It's doubtful that this is an entirely accurate depiction because the vertical rotisserie depicted (no matter how powered) apparently was only invented in the 19th century, or at least we don't have evidence before.
On the other hand, that method of transmission is more like a steam turbine... and amusingly "In 1551, Taqi al-Din in Ottoman Egypt described a steam turbine with the practical application of rotating a spit." But it was probably a horizontal one... although Wikipedia cites that from a book review, which says:
There is also a self-rotating spit of obvious importance to the history of steam
power: on the end of the axle is a vaned wheel driven by a jet of stem directed onto its vanes from the spout of a heated water-filled pitcher.
The original publication under review would be pretty hard find though, as it was published in Arabic in Syria. I don't know if there are other reproductions of Taqi al-Din's work, or if it was ever put in practice or was just a design. Also, the review gives the year as 1552; the manuscript is located at the Egyptian National Library in Cairo (item K3845), with a copy in Turkey and one at the Chester Beatty Library
in Dublin (item 5232).
The (vertical) reproduction you have pictured seems to be pretty similar to what's displayed at the Museum of Science & Technology in Islam at the King Abdullah University in Saudi Arabia.
I'm not sure how reliable this Wikia site is, but supposedly a fuller translation of what al-Din wrote on the matter is
"Part Six: Making a spit which carries meat over fire so that it will rotate by itself without the power of an animal. This was made by people in several ways, and one of these is to have at the end of the spit a wheel with vanes, and opposite the wheel place a hollow pitcher made of copper with a closed head and full of water. Let the nozzle of the pitcher be opposite the vanes of the wheel. Kindle fire under the pitcher and steam will issue from its nozzle in a restricted form and it will turn the vane wheel. When the pitcher becomes empty of water bring close to it cold water in a basin and let the nozzle of the pitcher dip into the cold water. The heat will cause all the water in the basin to be attracted into the pitcher and the [the steam] will start rotating the vane wheel again."