Transport Phenomena

Shell Momentum Balance

ByAmol Deshpande

11/08/2011

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

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

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

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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