**What is Passive Solar?**

Passive solar house design can be very valuable. Usually when thinking about energy efficiency, we think about insulation, which just KEEPS heat IN your home, but with a passive solar layout, the sun ADDS heat to your home. When the sun shines straight through a window, it radiates about 1000 watts of power into your house per square meter of window. Yes, that should be able to replace a 1000 watt heater, but the sun will do most of its direct winter shining through SOUTH-facing windows. Unfortunately, most homes don’t seem to be designed with many south facing windows and fail to take advantage of the sun’s heat in the winter. Also, the sun usually shines in neither through a direct vertical angle nor a direct horizontal angle.

**Passive solar savings?**

I will skip to the end result of my estimations and say that you could save, over the course of a **cloudless** winter month, **$225 from 10 square meters of south-facing windows**. Here is the good part, how I calculated that:

**Accounting for the vertical angle**

In mid-America, at the winter equinox, the angle of sun at noon is about 23 (Earth’s tilit) +37 (our latitude) = 60 degrees from above, or 30 degrees from the line straight in your window from the horizon. So we should multiply the 1000 watts/ m^2 of solar intensity by cos30 to account for that non-directness…. But only at noon. At sun rise and set the vertical angle is 0. I am going to guesstimate an average value for the cosine of the sun’s vertical angle between cos(0), which is 1, and cos(30), which is 0.866. Let’s be careful to not overestimate the sun’s power, and settle on a value less than midway, let’s say 0.9.

**Accounting for the horizontal angle**

Horizontal angle: We also need to account for the non-direct angle side to side as the sun cuts from one horizon to another during the day. In the winter in this hemisphere it will not quite traverse all the way through -90 deg. to +90 deg relative to the line straight out your window, but we will assume that is does for simplicity, and because this cut corner will only slightly cause us to UNDERestimate the power of passive solar. Let’s find the average value of cosX for X between -90 and 90. We will need to use radians, so our angle range becomes -0.5pi to +0.5pi. A formula we can use for the average is 1/(b-a) INTatob f(X)dX. With our function, f, as cos, then integral is sin, and we get: 1/(0.5pi – -0.5pi) * (sin(0.5pi) – sin(-0.5pi)) = 1/pi * (1 – -1) = 0.3183 * 2 = 0.637.

**So what IS the sun’s average daytime radiative power through south-facing windows in the winter? **

One could approximate the average power by just multiplying the direct sun’s 1000w by both 0.9 and 0.637: 1000w * 0.9 * 0.637 = **572w**.

**For an energy savings of…**

We can find energy per m^2 from power times time. 572w is 0.572 kw. Multiply that by the approximately 10hr of light per day. That gives 5.7kw*hr of energy per square meter per day. Assuming you are heating with electricity and pay the average 13.5 cents per kw*hr, your window can replace 5.7 kw*hr * $0.135 per kw*hr= **$0.77** per square meter per day.

A decent sized window can be a full square meter in area or bigger. It is not unreasonable to think that you could have 10 or more square meters of window on the broad south side of a house if you put in even a marginal effort to go passive solar. With 10 sq. m of windows, that becomes 57kw*hr of energy per day of solar heating, enough to replace over $7.50 per day of electricity that would otherwise be used by space heaters, or $225+ per month for maybe half the year.

**In reality…**

We are making a couple of assumptions here. One is that it is sunny and cloudless all day long. Another is that the light that enters is absorbed in the house, maybe by a dark masonry floor, and that a lot of heat from that hot floor doesn’t leak into the crawl space or wherever. Optimally, there would be a thermal mass under the absorbing surface. Obviously, you don’t want to have the drapes closed or anything silly like that. We are also assuming that you use electricity to heat, and do not have a heat pump. In reality it is likely that you are paying less than 13.5 cents per kw*hr for your energy to heat your house. So, accounting for these factors, and assuming it will be cloudy half the time, you maybe actually only save half or less of $225 per month with your 10m^2 of windows. I would estimate, in reality, **saving about $100 per winter month**… but I bet you could have a lot more than 10m^2 of windows if you really wanted.

** What about in the summer? **

You might be concerned that you would have a lot of sun bringing in heat in the SUMMER, but during that season, the sun will be nearly overhead at mid-day, and the amount of light entering the window will be very limited. If you are at high enough latitudes that the sun stays low enough in summer to shine some light in, eaves above your window can block it. Definitely don’t have lots of skylights, though. 🙂 And this is a good argument for limiting the amount of east and west facing windows, because they will allow a significant amount of heat in during the summer unless you put in some good blinds or something.