Koch Chart Formula

[abripl[_1_]]

Does anybody have the Koch Chart formula (equation)? Or know where to
find it?
I have a chart copy but want the function.

[Peter Duniho]

"abripl" > wrote in message
ups.com...
> Does anybody have the Koch Chart formula (equation)? Or know where to
> find it?

I don't think there is an actual equation that works generally. As an
obvious counter-proof to the idea that there is one, consider that density
altitude affects airplanes with normally aspirated engines differently from
those with turbocharged engines. The same Koch chart would not work for
both types of airplanes.

I haven't done a lot of research on the origin of the Koch chart, but I
believe that it's to be used as a general guideline, not as a precise
determination of how airplane performance is affected by density altitude.

If you do want to implement the Koch chart mathematically somehow, I'd
suggest that your best bet (in terms of ease of implementation) would be to
manually read off a range of pressure altitude and temperature combinations
to create tables giving the performance adjustment, and then interpolate
between the values for specific input of pressure altitude and temperature.

You could more accurately describe a Koch chart mathematically by actually
reverse engineering it (the scales on the middle portion of the chart appear
to be logarithmic and exponential for the takeoff distance and climb rate
reduction, respectively, so you simply need to measure the scales and
determine the base and power for those functions, and the pressure altitude
and temperature graphs appear to be linear), but that may be more trouble
than it's worth. Given that the chart isn't a precise way to determine the
performance change anyway, you may find it's overkill to analyze the chart
that way.

Pete

[Jim Macklin]

Google found several "aviation Koch chart" links including
this one
http://wind-drifter.com/technical/technical.htm scroll
down the page. They have some calculators listed, but the
Koch Chart is just the graph. But any high school math
teacher should be able to walk you through the solution of
the algebra problem to find the system of equations.





"Peter Duniho" > wrote in
message ...
| "abripl" > wrote in message
|
ups.com...
| > Does anybody have the Koch Chart formula (equation)? Or
know where to
| > find it?
|
| I don't think there is an actual equation that works
generally. As an
| obvious counter-proof to the idea that there is one,
consider that density
| altitude affects airplanes with normally aspirated engines
differently from
| those with turbocharged engines. The same Koch chart
would not work for
| both types of airplanes.
|
| I haven't done a lot of research on the origin of the Koch
chart, but I
| believe that it's to be used as a general guideline, not
as a precise
| determination of how airplane performance is affected by
density altitude.
|
| If you do want to implement the Koch chart mathematically
somehow, I'd
| suggest that your best bet (in terms of ease of
implementation) would be to
| manually read off a range of pressure altitude and
temperature combinations
| to create tables giving the performance adjustment, and
then interpolate
| between the values for specific input of pressure altitude
and temperature.
|
| You could more accurately describe a Koch chart
mathematically by actually
| reverse engineering it (the scales on the middle portion
of the chart appear
| to be logarithmic and exponential for the takeoff distance
and climb rate
| reduction, respectively, so you simply need to measure the
scales and
| determine the base and power for those functions, and the
pressure altitude
| and temperature graphs appear to be linear), but that may
be more trouble
| than it's worth. Given that the chart isn't a precise way
to determine the
| performance change anyway, you may find it's overkill to
analyze the chart
| that way.
|
| Pete
|
|

[Peter Duniho]

"Jim Macklin" > wrote in message
news:CGYGg.5789$SZ3.2892@dukeread04...
> Google found several "aviation Koch chart" links including
> this one
> http://wind-drifter.com/technical/technical.htm scroll
> down the page.

So what? No one was asking what a Koch chart is, and hopefully everyone
here knows how to use Google by now.

> They have some calculators listed, but the
> Koch Chart is just the graph.

There are no calculators to provide the information that the Koch chart
provides.

> But any high school math
> teacher should be able to walk you through the solution of
> the algebra problem to find the system of equations.

Doubtful. There is no single "system of equations" that provides the
general answer that a Koch chart attempts to provide. The best you can do
is to show what the Koch chart itself shows, and that chart is based on
general rules of thumb, not actual mathematically derived functions based on
actual density altitude effects.

Pete

[RST Engineering]

I'm sure you can combine equations as well as I, so ...


