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  1. Does Smeaton's coefficient, k, have a modern value or it is dependent of the air density?

  2. 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.

  1. "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

  2. $L = k \cdot S \cdot V^2 \cdot C_L$

    $L$ = 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)"

Source: The Wiki page of the Wright brothers

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 $1.2 kg/m^3$.

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    $\begingroup$ Smeaton's coefficient is only valid in Middle Earth. $\endgroup$
    – Koyovis
    Jun 9, 2017 at 20:34
  • $\begingroup$ @Koyovis I believe you're referring to Smeagle's (or Smeagol's) coefficient - which begs the question from Annie in The Martian (in the book but not the movie) about Project Elrond... but that's a whole different discussion! $\endgroup$
    – Ralph J
    Jan 31, 2021 at 16:48

2 Answers 2

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There is no modern equivalent to the Smeaton coefficient, therefore it has no "modern" value.

The Smeaton coefficient is supposed to represent a "coefficient of air pressure" (and thus indirectly density), but it is a false constant: There is no single value because air pressure is not a constant value, but rather a variable (depending on altitude, temperature, etc.).

As you noted lift is calculated by a different formula today, using actual density of the fluid (air) you're moving through in place of the Smeaton coefficient.

$ L = Cl \cdot \frac{(A \cdot \rho)}{2} \cdot V^2$

You can calculate the Smeaton coefficient at any given location/altitude with simple algebra if you know the air density: Solve the modern lift equation, plug the values into the "old" Wright Brothers era equation, and solve for $k$.

The density-based calculation also has the benefit of working in other fluids besides air. (The same equation can give you hydrodynamic lift for boats and submarines by plugging in the density of water - itself a dynamic value depending on temperature, salinity, etc.)

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    $\begingroup$ If the formula L = k x S x V^2 x CL is correct (I suspect it is not) then the Smeaton coefficient can not be 0.0033 unless the Wright brothers experimented in temperatures well below -25 C. $\endgroup$ Oct 21, 2015 at 21:11
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    $\begingroup$ @Energizer777 the formula is correct, my understanding is the coefficient is based on drag using a flat plate & their measurement was pretty close to what you'd get if you worked the math, just poorly compensated (temperature & altitude/pressure change the "constant") $\endgroup$
    – voretaq7
    Oct 22, 2015 at 2:17
  • $\begingroup$ And it's also useful if the air is, say, cytherean or jovian or solar air rather than terran air! $\endgroup$
    – Vikki
    Aug 14, 2018 at 11:47
  • $\begingroup$ Your grouping of the terms in the equation is a bit unusual. The terms for dynamic pressure, $\frac12\rho V^2$, and the dimensional coefficient of lift, $C_l A$. are usually kept separate. Note that the $\frac12$ comes from the definition of kinetic energy, which is divided by volume to get the dynamic pressure, so it makes sense to apply it to the velocity, not the area. $\endgroup$
    – Jan Hudec
    Feb 1, 2021 at 8:48
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Wright brothers used the following 1900s equation:

$$ L = k \cdot V^2 \cdot S \cdot C_L $$

$k$ = pressure force (drag) on a one foot square flat plate moving at one mile per hour through the air.

$$ D = k \cdot V^2 \cdot S \cdot C_D \tag{1} $$

Modern drag equation is as follows: $$ D = \frac{1}{2} \cdot \rho \cdot V^2 \cdot S \cdot C_D \tag{2} $$ Therefore, by equating $(1)$ and $(2)$, we get:

\begin{align} k &= \frac{1}{2} \cdot \rho \\\\ & = \frac{1}{2} \cdot 1.225 \\\\ & = 0.6125 \left[\frac{kg}{m^3}\right] \end{align}

Value of k was given in units of lb force/m.p.hr sq/ft sq

1 lb force/m.p.hr sq /ft sq = 0.45359237 Kg × 9.8066 m/sec sq /(0.44704 m/sec) sq / 0.0929 m sq =239.5939 kg/m3

Therefore k = 0.6125/239.5939 = 0.00256 lbf/m.p.hr sq./ft sq.

Thus modern value of k is 0.00256 if we take air density as 1.225 kg/m3

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