solu.on( consider venturi flow with a gaseous oxygen o...
TRANSCRIPT
1!MAE 5420 - Compressible Fluid Flow!
Homework(5a(Solu.on(
1
Consider Venturi Flow with a Gaseous Oxygen O2 with Measured Static Pressures p1, p2
p1 p2
2!MAE 5420 - Compressible Fluid Flow!
Homework(5a((2)(
2
A1 A2
flow(
P0p1
p2 P0p2<γ +12
"
#$
%
&'
γγ−1
Part%1:%Prove%that%for%unchoked%flow%:%%
m = A2 ⋅2γγ −1$
%&
'
()⋅
P02
Rg ⋅T0
$
%&&
'
())⋅
p2P0
$
%&
'
()
2γ
−p2P0
$
%&
'
()
γ+1γ
*
+
,,,
-
.
///
P0 =
A1A2
!
"#
$
%&
2
⋅ p1( )γ+1γ − p2( )
γ+1γ
A1A2
!
"#
$
%&
2
⋅ p1( )2γ − p2( )
2γ
*
+
,,,,,
-
.
/////
γγ−1
1%Point%Each%
&(
3!MAE 5420 - Compressible Fluid Flow!
Part 1, Proof Part i Mass constant throughout Venturi
Solve for Mass flow at Venturi inlet :
m = A1 ⋅P0 ⋅γ
Rg ⋅T0
⋅M1
1+ γ −12
M12$
%&
'
()
12γ+1γ−1$
%&
'
()→
Isentropic Flow→P0
p1
$
%&
'
()
γ−1γ
= 1+ γ −12
M12$
%&
'
()
M1 =2
γ −1P0
p1
$
%&
'
()
γ−1γ
−1+
,
---
.
/
000
4!MAE 5420 - Compressible Fluid Flow!
Part 1, Proof Part i (2)
Mass constant throughout Venturi
→ m = A1 ⋅P0 ⋅
γRg ⋅T0
$
%&&
'
())2
γ −1P0p1
$
%&
'
()
γ−1γ
−1+
,
---
.
/
0001
P0p1
$
%&
'
()
γ−1γ
⋅12γ+1γ−1$
%&
'
()
= A1 ⋅
2γγ −1$
%&
'
()
P0Rg ⋅T0
⋅P0P0p1
$
%&
'
()
γ−1γ
−1+
,
---
.
/
000
P0p1
$
%&
'
()
12γ+1γ
$
%&
'
()
= A1 ⋅2γγ −1$
%&
'
()ρ0 ⋅P0
P0p1
$
%&
'
()
γ−1γ
−1+
,
---
.
/
000
P0p1
$
%&
'
()
−γ+1γ
$
%&
'
()
= A1 ⋅2γγ −1$
%&
'
()ρ0 ⋅P0
p1P0
$
%&
'
()
2γ
−p1P0
$
%&
'
()
γ+1γ
$
%&
'
()+
,
---
.
/
000
m = A2 ⋅2γγ −1$
%&
'
()⋅
P02
Rg ⋅T0
$
%&&
'
())⋅
p2P0
$
%&
'
()
2γ
−p2P0
$
%&
'
()
γ+1γ
*
+
,,,
-
.
///
Since&Po,&To&are&constant&By&Analogy&
5!MAE 5420 - Compressible Fluid Flow!
Part 1, Proof, Part ii
UnChoked 1−D Compressible Massflow Equation :
m = A1 ⋅P0 ⋅γ
Rg ⋅T0
⋅M1
1+ γ −12
M12$
%&
'
()
12γ+1γ−1$
%&
'
()→Venturi Inlet
m = A2 ⋅P0 ⋅γ
Rg ⋅T0
⋅M 2
1+ γ −12
M 22$
%&
'
()
12γ+1γ−1$
%&
'
()→Venturi Throat
→
A1 ⋅M1
1+ γ −12
M12$
%&
'
()
12γ+1γ−1$
%&
'
()=
A2 ⋅M 2
1+ γ −12
M 22$
%&
'
()
12γ+1γ−1$
%&
'
()
Mass constant throughout Venturi
6!MAE 5420 - Compressible Fluid Flow!
Part 1, Proof Part ii (2) Divide A1 by A2
A1A2
=
M 2
1+ γ −12
M 22#
$%
&
'(
12γ+1γ−1#
$%
&
'(
M1
1+ γ −12
M12#
$%
&
'(
12γ+1γ−1#
$%
&
'(
=M 2
M1
⋅1+ γ −1
2M1
2#
$%
&
'(
12γ+1γ−1#
$%
&
'(
1+ γ −12
M 22#
$%
&
'(
12γ+1γ−1#
$%
&
'(→
Isentropic Flow→ P0
p1
"
#$
%
&'
γ−1γ
= 1+ γ −12
M12"
#$
%
&'
P0
p2
"
#$
%
&'
γ−1γ
= 1+ γ −12
M 22"
#$
%
&'
Substitute to eliminate Mach numbers
7!MAE 5420 - Compressible Fluid Flow!
