Make Square Opening by Drilling Five Round Holes

In making a speed control for an off-the-shelf electric fan, I needed to install a square power receptacle in the phenolic box I am using for the speed dial switches and 7-segment displays.  Phenolic is brittle and does not machine well with the woodworking tools that I have, chipping instead of cutting.

A straightforward way to cut the hole would be to drill a hole, disassemble a coping saw and reassemble it with the blade through the hole, and sawing.  But the small size of the box would limit the saw strokes to a few centimetres.

A reciprocating "Sabre" saw might do the job but is hard to control.

Twist drill bits seemed to work the best of my tools on the material, and I have a good selection of sizes.  How close to a square opening can one create by drilling a small number of round holes?  It turns out that I can come fairly close.  There are two ways the problem can be posed, the largest opening bounded within the square or the smallest opening just larger than the square.  I am interested in the latter; the other solution can be had by scaling.

The idea is to drill holes on the diagonals near the 4 corners such that the corner touches the rim of the hole.  One then drills a hole in the center which is enough larger than the square so that its points of contact with the square are points of contact with the 4 smaller holes.  If the diameter of the 4 corner holes is reduced, then the diameter of the center hole must be increased in order for it to intersect the smaller holes and the square.

Let L be the length of one side of the square, R be the radius of the center hole, and r be the radius of the corner holes.  The center of each corner hole is L/sqrt(2)-r from the center of the square so that the rim lies on the corner point.  The furthest that the corner holes exceed the desired square is

  r+(L/sqrt(2)-r)/sqrt(2)-L/2 = r-r/sqrt(2) = r*(1-sqrt(1/2))

The furthest that the large center hole exceeds the desired square is R-L/2.  Desired is:

  R-L/2 = r*(1-sqrt(1/2))

The other constraint is that the rim of the center hole and the rim of the corner hole intersect on the side of the square.  The distance from the middle of the side to the intersection of the side with the large hole is x

  R^2 = (L/2)^2 + x^2

The distance from the corner to the intersection is r*sqrt(2).  So:

  L/2 = r*sqrt(2) + x

R, r, and x scale with L.  Let L = 1.

  R^2 = 1/4 + x^2

  1/2 = r*sqrt(2) + x

  R-1/2 = r*(1-sqrt(1/2))

Solving this system:

  r = 1/(4+sqrt(2))

  r = L * 184.69903125906464e-3

  R = L * 554.097093777194e-3

The diameters when L = 1.125 are:

  d = 415.57282033289544e-3

  D = 1.2467184609986863

I made the holes with a 3/8 inch twist drill and a 1.25 inch hole saw.  A 7/16 bit  would have been closer in size.

The flange on the outlet covers the non-square parts of opening.

Wiring the Plate

For those new to these postings, in 2011 through 2013 I developed a complete theory for single-phase mixed convection from a rough (or smooth) rectangular plate at uniform temperature:
  http://people.csail.mit.edu/jaffer/SimRoof/Convection

In 2014 I started assembling the equipment necessary to test this
theory:
  http://people.csail.mit.edu/jaffer/SimRoof/Convection/Measurements

Previous blog postings document the progress in building the wind tunnel and measurement and control electronics:
  http://voluntocracy.blogspot.com/2014

John Cox milled the front of the plate to have a precision rough surface with 3 mm average height of roughness.  This particular surface also has 3 mm root-mean-squared height of roughness.



John Cox milled slots in the back of the plate for the heaters (TO-220 resistors).  I drilled 11.mm deep pilot holes and tried to hand tap the holes for #6-32 screws.  On the first hole, the tap started to lean.  Carl Mikkelsen suggested that I remove the drive belt from my drill press, clamp the tap in the chuck, and tap (turning by hand) with the drill press enforcing vertical alignment.  Having only two hands, I used an elastic cord to counteract the spring in the press, as can be seen in the photograph.

 Tapping an 11.mm blind (not through) hole is not reliably acheived by machine, but is quite doable by hand.  I tapped each hole in 3 stages, backing out the tap and blowing out the hole with a (sports) ball inflation pump after each stage.  The resistance to turning the tap when the hole is clogged with chips feels different than the hard stop when the tap has reached the bottom of the hole.  Per Carl's suggestion I used vegetable oil when tapping, which eased cleanup and degreasing.



I wired the heaters (resistors) in series with solid AWG 22 wire (red).  The external drive points (top terminal of the top left and top right heaters) are also connected to conductors of the 6-wire ribbon cable (gray).  This allows measurement of the voltage across the heater chain to be unaffected by voltage drop in the drive wires.  Three gray wires connect to an electronic temperature sensor in the hole right of center.  The gray wire connected to the center screw grounds the plate in order to shield the temperature sensor from electrical noise.

In my initial calculations I neglected the tap taper length, so I shortened each screw by 1.6 mm.  I painted "Silver Alumina" thermal grease on the back of each heater and screwed it and a compression washer into its slot.  I used the TO-220 recommended torque limit on the screws.  But the screws were tight in the holes even without the compression washers, so I turned until the grease squeezed out.

To test the thermal coupling of the heaters to the plate, I jumpered the electronics board to run 1.83 Amps (33 Watts) through the string of heaters.  Applying the "finger test", the warmth of the plastic on the heaters was barely discernable, yet the plate temperature sensor registered a 1.K rise after several minutes.

Once the electronics board is programmed to record measurements of the sensors and heater power, I will encase the plate in polyisocyanurate foam board (insulation) and measure the power required to maintain a 5 K or 10 K temperature between the plate and ambient air.  Half of the ratio of this power to temperature difference is the (leakage) thermal conductivity of the back side of the plate.  The back of the plate will remain insulated while the front will be bare for convection measurements.