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Shell Momentum Balances

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Outline 1.Flow Through a Vertical Tube 2.Flow Through an Annulus 3.Exercises

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Flow Through a Vertical Tube The tube is oriented vertically. What will be the velocity profile of a fluid whose direction of flow is in the +z- direction (downwards)?

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Flow Through a Vertical Tube Same system, but this time gravity will also cause momentum flux.

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Flow Through a Vertical Tube

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Flow through a circular tube Flow through a vertical tube

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Flow Through a Vertical Tube Hagen-Poiseuille Equation

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Outline 1.Flow Through a Vertical Tube 2.Flow Through an Annulus 3.Exercises

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Flow Through an Annulus Liquid is flowing upward through an annulus (space between two concentric cylinders) Important quantities: R : radius of outer cylinder κR : radius of inner cylinder

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Flow Through an Annulus Assumptions: 1.Steady-state flow 2.Incompressible fluid 3.Only V z component is significant 4.At the solid-liquid interface, no-slip condition 5.Significant gravity effects 6.V max is attained at a distance λR from the center of the inner cylinder (not necessarily the center)

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Flow Through an Annulus

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BOUNDARY CONDITION! At a distance λR from the center of the inner cylinder, V max is attained in the annulus, or zero momentum flux.

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Flow Through an Annulus

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Take out R/2 Multiply r in log term by R/R (or 1) Expand log term Lump all constants into C 2

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Flow Through an Annulus We have two unknown constants: C 2 and λ We can use two boundary conditions: No-slip Conditions At r = κR, v z = 0 At r = R, v z = 0

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Flow Through an Annulus

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Shell Balances 1.Identify all the forces that influence the flow (pressure, gravity, momentum flux) and their directions. Set the positive directions of your axes. 2.Create a shell with a differential thickness across the direction of the flux that will represent the flow system. 3.Identify the areas (cross-sectional and surface areas) and volumes for which the flow occurs. 4.Formulate the shell balance equation and the corresponding differential equation for the momentum flux.

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Shell Balances 5.Identify all boundary conditions (solid-liquid, liquid- liquid, liquid-free surface, momentum flux values at boundaries, symmetry for zero flux). 6.Integrate the DE for your momentum flux and determine the values of the constants using the BCs. 7.Insert Newton’s law (momentum flux definition) to get the differential equation for velocity. 8.Integrate the DE for velocity and determine values of constants using the BCs. 9.Characterize the flow using this velocity profile.

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Shell Balances Important Assumptions* 1.The flow is always assumed to be at steady- state. 2.Neglect entrance and exit effects. The flow is always assumed to be fully-developed. 3.The fluid is always assumed to be incompressible. 4.Consider the flow to be unidirectional. *unless otherwise stated

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Design Equations for Laminar and Turbulent Flow in Pipes

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Outline 1.Velocity Profiles in Pipes 2.Pressure Drop and Friction Loss (Laminar Flow) 3.Friction Loss (Turbulent Flow) 4.Frictional Losses in Piping Systems

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Velocity Profiles in Pipes Recall velocity profile in a circular tube: 1.What is the shape of this profile? 2.The maximum occurs at which region? 3.What is the average velocity of the fluid flowing through this pipe?

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Velocity Profiles in Pipes

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Velocity Profile in a Pipe: Average Velocity of a Fluid in a Pipe:

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Maximum vs. Average Velocity

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Outline 1.Velocity Profiles in Pipes 2.Pressure Drop and Friction Loss (Laminar Flow) 3.Friction Loss (Turbulent Flow) 4.Frictional Losses in Piping Systems

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Recall: Hagen-Poiseuille Equation Describes the pressure drop and flow of fluid (in the laminar regime) across a conduit with length L and diameter D

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Hagen-Poiseuille Equation Pressure drop / Pressure loss (P 0 – P L ): Pressure lost due to skin friction

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Friction Loss In terms of energy lost per unit mass: Mechanical energy lost due to friction in pipe (because of what?)

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Friction Factor Definition: Drag force per wetted surface unit area (or shear stress at the surface) divided by the product of density times velocity head

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Friction Factor Frictional 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

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Friction 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

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Outline 1.Velocity Profiles in Pipes 2.Pressure Drop and Friction Loss (Laminar Flow) 3.Friction Loss (Turbulent Flow) 4.Frictional Losses in Piping Systems

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Friction Factor for Turbulent Flow 1.Friction factor is dependent on N Re and the relative roughness of the pipe. 2.The value of f F is determined empirically.

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Friction Factor for Turbulent Flow How to compute/find the value of the friction factor for turbulent flow: 1.Use Moody diagrams. - Friction factor vs. Reynolds number with a series of parametric curves related to the relative roughness 2.Use correlations that involve the friction factor f. - Blasius equation, Colebrook formula, Churchill equation (Perry 8 th Edition)

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Moody Diagrams Important notes: 1.Both f F and N Re are plotted in logarithmic scales. Some Moody diagrams show f D (Darcy friction factor). Make the necessary conversions. 2.No curves are shown for the transition region. 3.Lowest possible friction factor for a given N Re in turbulent flow is shown by the smooth pipe line.

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1.Blasius equation for turbulent flow in smooth tubes: 2.Colebrook formula Friction Factor Correlations

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3.Churchill equation (Colebrook formula explicit in f D ) 4.Swamee-Jain correlation Friction Factor Correlations

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Materials of ConstructionEquivalent Roughness (m) Copper, brass, lead (tubing)1.5 E-06 Commercial or welded steel4.6 E-05 Wrought iron4.6 E-05 Ductile iron – coated1.2 E-04 Ductile iron – uncoated2.4 E-04 Concrete1.2 E-04 Riveted Steel1.8 E-03 Equivalent Roughness, ε

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Instead of deriving new correlations for f, an approximation is developed for an equivalent diameter, D eq, which may be used to calculate N Re and f. where R H = hydraulic radius S = cross-sectional area P w = wetted perimeter: sum of the length of the boundaries of the cross-section actually in contact with the fluid Frictional Losses for Non-Circular Conduits

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Determine the equivalent diameter of the following conduit types: 1.Annular space with outside diameter D o and inside diameter D i 2.Rectangular duct with sides a and b 3.Open channels with liquid depth y and liquid width b Equivalent Diameter (D eq )

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