|Summary of Davy's 1799 ice-rubbing experiment. |
The experiment was done while Davy was working as Superintendent of a Pneumatic Institution in Bristol, whose function was to establish the beneficial effects of inhaling gases, and the results were published in 1799 when Davy was 20. The significance of Davy’s ice-rubbing experiment helped to prove that heat was a mode of motion.  The experiment was nearly the same in purpose as American-born English physicist Benjamin Thompson's 1798 "cannon boring experiment", in which used the heat from the boring of a cannon to make water boil, to show that caloric was seeming and endlessly coming from no where.
In a vacuum?
There are a number of publications in which Davy's ice rubbing experiment are said to have occurred in a vacuum, and others reporting that he never performed the experiment at all, among other variants. As early as 1875, for instance, Davy's experimental results were being summarized by Scottish physicists Balfour Stewart and Peter Tait as having occurred via come type of clockwork mechanism "in a vacuum" as follows: 
“Rumford's boiling of water by the heat generated in the boring of a cannon, and Davy's melting of ice by friction in vacuo, were each conclusively demonstrative alike of the non-materiality of heat and of the ultimate fate of work spent in friction. The exact and formal enunciation of the equivalence of heat and work required to fill the lacuna in Newton's statement was first given by Davy in 1812.”
In 1922, Arthur Taber Jones, of Smith College, published a correspondence in Science entitled “Did Humphry Davy Melt Ice by Rubbing Two Pieces Together Under the Receiver of an Air Pump?”, in which he states: 
“In books which I happen to have at hand I find twelve different authors stating that Davy melted two pieces of ice by rubbing them together in a vacuum, and four of them stating in addition that the two pieces of ice were rubbed together by clockwork.”
In a review of the corpus of Davy’s collected works, Jones speculates that the “vacuum” assertion was mis-read conglomeration of two different experiments. Likewise, German-born Israeli geophysicist and mathematician Ari Ben-Menahem, in his 2009 Historical Encyclopedia of Natural and Mathematical Sciences, comments that: 
“Although this work has been accepted for many years, it seems very doubtful that Davy, then 19 years of age, could have carried out such a [clockwork mechanism in a vacuum] experiment, which would tax the ingenuity of any trained engineer.”
There may be some truth to this "no vacuum" argument, as it is very hard to make a true vacuum, as is evidenced by the famous "sealing problem" encountered by Otto Guericke in his beer keg vacuum attempts.
In 1971, Donald Cardwell, in his historical From Watt to Clausius, was reporting that "Boyle also reported that two blocks of ice melted more rapidly if rubbed together than if left to melt undisturbed".  This, however, likely seems to be Cardwell recalling the ice rubbing experiment, but confusing Boyle with Davy.
1. Weber, Robert L. (1973). A Random Walk in Science (pg. 39-41). Francis & Taylor.
2. Stewart, Balfour and Tait, Peter G. (1875). The Unseen Universe: or Physical Speculations on a Future State (§101). Macmillan.
3. Jones, Arthur T. (1922). “Did Humphry Davy Melt Ice by Rubbing Two Pieces Together Under the Receiver of an Air Pump?” (Ѻ), Science, LV(1428):514.
4. Ben-Menahem, Ari. (2009). Historical Encyclopedia of Natural and Mathematical Sciences, Volume 1 (pg. 1533). Springer.
5. Cardwell, Donald S.L. (1971). From Watt to Clausius: the Rise of Thermodynamics in the Early Industrial Age (pg. 4). Cornell University Press.
● Henry, William. (1803). “A Review of Some Experiments: Which Have been Supposed to Disprove the Materiality of Heat”, Philosophical Magazine, 15: 45-54.
● Buckingham, Edgar. (1900). An Outline of the Theory of Thermodynamics (pgs. 13-14). MacMillan.
● Milestones in Thermodynamics – Thermal Physics, University of Notre Dame.