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.

Forced Convection Success!

When I replaced the plate insulation, I covered the back surface (but not the sides) with aluminum foil.  An isothermal surface, I put a temperature sensor at the center of the back foil and added modeling for this back surface to the simulation, which is described in Measurements of Convection From a Rectangular Plate

After calibrating the fan speed in the wind-tunnel, the next task was to model the forced convective component of the heat transfer from the parts of the heated plate other than the surface under test: the back and sides.  The isothermal back can be modeled using the standard formula in series with the block of insulation.  Unlike natural convection, forced convection can be modeled for non-isothermal surfaces by integrating the local convection in the direction of the fluid flow.  There are three calculations for the four sides: the sides whose long dimensions are parallel to the flow, the side facing into the flow, and the side facing away from the flow.

For the sides parallel to the flow, the local convective surface conductance in series with the local insulation conductance is integrated in the direction of the flow.

For the side facing the flow, the fluid velocity along the long center-line line is zero, increasing to V at the long edges.  So I integrate from the center-line to the long edges.

For the side facing away from the flow, the average fluid velocity along the long center-line is also zero.  I integrate from the long edges to the center-line, but only the turbulent component (the chart below shows the difference from windward is insignificant).

ins_back    long_side   windward    leeward    total      rough
65.6mW/K + 2*77.2mW/K + 77.2mW/K + 77.2mW/K = 0.374W/K vs 0.572W/K @ 0.0m/s
73.9mW/K + 2*98.3mW/K + 0.108W/K + 0.101W/K = 0.479W/K vs 1.20W/K @ 1.0m/s
74.9mW/K + 2*0.111W/K + 0.120W/K + 0.117W/K = 0.535W/K vs 2.17W/K @ 2.0m/s
75.5mW/K + 2*0.123W/K + 0.132W/K + 0.131W/K = 0.584W/K vs 3.25W/K @ 3.0m/s
76.0mW/K + 2*0.133W/K + 0.145W/K + 0.144W/K = 0.631W/K vs 4.33W/K @ 4.0m/s

4 m/s simulations with the new model were predicting way too much convection when compared with measurements.  After checking all the calculations, it was clear that something basic was wrong.  The formula which the apparatus was built to test is the Scaled Colburn Analogy.  Originally (it has been updated) it was scaling only the characteristic length in the Colburn Analogy. Scaling both the characteristic length and mean-height-of-roughness yields this:

The green trace is the measured plate temperature; black is the ambient temperature; and blue is the temperature of the back foil. The thin red traces are the simulated plate and back temperatures. Because 4 m/s flow has high convective surface conductance, the back and ambient temperatures are nearly the same.  The ambient bumps occured at times when a dehumidifier in the room turned on.

To have such close match before tweaking is exciting!  A run at 3 m/s also shows excellent match:



Does the model work for natural convection?



This match was unexpected because I didn't have a natural convection model for the four sides.  But this match leads to an explanation: the air heated by the downward-facing plate rises past the four sides, so there is little temperature difference through the sides to drive additional convection.

This natural convection match is perhaps too good.  The measurements were taken in the wind tunnel whose top panel will somewhat impede the ascent of the heated air.  The missing difference could be because the simulation has the thermal radiation from the sides cooling the plate, but most of that radiation would be due to heat rising from the (bottom) test surface.

Sandstone Formation

Roberta and I recently returned from a fantastic Grand Canyon Expeditions 8-Day Motorized Trip down the Colorado River through the Grand Canyon.  The Geology is fascinating and quite unlike the glacier-scoured granite mountains in New Hampshire with which I am familiar.

  

The "Big Dune" sand bar at mile 119 where we camped is backed by Tapeats Sandstone carved into flowing forms.

Sandstone is obviously deposited in layers.  Is this process observable at human time scales; or would one need to spend years at the bottom of a shallow sea?

The sand banks on which we camped were deposited by varying water flows in the Colorado river.  The Glen Canyon Dam having reduced the Colorado's flow and variations, in 2013 the flow was increased fourfold for 6 days in an effort to replenish the sandbanks which have been diminishing.

 

These photos from the USGS show the Big Dune sandbar before and after the flood.  Sand was deposited.  While walking along a small vertical face in the sand bar, I noticed horizontal striations.  So the sandbar is deposited in layers.

This photo and the detail below show the striations and their resemblance to the sandstone layers.

 

This would have merely been an amusing coincidence had not our next camp site, "OC's" at mile 137, had an even finer example of sand layering.  Not only does the sandstone face show both highly parallel layers and more fluid flows, the sand displays them as well.

 Names and mileages from:

"Belknap's Waterproof Grand Canyon River Guide"; New Edition 2014;
Buzz Belnap - Loie Belknap Evans;
Westwater Books, Evergreen Colorado US;
Library of Congress Control Number: 2006937059;
ISBN 978-0-916370-16-9;
ISBN 10:0-916370-16-X