Download Chapter3

Transport Processes and
Separation Process Principles
(Geankoplis)
CHE312
Dr. Othman Alothman
Dr. Mohamed Kamel Omar Hadj-Kali
1
Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312
Chapter 3
Principles of Momentum
Transfer and Applications
 Flow past immersed objects and packed an fluidized beds
 Measurement of flow of fluids
 Pumps and gas-moving equipment
 …
2
Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312
Flow Past
Immersed
Objects
2 courses
Read and explain the introduction in one course
3
Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312
Flow Past Immersed Objects
3.1A Definitions of Drag Coefficient for Flow Past Immersed Objects
Introduction
 In chapter 2, momentum transfer and the frictional losses for flow of
fluids inside conduits and pipes were discussed.
 In chapter 3, the flow of fluids around solid immersed objects will be
discussed.
 The flow of fluids outside immersed bodies occurs in many chemical
applications such as: flow past spheres in settling, flow through packed
beds in drying and filtration, flow past tubes in heat exchangers and so on.
 In chapter 2, the transfer of momentum perpendicular to the surface
resulted in a tangential shear stress or drag on the smooth surface parallel
to the direction of the flow. The force exerted by the fluid on the solid in the
direction of the flow is called skin or wall drag.
Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312
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Flow Past Immersed Objects
3.1A Definitions of Drag Coefficient for Flow Past Immersed Objects
Introduction
 For any surface in contact with a flowing fluid, skin friction will exist.
 In addition to skin drag, if the fluid has to change its direction to pass
around a solid body such as sphere, significant additional frictional losses
will occur and this is called form drag.
 For a fluid flowing parallel to a solid plate, the force dF on an element
of the area dA of the plate is the wall shear stress times the area dA:
dF   w dA
v0
F     w dA
dA
Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312
5
Flow Past Immersed Objects
3.1A Definitions of Drag Coefficient for Flow Past Immersed Objects
Introduction
 In many cases, the immersed body is a blunt-shaped solid with
various angles.
 In approaching the body, v0 is uniform.
 Lines representing the path of the fluid
elements around the body are called
streamlines.
 The velocity at the stagnation point is
zero, and the boundary layer start to grow
at this point.
 The thin boundary layer is adjacent to the solid surface. The velocity at
the edge of the boundary layer is the same as the bulk velocity adjacent to it.
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Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312
Flow Past Immersed Objects
3.1A Definitions of Drag Coefficient for Flow Past Immersed Objects
Introduction

The tangential stress on the body because of the velocity gradient in
the boundary layer is the skin friction.

Outside the boundary layer, the fluid change direction to pass around
the body accelerating near the front then decelerating.

Thus, an additional force is exerted by the fluid on the body: This is the
form drag.

Separation of the boundary layer occurs and a wake covering the
entire rear of the body occurs where large eddies are present and
contribute to the form drag.

Form drag for bodies can be minimized by streamlining the body which
forces the separation point toward the rear of the body, reducing the size of
the wake.
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Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312
Flow Past Immersed Objects
3.1A Definitions of Drag Coefficient for Flow Past Immersed Objects
Drag coefficient

The geometry of the immersed solid is a main factor in determining
the amount of total drag force exerted on the body.

Similar to the f-NRe correlations for flow inside conduits, correlations for
the drag coefficient- NRe for flow past an immersed body can be obtained.
CD 
FD Ap
 v 2
Where, FD is the total drag force.
2
0
Ap is the area obtained by projecting the body on a plane perpendicular to
the line of flow.
For sphere: Ap 

4
D
2
p
For cylinder
Ap  LD p
(axis perpendicular to the flow):
Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312
8
Flow Past Immersed Objects
3.1A Definitions of Drag Coefficient for Flow Past Immersed Objects
Drag coefficient
So, solving the pervious equation for the total force:
CD 
FD Ap
v02
FD  CD Ap
2

 v 2
2
0
The Reynolds number for a given solid immersed in a flowing liquid is:
N Re 
Dp v0 


DpG0

G0  v0  
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Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312
Flow Past Immersed Objects
3.1B Flow past sphere, long cylinder and disk

Correlations of drag coefficient CD vs. NRe depends mainly on the shape
of the immersed body and its orientation. These correlations are shown in
Figure 3.1-2 for spheres, long cylinders and disks. Note that the face of the
disc and the axis of the cylinder are perpendicular to the direction of flow.
These curves are determined experimentally.

