1MAE 5420 - Compressible Fluid Flow
Section 2: Lecture 1
Integral Form of the Conservation Equations
for Compressible Flow
Anderson: Chapter 2 pp. 41-54
2MAE 5420 - Compressible Fluid Flow
Section 1 Review
• Equation of State:
• Relationship of Rg to specific heats,
• Internal Energy and Enthalpy
p = !RgT"> Rg =RuM
w
- Ru = 8314.4126 J/°K-(kg-mole)
- Rg (air) = 287.056 J/°K-(kg-mole)
cp= c
v+ R
g
h = e + Pv cv=
de
dT
!"#
$%&v
cp =dh
dT
!"#
$%&p
! =cp
cv
3MAE 5420 - Compressible Fluid Flow
Chapter 1 Review (cont’d)
• First Law of Thermodynamics, reversible process
• First Law of Thermodynamics, isentropic process(adiabatic, reversible)
de= !pdv
de= dq ! pdv dh = dq+ vdp
dh = vdp
4MAE 5420 - Compressible Fluid Flow
Chapter 1 Review (cont’d)
• Second Law of Thermodynamics, nonadiabatic,
reversible process
• Second Law of Thermodynamics, isentropic process(adiabatic, reversible)
s2! s
1= cp ln2
T2
T1
"
#$
%
&' ! Rg ln
p2
p1
"
#$
%
&'
p2
p1
=T2
T1
!
"#
$
%&
'' (1 p
2
p1
!
"#
$
%& =
'2
'1
!
"#
$
%&
(
5MAE 5420 - Compressible Fluid Flow
Chapter 1 Review (concluded)
• Speed of Sound for calorically Perfect gas
• Mathematic definition of Mach Number
c = ! RgT
M =V
! RgT
6MAE 5420 - Compressible Fluid Flow
Concept of a Control Volume
• Arbitrary Volume (C.V), fixed in space, with fluid flux across its boundaries
• Outer surface of C.V is known as Control Surface (C.S.)
• System defined as the Mass
Within the control volume at
Some Instant in time t
• Mass within control volume
Must OBEY Laws of mass
And momentum conservation
mout
•
min
•
7MAE 5420 - Compressible Fluid Flow
Conservation of Mass• At any instant in time, the mass contained within the C.V. is conserved
• Within an elemental piece dv of the control volume
• And the total Contained mass is
!
!tm
C .V( ) = m
•
C .V .
"""in
# m
•
C .V .
"""out
dmC .V
= !dv
moutmout
•
min
•
dvmC .V
= !dvc.v.
"""
8MAE 5420 - Compressible Fluid Flow
Conservation of Mass (continued)
• The mass flow out of the C.V. across an elemental piece of
the control surface, ds, is
• And the incremental
mass-flow into the C.V. is:
• Integrating over the entire
control surface
d m
•
out
!"
#$ = %V
outcos &( )ds
moutmout
•
min
•
dv
n
v
!
dS
m
•
C .V . = ! "V!>
• ds
!>#$
%&
C .S .
''
d m
•
in
!"
#$ = %&V
incos '( )ds
! "V!>
• ds
!>#$
%&
C .S .
'' =((t
"dvc.v.
'''#
$)%
&*
9MAE 5420 - Compressible Fluid Flow
Conservation of Mass (concluded)
• One Dimensional Steady Flow (mass flow in = mass flow out)
!!t
"dvc.v.
###$
%&'
()= 0 = * "V
*>
• ds
*>$%
'(
C .S .
## + "1V1A1= "
2V2A2
“continuity equation”
10MAE 5420 - Compressible Fluid Flow
Conservation of Momentum• Newton’s second law applied to control volume
• Momentum flow across control surface
• But from earlier analysis, across an incremental
Surface area
F
!>
" =d
dtmV
!>#$
%& =
Momentum flow across C.S. +
Rate of Change of Momentum within C.V.
vn
V
n
V
m
•!"
#$
C .S .
%% V
&>
m
•
= !V">
• ds
">
# m
•$%
&'
C .S .
(( V
">
= !V">
• ds
">$%
&'
C .S .
(( V
">
11MAE 5420 - Compressible Fluid Flow
Conservation of Momentum (cont’d)
• Rate of Momentum Change within Control Volume
• But from earlier (continuity) analysis
For an incremental volume, dv
• And total net rate of momentum change within the control volume is
dVn
V
n
V
!
!t(mV
">
)
m = !dv
!!t(mV
">
)C .V .
=!!t
#dvV">$
%&'
C .V .
((( =!!t
#V">$
%&' dv
C .V .
(((
12MAE 5420 - Compressible Fluid Flow
Conservation of Momentum (cont’d)
• Then Newton’s second law becomes
• Resolution of Forces Acting on Control Volume
- Body Forces (electo-magnetic, gravitational, buoyancy etc.)
- Pressure Forces
- Viscous or frictional Forces
F
!>
" =d
dtmV
!>#$
%& = 'V
!>
• ds
!>#$
%&
C .S .
(( V
!>
+))t
'V!>#
$%& dv
C .V .
(((
F!>
"#$%&body
= ' fb!>
dvC .V .
(((
F!>
"#$%&pressure
= ! p( )C .S .
'' dS!> (Minus sign
Because pressure
Acts inward)
F!>
"#$%&fric
= | f!>
' dS!>#
$%&
C .S .
(( |dS
SC .S .
