What you need to find is the hypothenuse, so if you can’t find your high school trigonometry text book you can use an online calculator, but this does require inputting the correct angles and distances. 'Garbage in, garbage out' as they say, the answer is only as accurate as the figures you punch in. That would require determining the angle of the sun based on your latitude, and knowing the exact overhang you want on your roof, and the height of the windows. Before your head explodes though, check out this page. I promise it will be more fun to read than a page on trigonometry -
The basic Concept Passive Solar Home Design
If it's mid-September and you are calculating this now, does that mean you are still in the design phase or have you started building? If so, and if you’re not a mathemagenius, AND you can wait until December 21st, then you could just do it on site by hand and not risk having made any mistakes in calculation.
First, to orient you home properly, go to your site any day at noon and pin the exact location by driving a stake in the ground (straight would be good), and the shadow will tell you the exact orientation you’d want for your south facing wall to maximise your solar gain. That way you won’t get confused by ‘true north’ and ‘magnetic north’.
You could then – at noon on December 21st – nail 3 2x4s together to create a horizontal board you could raise into the air to the height of your overhang, see where the shadow falls, and you will have your answer.
It’s a great question and I really wish we could help more, but in order to solve it not just for winter heat gain but for summer shading, you do need to have a whole bunch of variables on hand like window size, where it will be in the wall and the height and length of your overhang, and the exact angle of the sun.
And while you may pick up a few more BTUs here and there due to the exact location of thermal mass in relation to the windows, keep in mind that once the energy is in, it’s in. If you have a bright and reflective floor, it will reflect the light (and heat) towards other surfaces than may then absorb additional heat.
The best council we can offer you though, is not to scratch a hole in your head trying to figure it out to the centimeter, there are too many variables to make that a valuable use of your time unless you are really having a kick doing it. This page on how to design for passive solar heat gain may help, best of luck!
What do you think about the argument that a) IF the average annual temperature is below optimum for humans and b) IF you put enough thermal mass of sufficient conductivity into the building and isolate it from the outside enough, then there is no such thing as too much solar gain - even in mid-summer?
I'd prefer to minimise my use of cement and use tubular earthbag construction for most of it instead. Earthbag has been tested by international NGOs for rebuilding in earthquake and hurricane hit areas, and it proved very structurally stable, has almost as much specific heat capacity and density as concrete, but it's very cheap and quick and easy to build with. And it looks nice with lime plaster on. They recommend ground and external walls insulation but only around the perimeter.
Enough thermal mass = I'm thinking of earthbag (aka. super-adobe) for most of it, with a wall of water tanks stacked 4 high in a timber frame, behind rebar mesh covered with fruit vines on the south side, granite cobblestone or brick paths, and use car radiators used in the apex of the dome to transfer heat into the bottom of the water tanks more efficiently, and underground heat exchange water pipes under the soil in the growing areas.
The two places where I'm thinking to do it - one has a warm climate, average 18C, coastal with 500mm precipitation per year, so there potentially excess heat is more of an issue, and the other location I'm thinking about has average annual temperature 10C, 680mm precipitation.
I plan to make a thermal mass rocket stove, with some modifications I got from wood kilns designed for firing porcelain (very high temperatures), including pre-heated air and a downdraft firebox. So I don't mind if the passive solar gain is not enough by itself. In the colder place where I'm planning to build, firewood is cheap. And I'd try to get recycled wood pellets too and design my stove for both.
I used to live in a house with a solar thermal system, 600l thermal store tank, and quite often in summer it'd get to nearly 95C throughout the tank, so to avoid it boiling or risking damage to the collector the next day by too much pressure in the pipes if the heat exchange circuit was not pumping, I'd release some heat by having a huge hot bath in the evening. Not really a hardship :) I can think of many more good uses for spare hot water. I think it's better to release some excess heat via water outside (water is plentiful in the cooler location I'm mainly thinking of) than to limit solar gain all year and then have to make an expensively super insulated and MHRV box.
Hi Kester, that's an interesting theory you have. We removed the links you posted as we don't keep links in comments sections, but one of the ones you posted was from an Earthship. There is a lot of apeal to them as it is a great looking story, but they do have drawbacks. In case you are getting a lot of your inspiration from them I would suggest a look at this page about why Earthships don't work in cold climates as it explains some of the downsides and challenges that you may not be getting from the Earthship corporate websites or blog pages from enthusiasts. There is a lot in there about thermodynamics, and it explains some of the limits of thermal mass in regulating temperatures on a 24 hour cycle.
Here are a few other pages on building science related to the topics you are covering so have a look and let us know what you think-