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Koch Chart Formula



 
 
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  #11  
Old August 25th 06, 02:39 PM posted to rec.aviation.piloting
Edwin Johnson
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Posts: 29
Default Koch Chart Formula

On 2006-08-25, abripl wrote:
Your function was for Da = f(Pa, T) which in turn require various
approximations to get takeoff distance and climb performance - not a
finished product.


Jim's formula were correct. The Koch chart, itself, contains no takeoff
distance and climb performance. It is merely a translation, if you will,
expressed in chart form of density altitude given pressure altitude and
temperature. Any additions to any Koch chart, such as TO distance or climb
performance, _are_ additions and may hold true for only for certain
classes of aircraft.

Generally what you see in charts with those additions might have been
created and would probably work for the typical SE general aviation plane.
To this extent they may be very useful and helpful.

But, again, they are _not_ part of the Koch chart, they are _additions_.

....Edwin
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  #12  
Old August 25th 06, 11:56 PM posted to rec.aviation.piloting
abripl[_1_]
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Posts: 23
Default Koch Chart Formula


Edwin Johnson wrote:
On 2006-08-25, abripl wrote:
Jim's formula were correct. ....


Ed,

I am not debating Jim's prescription correctness, just mathematical
function usefulness. I posted the original post and asked for a math
function to represent the Koch chart. Jim did not do that but gave the
Da = f(T,Pa) function and then we need to add more to it.

The Koch chart, itself, contains no takeoff distance and climb performance. .......


Aw come on. All you have to know is your sea level T.O. and Climb and
just multiply by Koch chart factors. Very simple one step process.....
And it easily goes to 10K dens. alt.

Does anybody know Mr. Koch? Surely somebody made the chart up
originally?

  #13  
Old August 26th 06, 12:47 AM posted to rec.aviation.piloting
Peter Duniho
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Posts: 774
Default Koch Chart Formula

"abripl" wrote in message
ups.com...
[...]
Does anybody know Mr. Koch? Surely somebody made the chart up
originally?


You could make such a chart from scratch fairly easily, using empirical
methods. For an airplane that has published performance figures for
different density altitudes, it's especially easy. For one that does not,
you'll have to do some flight testing to obtain those figures, but once you
have them, the process is the same.

If you know the *exact* performance characteristics of a given airplane, it
is possible to come up with some pretty precise formulae describing that
airplane's performance at various altitudes. But getting that data is
difficult, and unless you have access to a vast engineering database of
engines, prop and wing airfoils, drag coefficients, etc. you're unlikely to
be able to. But empirical data is relatively easy to come by. Fly the
plane, take notes, viola.

Of course, someone already did all that, and they made a chart out of it. I
gather from your previous post that your approach was to attempt to
parametrically combine all of the factors into a single equation, but I'm
not convinced that's the right approach, at least not initially. You
actually have a couple of equations, based on the same line-intersection
equation that can be based on the chart that's already published. But that
equation isn't going to take the form "(aT + bP + c)^d".

You've got a line equation defined by the two endpoints (temperature and
pressure altitude), intersecting with the vertical axis at some point. That
gives you a vertical coordinate that can be used logarithmically (takeoff
distance) or exponentially (climb rate reduction) to determine the actual
correct factor found on the chart. The actual Cartesian coordinates for the
graph and the base or exponent (as appropriate) are derivable from the
existing chart, simply by measuring the chart and mapping it back to the
original numbers. But you're not going to get an equation of the form "(aT
+ bP + c)^d"...you'll get one linear equation that gives you the point of
intersection, and then two other equations (one log, one exp) to map that to
the actual performance adjustments.

Of course, even after you do all that, all you've got is a mathematical
description of the Koch chart. It's not going to tell you the *actual*
performance variations for a given airplane. It's just going to give you
the same (generally conservative) rules of thumb that the Koch chart
provides.

Pete


  #14  
Old August 26th 06, 01:24 AM posted to rec.aviation.piloting
RST Engineering
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Posts: 1,147
Default Koch Chart Formula

Sorry, I wasn't aware I was teaching freshman high school Algebra 1 here.
The "final product" is a series of four equations, two for fixed pitch and
two for constant speed propellers. I will derive/illustrate the first one.
It is left as an exercise for the student to complete the other three
(hint -- you only have to change one number in the equation).


D = Density Altitude (from the prior equation).

TD = Takeoff Distance at D

TS = Takeoff Distance at sea level

% = The percentage increase figures from the prior document.


For example, for a fixed pitch propeller:

TD = TS * (1 + ((D / 1000) * 0.15))

Jim


"abripl" wrote in message
oups.com...
Your function was for Da = f(Pa, T) which in turn require various
approximations to get takeoff distance and climb performance - not a
finished product.

I may try again later to get a better empirical function - not
impossible.

RST Engineering wrote:
did you not get the exact equations I posted here two days ago?

Jim




 




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