Gas Dynamics (Module-1 & 2)

8/4/2019 Gas Dynamics (Module-1 & 2)

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Introduction to Gas Dyanamics Fluid: A substance which continuously deforms when

shearing forces are applied. Eg: Liquids, gases, vapors etc.

Incompressible fluid:

Density of the fluid remains constant during the flow.

Compressible fluid:

Density of the fluid changes during the flow.

System: A prescribed quantity of matter or region in space

upon which attention is concentrated for study. This

quantity of matter or region is separated from its

surroundings by a boundary.

Control volume: Arbitary volume fixed in space through

which fluid flows.



Concept of Gas Dyanamics

Deals with the study of compressible flow when it is in motion.

Analyses the high speed flows of gases and vapors’ considering

its compressibility.

Considers thermal and chemical effects.

Applications:

In steam and gas turbines

High speed aero dynamics Jet and rocket propulsion

High speed turbo compressors



Module – 1 & 2



Energy equation



According to first law of TD: Q = W+(E2-E1)

But E = U + mgZ + ½mc2

Differential form: dE = dU + mgdZ + md(½c

2

)

Integrating; ∫1

2dE = ∫

1

2dU + mg ∫

1

2dZ + ½m ∫

1

2dc2

E2 - E1 = (U2 - U1) + mg (Z2 - Z1) + ½m(c22 - c1

2)

Q = W + (U2 - U1) + mg (Z2 - Z1) + ½m(c22 - c1

2)

In terms of specific values:

q = w + (u2 - u1) + g (Z2 - Z1) + ½(c22 - c12)



Energy equation for a non-flow process Non-flow process: change or series of changes in closed system.

Constant volume heating or cooling of gas.

Expansion or compression in reciprocating engines.

Here kinetic and potential energy is negligible

Hence Q = Ws + (U2 - U1)

In differential form: dQ = dWs + dU

On assumption of perfect gas: dQ = pdV + mcvdT

Integrating: dQ = p∫1

2dV + mcv ∫

1

2dT

Hence dQ = p(V2 –V1) + mcv(T2 –T1)



Energy equation for a flow process Flow process: change or series of changes in a open system.

Expansion of steam and gas in turbines.

Expansion of gas in turbo compressors.

Work done: W = Ws + (p2v2-p1v1)

Q = Ws + (p2v2-p1v1) + (U2 - U1) + mg (Z2 - Z1) + ½m(c22 - c1

2)

Q = Ws + (U2 + p2v2) - (U1 + p1v1) + mg (Z2 - Z1) + ½m(c22 - c12)

Putting U + PV = H,

Q = Ws + (H2 -H1) + mg (Z2-Z1)+ ½ m(c22 - c1

2)



Q = Ws + (H2 -H1) + mg (Z2-Z1)+ ½m(c22 - c1

2)

H1 + mgZ1 + ½ mc12 + Q = H2 + mgZ2 + ½ mc2

2 + Ws

In terms of specific values:

h1 + gZ1 + ½c12 + q = h2 + gZ2 + ½c2

2 + ws

For an adiabatic process involving only energy transformation:

h1 + ½c12 = h2 + ½c2

2



Stagnation state

State of fluid attained by isoentropically decelerating it to

zero velocity at zero elevation.

Stagnation enthalpy Stagnation temperature

Stagnation velocity of sound

Stagnation pressure

Stagnation density



Stagnation enthalpy

Enthalpy of a gas when it is isoentropically decelerated to

zero velocity at zero elevation.

We have the energy equation,h1 + ½ c1

2 = h2 + ½ c22

Put h1=h, c1=c for initial state

h2= h0, c2=0 for final state

Stagnation enthalpy, h0 = h+½c2



Stagnation temperature





Stagnation pressure



Stagnation density



Stagnation velocity of sound



Adiabatic energy equation



Various regions of flow

By plotting adiabatic energy equationon c and a co-ordinates, steady flowellipse is obtained.

1. Incompressible flow: c<<a, henceM is very low.

Eg: Flow through nozzles

2. Subsonic flow: c<a, hence M<1

Eg: Passenger air craft

3. Sonic flow: c=a, hence M=1

Eg: Nozzle throat



4. Transonic flow: Small region on

both sides of sonic point. M is

slightly lesser or higher than

unity.

5. Supersonic flow: c>a, hence M>1

Eg: Military air crafts

6. Hypersonic flow: c>>a, hence Mis very high.

Eg: Rockets



Wave propagation of incompressible flow

Velocity ‘u’ of source is

negligibly small compared to

velocity of sound ‘a’.

Displacement of point S isinsignificantly small compared

to distance traveled by

pressure waves.



Wave propagation of subsonic flow

Source of disturbance travels

with a velocity less than that of

pressure waves.

Wave front move ahead of pressure point source.



Wave propagation of sonic flow

Point source travels with thesame velocity as that of wave.

Wave fronts always exists at present position of source

point.

Zone lying left of wave front is‘Zone of Silence’, because wavefronts do not reach there.

Zone on right is traversed bywaves and is therefore ‘Zone of Action’.



Wave propagation of supersonic flow

Point source is moving at a

velocity more than that of

sound.

Point source is always ahead of wave fronts.

Tangents drawn from point S

on the spheres define conical

surface called ‘Mach Cone’.



All waves are confined to

region within Mach Cone, so it

is referred as ‘Zone of Action’.

