I turned the wind-tunnel on its side and hung the plate vertically as shown in the photograph.

These graphs show that my mixed convection model is successful from natural through forced convection for horizontal and vertical rough plates with forced flow perpendicular to the natural flow. The leftmost red dot in each graph is the natural convection (Re=0) for that orientation; it is placed at Re=1000 so that it can appear on the graph.

The L4-norm for downward convection indicates that the interaction between the downward mode and forced convection is more competitive than the fairly cooperative L2-norm of the vertical and upward modes.

Horizontal downward and vertical natural convection from the rough plate match that expected from a smooth plate. Horizontal upward convection matches assuming that the upper (rough) surface convection is reduced by 93% of the non-forced convection from the four adjoining sides. In order to test if horizontal upward convection is the same for rough and smooth, I will cover the rough surface with a flat sheet of aluminum and repeat the test.

I have started writing a paper titled "Mixed Convection from a Rough Plate". Which journal should I submit it to?

# Voluntocracy

Governance by those who do the work.

## Monday, June 13, 2016

## Saturday, June 4, 2016

### Fan Windspeed

With the wind-tunnel fan being phase-locked now, the speed variability which plagued earlier speed measurements should be reduced or eliminated.

The process of conducting the measurements for these graphs finds this to be the case. Although some variability remains above 3 m/s (1000 r/min), at slower speeds the anemometer readings are steady after the phase-lock-loop settles. The measured traces are in blue; the black curve is that used in convection calculations. This first graph is for the wind tunnel with horizontal plate.

This second graph is with the wind tunnel laying on its side with the plate vertical.

An earlier post found that the kink just above 2 m/s is due to the anemometer; it remains. Friction in the anemometer makes the measurements below fan speeds of 700 r/min unreliable; below fan speeds of 200 r/min the anemometer reads 0.0.

The process of conducting the measurements for these graphs finds this to be the case. Although some variability remains above 3 m/s (1000 r/min), at slower speeds the anemometer readings are steady after the phase-lock-loop settles. The measured traces are in blue; the black curve is that used in convection calculations. This first graph is for the wind tunnel with horizontal plate.

This second graph is with the wind tunnel laying on its side with the plate vertical.

An earlier post found that the kink just above 2 m/s is due to the anemometer; it remains. Friction in the anemometer makes the measurements below fan speeds of 700 r/min unreliable; below fan speeds of 200 r/min the anemometer reads 0.0.

## Saturday, May 7, 2016

### How to Phase-Lock a Fan

Using an auto-transformer to reduce the voltage to the wind-tunnel fan

in order to reduce its speed didn't work below 45 r/min (it ran for a

while and stopped). So I modified The Convection Machine to toggle

the fan power with a solid-state relay controlled by micro-processor.

Consider the shaft of the wind-tunnel fan. Every full rotation of the

fan results in 3 micro-processor interrupts. A phase-accumulator

register is incremented by the desired rotation rate (in r/min) 1200

times a second and decreased by 24000 every time a fan blade crosses a

light beam. If the fan rotates at the desired rotation rate, then the

average phase-accumulator value is constant. If the fan is too slow,

then the phase-accumulator value increases with time; if it is too

fast, then the phase-accumulator value decreases with time.

Phase-locking is the process of controlling the fan-speed so that the

average phase-accumulator value is 0. Such feedback systems are

tricky to stabilize. My fan controller operates in one mode when the

desired speed is less than 400 r/min and a different one otherwise.

At high speeds, the fan speed is roughly proportional to the average

voltage applied, which is proportional to the duty cycle of applied

voltage. The phase accumulator operates as described above. Its

instantaneous value is compared with a variable which decrements from

the upper phase range bound to 0 ten times a second. If greater, the

fan is turned on, otherwise it is turned off. Some of the

instabilities of the fan speed may be due to a centripetal switch

disconnecting the starter capacitor and hooking in the running

capacitor, which increases the loop gain of the system. The change in

gain causes the system to overshoot and undershoot the desired r/min

with long settling times.

