Post on 15-Feb-2017
ME 6604 - GAS DYNAMICS AND JET PROPULSION
UNIT – I
BASIC CONCEPTS AND FUNDAMENTALS OF COMPRESSIBLE FLOW
T.SURESHASSISTANT PROFESSOR
DEPT OF MECHANICAL ENGGKAMARAJ COLLEGE OF ENGINEERING
PART - A• FUNDAMENTALS OF COMPRESSIBLE FLOW
• Energy and momentum equations for compressible fluid flows, various regions of flows, reference velocities, stagnation state, velocity of sound, critical states, Mach number, critical Mach number, types of waves, Mach cone, Mach angle, effect of Mach number on compressibility.
•PART – B
• Flow through variable area duct • Isentropic flow through variable area ducts, T-
s and h-s diagrams for nozzle and diffuser flows, area ratio as a function of Mach number, mass flow rate through nozzles and diffusers, effect of friction in flow through nozzles
FLOW THROUGH VARIABLE AREA DUCTS
FLOW THROUGH VARIABLE AREA DUCTS
• As a gas is forced through a tube, the gas molecules are deflected by the walls of the tube. If the speed of the gas is much less than the speed of sound of the gas, the density of the gas remains constant and the velocity of the flow increases. However, as the speed of the flow approaches the speed of sound we must consider compressibility effects on the gas. The density of the gas varies from one location to the next. Considering flow through a tube, as shown in the figure, if the flow is very gradually compressed (area decreases) and then gradually expanded (area increases), the flow conditions return to their original values. We say that such a process is reversible. From a consideration of the second law of thermodynamics, a reversible flow maintains a constant value of entropy. Engineers call this type of flow an isentropicflow; a combination of the Greek word "iso" (same) and entropy.
FLOW THROUGH VARIABLE AREA DUCTS
FLOW THROUGH VARIABLE AREA DUCTS
• The conservation of mass is a fundamental concept of physics. Within some problem domain, the amount of mass remains constant; mass is neither created or destroyed. The mass of any object is simply the volume that the object occupies times the density of the object. For a fluid (a liquid or a gas) the density, volume, and shape of the object can all change within the domain with time and mass can move through the domain.
• The conservation of mass (continuity) tells us that the mass flow rate mdot through a tube is a constant and equal to the product of the density r, velocity V, and flow area A:
Conservation of mass
Conservation of mass• Solid Mechanics• The conservation of mass is a fundamental concept of physics along with the conservation of energy and
theconservation of momentum. Within some problem domain, the amount of mass remains constant--mass is neither created nor destroyed. This seems quite obvious, as long as we are not talking about black holes or very exotic physics problems. The mass of any object can be determined by multiplying the volume of the object by the density of the object. When we move a solid object, as shown at the top of the slide, the object retains its shape, density, and volume. The mass of the object, therefore, remains a constant between state "a" and state "b."
• Fluid Statics• In the center of the figure, we consider an amount of a static fluid , liquid or gas. If we change the fluid
from some state "a" to another state "b" and allow it to come to rest, we find that, unlike a solid, a fluid may change its shape. The amount of fluid, however, remains the same. We can calculate the amount of fluid by multiplying the density times the volume. Since the mass remains constant, the product of the density and volume also remains constant. (If the density remains constant, the volume also remains constant.) The shape can change, but the mass remains the same.
• Fluid Dynamics• Finally, at the bottom of the slide, we consider the changes for a fluid that is moving through our domain.
There is no accumulation or depletion of mass, so mass is conserved within the domain. Since the fluid is moving, defining the amount of mass gets a little tricky. Let's consider an amount of fluid that passes through point "a" of our domain in some amount of time t. If the fluid passes through an area A at velocity V, we can define the volume Vol to be:
• Vol = A * V * t
Conservation Laws for a Real Fluid
0. Vt
wqVete
.
gVVtV
ij ˆ..
iiij pij '
gpVVtV
ij ˆ.. '
Conservation of Mass Applied to 1 D Steady Flow
0. Vt
Conservation of Mass:
Conservation of Mass for Stead Flow:
0. V
Integrate from inlet to exit :
onstant. CVdVV
One Dimensional Stead Flow
A,V
A+dA,V+dV d
dl
onstant.. CdxdAVV
onstant. CdxdxVAd
0VAd
0AdA
VdVd
Conservation of Momentum For A Real Fluid Flow
pVVij '..
