Welcome in the last two lectures. We reviewed
fundamentals of thermodynamics.
So now let us review fundamentals of fluid
flow in this particular lecture.
So the objectives of this lesson are to present
equations for conservation of mass, conservation
of momentum and discuss Bernoulli equation
and its applications and discuss methods for
evaluating friction pressure drop and minor
losses.
At the end of this lesson you should be able
to state laws of conservation of mass and
momentum. Apply these laws to fluid flow problems
in refrigeration and air conditioning and
state Bernoulli equation and apply it to simple
problems and finally evaluate friction pressure
drops and minor losses using suitable correlations
and design data.
Now why do we have to study about fluid flow
in refrigeration and air conditioning? As
you have probably seen in a in a typical refrigeration
system or air conditioning system we will
be handling a large number of fluids for example
refrigerants air water etcetera. These fluids
will be flowing from one point to the other
that means fluid flow involved in all these
systems. Now these fluid flows are subjected
to certain fundamental laws of fluid mechanics.
So in order to understand or in order to design
refrigeration and air conditioning system
one must know the fundamentals of fluid mechanics
or fluid flow. Okay.
So that is what is mentioned here in refrigeration
and air conditioning systems. Various fluids
such as air water and refrigerants flow through
pipes and ducts and the flow of these fluids
is subjected to certain fundamental laws and
the subject of fluid mechanics deals with
the subjects with these aspects probably we
have studied the fluid mechanics. So it is
not possible to review the entire subject
here but we will be presenting the basic basics
of these things relevant to refrigeration
and air conditioning.
Now fluid flow in general can be compressible
or incompressible. What do we mean by compressible
flow? A compressible flow means the density
of the fluid varies along the flow direction.
That means basically fluid density varies
as the fluid flows. Such flows are known as
compressible flows and in incompressible flow.
The density remains constant it does not vary
in most of the refrigeration and air conditioning
applications. The fluid flows are can be assumed
to be compressible and why do we assume this
because this will simplify the problem considerably.
So most of the times we assume the flow to
be incompressible and the resulting mathematics
will be simpler. However you cannot just like
that apply the assumption of incompressible
flow everywhere. You have to see its validity
the assumption that the flow incompressible
is valid as long as the velocity of fluid
is much smaller than the local sonic velocity.
That means the velocity of the fluid should
be smaller than this velocity of sound or
in other words the Mach number should be less
than point three. And as you know the Mach
number is the ratio of velocity of fluid divided
by the sonic velocity. So when the Mach number
is less than point three you can assume the
flow to be incompressible and apply the suitable
laws.
Now to analyze fluid flow problems one has
to consider three conservative equations.
They are equation for conservation of mass
equation for conservation of momentum and
equation for conservation of energy. A conservation
of energy is nothing but the first law of
thermodynamics which we have discussed in
the review on thermodynamics. So in this pre
present lecture we will be discussing conservation
of mass and conservation of momentum. Now
depending upon the approach you can write
these equations either in integral form or
in differential form. That means if you are
taking control volume and apply these conservative
laws you get integral form of equations or
if you take a differential element inside
the control volume and apply the conservative
laws you get the equation governing equations
in differential form. In the present lesson
I will just give you only the integral forms
of conservation of mass and conservation of
momentum.
Now this equation shows the conservation of
mass. As the name implies conservation of
mass in simple terms is that mass can be neither
created nor destroyed just like energy you
cannot create mass nor you can destroy mass.
So if you apply this conservation of mass
to a control volume then you get a equation
of this form as you can see in the in this
equation you have two terms.
The 
first term here is, what is happening, Okay.
So the first term here I am sorry is the rate
at which the mass of the control volume is
varying with time. That means time rate of
change of mass of the control volume C V stands
here for control volume.
And the second term is nothing but the net
mass flux out of the control volume. So the
conservation of mass when applied to a control
volume gives you two terms, two integral terms
since we are applying integral form. So this
first term accounts for the rate at which
the mass of the control volume is varying
with time and the second term accounts for
the net mass flux out of the control volume.
In this equation rho is the density, t is
the time and V is the velocity vector with
reference to the control volume. Okay. And
when the flow is steady that means when the
flow does not vary with time this term will
not be there. So this equation boils down
to conservation of mass for a steady flow
situation. So these equations are also known
as continuity equations.
So basically for steady flows we will be having
this kind of an equation.
Now if you apply this conservation of mass
to a one dimensional steady flow through a
duct let us say that we take an arbitrarily
shaped duct and let us say that this is the
control volume.
And let the flow be steady and flow be one
dimensional since the flow is steady no mass
can be accumulated within the control volume.
That means the mass inside the control volume
is constant. That means whatever mass is entering
that must leave the control volume. Okay.
That is what is written here the rate at which
mass is entering into the control volume is
equal to the rate at which the mass is leaving
the control volume. And the rate at which
mass is entering the control volume can be
written as a product of density rho one cross
section area A one at the inlet and velocity
V one at the inlet.
