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Shell Momentum Balances

Shell Momentum Balances. Outline. Flow Through a Vertical Tube Flow Through an Annulus Exercises. 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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Shell Momentum Balances

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  1. Shell Momentum Balances

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

  3. 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)?

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

  5. Flow Through a Vertical Tube

  6. Flow Through a Vertical Tube

  7. Flow Through a Vertical Tube

  8. Flow Through a Vertical Tube Flow through a circular tube Flow through a vertical tube

  9. Flow Through a Vertical Tube Hagen-Poiseuille Equation

  10. Outline Flow Through a Vertical Tube Flow Through an Annulus Exercises

  11. 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

  12. Flow Through an Annulus Assumptions: Steady-state flow Incompressible fluid Only Vz component is significant At the solid-liquid interface, no-slip condition Significant gravity effects Vmax is attained at a distance λR from the center of the inner cylinder(not necessarily the center)

  13. Flow Through an Annulus

  14. Flow Through an Annulus

  15. Flow Through an Annulus BOUNDARY CONDITION! At a distance λR from the center of the inner cylinder, Vmax is attained in the annulus, or zero momentum flux.

  16. Flow Through an Annulus

  17. Flow Through an Annulus

  18. Flow Through an Annulus Take out R/2 Multiply r in log term by R/R (or 1) Expand log term Lump all constants into C2

  19. Flow Through an Annulus We have two unknown constants: C2 and λ We can use two boundary conditions: No-slip Conditions At r = κR, vz = 0 At r = R, vz = 0

  20. Flow Through an Annulus

  21. Flow Through an Annulus

  22. Shell Balances Identify 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.

  23. Shell Balances Identify 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.

  24. Shell 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

  25. Design Equations for Laminar and Turbulent Flow in Pipes

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

  27. Velocity 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?

  28. Velocity Profiles in Pipes

  29. Velocity Profiles in Pipes Velocity Profile in a Pipe: Average Velocity of a Fluid in a Pipe:

  30. Maximum vs. Average Velocity

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

  32. 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

  33. Hagen-Poiseuille Equation Pressure drop / Pressure loss (P0 – PL): Pressure lost due to skin friction

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

  35. 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

  36. 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

  37. 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

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

  39. Friction Factor for Turbulent Flow Friction factor is dependent on NReand the relative roughness of the pipe. The value of fF is determined empirically.

  40. Friction 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 roughness • Use correlations that involve the friction factor f. - Blasius equation, Colebrook formula, Churchill equation (Perry 8th Edition)

  41. Moody 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 givenNRein turbulent flow is shown by the smooth pipe line.

  42. Friction Factor Correlations • Blasius equation for turbulent flow in smooth tubes: • Colebrook formula

  43. Friction Factor Correlations • Churchill equation (Colebrook formula explicit in fD) • Swamee-Jain correlation

  44. Equivalent Roughness, ε

  45. Frictional 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 radius S = cross-sectional area Pw = wetted perimeter: sum of the length of the boundaries of the cross-section actually in contact with the fluid

  46. Equivalent Diameter (Deq) Determine the equivalent diameter of the following conduit types: • Annular space with outside diameter Doand inside diameter Di • Rectangular duct with sides a and b • Open channels with liquid depth y and liquid width b

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