Fluid Mechanics Applications/B25:Supersonic Flow In Convergent-Divergent Type of Nozzles

Nozzles are actually used to modify the flow of a fluid (i.e. by increasing kinetic energy of the flow in expense of its pressure). Convergent-divergent type of nozzles are mostly used for supersonic flows because it is impossible to create supersonic flows (mach number more than one) in convergent type of nozzle and therefore it restricts us to a limited amount of mass flow through a particular nozzle. In convergent-divergent type of nozzles we can increase the flow velocity much higher than sonic velocity that is why these type of nozzles have a wide applications such as propelling nozzles in jet engines or in air intake for engines working at high rotations per minute (rpm).

Principal Of Operation

For understanding the working principle of convergent-divergent type of nozzles, first we need to look the working principle of only convergent type of nozzles.In these type of nozzles the area of the nozzle reduces gradually in the direction of flow.The pressure at intake is called stagnation pressure and the pressure at exit is called back pressure.The value of back pressure can never be more than 1 in case of a nozzle. As we start reducing the back pressure we observe that flow velocity and mass flow rate also starts increasing, but this will happen up to a certain limit, after which no increase in velocity or mass flow rate takes place. This situation is known as choked i.e. no further increase in mass flow rate takes place whatever be the back pressure now. This situation takes place at a particular mach number i.e. at mach number '1'.

But the case is not the same when we use a divergent nozzle just after the convergent. Actually the principle reverses i.e. when we attach a divergent nozzle just after the convergent nozzle our flow speed starts increasing with the decrease in back pressure and also the mass flow rate. And therefore in this type of nozzles we can reach to the speeds above sonic i.e. supersonic.

Mach number::

It is the ratio of speed of flow in a medium to the speed of sound in that medium.

For mach numbers ≤ 0.3 we consider the flow to be incompressible because the density variation is below 5% and for flows having mach number ≥ 0.3 we consider the flow to be compressible because the density variation can not be neglected now. For supersonic flows increase in velocity causes flow velocity to increase . And therefore for our case i.e. supersonic flows we will be doing all he calculations considering compressible flow only.

Normal Shock:

It is a completely irreversible process takes place in the Convergent divergent type of nozzles (or in venturi) at the divergent section. A sudden change in pressure, temperature, and flow velocity takes place while supersonic flow was taking place. After shock flow becomes subsonic and stays subsonic till end. Width of this shock is very less i.e. about 4 times the mean free path of the gas molecules.

Formulas Used

m= mass flow rate
V= velocity
 $\rho$ = density
 $\gamma$ = specific heat ratio 
A= area 
M= mach number
a= speed of sound
 $p$ -difference in pressures on body
 $V$ -velocity of fluid surrounding the body
 $g$ -acceleration due to gravity
 $z$ -height of body
 ${p_{0}}$  -  stagnation pressure
 ${p_{e}}$  -  back pressure
 ${A}$  - Area at exit of nozzle 
 ${A^{*}}$  - Area at throat

Conservation of mass::

m=  $\rho$ *V*A

Conservation of momentum::

 $\rho$ *V*dV = -d $p$

Isentropic steady flow::

 ${\frac {dP}{P}}=\gamma *{\frac {d\rho }{\rho }}$

Bernoulli's principle::

 ${\frac {P}{\rho }}+{\frac {V^{2}}{2}}+gz=constant$

 ${\frac {p_{0}}{p_{e}}}=(1+{\frac {\gamma -1}{2}}.M^{2})^{\frac {\gamma }{\gamma -1}}$

 ${\frac {A}{A^{*}}}={\frac {1}{M}}\left({\frac {2+(k-1).M^{2}}{K+1}}\right)^{\frac {k+1}{2(1-k)}}$

Examples Of Problems Related To Supersonic Flows

Consider  a converging-diverging nozzles having throat area of  $0.002m^{2}$

Advantages and Applications Of Convergent-Divergent Types Of Nozzles

convergent-divergent(C-D) type of nozzles have a lot of application as a propelling nozzle in automobile and jets. Few examples of the application of convergent divergent type of nozzles in engineering are:

*Steam turbines : In power plants .

*Rockets : for providing sufficient thrust to move upwards.

*The supersonic gas turbine engine : for the air intake  when air requirement of engine is high.

C-D nozzles can be seen in water supply pumps or in formula car intake system or in jet engines for providing sufficient thrust to propel at high speeds mostly in supersonic jets.