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Theory of Compressible FlowsM.S. Mechanical ProcessEngineering4thSemester
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IntroductionGAS DYNAMICS
It is the branch of fluid mechanics concerned with causes and effects
arising from the motion of compressible fluids particularly gases.Fluid Mechanics
Statics Fluid Dynamics
Aerodynamics Hydrodynamics Gas Dynamics
In this subject we are concerned with following fundamental physical laws
1. Law of conservation of mass
2. Newtons second law of motion
3. First law of thermodynamics
4. Second law of thermodynamics
Above laws are applicable to all fluids and all flow processes
Above laws are applied to a fluid utilizing the continuum concept
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Continuum
Due to elastic collisions limits of velocity are between
zero and very high velocity
Continuum concept may not be applicable for a gas
at low pressure (High altitudes)
Distance b/w two consecutive collisions is mean free path
Mean free path is inversely related to molecular diameter
All materials, solid or fluid, are composed of molecules discretely
spread and in continuous motion.
However, in dealing with fluid-flow relations on a mathematical basis, itis necessary to replace the actual molecular structure by a hypothetical
continuous medium, called the continuum.
Continuum postulate assumes that every differential element of body of
fluid contains a large number of molecules.
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Continuum, contd.For continuum postulate to hold, mean free path () 0.01 gas is a combination of discrete particles
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CompressibilityAmount by which a substance can be compressed is given by a property
compressibility
Compressibility isfractional change in
volume per unit change in
pressure
Isothermal CompressibilityRise in temp is controlled by
some heat transfer mechanism
Isentropic Compressibility
Compressibility in
the form of density
Liquid dis small
Gas d
Is small for low speed flow
Is large for high speed flowAnother index
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where
Collisions between the fluid molecules causes the propagation of sound
waves
Propagation of sound waves is affected by molecular density of the
medium Where molecular density is low, dissipation of energy occurs without
colliding with any molecule (free particle flow)
It is an important quantity in gas dynamics
We shall study further details about speed of sound in Chapter 3
For Ideal Gas
The Acoustic SpeedThe speed at which a sound wave or small pressure disturbance is propagated
in the fluid medium
s
a
1
s
sdp
d
1
s
pa
2
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It is the ratio of local fluid speed Vto its acoustic speed
Mach Number
Properties of Atmosphere
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1-7 Simple thermodynamic system
1-8 Reversible and irreversible processes, 1-9 Work, 1-10 Heat
1-11 1stlaw of thermo
1-12 2ndlaw of thermo; 1-15(a) to (c)
1-15(d) to (g)
1-18 Properties of atmosphere (Detailed)
2-3 Mathematical description of continuum
2-5, 2-6, 2-7, 2-8 (equations)
Presentation Topics
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Simple Thermodynamic System
Equation of State
Determined by experiment zis dependent andx, y are independent properties
x, y are scalars but may be intensiveor extensive
A readily measurable characteristic showing the behavior of the system
and its interaction with the environment is called a Property
It depends on the state of a system In the case of a simple system, state is fully defined by any two
independent properties
If fis a differentiable
function ofxand y, dzis
called exact differential
fis not exact differential
or inexact its written as
z
For continuous M
and N, it must satisfy
It contains
Either a restricted region in space or finite portion of matter called System
A surface that system from other space or matter called Control Surface
Everything external to control system is Surrounding
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If the reciprocal relationship exists, then
Simple Thermodynamic System, contd.
Line integral of an exact differential expression is independent of the
paths connecting the end points of the integration
Line integral of an exact differential expression taken over a closed curve
is zero
Let
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Reversible and Irreversible ProcessesA process is reversibleif its direction can be reversed at any point by an
infinitesimal change in external conditions
Such process take infinite time to complete
Irreversible Effects
1. Friction
2. Heat transfer across finite temp difference
3. Free expansion
4. Mixing
5. Inelastic deformation
Irreversible Processes
1. Viscous Momentum
Flux2. Heat Conduction
3. Mass Diffusion
Reversible processes are infinitely slow processes with maximumefficiency
These processes are always in equilibrium
These processes provide the limiting results
All real processes are irreversible in nature
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Work and HeatWorkis the energy in transit across the boundary of the system where the
sole effect external to the system could have been raising or lowering of a
weight
Heatis the energy in transit across the boundary of the system caused by atemperature difference between system and its surrounding
Work done by
the system is
positive
Heat added to the system is positive
Heat and work both are path functions and depend upon the details of the
processes involved
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First Law of ThermodynamicsEnergy can neither be created nor be destroyed but it can be
changed from one form to another form.
