Maximum Lift-to-Drag Ratio

[sldauby]

Hey everyone again,

I am hoping you fine aviators can again help me in understanding an aviation equation...this time regarding the maximum lift-to-drag ratio.

According to an aviation text that I have, lift-to-drag (L/D) ratio is defined as lift divided by drag, or when simplified just lift coefficient divided by drag coefficient, where the drag coefficient has two terms: Cd = Cd0 + KCl^2 (I know the 2 is technically an x, but it's 2 for typical subsonic flight).

Now, to get the maximum L/D ratio I must take derivative of this aforementioned equation and set it equal to zero. Solving this derivative for lift coefficient at maximum L/D gives me Cl = (Cd0/K)^1/2 (the square root of the zero-lift drag coefficient divided by constant K).

Finally, to get the maximum L/D ratio, I need to substitute this solved Cl equation into the "regular" L/D equation and simplify. The result should be L/D max = 1/[2*(K*Cl)^1/2]. Everything is on the bottom of the fraction. This is the result in my book and some sources online.

However, I do not get this square root in the bottom...just 1/[2*K*Cl].

What am I doing incorrectly? I tried to find a full derivation online, but the ones I found were not clear.

Thank you kindly.

 

[Realms09]

L/D = C_L/C_D = C_L / (C_D0 + K*C_L^2)

d(L/D) / dC_L = ( (C_D0 + K*C_L^2) - C_L * (2*K*C_L) ) / (C_D0 + K*C_L^2)^2

For an extremum

d(L/D) / dC_L = 0 ==> (C_D0 + K*C_L^2) - C_L * (2*K*C_L) = 0

C_D0 - K*C_L^2 = 0

C_D0 = K*C_L^2

(This says parasite drag equals drag due to lift when at the maximum L/D - something commonly depicted in graphical form in basic flying textbooks.)

Solving for C_L gives

C_L = sqrt(C_D0 / K)

Inset this back into the L/D equation to get

(L/D)max = sqrt(C_D0/K) / (C_D0 + C_D0) = sqrt(C_D0/K)/(2*C_D0) = 1/(2*sqrt(K*C_D0))

Q.E.D.

 

[sldauby]

Oh my goodness...I am such a fool!

I confused C_l with C_D0!!

Thank you for your derivation...I was able to check my work and now it's correct!