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Does the old battery of an old laptop use a lot of energy when the laptop is connected to AC? Does the power supply waste electricity trying to continually charge an old battery? If I remove the battery, how many Watts of electricity would I save? Would the laptop stay cooler?

The power supply is very hot in both cases. The laptop is a Toshiba Satellite from 2004/2005.

This is a slightly different question than this: Should I remove my laptop battery?, so don't repeat the same answers. I've read about he UPS/power buffer effect, about the life of the battery, etc. I don't care much about the battery and I boot the laptop from DVD. There is no hard disk. I care about keeping the laptop cooler and spending less electricity.

user13097
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Yes, having less equipment under power will save some energy and decrease the overheat. However, the reduction will be so small you probably won't notice it. I don't have the exact numbers for your laptop, but I'd expect that you'll save 1 Watt tops, i.e. about 5% of your laptop power consuption in idle state.

According to Wikipedia the power consumed by a fully charged battery is even less. If we suppose that topping charge occurs once every 500 hours, lasts for 1 hour and uses 4 Watts of power (10% of the maximum charging power estimated at 40 Watts), then the annual consumption will be 365*24*4/500 = 0.07 kWh. Personally I find this estimation unrealistically small, but without the datasheet of your battery it's impossible to make a more precise estimation.

Is it worth the trouble to save 1 Watt and lose the UPS function your battery gives you?