VdpVdVdVVVVV
ij '..
No body forces
One Dimensional Steady flow
A,V
A+dA,V+dV d
dl
dAdxpdxdAdAdxVVV
wVV
ij '.
dxdxpAddx
dxAddx
dxAVd ww
2
pAdAdAVd ww 2
Conservation of Energy Applied to 1 D Steady Flow
wqVete
.
Steady flow with negligible Body Forces and no heat transfer is adiabatic real flow
wVe
.
For a real fluid the rate of work transfer is due to viscous stress and pressure. Neglecting the the effect of viscous dissipation.
VdAnpVe
.ˆ.
For a total change from inlet to exit :
AV
VdAnpVdVe
.ˆ.
Using gauss divergence theorem:
One dimensional flow
VV
VdVpVdVe
..
VV
dAdxVpdAdxVe
..
dx
dxpAVddx
dxeVAd
pAVdeVAd
2
2Vue
AVpdVuVAd
2
2
02
2
VpuVAd
02
2
VhVAd
Summary of Real Fluid Analysis
0AdA
VdVd
pAdAdAVd ww 2
02
2
VhVAd
Further Analysis of Momentum equation
pAdAdVAdVVAVd ww
pAdAdVAdV ww
pAdPxddVm w
pAdAddVm ww
pdAAdpPdxxdPPxddVm www
Frictional Flow in A Constant Area Duct
0VdVd
AdpPdxPxddVm ww
02
2
VhVd
Frictional Flow in A Constant Area Duct
AdpPdxdVm w
w
The shear stress is defined as and average viscous stress which is always opposite to the direction of flow for the entire length dx.
AdpPdxPxddVm ww
AdpPdxAVdV w
AdpPdxAVdV w
Divide by AV2
22 Vdpdx
AP
VVdV w
0VdVd
002
2
VdVdTCVhd p
One dimensional Frictional Flow of A Perfect Gas
0VdVd
0VdVdTC p
2Vdpdx
APf
VdV
TdT
VdV
pdp
TdTd
pdp
Sonic Equation
2
22
2
22 22
RTdTV
RTVdVMdM
RTV
cVM
Differential form of above equation:
TdT
VdV
MdM
2
TdT
VdV
pdp
TdT
MdM
pdp
2
M
dM
M
MTdT
2
2
211
1
Energy equation can be modified as:
TdT
MdM
pdp
2
M
dM
M
MMdM
pdp
2
2
211
121
1D steady real flow through constant area duct : momentum equation
022 Vdpdx
AP
VVdV w
022 pdp
Vpdx
AP
VVdV w
022 pdp
Vpdx
AP
VVdV w
022 pdp
Vpdx
AP
VVdV w
022 pdp
V
pdx
AP
VVdV w
0122 pdp
Mdx
AP
VVdV w
0122 pdp
Mdx
AP
VVdV w
M
dM
M
MTdT
2
2
211
1
M
dM
M
MMdM
pdp
2
2
211
121
TdT
VdV
MdM
2
Differential Equations for Frictional Flow Through Constant Area Duct
TdT
MdM
pdp
2
0122 pdp
Mdx
AP
VTdT
MdM w
0
211
1211
211
12
2
222
2
MdM
M
MMdM
Mdx
AP
VMdM
M
MMdM w
dxAP
VM
MM
MdM w
22
22
12
11
dxAP
VMM
TdT w
22
4
11
dxAP
VMMM
pdp w
22
22
111
dxAP
VM
MM
MdM w
22
22
12
11
Second Law Analysis
vdpdTCTds p
dpTv
TdTCds p
pdpR
TdTCds p
pdpR
TdTCds p
pdp
TdT
Cds
p 1
211
VTC
TdT
TdT
Cds p
p
TTT
TdT
TdT
Cds
p 0211
TTdT
TdT
Cds
p
02
11
T
T
T
T
s
s p iiiTT
dTTdT
Cds
0211
2
1
0
0
/1
lniip
i
TTTT
TT
Css
dxAP
VMM
TdT w
22
4
11
Fanno Line
Adiabatic flow in a constant area with friction is termed as Fanno flow.
