pressure and temperature. In general, gases are highly compressible and liquids
have a very low compressibility. Part one : Introduction of Compressible Flow. 1 ...
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
1
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
2
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
3
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
4
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
5
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
6
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
7
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
8
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
9
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
10
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
11
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
12
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
13
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
14
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
15
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
16
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
17
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
18
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
19
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
20
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
21
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
22
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
23
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
24
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
25
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
26