Download Introduction of compressible flow

COMPRESSIBLE FLOW
COMPRESSIBLE FLOW
Introduction
The compressibility of a fluid is, basically, a
measure of the change in density that will be
produced in the fluid by a specific change in
pressure and temperature.
In general, gases are highly compressible and
liquids have a very low compressibility.
Part one : Introduction of Compressible Flow
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COMPRESSIBLE FLOW
Application ;
Aircraft design
Gas and steam turbines
Reciprocating engines
Natural gas transmission lines
Combustion chambers
Compressibility effect ;
Supersonic – the flow velocity is relatively
high compared to the speed of sound in the gas.
Subsonic
Part one : Introduction of Compressible Flow
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COMPRESSIBLE FLOW
Fundamental assumptions
1. The gas is continuous.
2. The gas is perfect (obeys the perfect gas law)
3. Gravitational effects on the flow field are
negligible.
4. Magnetic and electrical effects are negligible.
5. The effects of viscosity are negligible.
Applied principles
1. Conservation of mass (continuity equation)
2. Conservation of momentum (Newton’s law)
3. Conservation
of
energy
(first
law
of
thermodynamics)
4. Equation of state
Part one : Introduction of Compressible Flow
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COMPRESSIBLE FLOW
Perfect gas law :
P
ρ
= RT
P : Pressure
ρ : Density
R : Universal gas constant
Rair = 287.04( kgJ⋅ K )
T : Temperature
Part one : Introduction of Compressible Flow
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COMPRESSIBLE FLOW
Conservation laws :
Conservation of mass
Rate of increase
of mass of fluid in
control volume
=
Rate mass enters
control volume
_
Rate mass leaves
control volume
Conservation of momentum :
Net force on gas
in control volume
in direction
considered
=
Rate of increase
of momentum in
direction
considered of
fluid in control
l
_
Part one : Introduction of Compressible Flow
+
Rate momentum
leaves control
volume in
direction
considered
Rate momentum
leaves control
volume in
direction
considered
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COMPRESSIBLE FLOW
Conservation of energy :
Rate of increase
in internal energy
and kinetic
energy of gas in
control volume
=
+
Rate enthalpy
and kinetic
energy leave
control volume
Rate heat is
transferred into
control volume
Part one : Introduction of Compressible Flow
_
_
Rate enthalpy
and kinetic
energy enter
control volume
Rate work is done
by gas in control
volume
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COMPRESSIBLE FLOW
Definition :
A control volume is a volume in space (geometric
entity, independent of mass) through which fluid
may flow
Enthalpy H, is the sum of internal energy U and
the product of pressure P and volume V appears.
H = U + PV
Part one : Introduction of Compressible Flow
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COMPRESSIBLE FLOW
COMPRESSIBLE FLOW
Introduction
Many of the compressible flows that occur in
engineering practice can be adequately modeled as a
flow through a duct or streamtube whose cross-sectional
area is changing relatively slowly in the flow direction.
A duct is a solid walled channel, whereas a streamtube
is defined by considering a closed curve drawn in a
fluid flow.
Part two : The Equation of Steady One-Dimensional Compressible Fluid Flow
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COMPRESSIBLE FLOW
Quasi-one-dimensional flow is flows in which the flow
area is changing but in which the flow at any section
can be treated as one-dimensional.
Part two : The Equation of Steady One-Dimensional Compressible Fluid Flow
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COMPRESSIBLE FLOW
CONTINUITY EQUATION
The continuity equation is obtained by applying the
principle of conservation of mass to flow through a
control volume.
One-dimensional flow is being considered.
There is no mass transfer across the control volume.
The only mass transfer occurs through the ends of the
control volume.
Part two : The Equation of Steady One-Dimensional Compressible Fluid Flow
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COMPRESSIBLE FLOW
Mass enters through the left hand face of the control
volume be equal to the rate at which mass leaves
through the right hand face of the control volume.
m& 1 = m& 2
We know that
m& = ρVA
We considered ;
ρ1V1 A1 = ρ 2V2 A2
For the differentially short control volume indicated,
Part two : The Equation of Steady One-Dimensional Compressible Fluid Flow
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COMPRESSIBLE FLOW
above equation gives ;
ρVA = ( ρ + dρ )(V + dV )( A + dA)
Neglecting higher order terms, we found ;
VAdρ + ρAdV + ρVdA = 0
Dividing this equation by
ρVA
then gives ;
dρ
dV dA
+
+
=0
V
A
ρ
This equation relates the fractional changes in density,
velocity and area over a short length of the control
volume.
Part two : The Equation of Steady One-Dimensional Compressible Fluid Flow
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COMPRESSIBLE FLOW
MOMENTUM EQUATION (Euler’s equation)
The flow is steady flow.
Gravitational forces are being neglected.
The only forces acting on the control volume are the
pressure forces and the frictional force exerted on the
surface of the control volume.
Part two : The Equation of Steady One-Dimensional Compressible Fluid Flow
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COMPRESSIBLE FLOW
The net force on the control volume in the x-direction
is ;
PA − ( p + dP )( A + dA) + 12 [( P + ( P + dP )][( A + dA) − A] − dFµ
Note :
dx is too small, dPdA have been neglected.
Mean pressure on the curved surface can be taken
as the average of the pressures acting on the two end
surfaces.
dFµ is the frictional force.