Isentropic Nozzle and Adiabatic Duct
C Nozzle Discharge Curve
CD Nozzle + Discharge Curve
Nature of Real FlowEntropy of an irreversible adiabatic system should always increase!
dxAP
VMCds w
p 221
dxAP
VM
MM
MdM w
22
22
12
11
dxAP
VMM
TdT w
22
4
11
dxAP
VMMM
pdp w
22
22
111
M dM dp dT dV<1 +ve -ve -ve +ve>1 -ve +ve +ve -ve
Compressible Real Flow
),(Re, Mdkfunctionf
Effect of Mach number is negligible….
)(Re,dkfunctionf
1Re
n
TT
2
1
2
1
Pressure drop in Compressible Flow
Laminar Flow
Turbulent Flows
22
2
211
1
MM
MMdMdx
APf
Re16
f
2
9.0Re74.5
7.3log
0625.0
hDk
f
Moody Chart
Compressible Flow Through Finite Length Duct
Integrate over a length l
22
2
211
14
MM
MMdM
Dfdx
h
MdM
MM
MDfdx e
i
M
M
l
h
22
2
0
211
14
22
2
211
14
MM
MMdM
Dfdx
h
22
22
22
211
211
ln2
11114
ie
ei
eih MM
MM
MMl
Df
is a Mean friction factor over a length l . f
Maximum Allowable Length
• The length of the duct required to give a Mach number of 1 with an initial Mach number Mi
Similarly
2
2
2max
211
12
11ln
211114
i
i
ih M
M
Ml
Df
1
2
2
*
211
21
1*
iM
p
p MdM
M
M
pdp
pp
1
2
2
211
1*
ii M
T
T MdM
M
MTdT
2/1
2*
211
21
1
i
i MMpp
2
*
211
21
iMTT
2
5.15 .
1111.384
293103.3
msN
TT
1/2
211
i
o Mpp
*0
*0
*0
0
pp
pp
pp
12
**0
0
21
211
iM
pp
pp
122
*0
0
21
2111
i
i
M
Mpp
Compressible Frictional Flow through Constant Area Duct
HDfL*4
0
*0
pp
pp*
TT *
VV *
M
Frictional Flow in A Variable Area Duct
0AdA
VdVd
A,V
A+dA,V+dV d
042
22
VdVMdx
DfM
pdp
h
TdTd
pdp
0AdA
VdVd
AdA
VdV
TdT
pdp
042
22
VdVMdx
DfM
AdA
VdV
TdT
h
TdT
MdM
VdV
TdT
VdV
MdM
22
02
422
22
TdT
MdMMdx
DfM
AdA
TdT
MdM
TdT
h
M
dM
M
MTdT
2
2
211
1
042
211
1 2
2
2
dx
DfM
MdM
M
MAdA
h
dxDfM
M
M
AdA
M
M
MdM
h
421
211
12
11 2
2
2
2
2
Constant Mach number frictional flow
hDfM
dxdA 22
dxDfMM
AdAM
MdMM
h
422
112
1112
222
Sonic Point : M=1
0422
112
11
dx
Df
AdA
h
0422
12
1
dx
Df
AdA
h
dxDf
AdA
h
42
54
Stagnation Properties
Consider a fluid flowing into a diffuser at a velocity , temperature T, pressure P, and enthalpy h, etc. Here the ordinary properties T, P, h, etc. are called the static properties; that is, they are measured relative to the flow at the flow velocity. The diffuser is sufficiently long and the exit area is sufficiently large that the fluid is brought to rest (zero velocity) at the diffuser exit while no work or heat transfer is done. The resulting state is called the stagnation state.
V
We apply the first law per unit mass for one entrance, one exit, and neglect the potential energies. Let the inlet state be unsubscripted and the exit or stagnation state have the subscript o.
q h V w h Vnet net o
o
2 2
2 2
55
Since the exit velocity, work, and heat transfer are zero,
h hV
o
2
2
The term ho is called the stagnation enthalpy (some authors call this the total enthalpy). It is the enthalpy the fluid attains when brought to rest adiabatically while no work is done.
If, in addition, the process is also reversible, the process is isentropic, and the inlet and exit entropies are equal.
s so
The stagnation enthalpy and entropy define the stagnation state and the isentropic stagnation pressure, Po. The actual stagnation pressure for irreversible flows will be somewhat less than the isentropic stagnation pressure as shown below.
56