So why is it that power is determined by true airspeed as opposed to indicated?

It seemed more intuitive to me that when flying high with the same IAS as at a lower height but higher TAS, power and the associated fuel flow would be unaffected. My assumptions are that drag, like lift, is IAS dependent and has not changed so neither has the required thrust. That leaves the engine whose output and fuel consumption I also assume aligns with density once leaned correctly. I'm definitely missing something.

Welcome to the world of learning about things aeronautical. Speeds probably give more student engineers and pilots more headaches than anything ATC or the weather can throw at them subsequently in their careers ....

So why is it that power is determined by true airspeed as opposed to indicated?

Probably something to do with TAS's being a real speed while IAS is only something to do with a bunch of air pressures ? G/S, likewise, is a real speed, albeit only of much use for getting from A to B, unless you incline to Paul Crickmore's views about being lost at Mach 3, but that's another story.

It seemed more intuitive to me that when flying high with the same IAS but, you're not actually flying at an IAS as at a lower height but higher TAS, while you are flying at a TAS. IAS has to do with dynamic and static pressures and is very useful when it comes to maintaining control of the aeroplane but TAS remains the real speed.

My assumptions are that drag, like lift, is IAS dependent this is another big point of confusion. You can set up the equations using either IAS (more generally, EAS, but IAS is close enough for this level) or TAS. As before, though, TAS is the real deal when it comes to kinematics and mechanics which is where you get to play with powers and so forth.

I'm definitely missing something.

A good read in some basic engineering books over coffee would help but, I wouldn't worry too much. You'll get the hang of things as you go along, I'm sure.

Cheers John. I'll add those books to the wishlist.

Let me take a stab at rephrasing my thought process to try to get to the bottom of my misunderstanding. If parasitic drag= C_D0.5ρv²A
as per a google search and notably including density. And Power can be phrased as Drag x TAS as per the book.
Situation A- I am cruising at a low height
Situation B- I am cruising at a high height
If in situation B I have a higher TAS due to increase in height (reduction in density) alone i.e. I have the same throttle setting, then looking back at the equation-
Power= Drag x TAS
Compared to situation A I can assume to have a higher power requirement due to a higher TAS. But wouldn't the equation be affected by that decrease in density so that the drag becomes less? Maybe even balancing it out as it increases one but decreases the other?

Perhaps the following explanation might help!

Power Required = TAS x Drag
P_R = V x C_D0.5ρV²A …. where V is TAS

rearranging we get
P_R = V³ C_D0.5ρA

Now, looking at the last equation we can judge the effect of altitude. As altitude increases two things happen
1) density decreases
2) TAS (V) increases

So, how will an increase in altitude affect power required (P_R) when density decreases and TAS (V) increases?
Ans: P_R will increase because the V³ factor has a bigger effect on Power than the density factor.

Too make my point, if we wanted to double our speed from V to 2V the power required would increase 8 fold
P_R = (2V)³ C_D0.5ρA
P_R = 8 x (V³ C_D0.5ρA)

So in summary, the V³ factor is the dominant factor in the calculation of P_R

I hope this helps!

Thanks Carello! That clears things up splendidly!!


However, staring at the equation too long has cost me another question.

For similar reasons that power required increases with height, wouldn't that mean that thrust required would increase aswell according to the equations?

Drag (and so thrust required)=C_D0.5ρV²A and like we have established- as height increases, density decreases and V increases. However as V increases more than density decreases, so too would thrust required increase. In this case it's squared rather than cubed but the difference is there. However the consensus is that higher is better for range.

Carello has given the usual story from the pilot text books but let's have a look at some real world numbers to get a check on all this theory stuff. If it gets a bit too much, don't worry but, for those who are interested, it puts some reality to the theory.

parasitic drag= CD0.5ρv2A

First, as an aside, keep in mind that this is but only one component of the aeroplane's drag. Generally, we figure drag from wind tunnel or flight test activities although we can torture ourselves by trying to figure out the various components analytically. However, it is important that we keep in mind that we need to look at all the drag, not just a bit here and there.

For these low speed aircraft with which you are involved in pilot training, the drag comes down to a bunch of components which are not tied up with the production of lift (lift-independent drag) and the lift-related drag (lift-dependent, or induced, drag). These are the two main drag elements seen in the usual drag curve graphs which, no doubt, Bob has in his text notes. In Carello's post, the Cd number must relate to the total drag at the test point, regardless of its origin or mix.

Power can be phrased as Drag x TAS as per the book.

Thinking power required here. The important thing is that the power required (which is tied up closely with fuel flow required ... the latter is of more interest to the pilot than power numbers) is related to speed cubed hence the reason why we don't just fly faster. That's all good fun but to go a little bit faster takes a whole lot more fuel and that irritates the bean counters in the back room no end !! More importantly, if you get the sums wrong, you might find yourself on fumes at some time during your flight and that is not a good idea.

Situation A- I am cruising at a low height
Situation B- I am cruising at a high height

OK, we have the picture in your scenario. So let's find some data and have a play with them.

