”The life is too short to drink bad wine.”
Common Interpretation of the
Equations of Flow
in Chemical Engineering
1
Contents of Lecture
•
•
•
•
•
•
•
•
1. Introduction
2. Classification of fluids
3. Fluids in motion
4. Transport equations
5. Energy balance of flow of fluids
6. Flow of fluids in simple systems
7. Flow of fluids in complex systems
8. Summary
2
Introduction
Fluid Kinematics deals with the motion of fluids
without considering the forces and moments which
create the motion.
Items discussed in this Chapter.


Introduction
 Velocity Field
 Acceleration Field
 Control Volume and System Representation
 Reynolds Transport Theorem
 Examples

Introduction
•
•
•
•
If a Fluids is subjected to shear force then it tends to flow
Fluids that are subjected to pressure imbalance resulted in
fluid flow
In kinematics we are not concerned with the force, but the
motion, thus, we are interested in flow visualization.
We can learn a lot about flows from watching.
Velocity field-Continuum
Hypothesis

Continuum Hypothesis: The flow is made of tightly packed fluid
particles that interact with each other. Each particle consists of
numerous molecules, and we can describe the field variables
velocity, acceleration, pressure, and density of these particles at a
given time.
Velocity field

Different particles in fluid flow, which move at different
velocities and may be subjected to different accelerations.
The velocity and acceleration of a fluid particle may change
both with respect to time and space.

In the study of fluid flow it is necessary to observe the motion
of the fluid particles at various points in space and at a
successive instants of time.

Velocity field - Lagarangian Vs Eulerian
method



Generally there are two methods by which the motion of a fluid
may be described, Lagarangian and Eulerian method.
In the Lagarangian method any individual fluid particle is selected ,
and observation is made about the behavior of this particle during
its course of motion through space (Control mass approach).
In the Eulerian method any point in the space occupied by the fluid
is selected and observation is made of whatever changes of
velocity, density and pressure which take place at that point (control
volume approach).
Lagrangian Description


Lagrangian description of fluid flow tracks the
position and velocity of individual particles.
Based upon Newton's laws of motion, it is
difficult to use Lagrangian description for
practical flow analysis as.



However, useful for specialized applications in



Fluids are composed of billions of molecules.
Interaction between molecules hard to describe/model.
Sprays, particles, bubble dynamics, rarefied gases.
Coupled Eulerian-Lagrangian methods.
Named after Italian mathematician Joseph Louis
Lagrange (1736- 1813).
Eulerian Description



In a Eulerian description of fluid flow a flow domain or control volume is
defined by which fluid flows in and out.
We define field variables which are functions of space and time.
Pressure field, P = P(x, y, z, t)
 
 Velocity field,
V  V x, y, z,t 




V  u x, y, z,t i  v x, y, z,t  j  w x, y, z,t k

Acceleration field,
 
a  a x, y, z,t 




a  a x x, y, z,t i  a y x, y, z,t  j  a z x, y, z,tk



These (and other) field variables define the flow field.
Well suited for formulation of initial boundary-value problems (PDE's).
Named after Swiss mathematician Leonhard Euler (1707-1783).
Lagarangian Vs Eulerian
method
Measurement of fluid temperature
Eulerian
Lagrangian
Eulerian methods are commonly
used in fluid experiments or
analysis—a probe placed in a
flow.
Velocity field- Uniform Vs Non-Uniform


Uniform flow: If the flow velocity is the same magnitude and direction
at every point in the fluid at the given instant of time, it is said to be
uniform.
V
0
s
Non-uniform: If at a given instant, the velocity is not the same at
every point the flow is non-uniform. (In practice, fluid that flows near
a solid boundary will be non-uniform – as the fluid at the boundary
must take the speed of the boundary, usually zero.
V
s
0
Velocity field- Steady Vs
Unsteady




Steady Flow: The velocity at a given point in space does not vary
with time.
Very often, we assume steady flow conditions for cases where there
is only a slight time dependence, since the analysis is “easier”
Unsteady Flow: The velocity at a given point in space does vary with
time.
Almost
all
flows have some unsteadiness. In
addition,
there are periodic flows, non-periodic flows,
and completely random flows.
Velocity field- Steady Vs
Unsteady

Periodic flow: “fuel injectors” creating a periodic swirling in the
combustion chamber. Effect occurs time after time.
Random flow: “Turbulent”, gusts of wind, splashing of water in the
sink

Steady or Unsteady only pertains to fixed measurements, i.e.
exhaust temperature from a tail pipe is relatively constant “steady”;
however, if we followed individual particles of exhaust they cool!

Velocity field- Steady Vs
Uniform


Combining the above we can classify any
flow into one of four type:
Steady uniform flow. Conditions do not
change with position in the stream and with
time at a point.


An example is the flow of water in a pipe of constant diameter
at constant velocity.
Steady non-uniform flow. Conditions change
from point to point in the stream but do not
change with time at a point.
Velocity field- Steady Vs
Uniform

Unsteady uniform flow. At a given instant in time the conditions at
every point are the same, but will change with time.


Example : An example is a pipe of constant diameter connected
to a pump pumping at a constant rate which is then switched off.
4.
Unsteady
non-uniform
flow. Every condition
of
the
flow
may change from point to point and
with time at every point.

For example waves in a channel.
Velocity Field- 1D, 2D, and 3D
Flows


Most fluid flows are complex three dimensional, time-dependent
phenomenon, however we can make simplifying assumptions allowing
an easier analysis or understanding without sacrificing accuracy. In
many cases we can treat the flow as 1D or 2D flow.
Three-Dimensional Flow:
All three velocity components are
important and of equal magnitude. Flow past a wing is complex 3D
flow, and simplifying by eliminating any of the three velocities would
lead to severe errors.

Velocity Field- 1D, 2D, and 3D
Flows
Two-Dimensional Flow : In many situations one of the velocity
components may be small relative to the other two, thus it is
reasonable in this case to assume 2D flow.

