I came home yesterday all wicked sore from skiing, and the big hot bath was pretty mandatory. You might think this is quite the waste of energy, all that hot water, but it isn’t at all if you use it to heat the house after you use it for bathing. And that’s easy to do: you just leave the plug in until the water cools to room temp.
Bubbling in the narcotic warmth of the tub, I got to doing mental calculations of the heat in the tub water, and how much it could warm up the bathroom (assuming the bathroom is a sealed system). That required some thought about the heat capacity of various components of the bathroom.
Here’s a pie chart that shows the findings, roughly. It is specific to my bathroom, which is about 7×9 ft with an 8ft ceiling, with tiled floor, and contains a heavy iron tub, small iron radiator, and ceramic toilet and sink. I’ve assumed that about 20 gallons of water has been run for a bath, and that the room is well-insulated beyond the sheetrock (which isn’t really the case, but, you know, just for illustration.)
So, once the temperature of the bathroom equilibrates, about 1/3 the heat will be contained in the tub water, 1/3 in the walls and tile floor, and 1/3 in the sink, toilet, radiator, and tub together. The actual air in the room is what “feels” warm or cold to a person walking around, but it actually contains only a tiny fraction of the room’s heat.
In practical terms, the fact that the tub water has about 1/3 the heat capacity of the rest of the room means that for every degree the water cools, the room will warm by 0.5 degrees. So if you start with tub water at 100F, and the room at 50F, you would eventually end up with both water and room at 50 + ((100-50)*.33) = 66.5F.
But in reality, you can extract more heat than that, because the bathroom is connected to the rest of the house, and given enough time and a coldish house, the bath water will cool further. If you keep the house close to the temperature of incoming cold tap water, you get back as house-heating almost all the energy you put into making the hot bath. Plus, you get a hot bath. Nice.