What a nozzle does
Hot gas leaves the combustion chamber, accelerates to the speed of sound in the narrowest point, the throat, and keeps accelerating to several times the speed of sound in the widening part behind it. The nozzle turns pressure and heat into directed velocity, and only the velocity along the engine axis produces thrust.
The trouble with cones
In a conical nozzle the gas leaves roughly like a spherical spray. The fraction of momentum that points forward is
λ = (1 + cos α) / 2
For a typical half-angle of 15° that is 0.983: 1.7 % of the thrust is lost to sideways flow. A smaller angle reduces the loss but makes the nozzle long and heavy.
How the bell solves it
A bell opens steeply right after the throat, where the gas is dense and easy to turn, and then curves back so that the wall ends at only a few degrees. Rao used the method of characteristics to find the contour that gives the most thrust for a given length. A bell 80 % as long as a 15° cone is the common compromise: shorter, lighter, and with a smaller divergence loss than the cone.
A worked example
For our 100 N study engine WD-01 with an area ratio of 40, the 80 % bell is 64 mm long instead of 80 mm, has 6 % less wall area to cool and to manufacture, and wins back roughly half of the cone’s divergence loss. The calculation is in the engineer’s log.