If in situation B I have a higher TAS due to increase in height (reduction in density) alone i.e. I have the same throttle setting,

That's probably not the case. When you go flying, and you do a little bit of cruising at low level, do you then use the same throttle setting if you climb and cruise at a higher level ? No ? In general, what you will do is either cruise at a desired IAS (not the usual scenario) or, for larger engines (especially) and more commonly, a desired power setting (ie %BHP or similar). The other option is to cruise at a G/S required to make good an ETA. However, for this question, we will need to keep the IAS (actually, EAS but IAS will do us just fine) the same so that we control the value of Cd reasonably well. That way, if Cd is kept more or less constant, we only need to look at rho (density) and TAS^3 as variables in the sums.

Now Bob's school being at Redcliffe, we'll take advantage of the nice folks at Redcliffe Aero Club who, very kindly, have put up a C310 POH on their website for other folks to have a look at. (If you haven't flown 310s, you really must do so .. they are great fun). You can download the POH at redcliffeaeroclub.com.au/index.php/aircr...cessna-310-r-vh-jtv/. With this manual, we can play with some numbers rather than try to talk theory .. and then see that the one relates reasonably well to the other anyway.

In the following story, I won't try to set up the equations as real world engineering sums (I don't have all the data, anyway). As we are only wanting to play with the numbers, and equations are more useful that tabular look ups, simple plot and regression will be more than adequate. It is worth noting that the curves for the 310 are entirely typical and look much the same as would the curves for the next aircraft Type/Model. Indeed, while GAMA POHs tend to have tables, many aircraft run the graphs which are much easier to use and play with anyway. Further, with the equations there for you to use, if any of you want to have a play with the data for yourself, you have some machinery to do so - MS Excel, or similar, and away you go.

As you would be well aware, for routine pilot line stuff, you just interpolate linearly between the tabulated point data and accept the resulting (generally acceptably small) errors. You could use some of the fancier mathematical interpolation routines but that is just way over the top.

Now, in general, for smaller aeroplanes (engines), the fuel flow is pretty closely related to the power setting and we can ignore RPMs, MPs, altitudes and so on. If you like to have a look at the 310 ISA cruise data (pp 5-35 and subs) then, were you to plot and run a regression on the data, you would find that the relationship between power setting and fuel flow functionally is a straight line. For interest, the equation is
fuel flow = 13.626605 + 2.3226622 x %BHP and, even with round off and so forth, this will give the cruise table data for ISA conditions to within one pound per hour regardless of altitude and specific engine settings.

In a similar fashion we can generate some power/TAS curves from the chart data. The curves are a bit more complex but can be modelled adequately as quadratics/cubics. Notice, if you try to fit a cubic to the 15000 ft data, the curve shape is not realistic (we really need a few more data points) so we stay with the quadratic albeit with a slightly greater error. Again, if you are interested, the regressions result in

2500 ft ISA TAS = 22.952676 + 3.5832094 x %BHP - 0.019469776 x %BHP^2

5000 ft ISA TAS = - 45.704052 + 7.113337 x %BHP - 0.078002748 x %BHP^2 + 0.00032863525 x %BHP^3

7500 ft ISA TAS = - 61.162574 + 7.9418326 x %BHP - 0.090693149 x %BHP^2 + 0.0003977951 x %BHP^3

10000 ft ISA TAS = - 179.48467 + 14.33308 x %BHP - 0.20338361 x %BHP^2 + 0.0010639547 x %BHP^3

15000 ft ISA TAS = - 348.49362 + 19.079932 x %BHP - 0.17583887 x %BHP^2

I've given you the explicit sums so that you can keep a check on me just to make sure that I am not telling porkies if you are so inclined. Alternatively, you could also run some sums yourself and learn a bit about this stuff.

As a sidenote, do be very careful with any regression-based extrapolation you might be tempted to do. Once you get much beyond a straight line, extrapolation can do quite funny things unless you do some testing to make sure that the equations behave sensibly outside the data range. Indeed, sometimes we have to force things a bit and fudge some extrapolated data to make the equations behave sensibly. Quadratics are not too bad for a little bit of extrapolation but cubics and above just have too many twists and turns waiting to catch out the unwary.

But wouldn't the equation be affected by that decrease in density so that the drag becomes less? Maybe even balancing it out as it increases one but decreases the other?

Now let's have a play with some 310 numbers and see what might come out of the exercise. I'll give you an example and leave it to those who might be interested to have a go at figuring some numbers yourself. If you have a go and get into trouble, please do make a post and I'll help you sort things out. The value of having a go yourself is that we all learn best by doing, not reading, not listening, not watching but doing (after having done the other things).

The basic consideration is that the power required equation suggests that the relevant variables are Cd, density, and speed. If we can constrain Cd to be reasonably constant, then we are looking only at density and speed as variables. The relevant resulting relationship is

power required = some constant x density x speed^3

We established before that the fuel flow is directly related to the power required so we can look at fuel flow instead of power required without changing things to any great extent. That is we should expect to see something along the lines of

fuel flow = some other constant x density x speed^3

and we can figure this as ratios of numbers at different data points to get rid of those pesky dimensions.

