Transport Processes Lec

7/30/2019 Transport Processes Lec

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Chapter 1. Viscosity and the Mechanisms ofMomentum Transport

Newtons law of viscosity (molecular momentumtransport)

Generalisation of Newtons law of viscosity

Pressure and temperature dependence of viscosity Molecular theory of the viscosity of gases at lowdensity, of liquids, and of suspensions andemulsions

Convective momentum transport


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Newtons law of viscosity(molecular transport of momentum)

At steady-state

Newtons law ofviscosity

F V

A Y

xyx

dv

dy


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Newtons law of viscosity

Molecular transport of momentum

= Viscosity,yx = Shear stress in the x-direction at constant y

Velocity gradient is the driving force formomentum transport

xyx

dv

dy


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Newtons law of viscosity

Another interpretation x-momentum is transmitted through the fluid in

the y-direction

yx = viscous flux of x-momentum in thepositive ydirection.


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Newtonian fluids

All gases and most simple liquids Viscosity is independent of -> =(T) Units: Pa.s

Non-Newtonian fluids Viscosity is dependent of Polymeric liquids, pastes, slurries

Kinematic viscosity, Units: m2/s


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Viscosity Definition: the measure of a fluids resistance to flow under applied

stress

Newtonian: origin, constant slope( = viscosity), not time dependent

Non-Newtonian: all others

The apparent viscosity is the ratio of the shearstress to the shear rate. It is not the slope.


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Example: Calculation of momentum flux Fluid between two parallel plates

Details: plate velocity V= 1 ft/s, fluid viscosity = 0.7 cP, Y=0.001 ft

Required: calculate steady state momentum flux yx

2

5 2 2/ 0 1.0 /

(0.7 )(2.0886 10 ) 1.46 10 /0.001 0

fx xyx f

lb s ftdv v ft scp lb ft

dy y cp ft

Solution


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Generalization of Newtons lawof viscosity

Velocity is a vector

Stress is a tensor with 9 components

, ,x y zv v v v

xx xy xz

yx yy yz

zx zy zz


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A half of the cube is removed. Forces onthe free surface:

x x xx xy xzpe and

Associated to the pressure(perpendicular to the surface)

Associated to the viscous forces.For a plane perpendicularto x-direction


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Pressure and viscous forces;


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Molecular stress

Molecular Stress : Normal stress :

Shear stress :

Interpretationsij force in the j-direction on a unit areaperpendicular to the i-direction or

ij flux of j-momentum in the positive ofi-direction

ij ij ijp ( )ij i j

( )ij i j


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= viscosity

= dilatational viscosity (commonly not used)

Stress tensor

2

3

2

3

j yi x zij ij

i j

T

v vv v v

x x x y z

v v v


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Viscosity of gases and liquids

Viscosity is a function of =(T,p) Viscosity may be obtained from Molecular theories

Empirical correlations

Correlation based in molecular theories

Reference The Properties of Gases and Liquids by Poling, Prausnitz,

and OConnell, 5th Ed.


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Pressure and temperaturedependence of viscosity

Reduced viscosity infunction of thereduced temperatureand reduced pressure


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Critical pressure andtemperature

The critical temperatureof a liquid is thattemperature beyond which the liquid cannot exist,no matter how much pressure is placed on it.

The pressure that is needed to cause the gas to

condense at the critical temperature is thecritical pressure.


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Pressure and temperaturedependence of viscosity

Viscosity of a gas at low density increases withincreasing temperature

Viscosity of a liquid decreases with increasingtemperature


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Molecular theory of the viscosityof gases at low density

At low pressure, average velocity (random) andmean free path

Viscosity

2

8 1

2

Tu

md n

2

2

3

m T

d


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Rigorous kinetic theory.Monoatomic gases Lennard-Jones potential

Viscosity for a pure mono-atomic gas (it can be usedfor poly-atomic gases aswell) in term of Lennard-

Jones parameters

12 6

( ) 4rr r

2

5

16

m T


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Convective momentumtransport

Momentum may be transported by the bulk flow ofthe fluid. Convective transport Momentum flux across the shaded area are

Convective momentum-flux tensor (2nd order)

Combined momentum flux

( ) ( ) ( )x y zv v v v v v

i j i ji j

vv vv ee

vv p vv


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Convective momentum transport.Convective momentum flux tensor

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