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In this article we will discuss about:- 1. Definition of Nozzle 2. Some Applications of a Nozzle 3. General-Flow Analysis 4. Velocity 5. Mass-Flow Rate 6. Critical Pressure Ratio 7. Effect of Friction 8. Velocity Coefficient 9. Super Saturated or Metastable Flow 10. Phenomenon in Nozzles Operating Off the Design Pressure Ratio.

**Contents:**

- Definition of Nozzle
- Some Applications of a Nozzle
- General-Flow Analysis of a Nozzle
- Velocity in a Nozzle
- Mass-Flow Rate in a Nozzle
- Critical Pressure Ratio of Nozzle
- Effect of Friction on Nozzle
- Velocity Coefficient of Nozzle
- Super Saturated or Metastable Flow through Nozzle
- Phenomenon in Nozzles Operating Off the Design Pressure Ratio

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**1. Definition of Nozzle: **

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Turbo machines like steam turbines, water turbines and gas turbines produce power by utilising the kinetic energy of the jets produced by passing high pressure steam, water and gas through the devices called nozzles. Corresponding to the fluids used, the nozzles are called steam nozzles, water nozzles and gas nozzles.

**These nozzles serve two purposes: **

(1) To convert pressure energy and thermal energy into kinetic energy and

(2) To direct the fluid jet at the specific angle known as nozzle angle.

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A nozzle is a device, a duct of varying cross-section area in which a steadily flowing fluid can be made to accelerate by a pressure drop along the duct.

So when a fluid flows through a nozzle, its velocity increases continuously and pressure decreases continuously.

** 2. Some Applications of a Nozzle: **

**There are following applications of a nozzle are: **

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1. Nozzles are used in steam turbines, gas turbines, water turbines and in jet engines, Jet propulsion.

2. Nozzles are used for flow measurement e.g. in venturimeter.

3. Nozzles are used to remove air from a condenser.

4. Injectors for pumping feed water to boilers.

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5. Artificial fountains.

**There are three types of nozzles: **

1. Convergent nozzle,

2. Divergent nozzle, and

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3. Convergent-Divergent nozzle.

If the cross-section of the nozzle decreases continuously from entrance to exit, it is called a convergent nozzle.

If the cross-section of the nozzle increases continuously from entrance to exit, it is called a divergent nozzle. If the cross-section of the nozzle first decreases and then increases, it is called a convergent-divergent nozzle. The section where cross-sectional area is minimum is called ‘throat’ of the nozzle.

**Diffuser: **

When a steadily flowing fluid is decelerated in a duct causing rise in pressure along the stream, then the duct is called a diffuser.

So in a diffuser, velocity of the fluid decreases continuously and pressure increases continuously.

**Similar to nozzle, there are three types of diffusers: **

1. Convergent diffuser

2. Divergent diffuser

3. Convergent-divergent diffuser.

The given duct will work as a diffuser or a nozzle depending upon the fluid velocity at the inlet of a duct.

**Some Applications: **

1. Diffusers are used in centrifugal compressor.

2. Diffusers are used in ram-jet engines to increase the pressure of incoming fresh-air.

** 3. General-Flow Analysis of a Nozzle: **

**Assumptions: **

1. The flow of the fluid is assumed to be one dimensional.

2. The change in area and curvature along the axis of the duct are gradual.

3. Thermodynamic and mechanical properties are uniform across planes normal to the axis of a duct.

4. The flow of the fluid is assumed to be steady flow.

The steady-flow equation is-

Q – W = ΔH + ΔPE + ΔKE

or differential form of the equation is-

∂q – ∂w = dh + d(PE) + d(KE)

We will assume heat the nozzle is horizontal,

**∴** d(PE) = 0

The fluid is just flowing through a duct. So obviously work-done is zero

**∴** ∂w = 0

The definition of enthalpy is

h = u + PV

dh = du + d(PV)

= du + pdv + Vdp

But from the first law of thermodynamics,

du + pdv = dq

**∴** dh = dq + vdp

The change in Kinetic energy for unit mass is-

This equation gives information whether the given duct will act as a nozzle or a diffuser if the inlet fluid velocity is known. It also gives information which type of duct should be used for a particular application.

For a nozzle, velocity of the fluid should increase continuously from entrance to exit.

**So if the inlet fluid condition is known, we can select the nozzle as below: **

1. When the velocity of fluid is less than sonic velocity (i.e. when the flow is sub-sonic), the match no. (M) is less than 1,

i.e. change in across sectional area along the duct should be negative.

This tells that, for sub-sonic flow, the duct must be convergent. By using this convergent nozzle, the flow of the fluid can be increased to sonic velocity.

But by using convergent nozzle we cannot obtain super-sonic flow.

2. When the flow is super-sonic, M > 1

i.e. for super-sonic flow, the duct must be divergent.

3. The convergent-divergent nozzle is used for convert sub-sonic flow into super-sonic flow.

In the convergent part the velocity of fluid is increased from sub-sonic to sonic condition. At throat, the velocity is sonic. In the divergent part, the velocity is increased from sonic to super-sonic.

