Good morning I welcome you to this session
of fluid machines, we have completed the
discussions on fluid machines, and now we
will switch over to a new topic introduction
to compressible flow. So, at the outset I
must start with the definition of a compressible
flow what is meant by compressible flow. So,
as you know the compressibility is a
property of fluid, and it is characterized
by a parameter known as bulk modulus of
elasticity, and physically the compressibility
the property of the fluid is a measure of
its
change in volume or density with respect to
the pressure.
.
Now, if we look to the definition of elasticity
the characteristic parameter for the
compressibility of a fluid. We will see the
weight is defined the bulk modulus of
elasticity is defined like this it is d p
into V by d V where the V with a cut I use
to
represent the volume to distinguish, it from
the velocity or is equal to rho d p by d rho.
So, therefore, you see the bulk modulus of
elasticity is defined this way. Now a large
value of bulk modulus of elasticity represents
a large change in pressure required to
cause a definite change in volume or density.
.And for fluids whose bulk modulus of elasticity
is very large are usually termed as
incompressible fluid, because a change in
volume or density is very low as compared
to
the change in pressure. Similarly for the
fluids whose bulk modulus of elasticity is
relatively very low; that means, which suffer
a considerable change in volume or density
for a given change in pressure are termed
as compressible fluids.
For an example I can tell you that for water
at atmospheric pressure, for water at
atmospheric pressure for water at atmospheric
pressure the value of e is equal to 2 into
ten to the power six kilonewton per meter,
square as compared to that for air at
atmospheric pressure for air at atmospheric
for air at atmospheric pressure e is equal
to
hundred 1 kilonewton per meter square. So,
you can very well see that water is almost
in
compressible practically, because a such value
of e indicates a very large change in
pressure is required to cause a little change
in volume or density as compared to that at
that of air.
Now, question comes this is the characteristic
property of a fluid, but what is a
compressible flow is it true that compressible
flow means the flow of compressible fluids
whose elasticity or coefficient of bulk modulus
of elasticity is very low, and flow of all
incompressible fluids are incompressible flow
it is not exactly. So, compressible fluids
are defined in this way, that the if the change
in density brought about by the change in
pressure due to the flow is very less those
flows we treat as compressible flow. Now the
concept comes like that even if the fluid
itself is compressible for example, air whose
bulk modulus of elasticity is very low.
If it flows in such conditions that the pressure
differences the maximum value of the
pressure difference due to the flow in such
that, it cannot change the density or volume
in
the flow very much, then the flow can be treated
as incompressible. So, therefore, a flow
is whether incompressible or compressible
depends upon whether the change in volume
or density encountered in the flow is small
or large. So, therefore, it is very much tag
with the flow condition, because the change
in pressure is not an arbitrary one.
So, if the change in pressure is very low
in the flow. So, that the change in volume,
and
density is low those flows can be considered
as incompressible. So, to have a criteria
for
an incompressible or compressible flow for
the fluids, we should confine ourself with
these directions we see that a rough order
of magnitude we can find out in this way that
.in any flow of fluid the pressure difference
delta p the order of pressure difference can
be
written as like this, it is in the order of
the dynamic head where V is the velocity of
fluid
we consider any flow through a duct any flow.
We can consider like that the delta p the
maximum pressure difference or the order of
the pressure difference in the flow which
will be encountered will be in the order of
the dynamic head half rho V square is true.
Now, it is very simple manipulation. Now if
we express this delta p in terms of the
coefficient of modulus of elasticity from
this expression. You see that which we can
rho
we can write like, this is e delta rho by
rho instead of delta p is in the order of
half rho V
square or we can write the order of del rho
by rho is in the order of half V square by
e by
rho well. Now after this therefore, I write
it again that order of so, we see that the
order
of del rho by rho is equal to the order of
half V square e by rho.
.
Now, in any flow this in any flow situation
through a duct, it may be through a duct or
it
may be a flow over a body. Now this value
e by rho represents the square of the velocity
of sound in that flow at that condition, where
a is this velocity of sound velocity of sound
velocity of sound or acoustic velocity velocity
of sound or another name is acoustic
velocity acoustic velocity in the fluid at
that particular condition.
So, this is the definition which probably
you know is already derived in classical physics
preliminary physics e by rho a square. So,
therefore, if I use this definition we see
that
the order of the change in density to the
density the ratio of change in density to
the
.instantaneous density or the initial density
whatever you call is half V square by a
square. Now this ratio of V square by a square
this is the ratio of the square of the
velocity of flow to the square of the velocity
of sound in that fluid at that condition.
