Not through Dell. We only offered the standard heatsink with fan. In theory only, any of the retail CPU coolers might work.
Here is how I found the answer...
* Went to Google
* Did a search for, "Inspiron 3847, CPU cooler upgrade"
* 5th link from the top was Toms Hardware. Opened that thread. User posted a link to the Arctic CPU cooler that he used in his Inspiron 3847
Dell does not use standard wiring or connectors for the cooling fans.
The result is a system fan failure message and shutdown when using NON DELL FAN. Worse if the wires are not in the correct position you FRY the new fan.
The 3847 uses DELL WDRTF heatsink/FAN
Dell wire color
Std wire color