Lect 5 - 7 - Shell Momentum Balance 46w68

Shell Momentum Balance By Amol Deshpande

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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 – – – – 11/08/2011

Flow of a falling film Flow through a circular tube Flow through an annulus Flow of two adjacent immiscible fluids 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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Transport Phenomena

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. 11/08/2011

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-direction 16/08/2011

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 16/08/2011

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) 16/08/2011

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

• 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) 18/08/2011

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 18/08/2011

Transport Phenomena

Momentum Flux/Velocity Distribution • Constants

• Momentum Flux / Velocity Distribution

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Transport Phenomena

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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) 18/08/2011

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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Transport Phenomena