Rate of Climb = (Thrust - Drag)/Weight * True Airspeed
Assuming that thrust is always the same (not depending on weight), we have to minimize the drag.
Drag can be described as:
Drag = [CD0 + k * CL^2] * Dynamic Pressure * Wing Area
CD0 and k are pretty much constant up Mach cruise.
CL = Weight/(Wing Area * Dynamic Pressure) [short: W/(Qdyn*S)]
So:
Drag = [CD0 + k * [W/(Qdyn*S)]^2] * Qdyn * S
Qdyn = True Airspeed ^2 * Air Density * 0.5
So, more true airspeed means less lift coefficient, but it increases dynamic pressure and so the total drag (not as coefficient, but as force).
When you put all into a spreadsheet (yes, sometimes I use Excel
) , the higher speeds will always show better rate of climb. I increases almost linearly with the true airspeed, with some exceptions at high weights and low speed (here the quadratic influence from the drag polar kicks in).
My conclusion: the fastest true airspeed (which is also the fastest indicated airspeed) is always quickest climb. It may not be minimum fuel. You might not have best climb gradient.
To sum all up:
Google: "getting to grips performance"
Open the PDF you'll hopefully find.
Go on PDF-page 158ff.
Read.
It says: most economic is always highest rate of climb. The best indicated airspeed depends on the aircraft design, higher wing loadings will see higher speeds.
And yes, please be critical with this post, not sure I got everything right.