Laminar Complex

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TK5102Ad d T t PhAdvanced Transport Phenomena

“Transports in Laminar Regimes:Complex Problems”

I Dewa Gede Arsa PutrawanChemical Engineering ITB

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Outcomes

• Students can simplify equations of change to derive• Students can simplify equations of change to derive mathematical models of complex problems in transport phenomena.

• Students can estimate property profiles for complex problems in transport phenomena.

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The Equation of Changefor Isothermal System

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Equation of Continuity

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Mass balance over a volume element x, y, z

x, y, z Rate of mass accumulation

z xyz /t

Rate of mass in( vxyz)x + ( vyxz)y

+ ( vzxy)z

x, y, zx

y Rate of mass out( vxyz)x+x + ( vyxz)y+y

+ ( vzxy)z+z28-Sep-09 DGA/5TK5102

Mass Balanceover a volume element x, y, z

y z v vx y z

x xx x x

y yy y y

z zz z z

y z v vx y z tx z v v

x y v v

( )( ) ( ) 0yx zvv vt x y z

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Mass Balanceover a volume element x, y, z

D 0 0Dv or vt Dt

(Rate of increase of mass per unit volume = Net rate of mass addition per unit volume by convection)

• Incompressible fluid

0v

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The equation of continuity

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The Equation of Motion

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Momentum concentration

• Three components of momentum (x y z• Three components of momentum (x, y, z components)

• Concentration of x-component of momentum : vx

• Accumulation rate of x component of• Accumulation rate of x-component of momentum ( )xvx y z t

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Flow of momentumby molecular diffusion

yyz

xx

yxzxx

yz

xx yx zxy z x z x y

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• Flow of momentum in by molecular

xx yxx x y yy z x z

• Flow of momentum out by molecular

zx z zx y

xx yxy z x z

xx yxx x x y y y

zx z z z

y

x y

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Flow of momentumby convective mechanism

xv y z

x

y

z

volumetric flow rate vv

x

yv x z

x y

x( v )( )xy z v Q

Qz

xy

y

z

( v )( )

( v )( )x

x

x z vx y v

Qx

Qy

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• Flow of momentum in by convective

x y( v )( ) ( v )( )x xx x y yy z v x z v

z( v )( )x z zx y v

• Flow of momentum out by convective

x y( v )( ) ( v )( )x xy z v x z v

x y

z

( )( ) ( )( )

( v )( )

x xx x x y y y

x z z z

y

x y v

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Forces acting on fluid

• Gravity• Gravity

• Pressure

( )( )xx y z g

( )( )y z p p

( )( )x x x x x

y p p

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The equation of motion in terms of

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Equation of motion for a newtonianfluid with constant and

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Tangential annular flow of a Newtonian fluid

• Outer cylinder rotates atOuter cylinder rotates at angular velocity of o

• Incompressible and Newtonian fluid with constant transport properties

• Laminar flow in direction only at steady state condition

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y y• 1D momentum transfer in

radial direction• Pressure varies in radial and

axial directions

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Continuity equation

( )( ) ( )1 1 0r zvrv vt r r r z

v

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0v

Equation of motion for r-component

2r r r r

r zv vv v v v pv v

t r r r z r

2 2( )1 1 2 vrv v v

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2 2 2 2

( )1 1 2r r rr

vrv v v gr r r r r z

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Equation of motion for -component

1rr z

v v v v v v v pv vt r r r z r

2 2( )1 1 2rv v vv

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2 2 2 2

( )1 1 2 rrv v vv gr r r r r z

Equation of motion for z-component

2z z z z

r zv vv v v v pv v

t r r r z z

2 21 1v v v

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2 2 2

1 1z z zz

v v v gr r r r z

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Simplified equations of motion

2vp (Effect of centrifugal• Eq. B.6.4vp

r r

(Effect of centrifugal force on pressure)