Similarly at the outlet you can write the
mass flow rate out of the control volume as
rho two into A two into V two. Also this is
the simple one dimensional steady flow equation
when you apply it to any duct now if the density
is constant.
That means when rho one is rho two is rho
then you get a conservation of mass. If conservation
of mass in a very simple form as A one V one
is equal to A two V two. Okay. Right so far
steady incompressible flow for a one dimensional
situation we get the conservation of mass
equation as A one V one is equal to A two
V two. Now when A one is greater than A two,
then the velocity V one should be less than
V two. This is what is known as a nozzle that
means the velocity increases in the direction
of flow as the cross sectional area reduces
in the direction of flow this is what is known
as nozzle.
And when reverse case, when A one is less
than A two that means cross sectional area
increases in the direction of flow then obviously
according to the conservation of mass equation.
The velocity should reduce in the direction
of flow and this is what is known as a diffuser.
Now let us look at the conservation of linear
momentum this is nothing but a mathematical
expression for Newton's second law applied
to a control volume and the general equation
in integral form is like this 
so here what we have is d P by d t applied
over a control volume this term. This term
is nothing but rate of change of linear momentum
of the control volume and this is equal to
according to Newton's second law is a net
force acting on the control volume this is
nothing but the Newton's second law. And when
you apply this to a control volume we can
write this and introduce two terms.
You have one term here another term here.
This first term accounts for the rate at which
the momentum of the control volume is increasing
with time. So this term accounts for the rate
of change of momentum of the control volume.
And the second term accounts for the net momentum
flux out of the control volume. So see in
simple terms the conservation of linear momentum
says that the net force acting in a system
is equal to sum of the rate at which the momentum
of the control volume is increasing plus net
momentum flux out of the control volume.
This is a vector equation. That means you
can write this equation for different directions.
For example you can have linear momentum equation
for x direction, linear momentum equation
for y direction and z direction. So this can
be written in this form. The net force acting
on the control volume can be split into sigma
F s plus sigma F b where this is the sum total
of all the surface forces and this is sum
total of all body forces acting on the control
volume. So finally the linear momentum equation
boils down to sigma F s plus sigma F b is
equal to momentum rate of momentum change
into the control volume plus net momentum
flux out of the control volume.
Now for steady state again just like conservation
of mass for steady state you do not have the
time terms. So the linear momentum equation
for steady state becomes like this, where
F s stands for the surface forces acting on
the control surface and F b accounts for all
the body forces acting on the control volume.
So the sum of these two forces is equal to
net momentum flux out of the control volume
and what are the typical surface forces the
typical surface forces are pressure forces
exerted by the physical boundary on the control
surface etcetera. Okay. And the most common
body force is the gravity force acting on
the mass inside the control volume.
Now some of the applications of linear momentum
equation are force exerted by the fluid flow
on nozzles bends in a pipe etcetera. That
means you want to find out what is the force
exerted due to fluid flow on nozzles and on
the walls of the pipe and the bends etcetera
then you have to apply the linear momentum
equation. A linear momentum equation is also
applied to find the motion of rockets in the
calculation of motion of rockets and it is
also applied for in water hammers etcetera.
Now let me give an example of the application
of linear momentum equation. This is a very
simple example.
What we have here, okay, let me read the problem.
We have here a straight rigid hundred meter
long pipe through which water flows, okay,
for water flows through this pipe. And it
has a diameter of point two five meters. Okay.
So diameter of the pipe is point two five
meter it is rigid it is horizontal and its
hundred meters long. And the water enters
at certain pressure and it leaves at a pressure
that is equal to atmospheric pressure. That
means here you have P two is equal to one
atmosphere.
So basically the pipe discharges water into
atmospheric pressure and at a given instant
the flow rate is point one met point zero
one meter cube per second that means the flow
rate of water through the pipe at a given
instant is point zero one meter cube per second.
But it increases at a rate of point one meter
cube per Second Square. That means the flow
rate through the pipe is not constant but
it increases with time and the rate at which
the flow rate is increasing is equal to point
one meter cube per second.
And we can assume the water to be incompressible
and we can also neglect frictional effects
and take the density of water as hundred kg
meter cube. And what we have to find out is
what should be the pressure at the inlet.
So that water under these conditions can flow
through this pipe. So first what we do is
we apply the conservation of mass even though
here the flow is not steady. Since we have
a rigid pipe, a rigid horizontal pipe and
the water is incompressible the mass of the
control volume cannot change because the volume
of the control volume is constant. So the
amount of mass within the control volume is
constant at any given instant. So it does
not change its time that means water mass
enters into the control volume must leave
the control volume. Okay.
So if you apply this then you get the conservation
of mass equation as A one V one is equal to
A two V two. That means, and, since the area
is constant at every place if its uniform
cross section pipe the velocity remains constant
throughout the pipe. So this is basically
the conservation of mass. Now let us apply
conservation of linear momentum as I have
already shown conservation of linear momentum
says that the net force acting on the control
volume, this is the control volume, here I
have shown the control volume by dash line.
