In our Wednesday’s Instagram post we discussed in brief on the shape of nozzles — mainly why rocket engines nozzles diverge and jet engine nozzles converge.



Today let’s get into the technical details of it



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By definition -



A nozzle is a device designed to control the direction or characteristics of a fluid flow (especially to increase velocity) as it exits (or enters) an enclosed chamber or pipe.



A nozzle is often a pipe or tube of varying cross-sectional area, and it can be used to direct or modify the flow of a fluid (liquid or gas). Nozzles are frequently used to control the rate of flow, speed, direction, mass, shape, and/or the pressure of the stream that emerges from them. In a nozzle, the velocity of fluid increases at the expense of its pressure energy.

A water nozzle

Now in our day to day lives, it is a common phenomenon that decreasing the cross-section of the nozzle increases the speed of the fluid going through it. A common experience when watering garden plants, where we decrease the cross-section of pipe at exit using our thumb, in order to increase the exit velocity of water and make it move through larger distance.



Question is: To what limit, can we increase the fluid velocity by decreasing the area?

The figure you see above, whereby fluid velocity is increased by decreasing the area (known as convergent nozzles) is valid only for subsonic fluids. Meaning if the fluid final velocity is less than the speed of sound in that medium, the converging nozzle will accelerate it — till it reaches the speed of sound or Mach number 1. Beyond that, if it is further converged, the fluid velocity will start decreasing because of a phenomenon called Choking.



You can imagine “chocking” like this — all the fluid molecules have to pass through narrowing space. Now space is too narrow for the motion to happen and choking occurs whereby each molecule is staring each other!



So, after choking occurs, the fluid has approached Mach 1. Beyond that to further increase the speed, the nozzle needs to diverge.



So the complete system works like this — First you converge the nozzle to increase it’s speed to Mach 1. Then diverge to increase the speed further. In this way, there will appear a section at which the cross-section is smallest — that is called throat, the point where choking occurs.



Note: Mach Number is a dimensionless number representing the ratio of flow velocity to the local speed of sound (in that medium). M<1 for Subsonic, M>1 for Supersonic, M = 1 for sonic flows.

A converging diverging nozzle.

Let us understand this mathematically.

Look at the above figure and the equation.



A. The term on left-hand side is : (Final Area — Initial Area)/Final Area

B. The first term on Right-hand side is: (Final Velocity — Initial Velocity)/Final Velocity

C. M*M — 1



Let us simply look at the signs of each term throughout the flow. Whether they are negative or positive.



For the converging section of the nozzle.



A = Negative, as the final area is less than the initial area.

C = Negative as M squared is less than 1, for subsonic flow.

This means Term B has to be positive which implies the final velocity to be greater than the initial velocity. Which happens to be true.



For the Throat



A = Zero. (From your differential calculus class you would know when a curve is switching direction, at point of the switch the tangent slope is zero. ;-) )

C = Zero. M Squared = 1

This means Term B is not defined in this case. Which is the phenomena of Choking.



For the Diverging section



A = Positive, as the final area is more than the initial area.

C = Positive as M squared is more than 1, for subsonic flow.

This means Term B has to be positive which implies the final velocity to be greater than the initial velocity. Which happens to be the exact case in rocket engines.