One-Dimensional Flow: In some situations two of the velocity
components may be small relative to the other one, thus it is
reasonable in this case to assume 1D flow. There are very few
flows that are truly 1D, but there are a number where it is a
reasonable approximation.
Flow Visualization



Flow visualization is the visual examination of flow-field
features.
Important for
both physical experiments and
numerical (CFD) solutions.
Numerous methods
 Streamlines and streamtubes
 Pathlines
 Streaklines
 Timelines
 Refractive techniques
 Surface flow techniques
Velocity field - Streamlines

A Streamline is an imaginary curve
drawn through the flowing fluid in such a
way that the tangent to it at any point
gives the direction of the velocity at that
point.

Because the fluid is moving in the
same direction as the streamlines,
fluid can not cross a streamline.
 Streamlines
can not cross each other. If
they were to cross this would indicate
two different velocities at the same
point. This is not physically possible.

The above point implies that any
particles of fluid starting on one
streamline will stay on that same
streamline throughout the fluid.
Streamline
s
A Streamline is a curve that is
everywhere tangent to the
instantaneous local velocity vector.


Consider an arc length




dr  dxi  dyj  dzk
dr

must be parallel to the local
velocity vector


 
V  ui  vj  wk

dr
Geometric arguments results in the
equation for a streamline

dx dy dz Vu
 
v
w
Velocity field - Streamlines
NASCAR surface pressure contours
and streamlines
Airplane surface pressure contours,
volume streamlines, and surface
streamlines
Velocity field - Pathlines


A Pathline is the actual path
traveled by an individual fluid
particle over some time period.
Same as the fluid particle's
material position vector
x
particle


t , yparticle t , zparticle t 
Particle location at time t:
Particle Image Velocimetry (PIV) is
a modern experimental technique
to measure velocity field over a
plane in the flow field.
t 
 
x  xstart   Vdt
tstart
Velocity field - Streakline


A Streakline is the locus of
fluid particles that have passed
sequentially
through
a
prescribed point in the flow.
Easy
to
generate
in
experiments: dye in a water
flow, or smoke in an airflow.
Comparison
s


For steady flow, streamlines, pathlines, and streaklines are
identical.
For unsteady flow, they can be very different.

Streamlines are an instantaneous picture of the flow field

Pathlines and
have
a
 Streakline:
Streaklines are
flow patterns
time history associated with them.
instantaneous snapshot of a time-integrated flow
pattern.

that
Pathline: time-exposed flow path of an individual particle.
Plots of Data


A Profile plot indicates how the value of a
scalar property varies along some desired
direction in the flow field.
A Vector plot is an array of arrows
indicating the magnitude and direction of a
vector property at an instant in time.
Lagrangian Frame:
Acceleration Field
Eulerian Frame: we describe the acceleration in terms of position and time
without following an individual particle. This is analogous to describing the
velocity field in terms of space and time.
A fluid particle can accelerate due to a change in velocity in time
(“unsteady”) or in space (moving to a place with a greater velocity).
Acceleration Field: Material (Substantial) Derivative
time dependence
spatial dependence
We note:
Then, substituting:
The above is good for any fluid particle, so we drop “A”:
Acceleration
Field: Material (Substantial) Derivative
Writing out these terms in vector components:
x-direction:
y-direction:
z-direction:
Writing these results in “shorthand”: where,
() 
 ˆ  ˆ  ˆ
i
j k ,
x
y
z
Fluid flows experience fairly
large accelerations
or
decelerations,
especially
approaching
stagnation
points.
Acceleration
field


The time dependant term in the acceleration field is called the local
acceleration and is nonzero only for unsteady flows.
The spacial dependant term in the acceleration field is called the
advective acceleration and accounts for the effect of the fluid
particle moving to a new location in the flow, where the velocity is
different.
The total derivative operator d/dt is called the material derivative
and is often given special notation, D/Dt.
 Advective acceleration is nonlinear: source of many phenomenon
and primary challenge in solving fluid flow problems.
 Advective
acceleration
provides
``transformation''
Lagrangian
and Eulerian frames.
between
 Other names for the material derivative include:
total, particle,
Lagrangian, Eulerian, and substantial derivative.

