quasi - one dimensional flow with heat addition p m v subbarao professor mechanical engineering...
TRANSCRIPT
Quasi - One Dimensional Flow with Heat Addition
P M V SubbaraoProfessor
Mechanical Engineering DepartmentI I T Delhi
A Gas Dynamic Model for Combustion Systems …..
Variable Area with Heat Transfer
pA,TV ,,M
dppdAA ,
dTTdVV , ddMM ,
Conservation of mass for steady flow:
0A
dA
V
dVd
0 VdVdp Conservation of momentum for ideal steady flow:
0''' dTCdqVdVdTC pp
T
dTd
p
dp
Ideal Gas law:
Conservation of energy for ideal steady flow:
Combining momentum and gas law:
02 T
dTd
V
dVM
Using conservation of massA
dA
V
dVd
02 T
dT
A
dA
V
dV
V
dVM
012 T
dT
A
dA
V
dVM
Mach number equation:M
dM
T
dT
V
dV
2
02
12
T
dT
A
dA
M
dM
T
dTM
0
2
11
22
A
dA
T
dTM
M
dMM
0''' dTCdqVdVdTC pp
21
1
1
2
2'''
M
MdM
MTC
dq
T
dT p
Energy Equation with Mach Equation:
0
2
11
22
A
dA
T
dTM
M
dMM
0
21
1
1
2
11
2
2'''
22
A
dA
M
MdM
MTC
dq
M
M
dMM p
Combined momentum,mass, gas & Mach Equations
0
2
11
2
1
2
11
1
2
11
2
'''
2
2
222
A
dA
M
TCdq
M
M
dM
M
MMM p
0
21
12
1
21
1
1 '''
2
2
2
2
A
dA
TC
dq
M
M
M
dM
M
M
p
0
21
12
1
21
1
1 02
2
2
2
A
dA
T
dT
M
M
M
dM
M
M
0
21
12
1
21
1
1 02
2
2
2
A
dA
T
dT
M
M
M
dM
M
M
Condition for M=1
0
21
12
1 0
A
dA
T
dT
0
21
12
1 '''
A
dA
TC
dq
p
0'''
A
dA
TC
dq
p
0'''
A
dA
TC
dq
p
For heat addition, M=1, dA will be positive.
For heat removal, M=1, dA will be negative.
Constant Mach Number Flow with Heat Transfer
0
21
12
1 02
2
A
dA
T
dT
M
M
0
21
12
1 '''
2
2
A
dA
TC
dq
M
M
p
TC
dq
M
M
A
dA
p
'''
2
2
12
1
Quasi - One Dimensional Flow with Heat Transfer & Friction
P M V SubbaraoProfessor
Mechanical Engineering DepartmentI I T Delhi
A Gas Dynamic Model for Gas Cooled High Heat Release Systems…..
Frictional Flow with Heat Transfer
Governing Equations
Nonreacting, no bodyforces, viscous work negligible
Conservation of mass for steady flow: 0A
dA
V
dVd
02 2
22 V
dVM
p
dpdx
A
L
ppx
Conservation of momentum for frictional steady flow:
Conservation of energy for ideal steady flow:
T
dT
TC
q
V
dVM
T
dT
p
0'''
21
01 2'''
V
dVM
T
dT
TC
q
p
0
2
12
22'''
V
dVM
T
dT
TC
q
p
0T
dTd
p
dp
Ideal Gas law:
Mach number equation: 02
2
2
2
V
dV
M
dM
T
dT
Into momentum equation
h
px
D
fdxMdx
D
f
p
Vdx
A
L
p2
2
2
4
22
1
02 2
22 V
dVM
p
dpdx
A
L
ppx
022 2
222 V
dVM
p
dp
D
fdxM
h
Combine conservation, state equations– can algebraically show
2
0
'''2
2
2
2
2
121
21
1M
TC
q
A
dA
D
fdxM
M
M
M
dM
ph
So we have three ways to change M of flow
– area change (dA): previously studied
– friction: f > 0, same effect as –dA
– heat transfer:heating, q’’’ > 0, like –dA cooling, q’’’ < 0, like +dA
Mach Number Variations
2
0
'''2
2
2
2
2
121
21
1M
TC
q
A
dA
D
fdxM
M
M
M
dM
ph
• Subsonic flow (M<1): 1–M2 > 0
– friction, heating, converging area increase M (dM > 0)
– cooling, diverging area decrease M (dM < 0)
• Supersonic flow (M>1): 1–M2 < 0
– friction, heating, converging area decrease M (dM < 0)
