Shouldn't turning the air conditioner off during the day save energy?

Question

Websites like This site state you shouldn't turn air-conditioning off when you leave for the day because:

So when you have your air conditioner set to a comfortable 68 degrees, and it’s 98 outside, it is your insulation that helps your air conditioner do this. However, this also means that the inside part of your insulation is 68 degrees and the insulation on the outside is 98 degrees. The 30 degree difference is what your insulation buys you. Unfortunately, when you raise the temperature of your thermostat to 85 when you leave the house, you are not only changing the temperature of your air conditioner’s thermostat, but you are raising the temperature of your insulation as well – raising the temperature that your air conditioner has worked hard to bring down to 68 degrees. Therefore, when you return home and change your thermostat to 68 again, your air conditioner now has to work overtime to not only lower the temperature of the air in your house, but of the insulation as well.

But just because the insulation is cool doesn't mean heat isn't flowing into it! The AC has to keep removing this heat. The energy saved during the day should be greater than the "must work harder when you go home" effect:

A simple resistor model

This model ignores the effects of different parts of the house being at different temperatures.

In the long run (weeks), even a slight average heat net flux would cause the house to get extremly hot or cold. Thus we can assume that total heat in = total heat out.

Assume that the houses heat gain is: $dE/dt = (T_{out}(t)-T_{in}(t))/R + H(t) - A(t)$

Where E is the thermal energy in the home, R is the thermal resistance, H(t) are heat sources that don't depend on the houses temperature all that much, such as people, ovens, or sunlight (see note 1), and A(t) is the air-conditioning heat removal rate.

The AC is always 100% on or 100% off (most AC's work this way). We assume that the AC uses no energy when off. When it is on it uses x Watts of electrical energy to remove y Watts of heat and costs z dollars/day. The COP = y/x is fixed (see note 2) and the electricity rate = z/y is also fixed (see note 3). Under these assumptions the average of the heat removed, A(t), is proportional to our electricity bill.

If we leave the AC off during the day and turn it on at night will mean $T_{in}$ is higher during the day and the same at night, so it is higher on average. The total heat flowing in, $\int (T_{out}-T_{in})/R$ will be lower and $\int H(t)$ doesn't change. Thus turning the AC off for a given time period instead of keeping it on will lower $\int A(t)$, so it should be turned off whenever we don't need it.

More complex models

Our result holds for any linear resistor and capacitor circuit model that accounts for different heat capacities and different temperatures of different parts of the house. We care about the average heat transfer, i.e. the DC component of the temperature. Capacitors become open-circuits so we are looking at a network of resistors, which can be reduced to the single-resistor case above.

We are looking at $T_{in, off at day}(t) <= T_{in, on at day} (t)$ for all time t. This is a pretty strong condition. It is hard to envision, even with more complex factors such as latent heat of condensation, radiation, etc, a realistic scenario where it can take more AC energy to maintain house at a temperature curve that is never cooler.

Human factors

Getting into a hot house that may take hours to cool is unpleasant. People can instinctively crank down the thermostat below their comfort zone (which doesn't make it cool down any faster) wasting energy making the house too cold.

Apart from human behavioral quirks, is my conclusion correct or is there a potentially important factor I am missing that may make my conclusion wrong?

Note 1: Ovens, the sun, etc are so hot that they shouldn't care about the difference between a 20C house and a 30C house. People have a relatively constant metabolism if they aren't generating extra energy to fight the cold; if the house is kept at 37C the body heat is released as latent sweat evaporation and would still have to be removed some how.

Note 2: The COP (coefficient of performance) has a thermodynamic limit of $COP < T_{in}/(T_{out}-T_{in})$, which would make keeping a house cool during the heat of the day even more expensive. The real COP (3-6.5) is much lower than this limit (rarely less than 15), so it is likely to be more constant.

Note 3: Peak hours tend to be during the day (this may reverse if we get wide-spread solar energy), so keeping the house cool during the day would cost even more.

Answer 1

I agree with you that it almost definitely uses more net electricity to keep the air conditioning on all day than to turn it off for part of the day, and that the website you quoted is nonsense, like you say. It's not entirely inconceivable that the website is correct--one can imagine a scenario where air conditioners are far less efficient at higher power or whatever--but I think it's unlikely (note: I don't have any specific knowledge in this area).

HOWEVER, if you're trying to be green or civic-minded rather than minimize total electricity usage, it's plausible (though still not certain) that it's a good idea to use more power overall but less power during peak usage hours. I know it differs from place to place, but personally, my electrical utility is always sending us notices begging us to reduce electricity usage at 3-6PM during hot days. In that case, keeping an AC on low all day would make life easier for your utility, reduce the need to build expensive peaker plants etc., versus letting the house get hot then turning the AC on full blast at the worst possible time. For some people, it's not just civic-mindedness: each joule of electricity may cost more during peak hours. But that depends on your particular type of electrical meter, and local policies.

But if you really want to be green and civic-minded, reduce AC usage altogether, by improving your insulation, closing shades during the day and opening them at night, opening a window or going outside or having squirt gun battles when appropriate, etc. :-D

Answer 2

You pretty much have it right. We have two scenarios:

1 - Leave air conditioning on all day. Say that outside temperature is 90 degrees and inside temperature is 70 degrees. Then, all day long, the air conditioning has to remove any heat that gets into the house continuously. Heat transfer depends on the difference in temperature between outside and inside the house. In this case, heat transfer per unit time is equal to a constant K times 20 degrees. Over a whole day of D units of time, the total energy removed by the air conditioner is $20KD$ This assumes that convection (ie air draft) can be neglected.

2 - Stop the air conditioning during the day. If you stop the air conditioning, the flow of heat in the house initially will be the same. However, as time goes by, the inside temperature will increase. This will gradually reduce heat flow as it is proportional to the difference in temperatures. If the house is small enough and badly insulated, you could even get to a point where the inside temperature is equal to the outside temperature, at which point no more heat enters the house. When you get back from work, or if you have a timer a bit before, air conditioning is restarted. You now have to remove all the accumulated heat. However, since less heat entered the house over the duration of the day than if you had kept the air conditioning on, you have less work to do to cool down the house. Yes, the heat will have accumulated in insulation, furniture, walls, etc, but it doesn't matter. There is less heat total to remove.

Answer 3

It could be important to take into account the humidity/waterdamp. The hotter the air, the more water it may contain in a given volume. After switching off the AC, the air gets warmer and might take on more and more water. If the AC is switched on again, not only the same volume of air and the building itself must be cooled down again, but also the additional amount of water that is now contained in the air. That is also the reason why there is a higher danger of molding and fungi if the AC is switched off periodically.

Answer 4

hmph. disagree from the perspective of thermodynamics. it is MUCH easier to KEEP a room cool than it is to CHANGE it to cool, as per q=mc(change T). Leaving the AC off means the universe gives the energy (in the form of heat) back to the room that has already been removed, at which point that energy will again have to be removed to get it to a cooler temp. In no way is that efficient.