At low speeds each pulse of power incrementally increases the fan

speed while friction continually slows it. The solid-state relay has

"zero-crossing" control, so only complete half-cycles of 60 Hz power

are applied to the fan motor. The combination of the motor windings

and phase capacitor stores energy, so the acceleration of the rotor is

delayed from the application of power. At low speeds the rotational

inertia of the rotor introduces 90 degrees of phase shift. The

microprocessor clock is not synchronized to the line voltage, so the

minimum pulse width varies with the relative phase, another source of

loop gain variation.

This photo shows the new fan-speed control. The number on the

7-segment displays is the rotation rate in r/min measured every

second. The right 3 dial switches set the desired rotation rate.

This video which shows the phased-locked fan in

operation at a variety of speeds. The low light level was necessary

so that the stroboscopic interaction of camera shutter with the

scanned 7-segment display didn't render the numbers unreadable. If

you turn up the audio volume you can hear the fan chugging as its

power is switched on and off.

in order to reduce its speed didn't work below 45 r/min (it ran for a

while and stopped). So I modified The Convection Machine to toggle

the fan power with a solid-state relay controlled by micro-processor.

Consider the shaft of the wind-tunnel fan. Every full rotation of the

fan results in 3 micro-processor interrupts. A phase-accumulator

register is incremented by the desired rotation rate (in r/min) 1200

times a second and decreased by 24000 every time a fan blade crosses a

light beam. If the fan rotates at the desired rotation rate, then the

average phase-accumulator value is constant. If the fan is too slow,

then the phase-accumulator value increases with time; if it is too

fast, then the phase-accumulator value decreases with time.

Phase-locking is the process of controlling the fan-speed so that the

average phase-accumulator value is 0. Such feedback systems are

tricky to stabilize. My fan controller operates in one mode when the

desired speed is less than 400 r/min and a different one otherwise.

At high speeds, the fan speed is roughly proportional to the average

voltage applied, which is proportional to the duty cycle of applied

voltage. The phase accumulator operates as described above. Its

instantaneous value is compared with a variable which decrements from

the upper phase range bound to 0 ten times a second. If greater, the

fan is turned on, otherwise it is turned off. Some of the

instabilities of the fan speed may be due to a centripetal switch

disconnecting the starter capacitor and hooking in the running

capacitor, which increases the loop gain of the system. The change in

gain causes the system to overshoot and undershoot the desired r/min

with long settling times.

At low speeds each pulse of power incrementally increases the fan

speed while friction continually slows it. The solid-state relay has

"zero-crossing" control, so only complete half-cycles of 60 Hz power

are applied to the fan motor. The combination of the motor windings

and phase capacitor stores energy, so the acceleration of the rotor is

delayed from the application of power. At low speeds the rotational

inertia of the rotor introduces 90 degrees of phase shift. The

microprocessor clock is not synchronized to the line voltage, so the

minimum pulse width varies with the relative phase, another source of

loop gain variation.

This photo shows the new fan-speed control. The number on the

7-segment displays is the rotation rate in r/min measured every

second. The right 3 dial switches set the desired rotation rate.

This video which shows the phased-locked fan in

operation at a variety of speeds. The low light level was necessary

so that the stroboscopic interaction of camera shutter with the

scanned 7-segment display didn't render the numbers unreadable. If

you turn up the audio volume you can hear the fan chugging as its

power is switched on and off.

Labels:
Phase-Lock,
Phased-Lock-Loop,
software PLL,
Wind-tunnel

## Saturday, March 19, 2016

### 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.

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.

## Thursday, February 18, 2016

### Vertical Natural Convection

Unexpected results for downward convection at small-angles raised
the question of whether vertical natural convection is the same for
rough and smooth plates. This photo shows the plate suspended
vertically by steel wire from the two boards above. The ambient
temperature sensor is taped to the table leg. I measured the
natural convection over three temperature ranges as was done in the
other natural convection runs.

