# Hot Tub Thermostat

June 14, 2008

It’s common to hear the claim that it’s more efficient to always keep a hot tub on, because it’s cheaper just to keep it hot than to heat it up from cold when you need it. There are variations, like the temperature of a water bed, or a house, or the humidity of a house with air conditioning. There are two ways to show why this is wrong, assuming that we’re trying to minimize overall energy usage.

The first is to consider energy loss, which is equal (more or less) to energy usage. In scenario A we leave the hot tub’s thermostat up all the time. In scenario B we turn it off, only turning it on the day before we need it (or however long we need for it to heat up from cold). In scenario B the temperature is closer, on average, to ambient temperature. Heat transfer between two objects is proportional to the difference in their temperatures. That’s why hot objects feel hotter than warm objects—because they’re transmitting heat faster to your hand. So if the average temperature is closer to ambient temperatures, then average heat transfer is less, so heat loss is less, so energy loss is less, and energy usage is less.

The second is to look at the curve of water temperature as it cools down and heats up. Again because heat transfer is proportional to temperature difference, the curve is exponential, steep at first and leveling asymptotically to the heating element’s temperature or ambient temperature. When the heating element turns off, the temperature of the water is furthest from the ambient temperature and the water cools off fastest. When that temperature drops below the thermostat’s setting, the heater kicks in. But at that point, the water’s temperature is still fairly close to the heating element’s temperature and the curve is mild. So with the thermostat on, you’re getting the worst of both curves: the steep part when it’s cooling and the flat part when it’s heating up. That’s not true when you turn the hot tub off. In that case you get the good and bad ends of both curves.