So,
there is non dimensional number known as mach
number. It is after the scientist mach
who first discovered it mach number or introduced
it is defined by V by a.
So, therefore, mach number is a dimensionless
number, which represents the ratio of the
velocity of fluid at any conditions to the
velocity of sound in the fluid medium at that
condition, this ratio of V by a is known as
mach number. So, therefore, we can write in
terms of a dimensionless number half m a square.
So, we see the change in density as a
fraction of the density itself rho is in the
order of half m a square. So, e power criteria
for
incompressible flow is delta rho by rho is
very less than 1 for incompressible flow for
flows to be for flows to be incompressible,
for flows to be incompressible for flows to
be
incompressible delta rho by rho should very
very less than one. So, therefore, the criteria
is that half m a square should be very very
less than one; that means, the mach number
of
flow should be such that half m a square should
be very very less than one. So, that to
make the delta rho by rho that is change in
density with respect to the density itself
is
very less.
.
Now, to have a definite quantitative criteria,
we set this delta rho by rho as or like this
less than equal to 0.05, which means that
we can neglect a density variation of 5 percent
.of the initial one.So, a change in density
of 5 percent a change in density of 5 percent
or
less than 5 percent can be ignored, and the
flow can be consider to be incompressible
if it
is. So, flow, then a quantitative criteria
can be defined that half m a square should
be
very less than point 0.05 or should be simply
here simply less than 0.05 from which we
can derive that m a should be less than is
equal to point three three.
So, this is the very important conclusion
you have to remember throughout your life
whenever you deal with flows of fluids that,
when the mach number of flow is less than
0.33 the variation in density is 5 percent
that 5 percent of the initial density or below
the
5 percent at 0.33 it becomes 5 percent. So,
mach number is equals or less than 0.33 the
change in density equals to or less than 5
percent of the initial density, and the flow
can
be considered to be incompressible flows are
incompressible.
So, therefore, we see whether a flow, will
be compressible or incompressible will depend
upon this dimensionless parameter mach number
just an example. I am telling that flow
of air, and normal pressure, and temperature
you know that the speed of sound at that
condition at n t p through air is three thirty
meter per second. So, we this criteria we
can
say that the velocity of air at this normal
condition temperature, and pressure if it
is less
than equal to 100 meter per second this is
a thumb rule we tell that the flow of air
is
incompressible; that means, in a situation
where there is a flow of air is 50 meter per
second we can tell the flow is incompressible
flow.
So, in that situation the pressure difference
associated with that flow, that is a flow
of air
at 50 meter per seconds at the atmospheric
condition cannot bring about a change in
volume or density, which is more than 5 percent,
and we can neglect that change in
volume, and and change in density in the flow,
and we can treat the flow to be
incompressible all right. Now, you see another
interesting feature is that we are found
out that del rho by rho del rho by rho is
the order wise is in the order of half V square
divided by e by rho.
Now the bulk modulus of elasticity for incompressible
flows are very large very large;
that means, for all liquids, which are treated
as incompressible fluids, because they are
bulk modulus of elasticity is very large otherwise
the velocity of sound through that
medium is very large. So, usually even for
a very small velocity we get the value of
delta
rho by rho is very high.
.So, therefore, a sorry very low sorry very
low. So, therefore, flows of all incompressible
fluids are usually incompressible, because
even with very high velocity encountered in
practice they cannot bring about a delta rho
by rho more than 5 percent this is, because
of
their very large values in e it is not practicable
theoretically you can consider infinitely
high velocity which can make,, but it is not
practicable.
So, under all practical conditions flow of
all incompressible fluids or flow of liquids
are
incompressible why the reverse is not the
true; that means, flow of compressible fluids
that is the flow of gases may be incompressible
provided its velocity is low, and that is
not in absolute velocity. It is related to
the velocity of sound, and the criteria is
the
dimensionless parameter mach number, that
if the velocity is such that it corresponds
to a
mach number of flow less than 0.33, then the
density change or volume change is lower
than the 5 percent. So, therefore, the flow
can be considered as incompressible flow all
right.
Now, before going to the next chapter we should
recapitulate little bit of
thermodynamics, because the knowledge of thermodynamics,
and the property relations
derived from thermodynamics will be very much
applicable in the directions of
compressible flows. So, first in first 1 or
2 lectures, we will be recapitulating the
basic
laws of thermodynamics first, and second law
thermodynamics, and important property
relations. So, therefore, we must first start
with the first law of thermodynamics. So,
what is first law of thermodynamics first
law of thermodynamics is basically the law
of
conservation energy as you know the first
law of thermodynamics is basically the law
of
conservation of energy.
