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- dswire10
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Studying altimetry, I see two rate figures commonly used in performance computations.

These are 30 feet per hectopascal for pressure height calculations and 120 feet per degree temperature deviation from ISA for density height calculations.

The textbook does not provide any indication from where these figures are obtained. Can you expand on the material in the textbook to fill in some detail, please?

These are 30 feet per hectopascal for pressure height calculations and 120 feet per degree temperature deviation from ISA for density height calculations.

The textbook does not provide any indication from where these figures are obtained. Can you expand on the material in the textbook to fill in some detail, please?

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- Stuart Tait
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The 30 ft/hPa is an approximation, The rate varies. 30 ft/hPa is accurate only for one altitude and will be less, at lower, or greater, at higher, levels. For light aircraft, 30 ft/hPa is a reasonable approximation.

The pressure rate varies from around 27 ft/hPa at sea level to around 37 ft/hPa at 10000 ft.

The pressure rate varies from around 27 ft/hPa at sea level to around 37 ft/hPa at 10000 ft.

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- dswire10
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Thank you for that reply and information Stuart. Are you able to cast some light onto the 120 ft/degree factor?

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- bobtait
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1° change in temperature equating to 12O ft change in height is also an approximation. As far as I know, it has no mathematical basis but rather is the result of actual measurement.

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- dswire10
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That has me confused. I would have expected both to have some relationship to the underlying ISA mathematical model.

However, thanks to both for your answers.

However, thanks to both for your answers.

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- John.Heddles
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If I may add a few words to Stuart’s and Bob’s posts.

**I would have expected both to have some relationship to the underlying ISA mathematical model.**

That is the case. We can derive the two rate figures without too much effort from the atmospheric models.

We start with a mathematical ISA model of how temperature varies with height (from which we get the well-known pilots’ approximation of 15 deg C – 2 C deg x height in thousands of feet). This then allows us to model a pressure variation with height and, in turn, a density variation with height.

Two important concepts follow from this:

(a) “pressure height” is the height in the (mathematical) ISA where a particular pressure exists

(b) “density height” is the height where a particular density exists.

So, if we know what the pressure or density is somewhere, then we can figure out the height at which this pressure or density would exist in ISA and, so, figure out the pressure or density height. This we can do simply by looking up a table of data or running the equations as sums.

**Pressure Height and 30 ft/hPa**

The pressure variation with height can be reworked to get the rate of pressure variation with height and, by looking to the reciprocal, the rate of height variation with pressure.

Traditionally, pilot calculations use a figure of 30 feet height change for each hPa variation in pressure. This, generally, is wrong as the actual figure varies very strongly with height. However, for the purposes of pilot calculations, where we are interested in takeoff and landing performance and low level terrain clearance, the calculations are reasonably fit for purpose.

The following graph gives you an idea of the rate variation for the troposphere. As the height increases, the rate (being exponential) ramps up very rapidly. For instance, at 50000 ft, we are looking at a figure a little below 180 ft/hPa.

While we might use the 30 ft/hPa rate value for all exam calculations, we should keep in mind that it is pretty silly and not at all appropriate for calculations relating to higher levels. The discrepancy between the actual exponential variation and the constant pilot calculation figure leads to much confusion when pilots run inappropriate calculations and then can't reconcile the difference between ISA tabular pressure data and their calculations.

**Density Height and 120 ft/C degree deviation**

If we know the pressure height, we can use standard equations in ISA-speak to work out the density at that height both for standard and non-standard temperatures. In turn, we can then use the density by height equation to figure out the height in “standard” ISA conditions at which a non-standard density occurs (ie density height).

We then can run some calculations of rates of density height variation with temperature deviation (temperature variation, or offset) from standard ISA conditions. The following graph shows the sort of variation for pressure heights to 15000 ft and for temperature deviations in the range ISA-30 to ISA+30.

