John, the copper block isn't a bad idea, but lets morph that:
If you lengthened the block of copper to get more flame area on it, and more water contact surface in the hole, you might carry that on until you reduced the width dimension so that you essentially had a long square tube. Trimming the corners off of that and making it round would make it easier to manufacture and further reduce the cost of the copper while increasing further the rate of heat transfer. Of course you'd need a very long flame for something like that. So you would coil it up to make it more compact. And so we've arrived at the coiled monotube.
But i have no doubt that for a very small engine, a fairly large block of copper would produce some steam. It doesn't have to be a single hole, but could be a gallery of holes, looking on end like a bee's honeycomb. Header blocks at the end could connect the tubes in series or parallel, however desired, and in fact you could alternate water tubes with hot gas tubes. It's a heat exchanger.
It is expensive and difficult to make however. A block of copper is dear, and you'd have to drill a lot of that expensive material away. And that's a lot of drilling work and header fabrication, too.
Still, maybe it has some advantages for some purpose. Seems robust and compact, and those are good things.
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