Acceleration Field: Material (Substantial) Derivative
Applied to the Temperature Field in a Flow:
The material derivative of any variable is the rate at which that variable
changes with time for a given particle (as seen by one moving along with the
fluid— Lagrangian description).
Acceleration Field: Unsteady Effects
If the flow is unsteady, its paramater values at any location may change with
time (velocity, temperature, density, etc.)
The local derivative represents the unsteady portion of the flow:
If we are talking about velocity, then the above term is local acceleration. In
steady flow, the above term goes to zero.
If we are talking about temperature, and V = 0, we still have heat transfer
because of the following term:
0
=
0
0
Acceleration Field: Unsteady Effects
Consider flow in a constant diameter pipe, where the flow is assumed to be
spatially uniform:
0
0
0
0 0
Acceleration Field: Convective Effects
The portion of the material derivative represented by the spatial derivatives is
termed the convective term or convective accleration:
It represents the fact the flow property associated with a fluid particle may
vary due to the motion of the particle from one point in space to another.
Convective effects may exist whether the flow is steady or unsteady.
Example 1:
Example 2:
Acceleration = Deceleration
Control Volume and System Representations
Systems of Fluid: a specific identifiable quantity of matter that may consist of a
relatively large amount of mass (the earth’s atmosphere) or a single fluid
particle. They are always the same fluid particles which may interact with their
surroundings.
Example: following a system the fluid passing through a compressor
We can apply the equations of motion to the fluid mass to describe their
behavior, but in practice it is very difficult to follow a specific quantity of matter.
Control Volume: is a volume or space through which the fluid may flow,
usually associated with the geometry.
When we are most interested in determining the forces put on a fan, airplane, or
automobile by the air flow past the object rather than following the fluid as it
flows along past the object.
Identify the specific volume in space and analyze the fluid flow within, through,
or around that volume.
Surface of the Pipe
Control Volume and System Representations
Surface of the Fluid
Fixed Control Volume:
Volume Around The
Engine
Inflow
Fixed or Moving
Control Volume:
Outflow
Deforming Control
Volume:
Outflow
Deforming Volume
Reynolds
Transport
Theorem:
Concepts
All
the laws of governing
the motion
of a fluidPreliminary
are stated in their
basic form in
terms of a system approach, and not in terms of a control volume.
The
Reynolds
Transport
to
shift
from
the
volume approach, and back.
Theorem
allows us
system approach to the control
General Concepts:
B represents any of the fluid properties, m represent the mass, and
represents the amount of the parameter per unit volume.
Examples:
Mass
b=1
b = V2/2
Kinetic Energy
b = V (vector)
Momentu
m
B is termed an extensive property, and b is an intensive property.
directly proportional to mass, and b is independent of mass.
B is
b
Reynolds Transport Theorem: Preliminary Concepts
For a System: The amount of an extensive property can be calculated by
adding up the amount associated with each fluid particle.
Now, the time rate of change of that system:
Now, for control volume:
For the control volume, we only integrate over the control volume, this is
different integrating over the system, though there are instance when
they could be the same.
Transport
Theorem:
Derivation
Consider Reynolds
a 1D flow through
a fixed control
volume between
(1) and (2):
System at t2
System at t2
CV, and system at t1
Writing equation in terms of the extensive parameter:
Originally,
At time 2: Divide by t:
Reynolds Transport Theorem: Derivation
Noting,
(1)
Let,
(2)
(3)
(4)
(1)
Time rate of change of mass within the control volume:
(2)
The rate at which the extensive property flows out of the control surface:
(4)
Reynolds Transport Theorem: Derivation
The rate at which the extensive property flows into the control surface:
(3)
Now, collecting the terms:
or
Restrictions for the above Equation:
1) Fixed control volume
2) One inlet and one outlet
3) Uniform properties
4) Normal velocity to section (1) and (2)
Reynolds
Theorem:
Derivation
The Reynolds
Transport Transport
Theorem can be
derived for more
general conditions.
Result:
This form is for a fixed non-deforming control volume.
Reynolds Transport Theorem: Physical Interpretation
(1)
(2)
(3)
(1)The time rate of change of the extensive parameter of a system, mass,
momentum, energy.
(2)The time rate of change of the extensive parameter within the control
volume.
(3)The net flow rate of the extensive parameter across the entire control
surface.
“outflow across the surface”
“inflow across the surface”
“no flow across the surface”
Mass flow rate:
Reynolds Transport Theorem: Analogous to Material Derivative
Time dependant Portion
Steady Effects:
Unsteady Effects (inflow = outflow):
Convective Portion
Kinematic
Description

In fluid mechanics, an element
may undergo four fundamental
types of motion.
a)
b)
c)
d)

Translation
Rotation
Linear strain
Shear strain
Fluids motion and deformation is
best described in terms of rates
a)
b)
c)
d)
velocity: rate of translation
angular velocity: rate of rotation
linear
strain
rate:
rate
of
linear strain
shear strain rate:
rate of
shear strain
Rate of Translation and Rotation


To be useful, these rates must be expressed in terms of velocity
and derivatives of velocity
The rate of translation vector is described as the velocity vector.
In Cartesian coordinates:


 
V  ui  vj  wk

Rate of rotation at a point is defined as the average rotation
rate of two initially perpendicular lines that intersect at that
point. The rate of rotation vector in Cartesian coordinates:
 1  w v   1  u w   1  v u  
    i     j    k
2  y z 
2  z x 
2  x y 
Linear Strain
Rate

Linear Strain Rate is defined as the rate of increase in length per unit length.

In Cartesian coordinates

u
v
w
 xx 
, yy  , zz

y
z
Volumetric strain rate inx
Cartesian coordinates
1 DV        u  v  w
xx
yy
zz
x y z
V Dt

Since the volume of a fluid element is constant for an incompressible flow, the
volumetric strain rate must be zero.
Shear Strain Rate


Shear Strain Rate at a point is defined as half of the rate of
decrease of the angle between two initially perpendicular lines
that intersect at a point.
Shear strain rate can be expressed in Cartesian coordinates
as:
1  u v 
1  w u 
1  v w 
 xy    ,  zx    ,  yz    
2  z y 
2  y x 
2  x z 
Vorticity and
Rotationality



The vorticity vector is defined as the curl of the velocity vector
  
   V
Vorticity is equal to twice the angular velocity of a fluid particle.


