factor). Make the necessary conversions.No curves are shown for the transition region.Lowest possible friction factor for a given NRe in turbulent flow is shown by the smooth pipe line.
44Friction Factor Correlations Blasius equation for turbulent flow in smooth tubes:Colebrook formula
45Friction Factor Correlations Churchill equation (Colebrook formula explicit in fD)Swamee-Jain correlation
46Equivalent Roughness, ε Materials of ConstructionEquivalent Roughness (m)Copper, brass, lead (tubing)1.5 E-06Commercial or welded steel4.6 E-05Wrought ironDuctile iron – coated1.2 E-04Ductile iron – uncoated2.4 E-04ConcreteRiveted Steel1.8 E-03
47Frictional Losses for Non-Circular Conduits Instead of deriving new correlations for f, an approximation is developed for an equivalent diameter, Deq, which may be used to calculate NRe and f.where RH = hydraulic radiusS = cross-sectional areaPw = wetted perimeter: sum of the length of the boundaries of the cross-section actually in contact with the fluid
48Equivalent Diameter (Deq) Determine the equivalent diameter of the following conduit types:Annular space with outside diameter Do and inside diameter DiRectangular duct with sides a and bOpen channels with liquid depth y and liquid width b