No waves reach region outsideMach Cone, so it is referred as

‘Zone of Silence’.

Semi angle of cone is known as

‘Mach Cone’, given by

a = sin-1 (1/M)



Reference velocities



Mach number, M*



Crocco Number



Bernoulli eqn. for compressible flow





Reynolds transport theorem



Continuity eqn. (Conservation of mass)









T-S diagram for expansion process

Initial stagnation pressure: P01s

Final stagnation pressure:

P02s (Isoentropic process)

P02a (Adiabatic process)

Stagnation temperature: T01 = T02

Initial kinetic energy: ½ c12

Final kinetic energy:

½ c2s2 (Isoentropic process)

½ c2a

2

(Adiabatic process)

Initial temperature: T1

Final temperature:

T2s (Isoentropic process)

T2a (Adiabatic process)



T-S diagram for compression process Initial stagnation pressure: P01s

Final stagnation pressure:

P02s (Isoentropic process)

P02a (Adiabatic process)

Stagnation temperature: T01 = T02

Initial kinetic energy: ½ c12

Final kinetic energy:

½ c2s2 (Isoentropic process)

½ c2a

2

(Adiabatic process)

Initial temperature: T1

Final temperature:

T2s (Isoentropic process)

T2a (Adiabatic process)



Flow through nozzles

A nozzle is a duct that increases the velocity of the flowing

fluid at the expense of pressure drop.

a) For M<1, dA= -ive: Nozzle area decreases b/w M = 0 to M = 1, giving

a convergent passage.b) For M=1, dA=0: There is no change in passage area. This section

is called throat. Mach no. is always unity.

c) For M>1, dA= +ive: Nozzle area continuously increases giving a

divergent passage.



Flow through diffusers

Diffusers are used to obtain pressure rise in flowing fluids

at the expense of velocity drop.

a) For M<1, dA= +ive: Nozzle area increases b/w M = 1 to M = 0,

giving a divergent passage.b) For M=1, dA=0: There is no change in passage area. This section

is called throat. Mach no. is always unity.

c) For M>1, dA= -ive: Nozzle area continuously decreases giving a

convergent passage.





Convergent – Divergent (De laval) Nozzle

Used to obtain a supersonic stream starting from low

speeds at the inlet.

Nozzle must converge in the subsonic portion and diverge

in the supersonic portion.

De laval Nozzle with M=1 at throat De laval Nozzle with De laval Nozzle with

subsonic flow supersonic flow



Effect of back pressure on nozzle

Fluid is allowed to flow from a

reservoir to an exhaust chamber

through a convergent nozzle.

Stagnation conditions in reservoir

are kept const, while back pressure

in exhaust chamber is varied.

Pressure distribution along the

nozzle for various values of

pressure ratio (Pe/P0) is plotted.



Curves a & b correspond to values of

pressure ratio more than critical.

Curve c correspond to critical

pressure ratio (Pb /P0 = 0.528)

For curves a, b & c, pe = pb

Nozzle exhaust pressure does not

decrease when exhaust pressure is

further reduced below critical value.

This condition is depicted by curves

d & e; here nozzle exit pressure is

still pe but pb < pe

Change of pe to pb takes place

outside nozzle exit through

expansion valve.



Effect of back pressure on De laval nozzle For curves a & b pressure ratio

across nozzle is such that flow is

accelerating only up to throat;

diverging part acts as diffuser

through which pressure rises to pb.

In curve c critical pressure ratio isreached at throat, but diverging part

acts as diffuser.

Curve h correspond to design value

of back pressure. All other curves

are for off-design conditions.



In curves d & e, when pb is further

lowered expansion takes place to

supersonic velocity beyond throat to

a point where discontinuity occurs.

For deceleration of supersonic flow,

passage should be convergent. But existing shape of nozzle downstream

is incompatible with required

process.

Hence flow readjusts itself to shape

of flow by suddenly becomingsubsonic.

Such a sudden change of supersonic

to subsonic flow occurs through

plane of discontinuity called ‘shock

wave’



When pb is further lowered shock

wave moves downstream till it

reaches exit as in curves f & g. Here

pb rises to pe through shock wave

outside nozzle exit.

Curves i & j correspond to pb < pe.Change of pe to pb takes place

outside nozzle exit through

expansion valve.



Under-expanding & Over-expanding nozzle

Under-expanding nozzle

When the back pressure of thenozzle Pb is below designpressure Pd, the nozzle is saidto be under-expanding.

Fluid enters at design pressurePd in nozzle and expandsviolently and irreversiblydown to the back pressure Pb after leaving the nozzle. The jet will exit in a diverging stream.

Over-expanding nozzle

When the back pressure of thenozzle Pb is above designpressure Pd, the nozzle is saidto be over-expanding.

For overexpansion in aconvergent nozzle exit pressure is greater than criticalpressure and effect is to reducemass flow rate through thenozzle.

For over-expansion in aconvergent-divergent nozzle,there is always an expansionfollowed by compression. Theflow will exit the jet in a

converging stream.



Flow through diffusers Diffusion occurs through a shock

wave (irreversible diffusion).

Pressure rise across shock wave is

sudden and goverened by upstream

Mach no.

Flow before and after shock is still

isentopic.



Impulse Function

Documents

Gas Dynamics (Module-1 & 2)