Geometry, logic and sun angles: help

[TaniaB]

I am trying to figure out what size eaves we need for our house extension.

I've tried a few calculations and now I'm posting here in the hope that a logical thinker will be able to tell me what I'm doing wrong.

We live in Adelaide, Australia, and we want to put in glass double-doors on a north-facing wall.

The glass doors are 2.7 metres high. The height from the floor to the underside of the eaves is 3.6 metres.

I've found 3 ways to calculate the desired eave length.

Method 1.

To use the sun angles for particular dates.

I've only been able to find 3 relevant sun angles - the elevation angles of the sun at a). the summer solstice (Dec 22), b). the equinoxes (March 21 and September 23), and c). the winter solstice (June 21).

The angles are a). 79deg (approx only, this is for Sydney), b.) 56deg (again, approx only, this is for Sydney) and c.) 31deg (apparently this is correct for Adelaide).

If we want to block the sun between the equinoxes, from Sept 23 until March 21, then the

eave length = 3.6/tan56 = 2.43 metres.

but this seems awfully long.

I can also calculate the penetration of winter sun if the eave was 2.43 metres, and that would be = ( 3.6 - (tan31*2.43) )/tan 31 = 3.56m

which = 3.56m



Method 2.

To use a rule of thumb a found on the internet.

This rule says that the eave length should equal 45% of the height from the bottom of the glass (in our case, the ground) to the underside of the eave

So this would be 45% of 3.6 = 1.62 metres.

This sounds closer, but I have no ideas what dates the sun would be blocked.


Method 3.

I found a chart used for calculating eave lengths here

http://www.concrete.net.au/publications/pdf/briefing09.pdf

The chart is Figure 15. It is towards the end.

I can calculate the eave length if we wanted to block the sun at 7 dates throughout the year.

If we look at blocking the sun between the equinoxes, the eaves length should be 2.52.

This matches closely with my calculations using sun angles, but again, it sounds huge.

I'm a bit stumped.

Thanks for reading this.

Tania

 

[cepheid]

Welcome to PF Tania!

You seem like you know what you're doing geometry-wise. In addition to the sun elevation, the sun azimuth (compass direction, essentially) matters as well. I mean, it's not a problem if the sun isn't shining through your door in the first place! However, since you're in the southern hemisphere, the sun will indeed be due north at local high noon, which is the time at which it also has its highest elevation angle. So if this is a north-facing door, then it'll be getting a lot of sun. I'm guessing that you already know that, which is why you are carrying out this exercise.

The diagram below shows my reading of the situation:

The two rays that are drawn are the extremal rays, the ones that hit the wall at its highest and lowest points respectively. In order to block out the sun, all rays in between these two must be blocked, necessitating a horizontal eaves of length "L" as shown in the figure. The elevation angle of the sun is indicated using the Greek letter theta (θ). From the right triangle formed by the eaves, the wall, and the lower ray, you can solve for L using simple trigonometry. So it would seem from the geometry that the equation you used is indeed correct.

Now, here's the thing: this assumes that you want to block out the sun entirely. However, in practice, this may not be necessary. The rays that are closer to horizontal are a problem, because they hit higher up on the door, and hence will penetrate farther into your house, which might fade your furniture etc. However, the steeper (more vertical) rays that hit lower down on the door will not penetrate nearly as far into your house. This means that it may not be necessary to block the sun from shining through the lower portion of the door. It's up to you to decide how far into the room is acceptable. For instance, if you decide that the unblocked rays shall hit the floor no farther than 1.0 m into your house, then the situation is as shown in the second diagram below:

The ray with the dashed line is now the steepest one that gets blocked. So now you have a couple of right triangles, and you can use them to solve for L. (I gather from your original post that you already knew how to do this as well). For θ = 56° and an illumination that extends no farther than 1.0 m along the floor, I get that L = 1.43 m, which saves you a whole metre of material! So that's something to take into consideration. Is it OK if a small portion of the floor just inside your door is illuminated?

EDIT: I might post again tomorrow with some thoughts on figuring out the sun elevation angles in the place where you live. A really easy way to do so is to consider looking into some astronomical software that simulates the sky. Stellarium is an excellent choice and is available for download online for free. With Stellarium, you can enter in your location on Earth, the date, and the time, and it will show you what the night (or daytime) sky will look like at that location and time. You can click on objects and information about them will pop up, including their azimuth and elevation. http://www.stellarium.org/

 

[TaniaB]

Thankyou cepheid.

Your reading is absolutely correct. I also like your point about whether I need to block the sun completely at the equinox or whether the sunlight can penetrate some distance at these dates. And thank you too for pointing me to Stellarium.

So now I think my calculations were ok and the issue is the dates I want the sun completely blocked.

I've tried re-doing the calculations.

Adelaide has really hot summers. February is usually the hottest month, and it is not unusual to get days that are 40 deg C in February so I think I'd want the sun blocked until at least March 1.

I used Stellarium and got a sun elevation angle of 66.95deg at noon on March 1, 2012 (tomorrow in fact!).

This gives me an eave length of 2.52m. Still very long.

So maybe there is no way around it with eaves alone. Perhaps I have to be happy with a shorter eave length, and then on hot days outside of the dates covered by the eaves I have to just put up with it and make sure i have thick, blockout curtains. Or get shutters or something.

Tania

 

[sophiecentaur]

This question has no definite answer, of course because, at sunset, the Sun's rays are arriving horizontally. The time of day at which the Sun is obscured will change as the year progresses and it will also depend on the direction in which the walls actually face.

It's a fascinating idea to discuss but I have a feeling that the best practical solution to this would be to look at what everyone else is doing. Local Architectual styles tend to respond to the local environment so they probably know best.

BTW, cepheid's diagram is fine except that all rays from the Sun should be drawn parallel (or as near as dammit). So you can't really include the Sun in the diagram.

If you look in a table of Ephemeris which is used for Astro Navigation, you can, after doing the Nav Course (a pain), work out what you want to know, once you've decided what you want in the way of Sun exposure. There are, alternatively, formulae for the Sun's azimuth and elevation at any point on the Earth, at any time in the year.

 

[TaniaB]

Hi Sophiecentaur,

You might be confusing your directions. I agree it is not possible to block the sun from the east or west with horizontal eaves alone. To block early morning eastern sun or late afternoon western sun you'd have to use vertical shutters or something.

But it is possible to block the northern sun (or the southern sun if you are in the northern hemisphere) with horizontal eaves or louvres etc.

>the best practical solution to this would be to look at what everyone else is doing

That does sound like a good idea but actually there doesn't seem to be a lot of consistency as far as I can tell. Even though we have a hot climate there are plenty of new places with absolutely no shading at all.

Tania

 

[sophiecentaur]

You are right about modern buildings. Architects should be made to live in what they design!
Perhaps it would be best to look at pictures of houses that were built in the past when people actually thought about these problems.
But I think you have the solution, in the form of slats or Venetian Blinds. You can adjust them at will, to suit the prevailing conditions.
If you happen to have a year to spare, you could stick a pole in the ground and record the shadow length for each day. That would tell you exactly what you need to know - but not now!

 

[sophiecentaur]

Hi Sophiecentaur,

You might be confusing your directions.
Tania

Which directions?