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Solar staircase design

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Posted by nicksanspam on April 4, 2008, 2:05 pm
PE Norman B. Saunders issued on 10/27/1981 describes a "solar staircase"
(like a solar pergola or the "brise soleil" slats over the flattish back
windows of sports cars) that admits most of the low-angle winter sun and
excludes most of the high-angle summer sun.

We might avoid overheating a 4/12 pitch 12'x32' 90% transparent R1 Dynaglas
corrugated polycarbonate roof over a sunspace in summertime and make winter
heat collection more efficient by raising the roof's average R-value with
something like this, viewed in a fixed font:

.   .   .   p        
.   .           o
    .               l
    .<-~2'-tall glazing y  
    .   .   .   .   .   p   c
    .   .   .   .   p /         a    
    .1x3.           .               r           south -->
                    .   -  ~4'  -       b          
                    .   .   .   .   .   p   o                
                    .foil-faced foamp / .       n
                    .1x3.           .               a
                                    .                   t
                                    .   .   .   .   .   .   e
                                    .   .   .   .   .   p
                                    .1x3.           p /

Horizontal reflective treads might be 1" foil-faced foamboard. Vertical
risers might be clear flat polycarbonate plastic, eg 0.007" GE HP92W Lexan,
which comes in 20-pound 4'-wide x 100' $00 rolls with an 8500 psi yield
strength and a 20-year expected outdoor lifetime. With springs to avoid
buckling, we can also make 2-layer 8'x12'-tall south wall Lexan glazing
panels, ie 8'x12' optically-clear "windows" with minimal framing.

At 40 N. latitude, the sun elevation at noon on an average 62.9 F Phila
May day is 90-40+18.8 = 68.6 degrees above the horizon.  The 8' 2x8 purlins
(0.6042' deep) p--p with beta degree elevation between doubled 12' 2x12
rafters might have foil-foamboard on the south face.  

If 16" Lexan risers need 3/4" to attach to the purlins and 1.75" to attach
to the treads with a horizontal 1x3 beneath, they would have 13.5" (1.125')
of transparent height. To shade the glazing at noon, we need:
                      .  .
            noon sun . .
                   . beta
                 p<--------- dawn sun
   a = 0.6042' pB.= 180-21.2-(90+beta)
             p beta
           p ---------
           g90 .
           a  .
b = 1.125' z  
           iA.= 90-68.8 = 21.2 degrees

Using the law of sines, beta = 68.8-sin^-1(1.125sin(21.2)/0.6042) = 26.5,
the same as the sun elevation at noon on 12/21, so the purlins won't shade
the glazing then. Dawn sun would hit the glazing 0.6042sin(26.5)/sin(37)
= 0.448' below the purlins. With an atn(4/12) = 18.4 degree roof elevation,
the purlins would tilt 90-18.4-26.5 = 45.1 degrees from an orientation
perpendicular to the rafters, with a 3.33/cos(18.4) = 3.51' purlin spacing
along the rafters.

The treads need to tilt up towards the south in order to bounce winter sun
into the glazing... 4' foamboard on 3.33' centers with a 0.6042cos(26.5)
= 0.5407 purlin overlap and a T degree tilt makes cos(T) = (3.33+0.5407)/4,
ie T = 14.4 degrees.

In dawn direct sun on an average 30 F January day in Phila, the roof might
receive 4'x32'x250cos(45) = 22.6K Btu/h. With no staircase, it could gain
0.9x22.6K = 20.4K Btu/h. If the air under the roof is 100 F, it would lose
(100-30)32x12/cos(18.4)/R1 = 28.3K Btu/h, for a net loss of 7.9K Btu/h.
With the staircase, it might gain 0.9x20.4K = 18.4K and lose about
(100-30)32(2+3x1.125)/R1 = 12K, for a net gain of 6.4K Btu/h.

If the top of the glazing is 0.5407tan(18.4)+0.6042sin(26.5) = 0.445' below
the top of the rafter above it, which is 4cos(T)tan(18.4) = 1.291' above
the purlin below and to the south, the tread would reflect sun below theta
degrees elevation into the upper edge of the glazing until tan(theta-2T)
= (1.291-0.445)/(4cos(T)), ie theta = 41.1 degrees, which happens at noon
when declination 23.45sin(360(284+N)/365) = 50-41.1 = 8.9 degrees on
the N = 104th day of the year, ie on April 14.

We'll be discussing sunspace heating at 2 PM tomorrow Saturday April 5
at Earthmart in Phoenixville, PA, (610) 935-1793, with a quiz at 2:30.


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