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.