ESA 368 Notes 03 - Isentropic Flow

7/28/2019 ESA 368 Notes 03 - Isentropic Flow

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ESA 368 High Speed Aerodynamics ZMK 2013 Aero USM

Isentropic Flow Formula Derivation

Applications & Sample Problems


2/15

Real flows are not entirely isentropic, but major

portions of the flow is isentropic

In practice, many flows can be assumed as

isentropic flow No shock waves, no heat transfer, no friction

Overview

Core flow is

isentropic

Example

flows in ducts or nozzles: Friction or heat transfer effects are important only

in the wall boundary layer

Use correction factors to handle non-isentropic

region


3/15

1. If area changes, how does if affect V, T, P, r?

2. For compressible flows, use M instead of V.

3. Examples: Jet or rocket nozzles

Convergent-divergent nozzles

Airfoils

Isentropic

core flow

Chapter Objectives


4/15

For isentropic flow:

And:

So:

1

1

2

1

1

2

1

2

r

r

P

P

T

T

RTa

1

1

2

1

1

2

2

1

2

2

1

2

r

r

P

P

a

a

T

T

Governing Equations

22

2

2

2

2

1

1

VTc

VTc pp

2

2

2

1

2

1

1

2

21

21

TcV

TcV

T

T

p

p

Applying energy equation to relate between T & M:

RT

V

a

VM

Mach Number:


5/15

Isentropic flow

So:

And:

1

1

2

1

1

2

2

1

2

2

1

2

r

r

P

P

a

a

T

T

2

22

2

11

22M

RT

V

Tc

V

p

1

2

Governing Equations

2

2

2

1

2

1

1

2

21

21

TcV

TcV

T

T

p

p

2

2

2

1

1

2

2

11

2

11

M

M

T

T

1

2

1

2

2

1

2

2

11

2

11

M

M

P

P

1

1

2

1

2

2

1

2

2

11

2

11

r

r

M

M


6/15

Sample Problem 1

Air flows through a convergent-divergent duct with an inlet area of 5 cm2 and an

exit area of 3.8 cm2. At the inlet section the air velocity is 100 m/s, the pressure is

680 kPa, and the temperature is 60oC.

Find the mass flow rate through the nozzle and the pressure and velocity at the

exit section. Assume that the flow is isentropic throughout the nozzle.

Sample Problems


7/15

222111VAVA rr

2

1

2

1

1

2

V

V

A

A

r

r

22

11

2

1

1

2

RTM

RTM

A

A

r

r

Using the relations r-M and T-M:

Where:

121

2

1

2

2

2

1

12

1

1

2

2

1

2

1

1

2

1

1

1

2

2

1

1

2

2

11

2

11

M

M

M

M

K

K

M

M

K

K

K

K

M

M

A

A

2

2

11 MK

Governing Equations

To find relation between A & M:


8/15

Flow stagnates when V changed isentropically to V = 0.

Can be used to relate points 0-1 and 0-2.

202

11 M

T

T

120

2

11

M

P

P

1

1

20

2

11

r

rM

2

2

2

1

1

2

2

11

2

1

1

M

M

T

T

1

2

1

2

2

1

2

2

1

1

2

11

M

M

P

P

1

1

2

1

2

2

1

2

2

11

2

11

r

r

M

M

0

0

0

0

rr

TT

PP

MV 1

20

Stagnation Conditions


9/15

1

0

2

Stagnation point is point 0

0

0

0

0

rr

TTPP

MV

Stagnation point is inside the chamber

Examples of Stagnation Conditions


10/15

Sample Problem 2

If Concorde is flying at a Mach number of 2.2, at an altitude of 10 km in the standard

atmosphere, find the stagnation pressure and temperature for the flow over the

aircraft.

Sample Problem 3

The pressure, temperature, and Mach number at the entrance to a duct through which

air is flowing are 250 kPa, 26oC, and 1.4 respectively. At some other point in the duct,

the Mach number is found to be 2.5.

Assuming isentropic flow, find the temperature, velocity, and pressure at the second

section. Also find the mass flow rate per square meter at the second section.

Sample Problems


11/15

V = 0

P0

P

Incompressible flow (Bernoulli equation):

Compressible flow:

2

0

2

1VPP r

r

PPV

0

2

120

2

11

M

P

P

11

2

1

0

P

PM

11

1

2

1

0

P

PP

a

V

Pitot Probe: Measuring Velocity


12/15

Critical condition is when flow

is isentropicallyaccelerated or

decelerated until M = M* = 1

M* = 1 T*, P*, r*, a*, A* =

2

1

1

1

2*M

T

T

12

1

1

1

2*

M

P

P

1

1

2

1

1

1

2*

r

rM

2

1

2

1

1

1

2*

Ma

a

1

Critical Conditions

2

3

4 A = A*

M = 1 only at throat

M < 1 M > 1

12

1

2

1

1

1

21

*

MMA

A

0

1

2

3

4

5

6

7

0 1 2 3

Mach number

A/A*

55


13/15

M T0/T P0/P r0/r a0/a A/A* q

0.50 1.05000 1.18621 1.12973 1.02470 1.33984 -

0.52 1.05408 1.20242 1.14073 1.02668 1.30339 -

2.40 2.15200 17.08589 7.59373 1.50000 2.63671 36.74650

2.42 2.17128 15.08357 6.94686 1.47353 2.44787 37.22883

20

2

11 M

T

T

120

2

11

M

P

P

1

1

20

2

11

r

rM

121

2

*1

1

1

21

M

MA

A

2

1

20

2

11

M

a

a

Table of Isentropic Flow


14/15

Air flow moves from a reservoir into a cylindrical converging-diverging nozzle. The

regions are labeled as reservoir, Region 1 (1), throat (t), exit (e), and the region just

outside the exit area (b). The flow condition is measured at Region 1 with M = 0.3,static P = 70 kPa, and diameter d = 10 cm.

If the exit Mach number is 3.4, calculate the throat and exit areas necessary to produce

that exit condition. Calculate also the pressure at the exit.

Sample Problem 4

Sample Problems


15/15

Sample Question 5A supersonic flow enters an inlet (Region 1) of a cylindrical converging-diverging

channel. The flow decelerates to the throat of the channel at Region 2, where the

diameter d2 = 0.5d1. The channel then diverges to a uniform-area tube at Region 3 with

a diameter d3 = 0.8d1. The flow then faces a moving normal shock wave, where the

density increases by 3.8 across the moving shock.

1. Calculate M3, P3, and V3.

2. If the flow decelerates from R1

R2, why is the flow at R3 still supersonic?

Sample Problems

ESA 368 Notes 03 - Isentropic Flow

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