When someone did invent coalburning steam engines in 19th century, the Mediterranean people found that they did not have much fossil coal. England had quite some, Belgium, Germany also. What Spain had was in Asturias, on the Atlantic side. Italy and Greece lacked large fossil coal reserves.
But where an interesting spot does exist is Zonguldak on Black Sea. Not awfully far from Bosporus, close to water transport to Constantinople.
Suppose that in 4th century, someone invents coal-powered water pump. And builds big baths heated by coal, and with water pumped uphill by steam pumps.
Roman aqueducts rely on bringing water to hilltops by long bridges. But these bridges are vulnerable to enemy action when the foes are outside the walls. Steam pumps, on the other hand, can pump water uphill to the fortress, from watercourses or groundwater low at the foot of the hill which enemies cannot easily block. The coal for pumps can be stored up in peacetime, and the available people can make themselves busy fighting enemy attacks rather than carrying water uphill.
Would this use of steam engine make sense in 4th century?