The net force acting on the control volume
consists of the sum total of all surface forces
plus sum total of all body forces. And this
is equal to the rate at which the momentum
of the control volume is increasing plus net
momentum flux out of the control volume. That
means this term is nothing but at any given
instant what is the difference between the
momentum out of the control volume minus momentum
entering into the control volume. That means
momentum at this point minus momentum at this
point. Now the mass flow rate is constant
at any given instant and the velocity at this
point is same as velocity at this point. So
their nets momentum flux out of the control
volume is zero that means this term is zero
for this particular problem. And since we
are neglecting frictional effects and if a.
since the pipe is also horizontal if there
are no body force also. So this term is also
not there. Then the surface forces in x direction
is simply because of the pressure difference
at the inlet and the outlet that means sigma
F s x is nothing but P one A minus P two A
that is equal to A into P one minus P two
which is equal to phi D square by four into
P one minus P two.
So this is the net force acting on the control
volume in x direction. So this should be equal
to the rate at which the momentum of the control
volume is changing. So the rate at which momentum
of the control volume is changing is this
and here this small v is the momentum per
unit area. Okay, which, I am sorry momentum
per unit area. And this can also be written
as its some kind of velocity and this can
be written as volumetric flow rate divided
by the cross section area. That means V is
equal to Q by A so if you substitute Q by
A for V you get this equation. And if you
integrate it over the entire volume this d
V is the differential volume and the total
volume of the pipe is equal to L into A.
So this equation becomes d by d t Q by A into
rho L A and A L A get cancelled. So ultimately
you have rho L V Q by d t and in the problem
it is given that d Q by d t is point one meter
cube per second and rho is given as hundred
kg per meter cube and length is one meter.
So if we substitute these values here and
then use the equations one to four you get
the answer P one is equal to three hundred
four point seven kilopascal absolute.
That means the pressure here should be three
naught four point seven kilopascal. So that
the flow can take place this is a very simple
application of linear momentum to a pipe flow.
And the same linear momentum principle is
used in many other applications and when we
discuss the fluid flow aspects through refrigerant
pipes and ducts. We will discuss these issues
now similar to linear momentum.
We can also write a conservative equation
for moment of momentum or angular momentum.
This equation is known as moment of momentum
equation. And this states that the net moment
applied to a system is equal to the rate of
change of angular momentum of the system.
This is known as a conservation of angular
momentum. And this is very widely used in
hydraulic machines such as pumps, turbines,
centrifugal compressors etcetera.
Now let us look at one of the very important
equations called Bernoulli's equation. This
equation is applied in a wide variety of fluid
flow related problems and in, it can be derived
either from the momentum equation or from
the first law of thermodynamics. That means
either you can use momentum equation or basically
the Euler's equation or you can also derive
the same equation from first law of thermodynamics
or conservation of energy equation. And this
equation in simple terms relates pressure
velocity and elevation along a streamline
or pressure velocity and elevation between
any two point in the flow field. Now I am
sure that you have studied about streamlines
in fluid mechanics.
Let me just give the definition of streamlines.
Streamlines are the lines drawn through the
flow field in such a manner that the velocity
vector of the fluid at each and every point
on the streamline is tangent to the streamline
at that instant. That means streamlines are
lines drawn through the flow field. If you
take a point on the streamline and draw tangent
that gives the direction of the velocity of
the fluid at that point. So this is the basic
definition of streamline.
Now the concept of streamline is very useful
in describing a flow field in terms of speed
and direction of flow, if want to describe
a flow field as you know that it is a vector
field you have to specify both the magnitude
and direction.
So the concept of streamline is very useful
for such flow field description and it can
be shown from continuity equation that when
the streamlines are very close to each other.
That means speed of flow is high that we or
in other words speed of flow is inversely
proportional to the spacing between streamlines.
Now Bernoulli equation is applicable along
a streamline under the assumptions of steady
flow one dimensional flow incompressible flow
and inviscid flow. So what do you mean by
steady flow? As you know steady flow means
it does not the flow, does not vary with time.
So whenever you have fluid flow situation
which do not vary with time you call that
flow as steady flow. And what do you mean
by one dimensional flow? One dimensional flow
means properties vary only in one dimension.
In the other two dimensions they remain constant.
That means at any cross section the fluid
properties remain constant.
If you are talking about one dimensional flow
in y x direction that means in y and z directions
there are no variations in the fluid properties
at any x. So this is the meaning of one dimensional
and what do you mean by inviscid? Inviscid
means the fluid should not have any viscosity
that means it is basically apply applicable
to ideal fluid. Because all real fluids have
viscosity so the moment you say that it is
applicable to inviscid flow. That means it
is applicable to ideal fluids and finally
incompressible as we have seen means that
the density does not vary along the fluid
flow direction.
So under these assumptions you can apply Bernoulli
equation to any two points on a streamline
and the Bernoulli equation also applicable
to any two points of the flow field when the
flow is irrotational in addition to one dimensional
inviscid steady and incompressible. What it
means is if the flow is irrotational in addition
to steady incompressible and inviscid. Then
you can also apply Bernoulli's equation to
any two points in the flow field is not necessarily
on the same streamline. What do you mean by
irrotational flow? Irrotational flow means
the fluid particles do not undergo net rotation.
