One Dimensional Flow of Real Fluid P M V Subbarao Associate Professor Mechanical Engineering Department I I T Delhi Real study is essential before using ……..
One Dimensional Flow of Real Fluid
Jan 03, 2016
One Dimensional Flow of Real Fluid. P M V Subbarao Associate Professor Mechanical Engineering Department I I T Delhi. Real study is essential before using ……. Conservation Laws for a Real Fluid. Conservation of Mass Applied to 1 D Steady Flow. Conservation of Mass:. - PowerPoint PPT Presentation
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One Dimensional Flow of Real Fluid
P M V SubbaraoAssociate Professor
Mechanical Engineering DepartmentI I T Delhi
Real study is essential before using ……..
Conservation Laws for a Real Fluid
0.
Vt
wqVet
e
.
gVVt
Vij ˆ..
iiij pij '
gpVVt
Vij 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. Cdx
dx
VAd
0VAd
0A
dA
V
dVd
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
w
VVij '.
dx
dx
pAddx
dx
Addx
dx
AVd ww 2
pAdAdAVd ww 2
Conservation of Energy Applied to 1 D Steady Flow
wqVet
e
.
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
dx
pAVddx
dx
eVAd
pAVdeVAd
2
2Vue
AV
pd
VuVAd
2
2
02
2
VpuVAd
0
2
2
VhVAd
Summary of Real Fluid Analysis
0A
dA
V
dVd
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
0V
dVd
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 V2
22 V
dpdx
A
P
VV
dV w
0V
dVd
002
2
VdVdTC
Vhd p
One dimensional Frictional Flow of A Perfect Gas
0V
dVd
0VdVdTC p
2V
dpdx
A
Pf
V
dV
T
dT
V
dV
p
dp
T
dTd
p
dp
Sonic Equation
2
22
2
22 2
2RT
dTV
RT
VdVMdM
RT
V
c
VM
Differential form of above equation:
T
dT
V
dV
M
dM
2
T
dT
V
dV
p
dp
T
dT
M
dM
p
dp
2
M
dM
M
M
T
dT
2
2
21
1
1
Energy equation can be modified as:
T
dT
M
dM
p
dp
2
M
dM
M
M
M
dM
p
dp
2
2
21
1
1
2
1
1D steady real flow through constant area duct : momentum equation
022
V
dpdx
A
P
VV
dV w
022
p
dp
V
pdx
A
P
VV
dV w
022
p
dp
V
pdx
A
P
VV
dV w
022
p
dp
V
pdx
A
P
VV
dV w
022
p
dp
V
p
dxA
P
VV
dV w
01
22
p
dp
Mdx
A
P
VV
dV w
01
22
p
dp
Mdx
A
P
VV
dV w
M
dM
M
M
T
dT
2
2
21
1
1
M
dM
M
M
M
dM
p
dp
2
2
21
1
1
2
1
T
dT
V
dV
M
dM
2
Differential Equations for Frictional Flow Through Constant Area Duct
T
dT
M
dM
p
dp
2
01
22
p
dp
Mdx
A
P
VT
dT
M
dM w
0
21
1
1
2
11
21
1
1
2
2
222
2
M
dM
M
M
M
dM
Mdx
A
P
VM
dM
M
M
M
dM w
dxA
P
VM
MM
M
dM w22
22
12
11
dxA
P
VM
M
T
dT w22
4
1
1
dxA
P
VM
MM
p
dp w22
22
1
11
dxA
P
VM
MM
M
dM w22
22
12
11
Second Law Analysis
vdpdTCTds p
dpT
v
T
dTCds p
p
dpR
T
dTCds p
p
dpR
T
dTCds p
p
dp
T
dT
C
ds
p 1
21
1
V
TC
T
dT
T
dT
C
ds p
p
TT
T
T
dT
T
dT
C
ds
p 021
1
TT
dT
T
dT
C
ds
p
02
11
T
T
T
T
s
s p iiiTT
dT
T
dT
C
ds
02
11
2
1
0
0
/1
lniip
i
TT
TT
T
T
C
ss
dxA
P
VM
M
T
dT w22
4
1
1
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 Flow
Entropy of an irreversible adiabatic system should always increase!
dxA
P
VMCds w
p 221
dxA
P
VM
MM
M
dM w22
22
12
11
dxA
P
VM
M
T
dT w22
4
1
1
dxA
P
VM
MM
p
dp w22
22
1
11
M dM dp dT dV
<1 +ve -ve -ve +ve
>1 -ve +ve +ve -ve
Flow in Conduits
P M V Subbarao
Associate Professor
Mechanical Engineering Department
IIT Delhi
An important infrastructure for Industrialization…..
Compressible Real Flow
),(Re, Md
kfunctionf
Effect of Mach number is negligible….
)(Re,d
kfunctionf
1
Ren
T
T
2
1
2
1
Pressure drop in Compressible Flow
Laminar Flow
Turbulent Flows
22
2
21
1
1
MM
M
M
dMdx
A
Pf
Re
16f
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
21
1
14
MM
M
M
dM
D
fdx
h
M
dM
MM
M
D
fdx e
i
M
M
l
h
22
2
0
21
1
14
22
2
21
1
14
MM
M
M
dM
D
fdx
h
22
22
22
21
1
21
1ln
2
11114
ie
ei
eih MM
MM
MMl
D
f
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
21
1
12
11
ln2
11
114
i
i
ih M
M
Ml
D
f
1
2
2
*
21
1
21
1
*
iM
p
p M
dM
M
M
p
dp
p
p
1
2
2
21
1
1*
ii M
T
T M
dM
M
M
T
dT
2/1
2*
21
1
21
1
i
i MMp
p
2
*
21
1
21
iMT
T
2
5.15 .
111
1.384
293103.3
m
sN
T
T
1/2
2
11
i
o Mp
p
*0
*0
*0
0
p
p
p
p
p
p
12
**0
0
21
21
1
iM
p
p
p
p
122
*0
0
21
21
11
i
i
M
Mp
p
Compressible Frictional Flow through Constant Area Duct
HD
fL*4
0
*0
p
p
p
p*
T
T *
V
V *
M
Frictional Flow in A Variable Area Duct
0A
dA
V
dVd
A,V
A+dA,V+dV d
04
22
2
V
dVMdx
D
fM
p
dp
h
T
dTd
p
dp
0A
dA
V
dVd
A
dA
V
dV
T
dT
p
dp
04
22
2
V
dVMdx
D
fM
A
dA
V
dV
T
dT
h
T
dT
M
dM
V
dV
T
dT
V
dV
M
dM
22
02
4
222
2
T
dT
M
dMMdx
D
fM
A
dA
T
dT
M
dM
T
dT
h
M
dM
M
M
T
dT
2
2
21
1
1
04
22
11
1 2
2
2
dx
D
fM
M
dM
M
M
A
dA
h
dxD
fM
M
M
A
dA
M
M
M
dM
h
4
212
11
12
11 2
2
2
2
2
Constant Mach number frictional flow
hD
fM
dx
dA 22
dxD
fMM
A
dAM
M
dMM
h
4
22
11
2
111
2222
Sonic Point : M=1
04
22
11
2
11
dx
D
f
A
dA
h
04
22
1
2
1
dx
D
f
A
dA
h
dxD
f
A
dA
h
4
2
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