Now, if we keep aside the physical phenomena
of conversion of mass to energy, and
energy to mass, we can tell that the conservation
of energy is that energy is neither create
nor destroyed that we know since our childhood.
This is the conservation of energy; that
means, if energy is transformed from 1 form
to other or if energy is transferred from
1
system to other system in the same form in
both the cases energy total energy remains
constant it is neither created nor destroyed;
that means, if the energy disappears in 1
form
it appears in other form this is simply the
conservation of the energy as simple as that,
and first law of thermodynamics is nothing,
but synonymous to this principle of
conservation of energy.
.But in the applications of fluid flow, and
classical thermodynamics as applied to
mechanical engineers or other engineering,
disciplines the same principle of conservation
of energy. We look from a view point where
the heat is being converted into work or
work is being converted into heat, because
heat, and work these 2 types of energies are
first described by classical thermodynamics
at the energy in transit energy in transit;
that
means, the energy quantities, which transfer
from 1 system to other system are either in
the form of heat or in the form of work. So,
therefore, we are interested to define or
reshape the conservation of energy while applied
to a system or applied to process where
the heat, and work energies are appearing
as the energies in transit.
.
So, therefore, if you recapitulate these you
know the first law of thermodynamics is
written like that in any cyclic process executed
by a system. So, this d q the cyclic
integral of heat transfer is equal to the
cyclic integral of work transfer, here this
cut I
mean maintain to distinguish this d from the
exact differential, because as you know, this
q is a path function heat flow, and work flow
or work transfer is also a path function.
So,
d cut. Where this d cut q I simply will pronounce
it is a d q is the infinite small heat
transfer d w represents the infinite small
work transfer hence forth.
So, cyclic integral of d q is d w; that means,
in any cyclic processes; that means, if a
system executes in a thermodynamic cycles
executes a processes in a thermodynamic
cycle; that means, in any thermodynamic property
diagram. There will be a close loop
.the total heat transfer during the cycle;
that means, heat may be coming out heat may
be
given in in some processes, there is no restriction
in the direction work may come out in
some processes may go in. So, as the whole
the if we make the accountability of the
energy will see the sum of all the heat transfer
process in a cycle must equal to the sum
of all the work transfer process in a cycle
this is a mere recapitulation of your basic
thing.
So, if we write it in a different that way
d q minus d w is zero; that means, we can
write
cyclic integral d of q minus w is 0 this gives
a very interesting thing that though the q,
and w are the path functions, but their difference
becomes a point functions, because this
cyclic integral of their difference is zero;
that means, if we represent this d q minus
d w
as some d x d cut x is zero, then we can tell
that cut is not required for that, because
perfect differential of any quantity integrated
over a cycle must be zero, that is the basic
definition from mathematics.
So, you see the difference between q, and
w over the cycle is 0 which means that q
minus w can be expressed by a point function
x. So, therefore, we can write that d cut
q
minus d cut w can be expressed as a perfect
differential of a point function x what is
that
x this comes straight from, the mathematical
concept that this minus this over this cyclic
integral is zero; that means, d q minus d
w can be expressed as a change of a point
function where x is a point function, and
this point function, and the property of a
system
you know, that any point function is known
as property of a system property of a system,
and this way the birth of internal energy
comes. So, this is the definition of internal
energy.
So, therefore, we can write d q minus d w
can be expressed as a change of a property
which is a point function known as internal
energy. So, the in physical implication of
this
mathematical statement comes like this that,
if we consider a process from 1 to two, then
this equation implying a infinite small process
can be written like, this if we integrate
this d q from 1 to 2 minus d w from, 1 to
2 is equal to d e from 1 to 2 as you know
this q,
and w are the path functions, and they cannot
be integrated like this.
So, therefore, we have represented this d
with a cut this is not a exact differential.
So,
therefore, this is written as q 1 two; that
means, the heat transfer in this process depends
upon the path of the process does not depend
only the state points, similarly the d w 1
to
.2 to be represented as the work transferred
during the process. 1 2 is usually written
as w
1 2 which depends upon path of the process
whereas, e being a point function which is
the internal energy by definition of the system
it can be written as e 2 minus e one.
So, simply the first law can be written as
q 1 2 is equal to w 1 2 plus e 2 minus e 1
this is
also the conservation of energy applied to
a system, that if we considered the direction
in
this way. That the heat added is positive
simultaneously you will have to take that
work
out as the positive, then we can interpret
this physical the amount of it added to the
system during its change from a state 1 to
state 2 by a process 1 2 is equal to the work
delivered by the system plus, the change in
its internal energy. So, this is precisely
the
first law which is written for a system.
