gas dynamics and nozzle

7/29/2019 Gas Dynamics and Nozzle

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Table of contents1 Generalized equations .......................................................................................................2

1.1 Continuity equation .......................................................................................................................................2

1.2 Momentum equation .....................................................................................................................................2

1.3 Energy equation ...........................................................................................................................................3

1.4 Isentropic Condition ......................................................................................................................................3

2 Gas Dynamics and Nozzle .................................................................................................4

2.1 Speed of sound and mach number ..............................................................................................................4

2.2 One-Dimensional flow ..................................................................................................................................52.2.1 Energy equation ....................................................................................................................................52.2.2 Sonic Properties ....................................................................................................................................62.2.3 General 1-D flow equations ...................................................................................................................62.2.4 Critical pressure ratio ............................................................................................................................92.2.5 Effect of friction ...................................................................................................................................102.2.6 Normal Shock Waves ..........................................................................................................................122.2.7 Diffuser ................................................................................................................................................13

2.3 Nozzle ........................................................................................................................................................142.3.1 Flow of steam through nozzle .............................................................................................................142.3.2 Condition of maximum discharge ........................................................................................................152.3.3 Supersaturated flow and effect of friction ............................................................................................15

Gas Dynamics

Nozzles

March 2013


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1 Generalized equations

1.1 Continuity equation

The time rate of change of the mass contained in the volume Vis equal to the (negative) value of the mass flux

across the surface. Mass flux is density x velocity.

= u.dS

(steady one-dimensional flow)

Where,

is mass flow rate

is density

dS is small change in area

Vis volume

u is velocity

Bold represents vector

1.2 Momentum equation

The time rate of change of the fluid momentum in volume Vequals the surface integral of the momentum flux due tofluid flow across S, plus the effect of the pressure Pacting on the fluid across the surface S, plus the contribution ofexternal forces (e.g. gravity) acting on every point ofV(causing an acceleration a)



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1.3 Energy equation

The time rate of change of the total fluid energy (kinetic energy of fluid motion plus internal energy, where e denotesthe specific internal energy) equals the surface integral of the energy flux (kinetic + internal), plus the surface integralof the work done by the pressure plus the volume integral of the work done by external forces (e.g. gravitation):

is rate of heat added per unit masse is internal energy


1.4 Isentropic Condition

Flow which is adiabatic, nonviscous, nonconducting and is in equilibrium condition is isentropic.

s=constant

Where, s=entropyh=enthalpy

The equilibrium condition cannot be attained in reality as flow is adjusting itself everytime to new condition. And therate of adjustment depends on gradient in the flow.


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2 Gas Dynamics and Nozzle

2.1 Speed of sound and mach number

Sound waves travel through the air at a velocity that is related to mean molecular velocity of air, as molecularcollisions cause sound waves to propagate. Sound waves are weak waves.Lets assume that velocity of sound is a through the air. Because there is continuous change in the flow properties,the flow behind has different velocity. However the change is very small, therefore change in velocity be da.

Applying continuity equation,For 1-D flow,

Applying momentum equation,

Change is isentropic, the fundamental equation for speed of sound:


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By substituting the ratio of pressure and density, could be found.

Mach number for calorically perfect ideal gas is

2.2.2 Sonic Properties

Let *" denote a property at the sonic state M2=1

2.2.3 General 1-D flow equations

Flow with area change is illustrated by the following sketch of a control volume:


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For this problem adopt the following conventions>surface 1 and 2 are open and allow fluxes of mass, momentum, and energy>surface w is a closed wall; no mass flux through the wall>external heat flux qw (Energy/Area/Time: Wm2 ) through the wall allowed-qw known fixedparameter>diffusive, longitudinal heat transfer ignored, qx = 0

>wall shear (Force/Area: Nm2 ) allowed{ known, fixed parameter

>diffusive viscous stress not allowed = 0

>cross-sectional area a known fixed function: A(x)

The amount of mass in a control volume after a time increment t is equal to the originalamount of mass plus that which came in, minus that which left. Using continuity equation:

For steady state case:

By momentum conservation equation:

Steady state case

By energy conservation equation:

Steady state case

Effect of area changeIn the isentropic limit the mass, momentum, and energy equation:


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Now, substituting energy and mass equation into momentum we get,

It is noted that>equation singular when M2 = 1

>if M2 = 1, one needs dA = 0>area minimum necessary to transition from subsonic to supersonic flow!!>can be shown area maximum not relevant.

