Does optimum IAS for climb vary with altitude?

Question

I was in an argument about aerodynamics and we eventually derived an equation as follows.

So I'm going by this equation:

$$RoC = \frac{\eta_p P-VD}{W}$$

Where $\eta$ represents propeller efficiency.

Assuming low Mach number:

$$TAS = IAS \cdot \sqrt{\frac{\rho_0}{\rho}}$$

$$ \text{Drag} = k(TAS)^2 \rho = k(IAS)^2 \rho_0$$

$$ \text{Specific Power} := P_s = \frac{\eta_p P}{W}$$

$$ \text{Drag Power} := P_d = \frac{TAS \cdot D}{W} = k \frac{\rho (TAS)^3}{W} = k \rho_0 \sqrt{\frac{\rho_0}{\rho}} \frac{(IAS)^3}{W}$$

The optimum climb IAS is given by:

$$\frac{dP_s}{d(IAS)}= \frac{dP_d}{d(IAS)}$$

Without loss of generality:

$$\frac{d\eta_p}{d(IAS)} = k \frac{(IAS)^2}{\sqrt{\rho}}$$

So does optimum IAS for climb vary with altitude?

Answer 1

Your equations aren't quite right. For example, you use IAS where CAS is intended. And drag power is V D, not V D / W. (Energy is force times distance and power is force times velocity.) Additionally, prop efficiency is more a function of true airspeed and RPM, so to take the derivative relative to IAS won't give you a good read on the effects of altitude.

Essentially this question is a complicated balance of several relationships. CAS vs TAS with airspeed, CL/CD curve, power available (are you normally aspirated, turbocharged, turboprop, electric?), prop efficiency, and even engine cooling. (Certification requirements require adequate cooling at Vy and if cooling is inadequate, one trick is to in rease Vy.) And with higher HP to weight, pitch angle becomes a factor as some thrust is providing lift.

A better approach, in my opinion, is to look at a couple flight manuals. What you'll generally find is that Vy decreases a little with altitude - maybe a knot every 2,000 ft or so.