Shell Momentum Balances

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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)?

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

Flow Through a Vertical Tube

Flow Through a Vertical Tube

Flow Through a Vertical Tube

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

Flow Through a Vertical Tube
Hagen-Poiseuille Equation

Outline Flow Through a Vertical Tube Flow Through an Annulus Exercises

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

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)

Flow Through an Annulus

Flow Through an Annulus

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.

Flow Through an Annulus

Flow Through an Annulus

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

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

Flow Through an Annulus

Flow Through an Annulus

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.

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.

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

Design Equations for Laminar and Turbulent Flow in Pipes

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

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?

Velocity Profiles in Pipes

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

Maximum vs. Average Velocity

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

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

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

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

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

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

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

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

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

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)

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 given NRe in turbulent flow is shown by the smooth pipe line.

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

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

Equivalent Roughness, ε
Materials of Construction Equivalent Roughness (m) Copper, brass, lead (tubing) 1.5 E-06 Commercial or welded steel 4.6 E-05 Wrought iron Ductile iron – coated 1.2 E-04 Ductile iron – uncoated 2.4 E-04 Concrete Riveted Steel 1.8 E-03

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

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

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