Consider A at a sonic state. From the mass equation:


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Substituting to the earlier equations, we get

Note:

> has a minimum value of 1 at M = 1

>For each > 1, there exist two values of M

> as M 0 or M .

2.2.4 Critical pressure ratio

The pressure at which area is minimum while the discharge per unit are is maximum is called critical pressure.

Infinite Reservoir 0 0 0 0, , ,P v T V 1 1 1 1, , ,P v T V

The critical pressure ratio is the value of P P1 0/ below which no further increase of m will occur.

Pv Ck = . Cp / CV=k

Our problem is now to find rP

P= 1

0

for which1

1

AVm

v=& is maximum.

v vP

Pv

ro

ok

o

k

1

1

1 1

1=

=

2 2

1

1 1 12 2

o

o o o

V Vu P v u Pv+ + = + +

Where, =volumeV=velocityu=internal energy

Let 0o

V =2

1

1 1 12

o o o

Vu u P v Pv= +

Noting that u uo

1 is equivalent to isentropic work:

( )

2

0 0 1 1 0 01 1 10 0 1 1 0 0 1 1

0 012 1 1 1

P v Pv kP vV k Pv

P v Pv P v Pvk k k P v

= + = =


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but,

1

01

0 1

kPv

v P

=

and r

P

P= 1

0

; therefore,

12

10

12 1

k

k

o

V kP v r

k

=

1

1 0

21

1

k

k

o

kV P v r

k

=

2 1 2 1

01

1

2

1

k k

k k k k

o

PAV km A r r A const r r

v k v

+ + = = =

&

( )

1

2 1 2 12 1 12 10

k k

k k k k

m kconst r r r r

r k k

+ +

+= =

&

( )2 1 02 1

r k r

k

k k

+ =

( )2 1 0

1

2 + =k

r

r

k

k

k

( )2 1 01

+ =

k r

k

k ; and, finally:

r

k

k

k

=+

12

1

rk

cr

k

k

=+

2

1

1

For air k=1.41; 0.527rc

r = For superheated steam 1.28;k r

cr= 0546.

Fordry, saturated steam k 1135. ; rcr 0577.

2.2.5 Effect of friction

In real life, all fluids are viscous, and this leads to friction between the walls of duct and the fluid causing changes in

the properties of flow along the duct. Frictional forces are considered as shear stress at the wall acting on the

fluid. Shear stress varies with distance x along the duct. Compressible adiabatic flow in constant area duct withfrictional effect is called fanno f low. Using momentum equation with shear stress included. Let the duct be cylinderwith diameter D and length L. So the momentum balance equation in control volume for small portion of duct insteady flow:

Here, C is perimeter and V is velocity of flow in duct and A is area.Now consider mass conservation equation:

Hence momentum eq result can be expressed as:

Heat transfer through the duct wall is negligible(adiabatic), energy equation is:


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Equation of state gives:

Combining all above equations:

Using

Above equation can be written as:

Since the wall shear stress , the velocity and the mach number are always positive, the above equation indicatesthat dM depends on sign of (1-M2). This equation states that if M1 friction cause mach number to decrease. Viscosity, therefore always causes mach number to tendtowards one. Once the M=1 is attained, changes in downstream condition cannot effect the upstream condition, itfollows that choking can occur due to friction.

