Max Range Relationship to Altitude for Propeller-Driven Aircraft

Chapter 11 of the PHAK states, "If a change in altitude causes identical changes in speed and power required, the proportion of speed to power required would be unchanged. The fact implies that the specific range of a propeller driver aircraft would be unaffected by altitude. Actually, this is true to the extent that specific fuel consumption and propeller efficiency are the principal factors that could cause a variation of specific range with altitude. If compressibility effects are negligible, any variation of specific range with altitude is strictly a function of engine/propeller performance. An aircraft equipped with a reciprocating engine experiences very little, if any, variation of specific range up to its absolute altitude."

I looked at the POH's of both a Cessna 172 S model, and a Piper Warrior, and it seems as though both airplanes experience a significant increase in range with higher cruising altitudes. Can anyone explain this?

• Related: aviation.stackexchange.com/questions/60762/…? , aviation.stackexchange.com/questions/4967/…? . See other questions in "related" sidebar also. Commented Mar 5 at 18:33
• Note that the graph you posted and the relevant text are perfectly correct as long as you consider the speed as being TAS. Might it be that the Cessna and Piper charts consider IAS instead? Commented Mar 5 at 21:20
• Wouldn't it be nice if they listed IAS in the range charts. IAS/TAS calculators are available on line. Commented Mar 8 at 13:24
• @RobertDiGiovanni Yes, having both would be very nice! All told, true airspeed is probably more helpful for cross country planning, but it would be nice to know what you should expect to see on the airspeed indicator in cruise! Commented Mar 8 at 19:36

A change in altitude doesn't cause identical changes in speed and power required. Assuming that the drag is dominated by parasitic drag (typically true at cruise speed), the drag force is proportional to the equivalent airspeed squared, and power required is drag force times true airspeed:

$$\text{Power required}\propto (\text{EAS})^2\times\text{TAS}$$

Equivalent airspeed is given by:

$$\text{EAS}^2=\frac{\text{density}}{\text{sea level density}}\text{TAS}^2$$

So power required is related to true airspeed by:

$$\text{Power required}\propto\text{density}\times(\text{TAS})^3$$ This means that, assuming fixed power, true airspeed, and thus specific range, is inversely proportional to the third root of the air density:

$$\text{Specific range}\propto\text{True airspeed} \propto \sqrt[3]{\frac{1}{\text{density}}}$$

As an example, the density at 8000 feet density altitude is 78.6% of that at sea level. This gives a specific range about 8% higher than at sea level for the same power.

This relation falls apart at high altitudes- as you get to a higher and higher altitude you need to pitch up you reach the minimum drag airspeed. If you keep going higher with the same power, the drag increases, the specific range decreases, and pretty quickly you end up near stall speed.

Note also that ranges in the POH's you're reading also include the fuel required to climb up to that altitude in the first place.

Here is the specific range of a Cessna 152, based on POH cruise data and plotted versus equivalent airspeed:

The blue curve is a fit to the data based on the model outlined above.

• "Assuming fixed power" implies a change in altitude does not affect IAS. More correctly, it's fixed thrust. As long as required thrust, therefor IAS, can be maintained, increased TAS results in greater range, literally miles per gallon. Commented Mar 8 at 1:15
• @RobertDiGiovanni No, I'm not assuming fixed thrust. I'm assuming fixed power, the same amount of fuel going in. Maintaining the same thrust at higher density altitudes requires more fuel/power and this complicates the analysis. With fixed power, IAS goes down with altitude. IAS goes down slowly enough that TAS goes up. But eventually IAS gets low enough that the induced drag starts to get significant and the IAS plummets and TAS decreases. Commented Mar 8 at 1:21
• @RobertDiGiovanni The engine also loses maximum power with altitude and at some point is unable to maintain a given power. But this is a separate effect. Even with a fixed power, specific range decreases past a certain altitude where induced drag becomes too significant. Commented Mar 8 at 1:31
• @RobertDiGiovanni Higher RPM doesn't necessarily mean that the engine is working harder. Higher power does. The same power at higher altitudes gives higher RPM- because the air is thinner it takes the engine less effort to spin the same speed. The RPM for fixed power follows basically the same $\text{RPM}^3\times\text{density}=\text{constant}$ relation. Commented Mar 8 at 3:24
• @RobertDiGiovanni Anyway, since what we're interested in is specific range, to compare apples to apples you need to look at the same fuel intake (i.e. power) at two different altitudes. Commented Mar 8 at 3:28