• Eq. B.6.51 ( ) 0d d rv

dr r dr

(Velocity distribution)

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• Eq. B.6.6p gz

(Effect of gravity force on pressure)

Distribution of angular velocity

1 ( ) 0d d

• Eq. B.6.5 1 ( ) 0d d rvdr r dr

• BC: v = 0 at r = R and v = oR at r = R• Velocity profile

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/( ) /1/o

r R R rv R

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Momentum flux and torque

vd • Momentum flux

2 2 22 ( / ) ( /(1 ))

r

o

vdr

dr r

R r

• Torque

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Torque

2 2 2

2

4 /(1 )z r r R

o

T RL R

R L

Concentric cylinder viscometer

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Flow near a slowly rotating sphere

22

1 vr

r r r

2

1 1( sin ) 0

sinv

r

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BC 1 : vφ = R Ω sin θ at r = RBC 2 : vφ = 0 at r = ∞

Cone and Cup Viscometer

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Cone and Cup Viscometer

2( / ) sinv R R r • Velocity distribution

• Torque

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38zT R

Equations of Changefor non isothermal system

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Energy Equation2 2ˆ ˆ(0,5 ) (0,5 )v U v U q (0,5 ) (0,5 )v U v U q

t

( ) ( [ ]) ( )pv v v g

Accumulation of energy

Convective mechanism

Conduction

( ) ( [ ]) ( )pv v v g

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Pressure forces

Gravitational forces

Viscous forces

Special Forms of Energy Equation

ˆ ˆ ( ) ( )U U ( ) ( : )U Uv q p v v

t

ˆ( ) ( : )DU q p v v

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( ) ( : )q p v vDt

(the equation of thermal energy)

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Special Forms of Energy Equation

• Incompressible fluid (constant )

ˆ( : )DU q v

Dt

• U = H – p/

( : )DH Dpq vDt Dt

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Energy equationin terms of fluid temperature

ˆ DT p ˆ

ˆ ( : ) ( )vV

DT pC q v T vDt T

(1/ )ˆ ( : )pp

DT DpC q v TDt T Dt

• For incompressible fluid : Cv = Cp

ˆ ( : )pDTC q vDt

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Energy equationsfor incompressible fluids

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Forced Convectionof Laminar Flow in Pipe

Fluid in at T1

z

r

fluks

q o• Steady state momentum and heat

transfer• 1D momentum transfer (vz = v(r))• T = T(r,z)

• Constant physical properties

Hea

t • Constant physical properties

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Continuity equation

( )( ) ( )1 1 0r zvrv vt r r r z

v

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0zvz

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Equation of motion for z-component

2z z z z

r z

v vv v v v pv v

t r r r z z

2 21 1v v v

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2 2 2

1 1z z zz

v v vg

r r r r z

Energy Equation

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Forced Convectionof Laminar Flow in Pipe

Fluid in at T1 Continuity equation

z

r

fluks

q oy q

dvz/dz = 0Equation of motion

10 zdvdP d r gdz r dr dr

Energy equation

viscous dissipation

Hea

t

22

2

1ˆ zp

dvdT dT TC k rdz r r dr z dr

28-Sep-09 DGA/51TK5102heat conduction

in z direction

Tangential flow in an annulus with viscous heat generation

• Incompressible and Newtonian fl id i hfluid with constant transport properties

• Outer cylinder rotates• Laminar flow in direction• Steady state condition • 1D momentum and heat transfers

in radial direction

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in radial direction• Pressure varies in radial and axial

directions

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Energy Equation

T T T T r z

vT T T TCp v vt r r z

2 21 1T T Tk #

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2 2 2 vk rr r r r z

#

Dissipation Function22 212 r zvv v

$ # %

2 2

2

1

r zv r

z z r

vr r z

v v v vz r r z

# % &

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21 r vv rr r r

$ % &

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Energy Equation

21 vT 1 vTk r rr r r r r

From momentum balance:

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/( ) /1/o

r R R rv R

Energy Equation

2 4 441 1RT 2 2 4

41 1 0(1 )

o RTk rr r r r

Boundary conditions :• T = T at r = R

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• T = Tb at r = R• T = To at r = R

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Energy Equation(dimensionless form)

1 14d d N' (

2 4 4

2 2( )(1 )o o

b o b o

T T Rr NR T T k T T

'

(

44Nd d

'' ' ' '

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( )( )b o b okBoundary conditions :

• = 1 at = 1• = 0 at = 0

Temperature Profiles

• Annulus

2 2

ln 1 1 ln1 1 1ln ln

N' ' '

(

1 (1 )( )Br' ' ' (

• Annulus

• Treating annulus as parallel plates

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21 2 (1 )Br

(

Note : N = Br 4 / (1 – 2)2

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• = 92 3 cP• = 92.3 cP• = 1.22 g/ml• k = 0.0055 cal/(s cm C)• Tb = 100 ºC• To = 70 ºC• R = 5.060 cm• = 0.99• = 7980 rpm

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• = 92 3 cP• = 92.3 cP• = 1.22 g/ml• k = 0.0055 cal/(s cm C)• Tb = 100 ºC• To = 70 ºC• R = 5.060 cm• = 0.50• = 7980 rpm

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Continuity Equation for Component

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Mass Balance of Aover volume element x, y, z

x, y, zz Rate of accumulation

y

zxyz A/t

Rate of A in

Rate of productionxyz rA

x, y, zx

Rate of A innAxxyz + nAyy xz+ nAzzxyRate of A out

nAxx+xyz + nAyy+yxz + nAzz+zxy28-Sep-09 DGA/64TK5102

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Equations of Continuityfor a Binary System

• Component A• Component A

AAA

AAzAyAxA rn

tataur

z

n

y

n

x

n

t

• Component B

BBB

BBzByBxA rnataur

nnn

BBB tzyxt

• Mixer

0)()(

vt

ataurrnnt BABA

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• Component A• Component A

AAA

AAzAyAxA RN

t

CatauR

z

N

y

N

x

N

t

c

• Component B

BBB

BBzByBxA RN

catauR

NNNc

BBB tzyxt

• Mixer

BABABA RRcvt

catauRRNN

t

c

)()( *

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• Component A • Average velocity• Component A

AAA rnt

• Fick’s law

AABBAAA wDnnwn )(

• Average velocityvnn BA

• Component A

AAABAA rwDvt

)()(

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• Component A • Average velocity• Component A

AAA RNt

c

• Fick’s law

AABBAAA xcDNNxN )(

• Average velocity*cvNN BA

• Component A

AAABAA RxcDvct

c

)()( *

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Equations of Continuityfor a Binary System (dilute component)

• Component A with constant ρ and DAB (weight base)Component A with constant ρ and DAB (weight base)

AAABAAA rDvvt

2

AAABAA rDvt

2

• Component A with constant ρ and DAB (molar base)Component A with constant ρ and DAB (molar base)

AAABAAA RcDcvvct

c

2**

)(2*BAAAAABA

A RRxRcDcvt

c

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Reaction in Pipe Reactor

• Fast reaction A → B at cylinder wall

zr

L

2RDilute solutionConcentration cA

• B in solution can be neglected

• Steady state, isothermal, and laminar flow

• Constant transport properties

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Reaction in Pipe Reactor

• Equation of motion• Equation of motion

dr

dvr

dr

d

rdz

dP z10

• Continuity equation of A

21 cc

Dc AAA

Boundary conditions:

Boundary conditions:• vz = 0 at r = R• dvz/dr = 0 at r = 0

2

1

z

c

r

cr

rrD

z

cv AA

ASA

z

y• cA = cA0 at z = 0• cA = 0 at r = R• dcA/dr = 0 at r = 0

neglected

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Laminar Complex

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