Part 1, Proof Part ii(3)
M1 =2
γ −1P0p1
#
$%
&
'(
γ−1γ
−1)
*
+++
,
-
.
.
.
M 2 =2
γ −1P0p2
#
$%
&
'(
γ−1γ
−1)
*
+++
,
-
.
.
.
A1A2
=M 2
M1
⋅
P0p1
"
#$
%
&'
γ−1γ
*
+,
-,
.
/,
0,
12γ+1γ−1"
#$
%
&'
P0p2
"
#$
%
&'
γ−1γ
*
+,
-,
.
/,
0,
12γ+1γ−1"
#$
%
&'=M 2
M1
⋅p2p1
"
#$
%
&'
γ+12γ=
2γ −1
P0p2
"
#$
%
&'
γ−1γ
−11
2
333
4
5
666
2γ −1
P0p1
"
#$
%
&'
γ−1γ
−11
2
333
4
5
666
⋅p2p1
"
#$
%
&'
γ+12γ
8!MAE 5420 - Compressible Fluid Flow!
Part 1, Proof Part ii(4)
Simplify
A1A2
!
"#
$
%&
2
=
P0p2
!
"#
$
%&
γ−1γ
−1)
*
+++
,
-
.
.
.
P0p1
!
"#
$
%&
γ−1γ
−1)
*
+++
,
-
.
.
.
⋅p2p1
!
"#
$
%&
γ+1γ=
1p2
!
"#
$
%&
γ−1γ
−1
P0( )γ−1γ
)
*
+++
,
-
.
.
.
1p1
!
"#
$
%&
γ−1γ
−1
P0( )γ−1γ
)
*
+++
,
-
.
.
.
⋅p2p1
!
"#
$
%&
γ+1γ
→A1A2
"
#$
%
&'
2
⋅p1p2
"
#$
%
&'
γ+1γ
=
1p2
"
#$
%
&'
γ−1γ
−1
P0( )γ−1γ
+
,
---
.
/
000
1p1
"
#$
%
&'
γ−1γ
−1
P0( )γ−1γ
+
,
---
.
/
000
9!MAE 5420 - Compressible Fluid Flow!
Part 1, Proof Part ii(5)
Collect Terms and Simplify
A1A2
!
"#
$
%&
2
⋅ p1( )γ+1γ ⋅
1p1
!
"#
$
%&
γ−1γ
−1
P0( )γ−1γ
*
+
,,,
-
.
///= p2( )
γ+1γ ⋅
1p2
!
"#
$
%&
γ−1γ
−1
P0( )γ−1γ
*
+
,,,
-
.
///
→A1A2
!
"#
$
%&
2
⋅ p1( )γ+1γ ⋅
1p1
!
"#
$
%&
γ−1γ
1−p1( )
γ−1γ
P0( )γ−1γ
*
+
,,
-
.
//= p2( )
γ+1γ ⋅
1p2
!
"#
$
%&
γ−1γ
1−p2( )
γ−1γ
P0( )γ−1γ
*
+
,,
-
.
//
A1A2
!
"#
$
%&
2
⋅ p1( )2γ 1−
p1( )γ−1γ
P0( )γ−1γ
*
+
,,
-
.
//= p2( )
2γ 1−
p2( )γ−1γ
P0( )γ−1γ
*
+
,,
-
.
//
→A1A2
!
"#
$
%&
2
⋅ p1( )2γ P0( )
γ−1γ − p1( )
γ−1γ
*
+,-
./= p2( )
2γ P0( )
γ−1γ − p2( )
γ−1γ
*
+,-
./
10!MAE 5420 - Compressible Fluid Flow!
Part 1, Proof Part ii (6)
Solve for P0
P0( )γ−1γ ⋅
A1A2
$
%&
'
()
2
⋅ p1( )2γ − p2( )
2γ
*
+,,
-
.//=
A1A2
$
%&
'
()
2
⋅ p1( )2γ ⋅ p1( )
γ−1γ − p2( )
2γ ⋅ p2( )
γ−1γ
*
+,,
-
.//
→ P0( )γ−1γ ⋅ A1
2 ⋅ p1( )2γ − A2
2 ⋅ p2( )2γ
*
+,-
./= A1
2 ⋅ p1( )γ+1γ − A2
2 ⋅ p2( )γ+1γ
*
+,-
./
→ P0 =A12 ⋅ p1( )
γ+1γ − A2
2 ⋅ p2( )γ+1γ
A12 ⋅ p1( )
2γ − A2
2 ⋅ p2( )2γ
*
+
,,
-
.
//
γγ−1
P0 =
A1A2
!
"#
$
%&
2
⋅ p1( )γ+1γ − p2( )
γ+1γ
A1A2
!