If there is less convection than expected, then it could be due to heat from one side reducing the convection of a side above it, as happens in the upward facing case.

But slightly more convection than expected was measured. As the plate is no longer in the wind tunnel, modeling the emissivity of the room as 0.9 (versus 0.8 for the wind tunnel) brings the simulation into reasonable agreement with measurement. It thus appears that, at least for laminar flows from rectangular plates, natural convection from a rough surface has the same magnitude as convection from a smooth surface.

The graph below is linked to a pdf of the measurements and simulations of natural convection in level and vertical orientations.

If there is less convection than expected, then it could be due to heat from one side reducing the convection of a side above it, as happens in the upward facing case.

But slightly more convection than expected was measured. As the plate is no longer in the wind tunnel, modeling the emissivity of the room as 0.9 (versus 0.8 for the wind tunnel) brings the simulation into reasonable agreement with measurement. It thus appears that, at least for laminar flows from rectangular plates, natural convection from a rough surface has the same magnitude as convection from a smooth surface.

The graph below is linked to a pdf of the measurements and simulations of natural convection in level and vertical orientations.

Labels:
Convection,
experiment,
Natural Convection

## Saturday, February 6, 2016

### Mixed Convection

As my blog post
Upward Natural Convection
details, there is no guarantee that natural convection of my plate
with the rough surface facing up behaves in a manner which can be
modeled. The air warmed by the insulated back and sides rises
adjacent to the heated rough surface which draws air towards its
center. So how strong is this effect? Measured with peak
temperature differences of 15 K, 10 K, and 5 K, the
measured upward natural convection is about half that predicted.
But looking at the natural convection components, the deficit is
roughly equal to sum of the back and side convective heat flows!

With the rough surface facing down, the mix of convective and radiative heat loss from the four sides matters little because both are subtracted from the overall heat loss. But when the rough surface faces upward it does matter; side convection reduces the rough surface convection while thermal radiation does not. Creating a plot of downward natural convection at a range of temperature differences allows evaluation of simulated mixtures. As the fraction of simulated radiative heat loss increases, the slope connecting the measured points increases. The graph below shows the fit when the effective radiative height of the side is 41% of its actual height, and the effective convective surface area is adjusted to fit:

With this rough estimate of the relative strengths of convective and radiative heat loss, we are now ready to see whether the effect of the sides on the top surface can be reasonably modeled.

The plot below compares upward convection correlations with total non-radiative heat flow minus 77% of the (modeled) sides and back natural convection. There is less variation from point to point because upward natural convection is three times stronger than downward natural convection. "Horizontal Hot Top" is my generalization of the four conventional upward convection correlations.

The unexpectedly close match above lends support to the idea that air heated by the sides is drawn over the upper surface of the plate, reducing the effective temperature difference between the plate and air, and can be modeled as a reduction of upward convection by an amount proportional to the side convection.

Now that I have convection measurements at low fan speeds which are comparable in magnitude to natural convection, the next step is to evaluate mixed convection (with the rough surface facing upward).

Just as there was no guarantee of a workable model for the interaction of the sides with the top in still air, there is no guarantee of a model of that interaction in forced air. Needed is a generalization matches the model developed for

The graph below shows the correlation I have arrived at for upward convection (see simulations). It assumes that mixed convection for the four short sides is the L

If the L

Here are the calculated values of convection from all six sides of the plate at windspeeds from 0 to 4 m/s and Δ

With the rough surface facing down, the mix of convective and radiative heat loss from the four sides matters little because both are subtracted from the overall heat loss. But when the rough surface faces upward it does matter; side convection reduces the rough surface convection while thermal radiation does not. Creating a plot of downward natural convection at a range of temperature differences allows evaluation of simulated mixtures. As the fraction of simulated radiative heat loss increases, the slope connecting the measured points increases. The graph below shows the fit when the effective radiative height of the side is 41% of its actual height, and the effective convective surface area is adjusted to fit:

With this rough estimate of the relative strengths of convective and radiative heat loss, we are now ready to see whether the effect of the sides on the top surface can be reasonably modeled.

