an easier way to calculate kerf locations for parabolas SOLAR.KnowledgePublications.com

an easier way to calculate kerf locations for parabolas
21 aug 2001
a while ago i posted an algorithm for calculating kerf (sawcut)
locations to approximate a parabola. here's an easier way...

a parabola y^2 = 4fx (f is the focus along the horizontal x axis) has
dy/dx = 2f/y, so it turns 90-atn(2f/y) degrees from the vertical on
reaching y. if the first segment from (0,0) turns alpha/2 degrees and
the next n-1 segments turn alpha degrees, (n-0.5)alpha = 90-atn(2f/y).

for example, if f=1' and y=4', the parabola turns 90-atn(0.5) = 63.43
degrees from the vertical, so alpha = 14.1 deg for an n=5-segment fit.

we might use 1/2" wide x 0.5/tan(14.1) = 2" deep kerfs in a 6' 2x4 and
rip a 3/4" x 1.5" x 6' cap strip above the kerfs and screw it back on
to hold some reflective film. a parabola from (0,0) to (x,y) has length
(a+y^2/(2x)ln((2x+a)/y))/2, where a = sqrt(4x^2+y^2), eg 5.9158' for
(0,0) to (4,4). a 2x4 can be bent with a comealong and held with kerfs
closed by attaching a galvanized strap with holes for deck screws into
the kerfed edge.

the 1st kerf has x1/y1 = tan(0.5alpha), and y1^2 = 4y1tan(0.5alpha),
so y1 = 4tan(0.5alpha) = 0.4946'. the 2nd kerf has (x2-x1)/(y2-y1)
= tan(1.5alpha) = (y2^2-y1^2)/(y2-y1)/4 = (y2+y1)/4 (since a^2-b^2
= (a+b)(a-b)), so y2 = 4tan(1.5alpha)-y1 = 1.0525'. the 3rd and 4th
have y3 = 4tan(2.5alpha)-y2 = 1.7735' and y4 = 4tan(3.5alpha)-y3
= 2.8832'... x1 = y1^2/4 = 0.0612, and so on.

l1 = y1/cos(0.5alpha) = 0.4984' from (0,0) to the 1st kerf. the distance
from the 1st to the 2nd kerf is l2 = (y2-y1)/cos(1.5alpha) = 0.5982'.
l3 = (y3-y2)/cos(2.5alpha) = 0.8828', and l4 = 1.7031', so the kerfs
are l1, l1+l2, l1+l2+l3, and l1+l2+l3+l4 feet from the end of the 2x4.
a surveyor's tape measure ("stick") marked in hundredths of feet is
helpful in measuring kerf distances.

caryn rogers and i built one of these out of 2" square steel tubing at
zomeworks, welding the kerfs closed, and the results were unfortunate.
 
steve baer said we were unkind to the dura-lar film (in 0.002"x54"-wide
rolls from graphics arts systems, inc. in cleveland, at (216) 581-9050),
since we didn't attach it well at the edges or use a compound curve, so
it had lots of big floppy wrinkles. we drilled holes vs slots for the top
and bottom rails, so we couldn't align and tighten the film well, and
it was only metalized on one side. we put that side out vs in, and the
metal dissolved in a day in a combination of rain and sun, leaving a
transparent reflector. i scratched some metal off one side with a knife,
leaving transparent film, and dissolved another metal film patch with
warm vinegar.

the next version will be wood, with the metal film side attached to
the underside of some uv poly film with a thin layer of axle grease,
as duane suggests. it seems reasonable to make an unpressurized target
with a flat uv poly film greenhouse duct on top of a dark sip on the
ground, with a float switch and a tank and a pump in a basement to keep
a 1" layer of water flowing through the duct when the sun is shining,
letting the water drain down out of the duct at dusk.

us patent no. 5,851,309 ("directing and concentrating solar energy
collectors" paavo kousa, issued 12/22/1996, with 66 drawing sheets
and 62 columns of text) describes an inverted version of this...

   as a preferred implementation for a horizontal solar energy collector, 
   one or more sloping, highly reflective, flexible sheets produce a
   concentrating reflector utilized under a transparent cover, such as
   a glass roof, above the collector.

   these sheets produce one or more sections of a curve called a catenary,
   which highly resembles a curve of a parabola [figs 5a and 5b show
   a catenary with y = acosh(x/a) = a/2(e^x/a+e^-x/a)] and when properly
   positioned under a horizontal energy converter, will produce the needed
   concentration... the reflecting sheets are manufactured from readily
   available, and cost effective, flexible polymer with a metal layer,
   such as flexible acrylic with highly reflective metallic layer, known
   as "mylar..."

   in the tracking version, each sheet hangs from a supporting device
   with one end of the sheet attached to a rotating roll allowing the
   sheet to be tightened and loosened, with the other end of the sheet
   attached directly to a supporting device... both turning the roll
   and moving the supporting devices are accomplished by stepper or
   other digitally controllable motors, controlled by a micro processor...
   in the stationary version, each sheet simply hangs from a stationary
   supporting device, at both ends.

   to direct radiation, in the direction parallel to the axis of the
   sloping sheet concentrating reflector, two stationary directing
   reflectors [endwalls] are mounted along each side of the concentrating
   reflector, perpendiculatr to it. to maximize the effective reflector
   area, these directing reflectors extend all the way from the
   concentrating reflector, at its lowest position, past the line of
   concentrating, to the transparent protective cover above the collector...
   additional directing reflectors cover the sides of the rafters (or
   other structural members) supporting the transparent sheeting above
   the collector... as with the sloping sheet(s), similar inexpensive
   reflective sheet is hung from one end with a weight attached to
   the other end, thus forming a section of straight vertical reflector.
   alternatively, the reflective sheet material is bonded with a rigid
   sheet of suitable building material (such as gypsum board), and cut
   to the required size and shape.

steve was also concerned about fire...

   the most serious mistake was making the outer container of the receiver
   of plywood. we thought that the plywood would be sufficiently insulated
   from the copper panel which was the receiver proper, that it would not
   get too hot. the copper panel was separated from the plywood by 4" of
   fiberglass insulation. nevertheless, the plywood caught fire and the unit
   was completely destroyed. we suppose this is a success, of sorts...

   from "a solar collector with no convection losses," (a downward-facing 
         receiver over a 4:1 concentrating parabolic mirror) written by
	 h. hinterberger and j. o'meara of fermilab, ca 1976

nick