Forced Convection on Convection Machine 2.0

The new back eliminated the dual modes seen earlier with peak ΔT=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.

Slightly Tilted Downward Convection

With the current configuration, I am measuring the natural convection from a plate inclined a few degrees from facing downward. Turning the front tuning pegs shortens the support wires and tilts the plate.

This graph shows that downward convection does not blend with other convective modes. With θ between 87° and 90° downward convection is in control; between 0° and 87° the vertical mode is dominant. I have modified the UCT3 formula in the Downward Natural Convection section with this new knowledge.

R = min(LH, LW)/2
h = k⋅max(Nu9.3(Ra(LH) cos θ)/LH , Nu45(Ra(R) sin θ)/R)	0°≤θ≤+90°	UCT3

The random variation in the plot above makes it less convincing than it might be. I modified the firmware to increase the heating threshold from 5 K to 15 K, which increases the average ΔT from 3.7 K to 11 K and reduces the relative variation by about a factor of 3.

Running the tilt tests at the higher temperature difference results in markedly reduced variation as can be seen by the proximity of the half-interval measurements (black dots) to the full-interval measurements (red dots). But the slope below 87.5° clearly does not match that derived from Nu_9.3, the natrual convection from a smooth vertical plate!  More about this below.
I also ran a series of forced convection trials with the 15 K peak temperature difference:

There are several things to notice about this graph. The tests where only part of the plate had roughness penetrating the boundary layer seemed to have two distinct breakpoints: Re=16666 and Re=22222. For nearly all the measurements, the half-interval measurements (black dots) are very close to the full-interval measurements (red dots). There are two which are not so close. The black dots around Re=7440 lie close to the line, indicating the wind speed change which occured partway through the run. This can be seen in the bottom panel on page 8 of the PDF. The dots at Re=8237 had the same windspeed, perhaps indicating that the system spent time in both breakpoint modes.
Below Re=7500 the measurements track a slope of turbulent convection (Re^4/5). Unfortunately the wind-tunnel can't produce wind speeds below 0.25 m/s (Re=4900), so we can't see the transition to natural downward convection.

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Returning to natural convection, either the natural convection from a rough vertical surface differs from that of a smooth vertical surface or the formula combining the downward and vertical convection is wrong. The former can be tested by suspending the plate vertically. I plan to do this by wrapping the wire suspending the plate around its (insulated) back. In order to protect the insulation from being cut by the wire, a sheet of aluminum should be glued to the back. I was planning to do this already in order to test upward natural convection.

Here is the back side of the plate ready to have the 0.65 mm sheet of aluminum glued to it. Notches have been cut (using the "nibbler" on the sheet) near the sheet corners to catch the wires which will suspend it.

Gluing completed.
 

Here the rough surface of the plate is facing up in the wind tunnel. But first I must rerun all the downward facing measurements.

Upward Natural Convection

There was a mistake in my simulation of the (insulated) back surface. It doesn't affect the measurement or simulation of the test surface. Here is a proper simulation of the plate with the rough side facing down:

The red lines are the simulations; the others are measured. The backside simulation expects more convection than was measured, resulting in the measured back face temperature being higher than its simulated temperature.

Because they are not thermally conductive over their whole surfaces, natural convection from the four vertical sides can't be modelled with established theory. My simulation models these sides as having 3.2 times the natural convection from the 52 mm × 305 mm metal surfaces not covered by insulation. The 3.2 factor was arrived at from natural convective runs on the Convection Machine.

The streamlines in Fujii and Imura[76]'s figure 14(f) show air from the edges of the plate moving toward the rising column at the plate's centre. But in the Convection Machine this air has already been heated by the rough (downward) face and four sides. Thus the convection from the (upward) back side would be reduced and its temperature higher than if the other faces were not convecting.

With the rough test surface facing downward its convection is not affected by the other faces. With the plate 5 K hotter than ambient, convection from the bottom face is about 0.6 W and about .348 W for each vertical face. Their combined 2 W dwarfs the .297 W expected through the insulation for the back face (upward). So it is not surprising that the back face convection is reduced.

If the rough test surface faces up, then its expected 20.3 W convection will experience reduction from the 1.63 W of convection from the other faces.  Unfortunately, in this case it does affect the measurements.

Scaled Colburn Analogy Asymptote

I have rewritten the Scaled Colburn Analogy Asymptote section to address the Convection Machine measurements at low Reynolds numbers:

The logarithmic scale amplifies the variances below Re=10000.  The graph below shows Nu/Pr^1/3 on a linear scale where variations can be seen as comparable in magnitude to the variation between experiments where Re is larger than 10000.

