An enterprising user has modded the stock cooler that comes with the Core i5 12400 to run quieter using nothing more than a sheet of paper and some tape. Not only that, but Patrick Bene has detailed his experience in a blog, experimenting with different paper heights to show why he's settled on the optimal 1.7-inches. Lovely stuff.
This was done after noticing that the stock cooler for the Alder Lake Core i9 was taller than the Core i5 unit and that it had a blue plastic cowl around the fan, while the Core i5 has an open-air design. The driving force behind the mod was not to improve the cooling, but rather to address the amount of noise produced—the stock cooler is indeed a bit of a noisy beast under load. There are sound files on his blog so you can hear just how loud it gets.
After some experimentation, Patrick discovered that a 1-inch cowl reduced the noise levels the most, coming in 8dB quieter than the stock cooler—definitely noticeable. Unfortunately, that did see the max CPU temperature increase to 85°C. After some more paper cutting, with cowls up to 3-inches tall, he settled on the optimal 1.7-inch cowl. This produced the same cooling as the stock fan at 80°C under load while also reducing noise levels by 6dB.
One potential side effect of this is that the VRMs around the CPU may have reduced airflow, and so could potentially run hotter, although at stock clocks this probably isn't a major problem. There's still plenty of airflow around the bottom of the cooler too, so it should be fine. It's something worth checking if you're thinking of modding your own Intel cooler though.
It's impressive work either way and easy enough for anyone to do themselves—the best kind of mod. You could argue that you'll get much better cooling and noise levels from a third-party cooler, and you'd be right, but if you're on a tight budget, this seems like a reasonable solution to the problem. And one that shouldn't cost you a penny.