.
Now, this can be again written in a differential
form rather, I will tell this is written in
a d
q is d e in an infinite small process differential
form many people tell you the differential
form, but I will tell for an infinite small
process, because q, and w cannot be expressed
as
any differential these are the path functions;
that means, either in a differential form
will
automatically mean in that case that d e is
the differential of internal energy, but q,
and w
are the infinite small amount of work, and
heat that takes place for an infinite small
process.
So, this is the outcome of the first law of
thermodynamics as applied to a process
executed by a system involving heat, and work
transfer now we come to a definition of a
.property enthalpy. Which is very important
enthalpy enthalpy how do you defined
enthalpy what is the definition of enthalpy
please what is the enthalpy how it is defined
h.
Property.
Good h enthalpy is a property which is defined
as u plus p into V good. So, the very first
line of definition of enthalpy is like, this
the first line of the very first definition
of
enthalpy comes from its mathematical statement,
that h is equal to u plus p V what is u u
is the intermolecular energy. Now before that
I tell you that this is the total internal
energy total internal energy. So, f I write
the internal energy of any system it comprises
several types of energies, that can be stored
in the system internal energy of a system
is
the energy that is stored in the system, and
it is a point of a function it depends upon
the
state of the system. So, it is a point function.
So, therefore, internal energy are those energy,
which can be stored in the system at a
given state. So, it comprises first the intermolecular
energy which is the kinetic energy,
and potential energy of the molecules, which
depends upon the state of this system
precisely the temperature. Similarly this
system itself may have velocities; that means,
the macroscopic particles of the system may
move within the system even, if the system
is a closed system. There may be a substantial
motion of the system that is the different
particles of the system may be in motion.
So, therefore, the kinetic energy is an energy.
Which may be contained by the system,
and another type of energy may be stored or
contained by the system, that is known as
potential energy what is that energy? this
is the energy by virtue of the stay of the
system
or the position of system in a conservative
force field. So, there may, be number of
conservative force fields magnetic force field
electrical force field in which the system
is
exposed the system is placed. If all conservative
force fields are relief the gravitational
force field is there show at least, there
will be gravitational potential energy or
simply
potential energy. So, this kinetic energy
of the particles of this system potential
energy,
and the intermolecular energy are the total
are the contributions are compressing the
total
internal energy.
So, if I write the internal energy general
symbol u is the intermolecular energy. So,
the
kinetic energy of the particles plus the,
let us consider only gravitational force field
that
.the potential energy. Let us write the mass
of the m g z the total potential energy total
kinetic energy m, and the total internal energy.
So, this is the internal energy total
internal energy now in a closed system in
equilibrium the kinetic energies are not
appearing, because the system is at rest the
particles is at rest, and if you neglect the
potential energy not, because of its absolute
value, because you know the absolute value
of potential energy to ascribe the absolute
value of potential energy is very difficult
we
also we always measure it in terms of its
change.
So, if you neglect the change in potential
energy of the systems between different states
we can neglect this m g z the potential energy
part. So, we can tell the internal energy
for
a close system or a stationary system simply
comprises the internal molecular energy.
So, u is the intermolecular energy. So, therefore,
this typical combinations of u p, and V
where p is the pressure, and V is the volume
defines the term enthalpy. Now you see u is
the point function p is the point function
V is a point function. So, therefore, enthalpy
is
a property, and it is a point function another
interesting thing is that the dimension of
enthalpy is the dimension of energy, because
u is the internal energy its dimension is
energy the product of p, and V this dimension
is energy.
So, in enthalpy is a property, and is dimension
is energy. So, it is something similar to
energy. So, very first line of definition
of h does not give by any physical concept,
but
immediately the query comes why such a combination
is defined as a property. So, you
know you start. It many properties first we
start with measurable properties first we
start
with observable properties that, we can see
the mass we can fill the temperature, then
comes with the measurable properties the volume
cannot see, but we can measure
pressure we cannot see, but we can measure.
So, therefore, you see that after wards in
thermodynamics, we mix several combinations
out of this preliminary properties or primary
properties to define other properties, but
why such definition is required, and why such
a particular combination is made. So, that
query is satisfied. If we go little further
to see the physical significance of such a
combination to yield the definition of another
property for example, this enthalpy if we
see the physical significance of this enthalpy
parameter or this enthalpy this property
enthalpy we have to extend our first law to
a to an open system or a steady flow system.