  2
Cartesian coordinates
  w v    u w    v u  
     i     j    k
 y z   z x 
 x y 


In regions where  = 0, the flow is called irrotational.
Elsewhere, the flow is called rotational.
Vorticity and
Rotationality
F.M
• There are two methods of motion of fluid
particles.
1. Lagrangian method:
 This method deals with the individual particles.
 Langrangian description of fluid flow tracks the
position and velocity of individual particles.
 (E.g.: Track the location of migrating bird.
 Motion is described based upon Newton's laws.
 Named after Italian mathematician Joseph Louis
 Lagrange (1736-1813).
Eulerian method.
• Eulerian
• Describes the flow field (velocity,acceleration,
pressure, temperature, etc.) as functions of
position and time.
• It deals with the flow pattern of all the
particles.
• Count the birds passing a particular location
• If you were going to study water flowing in a
pipeline
which approach would you use? Eulerian
Description
1. Introduction
• A substance in the liquid or gas phase is referred to
as a fluid.
• A large part of chemical operations are in
connection with the flow and mixing of fluids.
• These processes are described by the laws of flow.
• These laws determine the energy losses and
pressure drop of flow of fluids, the heat and mass
transfer processes and characteristics of chemical
reactions to a great extent.
• The goal of my lecture is to give a general form for
the determination of energy losses and pressure
drop of flow of fluids in simple and complex
systems.
52
Fluids in the Industry
Fluids vary considerably in their properties.
They include such materials as:
1. Thin liquids: water, alcohol…etc
2. Thick liquids: syrups, honey, oil…etc
3. Gases: air, nitrogen, carbon dioxide…etc
4. Fluidized solids: grains, flour, peas…etc
There is an increasing tendency to handle powered and
granular materials in a form in which they behave as
fluids. Fluidization, as this is called, has been developed
because of the relative simplicity of fluid handling
compared with the handling of solids .
53
2. Classification of Fluids
Fluids may be classified in two different ways;
either according to their behavior under externally
applied pressure, or according to the effects
produced by the action of a shear stress.
Stress is defined as the force per unit area.
Normal component: normal stress
In a fluid at rest, the normal stress is
called pressure
Tangential component: shear stress
Pressure and shear stress
Fluid can not resist an applied
shear stress by deforming, it
deforms continuously under the
influence of shear stress, no matter
how small.
54
Classification of Fluids
• In gases and in most pure liquids the ratio of the shear
stress to the rate of shear is constant and equal to the
viscosity of the fluid. These fluids are said to be
Newtonian in their behavior.
• In some liquids, particularly those containing a second
phase in suspension, the ratio is not constant and the
apparent viscosity of the fluid is a function of the rate
of shear. The fluids are said to be non-Newtonian and
to exhibit rheological properties.
• If the volume of fluid is independent of the action
under externally applied pressure, the density remains
nearly constant the fluid is said to be incompressible;
if its volume changes it is said to be compressible.
Liquid flows are typically incompressible. Gas flows
are often compressible, especially for high speeds.
55
Incompressible Flow
In fluid mechanics or more generally continuum
mechanics, incompressible flow (isochoric flow) refers to
a flow in which the material density is constant within a
fluid parcel an infinitesimal volume that moves with the
flow velocity. An equivalent statement that implies
incompressibility is that the divergence of the flow velocity
is zero.
Inviscid Flow
An inviscid flow is the flow of an ideal fluid that is
assumed to have no viscosity. In fluid dynamics there are
problems that are easily solved by using the simplifying
assumption of an inviscid flow. The flow of fluids with low
values of viscosity agree closely with inviscid flow
everywhere except close to the fluid boundary where the
boundary layer plays a significant role.
Cavitation
Cavitation is the formation of vapour cavities in liquid
i.e. small liquid free zones that are the consequence
of forces acting upon the liquid.
It usually occurs when a liquid is subjected to rapid
changes of pressure that cause the formation of
cavities where the pressure is relatively low.
When subjected to higher pressure, the voids
implode and can generate an intense shock wave.
Cavitating propeller model in
a water tunnel experiment
Newtonian fluids
• Viscosity is that property of a fluid that gives rise
to forces that resist the relative movement of
adjacent layers in the fluid. Viscous forces are of
the same character as shear forces in solids and
they arise from forces that exist between the
molecules
• If two parallel plane elements in a fluid are
moving relative to one another, it is found that a
steady force must be applied to maintain a
constant relative speed. This force is called the
viscous drag because it arises from the action of
viscous forces. Consider the system shown in Fig.
58
Viscosity
If the plane elements are at a distance Z apart, and if their relative velocity
is v, then the force F required to maintain the motion has been found,
experimentally, to be proportional to v and inversely proportional to Z for
many fluids. The coefficient of proportionality is called the viscosity of the
fluid, and it is denoted by the symbol  (mu).
where F is the force applied, A is the area over which force is applied, Z is
the distance between planes, v is the velocity of the planes relative to one
another, and  is the viscosity, and  is the shear stress.
F
v
dv
μ μ
τ
A
z
dz
59
The Boundary layer
At the interface between moving water and a stationary substrate,
the water velocity is 0, i.e. “no slip” condition
This means that there is a sharp shear or gradient in velocity near
the substrate
It is within this velocity gradient that viscosity exerts its friction
We call the gradient region a boundary layer
U
U
y
Boundary layer
thickness

Non-Newtonian Fluids
61
Non-Newtonian Fluids
Newtonian Fluid
dv
τ  μ
dz
Non-Newtonian Fluid
dv
τ  η
dz
η is the apparent viscosity and is not constant for
non-Newtonian fluids.
62
η - Apparent Viscosity
The shear rate dependence of η categorizes
non-Newtonian fluids into several types.
Power Law Fluids:
 Pseudoplastic – η (viscosity) decreases as shear rate
increases (shear rate thinning)
 Dilatant – η (viscosity) increases as shear rate increases
(shear rate thickening)
Bingham Plastics:
 η depends on a critical shear stress (0) and then
becomes constant
63
Non-Newtonian Fluids
Bingham Plastic: sludge, paint, blood, ketchup
Pseudoplastic: latex, paper pulp, clay solns.
Newtonian
Dilatant: quicksand
64
Modeling Power Law Fluids
 du z 
τ  K 

 dr 
n
where:
K = flow consistency index
n = flow behavior index
Note: Most non-Newtonian fluids are pseudoplastic n<1.
65
3. A fluidumok áramlása.
Stacionárius áramlás jellemzői:
.
•
•
•
•
•
Térfogatáram:
Tömegáram:
Átlagos áramlási sebesség:
Áramlási keresztmetszet:
A fluidum sűrűsége:
 
V m3 /s

m kg/s 
v m/s 
 
A m2
ρ kg/m 
3
66
Áramlás jellemzői közötti kapcsolat
• Az átlagos áramlási sebesség:
.


