Compressible flow , Nozzles , Diffusers, GATE , Power Plant , (New topic added), Mechanical engineering

 Nozzle:

It is  a device of variable cross section used to increase the kinetic energy at the expense of pressure energy . It should be design in such a manner that it will perform its functions with minimum losses.

Application of nozzle :

To produce a jet of working fluid of higher speed used to drive steam turbine , gas turbine , etc.

Diffuser:

It is a mechanical device of variable passage used to increase the pressure energy.

Application :

Centrifugal compressor , axial flow compressor.

Mach number:

It is defined as ratio of actual velocity of fluid to that of sonic velocity.

Stagnation state : 

Stagnation enthalpy:

It is the enthalpy of gas and vapour when it is adiabatically deaccelerated to zero velocity state. Whether the process is isentropic or adiabatic , the stagnation enthalpy remains constant between the two sections.

Stagnation temperature:

It is the temp. of gas or vapour when it is adiabatically deaccelerated to zero velocity state.

Stagnation pressure:

It is the pressure of gas or vapour when it is isentropically deaccelerated to zero velocity state.

Isentropic flow through nozzle:

Relationship between stagnation state and static state:

Relationship between stagnation state and critical state:

Relationship between critical state and static state:

Stagnation velocity of sound :

It is the velocity of sound corresponding to stagnation temperature.

Maximum velocity of fluid:

It is a hypothetical concept in all condition because as the enthalpy become zero , then we will get maximum velocity . When the fluid is accelerated to absolute zero temperature , in a imaginary adiabatic expansion process , it is termed as maximum velocity of the fluid.

Velocity of sound:

It is the speed at which infinity small pressure wave travelling through the medium and it depends on medium and temp.

The piston is given a infinity small dv movement due to which the whole medium get disturbed , the whole medium is shifted in opposite direction with velocity of sound for that medium.

For the isentropic process under equilibirium conditions:

1. Continuity equation:

2.  Momentum equation:

From 1 and 2 equations this equation generates:

The velocity of sound depends on its medium and temperature , for a given medium the velocity of sound is higher for higher temperature . Lower the molecular weight higher will be the velocity of sound.

Note:

• In both of the cases i.e. nozzle or diffuser the sonic conditions are achieved corresponding to minimum cross section area i.e. throat.
• In the case of incompressible fluid converging passage always acts as nozzle and diverging passage acts as diffuser.

Flow through a nozzle or critical pressure:

Note :

• Air - n = 1.4
• Superheated steam - n = 1.3
• Saturated steam - n = 1.35
• Wet steam - n = 1.035+0.1x , where x is dryness fraction

Mass flow rate per unit area is maximum when the area is minimum which is achieved corresponding to a throat condition where the value of mach number is 1 , is known as chocked nozzle.

Question: steam at 10 bar and 250 degree celcius enters a nozzle and discharge at a pressure of 2 bar (wet condition) . Find the final velocity of steam and neglecting inlet velocity .

Nozzle efficiency:

It is defined as the ratio of actual enthalpy drop to that of ideal enthalpy drop or it is defined as the ratio of actual gain in kinetic energy to that of ideal gain in kinetic energy.

Kn is the velocity coefficient of the nozzle .

Effects of nozzle friction:

• Actual enthalpy decrease 
• Actual velocity at exit decrease 
• Entropy increase 
• Dryness fraction increase
• Specific volume increase
• Density decrease
• Mass flow rate decrease

Question: a fluid at 6.9 bar and 93 degree celcius enters a convergent nozzle with negligible velocity and expands isentropically to 3.6 bar . Calculate the mass flow rate per metre square if the fluid is:

1.Helium having the value of Cp is 5.19 KJ/kg-k  and molecular weight is 4 . Assuming to be a perfect gas.

2. Ethane having the value of Cp is 1.88 KJ/kg-k and molecular weight is 30.

Solution:

Converging nozzle:

• When Pb=P0=Pe , there will be no flow through the nozzle i.e. m = 0 , v = 0.
• When Pc<Pb = Pe<P0 , the mass flow rate increases but the velocity remains sunsonic and flow is normal.
• When Pb = Pc = Pe, the mass flow rate is max called as chocked nozzle.
• When Pb< Pc=Pe, nozzle exit pressure can never be less than Pc , than the isentropic upto the nozzle exit and then after the nozzle exit plane their a irreversibility in the form of shockwaves and fluid will finally equal to Pb in a irreversibile manner.

Question : Air flow isentropically in a channel at section 1 , the mach no. is 0.3 and area is 0.001 m^2 . The absolute pressure and temperature are 650 kpa and 62 degree celcius . at section 2 the mach no. is 0.8 then find the properties at section 1.

Prandtyl velocity ellipse:

Vmax is possible in space where velocity of sound is 0.

Convergent and divergent nozzle:

• When Pb=P0, then there is no flow through the nozzle.
• When P*<Pb<P0 the flow remains subsonic through nozzle and mass flow rate is normal.
• When Pb=P* the sonic velocity exits at the throat and the mass flow rate is max i.e. chocked flow.
• Pd<Pb<P* the fluid continues to accelerate in diverging passage to the subsonic velocity and pressure decreases . This acceleration  comes  to a sudden end with a shock wave between the nozzle throat and exit plane, across the shock waves there is a sudden variation in pressure and velocity and the flow is irreversibile.
• Pd=Pb this is the best condition , the flow remains isentropic through the nozzle with supersonic velocity at the exit of nozzle with no shock.
• Pb<Pd just after nozzle exit plane a shock wave appears through which fluid becomes equal to back pressure . We tried to avoid under expansion as the shock will appears at the exit of nozzle. (entry of turbine ) means they creating the impact at the inlet of turbine .