The plot below compares upward convection correlations with total non-radiative heat flow minus 77% of the (modeled) sides and back natural convection. There is less variation from point to point because upward natural convection is three times stronger than downward natural convection. "Horizontal Hot Top" is my generalization of the four conventional upward convection correlations.

The unexpectedly close match above lends support to the idea that air heated by the sides is drawn over the upper surface of the plate, reducing the effective temperature difference between the plate and air, and can be modeled as a reduction of upward convection by an amount proportional to the side convection.

Now that I have convection measurements at low fan speeds which are comparable in magnitude to natural convection, the next step is to evaluate mixed convection (with the rough surface facing upward).

Just as there was no guarantee of a workable model for the interaction of the sides with the top in still air, there is no guarantee of a model of that interaction in forced air. Needed is a generalization matches the model developed for

*V*=0 and matches the forced correlation as*V*grows.The graph below shows the correlation I have arrived at for upward convection (see simulations). It assumes that mixed convection for the four short sides is the L

^{4}-norm of the natural and forced convections and that 77% of only the natural component of the back and sides is absorbed by the convection of the (upward-facing) rough surface.If the L

^{4}-norm mixing for the short sides is instead L^{2}, the increase in natural convection from the sides reduces the top surface convection, spoiling the upward-convection match with L^{2}-norm (gray dashed line) and other L-norm exponents (best is about L^{2.3}).Here are the calculated values of convection from all six sides of the plate at windspeeds from 0 to 4 m/s and Δ

*T*=11K:I had expected L

insulated

back+ 2 ⋅ parallel

side+ 2 ⋅ windward

leeward= total vs rough @ windspeed 53.1mW/K 55.3mW/K 55.3mW/K 0.272W/K 0.467W/K 0.0m/s 55.2mW/K 55.4mW/K 55.4mW/K 0.274W/K 0.484W/K 0.12m/s 60.4mW/K 55.5mW/K 55.4mW/K 0.280W/K 0.520W/K 0.25m/s 65.0mW/K 55.9mW/K 55.9mW/K 0.286W/K 0.666W/K 0.50m/s 68.5mW/K 57.3mW/K 57.6mW/K 0.296W/K 1.18W/K 1.0m/s 71.0mW/K 61.5mW/K 63.4mW/K 0.318W/K 2.22W/K 2.0m/s 72.6mW/K 72.4mW/K 78.4mW/K 0.371W/K 4.36W/K 4.0m/s

^{4}-norm mixing for upward convection. So these results will require modifications to my theory of mixed convection.
Labels:
Convection,
Mixed Convection

## Wednesday, January 27, 2016

### Forced Convection on Convection Machine 2.0

The new back eliminated the dual modes seen earlier with peak
Δ

The points at Re=3545 and Re=5231 could be following either the Smooth Turbulent Asymptote or the Transition Model; it's too close to tell. The reason that the Transition Model has less convection than the Smooth Turbulent Asymptote below Re=6000 is because the leading edge of the boundary layer is modeled as thinner than the roughness; but division by the local characteristic length, 0, is undefined at the leading edge.

*T*=15K. Reducing the r/min to m/s conversion by 2% (which is within the ± 3% error band of the anemometer) and adjusting the breakpoint results in a beautiful fit!The points at Re=3545 and Re=5231 could be following either the Smooth Turbulent Asymptote or the Transition Model; it's too close to tell. The reason that the Transition Model has less convection than the Smooth Turbulent Asymptote below Re=6000 is because the leading edge of the boundary layer is modeled as thinner than the roughness; but division by the local characteristic length, 0, is undefined at the leading edge.

Labels:
Convection,
correlation,
Forced-convection

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