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I wish to construct a sterling engine which can run with the heat off a cup of coffee. I have an advanced engineering workshop and can produce virtually anything in it but I need some basics. Unfortunately a lot of conflicting information.
Please i want to be corrected.

Displacer cylinder
I will start with the available displacer cylinder which is 146 mm internal diameter 24 mm height. The displacer piston shall be 144mm with a 1mm clearance on the sides. The displacer piston is thus 16mm high. I Am calculating that the dispacer sweeps through 18mm. The displacer is 2/3 of the total displacer cylinder.

The power piston

I read that the power piston diameter should be the same height of the power cylinder and the displacer volume shall be 1.5 x the power piston volume.

These are my calculations;-
The displacer area is 72 * 72 * pi * 18

Let X be the radius of the power piston
The power piston volume is pi * r * r * 2r (where 2 r is the height)

Therefore

72*72*18*pi = r * r *2r* 1.5 *pi
Pi cancels pi
Therefore
93312 = r * r * 2R * 1.5
62208 = r * r * 2r
62208 = r * r * r *2
31104= r cube
R= cube root of 31104 = 31.4

This means i have to construct the power cylinder 31.4 dia x 31.4 height.

Now my questions.

The displacer requires a 2mm each side clearance each side so I increase the cylinder height from 24mm to 28mm
The power cylinder will allow a stroke of 31.4 mm high and i will have it made 20mm high, so basically I shall have a 51.5 high power cylinder.

The crank positions I am still looking it up but appreciate some sound advise.

Unfortunately, a great friend of mine is James Rizzo who wrote many books and materials for model engineering but he is on his late eighties and unable to help me.

Many thanks for your help.

The best advice I can offer is to buy one - they are pretty cheap, but you do get what you pay for. I'd buy a Kontax (I have no connection with them whatsoever), but they are pricey. Once you have a good one, copy it or change the parameters to suit your interests.....in your nice workshop. Trying to learn how to make one work first time could be more expensive than buying and learning how they work from a good example that you can play with.

Yes, there is a lot of conflicting information! Which is why buying one that you like the look of (and which will work) is a good idea if you've never made a Stirling engine before.

One absolutely fundamental "rule" is that the LTD (Low Temperature Differential) type that you are interested in are extremely low power devices, therefore minimising all sources of friction that you can is essential, otherwise it may not run at all!

Once you have one that works that you can study you can design and build your own. Look at Youtube videos by "barumman" who has a superb workshop and superb skills to go with it - he made an LTD engine that only needs 2 degrees temperature difference (watch this video and see if you can make one!):-

(By the way, it is 'Stirling' not 'Sterling').

https://www.youtube.com/watch?v=1ycOyGMxAJk

With a regenerator, displacer clearance is unnecessary, it can be as tight as possible without actually making contact.

Such engines, with regenerative displacers can operate on an incredibly tiny ∆T.

Here are a couple:

https://youtu.be/ARD3ctp80ac

https://youtu.be/czVQVCIi7CU

Both utilize regenerative displacers of similar type.

As far as precise, exact dimensions, I think the above two examples alone, three, including the previous post, show, that can be somewhat variable.

There are other ways of incorporating a regenerator, if it is seen as useful.

The P-19 , that could run on a 0.5 temperature difference, incorporated such a regenerator within the displacer.

The P-19 would run on a sponge kept damp by a wick from a glass of water, indefinitely.

I have a notion that further improvement may be possible by making the regenerator independent of the displacer. That is, stationary, so it does not need to actually be lifted by the engine. And also by reducing weight through the use of a thin membrane or diaphragm to move air through the regenerator, rather than a bulky displacer, but that is still experimental and untested for the most part, at least in an LTD configuration, but seems to work well at higher temperatures.