NON-ISENTROPIC FLOW

2

MENU

1. Non-Isentropic flow : Real flow 2. Shockwave & Expansion waves Normal shockwave Oblique shockwave Prandtl Meyer expansion

3. Duct Flow with Friction without Heat Transfer (Fanno flow)

4. Duct Flow with Heat Transfer and Negligible Friction Force (Reyleigh Flow)

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Non-Isentropic flow : Real flow

What is Non-Isentropic Flow ?

1. Irreversible ( there is viscous effect) only 2. Non Adiabatic ( There is heat transfer) only 3. Combination of Both

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Non-Isentropic flow : Real flow

What is Non-Isentropic Flow ?

2 1 = ln0201

ln0201

1. Irreversible ( there is viscous effect) only 01 02 and 01 = 02

2. Non Adiabatic ( There is heat transfer) only 01 02 and 01 = 02

3. Combination of Both 01 02 and 01 02

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Isentropic flow

Flow through Convergent Divergent Duct

(1) 0= 1 ,

0= 1 ,

0= 1

Flow conditions

Is there an flow through the duct?

How is about the following flow through the duct?

(2) 0= 1 ,

0= 0.9725,

0= 0.7 5

(3) 0= 0.95 ,

0= 0.85,

0= 0.528

(4) 0= 0.95 ,

0= 0.1278,

0= 0.528

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Isentropic flow

Flow through Convergent Divergent Duct

Mach number and Mass flow rate

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Non-Isentropic flow

Flow through Convergent Divergent Duct

Normal shock wave

If the pressure at exit , pe is less than p3 and greater than pd so the normal shock wave will appear inside the divergent duct

The shock wave occurs in supersonic speed

The Mach number change from supersonic to subsonic

The flow properties change abruptly

pc

8

Non-Isentropic flow

Normal Shock Normal shockwave : shock waves that

occur in a plane / cross section normal to the direction of flow

A supersonic flow across a normal shock wave becomes subsonic

Total enthalpy remains constant across the shock (conservation energy principle)

01 = 02 = 1 + 1

2

2= 2 +

22

2

Continuity equation

= 11 = 22

Momentum equation

1 2 =

(2 1)

Entropy

2 1 > 0

01 = 02

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Non-Isentropic flow

Prandtl Relation

Flow relation between before shock wave and after shock wave

2 =

1

1

1 2

5.0

2

*

)1(2

1

MMM

where

22 =

2 + ( 1)12

2 12 ( 1)

()2 = 12

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Non-Isentropic flow

Prandtl Relation

Flow relation between before shock wave and after shock wave

1 2

22 =

2 + ( 1)12

2 12 ( 1)

What is happen if M1

11

Non-Isentropic flow

Prandtl Relation

Flow relation between before shock wave and after shock wave

2

1

12

Non-Isentropic flow

Static Properties Relation

Properties relation between before shock wave and after shock wave

Pressure

21

=2 1

2 ( 1)

( + 1)

Density

21

=12

=( + 1)1

2

2 + ( 1) 12

Temperature

21

= 2 1

2 1 [2 + 1 12]

( + 1)2 12

What is happen if M1

13

Non-Isentropic flow

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Non-Isentropic flow

Entropy

2 1

= ln0102

0201

=21

2 ( 1)

( + 1)

1/(1)( + 1)1

2

2 + ( 1) 12

/(1)

2 1

=1

( 1)

2 12

( + 1) 1

+ 1+

( 1)

2 + ( 1) 12

( + 1)12

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Non-Isentropic flow

Entropy

1

16

Non-Isentropic flow

Air Speed Measurement in Supersonic flight

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Non-Isentropic flow

1. A blunt nose missile is flying at Mach 2 at standard sea level. Calculate the temperature and pressure at the nose of the missile

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Non-Isentropic flow

Air Speed Measurement in Supersonic flight

Pressure ratio

1

)1(2

124

1 211

2

1

2

1

2

1

02

M

M

M

p

p

19

Non-Isentropic flow

2. Air with initial stagnation conditions of 700 kPa and 530 K passes through a frictionless CD nozzle Th troat area is 5 cm2 and the exit area is 12.5 cm2. The back pressure is 350 kPa, and a normal shock wave occurs within the diverging section. determine