.So, let us do that we consider a steady flow
system or a open system. So, before that I
think I should tell you the system different
types of system. So, how do you define a
system, and there are 2 types of systems 1
is the control mass system another is the
control volume system, how do you define a
system system is a definite quantity of mass
within a fixed boundary; that means, the 2
very interesting characteristic of system
a
definite quantity of mass, and a the content
or separated from the outside by a define
boundary.
So, quantity of mass quantity of mass quantity
of mass quantity of mass, and definite
boundaries definite boundaries are the 2 characteristic
feature of a system. So, system is
defined just like that it is some quantity
of mass. a definite quantity of mass at any
instant, and bounded by definite boundaries.
Now the system basically is divided into 2
ways 2 categories 1 is control mass system
another is control volume system what is a
control mass system in a control mass system
the mass, which is contained within the
system by its system boundaries. So, therefore,
we can see a system is a definite quantity
of mass, and this is the system bounded.
So, if the if the mass identity remains the
same within the system boundary; that means,
there is no flow of mass either in or out;
that means, d m z is 0 from the system
boundary; that means, the same mass not only
the amount, but with the identity remains
the same, then we call the system as the control
mass system which is usually told as
closed system. So, the characteristics of
closed system the additional characteristics
apart
from the definite quantity of mass within
definite boundaries that the identity of the
mass
remains same; that means, in other way there
is no mass flow in or out from the system.
So, if you take some mass flow out, and make
an inflow of the same amount to make the
mass of the system remains same that will
not satisfy the characteristics of a control
mass
system or closed system, because in that case
though the mass remains same, but the
identity of the mass changes; that means,
the same identity has to be there while on
the
other hand the control volume these the system.
Where we allow the mass flow; that
means, the d m not is equal to 0 that may
be mass in there may be mass out there may
be
mass in there may be mass out.
But the restriction is that there is the volume
of the system remains same; that means, the
boundary is fixed fixed boundary. Now you
can ask me sir, then what is the difference
.here the difference for the closed system
that system boundary may move system
boundary is not fixed at any instant there
should be a boundary of the system, but there
is
no restriction that the boundary of the system
has to be fixed; that means, the volume of
the system may change while the mass, and
the identity will remain the same that is
why
its control mass system whereas, in the control
volume system the fix boundary the
boundary will not move, there is no displacement
in the boundary the volume of the
system is controlled in a control mass system
the boundary remain fix the boundary may
move, but control volume system the boundary
remain the volume remain fixed the
boundary will be fixed the volume remains
same.
So, this is known as control volume system,
it is simply known as control volume or
open system. this is known as open system
or simply control volume the system we do
not use. So, we see there are 2 types of system
1 is the closed system another is the open
system or control volume an example, of closed
system is your reciprocating pump we
have seen that 1 of the boundaries that is
the piston which is moving. So, at any instant
the boundary is defined, but instant to instant
1 of the boundaries the piston which is
moving; that means, in a closed system there
is a there may be a displacement of the
boundary say boundary may expand or boundary
may contract. So, that the volume of
the system may change whereas, a control volume
system is a system what the boundary
is rigid; that means, the volume is same,,
but through the boundary the fluid can go
out
or the sorry the mass system mass can go out
can come in. So, this is the control volume
system.
So, now 1 difference is that if you see in
a control volume system if the same amount
of
mass comes in, and the same amount of mass
goes out; that means, the net rate of mass
inflow or mass outflow is zero, then the total
mass remains same. So, in which way it
differs from that of a closed system is that
the mass identity is changed. So, therefore,
at
steady state a control volume system differs
from, that of a closed system is that though
the mass remain same in both cases the identity
of the mass is same in closed system
whereas, the identity of the mass is not same
in the control volume system. So, control
volume system we simply tell us control volume,
and an open system whereas, the
control mass system we call as close system
or simply system when we call system this
is henceforth, you know that system means
is the closed system or control mass system
sometimes these are not used.
.So, system implies the control mass system
or the closed systems, and with all its
characteristics. Similarly a control volume
means a control volume system or another
name is the open system; that means, where
the mass flow out or mass flow in is
applicable or allow; that means, mass flux
across the system may bounded that is
possible, but the restriction is that the
boundary should be rigid; that means, the
boundary should be stiff; that means, there
is volume boundary should not move or
should not displace; that means, the volume
of the system should remains same that is
the control volume or open system. So, if
you see the application of first law to an
open
system, then you will come to the physical
significance of enthalpy I think the time
is up.
So, next class I will discuss any query please.
Thank you.
.
