laminar complex
TRANSCRIPT
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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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The equation of motion in terms of
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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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Equation of motion for a newtonianfluid with constant and
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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
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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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Tangential flow in an annulus with viscous heat generation
• = 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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Tangential flow in an annulus with viscous heat generation
• = 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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Equations of Continuityfor a Binary System
• 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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Equations of Continuityfor a Binary System
• Component A • Average velocity• Component A
AAA rnt
• Fick’s law
AABBAAA wDnnwn )(
• Average velocityvnn BA
• Component A
AAABAA rwDvt
)()(
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Equations of Continuityfor a Binary System
• 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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