So only in convergent-divergent nozzle, the sub-sonic flow is converted into super-sonic flow.

**For diffuser the velocity should decrease continuously so a diffuser is selected as below: **

1. For sub-sonic flow-

2. For super-sonic flow – M > 1-

The convergent-diffuser will decrease the velocity of fluid to sonic velocity.

3. The convergent-divergent diffuser is used to convert super-sonic flow into sub-sonic flow.

** 4. Velocity in a Nozzle: **

For unit mass,

The steady flow equation is,

q – w = Δ h + Δ PE + Δ KE

For a horizontal nozzle, Δ PE = 0

There is no work-done in nozzle therefore W = 0. In the nozzle, the velocity of the fluid is so high that there is hardly any time available for fluid to exchange heat with the surroundings. Therefore for nozzle, it is assumed that heat transfer is zero i.e., flow in Isentropic

Where, h_{t} = specific enthalpy at the throat conditions.

For steam nozzles the values of enthalpy (h_{1}, h_{2}, h_{t} etc.) are normally obtained by using Mollier Chart. If C_{1}, the initial or approach velocity is neglected, then

** 5. Mass-Flow Rate in a Nozzle: **

**The steady flow energy equation is:**

If P_{1} = P_{2} the mass-flow rate is zero.

Also if P_{2} = 0, the mass flow rate is zero.

If the graph is plotted for mass flow rate v_{s} pressure ratio, it will be as shown in the figure. It is observed that at some value of (P_{2}/P_{1}) the velocity and mass-flow rate reaches to its maximum value.

** 6. Critical Pressure Ratio of Nozzle:**

** 7. Effect of Friction on Nozzle: **

The flow of steam through nozzle is assumed to be isentropic. The Mollier Chart shows the isentropic flow (1 -1 – 2) of steam through a convergent-divergent nozzle. But in actual case, the friction losses occur.

**The frictional losses are due to: **

(i) Friction between sides of nozzle (wall of nozzle) and fluid.

(ii) Internal fluid friction and

(iii) Due to eddies in the flow.

Most of the friction in convergent-divergent nozzle is assumed to occur between the throat and exit. The expansion upto throat is taken to be isentropic. This is due to low initial velocity. In Mollier Chart, 1 -t- 2′ is the actual expansion of steam through nozzle. Figure 19.5 shows the actual expansion of steam through nozzle.

**Effects: **

1. Reduction in actual enthalpy drop.

2. Reduction in exit velocity of fluid.

3. Increase in the dryness-fraction of steam.

4. Increase in the specific volume and

5. Decrease in the mass-flow rate.

**Nozzle Efficiency: **

The nozzle efficiency is given as-

** 8. Velocity Coefficient of Nozzle:**

Velocity coefficient is defined as the ratio of actual velocity of steam to an isentropic or theoretical velocity.

** 9. Super Saturated or Metastable Flow through Nozzle:**

**Normal or Equilibrium Expansion: **

The ideal case of isentropic expansion of a superheated vapour to a state in the wet region is shown in T-S diagram and h-s diagram of Fig. 19.6. At the point in the expansion where the pressure is P_{s}, a change of phase should begin to occur. At this point, the random kinetic energy of the molecules has fallen to a level which is insufficient to overcome the attractive forces of the molecules and some of the slow moving molecules start to form tiny droplets to condensate.

Now although this process is rapid, it does not have time to occur in the nozzle where the flow velocity is very great. The achievement of equilibrium between the liquid and vapour phase is therefore delayed and vapour continues to expand in a superheated or dry state.

**Super-Saturated Expansion: **

Point A represents a steam in superheated region at pressure P_{1}. When at point s, due to the rapid expansion, instead of condensation commencing, the steam continues to behave as a super-heated vapour down to point B, at same intermediate pressure P_{2}. This non-equilibrium behaviour as a superheated vapour does not continue indefinitely and at point B, restoration of equilibrium quickly occurs and is after the throat in divergent portion of the nozzle. It is accompanied by a small increase in pressure.

In equilibrium flow, the energy released by condensing the molecules is provided for increasing the kinetic energy of the steam as it passes through the nozzle. If the steam does not condense, then the energy for this increase in kinetic energy come by reduction in the temperature and therefore the steam is called super-cooled.

Thus we see that condensation does not start immediately after S is passed, no drops of liquid are formed until some state B is reached, where condensation suddenly occurs, a phenomenon sometimes called Condensation Shock. The steam in states between S and B is supersaturated or a metastable state.

Since the collapse of the metastable state has been observed not in a converging nozzle, but always, in the diverging part of the De-Laval nozzle, one is probably safe assuming that the super-saturation, if it occurs at all, will persist to some point beyond throat. Of course the flow should be computed for the throat section since this is where it is limited.