So that is what is known as irrotational flow.
Now the Bernoulli equation is given by this
expression you can write it in different forms.
So for example if you write it in the form
of head, that means in the form of pressure
head velocity head static head and total head
you get this kind of a an expression. Here
this term as you can see is known as pressure
head and this term are known as velocity head
and this term is known as static head. And
here P is the static pressure rho is the density
and g is the acceleration due to gravity and
V is the velocity of the fluid and z is the
elevation with reference to a datum.
So basically what the Bernoulli equation written
in terms of the head says that the pressure
head plus velocity head plus static head between
any two points in the flow field or between
any two points in its streamline is constant.
That means the total head is constant.
You can also write this equation in terms
of pressures that means you can write this
in terms of pressures as static pressure P
velocity pressure rho V Square by two and
pressure due to datum.
So Bernoulli equation when written for in
terms of pressure states that, the static
pressure plus velocity pressure plus pressure
due to datum called as total pressure remains
constant under the assumptions stated earlier.
Now between any two points in the flow field
you can write Bernoulli equation in this form,
for example I am writing this between point
one and two. Then the summation of static
head plus velocity head plus, head due to
datum is equal to the static head plus velocity
head plus datum head at point two. And as
I was mentioning for real fluids the flow
is not inviscid. That means viscous effects
are there in real fluid so you can modify
Bernoulli equation and the by including what
is known as a head loss term.
So this is known as modified Bernoulli equation
that means it is nothing but Bernoulli equation
with provision for viscosity by including
head loss term. So in this expression all
these terms are known to you one additional
term is H l which is called as head loss.
So now let me show the a pressure variation
let us apply the Bernoulli equation to a pipe
flow and let's see how the pressures are varying.
Let us say what we have here is a uniform
cross section pipe through which a fluid is
flowing. Okay
And we would like to plot different pressures.
Now let us further to start with, let us assume
that the flow is inviscid that means the fluid
does not have any viscosity. Since the cross
section area is same here. That means area
at any point is same. So if we apply continuity
equation at any point V is constant. So velocity
does not change along the length since velocity
pressure is equal to rho V square by two and
rho is constant here velocity pressure is
also will remain constant because V is constant.
So you if you are plotting pressure different
pressures versus length you find that for
the velocity pressure you get a horizontal
line. Okay. So this is the line for velocity
pressure. And if the fluid is ideal then the
static pressure also remains same. So this
is the static pressure line for the ideal
fluid since the total pressure is summation
of velocity pressure plus static pressure
the total pressure line is also horizontal.
So total pressure at any point is equal to
velocity pressure plus static pressure and
it remains constant throughout the length.
So this is the application of Bernoulli's
not to an ideal fluid now what is the effect
if viscosity is non zero that means if the
fluid is viscid or it has finite viscosity.
For viscous flow the velocity pressure still
remains same because from the continuity equation
velocity at any point should remain same.
Okay. So this horizontal line applies to both
inviscid as well as viscous flows. So velocity
pressure remains horizontal only then what
happens to the static pressure. Static pressure
continuously reduces in the direction of flow
because of the viscous effects that means
there is a loss of static head due to viscosity
so you see the a sloped dash line for viscous
flow. Okay.
So the total pressure is nothing but the summation
of static pressure plus velocity pressure
so and the velocity pressure is remaining
constant and static pressure is reducing.
So the total pressure also reduces in a real
flow and this difference between the total
pressures at the inlet minus total pressure
at the outlet is nothing but your head loss.
So this is the application of Bernoulli equation
to a pipe flow.
Now let us say that we would like to maintain
the same pressure at the inlet and as well
as at the outlet and the flow is inviscid.
That means we want to maintain P one should
be same as P two and similarly static pressure
the inlet should be same as static pressure
at the outlet. But the fluid is viscid. That
means it has viscosity. Then how do we ensure
this? So whenever you have viscous flows and
if you want to have same pressures at the
inlet or outlet or if the inlet pressures
are greater than the outlet pressures then
what we have to do is we have to use some
kind of a pump or a fan for example if i am
using the pump here or a fan here. Okay.
Then it is possible to maintain the same static
pressures and the total pressures at the inlet
and the outlet. What happens when you have,
when you are putting a pump, when you are
keeping a pump or a fan in the pipe flow the
velocity pressure does not change because
it the fan or pump does not increase the mass
flow rate so velocity pressure remains same.
But it increases the static pressure because
it adds energy to the fluid so at the point
where you have added the pump the static pressure
increases.
Okay. Since the starting pressure increases
the velocity pressure also increases. So you
have the modified pressure lines with the
addition of pump or fan inside the fluid circuit.