 

The revised formula, which includes the Re range over which the surface should be treated as rough, is:

A Convection Surprise

Years ago I heard advice to experimental physicists to continue refining their experiments, even after they yield a hoped-for result.

Last week I realized that if I switched the wind-tunnel fan to the lower speed setting, it might be able to run at speeds lower than 100r/min.  I tried it and it worked!  This updated plot has points at Re values less than 10000 and they show significantly less convection than expected.  As in an earlier post, there would be many possible explanations for finding too much convection, but not for finding too little.

At these low wind speeds the fan speed wanders through a +/-10% range and the convection measurements are averaged over time.  I have rewritten my program to compute the convection over smaller intervals than the whole trial and included both the whole trial and smaller runs in the graph.  My program also averages measurements through a trapezoidal window, which reduces the variation between the smaller intervals within an experimental trial.

Here is a link to a pdf of plots splitting the measurements in 2 through 8 pieces.  Each trial's dots are distributed vertically, reflecting the noisy temperature readings.

The four lowest trials cluster near the asymptote for turbulent convection from a smooth surface, even though the rough surface asymptote would convect more heat.  In the "Measured over time split into 4 intervals" chart it is seen that the plate spends most of its time over the range between the two asymptotes.

If the switching between asymptotes were correlated with fan speed variations, we would expect to see the dots for each trial on a slanted line; but they are roughly vertical.  The transition between laminar and turbulent forced convective modes remains an unsolved problem for theory.  Modelling the transition between these two turbulent forced convective modes may be as intractable.

Simplified Convection Model

With the asymptote established, it was time to work on the lower speeds.  My models were stubbornly predicting too much convection for 0109rpm-20151031T023629.  It turns out that the apparent deficit for rough upward natural convection was nearly equal to the apparent deficit for rough downward convection.  Thus it was likely that I was underestimating the radiative cooling and natural convection from the four sides, which act in both orientations.  Increasing the modeled side surface conductances yields values for the rough surface matching smooth natural convection within the experimental variations.

I managed to coax the wind-tunnel fan to run at 93 r/min for a data run, extending test coverage down to Re=8000.  Where my earlier plot showed points clustering around the L4-norm of the Scaled-Colburn-Analogy-Asymptotes, the new point aligns with the others along the asymptotes themselves, as predicted by my SCAA formula:

NuSCAA = max(Nu8.9, Nu8.11, NuRS)

The small convection from 0093rpm-20151125T170700 and 0109rpm-20151031T023629 also confirm that mixed convection uses a high degree norm (the highest being the "max" function).  This essentially says that natural and forced convection do not mix; it is either one or the other.  When the Convection Machine gets fitted with a smaller fan, I will be able to test at the transition point.

Forced Convection From a Rough Plate

In October I collected data from experiments over the full range of wind speeds possible in my wind tunnel: 0.45m/s to 4.5m/s.  More time was consumed in making a simulation which matched the dozen datasets.  In this speed range all convection is a blend of natural and forced convection.  I built the simulation to support variable L-norms for combining the natural and forced convection components.  The L-norm that allowed match to the data was L4, which was also the only working L-norm for combining the three speed ranges of forced convection.  The L4-norm looks like: (N⁴+F⁴)^1/4.  The main goal was to test my correlation for forced convection from a rough plate:

This graph shows the experimental support for my "Scaled Colburn Analogy Asymptote" formula for forced convection from a rough plate.  Here is a link to a pdf of graphs comparing simulation with experiment.

The largest uncertainty in the measurements is the wind-speed (proportional to  Re), which the anemometer gives as 3% at 4m/s.  As the speed reduces, the uncertainty grows quickly.  The "fan law" says that wind-speed is proportional to fan rotation rate.  This turned out to not hold at speeds above 3m/s, which had a lesser slope than at low speeds.  At high speeds, the anemometer readings inside the wind-tunnel are erratic.  This link to the fan wind-speed curve shows the formula used in simulations versus 5 measured wind-speed data runs.

Some of my assumptions were wrong.  Natural convection from a rough surface turns out to be greater than convection from a smooth surface both upward and downward facing.  In order to obtain the matches (red versus blue, green) shown below, the downward convection is multiplied by 2.44; the upward convection by 1.56.  The rough plate surface has an area 2.44 times that of a smooth surface and 1.56 is its square-root.  There is latitude in these numbers and convection from the four sides; experiments with the rough surface covered by a sheet of aluminum would refine the model.

 The blue trace is the measured temperature from the smooth back.  The simulated red trace is a good match when the back is on top.   The poor match below when the smooth back is facing down is probably because the plate was not suspended, but sitting on small wooden blocks without much clearance.  I will perform the measurement again with more clearance in the future.