If we model these curves with straight lines (they aren't quite straight, but fairly close) it just so happens that the various (straight) lines intersect nearly at ISA + 0 at a value close to 118.8 ft/C deg so that would look to be a pretty good mean value to use following a similar reasoning philosophy to the pressure height figure discussed previously. You will see this approximate figure in various textbooks and other technical papers – eg the US National Weather Service.

Now, fortunately, whoever came up with the 120 ft/C deg rate figure we use, generally, for pilot calculations, had enough sense to realise that

(a) no-one in their right mind would want to use a figure like 118.8 ft/C deg when, for pilots, we like to facilitate mental calculations

(b) 120 ft/C deg is very nearly the same as 118.8 (about 1% difference)

(c) far more importantly, 120 = 12 x 10. Now, up until the early 1970s, when electronic calculators first started coming onto the scene, EVERYONE learned, and could parrot off, their multiplication tables. This meant that, by using 120 (rather than that silly 118.8) we could all do these sums in the back of our heads while doing ten other things at once in flight.

So, once again, like the 30 ft/hPa, 118.8 (or 120) ft/C deg is applicable for one (itself approximate) condition only. So far as we need to run sums in a manner which is fit for purpose (ie takeoff and landing), it is suitable for use, given the normal safety margins we build into our flying protocols.

**Accuracy and Precision in Calculations**

So, the calculations we use routinely are a bit rough and ready (although reasonably fit for purpose).**There is no point, or justification**, in running your calculation sums to the nth decimal place, is there ? Might I suggest that you just run your exam calculations to the nearer foot – and, even then, we really are having ourselves on a bit.

**When Did the Rate Figures Originate in Pilot Practice ?**

I don’t know the answer because I have never been able to locate a paper or text which claims an answer. However, it is unlikely that they predate WW1 military aviation or post-date the introduction of the sensitive altimeter in the late 1920s. The first generally accepted atmospheric model was introduced in the 1920s, although work on less rigorous models dates back quite some time prior to that decade.

That is the case. We can derive the two rate figures without too much effort from the atmospheric models.

We start with a mathematical ISA model of how temperature varies with height (from which we get the well-known pilots’ approximation of 15 deg C – 2 C deg x height in thousands of feet). This then allows us to model a pressure variation with height and, in turn, a density variation with height.

Two important concepts follow from this:

(a) “pressure height” is the height in the (mathematical) ISA where a particular pressure exists

(b) “density height” is the height where a particular density exists.

So, if we know what the pressure or density is somewhere, then we can figure out the height at which this pressure or density would exist in ISA and, so, figure out the pressure or density height. This we can do simply by looking up a table of data or running the equations as sums.

The pressure variation with height can be reworked to get the rate of pressure variation with height and, by looking to the reciprocal, the rate of height variation with pressure.

Traditionally, pilot calculations use a figure of 30 feet height change for each hPa variation in pressure. This, generally, is wrong as the actual figure varies very strongly with height. However, for the purposes of pilot calculations, where we are interested in takeoff and landing performance and low level terrain clearance, the calculations are reasonably fit for purpose.

The following graph gives you an idea of the rate variation for the troposphere. As the height increases, the rate (being exponential) ramps up very rapidly. For instance, at 50000 ft, we are looking at a figure a little below 180 ft/hPa.

While we might use the 30 ft/hPa rate value for all exam calculations, we should keep in mind that it is pretty silly and not at all appropriate for calculations relating to higher levels. The discrepancy between the actual exponential variation and the constant pilot calculation figure leads to much confusion when pilots run inappropriate calculations and then can't reconcile the difference between ISA tabular pressure data and their calculations.

If we know the pressure height, we can use standard equations in ISA-speak to work out the density at that height both for standard and non-standard temperatures. In turn, we can then use the density by height equation to figure out the height in “standard” ISA conditions at which a non-standard density occurs (ie density height).

We then can run some calculations of rates of density height variation with temperature deviation (temperature variation, or offset) from standard ISA conditions. The following graph shows the sort of variation for pressure heights to 15000 ft and for temperature deviations in the range ISA-30 to ISA+30.