At laminar flow (NRe ≤ 1.0), the total drag force (FD) can be obtained
from Stokes' law equation as follows:
So:
CD 
24
CD 
N Re
FD Ap
 v02 2

CD
FD  3D p v0

3D v  D 4 24


 v 2
Dv
2
p
p 0
2
0
p 0
 Example 3.1.1
Discuss figure 3.1-2
Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312
 Example 3.1.2
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Flow in Packed
and Fluidized
Beds
3 courses
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Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312
Flow in Packed Beds
v’
Introduction
 The packed bed (or packed column) is found in a
number of chemical processes including a fixed bed
v
catalytic reactor, filter bed, absorption and adsorption.
 The ration of diameter of the tower to packing
diameter should be at least 8:1 to neglect wall effects
Laminar flow in packed beds
 The void fraction, e, in a packed bed is defined as:
volume of voids in bed
e
total volume of bed (voids plus solids)
 The specific surface of a particular is:
av  S p V p
Where: Sp is the surface area of a particle and Vp its volume.
Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312
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Flow in Packed Beds (Laminar flow)
v’
 For a spherical particle: av = 6/Dp, (Sp=πDp2
Vp= πDp3/6)
 For a packed bed of non-spherical particles, the
effective particle diameter Dp is:
v
D p  6 av
 The volume fraction of the particles in the bed
is (1- ε) and, thus, the ratio of the total surface
area in the bed to total volume of the bed, a:
Sp
6
1  e 
a  av 1  e  
Dp
N  Sp
total surface area of particles
av 


Vp N  Vp
total volume of particles
total volume of particles
1  e  
total volume of bed
 Example 3.1.3
total surface area in the bed
a  av 1  e  
total volume of bed
Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312
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Flow in Packed Beds (Laminar flow)
v’ = ev
 The average interstitial velocity in the bed (v) is related
v  ev
to the superficial velocity (v’) by:
 The hydraulic radius rH for flow is modified to be:
v
cross - sectional area available for flow
rH 
wetted perimeter
void volume available for flow

total wetted surface of solids
volume of voids / volume of bed e


wetted surface / volume of bed
a
Combining equations
a  av 1  e  
6
1  e  and
Dp
e
rH 
a
gives:
rH 
e
61  e 
Dp
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Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312
Flow in Packed Beds (Laminar flow)
 Since the equivalent diameter for a channel is
v’ = ev
D  4rH
The Reynolds number for a packed bed is as follows:
N Re

4rH v



v
4 Dp v
61  e  
v  ev
For packed beds, Ergun defined NRe without 4/6:
N Re, p 
Dp v

D p G
1  e  1  e 
G  v
For laminar flow, the Hagen-Poiseuille equation can be expressed using rH
to give:
32vL 32 v e L 72vL1  e 
p 


2
2
D
e 3 Dp2
4rH 
2
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Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312
Flow in Packed Beds (Laminar flow)
v’ = ev
Experimental data show that the constant should be 150 (the
true L is larger because of the tortuous path and use of rH
predicts too large v’).
This leads to the Blake-Kozeny
equation for laminar flow, e < 0.5, Dp and NRe,p < 10:
2
150vL 1  e 
p 
Dp2
e3
Flow in Packed Beds (Turbulent flow)
For turbulent flow, we use the same procedure using equations:
L v 2
p  4 f
D 2
v  ev
D  4rH
3 f v L 1  e
To obtain:
p 
3
D
e
p
Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312
rH 
e
61  e 
Dp
2
16
Flow in Packed Beds (Turbulent flow)
v’ = ev
For highly turbulent flow, f should approach constant value.
Another assumption is that all packed beds have the same
relative roughness. Experimental data show that : 3 f  1.75
Hence, for turbulent flow (NRe,p>1000), the Burke-Plummer
eqnation is used:
1.75 v L 1  e
p 
Dp
e3
2
Adding Blake-Kozeny equation (for laminar flow) and Burke-Plummer
equation (for turbulent flow), Ergun proposed a general equation for low,
intermediate and high Reynolds number (NRe,p) as:
150vL 1  e  1.75 v L 1  e
p 

