How to Size a Jet Exhaust Cone – Math Method

By Steve B.

I needed to make a small conical shape from a flat piece of plastic.  I had replaced the small jet fan in my Kyosho Jet Vision since the original fan shucked a few blades on takeoff (FOD).  The new fan was bigger so it needed an adapter to fit between the fan and rear half of the exhaust cone.  I could do it with trial and error but I wondered if I could figure out the math and create it in my drawing program, and I did.

The part I made was the amber colored piece between the blue fan and the black tail cone. The fan assembly is pulled forward so you can see the cone. Normally it sits all the way back.

 

The part before installation.

I’m sure there are any number of places online that have this already figured out, where you can simply input your measurements and voila you get the answer.  I didn’t look.  I wanted to see how it was done.  My only external source was my lovely retired middle school math teacher wife, Alice, who provided some of the basic formulas I had forgotten.  Although, when I had a question, I did have to raise my hand and wait my turn. . .

Remember sitting in Middle School (Junior High) Math class and wondering what you’ll ever use this stuff for?  Well, this is a great example where you can use both Geometry and Algebra.  Sound like fun?

A typical exhaust cone will look something like these.  The long one fits my Avanti jet and the other is for my Jet Vision:

The cone template on the left is for the Avanti and the right one for my Jet Vision as seen above.

 

My drawings were done in a computer drafting program.  You can also do it by hand using a compass, protractor and straight edge.  You may need to commandeer the dining room table if your workbench is loaded with stuff, like mine.

Using my Avanti jet as an example, I created this drawing.  I know the Avanti does not need a tail cone, I just used it for an example.  The calculations are below:

This shows all the geometry involved. The part we are interested in is the small segment at the right.

 

This is the segment that gives us our cone template.

 

This is the math used to create the drawing:

What we know:

   <1 = <2  (Angle 1 = Angle 2)

   R2 = R1 + Lc  (Radius 2 = Radius 1 + Length of Cone).  (R1 & R2 are radii used for the layout above.)

   Arc1 (Arc length 1)  see just below

   Arc2 (Arc length 2)

The arcs lengths are the circumferences of the ends of the cone.  For their radii, just measure the diameters of the fan and outlet and divide by 2.  (ro1 = radius of outlet, rf2 = radius of fan.)  The arc lengths are:

eq. 1   Arc1 = 2π ro1 and Arc2 = 2π rf2

Now all we need to find are <1 & <2 and R1 & R2 in order to create the layout.  We know the relationships between the angles and between the radii.  This gives us only two unknowns.

Now we just need two equations.  Solve for one unknown in one equation, substitute that into the other equation and solve for the other unknown.

eq. 1a   (<1÷360) 2π R1 = Arc1

eq. 1b   (<2÷360) 2π (R1 + Lc) = Arc2

Lets solve for R1 in eq. 1a:

   Simplifying: 

eq. 1c   R1 = Arc1÷(.0175<1)

Substitute that into eq. 1b and we get: 

   (<2/360) 2π (Arc1÷(.0175<1) + Lc) = Arc2

Lets solve for the angles (<1 = <2):

   (.0175<2 (Arc1÷(.0175)<1 + Lc) = Arc2

   Arc1 + .0175<2 Lc = Arc2

eq. 2  <1 = <2 = (Arc2 – Arc1)÷(.0175Lc)

The Arcs and the Cone Length are measured quantities, so substitute them in to find the angle.

Now find R1 by substituting the angle into either eq1c.

eq. 3  R1 = Arc1÷(.0175<1)

Here are the numbers needed to make the drawing above:

Measured:

ro1 = 35mm, rf2 = 41.5mm

Lc = 365mm

Calculated:

From eq1:  Arc1 = 2π ro1 = 220mm, Arc2 = 2π rf2= 261mm

From eq2:  <1 = <2 = 6.419 Deg

From eq3:  R1 = 1958mm, R2 = R1 + Lc = 1958 + 365 = 2323mm

These highlighted values are what you need to draw your exhaust cone.