V
m
v 
A Aρ
• Kör keresztmetszetű cső esetén:

4V
4m
v 2  2
D π D πρ
• A fluidumok áramlását áramvonalakkal szemléltetjük.
Áramvonal a fluidum részecske mozgását leíró pályavonal,
mely adott pillanatban, az áramló közeg minden pontjában
az áramlás irányába mutat.
67
Az áramlástan alapjai
A folytonossági tétel
68
Az áramlás jellege
átmérő
69
70
71
Laminar and Turbulent Flow:
In Laminar Flow:
In Turbulent Flow:
•Fluid flows in separate layers
•No separate layers
•No mass mixing between fluid layers
•Continuous mass mixing
•Friction mainly between fluid layers •Friction mainly between fluid and pipe
•Reynolds’ Number (RN ) < 2000 walls
•Vmax.= 2Vmean
•Reynolds’ Number (RN ) > 4000
•Vmax.= 1.2 Vmean
Vmean
Vmax
Vmean
Vmax
3. Fluid in Motion
• There are two principal types of flow;
namely laminar and turbulent flow.
• Laminar: highly ordered fluid
motion with smooth streamlines.
• Turbulent: highly disordered fluid
motion characterized by velocity
fluctuations and eddies.
• Transitional: a flow that contains
both laminar and turbulent regions
• Reynolds number is the key
parameter in determining whether
or not a flow is laminar or turbulent:
inertia force
Re 
viscous force
Types of flow
O. Reynolds (1842-1912)
73
The Bernouille’s equation
This equation is the energy conservation of flow of unit
mass. The sum of the potential energy, flow energy and
kinetic energy is constant.
v2
mgh  pV  m  const.
2
pV v 2
pV v 2
gh 

 gh 

 const.
m
2
ρV 2
p v2
hg  
 const.
ρ 2
Without consideration of any losses, two points
on the same streamline satisfy:
p1 v12
p 2 v 22
z 1g    z 2 g  
ρ 2
ρ 2
 J m2 
  2 
 kg s 
74
Daniel Bernouille(1700 - 1782)
It can be rewritten in the form of pressure (SI unit: Pa), and in
the form of head (SI unit: m) as follows:
v12 ρ
v 22 ρ
z1ρg  p1 
 z 2 ρg  p 2 
2
2
 J


Pa
 m 3

Hydrostatic- +static- +dynamic pressure = a constant
p1 v12
p 2 v 22
z1 

 z2 

ρg 2g
ρg 2g
J


m
 N

Elevation- + pressure- + velocity head = a constant
The Bernoulli’s equation states that the sum of the
pressure, velocity, and elevation heads is constant.
75
Applications of Bernouille’s
Equations
• Emptying time from tank
• Pitot and Pitot-static Tube
• Ect.
76
Torricelli’s Discharge (Free jet)
▲ Fig. 5.5
ρ
ρ
p 0  0 2  pgh  p 0  U 2  ρg0
2
2
U 2  2gh ; U  2gh
5.10
5.11
77
Emptying
Time : Δt ?
1 A1 dh
 A1dh  UA 2dt  dt  
2g A 2 h
Δt
he
1 A1 dh
 Δt   dt  

A
h
2g
2 h a0
0
 Δt 
2 A1
g A2

ha  he

5.12
78
Static, Dynamic, and Stagnation
Pressures
The sum of the static, dynamic,
and hydrostatic pressures is
called the total pressure (a
constant along a streamline).
The sum of the static and
dynamic pressures is called the
stagnation pressure,
The fluid velocity at that location
can be calculated from
79
Pitot-static probe
The fluid velocity at that location
can be calculated from
A piezometer measures static pressure.
80
Pitot and Pitot-Static Tube
1) Pitot-Tube
▲ Fig. 5.8
p s  p 

2
U   pt
2
(5.22)
81
2) Pitot-Static Tube
▲ Fig. 5.9
▲ Fig. 5.10
U 
2  pM

(5.23)
82
The Bernouille’s equation
(for inviscid fluids)
This equation is the energy conservation of flow of fluids.
The sum of the potential energy, flow energy and kinetic
energy is constant.
2
p v
hg  
 const.
ρ 2
83
Without consideration of any losses (for ideal fluids: fluids
haven’t viscosity and density is zero), two points on the same
streamline satisfy:
p1 v12
p 2 v 22  J m 2 
z 1g    z 2 g  
  2 
ρ 2
ρ 2  kg s 
or
v12 ρ
v 22 ρ
z1ρg  p1 
 z 2 ρg  p 2 
2
2
 J


Pa
 m 3

Hydrostatic- +static- +dynamic pressure = a constant
p1 v12
p 2 v 22
z1 

 z2 

ρg 2g
ρg 2g
J


m
 N

Elevation head + pressure head + velocity head = a constant
The Bernoulli’s equation states that the sum of the
pressure, velocity, and elevation heads is constant.
84
Energy losses of Flow of Real Fluids
The real fluids have viscosity, therefore the Bernoulli equation
was extended by Fanning and Darcy with terms of frictional
losses : Δp pressure loss term (e=Δp/ρ energy loss term or
h=Δp/(ρg) head loss term):
v12ρ
v 22ρ
z1ρg  p1 
 z 2ρg  p 2 
 Δp L
2
2
Rearranged, we get a form of pressure loss, energy loss or head loss:
v12  v 22
ρgz1  z 2   p1  p 2 
ρ  Δp L
2
p1  p 2 v12  v 22
gz1  z 2  

 eL
ρ
2
p1  p 2 v12  v 22
z1  z 2 

 hL
ρg
2g
1
2
3
85
”The life is too short to drink bad wine.”
Flow of Fluids in Pipe
Dr. Lajos Gulyás, Ph.D.
college professor
86
Incompressible Flow
87
The Bernoulli’s equation
Bernouille’s law says the energy conservation of flowing ideal
fluid. The sum of the potential energy, flow energy and kinetic
energy is constant.
p1 v12
p 2 v 22  J m 2 
z 1g    z 2 g  
  2 
ρ 2
ρ 2  kg s 
Hydrostatic- +static- +dynamic pressure = a constant
v12 ρ
v 22 ρ
z1ρg  p1 
 z 2 ρg  p 2 
2
2
 J


