I think the coil as recommended is far from ideal for this application but it is one that is easy to make and the ability to trickle coolant though it allows for a relatively high duty cycle (whether you take advantage of that duty cycle is another matter).
A coil structure such as one using an air-gapped ferrite core is going to be much better. (The flux pattern is concentrated in the air gap into which the case is inserted.) The challenge with these is finding a suitable air gapped ferrite, winding around it vs retaining cooling (if that is desired) and supporting the coil structure.
The cylindrical coil fits the easy peasy lemon squeezy criteria.
Vf for these things is 1.6V. Are you sure you guys are heat sinking them appropriately?
Here's the updated drawing per my "as built". PM me if you'd like the TinyCad file. I'm also working the final parts list and can include that when complete. Still working on the hopper; need to machine a few parts on my mini CNC.
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