Now that is what is shown in this line. This
is the modified Bernoulli equation with friction
and a fan or pump. And I have shown here an
arbitrary cross section that means cross section
area and inlet to constant and the fluid enters
here and fluid leaks here. And this is the
fan or a pump so what is the modified Bernoulli
equation for this kind of situation this is
the modified Bernoulli equation only the difference
is this. So this is nothing but the head gain
due to the presence of the fan if p one is
equal to one by rho g is equal to p two by
rho g V one square by two g is equal to V
two square by two g and z one is equal to
z two then these terms get cancelled out.
And you have H p is equal to H loss that means
the head gain due to the fan should be sufficient
to overcome the head loss due to friction.
Now what is the power required to drive the
fan or pump under the case discussed just
now so this equation gives what is the power
requirement. When you are using a fan or a
pump and this is again is derived by using
Bernoulli's equation and what we have here
this m dot is nothing but the mass flow rate
of the fluid. And this is the head loss and
this is the static pressure at the outlet
the static pressure at the inlet V two square
minus V one square is the difference divided
by two is the difference in the velocity pressures
at the outlet and inlet.
And this is the difference in the datum heads
and here eta fan is the efficiency of the
fan that means if the fan has some inefficiency
then we have to take that into account. So
this expression gives you the fan power required
or pumps power required. So you can by applying
Bernoulli's equation you can find out what
should be the motor capacity to drive the
fan or pump. Now let us apply this Bernoulli
equation to a very simple case.
So this is an application of Bernoulli equation
to a Venturi now. What is the Venturi? Venturi
is a device for measuring flow rates basically
normally it is used for measuring air flow
rate or water flow rate. So as you can see
in the figure it consists of a straight portion
then a converging portion and a gradually
diverging portion. So this is a diverging
portion this is the converging and where the
diverging and converging portions meet we
call as that portion as the throat of the
Venturi.
And here this is the inlet to the converging
portion is point one and the throat portion
is point two and Venturi has a manometer here.
And one end of the manometer is connected
to the inlet to the converging section. That
means point one and the other end of the manometer
is converging is connected to the throat portion
that means point two. And this manometer is
filled with some manometric liquid. So basically
the air flow through the manometer and because
the air flow there will be some pressure difference
in the manometer. That means there will be
a manometric head here.
And from the characteristics of the Venturi
and from the by measuring the manometric head
you can calculate the air flow by applying
Bernoulli equation. So now let us look at
the problem statement it is given here that
the area of cross section area A one and A
two are point five meter square and point
four meter square. That means area at this
point one is point five meter square and area
at point two is point four meter square. And
the density of air is one point one two kg
per meter cube and manometric fluid used here
is water and it has a density of thousand
kg per meter cube.
And the measurement shows that the manometric
head is twenty mm and it is also given that
acceleration due to gravity is nine point
eight one meter per Second Square. And we
can neglect all frictional effects. That means
we will be applying the original Bernoulli
equation because we are assuming the fluid
to be incompressible steady irrotational and
inviscid. So frictional effect need not be
considered so we apply a Bernoulli equation
let us see what happens. First what is the,
okay, what is the required output we have
to find out put what the mass flow rate is
for this Venturi which shows a manometric
head of twenty mm?
So the equation for mass flow rate is rho
A V mass flow rate at any point is equal to
rho into cross sectional area at that point
and velocity at that point. So you can write
this expression for mass flow rate. And from
continuity equation nothing but conservation
of mass equation A one V one is equal to A
two V two because we are assuming the fluid
to be incompressible. So rho A at any point
is constant so from this equation you get
that A one V one is equal to A two V two or
velocity at the point two. That means velocity
at the throat can be expressed in terms of
the areas cross sectional areas at the throat
and at the inlet to the converging section
and in and velocity at the inlet to the converging
section.
So V two is equal to A one by A two into V
one. Now let us apply Bernoulli equation if
you are applying the Bernoulli equation you
can slightly write it in the different form
datum head is change in datum head is zero.
So we can write the Bernoulli equation in
this form P one minus P two is equal to rho
A into V two square minus V one square by
two and P one minus P two is nothing but the
pressure difference between the points one
and two which can be obtained from the manometer
reading as rho g h. Where rho, W here is the
density of the water g is acceleration due
to gravity and h is the manometric head.
So if you are use the input data and these
four equations you can you will find that
the mass flow rate of air is equal to eleven
point one eight kg per second so this is one
very useful application of Bernoulli equation
to a Venturi meter.
There are many other applications of Bernoulli
equation as I mentioned it is one of the most
useful equations and you will find its applications
particularly in air conditioning duct design
and in refrigeration piping design etcetera
will be using this equation repeatedly. But
you must keep certain things in mind even
though Bernoulli equation is very useful you
must be aware of the assumptions under which
the Bernoulli equation is valid.
For example Bernoulli equation is not valid
when you have a duct where there is a sudden
expansion when there is when there is a sudden
expansion what happens is the fluid gets separated
that means there will be separation of fluid.