In general

Here Re is reynold number and is wall roughness.Above equation can also be written as:

Could also be written in entropy equation:

These equations show that if M is less than 1, dT and dp are negative while if M is greater than 1, dT and dp arepositive. The entropy, however always increases. The maximum entropy point on the Fanno line is the point atwhich the mach number is 1.Now integrating and solving for M along the duct between two points distance L, f is assumed to be constant:

Setting M2=1 then gives:

The equation for pressure and temperature variation could be similarly integrated:

From the above equation it follows that:


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Now, the second law of thermodynamics requires that for an adiabatic process the entropy must remain unchangedor must increase, it is found that this will only occur if:

It therefore follows that the shock wave must always be compressive, i.e., that p2 / p1 must be greater than 1, i.e.,the pressure must always increase across the shock wave. Using the equations for the changes across a normalshock then shows that that the density always increases, the velocity always decreases and the temperature alwaysincreases across a shock wave. The entropy increase across the shock is, basically, the result of the fact that,because the shock wave is very thin, the gradients of velocity and temperature in the shock are very high. As aresult, the effects of viscosity and heat conduction are important within the shock leading to the entropy increaseacross the shock wave. Because the flow across a shock is adiabatic, the stagnation temperature does not changeacross a shock wave. However, because of the entropy increase across a shock, the stagnation pressure alwaysdecreases across a shock wave.While the relations derived in the previous section for the changes across a normal shock in terms of thepressure ratio across the shock, i.e., in terms of the shock strength, are the most useful form of the normal shockwave relations for some purposes, it is often more convenient to have these relations in terms of the upstream Machnumber M1 . To obtain these forms of the normal shock wave relations, it is convenient to start again with a controlvolume across the shock wave and to again apply conservation of mass, momentum and energy to this controlvolume but in this case to rearrange the resulting relations in terms of Mach number.Considering mass conservation we get:

Considering momentum conservation we get:

Considering energy conservation we get:

Combining the above equations we get:

By second law of thermodynamics, results show thatIt, therefore, follows that the Mach number ahead of a shock wave must always be greater than 1 and that the shockwave must, therefore, as discussed before, always be compressive, i.e., the pressure must always increase acrossthe shock wave.

2.2.7 Diffuser

Diffuser is used to slow the flow with smallest possible loss of total pressure. Ideal diffuser would compress the flowisentropically, hence no loss of total pressure.


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2.3 Nozzle

2.3.1 Flow of steam through nozzle

Steam flow through nozzle is adiabatic and no external work is done during the flow, therefore heat transfer and work

both are zero.

Applying the energy equation, velocity of steam at the exit of nozzle:

Velocity of steam entering the nozzle is very small, so could be neglected.

The isentropic flow of steam through the nozzle may be represented by an equation:

Where =1.135 for saturated steam and 1.3 for superheated steam.

Work done during expansion is , its equal to the change of internal energy.

Net work done for increasing kinetic energy of steam:

V=specific volume of steam and u=velocity of steam and H is enthalpy.

Solving the above equation we get,


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2.3.2 Condition of maximum discharge

The mass flow rate per unit area is maximum at the throat due to minimum area at throat. This is the condition of

maximum discharge. Discharge of nozzle is function of only.

Thus m/A is maximum when is minimum.

Differentiating above equation for maximum discharge with respect to pressure ratio will give:

Now using the above value in the maximum discharge condition we get value of maximum discharge:

2.3.3 Supersaturated flow and effect of friction

Due to expansion of superheated steam inside nozzle, the change of phase should start at a particular pressure. But

condensation does not occur as there is very limited time due to high velocity of steam. Thus this phenomenon is

delayed and vapour continues to expand and is said to be superheated. This type of flow is superheated flow.

The density of superheated steam exceeds that of equilibrium condition which in turn increases the mass of steam

discharge. It also increases the specific volume and entropy while reduces exit velocity.

Effect of friction in a nozzle

>It reduces the enthalpy, and therefore effects velocity.

>energy lost in friction is transformed as heat and makes the steam superheated, effecting the final condition of

steam.

coefficient of friction


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gas dynamics and nozzle

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