LESSON 25:
COMPRESSIBLE FLUID FLOW -STAGNATION PROPERTIES
STAGNATION PRESSURE, TEMPERATURE AND DENSITY
(P/Pk ) =(P1/P1k)=(P2/P2k) = Constant
FLUID MECHANICS
Compressible Fluid Flow
(Thermodynamic Relation) (P/wk) =(P1/w1k) = (p2/w2k) = Constant (ii)
In the above equation p is pressure, v is specific volume, p is
1. Introduction
mass density, w is weight density and K is adiabatic index. The
Hello Students today we are going to start the last and a very
adiabatic index is given by k = (Cp/Cv), the ratio of the
important topic or so to say last unit of fluid mechanics. This
specific heat at constant pressure Cp and the specific heat at
that we will discuss have wide practical ramifications from flow
constant volume Cv for air; and other diatomite gasses in the
of water through pipes to the flow of wind in atmosphere .By
usual ranges of temperature and pressure k = 1.4.
Having a good under standing of the compressible nature of
the fluid (i.e., liquid or gases) have helps in understanding The process is said to be reversible if the gas and its
various natural phenomena’s from the change in the properties surroundings could subsequently be completely restored to
to shockwaves their initial conditions by adding to (or extracting from) the gas
exactly the same amount of heat and work taken from (or
In general all the fluids are compressible, but in any fluid flow
added to) it during the process. A frictionless adiabatic process
system if the density of the fluid does not change appreciably.
is a reversible process and it is often termed as isentropic process
Then the fluid may be treated as incompressible. However,
because as indicated later in this process the entopic does not
when the density changes are appreciable, the compressibility of
change. However, a reversible process is ideal, i.e., it is nervier
the fluid needs to be taken into account in the analysis of such
achieved in practice.
fluid flow phenomena Significant density changes may usually
be produced in a gas if the velocity (either of the gas itself or of There are some actual processes which are not isentropic (so that
a body moving through it) approaches or exceeds the speed of pvk is not constant) b but in which the relation between p and
propagation of sound through the gas, if the gas, if the gas is v or ρ is given to reasonable approximation by pvn = (or p/
subjected to sudden acceleration: or if there are very large pn) constant (or p/pn= constant ) where n is a Such processes
changes in elevation. The compressibility is also significant in are termed as polytropic processes.
certain problems involve the variations of density of highly ii.Equation of state: The density p of a particular a gas is
compressible fluids such as gases due to the changes of related to its absolute pressure p and
pressure and velocity. P= ρRT; or PV=mRT
The density of a gas is related to both the pressure to both the In which R is gas constant and v is the volume occupied by the
pressure and certain laws of thermodynamics govern the mass m of the gas the absolute (or thermodynamic)
temperature and all the changes of density. As such in the next temperature is temperature is expressed in Kelvin (k)
paragraph some of the basic relationships of thermodynamics
When the temperature is measured infds0C and it is given by
are briefly reviewed
T0 (abs)= TK =273.15+t0
2. Basic Relationships Of Thermodyanmics
iii. Internal Energy: The energy posses by the molecules of a
i. Isothermal and Adiabatic Processes: When the physical fluid due to molecular activity is termed as internal energy of
properties of a gas such as pressure, density and temperature are microscopic energy it is to be distinguished from the external or
changed due to compression or expansion of the gas, it is said macroscopic energies vexes, Kinetic and potential energies
to undergo process. A gas may be compressed or expanded associated with the fluid mass the internal energy is a property
either by isothermal process or by adiabatic process. An of the fluid which cannot be measured directly but manifests in
isothermal process is that in which the temperature is held self in terms of quantity of heat is supplied to a certain mass
constant and it is governed by Boyle’s law according to which of gas , a part of it may be stored in the gas as internal energy
PV = P1 v1 = P2 V2 = Constant thus producing the rise din temperature of the gas and the
Or remaining part of the heat supplied may be utilized in
(P/P)= (P1/P1) =(P2/P2) = Constant increasing the volume of the gas thus doing some external
work.
Or
vi. First law of thermodynamics. The first law of
(P/w) =(P1/w1) =(P2/ w2) =(P2/ w2) = Constant (i) thermodynamics states that the total heat H supplied to a gas
On the other hand if during the course of a process, the gas must be equal to the increase in the internal energy
neither absorbs heat from, nor gives heat from, nor gives heat (I2-I1) of the gas plus the external work W done by the gas in
to, its surroundings then the process is said to be adiabatic an expanding. Thus
adiabatic process is governed by the following law:
Pvk =P1 v1k = P2 v2k = Constant
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H = (I 2-I 1)+W (iii) absolute temperature of the gas at that instant, then the change
FLUID MECHANICS
However if instead of heating,, the gas is cooled then the heat in the entropy Df is defined by the relation
will be rejected from the gas and hence H well be negative, the
value of (I2-I1) may also be negative if the internal energy of
the gas decreased. Further W is negative, if the gas is
compressed and consequently contracts in volume, which
means that work is done the gas from some external source
For isothermal process, since the temperature remains constant,
(I2-I1) = 0 and hence
H=W (iv) (x)
On the other hand, for adiabatic process there being neither
addition nor removal of heat, H=0,and
Or
W+(I2 –I1) = 0
For isothermal process since the temperature remains constant
W = (I1-I2) (v)
The work done by a gas in expanding (or on a gas in
contracting) is given by
(xi)
(vi)
But in adiabatic process since there is no transfer of heat,
∆H=0, hence
For the isothermal process introducing the value of p from
equation (i) in equation (vi) and integrating it we obtain Constant (xii)
P1 V1 log e (V2/V1)=P1V1 log e (P1/P2) A process in which the entropy does not change is termed as
isentropic process. However, the entropy does not change in an
W=K log e(V2/V1)=K log e (P1/P2) (vii)
adiabatic process, only if it is frictionless .A frictionless adiabatic
In which K is the constant of equation (i) Equation (vii) gives process is therefore isentropic, which is also reversible.
