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One Dimensional Flow with Heat Addition
P M V SubbaraoProfessor
Mechanical Engineering DepartmentI I T Delhi
A Gas Dynamic Model for Cross Country Gas Pipe Lines…..
Ideal Flow in A Constant Area Duct with heat Transfer
0V
dVd
AdpdVm
'''dqVdVdTC p T
dTd
p
dp
Mach equation givesM
dM
V
dVM
2
1 2
2
2
1
1
M
Md
p
dp
Mach equation in Momentum equation gives
M
dM
MM
M
T
dT
22
2
0
0
12
11
12
Mach equation in Energy equation gives
M
dM
M
M
T
dT2
2
1
12
Change over A Finite Length
1 2
22
21
1
2
1
1
M
M
p
p
121
22
122
21
01
02
21
11
21
11
MM
MM
p
p
222
222
1
2
12
21
1
1
MM
MM
T
T
22
22
1
2
12
21
1
1
MM
MM
V
V
Integrating from point 1 to point 2:
222
21
21
221
22
22
01
02
12
11
12
11
MMM
MMM
T
T
In subsonic flow, heat addition increases the Mach Number.In supersonic flow, heat addition decreases the Mach Number.Addition of heat leads the flow to move towards M=1.Removal of heat leads the flow to move away from M=1.Therefore T0 will be maximum when M=1.
221
21
221
01
max0
12
11
12
11
MM
M
T
T
21
21
221
01
max0
21
112
1
MM
M
T
T
Variation of Stagnation Temperature with Mach Number
M
T0
Heat Removal
Heat Addition Heat Addition
Heat Removal
M=1
0''' dTCdq p
Total heat transfer per unit mass flow rate
2
1
0
2
1
''' dTCdq p
0102'''
21 TTCq p
Total Heat Addition or Removal
1
01
0201
'''21 T
TTCq p
22
21
21
21
22
22
01
02
12
11
12
11
MMM
MMM
T
T
11
21
1
12
11
22
21
21
21
22
22
01'''
21
MMM
MMMTCq p
Relation Between M1 and M2
222
21
21
221
22
22
01
02
12
11
12
11
MMM
MMM
T
T
Adiabatic ideal flow:
222
21
21
221
22
22
12
11
12
11
1MMM
MMM
Similarly, for an ideal flow with heat addition
1
12
11
12
11
222
21
21
221
22
22
MMM
MMM
and for an ideal flow with heat removal
1
12
11
12
11
222
21
21
221
22
22
MMM
MMM
Relation Between M1 and M2
M2
M1
Adiabatic Heat AdditionHeat Removal
Entropy Change : Heat Addition or Removal
Integration from 1 to 2:
p
dp
T
dT
C
ds
p 1
pdTdC
ds
p
ln1
ln
121212 lnln
1lnln ppTT
C
ss
p
1
2
1
212 ln1
lnp
p
T
T
C
ss
p
2
12
2
12
12
1
1
2
2 1ln
1ln
M
M
M
M
C
ss
p
22
21
1
2
1
1
M
M
p
p
222
222
1
2
12
21
1
1
MM
MM
T
T
22
21
222
222
12
1
1ln
1
1
1ln
12
21
M
M
MM
MM
C
ss
p
1
2
2
2
2
12
1
2
2
1
1
1ln
M
M
M
M
C
ss
p
pC
ss 12
M1
Adiabatic
Heat Addition
Heat Removal
Maximum End Condition
• If heat is added to the flow, the Mach number tends towards one.
• If heat is removed from the flow, the Mach number tends away from one.
• All the properties of the flow can be conveniently written in terms of conditions that exist when M2 = 1.
22
22
*0
1
21
112
0 M
MM
T
T
2* 1
1
Mp
p
22
22
*1
1
M
M
T
T
1
12
2
*
2
M
M
12
122
*0
0
21
1
21
11
M
M
p
p
1
12
2
*
2
M
M
V
V
1
22
*
1
1ln
MM
C
ss
p
Variation of Normalized Properties
*0
0
p
p
*p
p
*0
0
T
T*T
T
*
M1
Nor
mal
ized
Val
ues
Temperature Entropy Relation
• Traditionally, heat addition or removal is characterized through relative temperature – entropy variations.
• Entropy signifies the quality of heat transfer process.
• An explicit relation between entropy and temperature is very useful in evaluating the heat transfer process.
p
dp
T
dT
C
ds
p 1
* **
1T
T
p
p
s
s p p
dp
T
dT
C
ds
On integration till maximum end point.
**
*
ln1
lnp
p
T
T
C
ss
p
2* 1
1
Mp
p
22
22
*1
1
M
M
T
T
The pressure ratio equation gives:
11
*2
p
pM
Substitute M2 in equation for temperature ratio :
01**
2
*
T
T
p
p
p
p
The roots of the equation are:
2
41
2
1 *
2
*
TT
p
p
2
41
2
1ln
1ln
*
2
*
* TT
T
T
C
ss
p
This allows the variation of temperature ratio with change in entropy to be found for any value of .
Rayliegh Line
One dimensional ideal flow with heat transfer is called as Rayliegh flow.
Maximum Entropy and Maximum Temperature Points
• Entropy will be maximum when M=1.
• Heat addition moves the Mach number towards 1 and vice versa.
• The point of maximum temperature occurs not at M=1.
• This value can be found by differentiating temperature ratio equation.
22
22
*1
1
M
M
T
T
01
14
1
1232
23
22
2
max
M
M
M
M
dM
dT
TT
M corresponding to Tmax:
1
M
4
1 2
*max
T
T
Tmax occurs at M<1 and
1*
max T
T
Summary of the Effect of Heat Transfer
q T0 M & V p s
M<1 + + + - +
M>1 + + - + +
M<1 - - - + -
M>1 - - + - -
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
One Dimensional Flow with Heat Transfer & Friction
P M V SubbaraoAssociate Professor
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
• 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