(a) The Mach number at the exit (b) The change in stagnation pressure (c) Mach number before and after the shock (d) the nozzle area at the point of shock (e) The back pressure if the flow were isentropic throughout

20

Non-Isentropic flow

QUIZ 3 Air with initial stagnation conditions of 700 kPa and 330 K passes

through a CD nozzle at the rate of 1 kg/s. At the exit are of the nozzle the stagnation pressure is 550 kPa and the stream pressure is 550 kPa. The nozzle is insulated and there is no irreversibility except for the occurrence of a shock

(a) What is the nozzle throat area ? (b) What is Mach number before and after the shock ? (c) What is the nozzle area at the point of shock and at the exit (d) What is the stream density at the exit (e) The back pressure if the flow were isentropic throughout

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Non-Isentropic flow

Rankine-Hugoniot Relations

Combining continuity equation and momentum equation

2 1 = 112 22

2 = 112(1

12)

21

=

+ 1 1

12

1

+ 1 1

12

21

=

+ 1 1

21

1

+ 1 1

21

Non-Isentropic flow

Duct Flow with Heat Transfer (Reyleigh Flow)

Combining continuity, momentum and energy equations

21 00TCqTC pp

Is the following isentropic gas law still valid ?

u1 u2

p1, r1 p2, r2

Fire

q

With Heat Addition

Non-Isentropic flow

Static Properties relations

2

2

2

1

1

2

1

1

M

M

p

p

Pressure Temperature

2

1

2

2

2

2

2

1

1

2

1

1

M

M

M

M

T

T

Density

2

2

1

2

1

2

2

1

2

1

1

M

M

M

M

r

r

Total Properties relations

1

2

12

1

2

22

1

2

2

2

1

10

20

1

1

1

1

M

M

M

M

p

p

Total Pressure Total Temperature

2

12

1

2

22

12

1

2

2

2

2

2

1

10

20

1

1

1

1

M

M

M

M

M

M

T

T

Non-Isentropic flow

Critical Static Properties relations

Pressure Temperature Density

Critical Total Properties relations

Total Pressure Total Temperature

2* 1

1

Mp

p

2

2

2

* 1

1

MM

T

T

r

r

1

112

2*

M

M

1

1

)1(2

1

12

2*

0

0

M

Mp

p

2

22

2

*

0

0 121

1M

M

M

T

T

Non-Isentropic flow

Graph of Total Temperature at Various Mach number

Adding heat will increase flow velocity or Mach number in Subsonic flow and decrease flow velocity of Mach number in Supersonic flow

Extracting heat (cooling of the flow) will decrease flow velocity or Mach number in Subsonic flow and increase flow velocity of Mach number in

Supersonic flow

0.0

1.0

2.0

3.0

0.0 0.5 1.0 1.5

T0/T0*

Mach

2

22

2

*

0

0 121

1M

M

M

T

T

Supersonic

Subsonic

Non-Isentropic flow

Graph of Total Pressure at Various Mach number

Supersonic

Subsonic

1

1

)1(2

1

12

2*

0

0

M

Mp

p

0

1

2

3

0 1 2 3 4 5 6

p0/p0*

Ma

ch

Non-Isentropic flow

Graph of Static Pressure at Various Mach number

0.0

1.0

2.0

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

p/p*

Ma

ch

2* 1

1

Mp

p

For subsonic flow (M1), adding heat will increases pressure

Supersonic

Subsonic

Non-Isentropic flow

Graph of Density at Various Mach number

Supersonic

Subsonic

r

r

1

112

2*

M

M

0.0

1.0

2.0

3.0

0.0 2.0 4.0 6.0 8.0 10.0 12.0

r /r *

Mach

Non-Isentropic flow

Graph of Static Temperature at Various Mach number

2

2

2

* 1

1

MM

T

T

0.0

1.0

2.0

3.0

0.0 0.5 1.0 1.5

T/T*

Mach

0.0

1.0

2.0

3.0

0.0 0.5 1.0 1.5

T/T*

Mach

Supersonic

Subsonic

30

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Non-Isentropic flow

Rankine-Hugoniot Relations

Combining continuity equation and momentum equation

1 2 + 12 =1

2 (1 2)(

1

2+

1

1)

1 2 + 12 =1

2 (1 2)(2 + 1)

1 2 + 12 =1

2 (1 + 2)(

1

1

1

2)

1 2 + 12 =1

2 (1 + 2)(1 2)