Altitude for the Maximum TAS
Regularly there seems to be some discussion regarding how one might handle a cargo fire which leads to discussion of the optimum TAS and at which altitude that might occur. I've heard that around F190 looks optimum. So I decided to figure it out! Foolish boy.: yes" There are some quite neat mathematical ways to derive the answer but most would have you reaching for a stiff drink before you started so I thought that the following less mathematical and more visual solution might work. I’m afraid it does involve the odd equation but I’ve tried to keep it simple.
Being a bus driver, I’ve used the Airbus A330 QRH and FCOM but I’m sure it translates to the WGA and you still fly in the same air that we lowly 'bus drivers use. Let’s start at Airbus QRH 4.17 which along with the VMO and MMO for the aircraft gives a nice simple picture of the high speed envelope. Now we need to convert both EAS and Mach to TAS. Brief refresher at this point: TAS is EAS corrected for density error. EAS, of course, comes from IAS via pressure, instrument and compressibility corrections. To derive TAS from EAS, we need to divide EAS by root sigma where sigma is the density ratio. Sounds horrid but the density ratio used to be given at FCOM 3.05.05 p2/3 (which moved to 2.03.05 p2) - now look here - and the rest we can do on a calculator. To derive TAS from Mach, we need the local speed of sounds. The speed of sound is simply 38.9 times the root of the absolute temperature, which is 273 plus the ambient temperature but fortunately these values are also given at here.
Going back to QRH 4.17 - for the 'bus - obviously the TAS at sea level equals the max EAS of 330kt. At F100, the TAS is 330kt divided by root sigma (.7385), which gives 384kt. Clearly TAS will increase with altitude until you reach limiting Mach. For the 'bus, this occurs at approx F295 where TAS is 534kt. Interestingly this shows the inaccuracy of the chart, as .86M where the speed of sound is 590kt gives a TAS of 507kt. Now with increasing altitude, the FCOM clearly shows a reduction in local speed of sound with altitude so equally the TAS at .86M must reduce.
Therefore the altitude for the maximum TAS occurs at the corner point where the maximum EAS intercepts the maximum Mach. Clearly this is in still air but my gut feeling is that even with altitude trade-off for head/tail wind, F295 is too high to be flying with a cargo fire but I also feel that 2000ft, such that you can ditch rapidly, is too low. Your call!
Now you reach for that stiff drink.........
For those of you who wish to persist for a few moments more, this also lends itself to a great method of selecting a FL when you are in a hurry. First you need to decide how fast you want to go. For example, let's decide that in the 'bus we want to be 10kt/0.02M off the limit. Now use the lines for 320kt and 0.84M and re-draw the curves as above. This gives you the optimum altitude for cruise at those speeds.
Cheers, Andy