Yes. It is, possible that you can cool a cpu without fans. You would need the i3 with the lowest tdp. Maybe an i7 but you need to get the one with the most thermal and heat savings. Even after you get the cpu, you would need a case with a decent airflow. If you are not going to put many fans, make sure that you do not install any high power graphic card or the least many sticks of ram as possible.
You would need to find the biggest ass cooler you can find on the market. make sure that the cooler is dust free as much as you can.
Now, for the testing part. Please get the best cooler and the best thermal compound you can find. UNDERCLOCK and UNDERVOLT the CPU until it reaches around 80 degrees max on load.
You do not need such performance because you are not going to do intensive stuff.
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WELL INSTEAD OF DOING ALL OF THOSE ***. You could go with a corsair 550D or any case which is quiet. Use SSDs as much as possible in your system. Use a fanless power supply. There are some good 500watt ones in the market which you can get like from FSP. Get silent fans for your system and turn down the rpm. Get the best air cooler or liquid cooler and get silent fans for them.
Now here for the important part. Get a automated fan controller. One that monitors your temperatures and adjusts the fan speed. You can set the speed to 0 rpm in the settings. Hook all your fans to that controller.
Undervolt and underclock your cpu until the fan barely turns on when idle. that would be a successful mod.
I would also recommend low power GPUs for your setup. I would personally go for the kepler lineup due to low TDP means low heat. Something like a gtx 650 or a 650ti or a gt640 would suffice for what you are doing. They are as good as a gtx 560.
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So in conclusion, YES. It is possible to make your cpu fanless. Although it would heat up your room. If you got the second method, It would be very very good and quiet and safe at the same time.
If you like my post, Please then select it as best answer as it took some time to write
Cheers