lect 5 - 7 - shell momentum balance
TRANSCRIPT
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Transport Phenomena
Shell Momentum Balance
ByAmol Deshpande
11/08/2011
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Transport Phenomena
Introduction
• Objective– To obtain velocity profiles for laminar flow of fluids
• Requirements– Definition of viscosity– Molecular and Convective momentum flux expressions– Concept of momentum balance
• Flow Systems to be studied– Flow of a falling film– Flow through a circular tube– Flow through an annulus– Flow of two adjacent immiscible fluids
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Transport Phenomena
Introduction
• Problems/Systems– Steady Flow– Laminar– Rectilinear Flow – Velocity – function of one spatial variable
• Momentum Balance - (Rate of Momentum In) – (Rate of Momentum out) +
(Force of Gravity) = 0
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Viscous Flow Problems – Solving Procedure
• Identify non-vanishing velocity components• Consider a shell and write a shell momentum balance• Use definition of first derivative to obtain differential equation
for momentum flux• Get momentum flux distribution• Insert Newton’s law viscosity and obtain a differential
equation for velocity.• Get velocity distribution• Use velocity distribution/profile to get other quantities such
as max velocity, avg velocity.
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Transport Phenomena
Boundary Conditions
• Solid – Fluid Interface– No Slip Condition
• Liquid – Liquid Interface– Continuity of velocity and stress-tensor components
• Liquid – Gas Interface– Shear Stress Tensor components are taken to be zero
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Transport Phenomena
Problem – Flow Of a Falling Film
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Transport Phenomena
Postulates - Assumptions
• vz= vz(x) , vx = 0 ; vy = 0 • p = p(x)• End effects are neglected• Steady Flow• Incompressible Fluid• Viscosity and Density are constant
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Transport Phenomena
Shell - Surface
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Transport Phenomena
Shell Momentum Balance
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Transport Phenomena
Momentum Flux Distribution
• First Derivative (Shell thickness approaches zero)
• Momentum Flux –
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Transport Phenomena
Velocity Distribution
• Newton’s Law Of Viscosity
• Velocity Distribution
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Transport Phenomena
Profiles
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Transport Phenomena
Other Quantities
• Maximum Velocity
• Average Velocity
• Mass Flow Rate
• Film Thickness
• Viscous Force in the z-direction16/08/2011
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Transport Phenomena
Analysis – Falling Film Problem
• Experimental Observations –– Three flow regimes (Based on Re)– Gives information about onset of instability
• Results obtained (Velocity, Momentum flux distributions) – Valid only for Re < 20
• Experiments play a vital role in Fluid Dynamics
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Transport Phenomena
Problem – Flow Through A Circular Tube
• Steady state, laminar flow of a fluid• Constant density and viscosity• Vertical tube of length L and radius R• L>>R End effects are neglected
• Postulates– vz= vz(r) , vr = 0 ; v = 0 – p = p(z)
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Transport Phenomena
Shell Surface
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Transport Phenomena
Momentum Balance
• Overall momentum balance
• Simplification – First Derivative
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Transport Phenomena
Momentum Flux Distribution
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• Boundary Condition
• Momentum flux Distribution
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Transport Phenomena
Velocity Distribution
• Newton’s Law of Viscosity
• Boundary Condition-– At r = R, vz = 0;
• Velocity Distribution
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Transport Phenomena
Other Quantities
• Maximum Velocity
• Average Velocity
• Mass Flow Rate
• Z- component of the force
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Transport Phenomena
Assumptions in Hagen-Poiseuille Equation
• Laminar Flow• Incompressible Flow• Steady Flow• Newtonian fluid (Newton’s law of viscosity– valid)• End effects are neglected• Fluid behaves as a continuum• No slip at the wall
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Transport Phenomena
Problem – Flow Through An Annulus
• Steady state axial flow• Incompressible liquid• System – Coaxial cylinders of radii kR and R, Liquid
flows through an annulus in upward direction
• Postulates– vz= vz(r) , vr = 0 ; v = 0 – p(z) = p(z)
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Transport Phenomena
Momentum Flux Distribution
• Boundary Conditions-– At r = R , momentum flux is zero.
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Transport Phenomena
Velocity Distribution
• Newton’s Law of Viscosity
• Velocity Distribution
• Boundary Conditions-– r = kR, vz = 0
– r = R, vz = 0
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Transport Phenomena
Momentum Flux/Velocity Distribution
• Constants
• Momentum Flux / Velocity Distribution
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Transport Phenomena18/08/2011
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Transport Phenomena
Other Quantities
• Maximum Velocity
• Average Velocity
• Mass Flow Rate
• Z- component of the force
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Transport Phenomena
Problem – Flow of two adjacent immiscible fluids
• Two Immiscible incompressible liquids• Fluid flow through a horizontal slit (z-direction) of
length L and width W and gap of ‘2b’• Fluid flow rates – Adjusted to have each fluid
filling half of the slit• Interface – Exactly planar• Postulates –– vz= vz(x) , vx = 0 ; vy = 0 – p(z) = p(z)
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Transport Phenomena
Shell Momentum Balance
• Momentum Flux –
• Boundary Condition
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Transport Phenomena
Velocity Distribution
• Velocity –
• Boundary Conditions– No Slip / Continuity of velocity
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Transport Phenomena
Momentum Flux and Velocity Distributions
• Momentum Flux and Velocity Profiles
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Transport Phenomena
Momentum Flux and Velocity Profiles
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Transport Phenomena
Problems
• Rederive the velocity profile and average velocity for a falling film problem by replacing x by a coordinate x1 measured away from the wall (i.e. x1 = 0 is the wall surface and x1= is the liquid gas interface.
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