Pa
 m 3

Elevation- + pressure- + velocity head = a constant
p1 v12
p 2 v 22  J  m 
z1 

 z2 



ρg 2g
ρg 2g  N
Daniel Bernoulli(1700 - 1782)
The Bernoulli’s equation states that the sum of the pressure,
velocity, and elevation heads is constant.
88
Extended Bernoulli’s Equations
• Up to this point we only considered ideal fluid where there is no loss
due to friction or any other factors. In reality, because fluids are
viscous, energy is lost by flowing fluids due to friction which must be
taken into account. The effect of friction shows itself as a pressure
(energy or head) loss. In a pipe with a real fluid flowing, the shear
stress at the wall retards the flow.
• The real fluid has viscosity, therefore the one part of energy of flowing
fluid has lost by frictional losses. The Bernoulli’s equation was
extended by Fanning and Darcy with terms of frictional losses : ΔpL
pressure loss term (eL=Δp/ρ energy loss term or hL=Δp/(ρg) head loss
term):
v 2ρ
v 2ρ
z1ρg  p1 
1
2
 z 2ρg  p 2 
2
2
 Δp L
p1 v12
p 2 v 22
z1g  
 z 2g 

 eL
ρ
2
ρ
2
p1 v12
p 2 v 22
z1 

 z2 

 hL
ρg 2g
ρg 2g
89
Determination of Pressure Loss in a Tube with
Dimensional Analysis
•
Goals: determination of friction losses of flowing fluids in pipes or ducts, and
of pumping power requirement.
•
Many important chemical engineering problems cannot be solved completely
by theoretical methods. For example, the pressure loss from friction losses in a
long, round, straight, smooth pipe. It is found, as a result of experiment, that
the pressure difference (∆p) between two end of a pipe in which is flowing is a
function of the following variables: pipe diameter d, pipe length L , fluid
velocity v, fluid density ρ, and fluid viscosity μ.
•
If a theoretical equation for this problem exist, it can be written in the general
form. Let the independent variable the pressure drop per unit length. In this
case the relationship may be written as:
pL
 f D, v,ρ, 
L
90
Determination of Pressure Loss in a Tube with Dimensional
Analysis
•
The form of the function is unknown, but since any function can be expanded
as a power series, the function can be regarded as the sum of a number of
terms each consisting of products of powers of the variables. Base on the rule
of dimensional analysis, the simplest form of relations will be where the
function consists simply of a single term, when:
pL
 constD a v b  c  d
L
•
•
•
The requirement of dimensional consistency is that the combined term on the
right hand side will have the same dimensions as that on the left, i.e. it must
have the dimensions of pressure per length.
Each of the variables in equation can be expressed in terms of mass, length,
and time. Thus dimensionally: Δp / L = ML-2T-2, d=L, v=LT-1, ρ=ML-3,
μ=ML-1T-1
b
c
d
i.e.
ML 2T  2  const.  La LT 1 ML3 ML1T 1







ML2T 2  const.  La LbT b M c L3c M d L dT  d

91
Determination of Pressure Loss in a Tube with Dimensional
Analysis



ML2T 2  const.  La LbT b M c L3c M d L dT  d
•

The conditions of dimensional consistency must be met for each of the
fundamentals of M, L, and T and the indices of each of these variables can be
equated. Thus:
M
1=c+d
L
-2= a+b-3c-d
T
-2 = -b-d
Thus three equations and four unknowns result and the equations may be
solved in terms of any one unknown. The problem is now deciding which
index not to solve. The best way is to use experience gained from doing
problems. Viscosity is the quantity that causes viscous friction so to identify.
We will resolve a, b, c in term of d:
c = 1-d (from equation M)
b = 2-d (from equation T)
92
Determination of Pressure Loss in a Tube with Dimensional
Analysis
•
Substituting in the equation L: -2= a+b-3c-d
-2= a+(2-d)-3(1-d)-d
i.e. -2= a+2-d-3+3d-d
i.e. -1= a+d
i.e. a = -1-d, (b = 2-d, and c = 1-d)
Thus, substituting into equation
p L
 const D 1d v 2d ρ1d μ d  const D 1D d v 2 v d ρ1 ρ d μ d
L
Δp L
L  vD ρ 



const
ρv2
D  μ 
•
d
 const
Let the const. = A/2, then we get
L d
Re
D
 Dvρ 

Δp L  A 
μ


A L v 2ρ
L ρv 2
L ρv 2
ρv 2
Δp L  d
 fD
 4f F
 Kf
Re D 2
D 2
D 2
2
d
L v 2ρ
A L v 2ρ
 d
D 2
Re D 2
f F  f F (Re)
f D  f D (Re)
93
The Friction Factor
w is not conveniently determined so the
dimensionless friction factor is introduced into
the equations.
f 
w
V
2
2
wall shear stress

density  velocity head
Friction Factor
The resulting pressure (energy and head) losses are
usually computed through the use of modified Fanning’s
Fk
friction factors:
f 
v2
Sρ
2
where Fk is the characteristic force, S is the friction surface
area. This equation is general and it can be used for all flow
processes.
p  p  D π
2
Used for a pipe:
f
1
2
Fk
4  p1  p 2 D  Δp D

2
v
v2
2Lρ v 2
L 2ρ v 2
Sρ
(Dπ L)ρ
2
2
where Fk is the press force,
S is the area of curved
surface. Rearranged, we get
a form of pressure loss:
L v 2ρ
L v 2ρ
v 2ρ
Δp L  4f F
 fD
 Kf
D 2
D 2
2
Kf=Loss coefficient
95
Friction Factor
• The friction factors were determined with dimensional
analysis for a smooth pipe
laminar
turbulent
turbulent
16
fF 
Re
f F  0.0791Re -1/4


