Not an expert but if there's 3 lines to the fan, then the motherboard might measure fan speed and complain loudly that a fan is missing. It's mostly on newer laptops, but perhaps check that before deciding?
how about a desktop cpu tower cooler? even though laptop cpus are low power, passive cooling is pretty difficult.
another alternative for mounting would be to 3D print a bracket but that would require a printer, basic cad skills and a heat resistant filament like polycarbonate.
look at how much metal there is on these fanless mini pcs which use low power laptop cpus
First of all, there's a good chance, you can get a replacement fan. There are not that many different fans.
If that's not what you want, you can probably just replace the fan with a regular fan, you just need an adapter for the laptop's fan header.
Honestly, cost wise you should just buy a cheap refurbished desktop with similar specs.
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