So let's have a quick look at some data points from the cruise charts to see how this might work out -

2500 ft ISA 40.5 %BHP gives 107.7 lb/hr TAS 136.1 kt IAS 131.2 kt
15000 ft ISA 49.7 %BHP gives 129.1 lb/hr TAS 165.4 kt IAS 131.2 kt - notice we are trying to keep the same IAS.

Now, if you look up some ISA data you will get

2500 ft ISA density 1.13786575 kg/m^3
15000 ft ISA density 0.77080675 kg/m^3

and multiply density by TAS^3 you will get

2500 ft ISA rho x TAS^3 2868570
15000 ft ISA rho x TAS^3 3487804

Now, if you take some ratios for 15000 ft to 2500 ft, you will get

density ratio 0.77080675 / 1.13786575 = 0.68, ie down by around 32 %
TAS^3 ratio 4524874 / 2521009 = 1.80, ie up by around 80 %

As Carello observed, TAS^3 wins out over density by a long shot so that answers one of your specific questions.

F/F ratio 129.1 / 107.7 = 1.20, ie fuel flow (and power required) is up by about 20 %.
rho x TAS^3 ratio 3487804 / 2868570 = 1.21, ie the theory says power required is up by about 20 %.

The slight discrepancy is due to round off errors and imprecision in the tabular data. However, for all intents and purposes, you can see that the theoretical variables (rho x TAS^3) gives the answer you were looking for ..

Now, why not have a go yourself with some other data points ?

For similar reasons that power required increases with height, wouldn't that mean that thrust required would increase aswell according to the equations?

Not quite the case. The earlier discussion required that drag be kept constant so that we didn't have to worry about Cd as a variable. If speed is varied so that drag changes at the higher, compared to the lower, level then thrust will need to change to accommodate whatever drag change might have occurred. Once you work out what you are going to discuss, the answers will follow but you are mixing apples and oranges here.

Drag (and so thrust required)=CD0.5ρV2A and like we have established- as height increases, density decreases and V increases. However as V increases more than density decreases, so too would thrust required increase.

Again, you are implying inconsistent assumptions without stating them explicitly. You need to define your assumptions before you draw your conclusions .. apples and oranges, I'm afraid. A specific point is that V doesn't, necessarily, increase more than rho decreases. For an example, at the data point I used in the previous example, rho decreased 30-odd percent while V increased around 20-odd percent. V^3, on the other hand, increased by around 80 percent.

You need to be a bit more careful when it comes to the details, I fear ...

For similar reasons that power required increases with height, wouldn't that mean that thrust required would increase aswell according to the equations?

An interesting question!
Most text books that I’ve read do not handle this subject well – I cannot comment on Bob’s books as I have not seen them. Most texts simply state that drag and therefore thrust required (Tr) is not affected by altitude (density) without any real explanation.

That being said, let me have a crack at some sort of explanation. Keep in mind that this is a pilot’s explanation, not an aeronautical engineer’s explanation.

See the attached pdf file

Thanks for the attention guys, it's much appreciated! It'll take me a bit of time to ponder the replies especially seeing as my actual theory for the topic started today.

Cheers

Yep that's a bit clearer to me now I reckon. I'm now content that drag and therefore thrust required in level flight, doesn't increase as height alone increases based on the following thought process:
T_R=D=C_D0.5pV²A
As height is gained, the value of p decreases. Assuming the thrust supplied has not changed, the thrust which now slightly exceeds drag will accelerate the aircraft back to a drag value that matches itself. The drag will be increased by an increase in TAS. V² will increase proportionally to the decrease in p. That is to say that the increase in TAS is not proportional to the decrease in density but the square root of it.

I hope that checks out. I’m pretty sure i’m back on track now.

It sure is great to see you pondering over this stuff and having a go at rationalising it all.

However, there are some problems when trying to simplify things - a general concern, in fact, with all early pilot training where complex things have to be simplified quite a bit to make them palatable for comprehension. Overall, the main problem is trying to make the story minimally complex but sufficient to facilitate comprehension to a level appropriate to the reasonable needs of the student at his/her stage of progress and development. Inevitably, that means we have to leave details out with the risk that the comprehension may go off the track a little. Just one of the joys of training, I guess.

I'm now content that drag and therefore thrust required in level flight, doesn't increase as height alone increases

Drag relates to Cd in our system so, to keep drag reasoanbly constant, you need to keep Cd reasonably constant. This requires that we keep EAS (near enough to IAS at low altitude and speed) reasonably constant. At the level changes, we also need to adjust engine settings to maintain something approaching a constant thrust output. Then there is the change in rho to keep track of as well, not to mention things like Mach and Reynolds Numbers which, fortunately, are not too much of a concern at low speeds and altitudes. If you dig down a bit, things get quite complex so we try to avoid that to whatever extent we might be able.

Assuming the thrust supplied has not changed

If you don't adjust the engine settings, though, thrust will have changed. Probably best at this stage to run with the generic pilot training textbook statements. At the end of the day, a pilot doesn't need to have an engineering understanding, rather an understanding sufficient to provide some background for the inflight pushing and pulling.