Does Smeaton's coefficient, k, have a modern value or it is dependent of the air density?
Why is the accepted value of k so high?
In various texts about the Wright brothers (see 1 and 2) one can read about Smeaton's coefficient that troubled them a lot and that they finally discovered the parameter had a much lower value reaching the conclusion $k = 0.0033 lbf/ft^2/mph^2 = 0.79 kg/m^3$ (instead of $k = 0.005$), a fact also noticed by others before them.
"the Wright brothers calculated a new average value of 0.0033. Modern aerodynamicists have confirmed this figure to be accurate within a few percent." Source: Correcting Smeaton's Coefficient
$L = k \cdot S \cdot V^2 \cdot C_L$
$L$ = lift in pounds
$k$ = coefficient of air pressure (Smeaton coefficient)
$S$ = total area of lifting surface in square feet
$V$ = velocity (headwind plus ground speed) in miles per hour
$C_L$ = coefficient of lift (varies with wing shape)"
However, knowing that the modern formula for lift is $$L = 0.5 \cdot \rho \cdot S \cdot V^2 \cdot C_L$$ Where $\rho$ = the air density.
It appears that $k = 0.5 \cdot \rho$ and so it does not have a standard average value. Also a $k = 0.0033 lbf/ft^2/mph^2 = 0.79 kg/m^3$ leads to a $\rho = k/0.5 = 1.58 kg/m^3$ that corresponds to a sea level air temperature well below -25 C, which is unusual.
If the two relations for lift are correct, the Smeaton's coefficient can not be 0.0033 but closer to 0.0025 a value corresponding to a standard air density at $20 ^\circ C$ close to $1.2 kg/m^3$.