Why is 391 used all over the place in aircraft calculations? I break things down today to find out why.

Henry, the aircraft specialist has been quite impressed with the work I’ve been putting in to this aircraft program, and is wanting to take things further. So, plans are for it to be updated so that different aircraft performance values can be entered, and for the impact on flight dynamics to be explored.

Meanwhile, major progress has been made on the value of 391 that turns up all over the place in flight dynamic calculations.

$\text{Glide angle }\theta_g = \frac{180}{\pi}\frac{\theta C_D S V^2}{391} \text{(degrees)}$

$\text{Minimum rate of sink }R_{S,min} = 88\sqrt{391} \frac{4}{(3\pi)^{\frac{3}{4}}}\sqrt{\frac{W}{\sigma}}\frac{A_D^{\frac{1}{4}}}{b_e^{\frac{3}{2}}} \text{(ft/min)}$

$\text{Minimum rate of sink airspeed } V_{minS} = \frac{\sqrt{391}}{(3\pi)^{\frac{1}{4}}} \frac{W/b_e}{\sigma A_D{\frac{1}{4}}} \text{(mph)}$

$\text{Minimum power for level flight } THP_{min} = \frac{88 \cdot 4}{33000} \frac{\sqrt{391}}{(3\pi)^{\frac{3}{4}}} \frac{A_D^{\frac{1}{4}}}{\sqrt{\sigma}} \left[\frac{W}{b_e}\right]^{\frac{3}{2}}$

### Why so many 391’s?

So where does this 391 come from? The 88 in the above formulas comes from converting mph to to feet per minute, so 391 is likely to result from other similar conversions.

Going back to basics we can figure this out:

$\text{Lift } L = C_L \frac{1}{2}\rho V^2 S \text{(V in ft/sec)} \equiv \frac{\sigma C_L SV^2}{391} \text{(V in mph)}$

We convert from mph to ft/sec by multiplying by 5280/3600, which needs to be done twice due to squaring V, and after cancelling common terms we have:

$\frac{1}{2}\rho = \frac{\sigma}{x} (\frac{3600}{5280})^2$

### The mystery of 391 is solved

$\text{With }\rho = 0.002377\text{ and }\sigma = 1 \text{ (at sealevel)}\text{ we can solve for x}$

$x = \frac{2}{0.002377} (\frac{3600}{5280})^2 = 391.1451687487179$

This is the last of the constants that I’m weeding out, so now I can move on to more interesting things.