Renown's main belt was of a uniform 9" thickness, only tapering at the very bottom edge.
According to GK DOS 100, 9" of KC n/A armor would be vulnerable up to 22km against 28.3cm L54 gunfire.
Of course Renown's armor plates where not
of KC n/A quality.
From Nathan's Okun table of metallurgical propeties,
"GERMAN THICK-PLATE IMPROVED KRUPP CEMENTED 'NEW TYPE' (KC n/A)
ARMOR QUALITY: Q=0.96 and QD=Q BLT: 59 TC=N CW=N SS=1"
"AVERAGE BRITISH WWI-ERA KC-TYPE ARMOR:
1919-1930:. . . . . . . . . . . . . . . . ARMOR QUALITY: Q=0.900 and QD=Q BLT: 65 TC=N CW=N SS=2"
=> Renown belt armor would be vulnerable up to 23,5km, assuming perfectly parrallel courses.
Koop in Battleships of the Scharnhorst class, quotes from Scharnhorst's war diary and the maximum speed recorded was 30 knots, and then the boiler failure occurred.
John Asmussen's analysis, drawing from primary German and British documents, states:
"At 17:27, Gneisenau opened fire against Ardent that was hit on the Nr. 1 Boiler room by the first salvo, speed got reduced on Ardent that started zig-zag in her own smoke while shooting her 4,7 inch (120 mm) guns.
At 17:28, speed on Scharnhorst was 29 knots.
Ardent kept on zig-zag turns and reacted to German shots firing her first salvo of torpedoes against the German ships, one of which was seen to pass close ahead of the Scharnhorst
At 17:34, on board Glorious first group of Swordfish was now on the main deck and getting ready to be launched, armed with torpedoes.
At 17:35, signal "P" received, course now 170°.
Scharnhorst's second salvo went too far over Glorious.
At 17:36, speed was 29 knots on board Scharnhorst. Gneisenau was making already 30,5 knots.
Fire was continuing with main and secondary artillery against the British ships.
At 17:38, speed was 30 knots on board Scharnhorst.
Gneisenau kept on speeding up at more than 32 knots moving in the wake of Scharnhorst moving from port side to the wake of the leading German battleship.
The first hit from Scharnhorst with her third salvo reached Glorious from 24.175 meters (26.450 yards), which is the longest gunfire hit on any enemy warship ever achieved.
The 280 mm (11 inch) shell penetrated the flightdeck and burst into the upper hangar starting a big fire, in the middle of the deck a big hole made it impossible to launch any other aircraft.
All the aircraft present on the main deck were seen falling over board (presumably the 2 Swordfish that reached the main deck earlier).
Splinters pierced a boiler casing and smoke entered air intakes. This caused a temporary drop in steam pressure from 2 boilers, but steam was built up again as the smoke cleared.
Gneisenau turned to starboard at full speed.
At 17:43, Gneisenau was moving fast crossing Scharnhorst wake.
At 17:44, Gneisenau was now sailing on Scharnhorst's starboard side at faster speed.
Scharnhorst Chief Engineer reported an engine room out of service, speed was decreasing.
At 18:02, Scharnhorst was making 27 knots. The ship was having engine problems, boiler 1.1 out, probably due to split tube."
Given the liquid loading required for her TDS, it is unlikely that S&G could operate at less than 37000 tonnes, which might give another ~.5 knots. Full load with full fuel was 39643 tonnes.
The "39643 tons" figure was a 1943 displacement, after wartime modifications.
In 1940, maximum load for Scharnhorst class was about 37.600 tons, and I haven't seen any
indication of a single time when they actualy carried even that kind of a load
However, we would have to make the same calculations for Ark Royal and Renown:
Ark Royal made 31.2 knots at deep displacement (27500 tons) with 103,000 shp ( no overload power). Full overload = ??.? knots. (my guess is ~32 knots at ~125000shp)
Renown made 30.1 knots at 34800 tons (full load = ~ 36500-37000 tons in 1941) with ~120,000 shp (no overload). One book on Renown states that she developed 160,000 shp and 32 knots when using full overload power in 1943.
We can make all the calculations you want,
fact is Renown made 29kts max at Stromvaer, and was gradualy left behind by the raiders, demonstrating quite clearly which was the faster ship class.
Ark Royal was tested in 1938
, and in the 3 years passing until her demise, the bottom fouling and machinery wear would take it's toll.
On the other hand, S&G demonstrated 31-32+ kts speeds in 1940 and 1941, so they were actualy capable of those speeds.
Moreover, the infamouse North Cape chase shows Scharnhorst at a 3-4kts advantage from Duke of York. Given DoY was the larger ship, I would expect it to behave better in heavy seas, so the speed reduction should have been smaller for her than for Schar.
Alas, even considering similar speed reductions, the situation showed again Schar capable of at least 31-32kts speeds.