NOZZLE AND DIFFUSER CLASS NOTES FOR MECHANICAL ENGINEERING

NOZZLE AND DIFFUSER

CLASS NOTES FOR MECHANICAL

ENGINEERING

 It is a passage of variable cross section

(area decreases from entrance onward).

Velocity increases and pressure decreases.

This is true as long as velocity at the inlet

of the nozzle is sub-sonic.  Velocity is less

the velocity of sound or Mach number <1.

 Diffuser is a duct  (passage) of variable cross

section (area increases from entrance

onward). Velocity decreases and

pressure increases. This is true as long

as velocity at the inlet of the diffuser

is sub-sonic.  Velocity is less the velocity

of sound or Mach number <1.

But a particular duct can be a nozzle

or a diffuser depending upon the

velocity at the inlet.

CONSERVATION OF MASS

 m^. =ρAv=constant=C

Take the log of both sides, we get 

ln ρ +ln A + ln v=ln C

Doing partial differentiation, we get

dρ/ρ +dA/A + dv/v=0

We know, for one dimensional isentropic flow

dA/A = (M² –1)dv/v

 dA/A  positive means area is increasing, then for M>1, quantity in bracket is also positive. Therefore dv/v will be positive. Hence with increase in area, velocity increases if M>1.

Diffuser becomes a nozzle and vice versa.

Case 1

When M>1 at the ENTRANCE

As flow area decreases, velocity decreases and pressure increases, thus a convergent section becomes a diffuser with M>1 at entrance. Nozzle becomes a diffuser.

Case 2

When M>1 at the ENTRANCE

As flow area increases, velocity increases and pressure decreases, thus a divergent section becomes a nozzle with M>1 at entrance. A diffuser becomes a nozzle.

Case 3

When M< 1 at the ENTRANCE

As flow area decreases, velocity increases and pressure decreases, thus a convergent section is a nozzle with M< 1 at entrance.

Case 4

When M< 1 at the ENTRANCE

As area increases, velocity decreases and pressure increases, thus a divergent section becomes a diffuser with M < 1 at entrance.

Thus a nozzle remains a nozzle only if M < 1.

 A diffuser remains a diffuser only if M < 1.

Thus flow is SUB-SONIC at entrance.

 

VARIABLE AREA EFFECTS FLOW

NOZZLE

A nozzle is a passage of variable cross-section. In this, pressure of  fluid converts into kinetic energy. The various practical applications of a nozzle are

1. Steam turbines
2. Water turbines
3. Gas Turbines
4. Flow measurements
5. Used in all aircraft’s
6. Jet engines
7. As ejectors for removing air from the condensers
8. As injectors for supplying feed water to boilers
9. Artificial fountains

DIFFUSER

It is also a passage of varying cross section which converts the kinetic energy of the fluid into pressure energy. It is just reverse of a nozzle.

Applications of a diffuser

1. Centrifugal compressors
2. axial flow compressors
3. Ramming of air in aircraft’s

In one dimensional flow, assume that the velocity and fluid properties change ONLY in the direction of flow. Therefore we can take mean value of velocity and other properties in one direction flow.

CONVERGENT DIVERGENT NOZZLE

This nozzle is a de Laval nozzle (or CD nozzle). In this, velocity increases in the convergent portion and becomes sonic velocity at the exit (throat). This exit of the of the nozzle has minimum area and maximum velocity. This exit is the THROAT. Now the fluid will enter the divergent portion with supersonic velocity. Here the velocity will further increase. It is necessary to run the turbine at very high speed to generate more power.  Thus, convergent divergent nozzles are used in gas / steam turbines & rocket engines.

Used in steam turbines and all super-sonic aircraft’s.

 

COMPARISON OF NOZZLES AND DIFFUSER

Nozzle                                                 DIFFUSER

Boundary layer is thin                      Boundary layer is thick.

Has negligible friction                      Has more friction.