the work done per unit mass or per unit weight of the gas
vi. Enthalpy: The sum of the internal energy and pressure-
respectively. Similarly for adiabatic process introducing the value
specific volume product (or pressure divided by mass density;
of p from equation (ii) in equation (vi) and integrating it we
or pressure divided by weight density) (converted into heat
obtain
units) is termed as enthalpy. It is a purely mathematical quantity,
the consideration of which simplifies certain calculations, since
in several problems of thermodynamics this sum occurs.
(Viii) 3. Continuity Dquation
The equation of continuity, which is application for
compressible as well as incompressible as well as incompressible
fluid, is expressed as
Equation (viii) gives the work done per unit mass of the gas if
the specific volume is considered as the volume per unit mass
of the gas. However, if the specific volume considered as the
volume per unit weight of the gas and it may then be expressed
as Further as we know for one-dimensional steady flow
compressible fluid the equation of continuity reduces to
PAV = Constant
(ix) Where ρ is the mass density of the fluid at any section of flow
passage having its cross-sectional area A and the average velocity
of flow V. Obviously for the flow of compressible fluid all
v. Entropy: Entropy of a gas may be defined as the measure of these viz, P, A, and V may vary along the flow passage. The one-
the maximum heat energy available for conversion into work. It dimensional equation of continuity for the flow of a
is a property of the gas and it varies with its absolute compressible fluid may also be expressed in differential
temperature and its state. If DH is the heat transferred per unit equation of continuity for the flow of a compressible fluid may
weight of gas in a small interval of time and if T is the also be expressed in differential form as
d(ρAV) =0
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4. Momentum Equation which for isothermal flow is equal to the work done and is
FLUID MECHANICS
The momentum equation for compressible fluids is similar to given by equation (vii) Now if there is no loss of energy i.e., Es
the one we have earlier derived for incompressible fluid. This is =0 then for a frictionless isothermal flow of fluid the energy
because in momentum equation the change in momentum flux equation becomes
is equated to the force required to cause this change. The
momentum flux is the product of the mass flux (ρAV) and the
velocity V. By the continuity equation the mass flux (ρAV) is
constant from section to section. As such the momentum (xvi)
equation is completely independent of the compressibility
effects and hence for compressible fluids too the momentum
equation (for say x direction) may be expressed as In the case of adiabatic flow there being neither addition nor
∑Fx = (ρAVVx) 2 –(ρAVVx)1. removal of heat energy Ea =H=0. Also for this case from
equations (v) and (ix)
5. Energy Equaion
The total energy Et at a point in a compressible fluid flow
system comprises the following.
1. Kinetic energy, Ek
2. Potential energy, Ee (xvii)
3. Flow or pressure energy, Ee
4. Molecular energy, Em
Thus for this case
Et =Ek+Ee+Ep+Em (xiii)
According to the law of conservation of energy. The total
energy at one point in a fluid flow system, plus the energy
added Ea minus the energy subtracted Es between two point (xviii)
must be equal to the total energy at the second point in the
system. That is
Again if there is no loss of energy i.e ., Es =0, then for a
(Ek+Ee+Ep+Em) 1+Ea-Es=(Ek+Ee+Ep+Em) 2 (xiv) frictionless adiabatic flow (or isentropic flow) of fluid the energy
Equation (xiv) shows that energy can be changed from one equation becomes
form to another on the two sides of the equation, but the total
must be the same. In other words energy can neither be created
nor destroyed.
For incompressible fluids, due to the resistance to the flow of (xix)
fluid, a part of the energy is converted into heat energy, which is
considered indirectly as an energy loss. However for
compressible fluids the heat energy produced or supplied may Equations 19.9 and 19.12 are same as same as those derived in
be stored as a molecular energy, which is termed ass internal chapter7.
energy, which is termed, as internal energy, and it must be
Further combining equations 19.2 and 19.12., we obtain
included as a specific part of the energy equation. Since the
molecular energy in of a compressible fluid is developed due to
activity of the molecules, it increases with the increase in the
temperature of the fluid, further as indicated earlier, in
thermodynamics the sum of the pressure energy (p/w) and the
internal energy I is known as enthalpy En, and hence (xx)
En=((p/w)+I.
For the flow of gas the potential head difference term (z2- z1) is
usually small as compared to the pressure head terms, and
(xv)
hence it may be neglected in equations
Points of Discuss
In which (V2/2g) is kinetic energy, z is potential energy and En
1. Isothermal process
is enthalpy, each of these being per unit weight of the fluid.
2. Adiabatic process
In the case of isothermal flow, since the temperature remains/
constant I1=Ie and from equation (i) (p1/w 2)= (p2/w 2), 3. Enthalpy
therefore E n1=E n2. Further Ea is usually the heat energy H, 4. Entropy
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5. Internal energy
FLUID MECHANICS
6. Equation of State
7. Continuity
Notes
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