Variables: D=(density altitude) P=(pressure altitude)
T=(temperature in degrees celsius)


D = (145426 * (1- ((( 288.16 - P * .001981)

/288.16)^5.2563 / ((273.16 + T) / 288.16))^0.235))


APPROXIMATIONS:

For fixed pitch prop, increase sea level standard day takeoff distance 15%
for each 1000 foot increase in density altitude. Approximation good to 8000
feet density altitude

For constant speed prop, replace 15% with 13% in the above equation

For fixed pitch prop, decrease sea level standard day climb rate 7.5% for
each 1000 foot increase in density altitude.

For constant speed prop, replace 7.5% with 7% in the above equation.

(Equations and approximations from "Axioms of Flight", James Embree, Flight
Information Publications, St. Louis MO, 1984. ISBN 0-9601062-7-8)

Density altitude equation not independently verified. Use and report
results please.


Jim



"abripl" > wrote in message
ups.com...
> Does anybody have the Koch Chart formula (equation)? Or know where to
> find it?
> I have a chart copy but want the function.
>

[Larry Dighera]

On Wed, 23 Aug 2006 09:37:50 -0700, "Peter Duniho"
> wrote in
>:

>There are no calculators to provide the information that the Koch chart
>provides.

Although not what you were talking about, there is this:

http://www.mountainflying.com/apr_denalt.htm
The APR DENALT (DENsity ALTitude) Performance Computer
To solve the takeoff distance and rate of climb at any particular
density altitude, dial in the outside air temperature (degrees
Fahrenheit) and read the "takeoff factor" and "rate-of-climb
factor" to apply to the normal sea level values obtained from the
Pilot's Operating Handbook.

[Jim Macklin]

For those who care, you can save the Koch Chart as a picture
file to your computer. Then open it with PAINT or some
other photo editing program. Locate the x,y coordinates for
each part of the graphs and then use those numbers to solve
the algebra problem to get the formula for the system of
equations. Or just use the chart and add a cushion for
safety.



"Larry Dighera" > wrote in message
...
| On Wed, 23 Aug 2006 09:37:50 -0700, "Peter Duniho"
| > wrote in
| >:
|
| >There are no calculators to provide the information that
the Koch chart
| >provides.
|
| Although not what you were talking about, there is this:
|
| http://www.mountainflying.com/apr_denalt.htm
| The APR DENALT (DENsity ALTitude) Performance Computer
| To solve the takeoff distance and rate of climb at any
particular
| density altitude, dial in the outside air temperature
(degrees
| Fahrenheit) and read the "takeoff factor" and
"rate-of-climb
| factor" to apply to the normal sea level values
obtained from the
| Pilot's Operating Handbook.

[abripl[_1_]]

I tried a (aT + bP + c)^d type of exponential empirical function
fitting but so far not very exact.

In any case I revised the koch chart into more details with both F and
C temp units and with take off / climb "factors" from SL - easier to
use. Everybody is welcome to the chart at
http://www.abri.com/sq2000/Koch-Chart.gif

My experimental with my engine/prop combo does not have POH values for
takeoff-climb values with altitude and I have flown some over
mountains. Thats why the chart is important to me.

[RST Engineering]

did you not get the exact equations I posted here two days ago?

Jim



"abripl" > wrote in message
ps.com...
>I tried a (aT + bP + c)^d type of exponential empirical function
> fitting but so far not very exact.
>
> In any case I revised the koch chart into more details with both F and
> C temp units and with take off / climb "factors" from SL - easier to
> use. Everybody is welcome to the chart at
> http://www.abri.com/sq2000/Koch-Chart.gif
>
> My experimental with my engine/prop combo does not have POH values for
> takeoff-climb values with altitude and I have flown some over
> mountains. Thats why the chart is important to me.
>

[abripl[_1_]]

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
>

[Edwin Johnson]

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
--
__________________________________________________ __________
"Once you have flown, you will walk the earth with your eyes
turned skyward, for there you have been, there you long to
return."-da Vinci http://bellsouthpwp2.net/e/d/edwinljohnson

[abripl[_1_]]

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?

[Peter Duniho]

"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

[RST Engineering]

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
>>
>