If we model these curves with straight lines (they aren't quite straight, but fairly close) it just so happens that the various (straight) lines intersect nearly at ISA + 0 at a value close to 118.8 ft/C deg so that would look to be a pretty good mean value to use following a similar reasoning philosophy to the pressure height figure discussed previously. You will see this approximate figure in various textbooks and other technical papers – eg the US National Weather Service.

Now, fortunately, whoever came up with the 120 ft/C deg rate figure we use, generally, for pilot calculations, had enough sense to realise that

(a) no-one in their right mind would want to use a figure like 118.8 ft/C deg when, for pilots, we like to facilitate mental calculations

(b) 120 ft/C deg is very nearly the same as 118.8 (about 1% difference)

(c) far more importantly, 120 = 12 x 10. Now, up until the early 1970s, when electronic calculators first started coming onto the scene, EVERYONE learned, and could parrot off, their multiplication tables. This meant that, by using 120 (rather than that silly 118.8) we could all do these sums in the back of our heads while doing ten other things at once in flight.

So, once again, like the 30 ft/hPa, 118.8 (or 120) ft/C deg is applicable for one (itself approximate) condition only. So far as we need to run sums in a manner which is fit for purpose (ie takeoff and landing), it is suitable for use, given the normal safety margins we build into our flying protocols.

So, the calculations we use routinely are a bit rough and ready (although reasonably fit for purpose).

I don’t know the answer because I have never been able to locate a paper or text which claims an answer. However, it is unlikely that they predate WW1 military aviation or post-date the introduction of the sensitive altimeter in the late 1920s. The first generally accepted atmospheric model was introduced in the 1920s, although work on less rigorous models dates back quite some time prior to that decade.

Engineering specialist in aircraft performance and weight control.

Last Edit: 3 weeks 4 days ago by John.Heddles.

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- dswire10
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That is the sort of information I was seeking, thank you. A further question.

I am aware that water vapour has a measurable effect on density. How does this fit into the ISA scenario, given that you have made no mention of water vapour in your comments ?

I am aware that water vapour has a measurable effect on density. How does this fit into the ISA scenario, given that you have made no mention of water vapour in your comments ?

Last Edit: 3 weeks 3 days ago by dswire10.

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- John.Heddles
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The ISA model is rather idealised and is based on a dry atmosphere which, of course, is pretty well never the case. That is to say, if we are talking ISA, humidity doesn't come into the discussion. This is a bit awkward when the real atmosphere always has some moisture content.

The met folk handle this by defining a not-quite-per-thermometer temperature, variously called virtual (or, sometimes, density) temperature. Using this fudged temperature figure, we can use the standard idealised equations for moist air calculations. A google search will provide you with lots of entertainment wading through the derivation of virtual temperature. You can save yourself some effort by having a read through this paper - commons.erau.edu/cgi/viewcontent.cgi?article=1124&context=ijaaa

As a side note, virtual temperature and humidity calculations don't come into pilot exams at any stage. About the extent the pilot needs to know is that increasing moisture decreases the actual density of air due to the difference in molecular masses.

The met folk handle this by defining a not-quite-per-thermometer temperature, variously called virtual (or, sometimes, density) temperature. Using this fudged temperature figure, we can use the standard idealised equations for moist air calculations. A google search will provide you with lots of entertainment wading through the derivation of virtual temperature. You can save yourself some effort by having a read through this paper - commons.erau.edu/cgi/viewcontent.cgi?article=1124&context=ijaaa

As a side note, virtual temperature and humidity calculations don't come into pilot exams at any stage. About the extent the pilot needs to know is that increasing moisture decreases the actual density of air due to the difference in molecular masses.

Engineering specialist in aircraft performance and weight control.

Last Edit: 1 week 5 days ago by John.Heddles.

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- dswire10
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Thank you for your response, John, I shall undertake some research to amplify the matter.

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