In general, the velocity of supersaturated steam is less than the value computed for the equilibrium flow. The density of supersaturated steam is greater than the equilibrium density at the same pressure. However, the density increases by a large percentage that the velocity decreases, with the result that the computed mass flow of supersaturated steam is greater than the computed flow of equilibrium steam, given the initial state and throat area. When discharge from a nozzle is actually measured it is found to be from 2 to 5% greater than the calculated discharge.

**Wilson Line: **

Calendar found that the Wilson line approximately follows the 97% dryness line.

**Degree of Under-Cooling: **

Difference in the temperature at point C and temperature at point B is known as degree of under-cooling or difference in saturation temperatures at pressure P_{2} end P_{B} is degree of under-cooling.

The temperature of the supersaturated steam at B will be below the normal temperature of steam for that pressure; this state is known as undercooled state, the amount of undercooling being the difference between the normal temperature and the actual temperature.

**Wilson’s Line: **

There is generally a limit to super-saturation. It is upto 96% dryness and beyond it, steam condensation occurs suddenly and irreversibly at constant enthalpy and remains in stable condition thereafter. This limit line is known as Wilson’s line.

**Effects of Super-Saturated Flow: **

1. Increase in discharge by 2 to 5% due to increase in density due to super cooling.

2. Decrease in exit velocity.

3. Increase in final dryness-fraction and increase in enthalpy.

** 10. Phenomenon in Nozzles Operating Off the Design Pressure Ratio: **

**(a) For Convergent Nozzle: **

Consider a convergent nozzle as shown in Fig. 19.8. The initial conditions are kept constant and exit pressure P_{2} is reduced gradually from the initial pressure P_{1} by a valve.

(i) When pressure P_{2} is equal to P_{t}, there is no decrease in pressure and therefore mass-flow rate is zero. This condition is shown by curve (I) in Fig. 19.8.

(ii) When pressure P_{2} is less than P_{1}, but more than critical pressure; distribution along the axis is shown by curve (II). The mass low rate increases as pressure P_{2} is reduced as shown in Fig. 19.8.

(iii) When exit pressure P_{2} is equal to critical pressure, the nozzle operates with maximum mass flow rate and the pressure distribution is shown by curve (III).

This state is called as ‘chocked flow’ or the nozzle is said to be ‘chocked’.

(iv) When pressure P_{2} is less than critical pressure, there is no change in mass-flow through nozzle and also pressure distribution along the nozzle is same. The pressure-drop from critical pressure to P_{2} takes place after the nozzle. This expansion is irreversible and gives rise to pressure oscillations as shown by curve (IV).

**(b) For Convergent-Divergent Nozzle: **

Let us consider a convergent-divergent nozzle as shown in Fig. 19.9

1. When pressure P_{e} is equal to inlet pressure P_{r} there is no flow. Refer Fig. 19.9 and Fig. 19.10.

2. When P_{e} is reduced to the pressure denoted by curve (II). So that P_{e}/P_{t} is less than 1 but greater than critical pressure ratio, the velocity increases in the convergent region of the nozzle, but mach number (m) is less than 1 at throat. So the divergent section acts as a sub-sonic diffuser in which the pressure increases and velocity decreases.

3. In case (IV), pressure is critical at throat and exit pressure P_{e} is design pressure. Therefore the flow is isentropic throughout the nozzle and velocity continuously increases along the nozzle. At throat, velocity is equal to sonic velocity. The divergent portion acts as a super-sonic nozzle with a continuous decrease in pressure and continuous increase in velocity.

4. If the exit pressure is more than the designed pressure but less than critical pressure, the flow is not isentropic in the divergent part and it is accompanied by highly irreversible phenomena known as shocks. Shocks occur only when the flow is supersonic and after the shock the flow becomes sub-sonic and the rest of the diverging portion acts as a diffuser. The condition is shown by case (a).

5. When the back pressure is increased the shock moves upstream and disappears at the nozzle throat where pressure P_{e} has some value P_{3}. Here mach number (m) is equal to one at the throat but divergent portion acts as a sub-sonic diffuser in which pressure increases and velocity decreases.

6. If the pressure P_{e} is less than the design pressure, no further decrease in exit pressure occurs and drop of pressure from design pressure to P_{e} occurs outside the nozzle giving pressure fluctuations as shown by case (V).

**Overexpansion and Under-Expansion: **

If the area of the exit section of a nozzle is such that the fluid expands to a pressure at this section less than that of the discharge region. Overexpansion has occurred. As this lower pressure stream emerges into the higher pressure discharge region, there is a sudden increase in pressure, an act that sets up compression pressure waves, much stronger than sound waves. This jump in pressure outside the nozzle occurs when the back-pressure is above the exit pressure.

If on the other hand the area of the exit section is such that the fluid expands to a pressure at this section greater than that in the discharge region, under-expansion has occurred. Since there is now a sudden decrease in pressure on the jet, expansion waves are initiated.

Both situations involve an increase in irreversibility’s and loss of efficiency. It is best for the expansion in the nozzle to occur to just the right (designed) discharge pressure.