Transport Phenomena

Equations of Change

ByAmol Deshpande

20/08/2011

Transport Phenomena

Introduction

• Shell momentum balance approach – Tedious for most of the problems (e.g. nonrectilinear motion problems)

• Generalized equations to deal with isothermal flow of a pure fluid.

– General mass balance - Equation of continuity

– General momentum balance - Equation of motion

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Transport Phenomena

Introduction

• Equation of continuity– Derivation in cartesian coordinates, Special cases

• Equation of motion– Derivation for the most generalized form, Significance

• Equation of change for mechanical energy– Another form of equation of motion

• “Substantial derivative” Concept• Use of equations of change to solve problems• Dimensional analysis

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Equation of Continuity• General mass balance -

Rate of increase of mass= (Rate of mass in) – (Rate of mass out)

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Transport Phenomena

Problem

• Normal stress at fluid solid interface for incompressible newtonian fluids

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Equation of Motion

• General momentum balance –(Rate of increase in momentum)= (Rate of momentum in) – (Rate of momentum out) +

(External force on the fluid)

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Transport Phenomena

Equation of Mechanical Energy

• Another form of equation of motion• Equation of change for kinetic energy

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Transport Phenomena

Equation of Change in terms of Substantial Derivative

• Substantial derivative/Material derivative /hydrodynamic derivative

• Equation of continuity– Tells how density changes as one moves along with the

fluid • Compression – density increases• Expansion – density decreases

• Equation of motion– Newton’s second law of motion

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Transport Phenomena

Equations of motion – Special cases

• Constant density and viscosity– Navier-Stokes Equation

• Acceleration terms are neglected– Stokes flow equation / Creeping flow equation

• Viscous forces are neglected – Euler Equation

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Transport Phenomena

Description of fluid flow

• Description of the flow of a Newtonian fluid at constant temperature– Equations

• Equation of continuity• Equation of motion• Expression for shear/viscous stress• Equations of state• Equations of viscosities

– Boundary/Initial conditions• Solution

– Pressure, Velocity and Density profiles– Other quantities important for engineering applications

(Max velocity, avg velocity, mass flow rate, etc)23/08/2011

Transport Phenomena

Description of fluid flow

• Flow of a fluid with constant density and viscosity– Equations

• Equation of continuity (Eq 3.1-4, Table B.4)• Navier-Stokes equation (Eq. 3.5-6, Table B.5, 6, 7)

– Initial/Boundary conditions• Solution – Velocity and pressure profiles– Other quantities important for engineering applications

(Max velocity, avg velocity, mass flow rate, etc)

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Transport Phenomena

Setting up a fluid flow problem (Mathematical modeling)

• Make suitable assumptions– Steady, isothermal, laminar flow, Newtonian fluid

• Make postulates about pressure and velocity distributions

• Using equations of change (Table B.1, 4, 5, 6)– Simplify equation of continuity and equation of

motion (Navier-Stokes) based on assumptions made – Modeling/Governing equations (Differential Eqs)

• Use appropriate initial/boundary conditions

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Transport Phenomena

Governing Equations - Solving Techniques • Analytical solution– Gives exact solutions– Can be obtained only for the simples flow regimes.

• Numerical Solution– Gives approximate solution – Need to be validated with experimental results.– CFD – Tool used to obtain results by using numerical

methods.

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Transport Phenomena

Example - Flow through a tube

• Assumptions – SS, Constants - T, density & viscosity, • Postulates – vz=vz(r,z), vr=0, vθ=0

• Equations of change (Appendix – B)

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Transport Phenomena

Example - Flow through a tube

• Pressure and Velocity Profiles

• Boundary Conditions

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Transport Phenomena

Example - Flow of a Falling Film• Assumptions• Postulates– vz = vz(x,z)

• Equations of change in Cartesian coordinates– Continuity (B.4)

– Motion (B.5)

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Transport Phenomena

Example – Couette Viscometer

• Determination of viscosity – By measuring the torque required to turn solid object

in contact with the fluid.• Assumptions• Postulates– vr=vz=0;

– vθ= vθ(r)– p = p (r,z)

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Transport Phenomena

Example – Couette Viscometer

• Continuity Equation – All terms are zero• Equation of motion

• Velocity Profile• Boundary Conditions

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Transport Phenomena

Example – Couette Viscometer

• Momentum Flux

• Torque

• Reynolds No

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Transport Phenomena

Example – Surface of Rotating Liquid• Liquid of constant density and viscosity in a cylindrical container

rotating with some angular velocity• Postulates

– vr=vz=0;

– vθ= vθ(r)– p = p (r,z)

• Boundary Conditions– r = R, vθ=RΩ

– r = 0, vθ=finite

– r = 0 & z = z0, p = patm

• Shape of the liquid- air interface (Obtained from pressure profile)25/08/2011

Transport Phenomena

Example – Flow around a rotating sphere

• Creeping Flow• Spherical Co-ordinates• Assumption/Postulates

• Equations of Change– Continuity 0 = 0– Motion

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Transport Phenomena

Example – Flow around a rotating sphere

• Boundary Conditions

• Solution– Need to guess velocity function– Need to assume some trial solutions

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Transport Phenomena

Dimensional Analysis – Equations of Change• Need • Similitude – Scaling Up /Down Experimental model– Geometric Similarity– Dynamic Similarity– Kinematic Similarity

• Scale Factors– Characteristic length– Characteristic velocity– Characteristic pressure

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Transport Phenomena

Dimensional Analysis – Equations of Change

• Equations of Change (Constant density & viscosity)

• Dimensionless variables

• Dimensionless operators

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Transport Phenomena

Dimensional Analysis – Equations of Change

• Equations of change in terms of dimensionless quantities

OR

• Limiting cases– Re Infinity Euler equation– Re 0; Creeping flow equation

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Transport Phenomena

Example – Transverse Flow around a Circular Cylinder

• Flow of an incompressible fluid past a circular cylinder (experimental study)– Need to find out effect of various parameters on flow

patterns and pressure distributions with minimum no of experiments

• Equation of continuity and motion (N-S)• Initial condition• Boundary conditions

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Transport Phenomena

Example - Flow past a cylinder

• Dimensionless Equations

• Initial/Boundary conditions

• Solution form

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Transport Phenomena

Example – Flow past a cylinder

• Analysis– Velocity and pressure depends only on Re, and L/D

ratio (dimensionless parameters)– Investigating the effects of L, D, velocity, density,

viscosity are not required– Saves lot of time and expense – For scaling -up

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Transport Phenomena

Problems

• Flow between two co-axial cylinders– Incompressible Fluid – Inner cylinder - rotating with angular velocity Ωi

– Outer cylinder – rotating with angular velocity Ωo

• Flow between two co-axial spheres– Incompressible Fluid – Inner sphere - rotating with angular velocity Ωi

– Outer sphere– rotating with angular velocity Ωo

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