web course 4

7/30/2019 Web Course 4

1/24

ME 1303 GAS DYNAMICS

AND JET PROPULSION

Presented by

G.Kumaresan


2/24

Flow Regime Classification

Subsonic Flow0.8 < M

Transonic Flow0.8 > M

> 1.2

Supersonic FlowM

> 1.2

Hypersonic FlowM

> 5

p

RTaRTRTTpch

conshch

and2but

1

02

2

1---------- (1)


3/24

Cont..

01

202

max2

12

2

1

1

2

1

20

0cons

0aa0,cAt;2max

2

10cons

maxand0,0At

;2

2

1

1

2

11

2

ha

cCa

ah

ch

cchT

consCa

pah

---------- (2), using 1&2

;Therefore from (1)

---------- (3)

Therefore from (3)

---------- (4); combining all


4/24

Problem-1

Example 1.

The jet of a gas at 593 K ( = 1.3 R = 469 J/kg K) has a Machnumber of 1.2. Determine the local and stagnation conditions

velocity of sound and enthalpy.

What is the Maximum attainable velocity of this jet ?

SolutionGIVEN:T = 593 K, = 1.3, R = 469 J / kg K, M = 1.2

For isentropic flow = 1.3, M = 1.2

T / To = 0.822 ; P / Po = 0.428

a = 601.29 m / s Ans


5/24

Cont..T / To = 0.822 To = 593 / 0.822 = 721.41 K

= 663.21 m / s

Ans

Now C* = a* &

= 618.44 m / s

Ans

Cmax = 1712.39 m / s Ans


6/24

Cont..

1R

pC

= ( 1.3 x 469) / 0.3 = 2032.333 Ans

ho = cp To = 3032.333 x 721.4112

= 1466.148 kJ / kg Ans

h = cp T = 2023.333 x 593

= 1205.174 kJ / kg Ans


7/24

SONIC VELOCITYGeneral

A disturbance at a given point creates a region of compressed

molecules that is passed along to its neighboring molecules and

in so doing creates a traveling wave.

Waves are measured by the amplitude of the disturbance.

The speed at which this disturbance is propagated through the

medium is called the wave speed.

Wave speed = f ( type of medium, thermodynamic state &strength of the wave)

The stronger the wave is, the faster it moves.


8/24

Cont..

Waves oflarge amplitude, which involve relatively large changes

in pressure and densityShock waves

Waves of very small amplitude, their speed is characteristic only

of the medium and its state- Sound waves

The presence of an object in a medium can only be felt by the

objects sending out or reflecting infinitesimal waves which

propagate at the characteristic sonic velocity.

For a compressible flow, the speed of propagation of smalldisturbances, called the sonic velocity (acoustic speed).

The ratio of the flow velocity to the sonic velocity Mach

number


9/24

Propagation of Sound waves

Piston

dV

p+dp+d

aV= 0P

wave front Tube

p+dpP

observer at rest Unsteady flow pattern

Pressure plot


10/24

Cont..

x

y

Moving observer Steady flow pattern

In this case the stagnant gas at pressure p on the left

appears to flow towards the right with a velocity a.

The wave can be considered as a stationary wave contained

within a control surface through which flow occurs from left

to right.


11/24

Writing continuity and momentum equation for the control volume

Continuity

For steady 1D flow, m = AV = cons. ; but A = cons.

V = cons. ; now for c.v

a = (+d) (a-dV)

expanding, then neglecting the higher-order and solving for dV

dV =da

Cont..

.

Momentum

Shear stress effect may be neglected because of infinitesimal

thickness of C.V

The x-component momentum equation is

(1)


12/24

ME 1303 GAS DYNAMICS

AND JET PROPULSION

Presented by

G.Kumaresan


13/24

ddP

a 2

Cont..

xxx i nVoutVmF PA ( P+dP ) A = Aa [ (a-dV) - a ] A dP = Aa dV

Cancelling the area and solving for dV, we have

dV = dP/ a (2)

By combining eqn. (1) & (2)(3)

Eqn. 3 shows that in compressible fluids in which there is a large

density change for a given pressure change the acoustic speed is

much lower compared to incompressible fluids.