1. Total Energy is conserved2. This Law is based on experience and no phenomenon contradicts it
3. Heat and work are mutually inter-convertible and fixed amount per
unit mass is needed for every degree rise of temperature
Line integral vanishes for the closed
curve so the integral defines a property
For a cyclic process of closed system of mass m
Eis the stored
energy
On per unit
mass basis
The equation is valid
for both Reversibleand Irreversible
processes
For Irreversible
processes
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Internal Energy
On unit mass basis
Work required to compress
volume dV intoa region
where pressure isp
It comprises of the
1. Translational kinetic energy of the gas molecule
2. Rotational kinetic energy of the gas molecule3. Vibrational kinetic energy of the gas molecule due to atoms
4. Electronic energy due to electronsFor Irreversible
processes
Flow Work
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EnthalpyDifferentiating both sides
For a reversibleprocess
By definition Enthalpy is
Where internal energy is
Putting into enthalpy
When process is
reversible and adiabaticSpecific HeatsSpecific heat at constant volume
Specific heat at constant pressure
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2ndLaw of Thermodynamics Heat is a relatively crude form f energy so it can not be converted
into work completely
Only a portion of heat can be converted into work
Every natural system will change spontaneously and reach
equilibrium and no further change will be possible
Hence
For a closed system
or
For a closed system that undergoes
a cyclic change, it may be shown
So
If heat is added reversibly then
But
For Reversible Adiabatic Process
For Irreversible
Adiabatic Process
For a Diabetic Process
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Thermodynamic Properties of a Perfect GasThermal Equation of State
A gas for which Ris nearly
constant is called thermally
perfect gas
Rdepends upon type of gas involved
It varies with tandpfor real gases
For a homogenous system composed of
single chemical specie of gas:
Thermodynamic stat is defined byp, v , t, u, h and s
Thermal Equation of State
Compressibility Factor For t/tcr> 2 andp/pcr< 0.05; Z~ 1
Caloric Equation of State
But So
Integrating
If cvis constant
and u0and t0are zero
Caloric Equation
of State
A gas having constant cvis called calorically
perfect gas
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Specific Heat Relationships
For thermally perfect gas
For a reversible process
of Thermally perfect gas
So
For an isobaric
process, we get
Combine the
two equations
Integrating
Also
cpis also
constant for
calorically
perfect gas
Where
Thermodynamic Properties of a Perfect Gas, contdDifferentiating thermal
equation state we get
Enthalpy Change
AsDifferentiating )(pvddudh
And Rdtpvd )(
dtRcdh v )( dtcdh p
If cvis constant and
h0and t0are zero
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Isentropic Relations
Integrating both equations
Considering calorically
perfect gas
Also
Combining all the relations
Where
Thermodynamic Properties of a Perfect Gas, contdEntropy Change
So
For a reversible process
of Thermally perfect gas vdpdhtds
1
2
1
212 lnln
v
vR
t
tcss v
1
2
1
212 lnln
p
pR
t
tcss p
1
2
1
2 lnln0p
pR
t
tcp
1
1
2
1
2
tt
pp
1
R
cp
1
2
1
2 lnln0v
vR
t
tcv
1
1
1
2
1
2
t
t
v
v
1
1
1
2
1
2
t
t
1
1
2
1
2
1
2
tt
pp
Where1
1
R
cv
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Process Equations
Following processes are most significant
1. Isothermal Change (dt= 0)2. Adiabatic Change (dQ= 0)
3. Isobaric Change (dp= 0)
4. Isochoric Change (dv= 0)
5. Isentropic Change (ds= 0)
where
All these may be written
in a general form
Processes occurring rapidly can easily be
considered adiabatic but these are seldom
reversible in nature. Reversibility may be assumed as a first
approximation
We shall assume flow through nozzle as
isentropic
n= 1 is for dt= 0
n= 0 for dp= 0n= for dv= 0
n= for ds= 0
Thermodynamic Properties of a Perfect Gas, contd
constantnpv
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EntropyEnthalpy Diagram
Curves of constantv are steeper than those of constantp For constant values of s,pincreases with twhile vdecreases with t
Reference conditions may be chosen arbitrarily
Same is h-sand u-splot for ideal gas
It is also called Mollier Diagram
Lines of constant specific
volume can be found from
Thermodynamic Properties of a Perfect Gas, contd
These come out to be
Lines of constant Pressure
can be found fromThese come out to be
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Quiz 1p v
c c R is valid for which of the following process of an ideal gas
1. Isobaric Process only2. Isochoric Process only
3. Isentropic Process only
4. All of the above
5. Only 1 and 2
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Governing Equations for Compressible FlowsFollowing are the Laws
1. Law of conservation of mass
2. Newtons second law of motion3. First law of thermodynamics
4. Second law of thermodynamics
Following are the Governing Equations
1. Continuity Equation
2. Momentum Equation3. Energy Equation
4. Entropy Equation
Continuity Equation
IntegralForm
DifferentialForm
Momentum Equation
Integral
Form
Differential
Form
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Energy Equation
Let us define:
B1= Rate of heat added to fluid in control volume from surroundingsB2= Rate of work done on fluid inside control volume
B3= Rate of change of energy of fluid as it flows through control volume
Governing Equations for Compressible Flows, contdIt is based on the physical
principle Energy can neither becreated nor be destroyed, it can
only change in form
dedwdq
321 BBB
Let us analyze B1term first
Heat may be added due to
1. Nuclear Reaction
2. Chemical Reaction
3. Radiation Heating etc
Let q
volumetric rate of heat addition per unit mass
Rate of heat addition to mass of small volume is
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Now let us tackle B2term:
Governing Equations for Compressible Flows, contd
Term B3:
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Energy Equation
IntegralForm
Differential
Form
Governing Equations for Compressible Flows, contd
Entropy Equation
t
qds
For S = ms the equation becomes
t
QdS
Time rate of change of entropy within the control volume is given byt
Q
Dt
DS
Integral Form
Differential Form