And the flow no longer remains one dimensional
so you cannot apply Bernoulli equation because
you remember that Bernoulli equation is a
scalar equation and it's applicable to one
dimensional flow. So whenever you have two
dimensional three dimensional effects you
cannot apply Bernoulli equation number one
and second thing is if you have a fluid flow
where there is lot of heat transfer as the
fluid flows through the duct then also you
cannot apply Bernoulli equation. Then because
what happens when there is the large amount
of heat transfer then the density may not
remain constant as the fluid is flowing through
the conduit.
That means the assumption of incompressible
flow will not be valid so you cannot again
apply the simple Bernoulli equation that we
have discussed here. So when you are applying
Bernoulli equation first make sure that the
assumptions or the conditions are met and
then apply the Bernoulli equation. Of course
Bernoulli equation are also have been developed
for unsteady flows and also for compressible
flows but we will be not discussing those
issues these will be discussed in applied
fluid mechanic courses.
Now let us look at the evaluation of pressure
loss during fluid flow. We have seen that
generally in one of the most common problems
or common requirements in the design of an
air conditioning that or in the design of
a refrigerant piping or in a chill water piping
is to find the capacity of the motor or capacity
of the pump. What should be the power consumption
or what is the power consumption of the pump.
So that you can select a suitable motor for
the pump or the fan. So you have seen from
the modified Bernoulli equation in the presence
of viscous effects and the fan that the pump
power depends upon the mass flow rate and
it depends upon the inlet and outlet pressures
velocities and datum heads and the friction
losses that means the head loss.
So if you want to find out what is the power
requirement you must know what is the head
loss okay, and normally in any in any given
problem the mass flow rate is an input and
the pressures and the velocities also in are
also inputs.
So if you know the head loss then you can
find out the power input. So the key problem
here is to find out what is the pressure drop
as the fluid flows through a conduit. Now
the loss in pressure is due to fluid friction
and turbulence change in fluid flow cross
sectional area and abrupt change in fluid
flow direction. That means the fluid undergoes
pressure loss due to these three effects friction
and turbulence change in fluid flow cross
sectional area and abrupt change in fluid
flow direction. A pressure loss due to friction
and turbulence is known as frictional pressure
drop.
And pressure drop due to change in cross sectional
area and abrupt change in fluid flow direction
is known as minor loss. That means frictional
losses all losses due to friction and turbulence
are known as frictional pressure drops and
losses due to change in cross sectional area
and change in direction are known as minor
losses. Sometimes name minor losses could
be a misnomer because in some situations the
minor losses can be must more than the frictional
pressure drops. So you have to keep it in
mind and you should not neglect that the minor
loss is thinking that they are minor.
Now the frictional pressure drop the frictional
pressure drop is given by Darcy Weisbach equation
for internal flows and it is given as delta
p f is equal to f L by D into rho V square
by two this is the frictional pressure drop.
And here L is the length of the pipe or tube
or the duct and D is the internal diameter
if it is circular pipe or tube or by an equivalent
hydraulic diameter. If the cross section is
non circular and rho as you know is the density
and V is the velocity. So if you know the
density velocity length diameter and this
factor then you can find out the frictional
pressure drops.
Now what is the factor f? Factor f is known
as Darcy friction factor it's a non dimensional
factor and it depends upon Reynolds number
and relative roughness of the internal surface.
Now what is Reynolds number? I am sure that
again you studied about Reynolds number in
your course and fluids mechanics.
But let me just give it for the sake of completion
Reynolds number is a dimensionless number
that quantitatively relates the viscous and
inertial forces and whose value determines
the transition from laminar to turbulent flow.
So basically you have to keep it in mind that
this is the dimensionless number and it relates
viscous and inertial forces and its value
determines whether the flow is laminar or
turbulent we will see litter later what is
a mean by laminar or turbulent flow. And mathematically
Reynolds number is expressed as R e is equal
to rho V D by mu where rho is the density
V is the fluid velocity and D is the length
scale parameter if it is a tube or circular
tube or pipe D is the diameter.
If it is a non circular tube or pipe D can
be could be a hydraulic diameter and if it
is a external flow that means the flow is
taking place over a flat plate something like
that then D can be the length and mu here
is the dynamic viscosity. So Reynolds number
is nothing but rho V D by mu and for internal
flows it is observed that the flow will be
laminar if Reynolds number is less than two
thousand three hundred and the flow becomes
turbulent if the Reynolds number increases
beyond two thousand three hundred.
Now this is only a very rough criteria strictly
speaking you really do not have a sudden transition
from laminar flow to turbulent flow. That
means basically we will see, I will show you
the boundary layer development generally you
have at low Reynolds number you have laminar
flow and at a certain critical Reynolds number
there will be transition. So that means flow
turn changes from laminar to a transition
flow and the transition flow continues to
a certain extent and beyond certain Reynolds
number the flow becomes turbulent. That means
you have laminar flow transition regime and
turbulent regime. But where most of the practical
purposes and most of the engineering calculations
we assume that when the Reynolds number particularly
for internal flows if it is less than two
thousand three hundred we take that the flow
is laminar and if it exceeds two thousand
three hundred we assume that the flow is turbulent.