14/24

ddp

RTd

dpP

d

dpP

TRpK

a

Cont..Eqn. 3 further simplified to

Where, K Bulk modulus of elasticity

ddp

d

dp

dv

dpv

volumei nchangerelative

pr essur ei nincreaseK

2

1

w.k.t adiabatic bulk modulus, pK

(a)

( b)

comparing (a) & (b)


15/24

Cont..

Speed of sound in different medium

Liquids Solids

where,

B- bulk modulus

E- youngs modulus


16/24

Cont..

Conclusion:

In a given fluid, a is higher at higher temperatures

Fluids with higher values of the K have higher a

a at a given temperature is higher for lower molecular weight ofgases vice versa

From the above it suggest that M plays an important role in the

design and working of machines using higher molecular weight

fluids.


17/24

a

CM

2

2

2

2

2;

22

2

a

C

a

C

aK

K

C

KA

AC

forceelastic

forceinert iaM

MACH NUMBER

RTV

a

VenergykinecticrandomenergykinecticdirectedM

2

2

2

2

V orCvelocity of the medium ; asonic velocity through the medium


18/24

Problem 2The temperature, pressure and Mach number at the entry of a

flow passage are 300 K, 3 bar and 1.4 respectively. If the exit

Mach number is 3 determine for adiabatic flow of a perfect gas (= 1.4, R=287 J/kg K):

(i) stagnation temperature (ii) temperature and velocity

of gas at exit, and (iii) the flow rate per square metre of the inlet

cross-section.

Solution:

For adiabatic flow 00201 TTT 1392.101392.12)4.1(

214.112

1211

101 TTMTT

3/484.3

300287

5103

1

11;6.417300392.101 mkg

RT

pKT

Ans


19/24

Cont..

2m-kg/s45.169306.486484.311

1

.m111

.

/06.4863002874.14.114.1111

CA

CAm

smTRaMC

smaMC

KTT

TT

MT

T

/4.73414.1492874.13222

14.1493008.2

392.1

2T011

022

1T

2T

8.22)3(

2

14.11222

11

2

02

Ans

Ans

Ans


20/24

Wave Propagation

Q.An airplane is traveling while you are observing from the

ground. How will you know whether it is subsonic or

supersonic?

Point disturbance is at rest Moving disturbance

M = 0 ( M = u/a = 0.5)

Always stays inside the family of circularsound waves


21/24

Wave fronts from Sonic disturbance

All the wave fronts coalesce on the left side and move along

with the disturbance.

No region upstream is forewarned of the disturbance as the

disturbance arrives at the same time as the wave front.

Zone of

Silence

Zone of action

W f f S i di b


22/24

Wave fronts from Supersonic disturbance

The wave fronts have coalesced to form a cone with the

disturbance at the apex.

The half angle at the apex is called Mach angle ( )

Always stays outside the family ofcircular sound waves

Di t b P ti


23/24

Disturbance PropagationIn Subsonic flow, bothRaj and Lisa can hear

Joy talking, since soundwaves travel from Joysmouth in all directions.

In Supersonic flow,sound waves (and otherdisturbances in the flow)travel only in the

downstream direction;thus, while Lisa can hearJoy talking, Raj cant.

Disturbances can nottravel upstream in asupersonic flow

P bl 3


24/24

Problem 3A needle nose projectile traveling at a speed of M=3 passes 200m

above an observer. Find the projectiles velocity and determine

how far beyond the observer the projectile will first be heard?

200 m

M=3

xo

M

smaMV

sm

TRa

5.193

11sin

11sin

/6.1041)2.347(3

/2.347

)300)(287(4.1

mxx 565

200

tan

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