Use a ceiling fan instead
For the reasons you discuss, ceiling fans are often installed and used in winter to draw air upward, from this answer on diy.SE:
This directs the relatively warm air above the fan outward and then
downward, making the rest of the room feel warmer.
But building it yourself is
cooler more awesome!
The cheapest PWM fan I could find on Digikey at a reasonable DC voltage is $8 USD. But if there's a university/college/school anywhere near you, it may be possible to find old PCs and salvage the fans from those. (My university, University of Wisconsin, has a warehouse with bins of non-working PCs where you can pull out parts for cheap.) Typically anything with four wires will work with PWM control.
Adafruit sells a custom micro-controller with PWM control--the Metro Mini 328--for just $12.50 in the US. For a bit more, an Arduino Nano will also do the trick.
After that you'll need a dc power supply, a thermistor (depending on the fan--some will include this), and assorted wires and connectors. I'd estimate $10 to $30 for all of this stuff.
Assuming you're able to scrounge at least a few of the parts you need, I estimate you could do this for $30 per radiator if you're going to make at least four or five for your home.
Doing a bit of back-of-the-envelope engineering, if you use four fans per radiator like the one I linked, and it's running for six hours per day on average for six months of the year, your electric usage per radiator will be about 10 kWh.
If we assume your electric cost is $0.15/kWh, and the system will last at least five years, then the annual cost will be about $8 per radiator per year. So as long as it saves you this much in heating costs per year, then it's a win!*
In reality, two or three fans would probably do, and they'll probably last longer than five years.
Other things to consider:
- If your electricity supply is from renewable sources, then by using electricity to reduce your need to burn gas for heating, you're saving the planet.
- While you're doing all of this work, you'll be burning calories, which will enable you to lower your thermostat and reduce heating costs.
- The time you spend doing this is inconsequential, because it's fun! and you'll be learning stuff!
*Figuring out how much this will actually save you per year will either require lots of work, or lots of assumptions, or both. I'll let someone else tackle that part of the problem.