2
3
Dp
e
Dp
e3
2
Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312
2
17
Flow in Packed Beds (Turbulent flow)
v’ = ev
By dimensional analysis, the general equation of Ergun can
be rewritten as:
p D p e 3
150

 1.75
2
G L 1  e N Re, p
G  v
This equation can be used for gases where density is taken at the
arithmetic average of the inlet and outlet pressures
 Example 3.1.4
Flow in Packed Beds (Shape factors)
Particles in packed beds are often irregular in shape. The shape factor
or sphericity (fs) is defined by:
Surface area of sphere having the same volume as tha particle
fs 
The actual surface area of the particle
Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312
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For a sphere:
 The surface area is
Sp=πDp2
 The volume is
Vp= πDp3/6
Hence, for any particle:
fs  Dp2 S p
Dp2 fs
6
av 


3
Vp Dp 6 fs Dp
Sp
and
For a sphere:
fs  1
For a cylinder (L=D):
fs  0.874
For a cube:
fs  0.806
(Sp the surface area of the particle)
a  av 1  e  
6
1  e 
fs D p
(See table 3.1-1)
Hw
 Mixtures of particles
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Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312
Flow in Fluidized Beds
Packed bed
Fluidized bed
Increased
L
Lmin
velocity
v’
v’min
Two general types of fluidization in beds can occur:
 Particulate fluidization
 Bubbling fluidization
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Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312
Flow in Fluidized Beds
 Minimum velocity and porosity
 At very low velocity, packed bed remains stationary
Lmin
 When fluid velocity is increased, the pressure
drop increases, according to Ergun (Eqn. 3.1-20).
 At a certain velocity, when the pressure drop force
(i.e. ∆P*A) equalizes the gravitational force on the mass of
v’min
particle (i.e. m*g), the particles begin to move (fluidize).
This velocity is called the minimum fluidization velocity (v’mf m/s)
(based on the superficial velocity).
At the minimum velocity, the porosity is called the minimum porosity of
fluidization, εmf (See Table 3.1-2 for ε for some materials )
Similarly, the new height of the bed is Lmf in m.
Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312
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Flow in Fluidized Beds
 Relation between bed height L and porosity e
The total volume of the solid particles is constant for a bed having a uniform
cross-sectional area A and is:
Therefore:
V  L  A  1  e 
L1  A  1  e1   L2  A  1  e 2 

L1 1  e 2

L2 1  e 1
 Pressure drop and minimum fluidizing velocity
At the onset of fluidization, the following is approximately true:
p A  Lmf A1  e mf  p   g

p
 1  e mf  p   g
Lmf
Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312
(SI)
22
Flow in Fluidized Beds
Modifying the general equation of Ergun (substitute Dp with Dpfs) to correct
for no spherical particles, this yield to:
2
2
p 150v 1  e  1.75 v 1  e
 2 2

3
L
fs D p e
fs D p
e3
This equation with the previous one can be combined to calculate the
minimum fluidization velocity:
 
2
'
mf
3
2
s mf
2
p
1.75D v
2
fe 

'
1501  e mf Dp vmf
fe 
2 3
s mf
Defining a Reynolds number as:
1.75N Re, mf 
2
We obtain:
fe
3
s mf

Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312

D3p   p   g
N Re, mf 
1501  e mf N Re, mf
fe
2 3
s mf

2
0
'
D p vmf



D 3p   p   g

2
0
23
Flow in Fluidized Beds
1.75N Re, mf 
2
fe
3
s mf

1501  e mf N Re, mf
fe
2 3
s mf

D 3p   p   g

2
0
=0
=0
For small particles For large particles
(NRe,mf < 20)
(NRe,mf > 1000)
fe
3
s mf
If the terms emf and/or fs are not known, Wen
and Yu proposed the following approximations:
N Re, mf

