I found this question by chance on the Internet and remembered that this problem has been discussed passionately and controversially for at least 40 years, even though the correct answer is obvious. Here it is.

### That little bit of warmth ..

About 30 years ago, a good friend of mine swore stone and stone that you always had to run the heating, even during longer absences, to save on heating costs. Because, he argues, if you turn off the heating, it cools down. The amount of heat you would need to bring the apartment back to room temperature would be bigger than keeping the apartment at room temperature with the little bit of heat you have. G. Wesener posted on Haustechnikdialog.de similar, almost even philosophical: "In relation to the overall heating system, switching off means in any case a step backwards, which as a renewed step forward additionally requires corresponding new energy. So in any case it is more correct to let the heating run through."Save heating costs by continuous heating? This is what I call the heating cost saving paradox, although it does sound kind of logical, doesn’t it?

That this argument cannot be fundamentally correct is shown by the following short consideration. Let’s assume you need 7 heat units to bring a cooled apartment back to room temperature and you only use 1 heat unit per day to keep it at room temperature. Then it is immediately clear that heating the apartment constantly for longer than a week is more expensive than bringing it back to room temperature from a cooled-down state. This consideration is independent of the exact values. It only says: At some point it must be worthwhile to turn off the heating instead of letting it run through. This is also what our common sense tells us. The only question is: When is it worthwhile??

### Lower a little, save a little

To figure this out, let’s take a closer look at an apartment. Let’s assume that you feel comfortable at home at 24°C, while it is 14°C cold outside. So the temperature difference is ?T = 10°C. Moreover, your apartment has the heat transfer coefficient U. This quantity, also known as the U-value, describes the ability of heat to pass through a wall. It is the most common parameter for describing the thermal insulation of a house. The exact size of U does not matter here. It is only important to know that the larger U, the worse your apartment is insulated. The heat W that escapes from the apartment is now determined from these two quantities as W = U×?T×A×t. Where A is the area of your home over which the heat can escape, i.e. walls, floor and ceiling, and t is the time period over which the heat escapes.

We now denote the amount of heat you need at ?T = 10°C over one hour over the walls lose as 1 heat unit. This is the amount of heat your heater must provide over 1 hour to keep the apartment at 24°C. Alternatively, let’s assume that you don’t heat the house for 1 hour and the temperature drops by 1°C to 23°C (admittedly a rather poorly insulated house, but it doesn’t matter for our consideration). What would be the amount of heat to bring the apartment back up to 24°C? Now, if you had continued to heat constantly, the apartment would have remained at 24°C. Therefore, the amount of heat up must be about this 1 unit of heat. In fact, it is a tiny bit lower: since your apartment is ½ h long ?T = 9.5°C and ½ h long ?T = 9.0°C had, you need because of W = U×?T×A×t together only ½ + 0.95×½ = 0.975 heat units to ramp up again. I leave it to you to calculate an even more accurate value for the amount of warm-up by further refinement.

### Lower a lot, save more

But now it comes. Since after 1 hour ?T = 9°C, you would need to heat the room because of W = U×?T×A×t only a heat quantity of 0.9 heat units is needed to keep the apartment at 23°C instead of 24°C. If they drop the temperature even further at night, you also save these 0.9 heat units and the temperature drops to 22°C after 2 hours. To get back to 24°C from there, you need 0.9 + 1 = 1.9 heat units. On the other hand, you need 2.0 to keep the temperature at 24°C all the time! So it is clear that if you let the temperature drop to 18°C at night, you save on the one hand 6 hours a 1 heat unit = 6 heat units and on the other hand you need 0.5 + 0.6 + 0.7 + 0.8 + 0.9 + 1.0 = 4.5 heat units to bring the apartment from 18°C back up to 24°C in the morning. You have saved 1.5 heat units!

### Saving is not always worth!

The answer to the question: "When do you start saving heat by turning off the heating or even just turning it down??" is therefore: "You always save!"In this respect, the heating cost saving paradox is not paradoxical at all, but simply wrong. But, the smaller the heat transfer coefficient U, i.e. the better the house is insulated, and the shorter the temperature drop, the less I save! Therefore, a well-insulated house has two advantages, I save over the year as a whole not only heating costs, but it is hardly worth it at all to turn down the heating when I am not in the house for half a day. Because I’ll come back and the apartment will be cozy and warm right away. The problem with an extremely well-insulated house, however, is good ventilation. On this, perhaps later .. .

*Guest article by Prof. Dr. Ulrich Walter, graduate physicist and well-known space expert and presents the documentary series "Spacetime" since September 2016.*