But in advance the heat transfer and fluid
mechanics problem the transition region is
also concerned but as far as our refrigeration
and air conditioning calculations are concerned
you may take the transition Reynolds number
or critical Reynolds number as two thousand
three hundred for internal flows.
One more thing is that it is possible by carefully
controlling the fluid flow conditions and
all to extend the critical Reynolds number.
That means you can have laminar flow even
at Reynolds number as high as say ten thousand
or twenty thousand under carefully controlled
conditions.
So this critical Reynolds number and the number
two thousand three hundred is a rough guide
line it is not a fixed number or any thing.
So we have, I have mentioned that the Darcy
friction factor is a function of Reynolds
number and relative roughness of the internal
surface.
So let us look at few cases for fully developed
laminar flow. The Darcy friction factor is
given by sixty-four divided by the Reynolds
number. That means for fully developed laminar
flow the friction factor is independent of
the roughness of the surface if it is only
a function of the Reynolds number. And for
fully developed turbulent flow the correlation
given here is known as Colebrook and White
correlation is Colebrook and White correlation
and it relates the friction factor with the
roughness and the Reynolds number.
This roughness has same dimensions as the
length scale or that means the diameter. So
this equation is very popular equation and
it is known as Colebrook and White equation
for the fully developed turbulent flow. One
thing you can notice here is that to solve,
to use this equation you have to use an iterative
procedure because f occurs both on the left
hand side as well as on the right hand side.
So you have to use the trial and aware iterative
method to find out f.
Now ASHRAE suggests another correlation very
useful correlation and explicit correlation
first ASHRAE suggests ASHRAE defines a parameter
f one f one is given as point one into k s
by D plus point six eight divided by Reynolds
number. So this factor is defined first and
if this factor is less than point zero one
eight then f one is equal to friction factor
f. And if the factor is less than point zero
one eight then friction factor is given by
this equation. That means first you have to
find out the factor f one from the roughness
and from the Reynolds number and see whether
the factor is greater than or equal to point
zero one eight or it is less than point zero
one eight.
If it is greater than or equal to point zero
one eight then f one is nothing but f, if
it is less than point zero one eight you have
to make a correction and f is equal to point
eight five f one plus point zero zero two
eight so this is the very useful equation
suggested by ASHRAE.
Now I was talking about laminar flow and turbulent
flow.
Let me just explain laminar and turbulent
flow and what you mean by fully developed
flow?
Let us say that we have a flat plate, a flat
plate over which fluid is flowing. So at this
point you have what you known as a free stream
and this fluid comes in contact with a solid
surface at this point let us say at this point
x is zero. So at this point the fluid comes
in contact with the solid once the fluid comes
in contact with solid what happens.
Let us consider real fluid. That means let
us say that the fluids had finite viscosity
as you know viscosity is one of the transport
properties of the fluid and when a real fluid
comes in contact with a let us say a stationary
surface. Then the velocity of the fluid adjacent
to the stationery surface will be same as
that of the stationery surface. That means
the velocity next to the solid surface will
be same as the solid surface temperature solid
surface velocity. And if the solid surface
is stationery solid surface velocity is zero.
That means the fluid layer adjacent to the
solid surface will be having a zero velocity
that means velocity at this point will be
zero okay. So suddenly the velocity changes
from the free stream velocity to zero velocity
on the surface.
Once the velocity is zero you have to satisfy
the continuity equation. That means the velocity
profile will develop the velocity profile
will develop because of viscous effects. Since
the velocity here is zero the fluid layer
on the surface will try to hold the fluid
layer adjacent to it.
That means it will try to decelerate the fluid
that is adjacent to the layer and the, that
layer will try to decelerate the fluid that
is flowing adjacent to it so like that a velocity
gradient develops.
And that is what is shown here so gradually
velocity gradient develops and at a certain
point the velocity will be same as the free
stream velocity and again let us say this
is the free stream velocity.
This is free stream on the plate and this
is the free stream velocity on the plate let
us say that U infinity on the plate okay.
So what basically has happened is because
of the presence of the solid surface a velocity
gradient has developed in this region. And
the region in which the velocity varies from
zero to the free stream velocity is known
as the boundary layer that means this is known
as the boundary layer. So because of the presence
of a surface and because of viscous effects
a boundary layer develops whenever a fluid
comes in contact with a solid surface okay.
Now initially this boundary layer will be
laminar. So what do you mean by laminar? Laminar
flow means the fluid particles will be flowing
in layers or in lamina and any mixing is only
due to molecular motion. That means in laminar
flow mixing is mixing between fluids layers
are purely because of molecular motion and
there is no bulk mixing such a flow is called
as laminar flow.
And for laminar flow appear very smooth and
it is steady and a typical example of laminar
flow is the flow of let us say thick liquid
like honey from a bottle that means you take
a honey bottle and pour it then the honey
comes out in a laminar manner okay. And laminar
flow is generally encountered when the Reynolds
number is small so at low Reynolds number
you generally have laminar flow. So in this
particular case to start with we have laminar
region and for this particular case we define
Reynolds number in terms of the length scale
that means the local Reynolds number is defined
as divided by U okay. So here this length
scale is x or the distance from the leading
edge so as the distance from the leading edge
increases Reynolds number increases. That
means Reynolds number increases in this direction.