Note : for supersonic speed at the exit of nozzle , sonic conditions should be exist at the throat i.e. M=1

Question: air flows steadily through a duct from a pressure of 350 kpa and 60 degree celcius and actual velocity is 183 m/sec . The outlet conditions are M=1.3 , stagnation pressure is 386 kpa , stagnation temprature is 350 k . Then find stagnation temp. and pressure at inlet , static temp. and static pressure at outlet.

Solution:

Fannow flow:

Flow in a constant area duct with friction and without heat transfer . It is a ideal condition and this type of flow may be assumed in aircralft duct , refrigeration system , etc. Assuming the 1 D flow of ideal gas between section 1 and 2 

Governing equations:

1. Continuity equation:

2. Momentum equation:

3. 2nd law  of thermodynamics:

4. 1st law of thermodynamics:

Note: in case of fannow flow , the stagnation temperature is constant.

5. Equation of state:

6.

• Now there are 7 variables (pressure, temp. , velocity , density , enthalpy , entropy , friction) but we have 6 equations therefore there are infinite no. of solutions , let us assume some value of temperature i.e. 200 k i.e. fixed so now there are 6 equations and 6 unknowns and the value of these can be found out using 6 equations , now repeat the same procedure and when they are plotted on the graph we will get:
  
  
  
• In the subsonic region due to duct wall friction irreversible action of fluid takes place with the decrease in static pressure and temp. stagnation temp. remains same but stagnation pressure decreases.
• In the supersonic region due to duct wall friction irreversible deacceleration of fluid takes place with the increase in static pressure and static temperature . stagnation pressure will decrease but stagnation temprature remains constant.

Question: Why the direction towards point C?

Solution : as the flow is takes place through a constant area duct with friction and without heat transfer means there is no heat interaction takes place between system and surroundings therefore the entropy change of the surroundings is equal to zero. From the principle of entropy we know that the entropy change of the universe can never have negative value therefore the entropy has to be increases.

Variation of mach with duct length:

Let is assume l is length of duct and d is dia of duct .

Temperature ratio of fannow flow:

Pressure ratio of fannow flow:

Density ratio of fannow flow:

question: air flow in an insulated pipe of 7.16 mm dia , stagnation pressure and stagnation temperature at inlet are 101 kpa and 23 degree celcius . the static pressure at inlet is 98.5 kpa and at section 2 located at same distance down stream in a constant area pipe the static temp is 14 degree celcius , then determine 

1. Mass flow rate 
2. Static and stagnation pressure at section 2
3. Friction and force on the wall between section 1 and 2
4. Entropy change 

Solution: 

Rayleigh flow :

Flow in a constant area duct without friction and with heat transfer is known as rayleigh flow . Under ideal conditions heat transfer in the combustion chamber and heat exchanger can be assumed as rayleigh flow.

considering the flow of ideal gas between section 1 and 2

Governing equations:

1. Continuity equation:

2. Momentum equation:

Impulse function:

Note: in case of rayleigh flow , the impulse function is constant between any two sections.

3. 2nd law  of thermodynamics:

4. 1st law of thermodynamics:

5. Equation of state:

6.

There are 6 equations and  7 variables so infinite no. of solutions are possible . In the case of rayleigh flow as there is a heat interaction takes place between system and surroundings therefore the system entropy may increase  or decrease but the entropy change of the universe can never have negative value from principle of entropy and due to the flow may takes place from 1 to 2 or 2 to 1.

Pressure ratio:

Temperature ratio:

Density ratio:

Velocity ratio:

Stagnation temperature ratio:

Question: air flow with a negligible flow in a constant area pipe . The properties at section 1 and 2 are given below. then find 

1. Entropy change 
2. Heat addition 
3. Static pressure  
4. Static temp
5. Static density 

at section 2?

Solution:

Shock waves:

They are the irreversible discontinuties formed in a very fast manner , which occurs in supersonic flow. The thickness of shockwaves nearly 0.2 um which is nearly 3 to 4 times the mean free path of gaseous molecules . As the flow occurs in a very fast manner therefore the value of heat interaction is zero and the thickness is very small therefore the friction is equal to zero.

Oblique shock waves:

When the shock waves occur at any angle other than 90 degree.

Normal shock waves:

When the shock waves appears in a direction perpendicular to the direction of flow.

Governing equations:

1. Continuity equation:

2. Momentum equation:

3. 2nd law  of thermodynamics:

4. 1st law of thermodynamics:

5. Equation of state:

6.

As there are 6 unknowns and 6 equations therefore we will get a unique solution which satisfies both rayleigh as well as fannow and the normal shock waves appears at the intersection of rayleigh curve and fannow curve . Therefore for a fixed mach no. at inlet , the outlet mach no. will get automatically fixed.  Normal shock waves are always occur from  point A to point B because of the entropy change of the surrounding is equal to zero.

Temperature ratio:

Pressure ratio:

Note : for the shock waves formation the initial state has to be in supersonic and after the formation the state changes to subsonic.

Density ratio:

Velocity ratio:

Effects of shock waves are:

Strength of shock wave:

Mach no. for constant enthalpy line :

Mach no. for constant entropy line :

Question : a normal shock wave appears in a duct and a fluid in air may be considered as ideal gas . The properties of shock wave are T1=5 degree celcius , P1=65kpa , V1=668m/sec . Determine the properties on the down stream side and change in entropy.

Solution: