TAS v Density altitude

[Finley Atherton]

Can anyone give me a formula for how TAS will change as density altitude changes if HP is kept constant?
For example, after careful testing I know that at 8,500 ft DA I have a TAS of about 150 kts at 2,250 rpm and 6 g/h leaned to around peak or just LOP (50% power from Lycomings Part Throttle Fuel Consumption chart).
If I fly at say, 1,000 ft DA at the same power settings (same HP) what would be the predicted TAS.

Fin
9A. 0-320, Hartzell

[elippse]

That's a hard one. For a given configuration, such as an RV-6, it will change about 1.1%/1000'. For my plane, it changes about 0.6%/1000'. From 4000' to 10,000' the density changes 20.3% Over that same span, the engine power changes 22.9%, so that the engine power, relative to the density, decreases 0.4% faster. That's one of the reasons planes with normally-aspirated engines fly slower with increasing altitude; their power falls off faster than their parasite drag. Why the difference between the two planes shown above? Induced drag! An RV-6 has a 4.8:1 aspect ratio, while my plane has 8.05:1. The higher you fly and the more weight you carry the more you will slow down due to induced drag, along with the power decrease.

[hevansrv7a]

I can't give you a formula, but I can suggest a work-around. Pick one of the spreadsheets linked below my signature (third link). Put in your airplane's drag data. Or use the 6A for an example.

Pick a few situations such as 2000', 4000' and so on. You can put in your assumptions about prop efficiency (in real life it is not a constant). Then you can quickly, iteratively solve for equal HP at each DA, changing IAS which then shows TAS. You can chart your results.

Cumbersome, but it will be the most correct answer. Again - be cautious about the prop efficiency which does change and not always the same because different props are optimized differently.

This is a good way to approach the problem because even your V l/d max changes with altitude. Lots of variables, all handled by the spreadsheet.

[Kevin Horton]

Originally Posted by Finley Atherton

Can anyone give me a formula for how TAS will change as density altitude changes if HP is kept constant?
For example, after careful testing I know that at 8,500 ft DA I have a TAS of about 150 kts at 2,250 rpm and 6 g/h leaned to around peak or just LOP (50% power from Lycomings Part Throttle Fuel Consumption chart).
If I fly at say, 1,000 ft DA at the same power settings (same HP) what would be the predicted TAS.

We can easily predict changes in TAS vs Density Altitude if we assume the propeller efficiency and drag coefficient do not change. This prop efficiency = constant assumption is probably reasonable for small changes in density altitude and rpm, but it is less and less accurate as we make significant changes in density altitude or rpm. The drag coefficient = constant assumption is true if the angle of attack is constant, but it gradually falls apart as the AOA changes. So, this prediction will not be completely accurate for large changes in CAS or weight.

If prop efficiency, power and AOA are constant, the TAS will vary with the cube root of the air density. If we insert the equation for density altitude, we end up with this ugly guy:

TAS2 = TAS1 *(((1-0.0000068755856*HD1)^4.2556)/((1-0.0000068755856*HD2)^4.2556))^0.33333



where:
TAS1 = TAS determined at density altitude 1
TAS2 = predicted TAS at density altitude 2
HD1 = Density Altitude 1 (units of feet)
HD2 = Density Altitude 2 (units of feet)

If we plug in TAS1 = 150 kt, HD1 = 8500 and HD2 = 1000, we get a predicted TAS of 139 kt.

Please note that with this much change in density altitude, the prop efficiency and AOA will both have changed, so this prediction will not be perfectly accurate. But it should be in the right ballpark.

[hevansrv7a]

The question assumed constant BHP. Given that constraint, using my own airplane's drag parameters for a starting point, I got this:
Prop=80%

DAlt CASmph TASmph BHP
2000 150.0 154.49 92.08
4000 147.8 156.84 92.02
6000 145.7 159.36 92.14
8000 143.4 161.75 92.07
10000 141.1 164.19 92.07
12000 138.7 166.59 92.02

This was all done quickly and iteratively using the model spreadsheet that can be found on the links below. No user-math required. All the previous caveats about prop efficiency and even engine efficiency still apply. This is hypothetical to that extent.

[DGlaeser]

You get pretty close by adding 2% per thousand feet of altitiude to your IAS. So at 5K you add 10%.
A lot easier to figure in your head than Kevin's formula And you don't need Howard's spreadsheet loaded on your PDA

[hevansrv7a]

Originally Posted by DGlaeser

You get pretty close by adding 2% per thousand feet of altitiude to your IAS. So at 5K you add 10%.
A lot easier to figure in your head than Kevin's formula And you don't need Howard's spreadsheet loaded on your PDA

Either this answers a different question or it conflicts pretty directly with my spreadsheet. In mine, for increasing altitude and equal BHP, the IAS (CAS) went down even though the TAS went up. In Dennis's rule, the IAS goes up. It can't all be right.

[RVbySDI]

Originally Posted by hevansrv7a

I can't give you a formula, but I can suggest a work-around. Pick one of the spreadsheets linked below my signature (third link). Put in your airplane's drag data. Or use the 6A for an example.

Pick a few situations such as 2000', 4000' and so on. You can put in your assumptions about prop efficiency (in real life it is not a constant). Then you can quickly, iteratively solve for equal HP at each DA, changing IAS which then shows TAS. You can chart your results.

Cumbersome, but it will be the most correct answer. Again - be cautious about the prop efficiency which does change and not always the same because different props are optimized differently.

This is a good way to approach the problem because even your V l/d max changes with altitude. Lots of variables, all handled by the spreadsheet.

I am looking at your "Glide Ratio & Angle with Cruise power" spreadsheet. Is this the spreadsheet you are recommending we use? On that spreadsheet, can you tell me what the "Ratio Ground/Vertical" field is representing? Also, am I assuming the "Glide Airspeed" is IAS? If so, how do I use this spreadsheet to determine TAS?

[hevansrv7a]

Originally Posted by RVbySDI

I am looking at your "Glide Ratio & Angle with Cruise power" spreadsheet. Is this the spreadsheet you are recommending we use? On that spreadsheet, can you tell me what the "Ratio Ground/Vertical" field is representing? Also, am I assuming the "Glide Airspeed" is IAS? If so, how do I use this spreadsheet to determine TAS?

Oops, I'm sorry!

There are two boxes on that page. The box on the left ("Workbooks") contains various versions of the same spreadsheet. Each version is for a different airplane. The airplanes were picked, mostly, because we have very solid data about their drag curves. The outlier is my own airplane. You can download and modify any of those spreadsheets. That is what I was recommending for this thread. Unfortunately, I don't have the RV-9 in there because CAFE did not use the Zero Thrust device on a "9". The spreadsheet and the presentation explain how to get one's own numbers to plug in.

The box on the right ("Documents and Aids") is misc. stuff, including the last one, a spreadsheet on "Glide Airspeed..". That one is pretty simplistic but solves other problems. Those speeds are necessarily TAS because you are dealing with what the airplane is doing, not what it says it is doing. Thus, ground/vertical is the actual ratio of the horizontal distance versus the vertical distance in a given time such as a minute. That's a true glide ratio, not necessarily L/D but certainly closely related to it. This one gives useful insights, but does not help you determine TAS, just uses it.

I'm thrilled that somebody looked. PM me or email me with a phone# if you want to get some personal help with anything on that page.

[Finley Atherton]

Thanks for the considered replies to my Post #1. It seems to be more complicated than I imagined. I was sort of hoping there may be a simple, approximate rule of thumb formula.
For flight planning purposes, I wanted to construct a table of true airspeeds at varying altitudes at three of my favored power settings. I realized I did not know this information recently when I was forced to cruise at low altitude due to weather.
I think I will go and do some flights to determine the actual true airspeeds and it will be interesting to see if they correlate to the answers calculated by the suggested methods.
Unfortunately I don't have anything that displays TAS directly so it will take some time using the NTPS method to determine TAS - sounds like a good excuse to do lots of flying.

Fin
9A