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## Density Height Calculations

• lostinspace2011
• Topic Author

### lostinspace2011 created the topic: Density Height Calculations

In my preparation for the PPL theory exam I came across this question:

Altitute: 7000ft
QNH: 1003 hPA
OAT: -4 C

and I am being asked to calculate pressure height, density height

PH=Elevation + (1013-QNH) x 30
PH=7000 + 10x30 = 7300ft (So far no problem)

ISA Temp at PH=15-7.3*2=0.4
ISA Deviation = OAT - ISA Temp
ISA Deviation = -4 - 0.4 = - 4.4

DH=PH+ISA Deviation x 120
DH=7300+(-4.4x120)
DH=7300-528 = 6772ft

However the work book suggest the correct answer to be 6820ft. Working backwards (7300-6820=480=4 x 120) this assumes an ISA deviation of 4, which suggest that instead of using 7.3 (7300ft) for the PH 7.5 was used.

So my question is what is the recommended way to round up/off the elevation to calculate the ISA temperature.
Should I always round up to the neared 500ft ?

• michel@deprez.id.au

### michel@deprez.id.au replied the topic: Density Height Calculations

From the examples I have done with Bobs exams round up or down to the nearest 500ft when calculating the ISA Temp.
Regards, Michel.

• arc0717

### arc0717 replied the topic: Density Height Calculations

Hi

But then to reach DA of 6820', the calculation will be using PH 7300' and ISA Dev using PH of 7,500 as follows:

DA = 7300 + (120 x (-4 - (15 - (7.5 x 2)))
DA = 6820'

Why is that?

Azhar

• michel@deprez.id.au

### michel@deprez.id.au replied the topic: Density Height Calculations

I'll defer to the more educated ones....(@Bob or @Stuart)... but if I can hazard a guess, it's probably accurate enough for our simple calculations. If you are really at the peak of weight and density limits then you should probably not fly the aircraft unless you are 100% certain you are under the maxes.
I'm sure we have all seen that youtube clip of that guy who took off from a high elevation aerodrome and almost clipped the trees?

• Posts: 2468

### bobtait replied the topic: Density Height Calculations

This method is used is to avoid using fractions of a degree in the calculations. That would be perfectly acceptable in practice. (After all we don't use fractions of hectopascals for the QNH - that's always given to the nearest whole).

Therefore, if the pressure height is 7300 feet, for the sake of establishing an ISA temperature, we use 7500 feet which will give us a temperature to the nearest whole degree.

Therefore, the ISA temperature is (15 - 2 x 7.5 = 15) which is zero. Since the actual temperature given is -4°C, the ISA deviation is -4.

120 times -4 = -480. So the density height is the pressure height of 7300 - 480 = 6820.
(Note that we only use the rounded-up figure to establish the ISA temperature, but we apply the ISA Deviation correction to the pressure height as calculated previously).

Also, the actual aircraft performance data is usually given in the form of tables or graphs. That makes it impossible to use entry arguments calculated to a fraction of a degree. Do you know the weight of the aircraft to a fraction of a kilo? You'd be lucky to know the gross weight to the nearest 10kg. That's why there are margins built into the performance figures.
The following user(s) said Thank You: arc0717, michel@deprez.id.au

• vuk

### vuk replied the topic: Density Height Calculations

Hi Bob,
I was just having similar problems with the rounding issue to get the correct answer. CASA are the kind of devils that would design a density height question in their exam with a pressure height worked out to something like 7250. In that case would you round up or down to workout the ISA temperature?
In the case at hand if we rounded up we would have 15-2x7.5=0 with a OAT of -4 so the ISA deviation is -4. However, if we rounded down then 15-2x7=1 with the same OAT of -4 the ISA Deviation would now be -5*C this in turn would give us a +/- 120ft discrepancy in our density height.
So you could say if we round up, we conservatively add a 120 ft margin of “safety” in the density height, but where would be the tipping point for rounding up the pressure height to calculate ISA temp calculation be?
I can see that this kind of thing will be critical problem in the exam. What say you?
Cheers
Andy

• John.Heddles
• Offline
• ATPL/consulting aero engineer
• Posts: 876

### John.Heddles replied the topic: Density Height Calculations

As Bob indicated, above, the pilot calculations are a tad rubbery.

Please keep in mind that the ISA calculations for pressure and density variation with altitude are (fairly straightforward and very similar) exponential functions but you don't want to go there, at all, I suggest. However, the slopes of the curves are not so weird that we can't reasonably approximate the particular exponential equations by linear (ie straight line) equations over small altitude variations and this is the basis of the techniques we use.

Be aware that 30 ft/hPa is "wrong" other than for the particular altitude where it is "right", but we use it anyway as a convenient approximation. It is reasonable for low altitudes, say, SL to 6000 ft but we still use it (in the exams) regardless of what the altitude might be. Similarly, 120 ft/deg is an approximation and so on.

The end result is that it is not worth getting excited about a few feet here and there with the sums. It is reasonable to presume that the examiner will avoid putting you in the 7250 ft style of quandary (and, even then, the mathematics convention is to round 7250 up and 7249 down).

So far as the performance side of things is concerned, while we endeavour, as engineers, to run the back room sums as accurately as we can and, as flight test personnel, to run the tests similarly, typically, we run to an accuracy of better than 2% which still leaves us with a fair bit of fiddle factor.

Bob suggests weight accuracies to 10 kg. As one of the most experienced weights engineers in Australia, I suggest that you will probably never have a gross weight to that accuracy - for a light aircraft, if you can get to, say, 20 kg, you will be doing very well on the day.

So long as you exercise reasonable care with the calculation housekeeping, your numbers will be fine in practice.

Engineering specialist in aircraft performance and weight control.