Has favorable gradient                     Unfavorable gradient

More efficient                                       Less efficient

No shock formation                          Chances of shock formation

VELOCITY OF STEAM AT ANY SECTION OF THE NOZZLE

v₂=C [k(H₁– H₂)]^0.5

Without friction C = 44.72  and k =1

With friction C = 44.72  and k is the friction factor =0.7 to 0.9

k is friction coefficient or nozzle efficiency

VELOCITY COEFFICIENT

It is the ratio of actual exit velocity to isotropic exit velocity for a certain same pressure drop

c_v = Actual velocity/ isotropic velocity

=√actual enthalpy drop/isotropic enthalpy drop

=√η = square root of efficiency√

 EFFICIENCY OF THE NOZZLE

It is a ratio of Actual enthalpy drop to  isotropic enthalpy drop between the same inlet and outlet pressures.

η = Actual enthalpy drop/  isentropic enthalpy drop

FACTORS ON WHICH THE EFFICIENCY OF A NOZZLE DEPENDS

1. Material of the nozzle
2. Inside surface condition of the nozzle
3. Nature of the fluid flowing through the nozzle
4. Size and shape of the nozzle
5. Reynolds number of the flow
6. Angle of divergence
7. Orientation of the nozzle

DESIGN OF A STEAM NOZZLE

Data required

1. Inlet pressure and the condition of the steam
2. Back pressure
3. Mass flow rate

STEPS

(i) Find the back pressure by the standard relation

p_{b =} p_i (2/(n+1))^(n/(n-1)

(ii) If this calculated back pressure is more than the given back pressure, then the nozzle is a convergent nozzle.

(iii) If this calculated back pressure is less than the given back pressure, then the nozzle is a convergent-divergent

Calculate nozzle dimensions from the

(i)  mass flow rate

(ii) the condition of the steam at inlet, throat and outlet of the nozzle.

Mass flow rate is constant at the inlet, throat and outlet.

Area = m^. sp. Volume/velocity

A = π d²/4

CONCLUSIONS

1. It  converts pressure into kinetic energy.

2. No work done in a nozzle.

3. There is no energy transfer in a nozzle.

4. Nozzle is horizontal.  Z1 = Z2

5.  Spray nozzles are convergent nozzle 

6.  Air crafts use convergent nozzles.

7. Steam turbines use convergent –divergent nozzles.

8. Inlet velocity is negligible.

9. Pressure decreases with decrease of area.

10. Velocity increases with decrease of area.

11. Flow is 1-dimensional.

12.  Properties of the flowing fluid change only in the direction of flow.

13. Analysis uses steady flow energy.

14. Flow is isentropic unless otherwise specified.

15. The minimum area in a nozzle is throat.

16. Velocity is sub –sonic or sonic in the convergent portion and becomes super-sonic in the divergent portion.

17. For liquids, specific volume is constant over a wide range of pressure and temperature. Liquids use convergent.

18. Critical pressure ratio is the ratio of pressure at inlet to pressure at throat. It gives sonic velocity at throat. 

19. At the critical pressure ratio, mass flow rate is maximum. Choking occurs.  

20. If the pressure at exit is higher than the designed pressure, it is an under expansion. It will cause a loss in thrust or power produced.

21. If the pressure at exit is lower than the designed pressure, it is an over expansion. It will cause a gain in thrust or power produced.

22. Angle of convergence is much greater than the angle of divergence.

23. The length of convergent portion is much smaller than that of the divergent portion.

24. Steam is a vapor. Properties can be read from enthalpy entropy chart. Properties are also available steam tables.

25. Sonic velocity = C* =√γRT

26. Critical pressure ratio fot

 

(a) Isentropic process

P^*/p = [2/(γ+1)]^γ/(γ—1)

 ^{(b) Actual process}

P^*/p = [2/(n+1)]^n/(n—1)

27. Mach number

M = actual velocity/sonic velocity

Sonic velocity = 332 m/s

28. Relation between stagnation and static (local ) temperature

T₀ /T = 1 + (γ—1) M²/2

.29. As steam is expanding, velocity at any section of the nozzle

C = 44.72     Without friction

C = 44.72     With friction, k is the friction factor =0.7 to 0.9

30. If the steam expands from pressure p₁ and volume v₁ to pressure p₂

c₂ =     2{[(n/(n-1)] [p₁ v₁ –p₂v₂]}^0.5

 

31. Friction reduces the enthalpy drop and hence the velocity at the outlet.

32. Assume no friction in the nozzle. Friction is  in the diffuser only.

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