– cooling, diverging area increase M (dM > 0)
Sonic Flow Trends
• Friction
– accelerates subsonic flow, decelerates supersonic flow
– always drives flow toward M=1
– (increases entropy)
• Heating
– same as friction - always drives flow toward M=1
– (increases entropy)
• Cooling
– opposite - always drives flow away from M=1
– (decreases entropy)
Nozzles : Sonic Throat
2
0
'''2
2
2
2
2
121
21
1M
TC
q
A
dA
D
fdxM
M
M
M
dM
ph
• Effect on transition point: sub supersonic flow
• As M1, 1–M20, need { } term to approach 0
• For isentropic flow, previously showed
– sonic condition was dA=0, throat
• For friction or heating, need dA > 0
– sonic point in diverging section
• For cooling, need dA < 0
– sonic point in converging section
Mach Number Relations
• Using conservation/state equations can get equations for each TD property as function of dM2
Constant Area, Steady Compressible Flow withFriction Factor and Uniform Heat Flux at the Wall Specified
• Choking limits and flow variables for passages are important parameters in one-dimensional, compressible flow in heated pipes.
• The design of gas cooled beam stops and gas cooled reactor cores, both usually having helium as the coolant and graphite as the heated wall.
• Choking lengths are considerably shortened by wall heating.
• Both the solutions for adiabatic and isothermal flows overpredict these limits.
• Consequently, an unchoked cooling channel configuration designed on the basis of adiabatic flow maybe choked when wall heat transfer is considered.
Gas Cooled Reactor Core
Beam Coolers
• The local Mach number within the passage will increase towards the exit for either of two reasons or a combination of the two.
• Both reasons are the result of a decrease in gas density with increasing axial position caused either by
• (1) a frictional pressure drop or• (2) an increase in static temperature as a
result of wall heat transfer.
2
0
'''2
2
2
2
2
121
21
1M
TC
q
A
dA
D
fdxM
M
M
M
dM
ph
Constant area duct:
2
0
'''2
2
2
2
2
11
21
1M
TC
q
D
fdxM
M
M
M
dM
ph
011
21
12
0
'''2
2
2
2
2
M
TC
q
D
fdxM
M
M
M
dM
ph
011
21
12
0
'''2
2
2
2
2
M
dxTC
q
D
fM
M
M
dxM
dM
ph
Divide throughout by dx
011
21
12
0
'''2
2
2
22
M
dxTC
q
D
fM
M
MM
dx
dM
ph
Multiply throughout by M2
011
21
12
0
22
2
22
M
dxTCm
q
D
fM
M
MM
dx
dM
ph
For a uniform wall heat flux q’’
011
21
12
0
''2
2
2
22
M
dxTCm
dxLq
D
fM
M
MM
dx
dM
p
p
h
Numerical Integration of differential Equation
011
21
12
0
''2
2
2
22
M
TCm
Lq
D
fM
M
MM
dx
dM
p
p
h
Choking Length
M1
K :non dimensional heat flux
2
0
'''2
2
2
2
2
11
21
1M
TC
q
D
fdxM
M
M
M
dM
ph
0
2
12
22'''
V
dVM
T
dT
TC
q
p
Mach number equation: 02
2
2
2
V
dV
M
dM
T
dT
0
2
12
22'''
M
dM
T
dTM
T
dT
TC
q
p
2
0
'''2
2
2
2
2
11
21
1M
TC
q
D
fdxM
M
M
M
dM
ph
2
222'''
2
1
2
11
M
dMMM
T
dT
TC
q
p
2
02
2222
2
2
2
2
12
1
2
11
12
11
MT
T
M
dMMM
T
dT
D
fdxM
M
M
M
dM
h
0
21
14
21
122
11
2
1
2
22
22
0
0
2
hDM
Mf
dx
dM
MM
T
dx
dT
T
M