"#
$
%&
2
⋅ p1( )2γ − p2( )
2γ
*
+
,,,,,
-
.
/////
γγ−1
11!MAE 5420 - Compressible Fluid Flow!
Homework(5a((6)(
11
Consider Venturi Flow with a Gaseous Oxygen O2 with Measured Static Pressures p1, p2
p1
D1
T0
!
"
###
$
%
&&&=
80 kPa1 cm
300 oK
!
"
###
$
%
&&&→
p2
D2
!
"##
$
%&&= 60kPa
0.5 cm
!
"#
$
%&
A = π4D2
• Part 2: Assuming Isentropic Flow .. Calculate the Massflow and Flow Velocity Through Venturi at stations (1) and (2)
γ = 1.4, Mw = 32.0 kg/kg-mol 3%Points%%
12!MAE 5420 - Compressible Fluid Flow!
Part 2 • Calculate Stagnation Pressure
P0 =
A1A2
!
"#
$
%&
2
⋅ p1( )γ+1γ − p2( )
γ+1γ
A1A2
!
"#
$
%&
2
⋅ p1( )2γ − p2( )
2γ
*
+
,,,,,
-
.
/////
γγ−1
=(
kPa
13!MAE 5420 - Compressible Fluid Flow!
Part 2 (2)
• Calculate Massflow through Venturi
m = A2 ⋅2γγ −1$
%&
'
()⋅
P02
Rg ⋅T0
$
%&&
'
())⋅
p2P0
$
%&
'
()
2γ
−p2P0
$
%&
'
()
γ+1γ
*
+
,,,
-
.
///
=(
= 3.48348 kPa
14!MAE 5420 - Compressible Fluid Flow!
Part 2 (2)
• Calculate Flow Velocities at station (1)
M1 =2
γ −1#
$%
&
'(⋅
P0p1
#
$%
&
'(
γ−1γ
−1#
$
%%%
&
'
(((
#
$
%%%
&
'
(((=
V1 =M1 ⋅ γ ⋅Rg ⋅T1 =M1 ⋅γ ⋅Rg ⋅T0
1+ γ −12
M1 2=
m/sec
15!MAE 5420 - Compressible Fluid Flow!
Part 2 (3)
• Calculate Flow Velocities at station (2)
m/sec
M 2 =2
γ −1#
$%
&
'(⋅
P0p2
#
$%
&
'(
γ−1γ
−1#
$
%%%
&
'
(((
#
$
%%%
&
'
(((=
V2 =M 2 ⋅ γ ⋅Rg ⋅T2 =M 2 ⋅γ ⋅Rg ⋅T0
1+ γ −12
M 22=
16!MAE 5420 - Compressible Fluid Flow!
Homework(5a((7)(
16
Compare results of compressible Venturi calculations to values calculated using incompressible Venturi Equation Massflow, V1, V2 1-point
V2 =2 ⋅ p1 − p2( )
ρ ⋅ 1− A2A1
$
%&
'
()
2$
%&&
'
())
→ m = A2 ⋅2 ⋅ ρ ⋅ p1 − p2( )
1− A2A1
$
%&
'
()
2$
%&&
'
())
Incompressible%Venturi%Equa?ons%
p1 p2Assume&T1&=&T0&=&300&K&
ρ&=&constant&&
17!MAE 5420 - Compressible Fluid Flow!
Part 3
• Calculate Massflow Using Incompressible Venturi Equations
m = A2 ⋅2 ⋅ ρ ⋅ p1 − p2( )
1− A2A1
$
%&
'
()
2$
%&&
'
())
=
g/sec
18!MAE 5420 - Compressible Fluid Flow!
Part 3 (1)
• Calculate Massflow Using Incompressible Venturi Equations
m/sec
V1 =m
ρ ⋅A1=A2A1⋅
2 ⋅ p1 − p2( )
ρ ⋅ 1− A2A1
$
%&
'
()
2$
%&&
'
())
=
19!MAE 5420 - Compressible Fluid Flow!
Part 3 (2)
• Calculate Massflow Using Incompressible Venturi Equations
m/sec
V2 =m
ρ ⋅A2=
2 ⋅ p1 − p2( )
ρ ⋅ 1− A2A1
$
%&
'
()
2$
%&&
'
())
=
20!MAE 5420 - Compressible Fluid Flow!
Compressible vs. Incompressible Comparisons
Flow Conditions Massflow Calculation
Mach Number Flow Velocity
Compressible Station 1 (inlet)
3.4835 g/sec 0.130597 43.069 m/sec
Station 2 (throat) 3.4835 g/sec 0.668471 211.574 m/sec
Incompressible Station 1 (inlet)
4.1333 g/sec -- 50.670 m/sec
Station 2 (throat) 4.1333 g/sec -- 202.681 m/sec
Quite%a%big!%Difference!%