D   p   g 
2
 33.7   0.0408

2



3
p
1  e mf
fe
Substituting in the equation above implies:
1

14
2 3
s mf
12
 11
 33.7 0.001  N Re, mf  4000 
 Example 3.1.6
Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312
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Flow in Fluidized Beds
 Expansion of fluidized beds
For
small particles
We can estimate the variation of porosity
D p v' 
and bed height L using equation a with the
N Re, f 

 20
first term taken equal to zero.
D p2  p   gfs2 e 3
e3
v' 
 K1
150
1 e
1 e
This equation can be used with liquids to estimate e with e < 0.8
 The maximum allowable velocity
For fine solids and NRe,f < 0.4
For large solids and NRe,f > 1000
'
vt'  90vmf
'
vt'  9vmf
 Example 3.1.7
Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312
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Measurement of
flow of fluids
1 course
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Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312
Measurement of flow of fluids (Pitot tube)

It is important to able to measure and control materials entering and
leaving any processing plants.

Many different types of devices are used to measure the flow of fluids
. Including Pitot tube, venture meter, orifice meter and open-channel weirs.
 It is widely used to determine the
 Pitot tube
airspeed on an aircraft and to measure
air and gas velocities in industrial
applications (AF447 (Rio-Paris) crash)
 It is used to measure the local
velocity at given point in the flow stream
h
and not the average velocity in the pipe
or conduit.
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Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312
Measurement of flow of fluids (Pitot tube)

The different between the stagnation pressure (at point 2) and the
static pressure (measured by the static tube) represents the pressure rise
associated with the deceleration of the fluid.
 For incompressible fluids, Bernoulli equation between point 1 and
point 2 can be written as:
 Pitot tube
v v
p1  p2
 
0
2 2

2
1
Setting v2 = 0
2
2

v  Cp
2 p1  p2 

h
Cp is a dimensionless coefficient to correct
deviation from the Bernoulli equation and
varies between 1.0 and 0.98.
 Example 3.2.1
Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312
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Measurement of flow of fluids (Venturi meter)
Venturi meter
A venturi meter is usually inserted
directly into a pipeline.
Since the narrowing and the expansion
are gradual, little friction loss is incurred.
A manometer or is connected to the two pressure taps to measure the
pressure difference between point 1 and 2.
For incompressible fluids, Bernoulli equation between point 1 and point 2
can be written as:
From the continuity equation:
Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312
v12 v22 p1  p2
 
0
2 2

v1
D12
4
 v2
D22
4
29
Measurement of flow of fluids (Venturi meter)
Combining these equations and
eliminating v1:
v2 
Cv
1  D2 D1 
4
2 p1  p2 

Cv is introduced to account for the small friction
 Venturi meter
loss. Its value is between 0.98 and 1.0.
for compressible gases , the adiabatic
expansion(??) from p1 to p2 must
be allowed for.
m
Cv A2Y
1  D2 D1 
4
2 p1  p2 1
The same equation is used where m is the mass flowrate (kg/s) and Y the
dimensionless expansion factor (shown in figure 3-2-3 for air).
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Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312
Measurement of flow of fluids (Orifice meter)
Orifice meter is similar to venturi meter, but
presents some advantages are:
 Adjustable throat,
 Cheaper price,
 Needs less space.
 Orifice meter
The equation for the orifice meter is similar to that of venturi and is:
v0 
C0
1  D0 D1 
4
2 p1  p2 

v0 and D0 are the velocity and
diameter at the point 0 and C0 the
dimensionless orifice coefficient
C0 = 0.61 (for NRe > 2x104 and D0/ D1 < 0.5)
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Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312
Measurement of flow of fluids (Orifice meter)
As for venturi meter, for compressible gases,
the adiabatic expansion(??) from p1 to p2
must be allowed for.
m
C0 A0Y
1  D0 D1 
4
2 p1  p2 1
 Orifice meter
The pressure loss is much higher than in venture meter because of the
eddies formed due to the jet expansion. This loss depends on D0/D1:
 73% of (p1-p2) for D0/D1 = 0.50
 56% of (p1-p2) for D0/D1 = 0.65
 38% of (p1-p2) for D0/D1 = 0.80
 Example 3.2.2
Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312
32