  Using boundary-layer theory, natural convection heat transfer formulas which
are accurate over a wide range of Rayleigh numbers ($Ra$) were developed in the
1970s and 1980s for vertical and downward-facing plates. A comprehensive
formula for upward-facing plates remained unsolved because they do not form
conventional boundary-layers.
  From the thermodynamic constraints on heat-engine efficiency, the novel
approach presented here derives formulas for natural convection heat transfer
from isothermal plates. The union of four peer-reviewed data-sets spanning
$1<Ra<10^{12}$ has 5.4% root-mean-squared relative error (RMSRE) from the new
upward-facing heat transfer formula.
  Applied to downward-facing plates, this novel approach outperforms the
Schulenberg (1985) formula's 4.6% RMSRE with 3.8% on four peer-reviewed
data-sets spanning $10^6<Ra<10^{12}$.
  The introduction of the harmonic mean as the characteristic-length metric for
vertical and downward-facing plates extends those rectangular plate formulas to
other convex shapes, achieving 3.8% RMSRE on vertical disk convection from
Hassani and Hollands (1987) and 3.2% from Kobus and Wedekind (1995).
