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.

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 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 L⁴-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 L⁴-norm mixing for the short sides is instead L², the increase in natural convection from the sides reduces the top surface convection, spoiling the upward-convection match with L²-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 ΔT=11K:

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

I had expected L⁴-norm mixing for upward convection. So these results will require modifications to my theory of mixed convection.

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.

---

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.