So to start with we have a laminar region
where the Reynolds number is small and as
I said that certain critical Reynolds number
the flow changes from laminar to transition.
So you have a transition regime here. So transition
regime continues to certain extent and again
at certain point flow transition takes place
to turbulent region. So you have turbulent
flow. So you to start with you have laminar
region you have transition region and turbulent
region. What happens in a turbulent region?
In a turbulent region you have molecular motion
and super imposed on that molecular motion
is a bulk mixing of fluid particles because
of eddies. That means in turbulent flow eddies
form and these eddies lead to mixing on a
bulk scale. That means on a microscopic scale.
So you have mixing due to molecular motion
as well as mixing due to eddies so this is
the characteristics of a turbulent flow. And
the unlike laminar flow turbulent flow is
highly unsteady and a typical example of turbulent
flow is let us say that you have a water tap
and you have opened the tap fully then water
comes out in very random irregular manner
and that is an example of turbulent flow.
And you slowly close the water tap then water
becomes more and more regular and at very
low flow rates the water becomes flow becomes
very smooth that means you have a laminar
region.
So this is how you can change the laminar
region to turbulent region by changing the
velocity in this case or in Reynolds number.
So this is the in short the physics behind
laminar and turbulent flows and as I said
the value of Reynolds number decides whether
the flow is laminar or turbulent and for internal
flows I mentioned that if the Reynolds number
is less than two thousand three hundred you
have laminar flow. And if it is greater than
two thousand three hundred you may consider
that as turbulent flow this is only the guide
line and for fluid flow over flat plate that
means the example that i have shown just now.
Or the critical Reynolds number is totally
different so in for flow over horizontal plate
the flow will be laminar as long as the Reynolds
number is less than five into ten to the power
of five this is again a guide line.
So as long as Reynolds number is less than
five into ten to the power of five we assume
that the flow is to be laminar and if exceeds
if it exceeds five into ten to the power of
five we assume the flow to be turbulent.
So now let us look at the determination of
minor losses as I mentioned minor losses are
due to change in cross section and also due
to change in the direction normally when the
cross section increases gradually or reduces
gradually there will not be any significant
losses. But when the cross section increases
steeply then there will be boundary layer
separation and there will be losses. So these
losses are known as minor losses in addition
to that whenever there is a change in flow
direction basically an abrupt change again
if the change takes place gradually then the
losses are negligible.
But if the change is abrupt for example you
have the sudden elbow then there is a sudden
change in the direction then you will have
again pressure losses. So the losses due to
change in direction and change in cross section
area are clubbed together and they are known
as minor losses. So generally the minor losses
are given as delta p m this is a sum total
of the minor losses and that is equal to K
a factor K into rho V square by two that means
the minor losses are always proportional to
the velocity pressure term okay. And this
factor K has to be determined experimentally
and little bit different for different types
of valves joints fittings bends etcetera.
And generally the values of K for different
fittings valves etcetera are available in
standard literature so what we have do if
we want to find out what is the minor loss.
For example you have a bend then you would
like to find out what is the pressure loss
due to bend. Then first you have to find out
what is the velocity pressure that means you
have to find out the velocity and you have
to find out the term rho m square by two and
multiply that into the factor K. So this factor
K has to be obtained from a published data
for that particular bend or valve or fitting
so this is generally the procedure for calculating
minor losses. So finally the total head loss
is nothing but the frictional loss plus minor
loss. So we have to find out these two from
that you have to find out the power required
for a fan or pump which is the typical problem
in any fluid flow related problems in refrigeration
and air conditioning.
Now let us conclude today's lecture.
In today's lecture we have reviewed the fundamentals
of fluids flow relevant to refrigeration and
air conditioning and we have presented mathematical
equations for conservation of mass and momentum.
Conservation of mass is also known as equation
for conservation of mass is also known as
continuity equation both is same. And we have
also presented and discussed Bernoulli equation
in its original form and also modified Bernoulli
equation for viscous flows and when fan or
pump is used in the fluid flow conduit. Then
we have presented methods for evaluating friction
losses and minor losses with typical correlations
for friction coefficients.
Let me mention here that the correlation shown
here for friction factor are typical correlations.
That means there are a large number of correlations
but different fluid flow situations and these
correlations are applicable generally they
are empirical. So they are applicable for
only that particular condition that means
every correlation has a range of application.
So when you are applying empirical correlations
you must see what its range of applicability
is and make sure that your problem falls into
that range of applicability only then apply
the correlation. And there are large numbers
of correlations available for friction factors
for different types of flows. So depending
up on the particular situation we have to
select a suitable correlation then find the
friction factor. Similarly the minor losses
we have to find out the factor K depending
up on the specific problem and from these
two we had finds out the total head loss.
So these aspects have been discussed here
and we will be applying these to refrigeration
and air conditioning in a later chapter okay.
Thank you.