Re  2100
4000  Re  105
1
 1.7372 ln Re f F  0.3946 4000  Re  107
fF
96
Friction Factor
Turbulent Flow
For turbulent flow f = f( Re , k/D ) where k is the roughness of
the pipe wall.
Material
Roughness, k
inches
Cast Iron
0.01
Galvanized Steel
0.006
Commercial Steel
Wrought Iron
0.0018
Drawn Tubing
0.00006
Note, roughness is not dimensionless. Here, the roughness is reported
in inches. MSH gives values in feet or in meters.
Friction Factor
Turbulent Flow
As and alternative to Moody Chart use Churchill’s correlation:
 8 

1
f  2   
32
 A  B  
 Re 
12
1 12



1

A  2.457 ln 
0.9





7
Re

0
.
27

D



 37530 
B

Re


16
16
Friction Factor
Turbulent Flow
A less accurate but sometimes useful correlation for estimates
is the Colebrook equation. It is independent of velocity or flow
rate, instead depending on a combined dimensionless quantity
Re
f.
 k D 1.255
1
 4 log 

f
 3.7 Re f



Fanning’s Friction Factors
100
101
Basic Equation of Laminar Flow
Pressure loss of laminar pipe flow
•
•
The pressure loss of laminar flow can be given theoretically by the basic equation of
laminar flow. In reality, because fluids are viscous, energy is lost by flowing fluids due
to friction which must be taken into account. The effect of friction shows itself as a
pressure (energy or head) loss. In a pipe with a real fluid flowing, the shear stress at the
wall and the shear stress between the layers of fluid retard the flow.
The shear stress will vary with velocity of flow and hence with Reynolds number. Many
experiments have been done with various fluids measuring the pressure loss at various
Reynolds numbers.
102
Basic Equation of Laminar Flow
Pressure loss of laminar pipe flow
•
Figure below shows a typical velocity distribution in a laminar pipe flow. It
can be seen the velocity increases from zero at the wall to a maximum in the
mainstream of the flow.
dA=2r π dr
Friction force of wall on fluid
•
In laminar flow the paths of individual particles of fluid do not cross, so the
flow may be considered as a series of concentric cylinders sliding over each
other – rather like the cylinders of a collapsible pocket telescope.
103
Pressure Drop and Head Loss
Let’s consider a cylinder of fluid with a length L, radius r, flowing steadily in
the center of pipe. Assume the elements isolated as a free body. Let the fluid
pressure on the upstream and downstream face of the cylinder be p1 and p2
respectively. Shear stress τ=F/S, and shearing force F=τS=τ2πrL. The fluid is
in equilibrium, shearing forces equal the pressure forces. (S=curved surface)
dA=2r π dr
Friction force of wall on fluid
2
2




τ2πrL

p

p
A

p

p

r


ΔPπr
2
1
2
1
Shearing force = Pressure force

P r
L 2
Taking the direction of measurement r (measured from the center of pipe),
rather than the use of y (measured from the pipe wall), the above equation can
be written as;
dv
τ  μ r
dr
104
Basic Equation of Laminar Flow
Pressure loss of laminar pipe flow
ΔP r
dv
 μ
L 2
dr
dv
ΔP r

dr
L 2μ
In an integral form this gives an expression for velocity,
with the values of r = 0 (at the pipe center) to
r = R (at the pipe wall)
r
r
ΔP 1
0 dv   L 2μ R rdr
vr  

Δp
R2  r2
4μ L

dA=2r π dr
Friction force of wall on fluid
Shear stress and velocity distribution in pipe for laminar flow
Where P = change in pressure (pressure lost), L = length of pipe, R = pipe radius
r = distance measured from the center of pipe.
The maximum velocity is at the center of the pipe, i.e. when r = 0.
v max
R 2 ΔP D2 ΔP


4μ L 16 L
105
Pressure Drop and Head Loss
The volume flow rate is
the Hagen-Poiseuille
equation which is given
by the following;


2
2
  vr dA  vr   2r π dr  R  r ΔP 2r π dr  2 π ΔP R 2  r 2 r dr
V
0
0
0 4 μ L
4Lμ 0
A
R
R

R
R
R

2π Δ P  2
R
rdr

r 3dr 


4Lμ  0
0

π ΔP  2 R 2 R 4  π ΔP R 4 π R 4 ΔP

R

V


2L μ 
2
4  2Lμ 4
8μ L


P  D 4

V
L 128
dA=2r π dr
Friction force of wall on fluid
Shear stress and velocity distribution in pipe for laminar flow
Hagen-Poiseuille equation for the average velocity:
ΔP πD 4

V
L 128μ ΔP D 2
v 

πD 2
A
L 32μ
4
And the fundamental equation of laminar flow for pressure loss:
∆P due to viscous effects represents an irreversible pressure loss,
and it is called pressure loss ∆PL to emphasize that it is a loss.
ΔPLoss 
32μ L v
D2
106
Pressure Drop and Head Loss
It can be shown that the mean velocity is half the maximum velocity,
i.e.
v = vavg=vmax/2
Friction force of wall on fluid
ΔP D 2
v avg
L 32μ 1


2
v max ΔP D
2
L 16μ
v
v  v avg  max
2
Therefore, the average velocity in fully developed laminar pipe flow is one half of the
maximum velocity
107
Basic Equation of Laminar Flow
Pressure loss of laminar pipe flow
•
Figure shows a horizontal pipe with concentric element marked ABCD. Since
the flow is steady, the net force on the element must be zero or
Pressure force = Shearing force
vr  

Δp
R2  r2
4μ L

Friction force of wall on fluid
The forces acting are the normal pressures over the ends and shear force over
the curved sides.
P r
dv

τ  μ r
τ2πrL  p 2  p1 A   p2  p1  r 2  ΔPπr 2
L 2
dr
ΔP r
dv
v
r
 μ
ΔP 1
dv


rdr
L 2
dr
0
L 2μ R
dv
ΔP r

dr
L 2μ
2
2
v max
R ΔP D ΔP


4μ L 16 L
108
Friction losses in pipe fittings
• Some average figures are given in Table for friction losses in various
pipe fittings for turbulent flow of fluid. They are expressed in terms of
the equivalent length of straights pipe with the same resistance and the
number of velocity heads.
109
Calculation of Pumping Power Requirement
The pressure loss is directly calculated from Hagen-Poiseuille’s equation for
laminar flow:
32μ Lv 32μ Lv  2ρ v 
16 L v 2ρ

