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!%