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
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:

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 L

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