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ADVANCED TRANSPORT

PROCESSES /

TRANSPORT PHENOMENA

CCB/CBB 3033

Momentum Transport Balance The Theory

Part 1

2

Course outcomeCLO1

Explain the theoretical aspect of momentum,

mass and energy transport

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Lesson outcomes

5. State two mechanism for momentum transport

6. Express the molecular transport term

7. Express and the viscous stress term in 3-

dimensional orientation (cartesan coordinates)

ijijij p +=

=vv +=vv +

4

Lesson outcome 5 and 65. State two mechanism for momentum transport

6. Express and explain the molecular transport

term

7. Express and explain the viscous stress term in

3-dimensional orientation (cartesancoordinates)

=vv +=vv +

ijijij p +=

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Momentum Transport Balance

There are two mechanisms :

1. Molecular transport.

2. Convective transport

Combined momentum flux

= molecular stress+ convective stress

6

Momentum Transport Balance

There are two mechanisms :

1. Molecular transport.

2. Convective transport

Convective transport

Molecular transport

vv

= p+

Combined momentum flux = vv +

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Molecular Transport Expression (1.2)

+= p

MolecularMolecularMolecularMolecular

stressesstressesstressesstresses

StaticStaticStaticStatic

pressurepressurepressurepressure

viscousviscousviscousviscous

stressstressstressstress

8

Molecular Transport Expression (1.2)

ijijij p +=i and j is the designation on x,y,z

Recall!

dy

dv xyx

ij

=

Force in thex-direction on a

unit area perpendicular to

y-direction

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Molecular Transport Expression (1.2)

+= p

Kronecker delta

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Molecular Transport Expression (1.2)

ij

i= j , =1; i not equal to j, =0

xx = 1, yy = 1, zz = 1

xy = 0, xz = 0, yz = 0 etc

Kronecker delta,

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Lesson outcome 7

5. State two mechanism for momentum transport

6. Express and explain the molecular transport

term

7. Express and explain the viscous stress term in

3-dimensional orientation (cartesan

coordinates)

12

Molecular Motion

Lets imagine a molecule as a particle

Particle

motion in

tornado

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Molecular Motion

Lets imagine a molecule as a particleFloating

ball

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Molecular Transport

Yes!

Split into 3D

components

x

z

y

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Molecular Transport Expression (1.2)

,Viscous Stress (Force) on a unit area perpendicular to x/y/z-direction

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Molecular Transport Expression (1.2)

There could be 3 components of the movement; vx, vy and vz

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Viscous Stress

ijijij p +=MolecularMolecularMolecularMolecular

stressesstressesstressesstresses StaticStaticStaticStatic

pressurepressurepressurepressure

i and j is the designation on x,y,z

xy xx xz

yy yx yz

zy zx zzx

z

y

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Viscous stress in cartesan coordinates

Viscous force per unit area in the x-direction on a plane

perpendicular to x-direction

xx

xy

Viscous force per unit area in the y-direction on a plane

perpendicular to x-direction

xz

Viscous force per unit area in the z-direction on a plane

perpendicular to x-direction

Area

perpendicular

to x

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Viscous stress in cartesan coordinates

Viscous force per unit area in the x-direction on a plane

perpendicular to y-direction

yy

yx

Viscous force per unit area in the y-direction on a plane

perpendicular to y-direction

yz

Viscous force per unit area in the z-direction on a planeperpendicular to y-direction

Areaperpendicular

to y

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Viscous stress in cartesan coordinates

Viscous force per unit area in the x-direction on a plane

perpendicular to z-direction

zz

zx

Viscous force per unit area in the y-direction on a plane

perpendicular to z-direction

zy

Viscous force per unit area in the z-direction on a plane

perpendicular to z-direction

Area

perpendicular

to z

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Viscous stress in cartesan coordinates

Velocity in thej-direction on a unit area perpendicular to i-direction

+

=

j

i

i

j

ijd

v

d

v

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Viscous stress in cartesan coordinates

Viscous force per unit area in the x-direction on a plane

perpendicular to x-direction

x

v

x

v

x

v xxxxx

=

+

= 2

+

=

y

v

x

vxy

xy

Viscous force per unit area in the y-direction on a plane

perpendicular to x-direction

+

=

z

v

x

v xzxz

Viscous force per unit area in the z-direction on a plane

perpendicular to x-direction

Area

perpendicular

to x

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Viscous stress in cartesan coordinates

Viscous force per unit area in the x-direction on a plane

perpendicular to y-direction

y

v

y

v

y

v yyyyy

=

+

= 2

+

=

x

v

y

v yxyx

Viscous force per unit area in the y-direction on a plane

perpendicular to y-direction

+

=

z

v

y

v yzyz

Viscous force per unit area in the z-direction on a planeperpendicular to y-direction

Areaperpendicular

to y

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Viscous stress in cartesan coordinates

Viscous force per unit area in the x-direction on a plane

perpendicular to z-direction

z

v

z

v

z

v zzzzz

=

+

= 2

+

=

x

v

z

v zxzx

Viscous force per unit area in the y-direction on a plane

perpendicular to z-direction

+

=

y

v

z

vzy

zy

Viscous force per unit area in the z-direction on a plane

perpendicular to z-direction

Area

perpendicular

to z

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Viscous stress in cartesan coordinates

Viscous Stress

Shear Stress Normal Stress

Direction!

xy

xx

xz

yy yx yz

zy zx zz

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Viscous stress in cartesan coordinates

Viscous Stress

Work out the

direction!

xy xx xz

yy yx yz

zy zx zz

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Viscous stress in cartesan coordinates

Viscous Stress

Shear Stress Normal Stress

xx

yy

zz

xy xz

yx yz

zy zx

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Viscous stress in cartesan coordinates

Shear Stress

xy =yx

zy =yz

xz =zx

Note that:

Look at thedirection!

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Lesson outcomes

5. State two mechanism for momentum transport

6. Express and explain the molecular transport

term

7. Express and explain the viscous stress term in

3-dimensional orientation (cartesan

coordinates)

ijijij p +=

=vv +=vv +

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Summary

There are two mechanisms :

1. Molecular transport.

2. Convective transport

Convective transport

Molecular transport

vv

= p+

Combined momentum flux = vv +

+

=

j

i

i

j

ijd

v

d

v

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Further understanding

Read and understand the textbook;Chapter 1,

page 16 to 17

page35 to 36

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