Rearranging above equation, we found the net force on
the control volume in the x-direction is ;
− AdP − dFµ
Part two : The Equation of Steady One-Dimensional Compressible Fluid Flow
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COMPRESSIBLE FLOW
Since the rate at which momentum crosses any section
of the duct is equal to
m& V , we found that ;
ρVA[(V + dV ) − V ] = ρVAdV
The above equation can be written as ;
− Adρ − dFµ = ρVAdV
Frictional force is assumed to be negligible. The Euler’s
equation for steady flow through a duct becomes;
−
dP
ρ
= VdV
Part two : The Equation of Steady One-Dimensional Compressible Fluid Flow
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COMPRESSIBLE FLOW
Integrating Euler’s equation ;
V2
dP
+∫
=C
2
ρ
(For compressible)
And if density can be assumed constant, Euler’s
equation become ;
V2 P
+ =C
2 ρ
(For incompressible)
Part two : The Equation of Steady One-Dimensional Compressible Fluid Flow
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COMPRESSIBLE FLOW
STEADY FLOW ENERGY EQUATION
For flow through the type of control volume
considered as before, we found ;
V22
V12
h2 +
= h1 +
+q−w
2
2
h = enthalpy per mass
V = velocity
q = heat transferred into the control volume per unit mass of fluid
flowing through it
w = work done by the fluid per unit mass
Part two : The Equation of Steady One-Dimensional Compressible Fluid Flow
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COMPRESSIBLE FLOW
Assumption ;
No work is done, w=0
Perfect gases is considered,
h = c pT
Steady flow energy equation ;
V22
V12
c pT2 +
= c pT1 +
+q
2
2
Applying this equation to the flow through the
differentially short control volume gives ;
V2
(V + dV ) 2
c pT +
+ dq = c p (T + dT ) +
2
2
Part two : The Equation of Steady One-Dimensional Compressible Fluid Flow
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COMPRESSIBLE FLOW
Neglecting higher order terms gives ;
c p dT + VdV = dq
This equation indicates that in compressible
flows, changes in velocity will, in general, induce
changes in temperature and that heat addition
can
cause
velocity
changes
as
well
as
temperature changes.
If the flow is adiabatic i.e., if there is no heat
transfer to of from the flow, it gives ;
V22
V12
= c pT1 +
c pT2 +
2
2
Part two : The Equation of Steady One-Dimensional Compressible Fluid Flow
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COMPRESSIBLE FLOW
Steady flow energy equation for adiabatic flow
becomes ;
c p dT + VdV = 0
This equation shows that in adiabatic flow, an increase
in velocity is always accompanied by a decrease in
temperature.
Part two : The Equation of Steady One-Dimensional Compressible Fluid Flow
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COMPRESSIBLE FLOW
EQUATION OF STATE
When applied between any two points in the flow ;
P1
P2
=
ρ1T1 ρ 2T2
When applied between the inlet and the exit of a
differentially short control volume, this equation
becomes ;
P
P + dP
=
ρT ( ρ + dρ )(T + dT )
Part two : The Equation of Steady One-Dimensional Compressible Fluid Flow
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COMPRESSIBLE FLOW
Higher order terms are neglected and it gives ;
P
P
(1 +
=
ρT ρT
dP
P
)(1 − dρρ )(1 − dTT )
dP dρ dT
−
−
=0
P
T
ρ
This equation shows how the changes in pressure,
density and temperature are interrelated in compressible
flow.
Part two : The Equation of Steady One-Dimensional Compressible Fluid Flow
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COMPRESSIBLE FLOW
ENTROPY CONSIDERATIONS
In studying compressible flows, another variable, the
entropy, s, has to be introduced. The entropy basically
places limitations on which flow processes are
physically possible and which are physically excluded.
The entropy change between any two points in the flow
is given by ;
s2 − s1 = c p ln
Since
[ ]− R ln[ ]
T2
T1
P2
P1
(1)
R = c p − cv , this equation can be written;
⎡
s2 − s1
= ln ⎢
cp
⎣
( )( )
T2
T1
P2
P1
−
γ −1
γ
⎤
⎥
⎦
If there is no change in entropy, i.e., if the flow is
Part two : The Equation of Steady One-Dimensional Compressible Fluid Flow
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COMPRESSIBLE FLOW
isentropic, this equation requires that :
T2 ⎛ P2 ⎞
= ⎜⎜ ⎟⎟
T1 ⎝ P1 ⎠
γ −1
γ
hence, since the perfect gas law gives ;
T2 P2 ρ1
=
T1 P1 ρ 2
it follows that in isentropic flow :
P2 ⎛ ρ 2 ⎞
= ⎜⎜ ⎟⎟
P1 ⎝ ρ1 ⎠
γ
in isentropic flows, then
P
ρ γ is a constant.
If equation (1) is applied between the inlet and the exit
Part two : The Equation of Steady One-Dimensional Compressible Fluid Flow
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COMPRESSIBLE FLOW
of a differentially short control volume, it gives ;
( s + ds ) − s = c p ln[T +TdT ] − R ln[ P +PdP ]
neglecting small value, the above equation gives;
ds = c p
dT
T
− R dPP
(2)
which can be written as ;
ds dT ⎛ γ − 1 ⎞ dP
⎟⎟
=
− ⎜⎜
cp
T ⎝ γ ⎠ P
lastly, it is noted that in an isentropic flow, equation (2)
gives;
RT
c p dT =
dP
P
using the perfect gas law ;
Part two : The Equation of Steady One-Dimensional Compressible Fluid Flow
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COMPRESSIBLE FLOW
c p dT =
dP
ρ
(3)
but the energy equation for isentropic flow, i.e., for flow
with no heat transfer, it gives ;
c p dT + VdV = 0
which using equation (3) gives ;
dP
ρ
+ VdV = 0
Part two : The Equation of Steady One-Dimensional Compressible Fluid Flow
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