4
Δp L 

Re D 2
D2
D 2  2ρ v 
When the fluid flows in a duct which is not circle in cross-section then we
have to use the hydraulic diameter, Dh:
Dh  4
Ac
(cross  section area)
4
P
(wetted perimeter)
The pumping power requirement (pump power equation):

L   L eq.  v 2ρ
1 
1 
1  

P  Vp pump  VΔp L  Δp h  Δp pres   V 1  4f
 z 2  z1 ρg  p 2  p1 
η
η
η 
D
 2

Where P is the power (W), V is the volume flow rate (m3/s), Leq is the
equivalent pipe length of fittings, η is the efficiency of the pump.
110
The real fluids have viscosity, therefore the Bernoulli equation
was extended by Fanning and Darcy with terms of frictional
losses : Δp pressure loss term (e=Δp/ρ energy loss term or
h=Δp/(ρg) head loss term):
v12ρ
v 22ρ
z1ρg  p1 
 z 2ρg  p 2 
 Δp L
2
2
Fanning constructed a general relationship, called modified
Fanning’s friction factor, from which the friction losses are
determined directly:
f 
Fk
v2
Sρ
2
where Fk is the characteristic force, S is the friction surface
area. This equation is general and it can be used for all flow
processes.
The resulting pressure (energy and head) losses are usually
computed through the use of modified Fanning’s friction
111
factors.
6. Flow of Fluids in Simple Systems
6.1. Fluid flow in pipes
Goals: determination of friction losses of fluids
in pipes and ducts, and of pumping power
requirement.
6.2. Motion of Particles in Fluids or Flow
around Immersed Objects.
Goals: determination of drag force and of
terminal velocity.
6.3. Mixing of liquids
Goal is the determination of power consumption
of agitators.
112
6.1. Fluid Flow in Pipes
Goals: determination of friction losses of fluids in pipes or ducts, and of
pumping power requirement.
The resulting pressure (energy and head) Δp L  z1  z 2 ρg  p1  p 2
loss
is usually computed through the use of the modified Fanning
friction factor:
D2 π
Used for a pipe:
p1  p 2 
Fk
4  p1  p 2 D  Δp D
f

v2
v2
2Lρ v 2
L 2ρ v 2
Sρ
(Dπ L)ρ
2
2
  v
2
1
f 

 v 22 ρ
2
Fk
v2
Sρ
2
where Fk is the press force, S is the area of curved surface. Rearranged, we
get a form of pressure loss:
L v 2ρ
L v 2ρ
v 2ρ
Δp L  4f
λ
ζ
D 2
D 2
2
The Funning’s friction factor is a function of Reynolds number, f = f(Re):
Re 
vD vDρ

ν
μ
113
Calculation of Pumping Power Requirement
The friction factors were determined with dimensional analysis for a smooth
pipe :
16
f
laminar
turbulent
turbulent
Re  2100
Re
f  0.0791Re -1/4

4000  Re  105

1
 1.7372 ln Re f  0.3946 4000  Re  107
f
The pressure loss is directly calculated from Hagen-Poiseuille’s equation for
laminar flow:
32μ Lv 32μ Lv  2ρ v 
16 L v 2ρ
Δp L 
D
2

D
2

  4
Re D 2
 2ρ
. v
When the fluid flows in a duct which is not circle in cross-section then we
have to use the hydraulic diameter, Dh: D  4 A c  4 (cross  section area)
h
P
(wetted perimeter)
The pumping power requirement (pump power equation):

L   L eq.  v 2ρ
1 
1 
1  

P  Vp pump  VΔp L  Δp h  Δp pres   V 1  4f
 z 2  z1 ρg  p 2  p1 
η
η
η 
D
 2

Where P is the power (Watt), V is the quantity of flow (m3/s), Leq is the
equivalent pipe length of fittings, η is the efficiency of the pump.
114
6.2. Motion of Particles in Fluids.
Flow Around Objects
There are many processes that involve the motion of
particles in fluids, or flow around objects:
• Sedimentation
• Liquid Mixing
• Food Industry
• Oil Reservoirs
Flow around objects
115
Sedimentation
The goal is the determination of drag force for the flow around an immersed
object , and the determination of terminal velocity for sedimentation.
In gravitational field an object reaches terminal velocity when the downward
force of gravity (Archimedesian weight) equals the upward force of drag.
Called the modified friction factor:
f 
Fk
v2
Sρ
2
where Fk =Fd is the drag force, S=Sp is the projected area of the particle, and
f=fd is the drag coefficient.
The drag force is
v 2ρ f
Fd  f dSp
2
Archimedesian weight in gravitational field is
Fnet  Fg  Fb  Vρ p g  Vρ f g  Vρ p  ρ f g
v
Principle of sedimentation
116
Terminal Velocity
At the terminal velocity:
For spherical objects:
v  v if,
v 2 ρ f
Vρ p  ρ f g  f dSp
2
Fd  Fnet
D3 π
D 2 π v 2
ρp  ρf g  f d
ρf
6
4 2
ρp  ρf 1
4
v 
Dg
3
ρf f d
The drag coefficient is a function of the Re-number: Re s 
24
laminar
fd 
Re s
18.5
transition al f d  0.6
Re s
turbulent
f d  0.44
Re s  0.6
v  D pρ f
μ
0.6  Re s  600
600  Re s  200,000
For laminar flow drag coefficient can be calculated directly from Stokes’ law:
Fd  3dπμv
fd 
Fd

v 2 

Sp  ρ f
2



3Dπμv
D 2 π  ρ f v 2

4  2




24μ
24

v  Dρ f
Re s
117