!>
13MAE 5420 - Compressible Fluid Flow
Conservation of Momentum (continued)
• Collected terms for conservation of Momentum
• Steady, In-viscous
Flow, Body Forces negligible
! fb">
dvC .V .
### " p( )C .S .
## dS">
+ | f">
$ dS">%
&'(
C .S .
## |dS
SC .S .
">
=
!V">
• ds">%
&'(
C .S .
## V">
+))t
!V">%
&'( dv
C .V .
###
! fb">
dvC .V .
### " p( )C .S .
## dS">
+ | f">
$ dS">%
&'(
C .S .
## |dS
SC .S .
">
=
!V">
• ds">%
&'(
C .S .
## V">
+))t
!V">%
&'( dv
C .V .
### ! p( )C .S .
"" dS!>
= #V!>
• ds!>$
%&'
C .S .
"" V!>
14MAE 5420 - Compressible Fluid Flow
Conservation of Momentum (cont’d)
• One-Dimensional Flow
! p( )C .S .
"" dS!>
= #V!>
• ds!>$
%&'
C .S .
"" V!>
p1
p2
A2
A2
V1
V2
• No flow across solid boundary
! p2A2! p
1A1( ) i_
x = "2V2A2V2i_
x! "1V1A1V1i_
x
# p1+ "
1V1
2( )A1 = p2+ "
2V2
2$% &'A2
15MAE 5420 - Compressible Fluid Flow
Conservation of Momentum (concluded)
• One-Dimensional Flow
! p( )C .S .
"" dS!>
= #V!>
• ds!>$
%&'
C .S .
"" V!>
p1
p2
A2
A2
V1
V2
• No flow across solid boundary
p1+ !
1V1
2( )A1 = p2+ !
2V2
2"# $%A2 &
! j =pj
RgTj
=' pj' RgTj
=' pjcj2& p
11+ '
V1
2
c1
2
(
)*+
,-A1= p
21+ '
V2
2
c2
2
(
)*+
,-A2
p2
p1
=A1
A2
1+ 'M1
2( )1+ 'M
2
2( )
16MAE 5420 - Compressible Fluid Flow
Conservation of Energy
• From First law of thermodynamics (reversible process)
(corresponding to inviscid fluid)
Rate of change of fluid energy as it flows thru C.V.
Rate of external heat addition to C.V.
Rate of work done to fluid inside of C.V.
dE
dt=dQ
dt+dW
dtdE
dt!
dQ
dt!
dW
dt!
17MAE 5420 - Compressible Fluid Flow
Conservation of Energy (cont’d)
Rate of change of fluid energy as it flows thru C.V. •
Earlier thermodynamic discussions were for static fluid… flow thru C.V. is not static
… Kinetic Energy terms must also Be captured … following process similar to
Momentum flow through C.V.
dE
dt!
dE
dt=
!!t
" e +V2
2
#
$%
&
'( dv
#
$%%
&
'((
C .V .
))) + "V*>
• d S
*>
e +V2
2
#
$%
&
'(
C .S .
))
Rate of change of energy with C.V Energy Flow Across C.S.
18MAE 5420 - Compressible Fluid Flow
Conservation of Energy (cont’d)
Rate of heat addition to C.V. •dQ
dt! ! q
•"#
$% dv
"#&
$%'
C .V .
(((
• dW
dt! Rate of work done to fluid inside of C.V.
dW
dt!
d
dt(F • dx) = F •V ! dx = direction " of " motion
Consider: 1) Pressure Forces, 2) Body Forces
19MAE 5420 - Compressible Fluid Flow
Conservation of Energy (cont’d)
1) Pressure Forces
2) Body Forces
! (pd S!>
) •V!>
C .S .
""
(! f">
dv) •V">
C .V .
###
MAE 5420 - Compressible Fluid Flow 20
Conservation of Energy (cont’d)
• Collected Energy EquationScalar equation
MAE 5420 - Compressible Fluid Flow 21
Conservation of Energy (concluded)
• Steady, Inviscid quasi 1-D Flow, Body Forces negligible
MAE 5420 - Compressible Fluid Flow 22
Conservation of Energy (concluded)
• Steady, Inviscid quasi-1-D Flow, Body Forces negligible
MAE 5420 - Compressible Fluid Flow 23
Conservation of Energy (concluded)
• Steady, Inviscid quasi 1-D Flow, Body Forces negligible
MAE 5420 - Compressible Fluid Flow 24
Conservation of Energy (concluded)
• Steady, Inviscid quasi 1-D Flow, Body Forces negligible, collected
• Divide through by massflow
MAE 5420 - Compressible Fluid Flow 25
Conservation of Energy (concluded)
• but from continuity, equation of state
MAE 5420 - Compressible Fluid Flow 26
Conservation of Energy (concluded)
• but from continuity, equation of state
h = e + Pv = e +RgT
MAE 5420 - Compressible Fluid Flow 27
Summary
• Continuity (conservation of mass)
• Steady quasi One-dimensional Flow
MAE 5420 - Compressible Fluid Flow 28
Summary (continued)
• Newton’s Second law-- Time rate of change of momentumEquals integral of external forces
• Steady, Inviscid quasi 1-D Flow, Body Forces negligible
MAE 5420 - Compressible Fluid Flow 29
Summary (concluded)
• Conservation of Energy--
• Steady, Inviscid quasi 1-D Flow, Body Forces negligible
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